ENGINEERING
TABLES AND DATA
A. M. Howatson, P. G. Lund and J. D. Todd
Edited by P. D. McFadden and P. J. Probert Smith
April 2009
ii
ENGINEERING TABLES AND DATA
Preface
Engineering Tables and Data, known affectionately as HLT after the initials of its authors, has been the primary reference for
generations of Oxford University engineering students. First published in 1973, and followed by a second edition in 1991, it
sought to provide a comprehensive collection of information covering all branches of engineering. This third edition, published
with the kind permission of the original authors, retains much of the core material, updated where necessary, while incorporating
several new sections thought to be important for a general engineering course.
We gladly acknowledge our gratitude to Drs B. Derby and C. R. Grovenor of the Department of Materials at Oxford, who very
kindly provided revised and extended data for large parts of the section Properties of Matter.
The tables on pages 36–40 are reproduced from Elementary Statistical Tables: Lindley and Scott, by permission of Cambridge
University Press.
The graphs on page 52 are based on data published in Handbook of Optical Constants of Solids: Palik, by permission of
Academic Press.
The graph on page 53 was provided by the Cable Products Division of STC Telecommunications, whose permission to publish it
is gratefully acknowledged.
The properties of water and steam tabulated on pages 54–71 inclusive are based on U.K. Steam Tables in S.I. Units published by
Edward Arnold.
The tables on page 72, the lower table on page 73, the upper table on page 75 and the thermochemical data on pages 82 and
83 are reproduced from Thermodynamic Tables: Haywood by permission of Cambridge University Press.
The table for refrigerant 134a on 73 is calculated using the Honeywell Refrigerants Properties Suite 2005.
Tables 1–3 on pages 93–102 inclusive are reproduced from Elements of Gas Dynamics: Liepmann and Roshko, by permission of
John Wiley.
Tables 4 and 5 on pages 102–121 inclusive are reproduced from Introduction to Gas Dynamics: Rotty, by permission of John
Wiley.
The tables of section properties on pages 136–153 inclusive are reproduced by permission of the Steel Construction Institute.
The graphs on pages 179 and 180 are reproduced from Electronic Circuits and Systems: King, by permission of Thomas Nelson.
We are grateful to these companies and authors for their collaboration, and to the Maurice Lubbock Memorial Fund for financial
support. We also express our thanks to our colleagues in the Department of Engineering Science at Oxford for their help and
advice. We shall be grateful to hear of those mistakes which, inevitably, will have escaped our notice.
P.D.M. & P.J.P.S.
April 2009
CONTENTS
iii
Contents
1 General
Greek alphabet . . . .
SI units . . . . . . .
Other metric units . . .
Multiples and sub-multiples
Conversion factors . . .
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1
2
2
3
3
4
2 Mathematics
Constants . . . . . . . . . . . . . .
Binomial coefficients . . . . . . . . . .
Series . . . . . . . . . . . . . . .
Fourier series for certain waveforms . . . . .
Trigonometric, hyperbolic and exponential functions
Trigonometric relations. . . . . . . . . .
Hyperbolic relations. . . . . . . . . . .
Differentials . . . . . . . . . . . . .
Indefinite coefficients . . . . . . . . . .
Definite integrals . . . . . . . . . . . .
Fourier transform . . . . . . . . . . .
Laplace transform . . . . . . . . . . .
z-transform . . . . . . . . . . . . .
Complex variable . . . . . . . . . . .
Algebraic equations. . . . . . . . . . .
Differential equations . . . . . . . . . .
Vector analysis . . . . . . . . . . . .
Matrices . . . . . . . . . . . . . .
Properties of plane curves and figures. . . . .
Moments of inertia, etc., of rigid bodies . . . .
Numerical analysis . . . . . . . . . . .
Statistics . . . . . . . . . . . . . .
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5
6
6
7
8
9
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10
11
12
13
16
17
17
18
19
20
22
23
29
34
35
3 Properties of matter
Physical constants . . . . . . . . . .
Periodic table . . . . . . . . . . . .
Atomic properties of the elements . . . . .
Physical properties of solids . . . . . . .
Mechanical properties of solids . . . . . .
Properties of reinforcing fibres . . . . . .
Work functions . . . . . . . . . . .
Properties of semiconductors . . . . . .
Properties of ferromagnetic materials . . . .
Superconducting materials . . . . . . .
Optical properties . . . . . . . . . .
Properties of liquids. . . . . . . . . .
Thermodynamic properties of fluids . . . .
Properties of gases . . . . . . . . . .
Thermochemical data for equilibrium reactions .
Psychrometric chart . . . . . . . . .
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41
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75
82
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iv
ENGINEERING TABLES AND DATA
4 Thermodynamics and fluid mechanics
Notation . . . . . . . . . . . . . . . . . .
Thermodynamic relations . . . . . . . . . . . . .
Equations for fluid flow . . . . . . . . . . . . . .
Dimensionless groups . . . . . . . . . . . . . .
Generalized compressibility chart . . . . . . . . . .
Nusselt numbers for convective heat transfer . . . . . .
Friction in pipes . . . . . . . . . . . . . . . .
Boundary-layer friction and drag . . . . . . . . . .
Open channel flow . . . . . . . . . . . . . . .
Black-body radiation . . . . . . . . . . . . . .
Tables for compressible flow of a perfect gas . . . . . .
Oblique shocks: shock-wave angle versus flow deflection angle
Oblique shocks: pressure ratio and downstream Mach number
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85
. 86
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. 92
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. 122
. 123
5 Solid mechanics and structures
Notation . . . . . . . . . . . . . . .
Two-dimensional stress and strain . . . . . . .
Three-dimensional stress and strain . . . . . .
Bending of laterally loaded plates . . . . . . .
Yield and failure criteria . . . . . . . . . .
Elastic behaviour of structural members . . . . .
Dimensions and properties of British Standard sections
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124
125
125
127
128
129
130
135
6 Mechanics
Statics . .
Kinematics
Dynamics .
Vibrations .
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154
155
155
156
158
7 Electricity
Electromagnetism . . . . . . . . . .
Analysis of circuits . . . . . . . . . .
Resonance and response . . . . . . . .
Electrical machines . . . . . . . . . .
Wave propagation . . . . . . . . . .
Antennas . . . . . . . . . . . . .
Transmission lines, optical fibres and waveguides
Communication systems . . . . . . . .
Components . . . . . . . . . . . .
Semiconductor devices . . . . . . . .
Instrumentation . . . . . . . . . . .
Digital logic . . . . . . . . . . . .
Verilog . . . . . . . . . . . . . .
State machines and computer architecture . .
Electrical properties of materials . . . . .
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161
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194
8 Miscellaneous
Standard screw threads .
Engineering drawing . .
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197
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GENERAL
Part 1
General
1
2
ENGINEERING TABLES AND DATA
Greek alphabet
α
β
γ
δ
ζ
η
θ
ι
κ
λ
μ
A
B
Γ
Δ
E
Z
H
Θ
I
K
Λ
M
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
ν
ξ
o
π
ρ
σ,ς
τ
υ
φ
χ
ψ
ω
N
Ξ
O
Π
P
Σ
T
Υ
Φ
X
Ψ
Ω
nu
xi
omicron
pi
rho
sigma
tau
upsilon
phi
chi
psi
omega
SI units
Quantity
Unit
Symbol
Equivalent
metre
kilogram
second
ampere
kelvin
candela
mole
m
kg
s
A
K
cd
mol
–
–
–
–
–
–
–
radian
steradian
rad
sr
–
–
hertz
newton
pascal
joule
watt
coulomb
volt
farad
ohm
siemens
weber
tesla
henry
lumen
lux
Hz
N
Pa
J
W
C
V
F
Ω
S
Wb
T
H
lm
lx
s−1
kg m s−2
N m−2
Nm
J s−1
As
J C−1
C V−1
V A−1
Ω−1
Vs
Wb m−2
Wb A−1
cd sr
lm m−2
Base units
Length
Mass
Time
Electric current
Temperature
Luminous intensity
Amount of substance
Supplementary units
Plane angle
Solid angle
Derived units
Frequency
Force
Pressure, stress
Work, energy, quantity of heat
Power
Electric charge
Electric potential, electromotive force
Electric capacitance
Electric resistance
Electric conductance
Magnetic flux
Magnetic flux density
Inductance
Luminous flux
Illumination
GENERAL
3
Other metric units
Quantity
Unit
Symbol
Equivalent
Length
Angstrom
micron
are
litre
tonne
dyne
bar
erg
poise
stokes
oersted
gilbert
gauss
maxwell (or line)
stilb
degree Celsius
Å
μm
a
l
t
–
bar
–
P
St
–
–
–
–
–
◦
C
10−10 m
10−6 m
102 m2
−3
3
10 m
3
10 kg
−5
10 N
5
10 Pa
−7
10 J
10−1 Pa s
10−4 m2 s−1
103 /4π A m−1
10/4π A
10−4 T
10−8 Wb
104 cd m−2
1K
Area
Volume
Mass
Force
Pressure
Energy
Viscosity (dynamic)
Viscosity (kinematic)
Magnetic field strength
Magnetomotive force
Magnetic flux density
Magnetic flux
Luminance
∗
Temperature
∗
The degree Celsius is used only for the temperature scale having zero
at the absolute temperature 273.15 K, the ice point of water.
Multiples and sub-multiples
Factor
Prefix
Symbol
24
10
1021
1018
1015
1012
109
106
3
10
102
10
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
Y
Z
E
P
T
G
M
k
h
da
10−1
−2
10
10−3
10−6
10−9
10−12
10−15
10−18
−21
10
10−24
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
m
μ
n
p
f
a
z
y
4
ENGINEERING TABLES AND DATA
Conversion factors
Length
Density
−3
1 mil = 25.4 μm (1 mil = 10 inch)
1 foot = 0.3048 m
1 yard = 0.9144 m
1 fathom = 1.829 m (1 fathom = 6 feet)
1 furlong = 0.2012 km (1 furlong = 220 yards)
1 mile = 1.609 km
1 nautical mile (n.m.) = 1.852 km
(1 nautical mile = 1.15 miles)
1 lb/in3 = 27.68 g cm−3
1 lb/ft3 = 16.02 kg m−3
1 lb/gallon = 99.78 kg m−3
1 ton/yd3 = 1.329 t m−3
Moments of mass
1 lb ft = 0.1383 kg m
1 lb ft2 = 421.4 kg cm2
Area
1 square inch = 645.2 mm2
1 square foot = 929.0 cm2
1 square yard = 0.8361 m2
2
1 acre = 4047 m (1 acre = 4840 sq. yd)
1 square mile = 2.590 km2
Force
1 lb force (lbf) = 4.448 N (standard gravity)
1 poundal (pdl) = 0.1383 N (1 pdl = 0.0311 lbf)
1 ton force = 9.964 kN
Pressure and stress
Volume
3
1 cubic inch = 16.39 cm
1 fluid ounce = 28.41 cm3
1 pint = 568.3 cm3
1 Imperial gallon = 4.546 l
1 U.S. gallon = 3.785 l
1 cubic foot = 28.32 l (1 cu. ft = 6.24 gallons)
1 cubic yard = 0.7646 m3
1 lb/ft2 = 47.88 Pa
2
1 lb/in (psi) = 6.895 kPa
1 ton/ft2 = 107.3 kPa
1 ton/in2 = 15.44 MPa
1 in Hg (0 ◦ C) = 3.386 kPa (1 in Hg = 0.491 psi)
1 ft H2 O (4 ◦ C) = 2.989 kPa (1 ft H2 O = 0.434 psi)
1 atmosphere (atm) = 1.013 25 bar
Energy and power
Angle
1 degree = 0.0175 rad
1
1 minute = 60
degree = 2.909 × 10−4 rad
1
1 second = 60
min = 4.848 × 10−6 rad
Speed
1 revolution/minute = 0.1047 rad s−1
1 mile/hour = 0.4470 m s−1
1 knot = 0.5148 m s−1 (1 knot = 1 nautical mile/h)
Mass
1 ounce = 28.35 g
1 pound (lb) = 0.4536 kg
1 slug = 14.59 kg (1 slug = 32.17 lb)
1 hundredweight (cwt) = 50.8 kg (1 cwt = 112 lb)
1 ton = 1.016 t (1 ton = 2240 lb)
1 ft lbf = 1.356 J
1 ft pdl = 42.14 mJ
1 horse-power = 745.7 W (1 hp = 550 ft lbf/s)
1 electron volt (eV) = 1.602 × 10−19 J
∗
1 degree Fahrenheit = 1 rankine (R) = 59 K
†
1 calorie = 4.187 J
†
1 Btu = 1.055 kJ (1 Btu = 252 cal)
1 Chu = 1.899 kJ (1 Chu = 95 Btu)
1 therm = 105.5 MJ (1 therm = 105 Btu)
1 Btu/hour = 0.293 W
1 Btu/lb = 2.325 kJ kg−1
∗
The degree Fahrenheit is used only for the temperature scale
having zero at the absolute temperature 459.67 R, on which
scale the ice point of the water is close to 32 ◦ F.
†
The calorie and Btu were defined in terms of heating water,
and their precise values depended on the temperature limits
specified. The figures given are based on agreed international
equivalents.
MATHEMATICS
Part 2
Mathematics
5
6
ENGINEERING TABLES AND DATA
Constants
Constant
π
π2
1/π
2
1/π
√
π
e
γ (Euler constant)
√
2
√
3
√
10
180/π
Value
log10
ln (loge )
3.141 59
9.869 60
0.318 31
0.101 32
1.772 45
2.718 28
0.577 22
2
3
10
1.414 21
1.732 05
3.162 28
57.295 78
0.497 15
0.994 30
1.502 85
1.005 70
0.248 57
0.434 29
1.761 34
0.301 03
0.477 12
1.000 00
0.150 51
0.238 56
0.500 00
1.758 12
1.144 73
2.289 46
−1.144 73
−2.289 46
0.572 36
1.000 00
−0.549 54
0.693 15
1.098 61
2.302 59
0.346 57
0.549 31
1.151 29
4.048 23
Binomial coefficients
n
n
n!
=
=
n −m
m
(n − m )! m !
m
n
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
1
3
6
10
15
21
28
36
45
1
4
10
20
35
56
84
120
1
5
15
35
70
126
210
1
6
21
56
126
252
1
7
28
84
210
⎫
⎪
⎪
⎬
⎪
⎪
⎭
etc.
Large factorials can be calculated from this version of Stirling’s formula which gives
log10 n ! ≈ 0.399 09 + n +
1
2
log10 n − 0.434 294 5 n +
and is correct to the fifth decimal place or better for n 70.
0.0362
n
MATHEMATICS
7
Series
π
1 1 1
+ − + ··· =
3 5 7
4
1
1
π2
1
+
+
+ ··· =
6
12 22 32
1
1
π2
1
+
+
+ ··· =
8
12 32 52
1
1
π3
1
−
+
− ··· =
32
13 33 53
n
n (n + 1)
1 + 2 + 3 + ··· + n =
r =
2
1
n
n (n + 1)(2n + 1)
12 + 22 + 32 + · · · + n 2 =
r2 =
6
1−
Trigonometric
3
x
x
+
− ···
3!
5!
cos x = 1 −
x2 x4
+
− ···
2!
4!
tan x = x +
3
2x 5 17x 7
π
x
+
+
+ · · · |x| <
3
15
315
2
13 + 23 + 33 + · · · + n 3 =
1
r3 =
2
2
n (n + 1)
4
Arithmetic
a + (a + d ) + (a + 2d ) + · · · + {a + (n − 1)d }
n
= {2a + (n − 1)d }
2
Geometric
1 + x + x 2 + x 3 + · · · + x n −1
⎧
n
1−x
⎪
⎪
(x = 1)
⎨
1−x
=
⎪
1
⎪
⎩
( |x| < 1, n → ∞)
1−x
Binomial
n (n − 1) 2 n (n − 1)(n − 2) 3
x +
x
2!
3!
n
+···+
xr + · · ·
r
x3 x5
− · · · ( |x| < 1)
+
5
3
sinh x = x +
x3 x5
+
+ ···
3!
5!
cosh x = 1 +
x2 x4
+
+ ···
2!
4!
tanh x = x −
3
2x 5 17x 7
π
x
+
−
+ · · · |x| <
3
15
315
2
Fourier
A periodic function f (x) with period T is represented by the
exponential series
∞
f (x) =
cn e j2πn x/T
n =−∞
where
1
cn =
T
T
f (x) e −j2πn x/T dx .
0
Alternatively, for real f (x) only,
f (x) =
a0
2πx
4πx
+ a2 cos
+ ···
+ a1 cos
2
T
T
2πx
4πx
+ b1 sin
+ b2 sin
+ ···
T
T
where
x
an =
2
T
T
f (x) cos
2πn x
dx
T
f (x) sin
2πn x
dx .
T
0
2
bn =
T
T
0
Maclaurin
x2
x3
f (x) = f (0) + xf (0) +
f (0) +
f (0) + · · ·
2!
3!
In these series, terms with n = 0 give the average of f (x),
those with n = 1 its fundamental component, and those with
n 2 its n th harmonic.
If f (x) = f (−x), f is even, bn = 0, and cn is real.
Taylor
f (a + h ) = f (a) + h f (a) +
7
tan−1 x = x −
Exponential and logarithmic
x2 x3
e =1+x +
+
+ ···
2!
3!
2
3
(x ln a)
(x ln a)
a x = 1 + x ln a +
+
+ ···
2!
3!
2
x3 x4
x
+
−
+ · · · ( |x| < 1)
ln(1 + x) = x −
2
3
4
5
1.3.5 x
1.3 x
1x
+
+ · · · ( |x| < 1)
+
2 3
2.4 5
2.4.6 7
(1 + x)n = 1 + n x +
( |x| < 1, all real n ; all x, n a positive integer)
3
sin−1 x = x +
1
n
5
sin x = x −
h2
h3
f (a) +
f (a) + · · ·
2!
3!
If f (x) = − f (−x), f is odd, bn = 0, and cn is imaginary.
T
, f has half-wave symmetry
If f (x) = − f x +
2
and an = bn = cn = 0 for all even n .
8
ENGINEERING TABLES AND DATA
Fourier series for certain waveforms
The series below are expressed in terms of the angular variable θ, the period of each waveform being 2π. Similar waveforms in
any variable x with period T can be represented by the same series with the substitution θ = 2πx/T . The origin of θ is chosen
so as to make the waveforms even functions (bn = 0) wherever possible. α and β are angles, k an integer.
θ=0
θ = 2π
π 3π
2 2
A
A
4A
π
2α
A
A
A
π
2
3π
2
A
π
2
3π
2
A
π
2
3π
2
π 3π
2 2
A
A
A
A
1
(sin α − α cos α ) + α − sin 2α cos θ
2
1
+ sin α + sin 3α − cos α sin 2α cos 2θ
3
1
1
2
+
sin 2α + sin 4α − cos α sin 3α cos 3θ + · · ·
2
4
3
8A
π2
4A
πβ
2A
π
A
2π
k
A
π
2
3π
2
2
A α
π
A2
4
1
1
cos θ + cos 3θ +
cos 5θ + · · ·
9
25
α
α2
+ 4 sin2 cos θ + sin2 α cos 2θ
2
2
4
3α
cos 3θ + · · ·
+ sin2
9
2
1
sin 3β cos 3θ
9
1
+
sin 5β cos 5θ − · · ·
25
sin β cos θ −
A
A
1
sin 2α cos 2θ
2
1
+ sin 3α cos 3θ + · · ·
3
sin α cos θ +
A
2
2
π
cos 4θ + · · ·
1 + cos θ + cos 2θ −
π
2
3
15
β
π 3π
2 2
α
2
+
π
π
A2
A2
2
A
πα
2α
1
1
cos θ − cos 3θ + cos 5θ − · · ·
5
3
2
2
2
2A
cos 4θ +
cos 6θ − · · ·
1 + cos 2θ −
π
3
15
35
A
π
2α
Mean square value
Series
sin θ +
1
1
sin 2θ + sin 3θ + · · ·
2
3
2
2
π
Ak
1+
cos k θ −
cos 2k θ
sin
π
k
k2 − 1
4k 2 − 1
2
+
cos 3k θ − · · ·
9k 2 − 1
2
3
A
α − sin 2α + 2α cos2 α
2π
2
A2
3
A2 α
3π
4β
A2 1 −
3π
2
A
3
MATHEMATICS
9
Trigonometric, hyperbolic and exponential functions
De Moivre’s theorem:
n
(cos x + j sin x) = cos n x + j sin n x
sin jx = j sinh x ;
cos jx = cosh x
sinh jx = j sin x ;
cosh jx = cos x
e jx = cos x + j sin x
sin−1 jx = j sinh−1 x ;
e x = cosh x + sinh x
sinh−1 jx = j sin−1 x ; cosh−1 x = j cos−1 x
sinh−1 x = ln x + x 2 + 1
sin x =
e
− e −jx
;
2j
jx
x
e −e
sinh x =
2
cos x =
−x
e
jx
+ e −jx
2
x
e +e
cosh x =
2
;
−x
cos−1 x = −j cosh−1 x
cosh−1 x = ln x + x 2 − 1
tanh−1 x =
1
2
ln
1+x
1−x
Trigonometric relations
sin (−A) = − sin A ;
cos (−A) = cos A
sin2 A + cos2 A = 1
2
1 + cot2 A = cosec2 A
sin (A ± B ) = sin A cos B ± cos A sin B
cos (A ± B ) = cos A cos B ∓ sin A sin B
tan (A ± B ) =
sin2 A =
cos A =
2
tan A =
tan A ± tan B
1 ∓ tan A tan B
(1 + cos 2A)
1
2
b
sin A + tan−1
a
A∓B
A±B
cos
2
2
A−B
A+B
cos
cos A + cos B = 2 cos
2
2
A−B
A+B
sin
cos A − cos B = − 2 sin
2
2
2 sin A cos B = sin (A + B ) + sin (A − B )
2 cos A cos B = cos (A + B ) + cos (A − B )
2 sin A sin B = cos (A − B ) − cos (A + B )
⎫
a
b
c
⎪
⎬
=
=
sin A
sin B
sin C
in a triangle ABC
⎪
a 2 = b 2 + c 2 − 2bc cos A ⎭
(1 − cos 2A)
1
2
1
2
sin A ± sin B = 2 sin
tan A + 1 = sec A
2
a sin A + b cos A = a 2 + b 2
sin 2A
1 + cos 2A
Hyperbolic relations
sinh (−A) = − sinh A ;
cosh (−A) = cosh A
sinh (A ± B ) = sinh A cosh B ± cosh A sinh B
cosh2 A − sinh2 A = 1
cosh (A ± B ) = cosh A cosh B ± sinh A sinh B
1 − tanh2 A = sech2 A
tanh (A ± B ) =
coth2 A − 1 = cosech2 A
sinh2 A =
1
2
cosh2 A =
tanh A =
1
2
(cosh 2A − 1)
(cosh 2A + 1)
sinh 2A
cosh 2A + 1
tanh A ± tanh B
1 ± tanh A tanh B
A∓B
A±B
cosh
2
2
A−B
A+B
cosh
cosh A + cosh B = 2 cosh
2
2
A−B
A+B
sinh
cosh A − cosh B = 2 sinh
2
2
2 sinh A cosh B = sinh (A + B ) + sinh (A − B )
sinh A ± sinh B = 2 sinh
2 cosh A cosh B = cosh (A + B ) + cosh (A − B )
2 sinh A sinh B = cosh (A + B ) − cosh (A − B )
10
ENGINEERING TABLES AND DATA
Differentials
In the following, u, v and w are functions of x and a is a constant.
f (x)
f (x)
uv
u
uv w
uv
u
v
dv
du
+v
dx
dx
dw
du
dv
+vw
+ wu
dx
dx
dx
du
1
dv
v
−u
dx
dx
v2
f (x)
f (x)
sin x
cos x
cos x
− sin x
tan x
sec x
cosec x
− cot x cosec x
2
sec x
tan x sec x
f (u,v )
∂f du
∂f dv
+
∂u dx
∂v dx
cot x
− cosec2 x
ax n
an x n −1
sinh x
cosh x
e ax
a e ax
cosh x
sinh x
tanh x
sech2 x
a
x
x
a ln a
xx
x x (1 + ln x)
cosech x
− coth x cosech x
1
x
sech x
− tanh x sech x
ln x
coth x
− cosech2 x
1
loga e
x
sin−1
loga x
x
a
−
x
a
a
a2 + x2
cos−1
tan−1
1
x
a
cosec−1
sec−1
x
a
cot−1
x
a
sinh−1
cosh−1
tanh−1
x
a
a2 − x2
1
a2 − x2
a
− x x2 − a2
a
x x2 − a2
−
a
a2 + x2
1
x
a
x
a
x
a
a
a2 − x2
cosech−1
x
a
a + x2
2
1
x − a2
2
a
− x x2 + a2
sech−1
x
a
a
− x a2 − x2
coth−1
x
a
−
x2
a
− a2
MATHEMATICS
11
Indefinite integrals
The constant of integration is omitted in each case.
f (x)
f (x) dx
n +1
(a + bx)n (n = −1)
(a + bx)
b(n + 1)
1
a + bx
1
ln(a + bx)
b
x
ax + b
ax + b − b ln(ax + b)
a2
x
a
ln a
ax
xa
− cos x
cos x
sin x
tan x
sec x
− ln cos x
x
ln tan
2
ln(sec x + tan x)
cot x
ln sin x
cosec x
sin x
x
a x
a
−
ln a
( ln a)2
2
cos x
2
xe
1
a + b e cx
1
x
−
ln(a + b e c x )
a ac
ln x
( ln x)
2
x n ln ax (n = −1)
( ln ax)
x
(n = −1)
1
x ln x
1
+ a2
1
1
x n +1
ln ax −
n +1
n +1
( ln ax)
n +1
sin−1 x
cos−1 x
−1
tan
−1
x tan
x
x−
1
2
cosh x
cosh x
sinh x
ln( ln x)
tanh x
1
x
tan−1
a
a
cosech x
sech x
ln cosh x
x
ln tanh
2
−1
tan (sinh x)
coth x
ln sinh x
cosh
x
a
−1
x
a
2
a2 − x2
1
2
x
x x 2 + a 2 + a 2 sinh−1
a
2
−1 x
x x 2 − a 2 − a cosh
a
x
x a 2 − x 2 + a 2 sin−1
a
sin−1
2
sec−1
x
a
x
a
+
1
(−x
2
sinh x
1
2
1
(x
2
2
tanh x
x − tanh x
cosech2 x
− coth x
cosh x
+
1
2
2
ln(1 + x )
sinh 2x)
2
2
1
2
x x2 − a2
tan x
sinh x
sinh
a
sec2 x
cos x + x sin x
x sin−1 x + 1 − x 2
x cos−1 x − 1 − x 2
x2 − a2
− cot x
x cos x
1
2
a −x
sin 2x)
x ( ln x)2 − 2 ln x + 2
x2 − a2
2
1
2
sin x − x cos x
x2 + a2
1
+
x sin x
x2 + a2
1
(x
2
x( ln x − 1)
−1
1
sin 2x)
−x − cot x
x − a 1
ln x +a
2a
1
x2 − a2
1
2
cosec x
2
n +1
n
−
cot x
x2
2
a2
1
(x
2
tan x − x
tan x
e ax (ax − 1)
ax
f (x) dx
sin x
2
x
x
f (x)
sinh 2x)
sech x
tanh x
coth2 x
x − coth x
x sinh x
x cosh x − sinh x
x cosh x
x
x sinh x − cosh x
−1
x sinh x − x 2 + 1
x cosh−1 x − x 2 − 1
x
x tanh
sinh
−1
x
−1
cosh
−1
tanh
−1
x+
1
2
2
ln(1 − x )
12
ENGINEERING TABLES AND DATA
Definite integrals
Legendre’s normal elliptic integrals include:
θ
F ( θ, k ) =
0
E ( θ, k ) =
∞
dθ
1−
k2
(first kind)
1/2
2
sin θ
0
θ 1/2
1 − k 2 sin2 θ
dθ
∞
(second kind)
0
π/2
E (π/2, k ) = E (k ) =
=
π
2
1/2
dθ
1 − k 2 sin2 θ
0
2
2 4
1
1.3
k
2
− ···
1−
(k 2 < 1)
k −
2
2.4
3
The error function is:
x
2
2
erf(x) = √
e −u du
π
0
5
7
1 x
1 x
2
x3
−
+ ···
+
= √
x−
3
2! 5
3! 7
π
∞
Γ(n ) =
x n −1 e −x dx =
0
1 ln
1
x
n −1
dx (Re n > 0)
∞
2
x e −x dx =
∞
2
x e
0
1
2
−x 2
√
dx =
π
4
0
∞
0
∞
e −ax sin bx dx =
b
a2 + b2
(a > 0)
e −ax cos bx dx =
a
a2 + b2
(a > 0)
0
∞
0
π
π
π
sin ax sin bx dx =
cos ax cos bx dx = 0
0
sin m x cos n x dx =
x p−1 (1 − x)q −1 dx =
0
Γ( p) Γ(q )
Γ( p + q )
0
(m − 1)(m − 3) · · · (2 or 1)(n − 1)(n − 3) · · · (2 or 1)
×C
(m + n )(m + n − 2) · · · (2 or 1)
π
for m and n even,
( m , n integers; C =
2
C = 1 for m or n odd )
(Re p, q > 0)
Stirling’s formula can be expressed as:
ln Γ(n ) ∼ (n − 12 ) ln n − n +
(a, b integers; a = b)
π/2
The beta function is:
B ( p, q ) =
⎫
(a > 0) ⎬
(a real)
(a = 0)
⎭
(a < 0)
tan x
π
dx =
x
2
π
1
(n > −1, Re a > 0)
a n +1
⎧
⎨ π/2
sin ax
dx =
0
⎩
x
−π/2
∞
0
1
2
ln (2π)
Euler’s constant is:
∞
γ = − e −x ln x dx = 0.57722
0
Γ(n + 1)
x n e −ax dx =
sin ax cos ax dx = 0
Also
=
(Re a > 0)
0
Γ(n ) = (n − 1) !
√
1
a
0
0
If n is a positive integer,
Γ( 12 )
e −ax dx =
(0 < m < n )
0
The gamma function is:
∞
π
mπ
x m −1
dx =
cosec
n
n
1 + xn
0
The ‘complete’ form of these is:
π/2
dθ
F (π/2, k ) = K (k ) =
1/2
1 − k 2 sin2 θ
0
2
2
1
π
1.3
1+
=
k2 +
k4 + · · ·
(k 2 < 1)
2
2
2.4
Other definite integrals
π
0
π sin n x
cos n u du
=
cos u − cos x
sin x
(n = 1, 2, 3 . . . )
(principal value)
MATHEMATICS
13
Fourier transform
The Fourier transform of a function of time x(t) may be written
as
∞
X (f ) =
x(t) e −j2πf t dt
Power spectral density
The power spectral density of x(t) is
W (f ) = lim
T →∞
−∞
where X (f ) is the Fourier transform of x(t) in the interval
−T /2 < t < T /2.
in terms of cyclic frequency f , or as
∞
X (ω) =
x(t) e −jωt dt
Autocorrelation
−∞
The autocorrelation of x(t) is the even function
in terms of angular frequency ω or 2πf . (Although in general
X (ω) = X (2πf ) = X (f ), the same symbol is conventionally
used for both forms since they differ only in scale.) The
inverse Fourier transform is then:
∞
x(t) =
1
2
|X (f )|
T
X (f ) e j2πf t df =
−∞
1
2π
∞
X (ω) e jωt dω .
−∞
1
Rx (τ ) = lim
T →∞ T
T/2
x(t) x(t + τ ) dt
−T /2
and Rx (0) is the mean square value of x(t). Rx (τ ) is the
inverse Fourier transform of the power spectral density of x(t),
i.e.
∞
Properties of the Fourier transform
Rx (τ ) =
Symmetry
W (f ) e j2πf τ df
−∞
If x(t) is real, then X (f ) = X ∗ (−f ).
and
If x(t) is even, then X (f ) = X (−f ).
If x(t) is odd, then X (f ) = −X (−f ).
If x(t) is real and even, then so is X (f ).
If x(t) is real and odd, then X (f ) is imaginary and odd.
∞
W (f ) =
Rx (τ ) e −j2πf τ dτ .
−∞
In all these cases X (ω) has the same properties as X (f ).
White noise
Duality
White noise is a random quantity having W (f ) constant and
Rx (τ ) an impulse at τ = 0.
If X (f ) or X (ω) is the transform of x(t), then x(−f ) or
2πx(−ω) is the transform of X (t).
Discrete Fourier transform
Convolution
If x(n ) is the n th of a sequence of N samples of a function of
time x, then the Fourier transform of these has N discrete
values of which the m th is given by
If x(t)∗y (t) represents the convolution
∞
x(τ ) y (t − τ ) dτ
X (m ) =
−∞
then the transform of x(t)∗y (t) is X (f )Y (f ) or X (ω)Y (ω) and
1
X (ω)∗Y (ω).
the transform of x(t) y (t) is X (f )∗Y (f ) or
2π
The inverse Fourier transform of X (m ) is then
x(n ) =
∞
2
2
|x(t)| dt =
−∞
|X (f )| df =
−∞
N
−1
m =0
Parseval-Rayleigh theorem
∞
N −1
1 x(n ) e −j2πn m /N .
N n =0
1
2π
∞
2
|X (ω)| dω
−∞
X (m ) e j2πn m /N .
14
ENGINEERING TABLES AND DATA
Fourier transforms of various functions
x(t)
X (f )
X (ω)
x(t − τ )
e −j2πf τ X (f )
e −jωτ X (ω)
x (n ) (t)
t
x(t) dt
( j2πf )n X (f )
( jω)n X (ω)
1
X (f ) + 12 X (0) δ(f )
j2πf
1
X (ω) + πX (0) δ(ω)
jω
k δ(f )
2πk δ(ω)
δ(f − f0 )
2π δ(ω − ω0 )
X (f − f0 )
X (ω − ω0 )
−∞
k , a constant
e
jω0 t
e
jω0 t
x(t)
{δ(f − f0 ) + δ(f + f0 )}
cos ω0 t
1
2
sin ω0 t
j 12 {−δ(f − f0 ) + δ(f + f0 )}
x(t) cos ω0 t
1
2
π {δ(ω − ω0 ) + δ(ω + ω0 )}
jπ {−δ(ω − ω0 ) + δ(ω + ω0 )}
{X (f − f0 ) + X (f + f0 )}
1
2
x(t)
{X (ω − ω0 ) + X (ω + ω0 )}
X (f )
X (ω)
1
1
Unit impulse δ(t)
t
0
Unit step u(t)
1
t
0
1
2
δ(f ) +
1
j2πf
πδ(ω) +
1
jω
Sign function (t)
1
t
0
1
jπf
2
jω
sin πf τ
= τ sinc f τ
πf
τ
2 sin(ωτ/2)
ωτ
= τ sinc
ω
2π
τ
-1
Rectangular pulse Π (t/τ )
1
-τ/2
0 τ/2
t
f
0
1/τ
2/τ
!
0
2π/τ 4π/τ 6π/τ
3/τ
Triangular pulse Λ(t/τ )
1
-τ
0 τ
t
sin2 πf τ
= τ sinc2 f τ
π 2f 2τ
2
4 sin (ωτ/2)
ω 2τ
= τ sinc2
ωτ
2π
MATHEMATICS
15
X (f )
x(t)
X (ω)
Carrier pulse Π (t/τ ) cos ω0 t
-τ/2
τ
{sinc (f − f0 ) τ + sinc (f + f0 ) τ }
2
t
τ/2
τ
2
sinc
(ω − ω0 ) τ
(ω + ω0 ) τ
+ sinc
2π
2π
Exponential pulse
u(t)e −at (a > 0)
1
t
0
1
a + j2πf
1
a + jω
2a
a 2 + 4π 2 f 2
2a
a2 + ω2
Two-sided exponential pulse
e −a|t| (a > 0)
1
t
0
2
Gaussian pulse e −π(t/τ )
1
τ e −π(f τ )
2
2
τ e −(ωτ ) /4π
t
0
Cosine-squared pulse
Π (t/τ ) cos2 (πt/2τ )
0
-τ
Impulse train
∞
t
τ
π 2 sin 2πf τ
2πf (π 2 − 4π 2 f 2 τ 2 )
π 2 sin ωτ
ω(π 2 − ω 2 τ 2 )
∞
n
1 δ f −
T −∞
T
∞
2πn
2π δ ω−
T −∞
T
δ(t − n T )
−∞
-2T
-T
0
T
2T
t
16
ENGINEERING TABLES AND DATA
Laplace transform
The Laplace transform of a function f (t) is
∞
F (s ) =
f (t) e −s t dt
0
and the inverse transform of F (s ) for t > 0 is then
1
f (t) =
2π j
σ +j∞
Function
Transform
t n −1
(n − 1)!
1
sn
e −at
1
s +a
1−e
a
F (s ) e s t ds
−at
1
s (s + a)
σ −j∞
where σ is a real constant greater than the real part of each
singularity of F (s ).
s
s 2 + a2
cos at
cosh at
s2
Properties of the Laplace transform
Convolution
If F (s ), G (s ) are the Laplace transforms of f (t), g(t) then
F (s ) G (s ) is the transform of the convolution of f and g from
0 to t, i.e. of
t
f ∗g =
f (τ ) g(t − τ ) dτ .
0
Initial-value theorem
sin at
a
s 2 + a2
sinh at
a
s 2 − a2
1 − cos at
a2
1
s (s 2 + a 2 )
at − sin at
a3
1
s 2 (s 2 + a 2 )
t e −at
1
(s + a)2
e −at (1 − at)
s
(s + a)2
sin at − at cos at
2a 3
1
(s 2 + a 2 )2
t sin at
2a
s
(s 2 + a 2 )2
t cos at
s 2 − a2
(s 2 + a 2 )2
e −at cos bt
s +a
(s + a)2 + b 2
lim s F (s ) = f (0)
s →∞
Final-value theorem
lim s F (s ) = lim f (t)
s →0
t→∞
Laplace transforms of various functions
Function
Transform
f (t − τ ) u(t − τ )
e
e −at f (t)
F (s + a)
f (t)
s F (s ) − f (0+)
f (t)
s 2 F (s ) − s f (0+) − f (0+)
f
(n )
(t)
−τs
F (s )
e
s n F (s ) − s n −1 f (0+)
−at
t
f (t) dt
F (s )
s
ae
∗
∗
0
Unit impulse at t = 0, δ(t)
1
Unit step at t = 0, u(t)
1
s
b
(s + a)2 + b 2
sin bt
e −bt − e −at
a −b
− s n −2 f (0+) − · · ·
− f (n −1) (0+)
s
− a2
∗
−at
− b e −bt
a −b
e −α t
e
1
(s + a)(s + b)
−α t
s
(s + a)(s + b)
sin β t
β
(cos β t −
1
s2
α
sin β t)
β
+ 2ζ ω0 s + ω02
s
s 2 + 2ζ ω0 s + ω02
where α = ζ ω0 , β = ω0 1 − ζ 2 and ζ < 1.
MATHEMATICS
17
z-transform
If f (i T ) represents samples of a function f (t) taken at intervals
T (i = 0,1,2 . . . ) then the z -transform of f is
F (z ) =
Final-value theorem
lim (z − 1) F (z ) = lim f (i T )
z →1
∞
i →∞
−i
z f (i T )
z-transforms of various functions
i =0
and the inverse z -transform of F is
f (i T ) =
1
2π j
z i −1 F (z ) dz .
In these the complex variable z is related to the complex
variable s of the Laplace transform by
z = e sT .
Properties of the z-transform
Convolution
If F (z ) and G (z ) are the z -transforms of f (i T ) and g(i T ) then
F (z ) G (z ) is the transform of
f ∗g =
i
f (k ) g(i − k ) .
Function sampled
Transform
f (t − k T ) (k is integer)
z −k F (z )
Unit impulse at t = 0, δ(t)
1
Unit step at t = 0, u(t)
z
z −1
Unit ramp at t = 0, f (t) = t
Tz
(z − 1)2
Unit acceleration at
2
t = 0, f (t) = t /2
T 2 z (z + 1)
e −at
z
z − e −aT
cos at
k =0
Initial-value theorem
2(z − 1)3
z (z − cos aT )
z2
− 2z cos aT + 1
z sin aT
z 2 − 2z cos aT + 1
sin at
lim F (z ) = f (0)
z →∞
Complex variable
Cauchy-Riemann relations
Residue theorem
If z = x + jy and the function f (z ) = u + jv , then for f (z ) to be
analytic it is necessary that
If f (z ) is analytic within and on C except at poles a, b, c . . .
enclosed by C ,
∂u
∂v
=
;
∂x
∂y
∂u
∂v
=−
.
∂y
∂x
f (z ) dz = 2π j (A + B + C . . . )
C
Cauchy’s theorem
where A, B , C . . . are the residues of the poles.
If f (z ) is analytic in a closed region bounded by a contour C ,
Nyquist criterion
f (z ) dz = 0 .
A consequence of the residue theorem is the following. If f (z )
is analytic within and on C except for P poles and Z zeros (a
pole or zero of order n being counted n times) within C , then
C
Cauchy’s integral
If a is a point inside C , f (z ) being analytic within and on C ,
f (a) =
where θc is the change in the argument of f (z ) for one circuit
of C ; or
f (z )
dz .
z −a
1
2π j
θc = 2π (Z − P )
C
N =Z −P
Also,
f (n ) (a) =
f (z )
n!
2π j
C
(z − a)n +1
dz .
where N is the number of times f (z ) encircles its origin
counter-clockwise for one counter-clockwise circuit of C . This
is the basis of the Nyquist stability criterion.
18
ENGINEERING TABLES AND DATA
Algebraic equations
The quadratic equation ax 2 + bx + c = 0 has roots
x=
−b ±
a0 x n + a1 x n −1 + · · · + an −1 x + an = 0
b 2 − 4ac
2a
which are
real and unequal if b 2 > 4ac ,
2
real and equal (and given by −b/2a) if b = 4ac ,
2
complex and conjugate if b < 4ac .
3
2
The cubic equation x + ax + bx + c = 0 is reduced to the
3
form y + py + q = 0, in which
p =−
a2
+b;
3
q =2
The general equation
a 3
3
−
ab
+c
3
by the substitution x = y − a/3. The roots are obtained from
√
1
3
y = A + B , and y = − (A + B ) ± j
(A − B )
2
2
of degree n has n roots (real, or complex and conjugate in
pairs) of which at least one is real for odd n . Their sum is
−a1 /a0 and the sum of their products taken r at a time is
(−1)r ar /a0 .
Routh-Hurwitz criterion
The number of roots of the general equation which are
positive or have positive real parts is the number of sign
changes in the sequence
D0 , D1 , D1 D2 , D2 D3 , . . . , where
D0 , D1 , D2 , D3 , . . .
a1
= a0 , a1 , a
3
where
⎡
q
B = ⎣− −
2
a0
a2
a4
0
a1
a3
,...
#
⎤1/3
q2 p3
⎦
+
4
27
All roots have negative real parts if there is no sign change
and no coefficient is zero.
#
⎤1/3
q2 p3
⎦
+
.
4
27
Simultaneous linear equations
q
A = ⎣− +
2
⎡
a1
a0 , a
3
a2 a
5
The set of n equations in n unknowns
n
If
q2 p3
+
> 0 , one root is real and two are
4
27
complex and conjugate,
= 0, all roots are real and two are equal,
< 0, all roots are real and different.
ai k xk = bi
(i = 1, 2, 3, . . . n )
k =1
has a unique solution if the determinant of the coefficients
det [ai k ] or is non-zero; the solution is given by Cramer’s rule
xk =
k/
(k = 1, 2, 3, . . . n )
where k is obtained by replacing the k th column of
column of bi .
by the
MATHEMATICS
19
Differential equations
Bessel’s equation
Laguerre’s equation
Bessel’s equation of order ν is
Laguerre’s equation of degree n has the form
z2
dw
d2w
+z
+ z2 − ν2 w = 0
dz
dz 2
z
and its solutions include the following Bessel functions.
d2w
dw
+ (1 − z )
+ nw = 0 .
dz
dz 2
Its solutions for positive integral n are the Laguerre
polynomials
First kind, order ν :
Jν (z ) =
∞
(−1)r
r =0
Ln (z ) =
z ν +2r&
r ! Γ (ν + r + 1) .
2
n
n −z
ez d z e
.
n!
dz n
Chebyshev’s equation
First kind, order n (an integer):
Chebyshev polynomials are solutions to equations of the form
Jn (z ) =
∞
r =0
(−1)r
z n +2r&
r ! (n + r )!
2
1 − z2
Second kind, order ν (non-integral):
for positive integral n , including:
&
Yν (z ) = { cos ν πJν (z ) − J−ν (z ) } sin ν π .
First kind Tn (z ) = cos n cos−1 z
Second kind Un (z ) = sin n cos−1 z .
Second kind, order n (an integer):
'
Yn (z ) =
2
dw
d w
−z
+ n 2w = 0
2
dz
dz
( ∂
∂
sin ν π
{cos ν πJν (z ) − J−ν (z )}
∂ν
∂ν
)
.
ν =n
Mathieu’s equation
Mathieu’s equation takes the form
Third kind (Hankel functions), order ν :
d2w
+ (a − 16q cos 2z ) w = 0
dz 2
(1)
Hν (z ) = Jν (z ) + jYν (z )
(2)
Hν (z ) = Jν (z ) − jYν (z )
in which a, q are real numbers.
Complete solutions may take the form, for any ν ,
Riccati’s equation
w = AJν (z ) + B Yν (z )
Riccati’s equation has general form
or
(1)
dy
= ay 2 + by + c
dx
(2)
w = AHν (z ) + B Hν (z )
in which a, b, c may be functions of x.
where A and B are constants.
Cauchy’s equation
Legendre’s equation
Cauchy’s equation has the form
Legendre’s equation of degree n has the form
1−z
2
2
2
dw
d w
− 2z
+ n (n + 1) w = 0
dz
dz 2
x2
and its solutions are Legendre functions of the first and
second kinds; for positive integral n the first are Legendre
polynomials. Associated Legendre functions of degree n and
order m are solutions to equations of the form
1 − z2
2
dw
d w
− 2z
+
2
dz
dz
n (n + 1) −
2
m
1 − z2
w = 0.
d y
dx 2
+ ax
dy
+ by = 0
dx
where a, b are constants; its solution is
y = Ax m1 + B x m2
where m1 , m2 are the roots of
m 2 + (a − 1) m + b = 0 .
20
ENGINEERING TABLES AND DATA
Wave equation
Poisson’s equation
The general form of the wave equation is
Poisson’s equation is
∇2 y − a
∂2 y
∂t 2
−b
∂y
− cy = 0
∂t
and has solutions in the form of attenuated travelling waves. If
b = c = 0, the solutions are undamped travelling waves of
√
phase velocity 1/ a. If a = c = 0, there results the equation
of diffusion or heat conduction which has solutions of
exponential form. If a = b = c = 0, the equation becomes
∇2 y = ρ
in which ρ may be a function of position; its solutions give the
potential y of a field whose divergence at any point is ρ.
∇2 y = 0
which is Laplace’s equation; its solutions give the potential y
of a field whose gradient is a vector field of zero divergence.
Vector analysis
For two vectors A, B with angle θ between them:
Scalar product: A · B = B · A = AB cos θ
In the following, V is a scalar field, F is a vector field:
grad V = ∇V = i
Vector product: A × B = −B × A
Scalar triple product: A × B · C = B × C · A = C × A · B
∂V
∂V
∂V
+j
+k
∂x
∂y
∂z
∂Fx
div F = ∇ · F =
Vector triple product: A × (B × C) = (A · C) B − (A · B) C
The vector product has magnitude AB sin θ and is normal to
the plane containing A and B.
curl F = ∇ × F = i
+j
For unit vectors i , j , k on right-handed orthogonal axes:
∂Fx
∇2 ≡
i·j = j·k = k·i =0
−
∂z
2
i·i = j·j = k·k =1
+
∂x
∂y
∂Fz
∂y
∂Fz
2
+
−
∂Fz
∂z
∂Fy
∂z
+k
∂x
∂Fy
∂x
−
∂Fx
∂y
2
∂
∂
∂
+
+
2
2
∂x
∂y
∂z 2
div grad V = ∇2 V =
i × i = j × j = k × k =0
∂Fy
∂2 V
∂2 V
∂2 V
+
+
∂x 2
∂y 2
∂z 2
∇2 F = i ∇2 Fx + j ∇2 Fy + k ∇2 Fz
i × j = k = − ( j × i ) etc.
In Cartesian coordinates:
In spherical coordinates (unit vectors u r , u θ , u φ ):
A · B = Ax Bx + Ay By + Az Bz
A × B = i (Ay Bz − By Az ) + j (Az Bx − Bz Ax )
i
=
Ax
B
x
j
Ay
By
k
Az
Bz
+ k (Ax By − Bx Ay )
A × B · C = Ax (By Cz − Cy Bz ) + Ay (Bz Cx − Cz Bx )
Ax
=
Bx
C
x
∇≡ i
+ Az (Bx Cy − Cx By )
Ay Az By Bz Cy Cz ∂
∂
∂
+j
+k
∂x
∂y
∂z
grad V = u r
div F =
∂V
1 ∂V
1 ∂V
+ uθ
+ uφ
r ∂θ
∂r
r sin θ ∂φ
1
1 ∂ 2
1 ∂Fφ
∂
r Fr +
(Fθ sin θ) +
2
r sin θ ∂θ
r sin θ ∂φ
r ∂r
'
)
∂Fθ
∂
1
F sin θ −
curl F = u r
r sin θ ∂θ φ
∂φ
)
'
∂F
∂
1
1
r
+ uθ
−
r Fφ
r sin θ ∂φ
∂r
)
'
∂Fr
1 ∂
+ uφ
(r Fθ ) −
r ∂r
∂θ
∇2 V =
1
∂V
1 ∂
∂
∂V
r2
+
sin θ
∂r
∂θ
r 2 ∂r
r 2 sin θ ∂θ
+
∂2 V
r 2 sin2 θ ∂φ 2
1
MATHEMATICS
21
In cylindrical coordinates (unit vectors u r , u φ , u z ):
grad V = u r
div F =
curl (V F) = V curl F − F × grad V
1 ∂Fφ
∂Fz
1 ∂
(r Fr ) +
+
r ∂r
r ∂φ
∂z
curl (F1 × F2 ) = F1 div F2 − F2 div F1
+ (F2 · ∇) F1 − (F1 · ∇) F2
∂Fφ
∂Fz
∂Fr
1 ∂Fz
−
+ uφ
−
r ∂φ
∂z
∂z
∂r
)
'
∂Fr
1 ∂
+ uz
(r Fφ ) −
r ∂r
∂φ
curl F = u r
∇2 V =
div (F1 × F2 ) = F2 · curl F1 − F1 · curl F2
1 ∂V
∂V
∂V
+ uφ
+ uz
r
∂r
∂φ
∂z
div (V F) = V div F + F · grad V
∂V
1 ∂2 V
1 ∂
∂2 V
r
+
+
r ∂r
∂r
r 2 ∂φ 2
∂z 2
General vector identities
curl grad V = ∇ × ∇V = 0
div curl F = ∇ · ∇ × F = 0
A vector field of zero divergence is said to be solenoidal. If the
line integral of F around any closed path is zero, then F has
zero curl, can always be expressed as grad V and is said to be
lamellar, conservative or irrotational.
Gauss’s divergence theorem
If dτ is an element of a volume T bounded by a surface S of
which dS is an element, then
div F dτ = F · dS .
T
s
curl curl F = ∇ × ∇ × F = grad div F − ∇2 F
Stokes’s theorem
grad (V1 V2 ) = V1 grad V2 + V2 grad V1
If dS is an element of a surface S bounded by a closed curve
C of which dl is an element, then
curl F · dS = F · dl .
grad (F1 · F2 ) = (F1 · ∇) F2 + (F2 · ∇) F1
+ F1 × curl F2 + F2 × curl F1
s
c
22
ENGINEERING TABLES AND DATA
Matrices
If Am,n represents a matrix A of order m × n (i.e. having m
rows and n columns) with the element aj k in the j th row and
k th column, then:
If A and B are the square non-singular matrices of the same
order, then
(AB )−1 = B −1 A−1 .
1. Am,n + Bm,n = Cm,n = Bm,n + Am,n and cj k = aj k + bj k .
(Only matrices of the same order may be added or
subtracted.)
A is orthogonal if A−1 = AT .
2. λ Am,n = Bm,n and bj k = λaj k , where λ is a scalar.
Jacobian
3. Am,n Bn,p = Cm,p and cj k =
n
*
l =1
aj l bl k .
(Two matrices can be multiplied only if they are
conformable, i.e. if the first has as many columns as the
second has rows; in general AB = B A.)
4. The transpose of Am,n is Bn,m = AT and bj k = ak j . (It
follows that (A + B )T = AT + B T and (AB )T = B T AT .)
5. A is a square matrix if m = n .
A is a row matrix if m = 1.
A is a column matrix if n = 1.
If u, v are functions of the variables x, y then the Jacobian of
x and y with respect to u and v is the determinant
∂x ∂x ∂u ∂v ∂(x, y )
J =
∂y ∂y = ∂(u, v ) .
∂u ∂v
For n functions ui of n variables xi the Jacobian is similarly
J =
∂ (x1 , x2 , . . . , xn )
∂ (u1 , u2 , . . . , un )
.
The functions are independent if J = 0.
Square matrices
A square matrix A is:
Matrix representation of vectors
symmetric if AT = A
skew-symmetric if AT = −A
diagonal if aj k = 0
( j = k )
A unit matrix U is a diagonal matrix with uj k = 1 ( j = k ).
An n -dimensional vector may be represented as a row matrix
of order 1 × n or as a column matrix of order n × 1. The
column matrix for a vector A may be written {A}. The scalar
product of A and B is then
T
The determinant of a square matrix A of order n has the value
det A = |A| =
n
aj k Cj k
for any j
aj k Cj k
for any k
k =1
=
n
j =1
Cj k = (−1) j +k Mj k
and Mj k is the determinant of order n − 1 obtained by deleting
the j th row and k th column of A. The determinants Cj k and
Mj k are the cofactor and minor respectively of the element
aj k . If |A| = 0 then A is a singular matrix.
The adjoint or adjugate matrix adj A of a square matrix A is
the transpose of the matrix of the cofactors of A. The inverse
A−1 of a non-singular square matrix A is a square matrix of
the same order such that
A × B = [A] {B }
where [A] is the skew-symmetric matrix
|A|
−Az
0
Ax
⎤
Ay
−Ax ⎦ .
0
Rotation of axes
If a vector is represented by {R} in a system of Cartesian
coordinates OXYZ , and by {r } in a second system Oxyz
having the same origin O, then
{r } = [C ] {R}
⎡
lxX
⎣ ly X
lz X
and is given by
adj A
0
⎣ Az
−Ay
where [C ] is the rotation matrix
AA−1 = U
A−1 =
The vector product of two three-dimensional vectors A and B
in Cartesian coordinates is
⎡
where
T
A · B = {A} {B } = {B } {A} .
.
lxY
ly Y
lz Y
⎤
lxZ
ly Z ⎦
lz Z
in which lxX = cos (xOX ), xOX being the angle between Ox
and OX , etc.
MATHEMATICS
23
Properties of plane curves and figures
Pappus’s theorems
1. The surface area generated by a curve of length l
revolving about an axis is
2. The volume generated by a plane surface of area A
rotating about an axis is
A = 2πl ȳ
V = 2πAȳ
where ȳ is the perpendicular distance of the centroid of
the curve from the axis.
where ȳ is the perpendicular distance of the centroid of
A from the axis.
Conic sections
Circle
Ellipse
Hyperbola
Parabola
y
y
a
b
O
a
O
y
y
l
a
b
F
x
F
O
l
x
O
l
F
x
x
l
4
Cartesian equation
x2 + y 2 = a2
Eccentricity
0
Focal distance OF
0
Latus rectum l
2
y2
x
+
=1
a2 b2
#
b2
1−
<1
a2
2a
a
2b 2
a
#
2b 2
a
2πa
Enclosed area
πa 2
Polar equation,
origin O
r =a
r2 =
Polar equation,
origin F
r =a
r =
a2 + b2
2
y2 = lx
1
l
4
a
4aE ≈ 2π
Circumference
2
y2
x
−
=1
a2 b2
#
b2
1+
>1
a2
†
3 ‡
†
‡
h
6l
πab
b2
1 − 2 cos2 θ
l
2 (1 −
E is complete elliptical integral of second kind (page 12).
Area enclosed by curve and vertical chord of length h .
cos θ)
r2 = −
r =
b2
1 − 2 cos2 θ
l
2 (1 −
cos θ)
r =
r =
l cos θ
1 − cos2 θ
l
2 (1 − cos θ)
24
ENGINEERING TABLES AND DATA
Other curves
y
Catenary
y = a cosh(x/a)
a
x
O
y
Cycloids
(ii)
x = aθ − b sin θ
y = a − b cos θ
(i)
(i) a = b (arc 8a, area 3πa 2 )
(iii)
θ
(ii) a < b (prolate)
O
aπ
(iii) a > b (curtate)
b is the generating radius on the circle of radius a.
Epicycloids
a +b
φ
x = (a + b) cos φ − b cos
b
a +b
y = (a + b) sin φ − b sin
φ
b
b
a
Φ
(i) 0 < b < a
(ii) a = b: cardioid
a
polar equation: r = 2a(1 − cos θ)
(iii) b → −b: hypocycloid
r
θ
a
b
a
Logarithmic spiral
r = a e bθ
r
2
Area between radii r1 , r2 =
2
θ
r2 − r1
4b
√
(r2 − r1 ) b 2 + 1
Length between radii r1 , r2 =
b
O
a
x
MATHEMATICS
25
Archimedean spiral
r = aθ
Area = a 2 θ 3 /6
θ
r
πa
O
Length → aθ /2 for large θ
2
Areas, centroids and second moments of area
The second moment of a plane area A about an axis x in its
plane is
Ixx = y 2 dA = Akx2
where kx is the radius of gyration about x. The product
moment of area is
Ixy = xy dA .
For any origin there is at least one pair of axes for which
Ixy = 0. These are principal axes; an axis of symmetry is a
principal axis for any origin lying on it. The above moments
are sometimes loosely called moment and product of inertia.
The centroid of A is defined by the coordinates
1
1
x dA ,
ȳ =
y dA .
x̄ =
A
A
A
A
Parallel-axis theorem
If I is the second moment of area about any axis through the
centroid of A, then that for a parallel axis at a perpendicular
distance d is
I = I + Ad 2
and if I is the product moment for any pair of Cartesian axes
through the centroid, that for a parallel pair at distances a, b is
I = I + Aab .
Rotation of axes
If Ixx , Iy y and Ixy are values for axes x, y then the
corresponding values for axes x , y at an angle α to x, y and
having the same origin are
Ix x =
Iy
Polar moment
The polar second moment for any point O in the plane of A is
the second moment of A about an axis z through O normal to
A. It is given by
Iz z = Ixx + Iy y
for any pair of orthogonal axes x, y in A with origin O.
y
=
Ix y =
1
2
1
2
1
2
Ixx + Iy y +
Ixx + Iy y +
1
2
1
2
Ixx − Iy y cos 2α − Ixy sin 2α
Iy y − Ixx cos 2α + Ixy sin 2α
Ixx − Iy y sin 2α + Ixy cos 2α .
o
o
b
c
a
c
b
a
c
o a
o
a
c
c
x
b
θ
o b
o
c
y
a
h
h
h
semicircle
circle
trapezium
triangle
parallelogram
rectangle
Figure
πa 2
2
πa 2
h
(a + b)
2
bh
2
ab sin θ
bh
A
a,
4a
3π
a, a
h (2a + b)
3(a + b)
36π 2
Aa 2 (9π 2 − 64)
Aa 2
4
18(a + b)2
Ah 2 (a 2 + 4ab + b 2 )
Aa 2
4
Aa 2
4
A 2
2
(b − ab + a )
18
Ah 2
18
a +b h
,
3
3
ȳ =
A 2
(b + a 2 cos2 θ)
12
Ab 2
12
Iy y
A
(a sin θ)2
12
Ah 2
12
Ixx
b + a cos θ a sin θ
,
2
2
b h
,
2 2
x̄, ȳ
In the following table, x̄, ȳ are the coordinates of the centroid C with respect to the origin O ; Ixx , etc., are moments for axes through C in the directions x, y .
0
0
Ah
(2a − b)
36
A 2
a sin θ cos θ
12
0
Ixy
26
ENGINEERING TABLES AND DATA
o
o
o
a
o
o
o
c
2a
c
2a
a
c
c
h
b
t
b
2b
semi-ellipse
ellipse
angle
section
ta
rectangle
segment
of circle
a
θ c
θ
Figure
sector
of circle
x
a
θ c
θ
y
1
2
sin 2θ)
πab
2
πab
2at
bh
a (θ −
2
a θ
2
A
1
2
sin 2θ)
,0
a,
4b
3π
a, b
a a
,
4 4
1√ 2
b + h 2, 0
2
3(θ −
3
2a sin θ
2a sin θ
,0
3θ
x̄, ȳ
Aa
4
2
3(θ −
2
1
2
sin 2θ)
sin θ sin 2θ
36π 2
2
2
Ab (9π − 64)
Ab 2
4
5Aa 2
48
Ab 2 h 2
6(h 2 + b 2 )
1−
+
Aa 2 (θ − sin θ cos θ)
4θ
Ixx
,
Aa
4
2
+
1
2
sin 2θ
4
4
A(h + b )
θ−
Aa
4
2
Aa 2
4
5Aa 2
48
12(h 2 + b 2 )
1+
2
sin θ sin 2θ
,
,
− Ax̄ 2
a 2 (θ + sin θ cos θ)
A
− x̄ 2
4θ
+
Iy y
0
0
Aa 2
16
12(h 2 + b 2 )
2
2
Abh (h − b )
0
0
Ixy
MATHEMATICS
27
o
o
o
c
y n
x
x n
a
c
y = b ( a)
a
c
x = a ( b)
a
y
b
b
b
parabola
Figure
ab
n +1
n ab
n +1
4ab
3
A
(n + 1)a (n + 1)b
,
n + 2 2(2n + 1)
(n + 1)a (n + 1)b
,
2n + 1 2(n + 2)
3a
,0
5
x̄, ȳ
Ab 2
5
Ixx
12Aa 2
175
Iy y
0
Ixy
28
ENGINEERING TABLES AND DATA
MATHEMATICS
29
Moments of inertia, etc., of rigid bodies
In the following, ρ is mass density, m total mass and V
volume.
The moment of inertia of a rigid body about the x-axis of a
Cartesian set is
Ixx = ρ y 2 + z 2 dV
V
and the product of inertia for axes x and y is
Ixy = ρxy dV .
If two axes, say x and y , lie in a plane of mass symmetry, then
only the product Ixy can be non-zero. If there are two
orthogonal planes of symmetry, their intersection is a principal
axis for any origin lying on it.
Parallel-axis theorem
If I is the moment of inertia about any axis through the centre
of mass C, then that for a parallel axis at a perpendicular
distance d is
I = I + md 2
V
The centre of mass is defined by the coordinates
1
ρx dV .
x̄ =
m
and if I is the product of inertia for any Cartesian pair through
C, that for a parallel pair at distances a, b is
I = I + m ab .
V
The inertia tensor or matrix is
⎤
⎡
Ixx −Ixy −Ixz
[I ] = ⎣ −Iy x
Iy y −Iy z ⎦ .
−Iz x −Iz y
Iz z
For any origin there exists at least one set of axes for which
the products of inertia are all zero. These are principal axes,
and the corresponding moments Ixx etc. are the principal
moments of inertia for that origin.
Rotation of axes
If I is the inertia matrix for certain axes, the matrix for a new
set having the same origin and a rotation matrix [ C ] is
[I ] = [C ] [I ] [C ]T .
For any origin, the sum of the moments Ixx + Iy y + Iz z is
invariant.
l
C
z
y
c
b
x
b
h
C
a
θ
θC
a
arc of hoop
O
uniform hoop
O
right rectangular
pyramid
a
C
O
rectangular prism
a
O C
uniform rod
O
Body
small
small
small
A
small
small
abh
3
abc
small
V
h
,0
4
a sin θ
, 0, 0
θ
0, 0, 0
0,
a b c
, ,
2 2 2
l
, 0, 0
2
x̄, ȳ , z̄
2
2
m a (θ − sin θ cos θ)
2θ
ma
2
m (4b 2 + 3h 2 )
80
m (b 2 + c 2 )
12
0
Ixx
ma
2
1 sin 2θ sin2 θ
+
−
2
4θ
θ2
m a2
2
m (a 2 + b 2 )
20
m (c 2 + a 2 )
12
ml 2
12
Iy y
ma
2
sin2 θ
1−
θ2
m a2
m (4a 2 + 3h 2 )
80
m (a 2 + b 2 )
12
ml 2
12
Iz z
In the following table for homogenous bodies x̄, ȳ , z̄ are the coordinates of the centre of mass C with respect to the origin O ; Ixx , etc., are the (principal) moments for axes through C in the
directions x, y , z ; A is the area of external curved surfaces only and V is the volume. Shells are assumed to have uniform thickness t.
30
ENGINEERING TABLES AND DATA
C
z
a
y
x
b
a
C
a
O
a
h
cylindrical shell
O
C C
a
hemisphere
C
sphere
O
hollow sphere
O
C
spherical shell
O
Body
2
2
2
2πah
2πa
4πa
4πa
4πa 2
A
At
2πa 3
3
4πa 3
3
4π 3
(a − b 3 )
3
At
V
0,
0,
h
,0
2
3a
,0
8
0, 0, 0
0, 0, 0
0, 0, 0
x̄, ȳ , z̄
2
2
m (6a + h )
12
83m a 2
320
2m a 2
5
5(a 3 − b 3 )
5
5
2m (a − b )
2m a 2
3
Ixx
ma
2
2m a 2
5
2m a 2
5
5(a 3 − b 3 )
5
5
2m (a − b )
2m a 2
3
Iy y
m (6a 2 + h 2 )
12
83m a 2
320
2m a 2
5
5(a 3 − b 3 )
5
5
2m (a − b )
2m a 2
3
Iz z
MATHEMATICS
31
a
z
h
y
x
O
h
O
h
R
C
c
ellipsoid
a O
b
torus
C
O
a
right circular cone
a
C
conical shell
a
C
right circular cylinder
O
C
Body
4π 2 Ra
√
πa a 2 + h 2
√
πa a 2 + h 2
2πah
A
4πabc
3
2π 2 Ra 2
πa 2 h
3
At
πa 2 h
V
h
,0
4
h
,0
3
h
,0
2
0, 0, 0
0, 0, 0
0,
0,
0,
x̄, ȳ , z̄
2
2
m (b + c )
5
2
2
m (5a + 4R )
8
3m (4a 2 + h 2 )
80
m (9a 2 + 2h 2 )
36
m (3a 2 + h 2 )
12
Ixx
2
2
m (c + a )
5
2
2
m (3a + 4R )
4
3m a 2
10
m a2
2
m a2
2
Iy y
2
2
m (a + b )
5
2
2
m (5a + 4R )
8
3m (4a 2 + h 2 )
80
m (9a 2 + 2h 2 )
36
m (3a 2 + h 2 )
12
Iz z
32
ENGINEERING TABLES AND DATA
C
O
z
y
h
x
C
h
∗
radius a
segment of sphere ∗
O
segment of spherical
shell ∗
Body
2πah
2πah
A
h
πh 2 a −
3
At
V
0,
h
,0
2
h (4a − h )
,0
4(3a − h )
0,
x̄, ȳ , z̄
m h (6a − h )
12
Ixx
m h (3a − h )
3
Iy y
m h (6a − h )
12
Iz z
MATHEMATICS
33
34
ENGINEERING TABLES AND DATA
Numerical analysis
Solution of algebraic equation f(x) = 0
Smoothing
Newton’s method
Third order, five point: a least-squares cubic for five
successive points y−2 . . . y2 is fixed by the points
xn +1 = xn − f (xn )/f (xn )
xn +1 =
3 4
δ y0 (similarly y11 , y21 , . . . )
35
⎫
2 4 ⎪
δ y0 ⎪
= y−1 +
⎬
35
end points
1 4 ⎪
⎭
δ y0 ⎪
= y−2 −
70
y01 = y0 −
Secant method
−xn f (xn −1 ) + xn −1 f (xn )
f (xn ) − f (xn −1 )
where xn is the n th estimate.
Approximations to derivatives
f (x + h ) − f (x − h )
f (x) =
2h
f (x + h ) − 2f (x) + f (x − h )
f (x) =
h2
f (x) =
f (x + 2h ) − 3f (x + h ) + 3f (x − h ) − f (x − 2h )
2h 3
1
y−1
1
y−2
Gaussian integration (second order)
1
f (x) dx = f
−1
1
−√
3
+f
1
√
3
Integration of ordinary differential equations
dy
= f(x,y)
dx
Runge-Kutta
2nd order:
f (4) (x) =
f (x + 2h ) − 4f (x + h ) + 6f (x) − 4f (x − h ) + f (x − 2h )
yn +1 = yn +
.
h f (xn , yn ) + f xn + h , yn + h f (xn , yn )
2
h4
4th order:
where h is an increment in x.
Numerical integration by equal intervals h
h
y (x) dx = ( y0 + y1 ) − 0 h 3 y0 /12
2
x0
Simpson’s rule
x2
y (x) dx =
1
(k + 2k2 + 2k3 + k4 )
6 1
where
Trapezoidal rule
x1
yn +1 = yn +
h
(4)
( y0 + 4y1 + y2 ) − 0 h 5 y1 /90
3
x0
where xn = x0 + n h , yn = y (xn ).
Everett’s interpolation formula for a table of y(x)
k1 = h f ( xn , yn )
k1
h
k2 = h f xn + , yn +
2
2
k2
h
k3 = h f xn + , yn +
2
2
k4 = h f (xn + h , yn + k3 )
Adams-Bashforth
Predictor:
yn +1 = yn +
If x = x0 + s (x1 − x0 ) and p = 1 − s , then
p
p +1 2
y0 +
δ y0 + · · ·
1
3
s
s +1 2
+
y1 +
δ y1 + · · ·
1
3
+ 37f (xn −2 , yn −2 ) − 9f (xn −3 , yn −3 )
Corrector:
y (x) ≈
where δ 2 y0 = y1 − 2y0 + y−1 , etc.
h 55f (xn , yn ) − 59f (xn −1 , yn −1 )
24
ynk+1 = yn +
h −1
9f (xn +1 , ynk+1
) + 19f (xn , yn )
24
− 5f (xn −1 , yn −1 ) + f (xn −2 , yn −2 )
k
Here xn = x0 + n h , yn is the k th estimate of y (xn ).
MATHEMATICS
35
Statistics
The variance of n values of a variable x is
s2 =
n
1 (xj − x̄)2
n −1
j =1
and the standard deviation is s .
If the value of a variable x has a probability function of density
f (x) then the mean of x is
The Gauss or normal distribution is defined by the probability
function
f (x) =
in which μ is the mean of x and σ the standard deviation.
The probability that in any one trial the variable will assume a
value ≤ x is the distribution function
∞
μ=
2
2
1
e −(x−μ ) /2σ
√
σ 2π
x
x f (x) dx
F (x) =
−∞
f (t) dt .
−∞
the variance of the distribution of x is
∞
σ2 =
(x − μ )2 f (x) dx
−∞
For the normal distribution with μ = 0, σ 2 = 1 this becomes
x
2
1
Φ(x) = √
e −t /2 dt
2π −∞
and σ is the standard deviation.
corresponding to a probability function (or frequency curve)
Distributions
If a certain event has a probability p of occurring in each of n
independent trials, then the occurrence of x events has the
probability
f (x) =
n
x
p x (1 − p)n −x .
x has the mean μ = n p and f (x) has the variance n p(1 − p).
This is the binomial distribution.
The limit of the binomial distribution for p → 0, n → ∞ is the
Poisson distribution with probability function
x
f (x) =
μ
e −μ
x!
(x = 0, 1, 2, ...)
when n p is defined as the mean, μ . The variance is then
σ2 = μ .
2
1
φ (x) = √
e −x /2 .
2π
The probability that a < x < b for a variable x with mean μ
and standard deviation σ is then
a − μ b −μ
p(a < x < b) = Φ
−Φ
.
σ
σ
The probability that (μ − n σ ) < x < (μ + n σ ) is
pn σ = Φ(n ) − Φ(−n ) .
For n = 2 this gives p2σ = 0.955; for n = 3, p3σ = 0.997.
36
ENGINEERING TABLES AND DATA
Percentage points of the t-distribution
This table gives percentage points tν (P ) defined by the
equation
1
1
P
1 Γ( 2 ν + 2 )
= √
100
ν π Γ( 12 ν )
∞
dt
(1 + t 2 /ν ) 2 (ν +1)
1
tν (P )
.
P/100
Let X 1 and X 2 be independent random variables having a
normal distribution with zero mean and unit variance, and a
2
χ -distribution
with ν degrees of freedom respectively; then
t = X 1 / X 2 /ν has Student’s t-distribution with ν degrees of
freedom, and the probability that t ≥ tν (P ) is P /100. The
lower percentage points are given by symmetry as −tν (P ), and
the probability that |t| ≥ tν (P ) is 2P /100.
0
tν(P )
The limiting distribution of t as ν tends to infinity is the normal
distribution with zero mean and unit variance. When ν is large
interpolation in ν should be harmonic.
P
40
30
25
20
15
10
5
2.5
1
0.5
0.1
0.05
ν=1
2
3
4
0.3249
.2887
.2767
.2707
0.7265
.6172
.5844
.5686
1.0000
0.8165
0.7649
0.7407
1.3764
1.0607
0.9785
0.9410
1.963
.386
.250
.190
3.078
1.886
1.638
1.533
6.314
2.920
2.353
2.132
12.71
4.303
3.182
2.776
31.82
6.965
4.541
3.747
63.66
9.925
5.841
4.604
318.3
22.33
10.21
7.173
636.6
31.60
12.92
8.610
5
6
7
8
9
0.2672
.2648
.2632
.2619
.2610
0.5594
.5534
.5491
.5459
.5435
0.7267
.7176
.7111
.7064
.7027
0.9195
.9057
.8960
.8889
.8834
1.156
.134
.119
.108
.100
1.476
.440
.415
.397
.383
2.015
1.943
1.895
1.860
1.833
2.571
.447
.365
.306
.262
3.365
3.143
2.998
2.896
2.821
4.032
3.707
3.499
3.355
3.250
5.893
5.208
4.785
4.501
4.297
6.869
5.959
5.408
5.041
4.781
10
11
12
13
14
0.2602
.2596
.2590
.2586
.2582
0.5415
.5399
.5386
.5375
.5366
0.6998
.6974
.6955
.6938
.6924
0.8791
.8755
.8726
.8702
.8681
1.093
.088
.083
.079
.076
1.372
.363
.356
.350
.345
1.812
.796
.782
.771
.761
2.228
.201
.179
.160
.145
2.764
.718
.681
.650
.624
3.169
3.106
3.055
3.012
2.977
4.144
4.025
3.930
3.852
3.787
4.587
.437
.318
.221
.140
15
16
17
18
19
0.2579
.2576
.2573
.2571
.2569
0.5357
.5350
.5344
.5338
.5333
0.6912
.6901
.6892
.6884
.6876
0.8662
.8647
.8633
.8620
.8610
1.074
.071
.069
.067
.066
1.341
.337
.333
.330
.328
1.753
.746
.740
.734
.729
2.131
.120
.110
.101
.093
2.602
.583
.567
.552
.539
2.947
.921
.898
.878
.861
3.733
.686
.646
.610
.579
4.073
4.015
3.965
3.922
3.883
20
21
22
23
24
0.2567
.2566
.2564
.2563
.2562
0.5329
.5325
.5321
.5317
.5314
0.6870
.6864
.6858
.6853
.6848
0.8600
.8591
.8583
.8575
.8569
1.064
.063
.061
.060
.059
1.325
.323
.321
.319
.318
1.725
.721
.717
.714
.711
2.086
.080
.074
.069
.064
2.528
.518
.508
.500
.492
2.845
.831
.819
.807
.797
3.552
.527
.505
.485
.467
3.850
.819
.792
.768
.745
25
26
27
28
29
0.2561
.2560
.2559
.2558
.2557
0.5312
.5309
.5306
.5304
.5302
0.6844
.6840
.6837
.6834
.6830
0.8562
.8557
.8551
.8546
.8542
1.058
.058
.057
.056
.055
1.316
.315
.314
.313
.311
1.708
.706
.703
.701
.699
2.060
.056
.052
.048
.045
2.485
.479
.473
.467
.462
2.787
.779
.771
.763
.756
3.450
.435
.421
.408
.396
3.725
.707
.690
.674
.659
30
32
34
36
38
0.2556
.2555
.2553
.2552
.2551
0.5300
.5297
.5294
.5291
.5288
0.6828
.6822
.6818
.6814
.6810
0.8538
.8530
.8523
.8517
.8512
1.055
.054
.052
.052
.051
1.310
.309
.307
.306
.304
1.697
.694
.691
.688
.686
2.042
.037
.032
.028
.024
2.457
.449
.441
.434
.429
2.750
.738
.728
.719
.712
3.385
.365
.348
.333
.319
3.646
.622
.601
.582
.566
40
50
60
120
0.2550
.2547
.2545
.2539
0.5286
.5278
.5272
.5258
0.6807
.6794
.6786
.6765
0.8507
.8489
.8477
.8446
1.050
.047
.045
.041
1.303
.299
.296
.289
1.684
.676
.671
.658
2.021
2.009
2.000
1.980
2.423
.403
.390
.358
2.704
.678
.660
.617
3.307
.261
.232
.160
3.551
.496
.460
.373
∞
0.2533
0.5244
0.6745
0.8416
1.036
1.282
1.645
1.960
2.326
2.576
3.090
3.291
MATHEMATICS
37
The normal distribution function
The function tabulated is
x
1 2
1
Φ(x) = √
e − 2 t dt .
2π −∞
Φ(x)
Φ(x) is the probability that a random variable, normally
distributed with zero mean and unit variance, will be less than
or equal to x. When x < 0, use Φ(x) = 1 − Φ(−x), as the
normal distribution with zero mean and unit variance is
symmetric about zero.
0
x
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
0.00
.01
.02
.03
.04
0.5000
.5040
.5080
.5120
.5160
0.40
.41
.42
.43
.44
0.6554
.6591
.6628
.6664
.6700
0.80
.81
.82
.83
.84
0.7881
.7910
.7939
.7967
.7995
1.20
.21
.22
.23
.24
0.8849
.8869
.8888
.8907
.8925
1.60
.61
.62
.63
.64
0.9452
.9463
.9474
.9484
.9495
2.00
.01
.02
.03
.04
0.97725
.97778
.97831
.97882
.97932
0.05
.06
.07
.08
.09
0.5199
.5239
.5279
.5319
.5359
0.45
.46
.47
.48
.49
0.6736
.6772
.6808
.6844
.6879
0.85
.86
.87
.88
.89
0.8023
.8051
.8078
.8106
.8133
1.25
.26
.27
.28
.29
0.8944
.8962
.8980
.8997
.9015
1.65
.66
.67
.68
.69
0.9505
.9515
.9525
.9535
.9545
2.05
.06
.07
.08
.09
0.97982
.98030
.98077
.98124
.98169
0.10
.11
.12
.13
.14
0.5398
.5438
.5478
.5517
.5557
0.50
.51
.52
.53
.54
0.6915
.6950
.6985
.7019
.7054
0.90
.91
.92
.93
.94
0.8159
.8186
.8212
.8238
.8264
1.30
.31
.32
.33
.34
0.9032
.9049
.9066
.9082
.9099
1.70
.71
.72
.73
.74
0.9554
.9564
.9573
.9582
.9591
2.10
.11
.12
.13
.14
0.98214
.98257
.98300
.98341
.98382
0.15
.16
.17
.18
.19
0.5596
.5636
.5675
.5714
.5753
0.55
.56
.57
.58
.59
0.7088
.7123
.7157
.7190
.7224
0.95
.96
.97
.98
.99
0.8289
.8315
.8340
.8365
.8389
1.35
.36
.37
.38
.39
0.9115
.9131
.9147
.9162
.9177
1.75
.76
.77
.78
.79
0.9599
.9608
.9616
.9625
.9633
2.15
.16
.17
.18
.19
0.98422
.98461
.98500
.98537
.98574
0.20
.21
.22
.23
.24
0.5793
.5832
.5871
.5910
.5948
0.60
.61
.62
.63
.64
0.7257
.7291
.7324
.7357
.7389
1.00
.01
.02
.03
.04
0.8413
.8438
.8461
.8485
.8508
1.40
.41
.42
.43
.44
0.9192
.9207
.9222
.9236
.9251
1.80
.81
.82
.83
.84
0.9641
.9649
.9656
.9664
.9671
2.20
.21
.22
.23
.24
0.98610
.98645
.98679
.98713
.98745
0.25
.26
.27
.28
.29
0.5987
.6026
.6064
.6103
.6141
0.65
.66
.67
.68
.69
0.7422
.7454
.7486
.7517
.7549
1.05
.06
.07
.08
.09
0.8531
.8554
.8577
.8599
.8621
1.45
.46
.47
.48
.49
0.9265
.9279
.9292
.9306
.9319
1.85
.86
.87
.88
.89
0.9678
.9686
.9693
.9699
.9706
2.25
.26
.27
.28
.29
0.98778
.98809
.98840
.98870
.98899
0.30
.31
.32
.33
.34
0.6179
.6217
.6255
.6293
.6331
0.70
.71
.72
.73
.74
0.7580
.7611
.7642
.7673
.7704
1.10
.11
.12
.13
.14
0.8643
.8665
.8686
.8708
.8729
1.50
.51
.52
.53
.54
0.9332
.9345
.9357
.9370
.9382
1.90
.91
.92
.93
.94
0.9713
.9719
.9726
.9732
.9738
2.30
.31
.32
.33
.34
0.98928
.98956
.98983
.99010
.99036
0.35
.36
.37
.38
.39
0.6368
.6406
.6443
.6480
.6517
0.75
.76
.77
.78
.79
0.7734
.7764
.7794
.7823
.7852
1.15
.16
.17
.18
.19
0.8749
.8770
.8790
.8810
.8830
1.55
.56
.57
.58
.59
0.9394
.9406
.9418
.9429
.9441
1.95
.96
.97
.98
.99
0.9744
.9750
.9756
.9761
.9767
2.35
.36
.37
.38
.39
0.99061
.99086
.99111
.99134
.99158
0.40
0.6554
0.80
0.7881
1.20
0.8849
1.60
0.9452
2.00
0.9772
2.40
0.99180
38
ENGINEERING TABLES AND DATA
The normal distribution function (continued)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
x
Φ(x)
2.40
.41
.42
.43
.44
0.99180
.99202
.99224
.99245
.99266
2.55
.56
.57
.58
.59
0.99461
.99477
.99492
.99506
.99520
2.70
.71
.72
.73
.74
0.99653
.99664
.99674
.99683
.99693
2.85
.86
.87
.88
.89
0.99781
.99788
.99795
.99801
.99807
3.00
.01
.02
.03
.04
0.99865
.99869
.99874
.99878
.99882
3.15
.16
.17
.18
.19
0.99918
.99921
.99924
.99926
.99929
2.45
.46
.47
.48
.49
0.99286
.99305
.99324
.99343
.99361
2.60
.61
.62
.63
.64
0.99534
.99547
.99560
.99573
.99585
2.75
.76
.77
.78
.79
0.99702
.99711
.99720
.99728
.99736
2.90
.91
.92
.93
.94
0.99813
.99819
.99825
.99831
.99836
3.05
.06
.07
.08
.09
0.99886
.99889
.99893
.99896
.99900
3.20
.21
.22
.23
.24
0.99931
.99934
.99936
.99938
.99940
2.50
.51
.52
.53
.54
0.99379
.99396
.99413
.99430
.99446
2.65
.66
.67
.68
.69
0.99598
.99609
.99621
.99632
.99643
2.80
.81
.82
.83
.84
0.99744
.99752
.99760
.99767
.99774
2.95
.96
.97
.98
.99
0.99841
.99846
.99851
.99856
.99861
3.10
.11
.12
.13
.14
0.99903
.99906
.99910
.99913
.99916
3.25
.26
.27
.28
.29
0.99942
.99944
.99946
.99948
.99950
2.55
0.99461
2.70
0.99653
2.85
0.99781
3.00
0.99865
3.15
0.99918
3.30
0.99952
The critical table below gives on the left the range of vaues of x for which Φ(x) takes the value on the right, correct to the last
figure given; in critical cases, take the upper of the two values for Φ(x) indicated.
0.9994
3.075
3.263
0.9990
0.99990
3.731
0.9995
3.105
3.320
0.9991
0.99991
3.759
0.9996
3.138
3.389
0.9992
3.480
0.9993
0.99992
3.615
0.9994
0.99997
4.055
0.99993
3.826
0.9998
3.215
0.99996
3.976
3.791
0.9997
3.174
0.99995
3.916
0.99998
4.173
0.99994
3.867
0.99999
4.417
0.9999
0.99995
1.00000
)
3
15 105
1
e
is very accurate, with relative error less than 945/x 10 .
+
−
+
1−
When x > 3.3 the formula 1 − Φ(x) √
2
4
x
x
x6
x8
x 2π
− 12 x 2
'
Percentage points of the normal distribution
This table gives percentage points x(P ) defined by the
equation
1
P
= √
100
2π
∞
P/100
1 2
e − 2 t dt .
x(P )
If X is a variable, normally distributed with zero mean and unit
variance, P /100 is the probability that X ≥ x(P ). The lower P
per cent points are given by asymmetry as −x(P ), and the
probability that |X | ≥ x(P ) is 2P /100.
0
x(P )
P
x(P )
P
x(P )
P
x(P )
P
x(P )
P
x(P )
P
x(P )
50
45
40
35
30
0.0000
0.1257
0.2533
0.3853
0.5244
5.0
4.8
4.6
4.4
4.2
1.6449
1.6646
1.6849
1.7060
1.7279
3.0
2.9
2.8
2.7
2.6
1.8808
1.8957
1.9110
1.9268
1.9431
2.0
1.9
1.8
1.7
1.6
2.0537
2.0749
2.0969
2.1201
2.1444
1.0
0.9
0.8
0.7
0.6
2.3263
2.3656
2.4089
2.4573
2.5121
0.10
0.09
0.08
0.07
0.06
3.0902
3.1214
3.1559
3.1947
3.2389
25
20
15
10
5
0.6745
0.8416
1.0364
1.2816
1.6449
4.0
3.8
3.6
3.4
3.2
1.7507
1.7744
1.7991
1.8250
1.8522
2.5
2.4
2.3
2.2
2.1
1.9600
1.9774
1.9954
2.0141
2.0335
1.5
1.4
1.3
1.2
1.1
2.1701
2.1973
2.2262
2.2571
2.2904
0.5
0.4
0.3
0.2
0.1
2.5758
2.6521
2.7478
2.8782
3.0902
0.05
0.01
0.005
0.001
0.0005
3.2905
3.7190
3.8906
4.2649
4.4172
MATHEMATICS
39
2
Percentage points of the χ distribution
This table gives percentage points χν2 (P ) defined by the
equation
P
1
=
ν
/2
100
2 Γ( ν2 )
∞
P/100
x 2 ν −1 e − 2 x dx .
1
1
χν2 (P )
0
2
If X is a variable distributed as χ with ν degrees of freedom,
√
2
P /100 is the probability that X ≥ χν (P ). For ν > 100, 2X is
√
approximately normally distributed with mean 2ν − 1 and
unit variance.
P
99.95
ν=1
2
3
4
0.0 3927
0.001000
0.01528
0.06392
0.05 1571
0.002001
0.02430
0.09080
0.04 3927
0.01003
0.07172
0.2070
0.03 1571
0.02010
0.1148
0.2971
5
6
7
8
9
0.1581
0.2994
0.4849
0.7104
0.9717
0.2102
0.3811
0.5985
0.8571
1.152
0.4117
0.6757
0.9893
1.344
1.735
10
11
12
13
14
1.265
1.587
1.934
2.305
2.697
1.479
1.834
2.214
2.617
3.041
15
16
17
18
19
3.108
3.536
3.980
4.439
4.912
20
21
22
23
24
99.5
99
The above shape applies for ν ≥ 3 only. When ν < 3 the mode
is at the origin.
95
90
80
70
60
0.03 9821
0.05064
0.2158
0.4844
0.003932
0.1026
0.3518
0.7107
0.01579
0.2107
0.5844
1.064
0.06418
0.4463
1.005
1.649
0.1485
0.7133
1.424
2.195
0.2750
1.022
1.869
2.753
0.5543
0.8721
1.239
1.646
2.088
0.8312
1.237
1.690
2.180
2.700
1.145
1.635
2.167
2.733
3.325
1.610
2.204
2.833
3.490
4.168
2.343
3.070
3.822
4.594
5.380
3.000
3.828
4.671
5.527
6.393
3.655
4.570
5.493
6.423
7.357
2.156
2.603
3.074
3.565
4.075
2.558
3.053
3.571
4.107
4.660
3.247
3.816
4.404
5.009
5.629
3.940
4.575
5.226
5.892
6.571
4.865
5.578
6.304
7.042
7.790
6.179
6.989
7.807
8.634
9.467
7.267
8.148
9.034
9.926
10.82
8.295
9.237
10.18
11.13
12.08
3.483
3.942
4.416
4.905
5.407
4.601
5.142
5.697
6.265
6.844
5.229
5.812
6.408
7.015
7.633
6.262
6.908
7.564
8.231
8.907
7.261
7.962
8.672
9.390
10.12
8.547
9.312
10.09
10.86
11.65
10.31
11.15
12.00
12.86
13.72
11.72
12.62
13.53
14.44
15.35
13.03
13.98
14.94
15.89
16.85
5.398
5.896
6.404
6.924
7.453
5.921
6.447
6.983
7.529
8.085
7.434
8.034
8.643
9.260
9.886
8.260
8.897
9.542
10.20
10.86
9.591
10.28
10.98
11.69
12.40
10.85
11.59
12.34
13.09
13.85
12.44
13.24
14.04
14.85
15.66
14.58
15.44
16.31
17.19
18.06
16.27
17.18
18.10
19.02
19.94
17.81
18.77
19.73
20.69
21.65
25
26
27
28
29
7.991
8.538
9.093
9.656
10.23
8.649
9.222
9.803
10.39
10.99
10.52
11.16
11.81
12.46
13.12
11.52
12.20
12.88
13.56
14.26
13.12
13.84
14.57
15.31
16.05
14.61
15.38
16.15
16.93
17.71
16.47
17.29
18.11
18.94
19.77
18.94
19.82
20.70
21.59
22.48
20.87
21.79
22.72
23.65
24.58
22.62
23.58
24.54
25.51
26.48
30
32
34
36
38
10.80
11.98
13.18
14.40
15.64
11.59
12.81
14.06
15.32
16.61
13.79
15.13
16.50
17.89
19.29
14.95
16.36
17.79
19.23
20.69
16.79
18.29
19.81
21.34
22.88
18.49
20.07
21.66
23.27
24.88
20.60
22.27
23.95
25.64
27.34
23.36
25.15
26.94
28.73
30.54
25.51
27.37
29.24
31.12
32.99
27.44
29.38
31.31
33.25
35.19
40
50
60
70
80
16.91
23.46
30.34
37.47
44.79
17.92
24.67
31.74
39.04
46.52
20.71
27.99
35.53
43.28
51.17
22.16
29.71
37.48
45.44
53.54
24.43
32.36
40.48
48.76
57.15
26.51
34.76
43.19
51.74
60.39
29.05
37.69
46.46
55.33
64.28
32.34
41.45
50.64
59.90
69.21
34.87
44.31
53.81
63.35
72.92
37.13
46.86
56.62
66.40
76.19
90
100
52.28
59.90
54.16
61.92
59.20
67.33
61.75
70.06
65.65
74.22
69.13
77.93
73.29
82.36
78.56
87.95
82.51
92.13
85.99
95.81
6
99.9
χν2(P )
97.5
40
ENGINEERING TABLES AND DATA
2
Percentage points of the χ distribution (continued)
P
50
ν=1
2
3
4
0.4549
1.386
2.366
3.357
5
6
7
8
9
4.351
5.348
6.346
7.344
8.343
10
11
12
13
14
9.342
10.34
11.34
12.34
13.34
15
16
17
18
19
40
30
20
10
0.7083
1.833
2.946
4.045
1.074
2.408
3.665
4.878
1.642
3.219
4.642
5.989
2.706
4.605
6.251
7.779
5.132
6.211
7.283
8.351
9.414
6.064
7.231
8.383
9.524
10.66
7.289
8.558
9.803
11.03
12.24
9.236
10.64
12.02
13.36
14.68
10.47
11.53
12.58
13.64
14.69
11.78
12.90
14.01
15.12
16.22
13.44
14.63
15.81
16.98
18.15
14.34
15.34
16.34
17.34
18.34
15.73
16.78
17.82
18.87
19.91
17.32
18.42
19.51
20.60
21.69
20
21
22
23
24
19.34
20.34
21.34
22.34
23.34
20.95
21.99
23.03
24.07
25.11
25
26
27
28
29
24.34
25.34
26.34
27.34
28.34
30
32
34
36
38
2.5
1
0.5
0.1
0.05
5.024
7.378
9.348
11.14
6.635
9.210
11.34
13.28
7.879
10.60
12.84
14.86
10.83
13.82
16.27
18.47
12.12
15.20
17.73
20.00
11.07
12.59
14.07
15.51
16.92
12.83
14.45
16.01
17.53
19.02
15.09
16.81
18.48
20.09
21.67
16.75
18.55
20.28
21.95
23.59
20.52
22.46
24.32
26.12
27.88
22.11
24.10
26.02
27.87
29.67
15.99
17.28
18.55
19.81
21.06
18.31
19.68
21.03
22.36
23.68
20.48
21.92
23.34
24.74
26.12
23.21
24.72
26.22
27.69
29.14
25.19
26.76
28.30
29.82
31.32
29.59
31.26
32.91
34.53
36.12
31.42
33.14
34.82
36.48
38.11
19.31
20.47
21.61
22.76
23.90
22.31
23.54
24.77
25.99
27.20
25.00
26.30
27.59
28.87
30.14
27.49
28.85
30.19
31.53
32.85
30.58
32.00
33.41
34.81
36.19
32.80
34.27
35.72
37.16
38.58
37.70
39.25
40.79
42.31
43.82
39.72
41.31
42.88
44.43
45.97
22.77
23.86
24.94
26.02
27.10
25.04
26.17
27.30
28.43
29.55
28.41
29.62
30.81
32.01
33.20
31.41
32.67
33.92
35.17
36.42
34.17
35.48
36.78
38.08
39.36
37.57
38.93
40.29
41.64
42.98
40.00
41.40
42.80
44.18
45.56
45.31
46.80
48.27
49.73
51.18
47.50
49.01
50.51
52.00
53.48
26.14
27.18
28.21
29.25
30.28
28.17
29.25
30.32
31.39
32.46
30.68
31.79
32.91
34.03
35.14
34.38
35.56
36.74
37.92
39.09
37.65
38.89
40.11
41.34
42.56
40.65
41.92
43.19
44.46
45.72
44.31
45.64
46.96
48.28
49.59
46.93
48.29
49.64
50.99
52.34
52.62
54.05
55.48
56.89
58.30
54.95
56.41
57.86
59.30
60.73
29.34
31.34
33.34
35.34
37.34
31.32
33.38
35.44
37.50
39.56
33.53
35.66
37.80
39.92
42.05
36.25
38.47
40.68
42.88
45.08
40.26
42.58
44.90
47.21
49.51
43.77
46.19
48.60
51.00
53.38
46.98
49.48
51.97
54.44
56.90
50.89
53.49
56.06
58.62
61.16
53.67
56.33
58.96
61.58
64.18
59.70
62.49
65.25
67.99
70.70
62.16
65.00
67.80
70.59
73.35
40
50
60
70
80
39.34
49.33
59.33
69.33
79.33
41.62
51.89
62.13
72.36
82.57
44.16
54.72
65.23
75.69
86.12
47.27
58.16
68.97
79.71
90.41
51.81
63.17
74.40
85.53
96.58
55.76
67.50
79.08
90.53
101.9
59.34
71.42
83.30
95.02
106.6
63.69
76.15
88.38
100.4
112.3
66.77
79.49
91.95
104.2
116.3
73.40
86.66
99.61
112.3
124.8
76.09
89.56
102.7
115.6
128.3
90
100
89.33
99.33
92.76
102.9
96.52
106.9
113.1
124.3
118.1
129.6
124.1
135.8
128.3
140.2
137.2
149.4
140.8
153.2
101.1
111.7
107.6
118.5
5
3.841
5.991
7.815
9.488
PROPERTIES OF MATTER
Part 3
Properties of matter
41
42
ENGINEERING TABLES AND DATA
Physical constants
Universal gas constant
Boltzmann constant
Universal gravitational
constant
Mean radius of earth
Mass of earth
Gravitational acceleration
(standard gravity)
Stefan-Boltzmann constant
Avogadro number
Loschmidt number
Molar volume†
Planck constant
∗
†
8.315 × 103 J kmol−1 K−1∗
−23
−1
1.38 × 10
JK
−5
8.617 × 10 eV K−1
R0
k
G
R
M
6.67 × 10−11 N m2 kg−2
6371 km
5.976 × 1024 kg
g
σ
N
9.807 m s−2
5.670 × 10−8 W m−2 K−4
6.022 × 1026 kmol−1
25
−3
2.687 × 10 m
3
−1
22.41 m kmol
6.626 × 10−34 J s
4.136 × 10−15 eV s
h
−27
Atomic mass unit (a.m.u.)
Velocity of light in vacuum
Absolute permittivity of
free space
Absolute permeability of
free space
Charge of an electron
Rest mass of an electron
Charge/mass ratio of an
electron
Mass of a proton
Impedance of free space
Bohr magneton
Wavelength of 1 eV photon
Faraday constant
kg
1.661 × 10
8
−1
2.998 × 10 m s
c
8.854 × 10−12 F m−1
0
μ0
e
me
4π × 10−7 H m−1
1.602 × 10−19 C
−31
9.110 × 10
kg
e/me
mp
1.759 × 1011 C kg−1
−27
1.673 × 10
kg
376.7 Ω
9.274 × 10−24 A m2
1.240 μm
7
−1
9.649 × 10 C kmol
μB
F
One kilomole of a substance is that quantity which contains as many of
its (specified) particles as there are atoms in 12 kg of carbon 12.
of a specified gas at 0 ◦ C, 1 atm (1.01325 bar).
Periodic table
IA
IIA
1
2
H
He
IIIB
IVB
VB
VIB VIIB
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
13
14
15
16
17
18
Al
Si
P
S
Cl
Ar
11
12
Na
Mg
IIIA
IVA
VA
VIA VIIA
| ←− VIII −→ |
IB
IIB
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
55
56
57
Cs
Ba
La
87
88
89
Fr
Ra
Ac
φ
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
⎫
⎪
⎪
⎪
⎪
⎬
θ
φ
58
59
60
61
62
63
64
65
66
67
68
69
70
71
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
θ
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
⎪
⎪
⎪
⎪
⎭
IIIA
PROPERTIES OF MATTER
43
Atomic properties of the elements
(for free neutral atoms in the ground state)
Z
Atomic number
AW
Atomic weight in a.m.u. (6 C12 = 12.000)
Vi
First ionization potential in eV
K, L, M, N, O, P, Q
Principal quantum number = 1,2,3,4,5,6,7
s, p, d, f, g, h
Azimuthal quantum number = 0,1,2,3,4,5
K
L
Z Element
AW
Vi
1s 2s 2p 3s
1 H
Hydrogen
1.008 13.5 1
2 He Helium
4.003 24.5 2
3 Li
Lithium
6.941
5.4 2
1
4 Be Beryllium
9.012
9.3 2
2
5 B
Boron
10.811
8.3 2
2
1
6 C
Carbon
12.011 11.2 2
2
2
7 N
Nitrogen
14.007 14.5 2
2
3
8 O
Oxygen
15.999 13.6 2
2
4
9 F
Fluorine
19.998 17.3 2
2
5
10 Ne Neon
20.179 21.5 2
2
6
11 Na Sodium
22.990
5.1 2
2
6
1
12 Mg Magnesium
24.305
7.6 2
2
6
2
13 Al
Aluminium
26.982
6.0 2
2
6
2
14 Si
Silicon
28.086
8.1 2
2
6
2
15 P
Phosphorus
30.974 10.9 2
2
6
2
16 S
Sulphur
32.066 10.3 2
2
6
2
17 Cl
Chlorine
35.453 13.0 2
2
6
2
18 Ar Argon
39.948 15.7 2
2
6
2
19 K
Potassium
39.098
4.3 2
2
6
2
20 Ca Calcium
40.078
6.1 2
2
6
2
21 Sc Scandium
44.956
6.7 2
2
6
2
22 Ti
Titanium
47.88
6.8 2
2
6
2
23 V
Vanadium
50.942
6.7 2
2
6
2
24 Cr Chromium
51.996
6.7 2
2
6
2
25 Mn Manganese
54.938
7.4 2
2
6
2
26 Fe Iron
55.847
7.8 2
2
6
2
27 Co Cobalt
58.933
7.8 2
2
6
2
28 Ni
Nickel
58.69
7.6 2
2
6
2
29 Cu Copper
63.546
7.7 2
2
6
2
30 Zn Zinc
65.39
9.4 2
2
6
2
31 Ga Gallium
69.723
6.0 2
2
6
2
32 Ge Germanium
72.59
8.1 2
2
6
2
33 As Arsenic
74.922 10.5 2
2
6
2
34 Se Selenium
78.96
9.7 2
2
6
2
35 Br Bromine
79.904 11.8 2
2
6
2
36 Kr Krypton
83.80
13.9 2
2
6
2
37 Rb Rubidium
85.468
4.2 2
2
6
2
38 Sr Strontium
87.62
5.7 2
2
6
2
39 Y
Yttrium
88.906
6.5 2
2
6
2
40 Zr
Zirconium
91.224
6.9 2
2
6
2
41 Nb Niobium
92.906
2
2
6
2
42 Mo Molybdenum
95.94
7.4 2
2
6
2
43 Tc Technetium
2
2
6
2
44 Ru Ruthenium
101.07
7.7 2
2
6
2
45 Rh Rhodium
102.906
7.7 2
2
6
2
46 Pd Palladium
106.42
8.3 2
2
6
2
47 Ag Silver
107.868
7.5 2
2
6
2
48 Cd Cadmium
112.41
9.0 2
2
6
2
49 In
Indium
114.82
5.8 2
2
6
2
50 Sn Tin
118.710
7.3 2
2
6
2
51 Sb Antimony
121.75
8.5 2
2
6
2
52 Te Tellurium
127.60
9.0 2
2
6
2
53 I
Iodine
126.905 10.6 2
2
6
2
54 Xe Xenon
131.29
12.1 2
2
6
2
M
3p
1
2
3
4
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
N
3d
4s
4p
1
2
3
5
5
6
7
8
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
1
2
2
2
2
1
2
2
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
3
4
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
O
4d
1
2
4
5
6
7
8
10
10
10
10
10
10
10
10
10
4f
5s
5p
1
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
2
1
2
3
4
5
6
5d
5f
44
Z
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
ENGINEERING TABLES AND DATA
Element
Cs Caesium
Ba Barium
La Lanthanum
Ce Cerium
Pr
Praseodymium
Nd Neodymium
Pm Promethium
Sm Samarium
Eu Europium
Gd Gadolineum
Tb Terbium
Dy Dysprosium
Ho Holmium
Er
Erbium
Tm Thulium
Yb Ytterbium
LU Lutetium
Hf
Hafnium
Ta Tantalum
W
Tungsten
Re Rhenium
Os Osmium
Ir
Iridium
Pt
Platinum
Au Gold
Hg Mercury
Tl
Thallium
Pb Lead
Bi
Bismuth
Po Polonium
At
Astatine
Rn Radon
Fr
Francium
Ra Radium
Ac Actinium
Th Thorium
Pa Protactinium
U
Uranium
Np Neptunium
Pu Plutonium
Am Americium
Cm Curium
Bk Berkelium
Cf
Californium
Es Einsteinium
Fm Fermium
Md Mendelevium
No Nobelium
Lr
Lawrencium
Rf
Rutherfordium
Db Dubnium
Sg Seaborgium
Bh Bohrium
Hs Hassium
Mt Meitnerium
AW
132.905
137.33
138.906
140.12
140.908
144.240
Vi
3.9
5.2
5.6
6.5
5.8
6.3
150.36
151.96
157.25
158.925
162.50
164.930
167.26
168.934
173.04
174.967
178.49
180.948
183.85
186.2
190.2
192.22
195.08
196.967
200.59
204.383
207.2
208.98
6.6
5.6
6.7
6.7
6.8
232.038
238.029
7.1
8.1
8.9
9.2
10.4
6.1
7.4
8.0
K
L
M
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
N
4s 4p 4d 4f
2
6 10
2
6 10
2
6 10
2
6 10
2
2
6 10
3
2
6 10
4
2
6 10
5
2
6 10
6
2
6 10
7
2
6 10
7
2
6 10
9
2
6 10 10
2
6 10 11
2
6 10 12
2
6 10 13
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
O
5s 5p 5d 5f
2
6
2
6
2
6
1
2
6
2
6
2
6
2
6
2
6
2
6
2
6
1
2
6
2
6
2
6
2
6
2
6
2
6
2
6
1
2
6
2
2
6
3
2
6
4
2
6
5
2
6
6
2
6
9
2
6
9
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
6 10
2
2
6 10
3
2
6 10
4
2
6 10
6
2
6 10
7
2
6 10
7
2
6 10
9
2
6 10 10
2
6 10 11
2
6 10 12
2
6 10 13
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
2
6 10 14
P
6s 6p 6d 6f 6g 6h
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
3
4
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
1
2
1
1
1
1
1
2
3
4
5
6
7
Q
7s
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
PROPERTIES OF MATTER
45
Physical properties of solids
CS
ρ
tm
tb
hif
hfg
Crystal structure:
BCC body-centred cubic
FCC face-centred cubic
CPH close-packed hexagonal
Mass density (kg dm−3 ) ≈ specific gravity
Melting point ( ◦ C)
Boiling point ( ◦ C)
Latent heat of fusion (kJ kg−1 )
Latent heat of vaporization (kJ kg−1 )
CS
Metallic elements
Aluminium
FCC
Carbon (Diamond) Diamond
Carbon (Graphite)
Hexagonal
Copper
FCC
Gold
FCC
Iron
BCC / FCC
Lead
FCC
Nickel
FCC
Platinum
FCC
Silicon
Diamond
Silver
FCC
Tantalum
BCC
Tin
Diamond/tetragonal
Titanium
CPH / BCC
Tungsten
BCC
Zinc
CPH
∗
Sublimation temperature
r
tan δ
ρ
2.7
3.51
2.25
8.96
19.3
7.9
11.3
8.9
21.5
2.3
10.5
16.6
7.3
4.5
19.3
7.1
ρ
Thermal conductivity at or near 0 ◦ C (W m−1 K−1 )
◦
−1 −1
Specific heat capacity at or near 0 C (J kg K )
−1
Coefficient of linear thermal expansion (K ) × 106
Electrical resistivity at 20 ◦ C (units as shown)
Temperature coefficient of resistance,
0–100 ◦ C (K−1 ×103 )
Dielectric constant (relative permittivity)
at 1 MHz, 20 ◦ C
◦
−4
Loss factor at ∼ 1 MHz, 20 C (units of 10 )
k
c
α
ρe
αe
tm
tb
hif
hfg
660
2400 387 9460
transforms to graphite
−
3652∗
1083
2580 205 5230
1063
2660
66 1750
1535
2900 270 6600
327
1750
24
850
1453
2820 305 5850
1769
3800 113 2400
1412
2355
961
2180 105 2330
3000
5300 160
232
∼ 2500
59 2400
1680
3300 435
3380
∼ 6000 185
420
907 110 1750
tm ∗
k
c
k
c
α
ρe (nΩ m)
αe
205
70
880
430
610
380
145
437
126
444
125
700
232
140
224
500
130
384
23
1.2
27
4.2
390
310
76
35
91
69
84
418
54
64
17
190
113
α
Alloys
Aluminium 2024 (4.5% Cu)
2.8
640
147
900
22.5
Brass (70/30)
8.55
965
121
370
20
Cast iron: grey
7.0
1250
180
13
Cast iron: nodular
7.1
1150
65
461
11
Constantan (60% Cu)
8.9
1280
22
410
16
Manganin (84% Cu)
8.5
950
22
405
19
Nimonic 80A (superalloy)
8.19
1400
11.2
460
12.7
Nichrome (80/20)
8.36
1420
13
430
12.5
Phosphor-bronze (5% Sn)
8.85
1060
∼ 75
380
18
Solder (soft) (50% Sn)
8.9
215
50
210
23
Steel (mild)
∼ 7.85
∼ 1500
∼ 50
∼ 450
∼ 11
Steel (austenitic stainless)
7.9
1500
16
∼ 500
16
Titanium-6Al-4V
4.42
1700
5.8
610
8
∗
Alloys generally do not have a unique melting point; solid and liquid co-exist over a freezing range.
The temperature tm given here is the maximum temperature of this range.
17
14
12
29
13
9
7.6
19
6
23
9
4.5
31
16.8
4.3
23
3.9
97
6.5
206
4.3
68
6.8
106
3.9
104 –107
16
4.1
135
∼ 3.5
120
∼ 4.5
550
∼ 3.5
55
4.6
59
4.2
ρe (nΩ m)
∼ 52
62
500
150
490
440
1170
1030
105
150
∼ 120
850
1680
αe
∼ 2.3
1.6
∼ 0.02
∼0
0.1
3.5
∼ 3.0
0.4
46
ENGINEERING TABLES AND DATA
ρ
tm
k
c
α
39
0.4−0.8
1.0−1.5
1050
800
1100
8
3−9
10−14
1.0
1.0
2−4
2.3
∼ 0.5
0.8−1.85
5−9
1.3
1.1−2.3
84
17
84
1.5
990
800
800
2100
840
1100
730
840
900
1422
627
−
670
8.5
4.0
6−9
ρe (MΩ m)
r
tan δ
Non-metals
Ceramics and brittle materials
Alumina
3.9
Brick
1.4−2.2
Concrete
2.4
Dry ground
∼ 1.6
Glass (soda)
2.48
Glass (borosilicate) 2.23
Granite
2.7
†
Ice
0.92
Mica
2.8
Porcelain
2.4
2.65
Quartz (crystal)
2.2
Quartz (fused)
Sandstone
2.4
3.2
Silicon carbide
3.2
Silicon nitride
Tungsten carbide 15.7
Zirconia
5.6
2050
750∗
950
0
1550
2840
1900
2777
2570
2.2
7.5−13.7
0.5
5−12
4.3
3.2
4.9
8
3
6
10 −10
1−2
0.01−0.1
5 × 104
4.5−8.4
5.8
2−100
13−100
7−9
105 −109
104 −107
106 −2 × 108
10
10
5−7
5.5−7
4.5−5
3.8
10
1−2
60−100
2
Polymers, composites and natural materials
Carbon-fibre
1.5−2.0
reinforced plastic
Epoxy resin
1.2−1.4
Glass-fibre
1.5−2.5
reinforced plastic
Nylon 6
1.14
Paper (dry)
∼ 1.0
Perspex
1.2
Polybutadiene
1.5
(synthetic rubber)
Polypropylene
0.91
Polystyrene
1.06
Polythene
0.91−0.94
(low density)
Polythene
0.95−0.98
(high density)
PTFE
2.2
PVC (plasticized)
1.7
PVC (unplasticized) 1.4
Rubber (natural,
1.1−1.2
vulcanized)
Timber (along grain) 0.4−0.8
∗
Softening temperature
†
hif = 333 kJ kg−1
−
0.3−1.0
130−170∗
−
0.2−0.5
0.3−1.0
−
1700–2000
−
200−220∗
0.25−0.33
1600
0.06
85−115∗ 0.19−0.23
1450
∼ 90∗
∼ 0.15
∼ 2500
40∗
80−105∗
∗
80
110
∗
∗
70 − 80
70 − 80∗
125
3−5
55−90
15−20
80−130
1900
1300
2250
100−300
60−80
160−190
0.52
2100
150−300
1050
90−130
50−250
50−70
∼ 200
∼ 0.15
∼ 1600
3−7
1.9−2.9
2.5−3.5
200−1300
20−45
160−300
1010
105
2.4−3.5
2.3
< 20
2−5
2.1
4−6
4−6
2−3.5
<3
600
600
280
2−9
350−600
50−80
∼ 600
0.2
0.08 − 0.2
0.35
0.23−0.27
0.16−0.19
0.15
∼ 0.15
104 −107
104
3−5
109
104 −107
104 −107
7
∼ 10
PROPERTIES OF MATTER
47
Mechanical properties of solids
E
G
K
ν
Young’s modulus (GPa)
Shear modulus (GPa)
Bulk modulus (GPa)
Poisson’s ratio
σy
σf
f
KIc
Proof or yield stress (MPa)
Ultimate (failure) stress or tensile strength (MPa)
Tensile strain to failure (%)
1
Fracture toughness (MPa m 2 )
Values of σy and σf , particularly, usually depend strongly on the preparation and condition of a material. The ranges given are
typical but not necessarily exhaustive, and unless otherwise stated those for metals refer to drawn or wrought, rather than cast,
material of commercial purity.
E
G
K
Metallic elements in the annealed state
Aluminium
70
Copper
130
Gold
79
Iron
211
Lead
16
Nickel
200
Platinum
170
Silicon
107
Silver
83
Tantalum
186
Tin
47
Titanium
120
Tungsten
411
Zinc (wrought)
105
25
48
26
82
6
76
61
62
30
69
17
46
161
42
75
138
171
170
46
176
280
Alloys
Aluminium 2024 (age-hardened)
Brass (70/30): annealed
Brass (70/30): rolled
Cast iron: grey
Cast iron: nodular
Constantan (60% Cu)
Manganin (84% Cu)
Mumetal (77% Ni)
Nimonic 80A (superalloy)
Nichrome (80/20)
Phosphor-bronze (5% Sn)
Solder (soft) (50% Sn)
Steel: mild
Steel: high-yield structural
Steel: ultra high strength
Steel: austenitic stainless
Titanium-6Al-4V
72
101
101
100−145
169−172
163
124
220
214
186
100
40
210
210
28
37
37
40−58
66
61
47
190−200
115
74−86
104
196
52
108
311
69
75
112
112
157
ν
0.34
0.34
0.42
0.29
0.44
0.31
0.39
0.42
0.37
0.34
0.36
0.36
0.28
0.25
160−170
170
σf
15−20
33
40−50
210
100
350
12
310
140−195
80−100
60
14−35
5000−9000
180
9−14
100−225
550
0.33
0.35
0.35
0.26
0.28
0.33
395
115
390
100−260
230−460
200−440
0.35
800
100−400
110−670
33
240
400
1600
255
800−900
0.38
81
81
σy
0.27−0.30
0.30
0.25−0.29
170
200
15−200
240−370
550−620
110−200
475
320
460
150−400
370−800
400−570
465
500−900
1300
170−900
330−750
42
400−500
600
2000
660
900−1000
f
50−70
60
30
40
30−40
0
50
35−45
70−80
0
40−80
10
67
20
2−17
20
2−50
60
10−20
20
10
45
10−20
48
ENGINEERING TABLES AND DATA
E
ν
380
10−50
10−17
1050
74
65
80
40−70
9.1
70
14−55
107
410
310
200
0.24
σf (tension)
σf (compression)
KIc
Non-metals
Ceramics and brittle materials
Alumina
Brick (grade A)
Concrete (28 day)
Diamond
Glass (soda)
Glass (borosilicate)
Gallium arsenide
Granite
Ice
Porcelain
Sandstone
Silicon
Silicon carbide
Silicon nitride
Zirconia
300−400
0.1−0.21
0.13
0.23
0.25
0.24
−
50
55
−
−
0.25
1.7
45
0.42
0.13
0.15
0.30
−
200−500
300−850
500−830
3000
69−140
27−55
5000−10000
1000
1200
1000−3000
90−235
6
350
30−135
5000−9000
2000
1200
2000
3−5
∼ 1−2
0.2−0.4
0.7
0.8
∼1
∼1
0.12
1.1
3−5
4
12
Polymers, composites and natural materials
Carbon-fibre reinforced plastic
Epoxy resin
Glass-fibre reinforced plastic
Nylon 6
Perspex
Polybutadiene
Polypropylene
Polystyrene
Polythene (low density)
Polythene (high density)
PTFE
PVC (plasticized)
PVC (unplasticized)
Timber (along grain)
Pine, oak, ash
Balsa
Timber (across grain)
Pine, oak, ash
Balsa
150−250
2.1−5.5
80−100
2−3.5
3.3
0.004−0.1
1.2−1.7
3.0−3.3
0.15−0.24
0.1−1.0
0.35
∼ 0.3
2.4−3.0
−
0.38−0.4
−
1000−1500
40−85
∼ 1000
50−100
80−90
∼ 10
50−70
35−68
7−17
20−37
17−28
75−100
40−60
7−20
2−5
80−150
20−40
0.7−1.5
0.1−0.3
10−40
3−10
−
100−200
−
60−110
80−140
∼ 60
0.6−1.0
∼ 90
3−5
1.6
10−110
15−20
3.5
2
1−2
2−5
75−100
2.4
20−80
10−30
7−10
2−3
0.7−1
0.1−0.2
PROPERTIES OF MATTER
49
Properties of reinforcing fibres
r
ρ
E
σf
fibre radius (μm)
mass density (kg dm−3 )
Young’s modulus (GPa)
failure stress (MPa)
α -alumina
Alumino-silicate (Saffil)
Carbon (high modulus)
Carbon (high strength)
E-glass
Kevlar
Silicon carbide (Nicalon)
Silicon carbide
(monofilament)
axial coefficient of thermal expansion (K−1 ) ×106
radial coefficient of thermal expansion (K−1 ) ×106
αl
αr
r
ρ
E
σf
10−20
∼3
7−10
8−9
10−12
12
10−15
3−4
2.8
1.95
1.75
2.5
200−380
103
390
250
76
130
170−200
1300−1700
1030
2200
2700
1400−2500
2800−3600
5300−6500
100−150
3
400−410
8000−9000
αl
αr
4−9
4−9
∼ −1.0
∼ −0.3
4.9
−2
7−12
7−12
4.9
59
Work functions
φt and φp are the least energies, in electron volts, to extract an electron from the element by thermionic and photoelectric
emission respectively.
Element
φt
φp
Element
φt
φp
Element
φt
φp
Aluminium
Barium
Bismuth
Caesium
Calcium
Carbon
Chromium
Copper
4.3
2.7
4.3
2.2
3.0
5.0
4.6
4.7
4.1
2.5
4.3
1.9
2.7
4.8
4.4
4.3
Gold
Iron
Mercury
Molybdenum
Nickel
Platinum
Potassium
Silver
5.2
4.5
4.5
4.7
5.2
5.7
2.3
4.3
4.8
4.6
4.5
4.2
5.0
6.3
2.2
4.7
Sodium
Tantalum
Thorium
Titanium
Tungsten
Zinc
2.8
4.3
3.4
4.3
4.6
4.4
2.3
4.1
3.5
4.2
4.5
4.2
50
ENGINEERING TABLES AND DATA
Properties of semiconductors
(at room temperature)
W
Atomic or molecular weight
N
Number density of atoms (m−3 ) ×10−28
a
Lattice constant (Å)
−3
ρ
Mass density (kg dm )
∗
me /mo Effective mass of electron
relative to free rest mass mo
mh∗ /mo Effective mass of hole (light, heavy
and split-off) relative to mo
Ni
μe
μh
r
EB
me∗ /mo mh∗ /mo Eg
W
N
Carbon C
12.01
17.60
3.57
3.51
0.2
Germanium Ge
72.60
4.42
5.65
5.33
0.12
Silicon Si
28.09
5.00
5.43
2.33
0.26
Aluminium arsenide
101.90
AlAs
Aluminium antimonide 148.73
AlSb
Gallium phosphide
100.69
GaP
4.50
5.66
3.81
0.35
3.42
6.14
4.22
0.1
4.94
5.45
4.13
0.35
a
Energy gap (eV)
Effective density of states in conduction
and valence bands (m−3 ) ×10−25
Intrinsic carrier concentration (m−3 )
2 −1 −1
Mobility of electrons (cm V s )
2 −1 −1
Mobility of holes (cm V s )
Dielectric constant at low frequencies
Breakdown field (V μm−1 )
Eg
Nc ,Nv
ρ
0.25
5.47
⎫
0.04 ⎬
0.67
0.28
⎭
0.08
⎫
0.16 ⎬
1.11
0.50
⎭
0.24
Nc
Nv
Ni
μe
μh
1800 1200
19
1.04
0.60 2.4×10
2.8
1.04 1.45×1016 1500
r
5.7
3900 1900 16.3
1200
420 10.1
0.5
1.6
200
500 14.4
0.6
2.25
190
120 11.1
Gallium arsenide
GaAs
144.63
4.42
5.65
5.32
⎫
0.082 ⎬
0.067 0.5
1.42
⎭
0.16
Gallium antimonide
GaSb
Indium phosphide
InP
Indium arsenide
InAs
Indium antimonide
InSb
Cadmium sulphide
CdS
Cadmium selenide
CdSe
191.47
3.53
6.10
5.62
0.04
0.4
0.7
5000 1400 15.7
145.79
3.96
5.87
4.79
0.07
0.8
1.3
4600
150 12.4
189.74
3.59
6.06
5.66
0.03
0.4
0.36
33000
460 14.6
236.58
2.94
6.48
5.80
0.013 0.4
0.17
80000 1250 17.7
144.48
4.04
5.83
4.82
0.21
0.8
2.42
340
191.36
3.65
6.05
5.8
0.13
0.5
1.8
650
10.0
5.9
0.13
0.35
1.6
1100
80 10.2
5.7
4.1
7.6
8.3
8.2
0.38
0.4
3.3
3.8
0.37
0.26
0.25
200 180 9.0
150
5.2
800 1000 17.0
1500 1500
1600 750 30.0
Cadmium telluride
CdTe
240.01
2.96
6.48
3.25
Zinc Oxide ZnO
Zinc sulphide ZnS
Lead sulphide PbS
Lead selenide PbSe
Lead telluride PbTe
81.37
97.43
239.3
286.2
334.8
8.43
5.07
3.82
3.49
2.95
5.21
5.41
5.94
6.12
6.45
8500
∼8
450 11.7 ∼30
2.16
0.047 0.70 1.8×1012
EB
400 13
50
5.4
∼40
PROPERTIES OF MATTER
51
Properties of ferromagnetic materials
Soft materials
Bsat
(μr )max
ρe
w
Saturated flux density (T)
Maximum relative permeability
Resistivity (nΩ m)
Hysteresis loss per cycle (mJ kg−1 )
Iron
Iron (4% Si)
Grain-oriented silicon-iron
Permalloy B, Radiometal (50% Ni)
Supermalloy (79% Ni, 5% Mo)
Permalloy C, Mumetal (77% Ni, 5% Cu)
Ferrite MnZn (Fe2 O3 )2
∗
Bsat
(μr )max
ρe
2.15
1.97
2.00
1.5
0.79
0.65
∼ 0.4
5000
10000
45000
30000
106
5
10
2500
100
600
550
500
600
620
2 × 108
w
30∗
30∗
25∗
0.1
1.0
Approximate values at Bmax = 1.3 T
Hard materials
Br
Hc
(BH)max
Remanent flux density (T)
Coercive force (kA m−1 )
is in units of kJ m−3
Steel (1% C)
Steel (6% W)
Steel (35% Co)
Alnico 5 (14% Ni, 24% Co, 8% Al)
Ferroxdur, BaO (Fe2 O3 )6
Fe-Co Powder
Alnico 9
Br
Hc
(BH)max
0.80
1.05
0.90
1.25
0.35
0.90
1.05
4.8
5.2
18.5
46
160
82
130
1.44
2.40
7.20
20
12
40
100
Superconducting materials
The critical field Hc for a superconducting material at an absolute temperature T depends on the critical field at absolute zero H0 ,
according to the relation Hc = H0 {1 − (T /Tc )2 } where Tc is the critical temperature. Values of H0 and Tc for some
superconductors are given below.
Aluminium
Gallium
Indium
Lead
Mercury, α
Mercury, β
Niobium
Tantalum
Thorium
Tin
Vanadium
Tc (K)
H0 (kA m−1 )
1.19
1.09
3.41
7.18
4.15
3.95
9.46
4.48
1.37
3.72
5.30
8
4
23
65
33
27
156
67
13
25
105
Zinc
Nb3 Sn
NbN
Nb3 (Al0.8 Ge0.2 )
V3 Si
V3 Ga
La2−x Srx CuO4−x
Y1 Ba2 Cu3 O7−x
Bi2 Sr2 Ca2 Cu3 Ox
Tl2 Ba2 Ca2 Cu3 Ox
Tc (K)
H0 (kA m−1 )
0.92
18.5
16
20
17
16.8
30
92
108
125
4
0.2
2
4
6 8 10
Wavelength (μm)
0.1
0.1
2
0.1
0.1
0.4 0.6 0.81
0.2
0.2
2
4
10
8
6
20
40
100
80
60
0.4
n
6 8 10
0.1
0.1
0.2
0.4
k
4
k < 10
-4
0.4
1
0.8
0.6
2
4
1
0.8
0.6
0.2
n
Silicon
Wavelength (μm)
0.4 0.6 0.81
Aluminium
k
10
8
6
1
0.8
0.6
2
4
10
8
6
20
40
100
80
60
0.1
0.1
0.2
0.4
1
0.8
0.6
2
4
Refractive Index
Refractive Index
Refractive Index
Refractive Index
0.2
0.2
Gold
k
2
2
Wavelength (μm)
0.4 0.6 0.81
Wavelength (μm)
0.4 0.6 0.81
k < 10
-4
n
n
k
Silicon Dioxide (glass)
4
4
6 8 10
6 8 10
k
Refractive Index
Refractive Index
10
8
6
0.1
0.1
0.2
0.4
1
0.8
0.6
2
4
10
8
6
20
40
100
80
60
0.1
0.1
0.2
0.4
1
0.8
0.6
2
4
10
8
6
0.2
Silver
0.2
2
2
Wavelength (μm)
0.4 0.6 0.81
n
k
4
-4
4
k < 10
6 8 10
6 8 10
Gallium Arsenide
Wavelength (μm)
0.4 0.6 0.81
k
n
52
ENGINEERING TABLES AND DATA
Optical properties
Refractive index
The following curves show the variation with wavelength of the real part (n ) and imaginary part (k , the extinction coefficient) of
the refractive index.
PROPERTIES OF MATTER
53
Attenuation of an optical fibre
Attenuation (dB km-1)
The curve below shows the attenuation of a commercial single-mode silica fibre, with germanium-doped core and matched
cladding doped with phosphorus and fluorine.
1.0
0.5
0
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Wavelength (μm)
Properties of liquids
ρ
K
tf ,tb
hfg
k (t)
Mass density (kg dm−3 ) ≈ specific gravity
Bulk modulus (GPa)
Freezing, boiling points ( ◦ C)
Latent heat of boiling (kJ kg−1 ) at ∼ 1 atm
Thermal conductivity (at t ◦ C) (W m−1 K−1 )
ρ
Mercury
Sodium
Water
Sea-water
Mineral oil
Carbon tetrachloride, CCl4
Acetone, C3 H5 OH
Ethyl alcohol, C2 H5 OH
Nitrogen
∗
†
◦
At 100 C
Distilled water
13.5
1.00
1.03
∼0.9
1.6
0.79
0.79
0.81
K
25
tf
−39
98
0
−2.5
2.3
2.3
1−2
1.1
−23
1.2
−95
1.3 −120
−210
c
β
μ
ρe
r
tb
357
883
100
103
77
57
78
−196
hfg
292
2260
215
560
850
Specific heat capacity at 20 ◦ C (J kg−1 K−1 )
Coefficient of volume thermal expansion (K−1 ×10−3 )
◦
Dynamic viscosity at 20 C (mPa s)
Electrical resistivity at or near 20 ◦ C (Ω m)
Dielectric constant (relative permittivity)
at 1 MHz, 20 ◦ C
k (t)
8 (20)
85 (130)
0.61 (20)
0.13 (100)
0.11 (20)
0.16 (20)
0.17 (20)
0.15 (−203)
c
β
μ
139
0.18
1.55
4180
3930
1700
840
2210
2500
0.21
1.00
0.7−0.9
1.22
0.97
1.43
0.32
1.08
1.20
ρe
r
−7
9.6×10
10−7∗
†
5000
0.2−0.3
3×1010 −1015
3000
81
∼2.2
2.2
21
26
54
ENGINEERING TABLES AND DATA
Thermodynamic properties of fluids
Saturated water and steam, to 100 ◦ C
t
p
v
u
h
s
◦
Temperature ( C)
Absolute pressure (bar)
Specific volume (m3 kg−1 ) ×103
Specific internal energy (kJ kg−1 )
Specific enthalpy (kJ kg−1 )
Specific entropy (kJ kg−1 K−1 )
The subscripts s, f, g and fg
indicate respectively saturation,
liquid, vapour and liquid-vapour
phase change.
ts
p
vf
vg
0.01
0.006112
1.0002
206163
1
2
3
4
5
0.006566
0.007055
0.007575
0.008129
0.008718
1.0001
1.0001
1.0001
1.0000
1.0000
6
7
8
9
10
0.009345
0.010012
0.010720
0.011472
0.012270
12
14
16
18
20
uf
ufg
ug
0.0
2375.6
2375.6
192607
179923
168169
157272
147163
4.2
8.4
12.6
16.8
21.0
2372.7
2369.9
2367.1
2364.3
2361.4
2376.9
2378.3
2379.7
2381.1
2382.4
1.0000
1.0001
1.0001
1.0002
1.0003
137780
129064
120966
113435
106430
25.2
29.4
33.6
37.8
42.0
2358.6
2355.8
2352.9
2350.2
2347.3
2383.8
2385.2
2386.5
2388.0
2389.3
0.014014
0.015973
0.018168
0.020624
0.023366
1.0004
1.0007
1.0010
1.0013
1.0017
93835
82900
73384
65087
57838
50.4
58.8
67.1
75.5
83.9
2341.7
2336.0
2330.5
2324.8
2319.2
2392.1
2394.8
2397.6
2400.3
2403.1
22
24
26
28
30
0.026422
0.029821
0.033597
0.037782
0.042415
1.0022
1.0026
1.0032
1.0037
1.0043
51492
45926
41034
36728
32929
92.2
100.6
108.9
117.3
125.7
2313.6
2307.9
2302.3
2296.6
2291.0
2405.7
2408.5
2411.2
2413.9
2416.7
32
34
36
38
40
0.047534
0.053180
0.059400
0.066240
0.073750
1.0049
1.0056
1.0063
1.0070
1.0078
29572
26601
23967
21627
19546
134.0
142.4
150.7
159.1
167.5
2285.4
2279.7
2274.1
2268.4
2262.8
2419.4
2422.1
2424.8
2427.5
2430.2
42
44
46
48
50
0.081985
0.091001
0.100860
0.111620
0.123350
1.0086
1.0094
1.0103
1.0112
1.0121
17692
16036
14557
13233
12046
175.8
184.2
192.5
200.9
209.3
2257.1
2251.4
2245.8
2240.0
2234.3
2432.9
2435.6
2438.3
2440.9
2443.6
52
54
56
58
60
0.136130
0.150020
0.165110
0.181470
0.199200
1.0131
1.0140
1.0150
1.0161
1.0171
10980
10022
9159
8381
7679
217.6
226.0
234.4
242.7
251.1
2228.6
2222.9
2217.1
2211.4
2205.7
2446.2
2448.8
2451.5
2454.1
2456.7
62
64
66
68
70
0.218380
0.239120
0.261500
0.285630
0.311620
1.0182
1.0193
1.0205
1.0217
1.0228
7044
6469
5948
5476
5046
259.5
267.8
276.2
284.6
293.0
2199.9
2194.1
2188.4
2182.5
2176.7
2459.4
2461.9
2464.6
2467.1
2469.6
72
74
76
78
80
0.339580
0.369640
0.401910
0.436520
0.473600
1.0241
1.0253
1.0266
1.0279
1.0292
4656
4300
3976
3680
3409
301.4
309.7
318.1
326.5
334.9
2170.8
2165.1
2159.3
2153.3
2147.5
2472.2
2474.8
2477.3
2479.8
2482.3
82
84
86
88
90
0.513290
0.555730
0.601080
0.649480
0.701090
1.0305
1.0319
1.0333
1.0347
1.0361
3162
2935
2727
2537
2361
343.2
351.6
360.0
368.4
376.8
2141.6
2135.7
2129.6
2123.7
2117.7
2484.8
2487.3
2489.7
2492.2
2494.6
92
94
96
98
100
0.756080
0.814610
0.876860
0.943010
1.013250
1.0376
1.0391
1.0406
1.0421
1.0437
2200
2052
1915
1789
1673
385.3
393.7
402.1
410.5
419.0
2111.7
2105.7
2099.6
2093.7
2087.5
2497.0
2499.5
2501.8
2504.2
2506.5
PROPERTIES OF MATTER
hf
55
hfg
hg
sf
sfg
sg
ts
0.0
2501.6
2501.6
0.0000
9.1575
9.1575
0.01
4.2
8.4
12.6
16.8
21.0
2499.2
2496.8
2494.5
2492.1
2489.7
2503.4
2505.2
2507.1
2508.9
2510.7
0.0153
0.0306
0.0459
0.0611
0.0762
9.1158
9.0741
9.0326
8.9915
8.9507
9.1311
9.1047
9.0785
9.0526
9.0269
1
2
3
4
5
25.2
29.4
33.6
37.8
42.0
2487.4
2485.0
2482.6
2480.3
2477.9
2512.6
2514.4
2516.2
2518.1
2519.9
0.0913
0.1063
0.1213
0.1362
0.1510
8.9102
8.8699
8.8300
8.7903
8.7510
9.0015
8.9762
8.9513
8.9265
8.9020
6
7
8
9
10
50.4
58.8
67.1
75.5
83.9
2473.2
2468.4
2463.8
2459.0
2454.3
2523.6
2527.2
2530.9
2534.5
2538.2
0.1805
0.2098
0.2388
0.2677
0.2963
8.6731
8.5962
8.5205
8.4458
8.3721
8.8536
8.8060
8.7593
8.7135
8.6684
12
14
16
18
20
92.2
100.6
108.9
117.3
125.7
2449.6
2444.9
2440.2
2435.4
2430.7
2541.8
2545.5
2549.1
2552.7
2556.4
0.3247
0.3530
0.3810
0.4088
0.4365
8.2994
8.2276
8.1569
8.0871
8.0181
8.6241
8.5806
8.5379
8.4959
8.4546
22
24
26
28
30
134.0
142.4
150.7
159.1
167.5
2426.0
2421.2
2416.5
2411.7
2406.9
2560.0
2563.6
2567.2
2570.8
2574.4
0.4640
0.4913
0.5184
0.5453
0.5721
7.9500
7.8827
7.8164
7.7509
7.6862
8.4140
8.3740
8.3348
8.2962
8.2583
32
34
36
38
40
175.8
184.2
192.5
200.9
209.3
2402.1
2397.3
2392.6
2387.7
2382.9
2577.9
2581.5
2585.1
2588.6
2592.2
0.5987
0.6252
0.6514
0.6776
0.7035
7.6222
7.5590
7.4967
7.4349
7.3741
8.2209
8.1842
8.1481
8.1125
8.0776
42
44
46
48
50
217.6
226.0
234.4
242.7
251.1
2378.1
2373.2
2368.3
2363.5
2358.6
2595.7
2599.2
2602.7
2606.2
2609.7
0.7293
0.7550
0.7804
0.8058
0.8310
7.3139
7.2543
7.1955
7.1373
7.0798
8.0432
8.0093
7.9759
7.9431
7.9108
52
54
56
58
60
259.5
267.8
276.2
284.6
293.0
2353.7
2348.8
2343.9
2338.9
2333.9
2613.2
2616.6
2620.1
2623.5
2626.9
0.8560
0.8809
0.9057
0.9303
0.9548
7.0230
6.9668
6.9111
6.8561
6.8017
7.8790
7.8477
7.8168
7.7864
7.7565
62
64
66
68
70
301.4
309.7
318.1
326.5
334.9
2328.9
2324.0
2319.0
2313.9
2308.9
2630.3
2633.7
2637.1
2640.4
2643.8
0.9792
1.0034
1.0275
1.0514
1.0753
6.7478
6.6945
6.6418
6.5896
6.5379
7.7270
7.6979
7.6693
7.6410
7.6132
72
74
76
78
80
343.3
351.7
360.1
368.5
376.9
2303.8
2298.7
2293.5
2288.4
2283.2
2647.1
2650.4
2653.6
2656.9
2660.1
1.0990
1.1225
1.1460
1.1693
1.1925
6.4868
6.4363
6.3861
6.3365
6.2874
7.5858
7.5588
7.5321
7.5058
7.4799
82
84
86
88
90
385.4
393.8
402.2
410.6
419.1
2278.0
2272.8
2267.5
2262.3
2256.9
2663.4
2666.6
2669.7
2672.9
2676.0
1.2156
1.2386
1.2615
1.2842
1.3069
6.2387
6.1905
6.1427
6.0954
6.0485
7.4543
7.4291
7.4042
7.3796
7.3554
92
94
96
98
100
56
ENGINEERING TABLES AND DATA
Saturated water and steam, to 221 bar
p
vf
vg
ufg
ug
0.01
3.77
1.0002
1.0000
206162.9
159668.5
0.0
15.9
2375.6
2364.9
2375.6
2380.8
0.01
0.02
0.03
0.04
0.05
6.98
17.51
24.10
28.98
32.90
1.0001
1.0012
1.0027
1.0040
1.0052
129210.7
67011.6
45670.0
34803.3
28194.5
29.3
73.5
101.0
121.4
137.8
2355.9
2326.1
2307.6
2293.9
2282.8
2385.2
2399.6
2408.6
2415.3
2420.6
0.06
0.07
0.08
0.09
0.10
36.18
39.03
41.54
43.79
45.83
1.0064
1.0074
1.0084
1.0094
1.0102
23740.6
20530.4
18103.8
16203.4
14673.7
151.5
163.4
173.9
183.3
191.8
2273.6
2265.5
2258.4
2252.0
2246.3
2425.1
2428.9
2432.3
2435.3
2438.1
0.11
0.12
0.13
0.14
0.15
47.71
49.45
51.06
52.58
54.00
1.0111
1.0119
1.0126
1.0133
1.0140
13415.2
12361.0
11464.9
10693.4
10022.1
199.7
206.9
213.7
220.0
226.0
2240.8
2236.0
2231.3
2227.0
2222.9
2440.5
2442.9
2445.0
2447.0
2448.9
0.16
0.17
0.18
0.19
0.20
55.34
56.62
57.83
58.98
60.09
1.0147
1.0154
1.0160
1.0166
1.0172
9432.4
8910.3
8444.6
8026.6
7649.2
231.6
236.9
242.0
246.8
251.5
2219.1
2215.4
2211.9
2208.6
2205.4
2450.7
2452.3
2453.9
2455.4
2456.9
0.22
0.24
0.26
0.28
0.30
62.16
64.08
65.87
67.55
69.13
1.0183
1.0194
1.0204
1.0214
1.0223
6994.6
6446.2
5979.9
5578.4
5229.0
260.1
268.2
275.7
282.7
289.3
2199.5
2193.9
2188.7
2183.8
2179.3
2459.6
2462.1
2464.4
2466.5
2468.5
0.32
0.34
0.36
0.38
0.40
70.62
72.03
73.38
74.66
75.89
1.0232
1.0241
1.0249
1.0257
1.0265
4922.0
4650.1
4407.6
4189.8
3993.2
295.6
301.5
307.1
312.5
317.7
2174.9
2170.8
2167.0
2163.1
2159.5
2470.5
2472.3
2474.0
2475.6
2477.2
0.42
0.44
0.46
0.48
0.50
77.06
78.19
79.28
80.33
81.35
1.0273
1.0280
1.0287
1.0294
1.0301
3814.8
3652.2
3503.2
3366.3
3240.1
322.6
327.3
331.9
336.3
340.5
2156.1
2152.7
2149.6
2146.5
2143.4
2478.7
2480.0
2481.5
2482.7
2484.0
0.52
0.54
0.56
0.58
0.60
82.33
83.28
84.19
85.09
85.95
1.0308
1.0314
1.0320
1.0327
1.0333
3123.3
3014.8
2913.9
2819.7
2731.7
344.6
348.6
352.4
356.2
359.8
2140.5
2137.8
2135.1
2132.3
2129.9
2485.2
2486.4
2487.5
2488.6
2489.7
0.62
0.64
0.66
0.68
0.70
86.80
87.62
88.42
89.20
89.96
1.0339
1.0344
1.0350
1.0356
1.0361
2649.1
2571.5
2498.5
2429.7
2364.7
363.4
366.8
370.2
373.5
376.7
2127.2
2124.9
2122.5
2120.1
2117.8
2490.7
2491.7
2492.7
2493.6
2494.6
0.72
0.74
0.76
0.78
0.80
90.70
91.43
92.14
92.83
93.51
1.0367
1.0372
1.0377
1.0382
1.0387
2303.1
2244.8
2189.5
2136.9
2086.9
379.8
382.9
385.8
388.6
391.6
2115.7
2113.4
2111.4
2109.2
2107.2
2495.5
2496.3
2497.2
2498.0
2498.8
0.00611
0.008
For definitions see page 54
ts
uf
PROPERTIES OF MATTER
hf
57
hfg
hg
sf
sfg
sg
p
0.0
15.9
2501.6
2492.6
2501.6
2508.5
0.0000
0.0576
9.1575
9.0008
9.1575
9.0584
0.00611
0.008
29.3
73.5
101.0
121.4
137.8
2485.1
2460.1
2444.6
2433.1
2423.8
2514.4
2533.6
2545.6
2554.5
2561.6
0.1060
0.2606
0.3543
0.4225
0.4763
8.8707
8.4640
8.2242
8.0530
7.9197
8.9767
8.7246
8.5785
8.4755
8.3960
0.01
0.02
0.03
0.04
0.05
151.5
163.4
173.9
183.3
191.8
2416.0
2409.2
2403.2
2397.8
2393.0
2567.5
2572.6
2577.1
2581.1
2584.8
0.5209
0.5591
0.5926
0.6224
0.6493
7.8103
7.7176
7.6369
7.5657
7.5018
8.3312
8.2767
8.2295
8.1881
8.1511
0.06
0.07
0.08
0.09
0.10
199.7
206.9
213.7
220.0
226.0
2388.4
2384.3
2380.3
2376.7
2373.2
2588.1
2591.2
2594.0
2596.7
2599.2
0.6738
0.6964
0.7172
0.7367
0.7549
7.4438
7.3908
7.3420
7.2966
7.2544
8.1176
8.0872
8.0592
8.0333
8.0093
0.11
0.12
0.13
0.14
0.15
231.6
236.9
242.0
246.8
251.5
2370.0
2366.9
2363.9
2361.1
2358.4
2601.6
2603.8
2605.9
2607.9
2609.9
0.7721
0.7883
0.8036
0.8182
0.8321
7.2147
7.1775
7.1423
7.1090
7.0773
7.9868
7.9658
7.9459
7.9272
7.9094
0.16
0.17
0.18
0.19
0.20
260.1
268.2
275.7
282.7
289.3
2353.4
2348.6
2344.2
2340.0
2336.1
2613.5
2616.8
2619.9
2622.7
2625.4
0.8581
0.8820
0.9041
0.9248
0.9441
7.0183
6.9644
6.9147
6.8684
6.8254
7.8764
7.8464
7.8188
7.7932
7.7695
0.22
0.24
0.26
0.28
0.30
295.6
301.5
307.1
312.5
317.7
2332.4
2328.9
2325.6
2322.3
2319.2
2628.0
2630.4
2632.7
2634.8
2636.9
0.9623
0.9795
0.9958
1.0113
1.0261
6.7850
6.7470
6.7111
6.6771
6.6448
7.7473
7.7265
7.7069
7.6884
7.6709
0.32
0.34
0.36
0.38
0.40
322.6
327.3
331.9
336.3
340.6
2316.3
2313.4
2310.7
2308.0
2305.4
2638.9
2640.7
2642.6
2644.3
2646.0
1.0402
1.0538
1.0667
1.0792
1.0912
6.6140
6.5845
6.5564
6.5294
6.5035
7.6542
7.6383
7.6231
7.6086
7.5947
0.42
0.44
0.46
0.48
0.50
344.7
348.7
352.5
356.3
359.9
2302.9
2300.5
2298.2
2295.8
2293.7
2647.6
2649.2
2650.7
2652.1
2653.6
1.1028
1.1140
1.1248
1.1353
1.1455
6.4786
6.4545
6.4313
6.4089
6.3872
7.5814
7.5685
7.5561
7.5442
7.5327
0.52
0.54
0.56
0.58
0.60
363.5
366.9
370.3
373.6
376.8
2291.4
2289.4
2287.3
2285.2
2283.3
2654.9
2656.3
2657.6
2658.8
2660.1
1.1553
1.1649
1.1742
1.1832
1.1921
6.3663
6.3459
6.3261
6.3070
6.2883
7.5216
7.5108
7.5003
7.4902
7.4804
0.62
0.64
0.66
0.68
0.70
379.9
383.0
385.9
388.9
391.7
2281.4
2279.4
2277.7
2275.8
2274.1
2661.3
2662.4
2663.6
2664.7
2665.8
1.2007
1.2090
1.2172
1.2252
1.2330
6.2701
6.2526
6.2353
6.2185
6.2022
7.4708
7.4616
7.4525
7.4437
7.4352
0.72
0.74
0.76
0.78
0.80
58
ENGINEERING TABLES AND DATA
Saturated water and steam, to 221 bar
p
For definitions see page 54
ts
vf
vg
uf
ufg
ug
0.82
0.84
0.86
0.88
0.90
94.18
94.83
95.47
96.10
96.71
1.0392
1.0397
1.0402
1.0407
1.0412
2039.2
1993.8
1950.4
1908.9
1869.1
394.4
397.2
399.9
402.5
405.1
2105.2
2103.2
2101.3
2099.4
2097.6
2499.6
2500.4
2501.2
2501.9
2502.7
0.92
0.94
0.96
0.98
1.00
97.32
97.91
98.50
99.07
99.63
1.0416
1.0421
1.0425
1.0430
1.0434
1831.1
1794.6
1759.6
1726.0
1693.7
407.7
410.2
412.6
415.0
417.4
2095.6
2093.8
2092.2
2090.5
2088.6
2503.3
2504.0
2504.8
2505.5
2506.0
1.10
1.20
1.30
1.40
1.50
102.32
104.81
107.13
109.32
111.37
1.0455
1.0476
1.0495
1.0513
1.0530
1549.2
1428.1
1325.0
1236.3
1159.0
428.7
439.3
449.1
458.3
466.9
2080.5
2072.8
2065.7
2059.0
2052.6
2509.2
2512.0
2514.8
2517.2
2519.6
1.60
1.70
1.80
1.90
2.00
113.32
115.17
116.93
118.62
120.23
1.0547
1.0563
1.0579
1.0594
1.0608
1091.1
1030.9
977.18
928.95
885.40
475.2
483.0
490.5
497.7
504.5
2046.4
2040.7
2035.1
2029.8
2024.7
2521.6
2523.7
2525.6
2527.5
2529.2
2.20
2.40
2.60
2.80
3.00
123.27
126.09
128.73
131.21
133.54
1.0636
1.0663
1.0688
1.0712
1.0735
809.80
746.41
692.47
646.00
605.53
517.4
529.3
540.6
551.2
561.1
2015.1
2006.0
1997.5
1989.4
1982.0
2532.4
2535.4
2538.2
2540.6
2543.0
3.20
3.40
3.60
3.80
4.00
135.76
137.86
139.87
141.79
143.63
1.0757
1.0779
1.0799
1.0819
1.0839
569.95
538.43
510.29
485.02
462.20
570.6
579.5
588.1
596.4
604.3
1974.7
1967.7
1961.1
1954.6
1948.5
2545.2
2547.2
2549.2
2551.0
2552.7
4.20
4.40
4.60
4.80
5.00
145.39
147.09
148.73
150.31
151.85
1.0858
1.0876
1.0894
1.0911
1.0928
441.47
422.57
405.26
389.34
374.66
611.8
619.1
626.2
633.0
639.6
1942.5
1936.8
1931.3
1925.8
1920.6
2554.4
2556.0
2557.5
2558.8
2560.2
5.20
5.40
5.60
5.80
6.00
153.33
154.77
156.16
157.52
158.84
1.0945
1.0961
1.0977
1.0993
1.1009
361.06
348.44
336.69
325.72
315.46
645.9
652.2
658.2
664.1
669.7
1915.6
1910.5
1905.8
1901.0
1896.5
2561.5
2562.7
2564.0
2565.1
2566.2
6.20
6.40
6.60
6.80
7.00
160.12
161.38
162.60
163.79
164.96
1.1024
1.1039
1.1053
1.1068
1.1082
305.84
296.80
288.29
280.26
272.68
675.3
680.8
686.1
691.2
696.3
1892.0
1887.5
1883.2
1879.0
1874.8
2567.3
2568.2
2569.2
2570.2
2571.1
7.20
7.40
7.60
7.80
8.00
166.10
167.21
168.30
169.37
170.41
1.1096
1.1110
1.1123
1.1137
1.1150
265.50
258.70
252.24
246.10
240.26
701.2
706.1
710.9
715.4
720.0
1870.8
1866.8
1862.8
1859.0
1855.3
2572.0
2572.9
2573.7
2574.4
2575.3
PROPERTIES OF MATTER
59
hf
hfg
hg
sf
sfg
sg
p
394.5
397.3
400.0
402.6
405.2
2272.3
2270.6
2268.9
2267.3
2265.7
2666.8
2667.9
2668.9
2669.9
2670.9
1.2407
1.2481
1.2554
1.2626
1.2696
6.1861
6.1706
6.1553
6.1404
6.1258
7.4268
7.4187
7.4107
7.4030
7.3954
0.82
0.84
0.86
0.88
0.90
407.8
410.3
412.7
415.1
417.5
2264.0
2262.4
2261.0
2259.5
2257.9
2671.8
2672.7
2673.7
2674.6
2675.4
1.2765
1.2832
1.2898
1.2963
1.3027
6.1114
6.0975
6.0838
6.0703
6.0571
7.3879
7.3807
7.3736
7.3666
7.3598
0.92
0.94
0.96
0.98
1.00
428.8
439.4
449.2
458.4
467.1
2250.8
2244.0
2237.8
2231.9
2226.3
2679.6
2683.4
2687.0
2690.3
2693.4
1.3330
1.3609
1.3868
1.4109
1.4336
5.9947
5.9375
5.8847
5.8356
5.7898
7.3277
7.2984
7.2715
7.2465
7.2234
1.10
1.20
1.30
1.40
1.50
475.4
483.2
490.7
497.9
504.7
2220.8
2215.8
2210.8
2206.1
2201.6
2696.2
2699.0
2701.5
2704.0
2706.3
1.4550
1.4752
1.4944
1.5127
1.5301
5.7467
5.7061
5.6678
5.6313
5.5967
7.2017
7.1813
7.1622
7.1440
7.1268
1.60
1.70
1.80
1.90
2.00
517.6
529.6
540.9
551.5
561.4
2193.0
2184.9
2177.3
2170.0
2163.3
2710.6
2714.5
2718.2
2721.5
2724.7
1.5628
1.5929
1.6209
1.6471
1.6717
5.5321
5.4728
5.4180
5.3669
5.3192
7.0949
7.0657
7.0389
7.0140
6.9909
2.20
2.40
2.60
2.80
3.00
570.9
579.9
588.5
596.8
604.7
2156.7
2150.4
2144.4
2138.5
2132.9
2727.6
2730.3
2732.9
2735.3
2737.6
1.6948
1.7168
1.7376
1.7575
1.7764
5.2744
5.2321
5.1921
5.1540
5.1179
6.9692
6.9489
6.9297
6.9115
6.8943
3.20
3.40
3.60
3.80
4.00
612.3
619.6
626.7
633.5
640.1
2127.5
2122.3
2117.2
2112.2
2107.4
2739.8
2741.9
2743.9
2745.7
2747.5
1.7946
1.8120
1.8287
1.8448
1.8604
5.0833
5.0502
5.0186
4.9881
4.9588
6.8779
6.8622
6.8473
6.8329
6.8192
4.20
4.40
4.60
4.80
5.00
646.5
652.8
658.8
664.7
670.4
2102.8
2098.1
2093.7
2089.3
2085.1
2749.3
2750.9
2752.5
2754.0
2755.5
1.8754
1.8899
1.9040
1.9176
1.9308
4.9305
4.9033
4.8769
4.8514
4.8267
6.8059
6.7932
6.7809
6.7690
6.7575
5.20
5.40
5.60
5.80
6.00
676.0
681.5
686.8
692.0
697.1
2080.9
2076.7
2072.7
2068.8
2064.9
2756.9
2758.2
2759.5
2760.8
2762.0
1.9437
1.9562
1.9684
1.9803
1.9918
4.8027
4.7794
4.7568
4.7347
4.7134
6.7464
6.7356
6.7252
6.7150
6.7052
6.20
6.40
6.60
6.80
7.00
702.0
706.9
711.7
716.3
720.9
2061.2
2057.4
2053.7
2050.1
2046.6
2763.2
2764.3
2765.4
2766.4
2767.5
2.0031
2.0141
2.0249
2.0354
2.0457
4.6925
4.6721
4.6522
4.6329
4.6139
6.6956
6.6862
6.6771
6.6683
6.6596
7.20
7.40
7.60
7.80
8.00
60
ENGINEERING TABLES AND DATA
Saturated water and steam, to 221 bar
p
For definitions see page 54
ts
vf
vg
8.20
8.40
8.60
8.80
9.00
171.44
172.45
173.43
174.40
175.36
1.1163
1.1176
1.1188
1.1201
1.1213
234.69
229.38
224.31
219.46
214.82
9.20
9.40
9.60
9.80
10.00
176.29
177.21
178.12
179.01
179.88
1.1226
1.1238
1.1250
1.1262
1.1274
11.00
12.00
13.00
14.00
15.00
184.06
187.96
191.60
195.04
198.28
16.00
17.00
18.00
19.00
20.00
uf
ufg
ug
724.5
729.0
733.2
737.5
741.6
1851.6
1847.8
1844.3
1840.7
1837.2
2576.1
2576.7
2577.5
2578.2
2578.8
210.37
206.10
202.01
198.08
194.30
745.8
749.7
753.7
757.6
761.5
1833.7
1830.3
1827.0
1823.7
1820.4
2579.5
2580.1
2580.7
2581.3
2581.9
1.1331
1.1386
1.1438
1.1489
1.1538
177.39
163.21
151.14
140.73
131.67
779.9
797.0
813.2
828.5
842.9
1804.7
1789.8
1775.7
1762.3
1749.5
2584.6
2586.8
2588.9
2590.8
2592.4
201.37
204.30
207.11
209.79
212.37
1.1586
1.1633
1.1678
1.1723
1.1766
123.70
116.64
110.33
104.67
99.55
856.6
869.8
882.4
894.6
906.2
1737.1
1725.3
1713.8
1702.7
1691.9
2593.8
2595.1
2596.2
2597.2
2598.1
21.00
22.00
23.00
24.00
25.00
214.85
217.24
219.55
221.78
223.94
1.1809
1.1850
1.1891
1.1932
1.1972
94.90
90.66
86.78
83.21
79.92
917.4
928.3
938.9
949.0
958.9
1681.5
1671.3
1661.3
1651.7
1642.2
2598.9
2599.6
2600.2
2600.7
2601.1
26.00
27.00
28.00
29.00
30.00
226.03
228.06
230.04
231.96
233.84
1.2011
1.2050
1.2088
1.2126
1.2163
76.87
74.03
71.40
68.94
66.63
968.6
977.9
987.1
996.0
1004.7
1633.0
1623.9
1615.0
1606.3
1597.8
2601.6
2601.8
2602.1
2602.3
2602.4
31.00
32.00
33.00
34.00
35.00
235.66
237.44
239.18
240.88
242.54
1.2200
1.2237
1.2273
1.2310
1.2345
64.47
62.44
60.53
58.73
57.03
1013.2
1021.5
1029.6
1037.6
1045.4
1589.2
1581.0
1572.9
1564.8
1557.0
2602.4
2602.5
2602.5
2602.4
2602.4
36.00
37.00
38.00
39.00
40.00
244.16
245.75
247.31
248.84
250.33
1.2381
1.2416
1.2451
1.2486
1.2521
55.42
53.89
52.44
51.06
49.75
1053.0
1060.6
1068.0
1075.2
1082.4
1549.2
1541.4
1533.9
1526.4
1518.9
2602.2
2602.0
2601.8
2601.7
2601.3
42.00
44.00
46.00
48.00
50.00
253.24
256.05
258.76
261.38
263.92
1.2589
1.2657
1.2725
1.2792
1.2858
47.31
45.08
43.04
41.16
39.43
1096.3
1109.8
1122.9
1135.7
1148.1
1504.4
1490.1
1476.1
1462.5
1449.0
2600.7
2600.0
2599.0
2598.1
2597.1
52.00
54.00
56.00
58.00
60.00
266.38
268.77
271.09
273.36
275.56
1.2925
1.2990
1.3056
1.3122
1.3187
37.82
36.33
34.94
33.65
32.43
1160.2
1172.0
1183.5
1194.8
1205.8
1435.8
1422.6
1409.8
1397.1
1384.6
2595.9
2594.6
2593.3
2591.9
2590.4
PROPERTIES OF MATTER
hf
61
hfg
hg
sf
sfg
sg
p
725.4
729.9
734.2
738.5
742.6
2043.1
2039.5
2036.2
2032.8
2029.5
2768.5
2769.4
2770.4
2771.3
2772.1
2.0558
2.0657
2.0753
2.0848
2.0941
4.5953
4.5772
4.5595
4.5421
4.5251
6.6511
6.6429
6.6348
6.6269
6.6192
8.20
8.40
8.60
8.80
9.00
746.8
750.8
754.8
758.7
762.6
2026.2
2023.0
2019.8
2016.7
2013.6
2773.0
2773.8
2774.6
2775.4
2776.2
2.1033
2.1122
2.1210
2.1297
2.1382
4.5083
4.4920
4.4759
4.4601
4.4446
6.6116
6.6042
6.5969
6.5898
6.5828
9.20
9.40
9.60
9.80
10.00
781.1
798.4
814.7
830.1
844.6
1998.6
1984.3
1970.7
1957.7
1945.3
2779.7
2782.7
2785.4
2787.8
2789.9
2.1786
2.2160
2.2509
2.2836
2.3144
4.3712
4.3034
4.2404
4.1815
4.1262
6.5498
6.5194
6.4913
6.4651
6.4406
11.00
12.00
13.00
14.00
15.00
858.5
871.8
884.5
896.8
908.6
1933.2
1921.6
1910.3
1899.3
1888.6
2791.7
2793.4
2794.8
2796.1
2797.2
2.3436
2.3712
2.3976
2.4227
2.4468
4.0740
4.0246
3.9775
3.9328
3.8899
6.4176
6.3958
6.3751
6.3555
6.3367
16.00
17.00
18.00
19.00
20.00
919.9
930.9
941.6
951.9
961.9
1878.3
1868.2
1858.2
1848.5
1839.0
2798.2
2799.1
2799.8
2800.4
2800.9
2.4699
2.4921
2.5136
2.5342
2.5542
3.8488
3.8094
3.7714
3.7348
3.6995
6.3187
6.3015
6.2850
6.2690
6.2537
21.00
22.00
23.00
24.00
25.00
971.7
981.2
990.5
999.5
1008.3
1829.7
1820.5
1811.5
1802.7
1794.0
2801.4
2801.7
2802.0
2802.2
2802.3
2.5736
2.5923
2.6105
2.6282
2.6455
3.6652
3.6321
3.6000
3.5687
3.5383
6.2388
6.2244
6.2105
6.1969
6.1838
26.00
27.00
28.00
29.00
30.00
1017.0
1025.4
1033.7
1041.8
1049.7
1785.3
1776.9
1768.6
1760.3
1752.3
2802.3
2802.3
2802.3
2802.1
2802.0
2.6622
2.6785
2.6945
2.7100
2.7252
3.5088
3.4800
3.4518
3.4245
3.3977
6.1710
6.1585
6.1463
6.1345
6.1229
31.00
32.00
33.00
34.00
35.00
1057.5
1065.2
1072.7
1080.1
1087.4
1744.2
1736.2
1728.4
1720.7
1712.9
2801.7
2801.4
2801.1
2800.8
2800.3
2.7401
2.7547
2.7689
2.7828
2.7965
3.3714
3.3457
3.3207
3.2961
3.2720
6.1115
6.1004
6.0896
6.0789
6.0685
36.00
37.00
38.00
39.00
40.00
1101.6
1115.4
1128.8
1141.8
1154.5
1697.8
1682.9
1668.2
1653.9
1639.7
2799.4
2798.3
2797.0
2795.7
2794.2
2.8231
2.8487
2.8735
2.8974
2.9207
3.2251
3.1799
3.1362
3.0939
3.0528
6.0482
6.0286
6.0097
5.9913
5.9735
42.00
44.00
46.00
48.00
50.00
1166.9
1179.0
1190.8
1202.4
1213.7
1625.7
1611.8
1598.2
1584.6
1571.3
2792.6
2790.8
2789.0
2787.0
2785.0
2.9432
2.9651
2.9864
3.0071
3.0274
3.0129
2.9741
2.9363
2.8994
2.8632
5.9561
5.9392
5.9227
5.9065
5.8906
52.00
54.00
56.00
58.00
60.00
62
ENGINEERING TABLES AND DATA
Saturated water and steam, to 221 bar
p
For definitions see page 54
ts
vf
vg
uf
ufg
ug
62.00
64.00
66.00
68.00
70.00
277.71
279.80
281.85
283.85
285.80
1.3252
1.3318
1.3383
1.3448
1.3514
31.29
30.23
29.22
28.27
27.37
1216.7
1227.3
1237.7
1248.0
1258.0
1372.2
1359.9
1347.8
1335.7
1323.8
2588.9
2587.2
2585.5
2583.7
2581.8
72.00
74.00
76.00
78.00
80.00
287.71
289.59
291.42
293.22
294.98
1.3579
1.3645
1.3711
1.3777
1.3843
26.52
25.71
24.94
24.22
23.52
1267.9
1277.7
1287.3
1296.8
1306.1
1312.1
1300.2
1288.6
1277.1
1265.6
2580.0
2577.9
2575.9
2573.8
2571.7
82.00
84.00
86.00
88.00
90.00
296.71
298.40
300.07
301.71
303.31
1.3909
1.3976
1.4043
1.4111
1.4179
22.86
22.23
21.62
21.05
20.49
1315.3
1324.4
1333.3
1342.3
1351.0
1254.3
1242.9
1231.6
1220.3
1209.1
2569.5
2567.3
2564.9
2562.6
2560.2
92.00
94.00
96.00
98.00
100.00
304.89
306.45
307.98
309.48
310.96
1.4247
1.4316
1.4385
1.4455
1.4526
19.96
19.45
18.96
18.49
18.04
1359.7
1368.2
1376.8
1385.2
1393.6
1198.0
1186.9
1175.9
1164.7
1153.7
2557.6
2555.1
2552.6
2550.0
2547.3
102.00
104.00
106.00
108.00
110.00
312.42
313.86
315.27
316.67
318.04
1.4597
1.4668
1.4740
1.4813
1.4887
17.60
17.18
16.78
16.39
16.01
1401.8
1409.9
1418.1
1426.2
1434.2
1142.8
1131.9
1121.0
1109.9
1099.0
2544.6
2541.9
2539.1
2536.1
2533.2
112.00
114.00
116.00
118.00
120.00
319.40
320.73
322.05
323.35
324.64
1.4961
1.5037
1.5113
1.5189
1.5267
15.64
15.29
14.94
14.61
14.29
1442.1
1450.1
1457.9
1465.7
1473.4
1088.1
1077.2
1066.3
1055.3
1044.4
2530.2
2527.2
2524.2
2521.0
2517.8
122.00
124.00
126.00
128.00
130.00
325.90
327.15
328.39
329.61
330.81
1.5346
1.5425
1.5506
1.5588
1.5671
13.97
13.67
13.37
13.08
12.80
1481.1
1488.8
1496.4
1503.9
1511.5
1033.4
1022.4
1011.4
1000.2
989.1
2514.5
2511.1
2507.7
2504.2
2500.6
132.00
134.00
136.00
138.00
140.00
332.00
333.18
334.34
335.49
336.63
1.5755
1.5841
1.5927
1.6015
1.6105
12.53
12.26
12.00
11.75
11.50
1519.1
1526.6
1534.0
1541.5
1549.0
977.9
966.6
955.4
944.0
932.5
2497.0
2493.2
2489.4
2485.5
2481.4
142.00
144.00
146.00
148.00
150.00
337.75
338.86
339.96
341.04
342.12
1.6196
1.6289
1.6383
1.6480
1.6578
11.26
11.02
10.79
10.56
10.34
1556.4
1563.8
1571.3
1578.6
1586.0
920.9
909.4
897.6
885.8
873.9
2477.4
2473.2
2468.9
2464.5
2460.0
152.00
154.00
156.00
158.00
160.00
343.18
344.23
345.27
346.30
347.32
1.6678
1.6780
1.6885
1.6992
1.7102
10.13
9.92
9.71
9.51
9.31
1593.4
1600.9
1608.3
1615.7
1623.0
861.9
849.8
837.6
825.3
812.9
2455.4
2450.7
2445.8
2441.0
2435.9
PROPERTIES OF MATTER
63
hf
hfg
hg
sf
sfg
sg
p
1224.9
1235.8
1246.5
1257.1
1267.5
1558.0
1544.8
1531.8
1518.8
1505.9
2782.9
2780.6
2778.3
2775.9
2773.4
3.0472
3.0665
3.0854
3.1039
3.1220
2.8281
2.7936
2.7598
2.7266
2.6941
5.8753
5.8601
5.8452
5.8305
5.8161
62.00
64.00
66.00
68.00
70.00
1277.7
1287.8
1297.7
1307.5
1317.2
1493.2
1480.4
1467.8
1455.2
1442.7
2770.9
2768.2
2765.5
2762.7
2759.9
3.1398
3.1572
3.1743
3.1912
3.2077
2.6621
2.6307
2.5998
2.5693
2.5393
5.8019
5.7879
5.7741
5.7605
5.7470
72.00
74.00
76.00
78.00
80.00
1326.7
1336.1
1345.4
1354.7
1363.8
1430.3
1417.9
1405.5
1393.1
1380.8
2757.0
2754.0
2750.9
2747.8
2744.6
3.2240
3.2400
3.2558
3.2714
3.2867
2.5097
2.4806
2.4518
2.4233
2.3953
5.7337
5.7206
5.7076
5.6947
5.6820
82.00
84.00
86.00
88.00
90.00
1372.8
1381.7
1390.6
1399.4
1408.1
1368.5
1356.3
1344.1
1331.8
1319.6
2741.3
2738.0
2734.7
2731.2
2727.7
3.3019
3.3168
3.3316
3.3462
3.3606
2.3674
2.3400
2.3128
2.2858
2.2592
5.6693
5.6568
5.6444
5.6320
5.6198
92.00
94.00
96.00
98.00
100.00
1416.7
1425.2
1433.7
1442.2
1450.6
1307.5
1295.4
1283.2
1270.9
1258.7
2724.2
2720.6
2716.9
2713.1
2709.3
3.3748
3.3889
3.4029
3.4167
3.4304
2.2328
2.2066
2.1806
2.1548
2.1292
5.6076
5.5955
5.5835
5.5715
5.5596
102.00
104.00
106.00
108.00
110.00
1458.9
1467.2
1475.4
1483.6
1491.7
1246.5
1234.3
1222.1
1209.8
1197.5
2705.4
2701.5
2697.5
2693.4
2689.2
3.4439
3.4574
3.4707
3.4840
3.4971
2.1038
2.0784
2.0533
2.0281
2.0032
5.5477
5.5358
5.5240
5.5121
5.5003
112.00
114.00
116.00
118.00
120.00
1499.8
1507.9
1515.9
1523.9
1531.9
1185.1
1172.7
1160.3
1147.7
1135.1
2684.9
2680.6
2676.2
2671.6
2667.0
3.5101
3.5231
3.5359
3.5487
3.5614
1.9784
1.9535
1.9289
1.9042
1.8795
5.4885
5.4766
5.4648
5.4529
5.4409
122.00
124.00
126.00
128.00
130.00
1539.9
1547.8
1555.7
1563.6
1571.5
1122.4
1109.7
1096.9
1084.0
1070.9
2662.3
2657.5
2652.6
2647.6
2642.4
3.5741
3.5867
3.5992
3.6116
3.6241
1.8549
1.6302
1.8056
1.7811
1.7563
5.4290
5.4169
5.4048
5.3927
5.3804
132.00
134.00
136.00
138.00
140.00
1579.4
1587.3
1595.2
1603.0
1610.9
1057.8
1044.6
1031.2
1017.8
1004.2
2637.2
2631.9
2626.4
2620.8
2615.1
3.6365
3.6488
3.6611
3.6734
3.6857
1.7316
1.7069
1.6821
1.6573
1.6323
5.3681
5.3557
5.3432
5.3307
5.3180
142.00
144.00
146.00
148.00
150.00
1618.8
1626.7
1634.6
1642.5
1650.4
990.5
976.7
962.7
948.7
934.5
2609.3
2603.4
2597.3
2591.2
2584.9
3.6979
3.7102
3.7224
3.7347
3.7470
1.6074
1.5822
1.5571
1.5317
1.5063
5.3053
5.2924
5.2795
5.2664
5.2533
152.00
154.00
156.00
158.00
160.00
64
ENGINEERING TABLES AND DATA
Saturated water and steam, to 221 bar
For definitions see page 54
p
ts
vf
vg
uf
ufg
ug
162.00
164.00
166.00
168.00
170.00
348.32
349.32
350.31
351.29
352.26
1.7214
1.7330
1.7446
1.7569
1.7695
9.12
8.93
8.74
8.55
8.37
1630.5
1638.0
1645.4
1653.4
1661.5
800.4
787.7
775.1
761.6
747.8
2430.9
2425.7
2420.5
2415.0
2409.3
172.00
174.00
176.00
178.00
180.00
353.22
354.16
355.11
356.04
356.96
1.7825
1.7961
1.8101
1.8247
1.8399
8.19
8.01
7.84
7.67
7.50
1669.6
1677.7
1685.7
1693.7
1701.7
733.9
719.9
705.8
691.6
677.3
2403.5
2397.6
2391.5
2385.3
2378.9
182.00
184.00
186.00
188.00
190.00
357.87
358.78
359.67
360.56
361.44
1.8557
1.8722
1.8894
1.9074
1.9262
7.33
7.16
7.00
6.84
6.68
1709.7
1717.8
1725.8
1733.9
1742.1
662.6
647.7
632.7
617.3
601.6
2372.3
2365.5
2358.5
2351.3
2343.7
192.00
194.00
196.00
198.00
200.00
362.31
363.17
364.03
364.87
365.71
1.9460
1.9669
1.9889
2.0124
2.0374
6.52
6.36
6.20
6.04
5.87
1750.5
1758.9
1767.7
1776.7
1785.9
585.3
568.8
551.4
533.5
514.9
2335.8
2327.7
2319.2
2310.2
2300.7
202.00
204.00
206.00
208.00
210.00
366.54
367.37
368.18
368.99
369.79
2.0643
2.0935
2.1256
2.1612
2.2018
5.71
5.55
5.38
5.20
5.02
1795.5
1805.6
1816.2
1827.6
1840.1
495.2
474.4
452.2
428.2
402.0
2290.7
2280.0
2268.4
2255.9
2242.0
212.00
214.00
216.00
218.00
220.00
370.58
371.37
372.14
372.92
373.68
2.2489
2.3059
2.3788
2.4819
2.6675
4.83
4.62
4.39
4.12
3.73
1853.9
1869.7
1888.4
1912.8
1951.6
372.6
338.8
298.4
245.8
162.8
2226.5
2208.4
2186.8
2158.6
2114.4
221.20
374.15
3.1700
3.17
2037.3
0.0
2037.3
PROPERTIES OF MATTER
65
hf
hfg
hg
sf
sfg
sg
p
1658.4
1666.4
1674.4
1682.9
1691.6
920.2
905.7
891.2
875.8
860.0
2578.6
2572.1
2565.6
2558.7
2551.6
3.7592
3.7716
3.7841
3.7973
3.8106
1.4809
1.4552
1.4292
1.4022
1.3750
5.2401
5.2268
5.2133
5.1995
5.1856
162.00
164.00
166.00
168.00
170.00
1700.3
1709.0
1717.6
1726.2
1734.8
844.1
828.1
811.9
795.6
779.1
2544.4
2537.1
2529.5
2521.8
2513.9
3.8239
3.8372
3.8502
3.8635
3.8766
1.3475
1.3199
1.2923
1.2643
1.2361
5.1714
5.1571
5.1425
5.1278
5.1127
172.00
174.00
176.00
178.00
180.00
1743.5
1752.2
1760.9
1769.8
1778.7
762.2
745.1
727.8
710.0
691.6
2505.7
2497.3
2488.7
2479.8
2470.5
3.8897
3.9029
3.9161
3.9295
3.9430
1.2078
1.1790
1.1499
1.1202
1.0900
5.0975
5.0819
5.0660
5.0497
5.0330
182.00
184.00
186.00
188.00
190.00
1787.9
1797.1
1806.7
1816.5
1826.6
673.0
653.9
633.9
613.2
591.6
2460.9
2451.0
2440.6
2429.7
2418.2
3.9568
3.9708
3.9851
3.9998
4.0151
1.0590
1.0273
0.9947
0.9610
0.9259
5.0158
4.9981
4.9798
4.9608
4.9410
192.00
194.00
196.00
198.00
200.00
1837.2
1848.3
1860.0
1872.6
1886.3
568.9
544.8
519.2
491.5
461.2
2406.1
2393.1
2379.2
2364.1
2347.5
4.0310
4.0476
4.0653
4.0843
4.1049
0.8891
0.8506
0.8095
0.7653
0.7173
4.9201
4.8982
4.8748
4.8496
4.8222
202.00
204.00
206.00
208.00
210.00
1901.6
1919.0
1939.8
1966.9
2010.3
427.3
388.4
341.9
281.5
186.3
2328.9
2307.4
2281.7
2248.4
2196.6
4.1279
4.1543
4.1858
4.2271
4.2934
0.6638
0.6027
0.5299
0.4357
0.2880
4.7917
4.7570
4.7157
4.6628
4.5814
212.00
214.00
216.00
218.00
220.00
2107.4
0.0
2107.4
4.4429
0.0000
4.4429
221.20
66
ENGINEERING TABLES AND DATA
Superheated steam, to 220 bar and 800 ◦ C
p
(ts )
t:
ts
50
100
150
200
250
2571.7
2594.8
2642.5
2688.7
2714.6
2783.8
2787.8
2880.1
2862.4
2977.8
0
u
h
0.02
(17.50)
v
u
h
s
67012
2399.6
2533.6
8.7246
74524
2445.4
2594.4
8.9226
86080
2516.3
2688.5
9.1934
97628
2588.4
2783.7
9.4327
109171
2661.7
2880.0
9.6479
120711
2736.3
2977.7
9.8441
0.06
(36.20)
v
u
h
s
23741
2425.1
2567.5
8.3312
24812
2444.6
2593.5
8.4135
28676
2515.9
2688.0
8.6854
32532
2588.2
2783.4
8.9251
36383
2661.5
2879.8
9.1406
40232
2736.2
2977.6
9.3369
0.10
(45.80)
v
u
h
s
14674
2438.1
2584.8
8.1511
14869
2444.0
2592.7
8.1757
17195
2515.6
2687.5
8.4486
19512
2588.0
2783.1
8.6888
21825
2661.4
2879.6
8.9045
24136
2736.0
2977.4
9.1010
0.50
(81.30)
v
u
h
s
3240.1
2484.0
2646.0
7.5947
3418.1
2511.7
2682.6
7.6953
3889.3
2585.6
2780.1
7.9406
4356.0
2659.9
2877.7
8.1587
4820.5
2735.1
2976.1
8.3564
1.00
(99.60)
v
u
h
s
1693.7
2506.0
2675.4
7.3598
1695.5
2506.7
2676.2
7.3618
1936.3
2582.7
2776.3
7.6137
2172.3
2658.2
2875.4
7.8349
2406.1
2733.9
2974.5
8.0342
1.50
(111.4)
v
u
h
s
1159.0
2519.6
2693.4
7.2234
1285.2
2579.7
2772.5
7.4194
1444.4
2656.2
2872.9
7.6439
1601.3
2732.7
2972.9
7.8447
2
(120.2)
v
u
h
s
885.40
2529.2
2706.3
7.1268
959.54
2576.6
2768.5
7.2794
1080.4
2654.4
2870.5
7.5072
1198.9
2731.4
2971.2
7.7096
3
(133.5)
v
u
h
s
605.53
2543.0
2724.7
6.9909
633.74
2570.3
2760.4
7.0771
716.35
2650.6
2865.5
7.3119
796.44
2729.0
2967.9
7.5176
4
(143.6)
v
u
h
s
462.20
2552.7
2737.6
6.8943
470.66
2563.7
2752.0
6.9285
534.26
2646.7
2860.4
7.1708
595.19
2726.4
2964.5
7.3800
5
(151.8)
v
u
h
s
374.66
2560.2
2747.5
6.8192
424.96
2642.6
2855.1
7.0592
474.43
2723.9
2961.1
7.2721
6
(158.8)
v
u
h
s
315.46
2566.2
2755.5
6.7575
352.04
2638.5
2849.7
6.9662
393.91
2721.3
2957.6
7.1829
7
(165.0)
v
u
h
s
272.68
2571.1
2762.0
6.7052
299.92
2634.3
2844.2
6.8859
336.37
2718.5
2954.0
7.1066
For definitions see page 54
PROPERTIES OF MATTER
67
p
(ts )
300
400
500
600
700
800
2938.4
3076.9
3095.1
3279.7
3258.4
3489.2
3428.7
3705.6
3605.8
3928.9
3789.5
4158.7
132251
2812.3
3076.8
10.025
155329
2969.0
3279.7
10.351
178405
3132.4
3489.2
10.641
201482
3302.6
3705.6
10.904
224558
3479.7
3928.8
11.146
247634
3663.4
4158.7
11.371
0.02
(17.50)
44079
2812.2
3076.7
9.5179
51773
2969.0
3279.6
9.8441
59467
3132.4
3489.2
10.134
67159
3302.6
3705.6
10.397
74852
3479.7
3928.8
10.639
82544
3663.4
4158.7
10.864
0.06
(36.20)
26445
2812.2
3076.6
9.2820
31062
2969.0
3279.6
9.6083
35679
3132.3
3489.1
9.8984
40295
3302.6
3705.5
10.162
44910
3479.7
3928.8
10.404
49526
3663.4
4158.7
10.628
0.10
(45.80)
5283.9
2811.5
3075.7
8.5380
6209.1
2968.5
3279.0
8.8649
7133.5
3132.0
3488.7
9.1552
8057.4
3302.3
3705.2
9.4185
8981.0
3479.5
3928.6
9.6606
9904.4
3663.3
4158.5
9.8855
0.50
(81.30)
2638.7
2810.6
3074.5
8.2166
3102.5
2968.0
3278.2
8.5442
3565.3
3131.6
3488.1
8.8348
4027.7
3302.0
3704.8
9.0982
4489.8
3479.2
3928.2
9.3405
4951.7
3663.1
4158.3
9.5654
1.00
(99.60)
1757.0
2809.8
3073.3
8.0280
2066.9
2967.5
3277.5
8.3562
2375.9
3131.2
3487.6
8.6472
2684.5
3301.7
3704.4
8.9108
2992.7
3479.0
3927.9
9.1531
3300.8
3662.9
4158.0
9.3781
1.50
(111.4)
1316.2
2808.9
3072.1
7.8937
1549.2
2966.9
3276.7
8.2226
1781.2
3130.8
3487.0
8.5139
2012.9
3301.4
3704.0
8.7776
2244.2
3478.8
3927.6
9.0201
2475.4
3662.7
4157.8
9.2452
2
(120.2)
875.29
2807.1
3069.7
7.7034
1031.4
2965.8
3275.2
8.0338
1186.5
3130.1
3486.0
8.3257
1341.2
3300.8
3703.2
8.5898
1495.7
3478.3
3927.0
8.8325
1649.9
3662.3
4157.3
9.0577
3
(133.5)
654.85
2805.3
3067.2
7.5675
772.50
2964.6
3273.6
7.8994
889.19
3129.2
3484.9
8.1919
1005.4
3300.1
3702.3
8.4563
1121.4
3477.8
3926.4
8.6992
1237.2
3662.0
4156.9
8.9246
4
(143.6)
522.58
2803.5
3064.8
7.4614
617.16
2963.5
3272.1
7.7948
710.78
3128.4
3483.8
8.0879
803.95
3299.5
3701.5
8.3526
896.85
3477.4
3925.8
8.5957
989.56
3661.6
4156.4
8.8213
5
(151.8)
434.39
2801.7
3062.3
7.3740
513.61
2962.4
3270.6
7.7090
591.84
3127.6
3482.7
8.0027
669.63
3298.9
3700.7
8.2678
747.14
3476.8
3925.1
8.5111
824.47
3661.2
4155.9
8.7368
6
(158.8)
371.39
2799.8
3059.8
7.2997
439.64
2961.3
3269.0
7.6362
506.89
3126.8
3481.6
7.9305
573.68
3298.3
3699.9
8.1959
640.21
3476.4
3924.5
8.4395
706.55
3660.9
4155.5
8.6653
7
(165.0)
0
68
ENGINEERING TABLES AND DATA
Superheated steam, to 220 bar and 800 ◦ C
p
(ts )
t:
ts
50
For definitions see page 54
100
150
200
250
8
(170.4)
v
u
h
s
240.26
2575.3
2767.5
6.6596
260.79
2630.0
2838.6
6.8148
293.21
2715.8
2950.4
7.0397
9
(175.4)
v
u
h
s
214.82
2578.8
2772.1
6.6192
230.32
2625.4
2832.7
6.7508
259.63
2713.1
2946.8
6.9800
10
(179.9)
v
u
h
s
194.30
2581.9
2776.2
6.5828
205.92
2620.9
2826.8
6.6922
232.75
2710.3
2943.0
6.9259
15
(198.3)
v
u
h
s
131.67
2592.4
2789.9
6.4406
132.38
2596.1
2794.7
6.4508
151.99
2695.5
2923.5
6.7099
20
(212.4)
v
u
h
s
99.549
2598.1
2797.2
6.3367
111.45
2679.5
2902.4
6.5454
30
(233.8)
v
u
h
s
66.632
2602.4
2802.3
6.1838
70.551
2643.1
2854.8
6.2857
40
(250.3)
v
u
h
s
49.749
2601.3
2800.3
6.0685
50
(263.9)
v
u
h
s
39.425
2597.1
2794.2
5.9735
60
(275.6)
v
u
h
s
32.433
2590.4
2785.0
5.8907
70
(285.8)
v
u
h
s
27.368
2581.8
2773.4
5.8161
80
(295.0)
v
u
h
s
23.521
2571.7
2759.9
5.7470
90
(303.3)
v
u
h
s
20.493
2560.2
2744.6
5.6820
PROPERTIES OF MATTER
69
p
(ts )
300
400
500
600
700
800
324.14
2798.0
3057.3
7.2348
384.16
2960.2
3267.5
7.5729
443.17
3126.0
3480.5
7.8678
501.72
3297.7
3699.1
8.1336
560.01
3475.9
3923.9
8.3773
618.11
3660.5
4155.0
8.6033
8
(170.4)
287.39
2796.0
3054.7
7.1771
341.01
2959.1
3266.0
7.5169
393.61
3125.2
3479.4
7.8124
445.76
3297.0
3698.2
8.0785
497.63
3475.4
3923.3
8.3225
549.33
3660.1
4154.5
8.5486
9
(175.4)
257.98
2794.1
3052.1
7.1251
306.49
2957.9
3264.4
7.4665
353.96
3124.3
3478.3
7.7627
400.98
3296.4
3697.4
8.0292
447.73
3475.0
3922.7
8.2734
494.30
3659.8
4154.1
8.4997
10
(179.9)
169.70
2784.4
3038.9
6.9207
202.92
2952.2
3256.6
7.2709
235.03
3120.3
3472.8
7.5703
266.66
3293.3
3693.3
7.8385
298.03
3472.6
3919.6
8.0838
329.21
3657.9
4151.7
8.3108
15
(198.3)
125.50
2774.0
3025.0
6.7696
151.13
2946.4
3248.7
7.1296
175.55
3116.2
3467.3
7.4323
199.50
3290.2
3689.2
7.7022
223.17
3470.2
3916.5
7.9485
246.66
3656.1
4149.4
8.1763
20
(212.4)
81.159
2751.6
2995.1
6.5422
99.310
2934.6
3232.5
6.9246
116.08
3108.0
3456.2
7.2345
132.34
3284.0
3681.0
7.5079
148.32
3465.3
3910.3
7.7564
164.12
3652.3
4144.7
7.9857
30
(233.8)
58.833
2726.7
2962.0
6.3642
73.376
2922.2
3215.7
6.7733
86.341
3099.6
3445.0
7.0909
98.763
3277.7
3672.8
7.3680
110.90
3460.5
3904.1
7.6187
122.85
3648.6
4140.0
7.8495
40
(250.3)
45.301
2699.0
2925.5
6.2105
57.791
2909.3
3198.3
6.6508
68.494
3091.2
3433.7
6.9770
78.616
3271.4
3664.5
7.2578
88.446
3455.7
3897.9
7.5108
98.093
3644.8
4135.3
7.7431
50
(263.9)
36.145
2668.1
2885.0
6.0692
47.379
2895.8
3180.1
6.5462
56.592
3082.6
3422.2
6.8818
65.184
3265.1
3656.2
7.1664
73.478
3450.8
3891.7
7.4217
81.587
3641.2
4130.7
7.6554
60
(275.6)
29.457
2633.2
2839.4
5.9327
39.922
2881.7
3161.2
6.4536
48.086
3074.0
3410.6
6.7993
55.590
3258.8
3647.9
7.0880
62.787
3445.9
3885.4
7.3456
69.798
3637.4
4126.0
7.5808
70
(285.8)
24.264
2592.7
2786.8
5.7942
34.310
2867.1
3141.6
6.3694
41.704
3065.2
3398.8
6.7262
48.394
3252.3
3639.5
7.0191
54.770
3441.0
3879.2
7.2790
60.956
3633.7
4121.3
7.5158
80
(295.0)
29.929
2851.8
3121.2
6.2915
36.737
3056.2
3386.8
6.6600
42.798
3245.9
3631.1
6.9574
48.534
3436.2
3873.0
7.2196
54.080
3630.0
4116.7
7.4579
90
(303.3)
70
ENGINEERING TABLES AND DATA
Superheated steam, to 220 bar and 800 ◦ C
p
(ts )
t:
ts
400
For definitions see page 54
500
600
700
800
100
(311.0)
v
u
h
s
18.041
2547.3
2727.7
5.6198
26.408
2835.8
3099.9
6.2182
32.760
3047.0
3374.6
6.5994
38.320
3239.5
3622.7
6.9013
43.546
3431.3
3866.8
7.1660
48.580
3626.2
4112.0
7.4058
110
(318.0)
v
u
h
s
16.007
2533.2
2709.3
5.5596
23.512
2819.2
3077.8
6.1483
29.503
3037.7
3362.2
6.5432
34.656
3233.0
3614.2
6.8499
39.466
3426.4
3860.5
7.1170
44.081
3622.4
4107.3
7.3584
120
(324.6)
v
u
h
s
14.285
2517.8
2689.2
5.5003
21.084
2801.8
3054.8
6.0810
26.786
3028.2
3349.6
6.4906
31.603
3226.5
3605.7
6.8022
36.066
3421.5
3854.3
7.0718
40.332
3618.7
4102.7
7.3147
130
(330.8)
v
u
h
s
12.800
2500.6
2667.0
5.4409
19.015
2783.5
3030.7
6.0155
24.485
3018.5
3336.8
6.4409
29.019
3219.9
3597.1
6.7577
33.189
3416.5
3848.0
7.0298
37.160
3614.9
4098.0
7.2743
140
(336.6)
v
u
h
s
11.498
2481.4
2642.4
5.3804
17.227
2764.4
3005.6
5.9513
22.509
3008.7
3323.8
6.3937
26.804
3213.2
3588.5
6.7159
30.723
3411.6
3841.7
6.9906
34.441
3611.1
4093.3
7.2367
150
(342.1)
v
u
h
s
10.343
2460.0
2615.1
5.3180
15.661
2744.2
2979.1
5.8876
20.795
2998.7
3310.6
6.3487
24.884
3206.5
3579.8
6.6764
28.587
3406.6
3835.4
6.9536
32.086
3607.3
4088.6
7.2013
160
(347.3)
v
u
h
s
9.3099
2435.9
2584.9
5.2533
14.275
2722.9
2951.3
5.8240
19.293
2988.4
3297.1
6.3054
23.204
3199.7
3571.0
6.6389
26.717
3401.6
3829.1
6.9188
30.025
3603.6
4084.0
7.1681
170
(352.3)
v
u
h
s
8.3721
2409.3
2551.6
5.1856
13.034
2700.1
2921.7
5.7599
17.966
2978.1
3283.5
6.2636
21.721
3192.9
3562.2
6.6031
25.068
3396.6
3822.8
6.8857
28.207
3599.8
4079.3
7.1366
180
(357.0)
v
u
h
s
7.4973
2378.9
2513.9
5.1127
11.913
2675.9
2890.3
5.6947
16.785
2967.5
3269.6
6.2232
20.403
3186.1
3553.4
6.5688
23.603
3391.6
3816.5
6.8542
26.591
3596.0
4074.6
7.1067
190
(361.4)
v
u
h
s
6.6759
2343.7
2470.5
5.0330
10.889
2649.8
2856.7
5.6278
15.726
2956.6
3255.4
6.1839
19.223
3179.3
3544.5
6.5360
22.291
3386.7
3810.2
6.8241
25.146
3592.2
4070.0
7.0783
200
(365.7)
v
u
h
s
5.8745
2300.7
2418.2
4.9410
9.9470
2621.6
2820.5
5.5585
14.771
2945.7
3241.1
6.1456
18.161
3172.3
3535.5
6.5043
21.111
3381.6
3803.8
6.7953
23.845
3588.4
4065.3
7.0511
210
(369.8)
v
u
h
s
5.0225
2242.0
2347.5
4.8222
9.0714
2590.8
2781.3
5.4863
13.907
2934.5
3226.5
6.1082
17.201
3165.3
3526.5
6.4737
20.044
3376.6
3797.5
6.7677
22.669
3584.6
4060.6
7.0251
220
(373.7)
v
u
h
s
3.7347
2114.4
2196.6
4.5814
8.2510
2557.3
2738.8
5.4102
13.119
2923.1
3211.7
6.0716
16.327
3158.2
3517.4
6.4441
19.074
3371.5
3791.1
6.7410
21.599
3580.7
4055.9
7.0001
PROPERTIES OF MATTER
71
Supercritical steam, to 1000 bar and 800 ◦ C
p
(ts )
t:
400
500
600
700
800
240
v
h
s
6.7392
2641.2
5.2430
11.737
3181.4
6.0003
14.798
3499.1
6.3876
17.377
3778.3
6.6905
19.729
4046.6
6.9529
260
v
h
s
5.2812
2511.7
5.0326
10.565
3150.2
5.9311
13.505
3480.6
6.3340
15.941
3765.5
6.6431
18.147
4037.2
6.9090
280
v
h
s
3.8219
2330.7
4.7503
9.5566
3118.1
5.8636
12.397
3461.9
6.2829
14.712
3752.6
6.5984
16.792
4027.8
6.8677
300
v
h
s
2.8306
2161.8
4.4896
8.6808
3085.0
5.7972
11.436
3443.0
6.2340
13.647
3739.7
6.5560
15.619
4018.5
6.8288
350
v
h
s
2.1108
1993.1
4.2214
6.9253
2998.3
5.6349
9.5194
3395.1
6.1194
11.520
3707.3
6.4584
13.275
3995.1
6.7400
400
v
h
s
1.9091
1934.1
4.1190
5.6156
2906.8
5.4762
8.0884
3346.4
6.0135
9.9302
3674.8
6.3701
11.521
3971.7
6.6606
450
v
h
s
1.8013
1900.6
4.0554
4.6249
2813.5
5.3226
6.9842
3297.4
5.9143
8.6988
3642.4
6.2890
10.160
3948.4
6.5885
500
v
h
s
1.7291
1877.7
4.0083
3.8822
2723.0
5.1782
6.1113
3248.3
5.8207
7.7197
3610.2
6.2138
9.0759
3925.3
6.5222
600
v
h
s
1.6324
1847.3
3.9383
2.9515
2570.6
4.9374
4.8350
3151.6
5.6477
6.2690
3547.0
6.0775
7.4603
3879.6
6.4031
700
v
h
s
1.5671
1827.8
3.8855
2.4668
2467.1
4.7688
3.9719
3060.4
5.4931
5.2566
3486.3
5.9562
6.3208
3835.3
6.2979
800
v
h
s
1.5180
1814.2
3.8425
2.1881
2397.4
4.6488
3.3792
2980.3
5.3595
4.5193
3428.7
5.8470
5.4805
3792.8
6.2034
900
v
h
s
1.4788
1804.6
3.8059
2.0129
2349.9
4.5602
2.9668
2913.5
5.2468
3.9642
3374.6
5.7479
4.8407
3752.4
6.1179
1000
v
h
s
1.4464
1797.6
3.7738
1.8934
2316.1
4.4913
2.6681
2857.5
5.1505
3.5356
3324.4
5.6579
4.3411
3714.3
6.0397
72
ENGINEERING TABLES AND DATA
Saturated water and steam
Specific volume (m3 kg−1 )
Specific heat capacity at constant
pressure (kJ kg−1 K−1 )
v
cp
−1 −1
Thermal conductivity (W m K )
Dynamic viscosity (mPa s, or cP)
Prandtl number, cp μ /k
k
μ
Pr
For other definitions see page 54.
ts
vf
0.01
10
20
30
40
0.00100
0.00100
0.00100
0.00100
0.00101
50
60
70
80
90
0.00101
0.00102
0.00102
0.00103
0.00104
100
125
150
175
200
0.00104
0.00107
0.00109
0.00112
0.00116
225
250
275
300
325
350
360
374.15
vg
206.2
106.4
57.8
32.9
19.5
cpf
cpg
kf
kg
μf
μg
Prf
Prg
ts
4.217
4.193
4.182
4.179
4.179
1.854
1.860
1.866
1.875
1.885
0.569
0.587
0.603
0.618
0.632
0.0173
0.0185
0.0191
0.0198
0.0204
1.755
1.301
1.002
0.797
0.651
0.0088
0.0091
0.0094
0.0097
0.0101
13.02
9.29
6.95
5.39
4.31
0.942
0.915
0.918
0.923
0.930
0.01 Triple point
10
20
30
40
4.181
4.185
4.190
4.197
4.205
1.899
1.915
1.936
1.962
1.992
0.643
0.653
0.662
0.670
0.676
0.0210
0.0217
0.0224
0.0231
0.0240
0.544
0.462
0.400
0.350
0.311
0.0104
0.0107
0.0111
0.0114
0.0117
3.53
2.96
2.53
2.19
1.93
0.939
0.947
0.956
0.966
0.976
50
60
70
80
90
1.673
0.770
0.392
0.217
0.127
4.216
4.254
4.310
4.389
4.497
2.028
2.147
2.314
2.542
2.843
0.681
0.687
0.687
0.679
0.665
0.0249
0.0272
0.0300
0.0334
0.0375
0.278
0.219
0.180
0.153
0.133
0.0121
0.0133
0.0144
0.0156
0.0167
1.723
1.358
1.133
0.990
0.902
0.986
1.047
1.110
1.185
1.270
100
125
150
175
200
0.00120
0.00125
0.00132
0.00140
0.00153
0.0783
0.0500
0.0327
0.0216
0.0142
4.648
4.867
5.202
5.762
6.861
3.238
3.772
4.561
5.863
8.440
0.644
0.616
0.582
0.541
0.493
0.0427
0.0495
0.0587
0.0719
0.0929
0.1182
0.1065
0.0972
0.0897
0.0790
0.0179
0.0191
0.0202
0.0214
0.0230
0.853
0.841
0.869
0.955
1.100
1.36
1.45
1.56
1.74
2.09
225
250
275
300
325
0.00174
0.00190
0.00317
0.00880
0.00694
0.00317
0.437
0.400
0.24
0.1343
0.168
0.24
0.0648
0.0582
0.045
0.0258
0.0275
0.045
1.50
2.11
∞
3.29
3.89
∞
350
360
374.15 Critical point
12.05
7.68
5.05
3.41
2.36
10.10
14.6
∞
17.15
25.1
∞
Ammonia, NH3
p = pressure (MPa), v = specific volume (m3 kg−1 ); for other definitions see page 54.
Saturated
hf
hg
ts
p
vf
−40
−35
−30
−25
0.0718
0.0932
0.1196
0.1516
0.00145
0.00146
0.00148
0.00149
1.552
1.216
0.963
0.772
0
22.3
44.7
67.2
−20
−15
−10
−5
0
0.190
0.236
0.291
0.355
0.429
0.00150
0.00152
0.00153
0.00155
0.00157
0.624
0.509
0.418
0.347
0.289
5
10
15
20
25
0.516
0.615
0.728
0.857
1.001
0.00158
0.00160
0.00162
0.00164
0.00166
30
35
40
45
50
1.167
1.350
1.554
1.782
2.033
0.00168
0.00170
0.00173
0.00175
0.00178
Superheated
By 50 K
By 100 K
h
s
h
s
sf
sg
1390
1398
1406
1413
0
0.095
0.188
0.279
5.963
5.872
5.785
5.703
1499
1508
1517
1526
6.387
6.292
6.203
6.119
1606
1616
1626
1636
6.736
6.639
6.547
6.461
89.8
112.3
135.4
158.2
181.2
1420
1426
1433
1439
1444
0.368
0.457
0.544
0.630
0.715
5.624
5.549
5.477
5.407
5.340
1535
1543
1552
1560
1568
6.039
5.963
5.891
5.822
5.756
1646
1656
1665
1675
1684
6.379
6.301
6.227
6.157
6.090
0.243
0.206
0.175
0.149
0.128
204.5
227.7
251.4
275.2
298.9
1450
1454
1459
1463
1466
0.799
0.881
0.963
1.044
1.124
5.276
5.214
5.154
5.095
5.039
1576
1583
1590
1597
1604
5.694
5.634
5.576
5.521
5.468
1693
1702
1711
1719
1728
6.027
5.967
5.909
5.853
5.800
0.111
0.096
0.083
0.073
0.063
323.1
347.5
371.5
396.8
421.9
1469
1471
1473
1474
1475
1.204
1.282
1.360
1.437
1.515
4.984
4.930
4.877
4.825
4.773
1610
1616
1622
1628
1633
5.417
5.368
5.321
5.275
5.230
1736
1744
1752
1760
1767
5.750
5.702
5.655
5.610
5.567
vg
Freezing point at 1 atm = −77.7 ◦ C; Critical point: Celsius temp. = 132.4 ◦ C, pressure = 11.30 MPa (113.0 bar)
PROPERTIES OF MATTER
73
Tetrafluoroethane, CH2 FCF3 (Refrigerant 134a)
3
−1
p = pressure (MPa), v = specific volume (m kg ); for other definitions see page 54.
Saturated
hf
hg
ts
p
vf
−40
−35
−30
−25
0.051
0.066
0.084
0.106
0.00071
0.00071
0.00072
0.00073
0.3611
0.2840
0.2259
0.1816
48.1
54.4
60.8
67.2
−20
−15
−10
−5
0
0.133
0.164
0.201
0.243
0.293
0.00074
0.00074
0.00075
0.00076
0.00077
0.1474
0.1207
0.0996
0.0828
0.0693
5
10
15
20
25
0.350
0.415
0.488
0.572
0.665
0.00078
0.00079
0.00080
0.00082
0.00083
30
35
40
45
50
0.770
0.887
1.017
1.160
1.318
0.00084
0.00086
0.00087
0.00089
0.00091
Superheated
By 15 K
By 30 K
h
s
h
s
sf
sg
274.0
277.2
280.3
283.5
0.7956
0.8223
0.8486
0.8746
1.7643
1.7575
1.7515
1.7461
285.3
288.7
292.1
295.5
1.8115
1.8046
1.7984
1.7931
297.0
300.5
304.1
307.7
1.8569
1.8500
1.8437
1.8382
73.6
80.1
86.7
93.3
100.0
286.6
289.6
292.7
295.7
298.6
0.9002
0.9256
0.9506
0.9754
1.0000
1.7413
1.7371
1.7334
1.7300
1.7271
298.8
302.1
305.4
308.7
312.0
1.7884
1.7842
1.7806
1.7774
1.7747
311.2
314.8
318.3
321.8
325.3
1.8335
1.8293
1.8257
1.8225
1.8199
0.0584
0.0494
0.0421
0.0360
0.0309
106.8
113.6
120.5
127.5
134.6
301.5
304.3
307.1
309.8
312.3
1.0243
1.0485
1.0724
1.0962
1.1199
1.7245
1.7221
1.7200
1.7180
1.7162
315.2
318.3
321.4
324.5
327.4
1.7723
1.7702
1.7685
1.7670
1.7656
328.8
332.2
335.6
338.9
342.2
1.8176
1.8157
1.8141
1.8128
1.8117
0.0266
0.0230
0.0200
0.0173
0.0151
141.7
149.0
156.4
163.9
171.6
314.8
317.2
319.4
321.5
323.4
1.1435
1.1670
1.1905
1.2139
1.2375
1.7145
1.7128
1.7111
1.7092
1.7072
330.3
333.2
335.9
338.6
341.1
1.7645
1.7634
1.7625
1.7616
1.7607
345.4
348.6
351.7
354.8
357.8
1.8109
1.8102
1.8097
1.8092
1.8089
vg
Triple point temp. = −169.9 ◦ C
Freezing point at 1 atm = −96.6 ◦ C
Critical point: Celsius temp. = 101.1 ◦ C, pressure = 4.059 MPa (40.59 bar)
Carbon dioxide, CO2
p = pressure (MPa), v = specific volume (m3 kg−1 ); for other definitions see page 54.
ts
Saturated
hf
hg
Superheated
By 30 K
By 60 K
h
s
h
s
sf
sg
321.1
322.2
323.1
323.7
0
0.039
0.079
0.119
1.377
1.352
1.328
1.304
355.4
356.9
358.7
360.4
1.507
1.485
1.464
1.442
383.0
385.6
388.0
390.3
1.611
1.588
1.566
1.545
39.7
50.2
60.9
72.0
83.7
323.7
323.2
322.3
320.5
318.1
0.158
0.198
0.238
0.278
0.320
1.280
1.256
1.231
1.205
1.178
361.8
363.0
363.9
364.6
364.9
1.421
1.401
1.381
1.361
1.342
392.5
394.5
396.2
397.8
399.3
1.525
1.505
1.486
1.467
1.449
0.00879
0.00743
0.00623
0.00516
0.00413
96.0
109.1
123.3
139.1
159.7
312.9
307.2
301.0
292.3
279.9
0.364
0.407
0.454
0.506
0.573
1.143
1.107
1.071
1.028
0.976
364.9
364.7
364.0
362.9
361.5
1.322
1.302
1.282
1.261
1.241
400.4
401.4
402.2
402.7
403.0
1.431
1.414
1.396
1.379
1.362
0.00294
0.00214
191.2
223.0
253.1
223.0
0.682
0.780
0.886
0.780
359.6
359.1
1.220
1.216
402.9
402.9
1.345
1.341 Critical point
p
vf
vg
−40
−35
−30
−25
1.005
1.20
1.43
1.68
0.00090
0.00091
0.00093
0.00095
0.0382
0.0320
0.0270
0.0229
0
9.7
19.5
29.5
−20
−15
−10
−5
0
1.97
2.29
2.65
3.04
3.48
0.00097
0.00099
0.00102
0.00105
0.00108
0.0195
0.0166
0.0142
0.0122
0.0104
5
10
15
20
25
3.97
4.50
5.08
5.73
6.44
0.00111
0.00116
0.00121
0.00129
0.00140
30
31.05
7.21
7.38
0.00169
0.00214
Triple point temp. = −56.6 ◦ C, pressure = 0.518 MN m−2
Sublimation point at 1 atm = −78.5 ◦ C
74
ENGINEERING TABLES AND DATA
International Standard Atmosphere
Altitude above sea level (km)
Absolute pressure (bar)
Absolute temperature (K)
Mass density (kg m−3 )
(at sea level ρ = ρ0 = 1.225)
z
p
T
ρ
2 −1
Kinematic viscosity (mm s , or cSt)
−1 −1
Thermal conductivity (W m K )
Mean free path (μm)
Speed of sound (m s−1 )
For other definitions see page 54
ν
k
λ
a
z
p
T
ρ/ρ0
ν
k
λ
a
−2.5
−2.0
−1.5
−1.0
−0.5
1.352
1.278
1.207
1.139
1.075
304.4
301.2
297.9
294.7
291.4
1.263
1.207
1.152
1.100
1.049
12.07
12.53
13.01
13.52
14.05
0.0266
0.0264
0.0261
0.0259
0.0256
0.053
0.055
0.058
0.060
0.063
349.8
347.9
346.0
344.1
342.2
0
1.013
288.2
1.000
14.61
0.0253
0.066
340.3
0.5
1.0
1.5
2.0
2.5
0.955
0.899
0.846
0.795
0.747
284.9
281.7
278.4
275.2
271.9
0.953
0.907
0.864
0.822
0.781
15.20
15.81
16.46
17.15
17.87
0.0251
0.0248
0.0246
0.0243
0.0241
0.070
0.073
0.077
0.081
0.085
338.4
336.4
334.5
332.5
330.6
3.0
3.5
4.0
4.5
5.0
0.701
0.658
0.617
0.578
0.540
268.7
265.4
262.2
258.9
255.7
0.742
0.705
0.669
0.634
0.601
18.63
19.43
20.28
21.17
22.11
0.0238
0.0235
0.0233
0.0230
0.0228
0.089
0.094
0.099
0.105
0.110
328.6
326.6
324.6
322.6
320.5
6.0
7.0
8.0
9.0
10.0
0.472
0.411
0.357
0.308
0.265
249.2
242.7
236.2
229.7
223.3
0.539
0.482
0.429
0.381
0.338
24.16
26.46
29.04
31.96
35.25
0.0222
0.0217
0.0212
0.0206
0.0201
0.123
0.138
0.155
0.174
0.196
316.5
312.3
308.1
303.8
299.5
11.0
12.0
13.0
14.0
15.0
0.227
0.194
0.166
0.142
0.121
216.8
216.7
216.7
216.7
216.7
0.298
0.255
0.218
0.186
0.159
38.99
45.57
53.33
62.39
73.00
0.0195
0.0195
0.0195
0.0195
0.0195
0.223
0.260
0.305
0.357
0.417
295.2
295.1
295.1
295.1
295.1
16.0
17.0
18.0
19.0
20.0
0.104
0.088
0.076
0.065
0.055
216.7
216.7
216.7
216.7
216.7
0.136
0.116
0.099
0.085
0.073
85.40
99.90
116.9
136.7
159.9
0.0195
0.0195
0.0195
0.0195
0.0195
0.488
0.571
0.668
0.781
0.914
295.1
295.1
295.1
295.1
295.1
21.0
22.0
23.0
24.0
25.0
0.047
0.040
0.035
0.030
0.025
217.6
218.6
219.6
220.6
221.6
0.062
0.053
0.045
0.038
0.033
188.4
222.0
261.4
307.4
361.4
0.0196
0.0197
0.0198
0.0199
0.0199
1.073
1.260
1.477
1.731
2.027
295.7
296.4
297.0
297.7
298.4
26.0
27.0
28.0
29.0
30.0
0.022
0.019
0.016
0.014
0.012
222.5
223.5
224.5
225.5
226.5
0.028
0.024
0.020
0.018
0.015
424.4
498.1
584.1
684.4
801.3
0.0200
0.0201
0.0202
0.0203
0.0203
2.372
2.773
3.240
3.783
4.413
299.1
299.7
300.4
301.0
301.7
At sea level the normal composition of clean dry air, by volume and (weight) percent, is:
Nitrogen
Oxygen
Argon
Carbon dioxide
78.08 (75.52)
20.95 (23.14)
0.93 (1.28)
0.03 (0.05)
Neon
Helium
Methane
Krypton
0.002 (0.001)
0.0005 (7×10−5 )
0.0002 (0.0001)
0.0001 (0.0003)
The approximate composition may be taken to be: nitrogen/oxygen = 79/21 (77/23)
Hydrogen
Nitrous oxide
Xenon
5×10−5 (3×10−6 )
5×10−5 (8×10−5 )
−6
−5
9×10 (4×10 )
PROPERTIES OF MATTER
75
Air at atmospheric pressure
t
v
cp
Temperature ( ◦ C)
Specific volume (m3 kg−1 )
Specific heat capacity at constant
pressure (kJ kg−1 K−1 )
−1 −1
Thermal conductivity (W m K )
Dynamic viscosity (mPa s, or cP)
Prandtl number, cp μ /k
k
μ
Pr
t
v
cp
k
μ
Pr
t
−100
0
0.488
0.773
1.01
1.01
0.016
0.024
0.012
0.017
0.75
0.72
−100
0
100
200
300
400
1.057
1.341
1.624
1.908
1.02
1.03
1.05
1.07
0.032
0.039
0.045
0.051
0.022
0.026
0.030
0.033
0.70
0.69
0.69
0.70
100
200
300
400
500
600
700
800
2.191
2.473
2.756
3.039
1.10
1.12
1.14
1.16
0.056
0.061
0.066
0.071
0.036
0.039
0.042
0.044
0.70
0.71
0.72
0.73
500
600
700
800
This table may be used with reasonable accuracy for values of cp , k , μ and Pr of N2 , O2 and CO.
Properties of gases
MW
ρ
R
tf ,tb
tc ,pc ,ρc
k
cp
γ
μ
λ
r
∗
Molecular weight or molar mass (a.m.u. or kg kmol−1 )
∗
−3
Mass density at s.t.p. (kg m )
Gas constant (kJ kg−1 K−1 )
Freezing, boiling points at 1 atm ( ◦ C)
Critical temperature, pressure, density ( ◦ C, bar, kg m−3 )
∗
−1 −1
Thermal conductivity at s.t.p. (mW m K )
Specific heat capacity at constant pressure, for s.t.p.∗ (kJ kg−1 K−1 )
Ratio of specific heat capacities cp /cv at s.t.p.∗
Dynamic viscosity at 1 atm, 20 ◦ C (μPa s)
∗
Mean free path at s.t.p. (nm)
Dielectric constant at s.t.p.∗ and 10 GHz
Standard temperature and pressure: 0 ◦ C, 1 bar; but the figures are for the former standard pressure of 1 atm (1.013 bar).
Air
Oxygen
Nitrogen
Hydrogen
Carbon monoxide
Carbon dioxide
Helium
Neon
Argon
MW
ρ
R
28.96
32.00
28.01
2.02
28.01
44.01
4.00
20.18
39.95
1.293
1.429
1.250
0.090
1.250
1.977
1.178
0.900
1.784
0.287
0.260
0.297
4.124
0.297
0.189
2.077
0.412
0.208
tf
tb
−219
−210
−259
−205
−57
−183
−196
−253
−192
−249
−189
−269
−246
−186
tc
pc
ρc
k
cp
γ
μ
λ
−141
−119
−147
−240
−140
31
−268
−229
−122
37.7
50.4
33.9
13.0
35.0
74.0
2.3
27.2
48.6
430
311
31
300
460
69
484
531
24.1
24.4
24.3
168.4
22.9
14.5
141.5
46.5
16.2
1.004
0.915
1.039
14.2
1.040
0.819
5.19
1.40
1.40
1.40
1.41
1.40
1.30
1.63
1.64
1.67
18.1
20.0
17.4
8.8
17.4
14.6
19.4
31.0
22.2
63
59
111
59
40
174
124
63
0.520
r
1.00058
1.00053
1.00059
1.00026
1.00070
1.00099
1.00070
1.00127
1.00056
Variation of specific heat cp with temperature
These values apply to a wide range of pressure.
◦
t( C)
Air
Oxygen
Nitrogen
Hydrogen
Carbon monoxide
Carbon dioxide
100
200
300
400
500
1000
1500
2000
2500
3000
1.010
0.934
1.042
14.45
1.045
0.914
1.025
0.963
1.052
14.50
1.058
0.993
1.045
0.995
1.069
14.53
1.080
1.057
1.069
1.024
1.092
14.58
1.106
1.110
1.092
1.048
1.115
14.66
1.132
1.155
1.184
1.123
1.215
15.52
1.231
1.290
1.235
1.164
1.269
16.56
1.280
1.350
1.265
1.200
1.298
17.39
1.307
1.378
1.287
1.234
1.316
18.01
1.323
1.388
1.302
1.260
1.331
18.55
1.336
1.394
76
ENGINEERING TABLES AND DATA
−1
Absolute molar internal energy (MJ kmol )
T (K)
Air
N2
O2
H2
CO
CO2
H2 O
NO
Ar
OH
O
H
0
100
200
−8.774
−6.615
−4.518
−8.823
−6.643
−4.527
−8.709
−6.586
−4.521
−8.222 −119.363 −402.239 −251.763
−6.415 −117.167 −400.729 −249.246
−4.467 −115.050 −398.597 −246.759
81.717 −6.201
83.592 −4.953
85.569 −3.704
32.721 243.041 211.787
34.349 244.322 213.034
36.053 245.601 214.281
298.15 −2.480
−2.480
−2.479
−2.476 −113.004 −395.991 −244.303
87.601 −2.479
37.798 246.854 215.505
300
400
−2.441
−0.352
−2.441
−0.356
−2.440
−0.298
−2.438 −112.966 −395.937 −244.256
−0.367 −110.876 −392.835 −241.697
87.640 −2.456
89.799 −1.207
37.831 246.877 215.528
39.680 248.151 216.776
500
600
700
800
900
1.773
3.953
6.198
8.513
10.896
1.752
3.904
6.114
8.390
10.735
1.931
4.258
6.680
9.186
11.761
1.723
3.822
5.930
8.053
10.198
−108.751
−106.572
−104.325
−102.004
−99.611
−389.366
−385.594
−381.574
−377.351
−372.960
−239.057 92.037
−236.314 94.349
−233.459 96.727
−230.483 99.166
−227.382 101.660
1000
1100
1200
1300
1400
13.341
15.838
18.380
20.963
23.584
13.145
15.610
18.123
20.680
23.276
14.393
17.067
19.777
22.519
25.294
12.371
14.583
16.837
19.133
21.472
−97.153
−94.642
−92.084
−89.485
−86.849
−368.427
−363.770
−359.028
−354.204
−349.304
1500
1600
1700
1800
1900
26.239
28.925
31.639
34.379
37.143
25.908
28.572
31.265
33.983
36.724
28.099
30.934
33.796
36.686
39.601
23.851
26.271
28.730
31.228
33.762
−84.179
−81.479
−78.753
−76.003
−73.233
2000
2100
2200
2300
2400
39.927
42.730
45.551
48.387
51.237
39.485
42.264
45.059
47.868
50.689
42.542
45.507
48.495
51.507
54.540
36.333
38.938
41.578
44.250
46.954
2500
2600
2700
2800
2900
54.101
56.977
59.864
62.761
65.667
53.521
56.364
59.214
62.073
64.939
57.595
60.670
63.767
66.883
70.018
3000
3100
3200
3300
3400
68.583
71.507
74.440
77.380
80.328
67.811
70.689
73.573
76.462
79.356
3500
3600
3700
3800
3900
83.283
86.245
89.214
92.191
95.174
0.041
1.290
2.538
3.787
5.035
41.595
43.575
45.617
47.718
49.874
249.423
250.692
251.959
253.225
254.487
218.023
219.270
220.517
221.764
223.012
−224.155
−220.798
−217.315
−213.713
−210.000
104.204 6.284
106.793 7.532
109.422 8.781
112.088 10.029
114.787 11.278
52.085
54.346
56.655
59.011
61.410
255.749
257.008
258.265
259.521
260.776
224.259
225.506
226.753
228.000
229.247
−344.339
−339.315
−334.239
−329.117
−323.955
−206.183
−202.270
−198.266
−194.178
−190.013
117.515
120.268
123.045
125.842
128.657
12.526
13.775
15.023
16.272
17.520
63.851
66.330
68.847
71.399
73.983
262.029
263.282
264.534
265.785
267.035
230.495
231.742
232.989
234.236
235.483
−70.444
−67.638
−64.819
−61.987
−59.144
−318.757
−313.529
−308.273
−302.993
−297.692
−185.775
−181.469
−177.102
−172.676
−168.198
131.487
134.332
137.188
140.055
142.932
18.769
20.017
21.266
22.514
23.763
76.599
79.244
81.916
84.615
87.337
268.286
269.536
270.787
272.039
273.291
236.730
237.978
239.225
240.472
241.719
49.688
52.451
55.242
58.060
60.904
−56.291
−53.429
−50.560
−47.683
−44.801
−292.373
−287.038
−281.688
−276.325
−270.950
−163.670
−159.096
−154.480
−149.825
−145.134
145.816
148.708
151.606
154.510
157.419
25.011 90.082
26.260 92.848
27.508 95.633
28.757 98.438
30.005 101.259
274.544
275.798
277.054
278.312
279.571
242.966
244.213
245.461
246.708
247.950
73.172
76.345
79.535
82.744
85.969
63.772
66.664
69.579
72.515
75.472
−41.913
−39.019
−36.120
−33.216
−30.308
−265.564
−260.168
−254.762
−249.345
−243.919
−140.410
−135.656
−130.872
−126.063
−121.229
160.333
163.251
166.173
169.100
172.031
31.254
32.502
33.751
34.999
36.248
104.096
106.949
109.815
112.694
115.585
280.833
282.098
283.364
284.634
285.907
249.200
250.449
251.697
252.944
254.191
82.255 89.211
85.159 92.469
88.068 95.743
90.981 99.032
93.900 102.336
78.448
81.444
84.459
87.491
90.541
−27.395
−24.477
−21.555
−18.627
−15.696
−238.484
−233.038
−227.583
−222.117
−216.642
−116.372
−111.494
−106.595
−101.678
−96.743
174.965
177.904
180.847
183.794
186.744
37.496
38.745
39.993
41.242
42.490
118.488
121.402
124.325
127.258
130.200
287.184
288.464
289.747
291.035
292.327
255.438
256.685
257.932
259.180
260.427
4000
4100
4200
4300
4400
98.163 96.823 105.653 93.608
101.160 99.751 108.984 96.692
104.162 102.684 112.328 99.792
107.171 105.621 115.684 102.909
110.186 108.563 119.050 106.043
−12.759
−9.818
−6.872
−3.922
−0.967
−211.156
−205.659
−200.153
−194.636
−189.109
−91.791
−86.823
−81.839
−76.840
−71.826
189.700
192.659
195.622
198.590
201.561
43.739
44.987
46.236
47.484
48.733
133.151
136.110
139.077
142.053
145.036
293.623
294.924
296.229
297.540
298.855
261.674
262.921
264.168
265.415
266.663
4500
4600
4700
4800
4900
113.206
116.232
119.262
122.296
125.334
1.993
4.956
7.922
10.892
13.863
−183.574
−178.030
−172.479
−166.921
−161.360
−66.798
−61.756
−56.699
−51.629
−46.544
204.536
207.515
210.496
213.480
216.465
49.981
51.230
52.478
53.727
54.975
148.027
151.027
154.034
157.050
160.075
300.175
301.500
302.831
304.167
305.508
267.910
269.157
270.404
271.651
272.898
5000
128.374 126.296 139.429 125.228
111.509
114.460
117.414
120.372
123.333
122.427
125.813
129.208
132.609
136.017
109.195
112.363
115.550
118.756
121.981
16.836 −155.797
−41.446 219.452 56.224 163.110 306.854 274.146
The term “absolute” molar internal energy adopted here uses the same datum as the enthalpy table, namely a datum of zero
enthalpy for elements when they are in their standard state at a temperature of 25 ◦ C. The tables have been extrapolated below
300 K, and these data should be used with caution.
The difference in internal energy of reactants and products at 25 ◦ C will thus correspond to the constant volume calorific value of
the reaction. When there is a difference in the number of kmols of gaseous reactants and products, the constant pressure
calorific value (the difference in enthalpy of reactants and products at 25 ◦ C) will differ from the constant volume calorific value.
PROPERTIES OF MATTER
77
−1
Absolute molar enthalpy (MJ kmol )
T (K)
Air
N2
O2
H2
CO
CO2
H2 O
NO
Ar
OH
O
H
0
100
200
−8.774
−5.784
−2.855
−8.823
−5.812
−2.864
−8.709
−5.755
−2.858
−8.222 −119.363 −402.239 −251.763
−5.584 −116.336 −399.897 −248.414
−2.804 −113.387 −396.934 −245.097
298.15
0.000
0.000
0.000
300
400
0.053
2.973
0.053
2.969
0.054
3.028
500
600
700
800
900
5.930
8.942
12.018
15.164
18.379
5.909
8.893
11.934
15.042
18.219
6.089
9.247
12.500
15.837
19.244
5.880 −104.594 −385.208 −234.899 96.195 4.198
8.811 −101.584 −380.605 −231.326 99.337 6.278
11.750 −98.505 −375.753 −227.639 102.547 8.358
14.704 −95.353 −370.699 −223.831 105.817 10.438
17.681 −92.128 −365.477 −219.899 109.143 12.518
1000
1100
1200
1300
1400
21.655
24.984
28.357
31.772
35.224
21.459
24.756
28.100
31.489
34.917
22.708
26.213
29.754
33.328
36.934
20.686
23.729
26.814
29.942
33.112
−88.838
−85.496
−82.107
−78.676
−75.208
−360.112
−354.624
−349.051
−343.395
−337.664
−215.841
−211.652
−207.337
−202.904
−198.359
112.518
115.939
119.400
122.897
126.427
1500
1600
1700
1800
1900
38.711
42.228
45.774
49.345
52.940
38.380
41.876
45.400
48.949
52.522
40.571
44.237
47.931
51.652
55.399
36.323
39.574
42.865
46.194
49.560
−71.707
−68.176
−64.618
−61.037
−57.435
−331.867
−326.012
−320.105
−314.151
−308.158
−193.711
−188.966
−184.131
−179.212
−174.215
2000
2100
2200
2300
2400
56.556
60.190
63.842
67.510
71.192
56.114
59.725
63.351
66.991
70.644
59.171
62.967
66.787
70.630
74.495
52.962
56.399
59.870
63.373
66.908
−53.815
−50.178
−46.527
−42.863
−39.189
−302.128
−296.068
−289.981
−283.870
−277.737
2500
2600
2700
2800
2900
74.887
78.595
82.313
86.041
89.779
74.308
77.981
81.664
85.354
89.051
78.381
82.288
86.216
90.163
94.130
70.474
74.068
77.691
81.341
85.016
−35.504
−31.811
−28.111
−24.403
−20.689
−271.587
−265.420
−259.239
−253.045
−246.838
295.330
297.416
299.503
301.592
303.684
263.753
265.831
267.910
269.988
272.067
3000
3100
3200
3300
3400
93.527 92.754 98.116 88.716
97.282 96.464 102.120 92.439
101.046 100.179 106.142 96.185
104.818 103.900 110.182 99.953
108.597 107.625 114.238 103.741
−16.969
−13.244
−9.514
−5.778
−2.039
−240.621 −115.467 185.276 56.197 129.040 305.777
−234.393 −109.881 189.026 58.277 132.724 307.872
−228.155 −104.266 192.780 60.357 136.421 309.971
−221.907 −98.625 196.538 62.437 140.132 312.072
−215.650 −92.959 200.300 64.517 143.855 314.177
274.146
276.224
278.303
280.382
282.460
3500
3600
3700
3800
3900
112.383
116.177
119.978
123.786
127.600
111.356
115.091
118.832
122.577
126.326
118.312
122.402
126.507
130.627
134.762
107.549
111.377
115.223
119.086
122.968
1.706
5.455
9.209
12.968
16.731
−209.383
−203.106
−196.819
−190.522
−184.215
−87.271
−81.561
−75.832
−70.083
−64.317
204.066
207.836
211.610
215.389
219.171
66.597
68.677
70.757
72.837
74.917
147.589
151.334
155.089
158.853
162.627
316.284
318.396
320.511
322.630
324.754
284.539
286.617
288.696
290.775
292.853
4000
4100
4200
4300
4400
131.421
135.249
139.083
142.923
146.769
130.081
133.840
137.604
141.373
145.147
138.911
143.074
147.249
151.436
155.634
126.866
130.781
134.713
138.662
142.627
20.499
24.272
28.049
31.831
35.617
−177.898
−171.570
−165.232
−158.884
−152.526
−58.533
−52.733
−46.918
−41.088
−35.242
222.958
226.748
230.543
234.342
238.145
76.997
79.077
81.157
83.237
85.317
166.409
170.200
173.998
177.805
181.620
326.881
329.014
331.150
333.292
335.439
294.932
297.011
299.089
301.168
303.246
4500
4600
4700
4800
4900
150.621
154.478
158.340
162.206
166.075
148.925
152.707
156.493
160.282
164.074
159.842
164.060
168.286
172.519
176.758
146.610
150.610
154.628
158.665
162.722
39.408
43.202
47.000
50.801
54.604
−146.159
−139.783
−133.400
−127.012
−120.619
−29.383
−23.509
−17.621
−11.719
−5.803
241.951
245.761
249.574
253.390
257.207
87.397
89.477
91.557
93.636
95.716
185.443
189.273
193.112
196.960
200.816
337.590
339.747
341.909
344.076
346.249
305.325
307.404
309.482
311.561
313.640
5000
169.947 167.868 181.001 166.800
81.717 −6.201
84.423 −4.121
87.231 −2.041
32.721 243.041 211.787
35.180 245.154 213.866
37.716 247.264 215.944
0.000 −110.525 −393.512 −241.824
90.080
0.000
40.277 249.333 217.984
0.056 −110.471 −393.443 −241.761
2.959 −107.550 −389.509 −238.371
90.135
93.125
0.038
2.118
40.326 249.372 218.023
43.005 251.477 220.101
58.409 −114.224
45.752
48.564
51.437
54.369
57.358
253.580
255.681
257.780
259.876
261.971
222.180
224.259
226.337
228.416
230.495
14.598
16.678
18.758
20.838
22.918
60.399
63.492
66.633
69.820
73.050
264.063
266.154
268.243
270.330
272.416
232.573
234.652
236.730
238.809
240.888
129.986
133.572
137.180
140.808
144.454
24.998
27.078
29.158
31.238
33.318
76.322
79.634
82.982
86.365
89.781
274.501
276.585
278.668
280.751
282.833
242.966
245.045
247.124
249.202
251.281
−169.146
−164.009
−158.810
−153.553
−148.243
148.116
151.792
155.480
159.179
162.886
35.398 93.228 284.915 253.359
37.478 96.705 286.997 255.438
39.558 100.208 289.079 257.517
41.638 103.738 291.162 259.595
43.718 107.292 293.246 261.674
−142.883
−137.478
−132.031
−126.545
−121.022
166.602
170.326
174.055
177.790
181.531
45.797
47.877
49.957
52.037
54.117
110.868
114.465
118.083
121.718
125.371
0.127 261.025 97.796 204.682 348.427 315.718
The term “absolute” molar enthalpy adopted here uses a datum of zero enthalpy for elements when they are in their standard
◦
◦
0
state at a temperature of 25 C. The enthalpy of any molecule at 25 C will thus correspond to its enthalpy of formation ΔHf . The
tables have been extrapolated below 300 K, and these data should be used with caution.
The enthalpies have been evaluated by the integration of a polynomial function that describes the molar specific heat capacity cp
◦
variation with temperature. The difference in enthalpy of reactants and products at 25 C will thus correspond to the constant
pressure calorific value of the reaction.
78
ENGINEERING TABLES AND DATA
−1
Absolute molar entropy (kJ kmol
T (K)
0
100
200
Air
N2
O2
−1
K )
H2
CO
CO2
H2 O
NO
Ar
OH
O
H
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
162.167 159.512 173.442 100.054 165.543 179.639 152.512 179.394 132.087 157.595 138.332 92.008
182.484 179.963 193.531 119.277 186.001 199.953 175.510 198.821 146.504 175.143 152.960 106.416
298.15 194.096 191.614 205.150 130.689 197.646 213.811 188.820 210.398 154.809 185.550 161.376 114.715
300
400
194.276 191.794 205.332 130.868 197.826 214.041 189.028 210.580 154.938 185.713 161.507 114.844
202.675 200.182 213.880 139.216 206.229 225.329 198.775 219.176 160.921 193.418 167.564 120.824
500
600
700
800
900
209.272
214.760
219.502
223.701
227.487
206.741
212.179
216.866
221.015
224.756
220.707
226.463
231.476
235.931
239.943
145.734
151.077
155.608
159.553
163.058
212.823
218.310
223.054
227.262
231.060
234.913
243.299
250.773
257.519
263.668
206.519
213.032
218.713
223.795
228.425
226.023
231.751
236.698
241.064
244.980
165.563
169.355
172.561
175.339
177.788
199.545
204.669
209.097
213.012
216.531
172.257
176.087
179.322
182.122
184.589
125.462
129.252
132.456
135.232
137.680
1000
1100
1200
1300
1400
230.938
234.111
237.046
239.779
242.337
228.170
231.312
234.222
236.934
239.474
243.591
246.935
250.015
252.876
255.548
166.224
169.127
171.811
174.315
176.664
234.525
237.712
240.661
243.407
245.976
269.319
274.541
279.390
283.917
288.163
232.700
236.693
240.447
243.995
247.362
248.536
251.796
254.807
257.606
260.222
179.980
181.962
183.772
185.437
186.978
219.735
222.682
225.415
227.965
230.359
186.793
188.786
190.604
192.275
193.821
139.870
141.851
143.660
145.324
146.864
1500
1600
1700
1800
1900
244.743
247.013
249.162
251.204
253.147
241.863
244.119
246.255
248.284
250.216
258.057
260.423
262.662
264.789
266.815
178.879
180.977
182.972
184.875
186.694
248.392
250.671
252.827
254.874
256.822
292.162
295.941
299.522
302.925
306.166
250.569
253.631
256.562
259.373
262.075
262.678
264.991
267.179
269.253
271.224
188.413
189.756
191.017
192.206
193.330
232.617
234.754
236.783
238.717
240.564
195.259
196.604
197.867
199.057
200.183
148.298
149.640
150.900
152.088
153.212
2000
2100
2200
2300
2400
255.002
256.775
258.474
260.104
261.671
252.058
253.820
255.507
257.125
258.680
268.750
270.602
272.379
274.087
275.732
188.439
190.116
191.731
193.288
194.793
258.679
260.453
262.152
263.780
265.344
309.258
312.215
315.047
317.763
320.373
264.675
267.181
269.600
271.936
274.196
273.102
274.896
276.611
278.255
279.833
194.397
195.412
196.379
197.304
198.189
242.332
244.028
245.658
247.227
248.739
201.251
202.267
203.235
204.161
205.048
154.278
155.292
156.259
157.183
158.068
2500
2600
2700
2800
2900
263.180
264.634
266.037
267.393
268.705
260.175
261.616
263.006
264.348
265.645
277.318
278.851
280.333
281.768
283.160
196.248
197.658
199.025
200.352
201.642
266.848
268.296
269.693
271.041
272.345
322.884
325.302
327.635
329.888
332.066
276.384
278.504
280.560
282.555
284.493
281.350
282.810
284.218
285.576
286.889
199.038
199.854
200.639
201.395
202.125
250.199
251.610
252.975
254.297
255.579
205.899
206.717
207.505
208.264
208.998
158.916
159.732
160.516
161.272
162.001
3000
3100
3200
3300
3400
269.975
271.206
272.401
273.562
274.690
266.901
268.117
269.296
270.441
271.554
284.512
285.825
287.102
288.345
289.556
202.896
204.117
205.306
206.466
207.597
273.606
274.827
276.012
277.161
278.277
334.173
336.216
338.196
340.118
341.987
286.376
288.208
289.990
291.726
293.417
288.159
289.388
290.580
291.736
292.860
202.830
203.513
204.173
204.813
205.434
256.823
258.031
259.205
260.347
261.458
209.708
210.395
211.061
211.708
212.336
162.706
163.388
164.048
164.687
165.308
3500
3600
3700
3800
3900
275.788
276.857
277.898
278.913
279.904
272.635
273.687
274.712
275.711
276.685
290.736
291.889
293.013
294.112
295.186
208.701
209.779
210.832
211.863
212.871
279.363
280.419
281.448
282.450
283.427
343.803
345.571
347.294
348.973
350.612
295.066
296.675
298.245
299.778
301.275
293.951
295.013
296.047
297.055
298.037
206.037
206.623
207.193
207.747
208.288
262.540
263.595
264.624
265.628
266.608
212.947
213.542
214.121
214.687
215.238
165.910
166.496
167.065
167.620
168.160
4000
4100
4200
4300
4400
280.872
281.817
282.741
283.644
284.529
277.635
278.564
279.471
280.357
281.225
296.237
297.264
298.271
299.256
300.221
213.858
214.825
215.772
216.701
217.613
284.381
285.313
286.223
287.113
287.984
352.211
353.773
355.301
356.795
358.256
302.740
304.172
305.573
306.945
308.289
298.996
299.932
300.847
301.741
302.615
208.814
209.328
209.829
210.318
210.797
267.566
268.502
269.417
270.313
271.190
215.777
216.303
216.818
217.322
217.816
168.686
169.199
169.700
170.189
170.667
4500
4600
4700
4800
4900
285.394
286.242
287.072
287.886
288.684
282.074
282.905
283.719
284.517
285.299
301.167
302.094
303.002
303.894
304.768
218.508
219.387
220.251
221.101
221.938
288.836
289.669
290.486
291.286
292.071
359.687
361.088
362.461
363.806
365.124
309.606
310.897
312.163
313.406
314.625
303.470
304.308
305.128
305.931
306.718
211.264
211.721
212.168
212.606
213.035
272.049
272.891
273.717
274.527
275.322
218.299
218.773
219.238
219.695
220.142
171.134
171.591
172.038
172.476
172.904
5000
289.466 286.066 305.625 222.762 292.839 366.416 315.823 307.489 213.455 276.103 220.582 173.324
The entropy is evaluated by the integration of ds = (cp /T ) dT . When cp is described by a polynomial function there is a
singularity at 0 K. The datum is provided by the values of entropy for substances at 25 ◦ C in their standard state at a pressure of
1 bar (WARNING: many sources use a datum pressure of 1 atm). The entropy can be evaluated at other pressures from
s = s ◦ − R ln(p/p ◦ ) with the superscript ◦ referring to the datum pressure p ◦ of 1 bar.
When the entropy of a mixture is being evaluated, then the properties of the individual constituents are summed, but the
pressure p now refers to the partial pressure of each constituent.
PROPERTIES OF MATTER
79
−1
Absolute molar Gibbs energy (MJ kmol )
T (K)
Air
N2
O2
H2
CO
CO2
H2 O
NO
Ar
OH
O
H
0
100
200
−8.77
−22.00
−39.35
−8.82
−21.76
−38.85
−8.70 −8.22 −119.36 −402.23 −251.76
−23.09 −15.58 −132.89 −417.86 −263.66
−41.56 −26.66 −150.58 −436.92 −280.19
81.71 −6.20
66.48 −17.33
47.46 −31.34
32.72
19.42
2.68
243.04
231.32
216.67
211.78
204.66
194.66
298.15 −57.87
−57.13
−61.16 −38.96 −169.45 −457.25 −298.12
27.35 −46.15
−15.04
201.21
183.78
300
400
−58.23
−78.09
−57.48
−77.10
−61.54 −39.20 −169.81 −457.65 −298.47
−82.52 −52.72 −190.04 −479.64 −317.88
26.96 −46.44
5.45 −62.25
−15.38
−34.36
200.92
184.45
183.57
171.77
500
600
700
800
900
−98.70
−119.91
−141.63
−163.79
−186.35
−97.46
−118.41
−139.87
−161.77
−184.06
−104.26
−126.63
−149.53
−172.90
−196.70
−66.98
−81.83
−97.17
−112.93
−129.07
−211.00
−232.57
−254.64
−277.16
−300.08
−502.66
−526.58
−551.29
−576.71
−602.77
−338.15 −16.81
−359.14 −39.71
−380.73 −63.14
−402.86 −87.03
−425.48 −111.33
−78.58 −54.02
−95.33 −74.23
−112.43 −94.93
−129.83 −116.04
−147.49 −137.52
167.45
150.02
132.25
114.17
95.84
159.44
146.70
133.61
120.23
106.58
1000
1100
1200
1300
1400
−209.28
−232.53
−256.09
−279.94
−304.04
−206.71
−229.68
−252.96
−276.52
−300.34
−220.88
−245.41
−270.26
−295.41
−320.83
−145.53
−162.31
−179.35
−196.66
−214.21
−323.36
−346.97
−370.90
−395.10
−419.57
−629.43
−656.61
−684.31
−712.48
−741.09
−448.54
−472.01
−495.87
−520.09
−544.66
−136.01
−161.03
−186.36
−211.99
−237.88
−165.38
−183.48
−201.76
−220.23
−238.85
−159.33
−181.45
−203.86
−226.53
−249.45
77.27
58.48
39.51
20.37
1.06
92.70
78.61
64.33
49.88
35.27
1500
1600
1700
1800
1900
−328.40
−352.99
−377.80
−402.82
−428.03
−324.41
−348.71
−373.23
−397.96
−422.88
−346.51
−372.44
−398.59
−424.96
−451.54
−231.99
−249.98
−268.18
−286.58
−305.16
−444.29
−469.24
−494.42
−519.81
−545.39
−770.11
−799.51
−829.29
−859.41
−889.87
−569.56
−594.77
−620.28
−646.08
−672.15
−264.03
−290.41
−317.02
−343.84
−370.87
−257.62
−276.53
−295.57
−314.73
−334.00
−272.60
−295.97
−319.55
−343.32
−367.29
2000
2100
2200
2300
2400
−453.44
−479.03
−504.80
−530.73
−556.81
−448.00
−473.29
−498.76
−524.39
−550.18
−478.32
−505.29
−532.44
−559.77
−587.26
−323.91
−342.84
−361.93
−381.18
−400.59
−571.17
−597.13
−623.26
−649.55
−676.01
−920.64
−951.71
−983.08
−1014.72
−1046.63
−698.49
−725.08
−751.92
−779.00
−806.31
−398.08
−425.48
−453.06
−480.80
−508.71
−353.39
−372.88
−392.47
−412.16
−431.93
−391.43
−415.75
−440.23
−464.88
−489.68
−117.58
−137.76
−158.03
−178.40
−198.86
−55.19
−70.67
−86.25
−101.92
−117.68
2500
2600
2700
2800
2900
−583.06
−609.45
−635.98
−662.65
−689.46
−576.13
−602.22
−628.45
−654.82
−681.32
−614.91
−642.72
−670.68
−698.78
−727.03
−420.14
−439.84
−459.67
−479.64
−499.74
−702.62
−729.38
−756.28
−783.31
−810.48
−1078.79
−1111.20
−1143.85
−1176.73
−1209.82
−833.84
−861.58
−889.54
−917.69
−946.05
−536.77
−564.98
−593.33
−621.82
−650.44
−451.79
−471.74
−491.76
−511.87
−532.04
−514.63
−539.72
−564.95
−590.31
−615.80
−219.41
−240.04
−260.75
−281.54
−302.41
−133.53
−149.47
−165.48
−181.57
−197.73
3000
3100
3200
3300
3400
−716.39
−743.45
−770.63
−797.93
−825.35
−707.94
−734.69
−761.56
−788.55
−815.65
−755.41
−783.93
−812.58
−841.35
−870.25
−519.97
−540.32
−560.79
−581.38
−602.08
−837.78
−865.20
−892.75
−920.41
−948.18
−1243.14
−1276.66
−1310.38
−1344.29
−1378.40
−974.59
−1003.32
−1032.23
−1061.32
−1090.57
−679.20
−708.07
−737.07
−766.19
−795.42
−552.29
−572.61
−592.99
−613.44
−633.95
−641.42
−667.17
−693.03
−719.01
−745.10
−323.34
−344.35
−365.42
−386.56
−407.76
−213.97
−230.27
−246.65
−263.08
−279.58
3500
3600
3700
3800
3900
−852.87
−880.50
−908.24
−936.08
−964.02
−842.86
−870.18
−897.60
−925.12
−952.74
−899.26
−928.39
−957.64
−986.99
−1016.46
−622.90
−643.82
−664.85
−685.99
−707.22
−976.06
−1004.05
−1032.14
−1060.34
−1088.63
−1412.69
−1447.16
−1481.80
−1516.62
−1551.60
−1120.00
−1149.59
−1179.33
−1209.23
−1239.29
−824.76
−854.21
−883.76
−913.42
−943.17
−654.53
−675.16
−695.85
−716.60
−737.40
−771.30
−797.60
−824.02
−850.53
−877.14
−429.03
−450.35
−471.73
−493.17
−514.67
−296.14
−312.76
−329.44
−346.18
−362.97
4000
4100
4200
4300
4400
−992.06
−1020.20
−1048.42
−1076.74
−1105.15
−980.46
−1008.27
−1036.17
−1064.16
−1092.24
−1046.03
−1075.71
−1105.48
−1135.36
−1165.33
−728.56
−750.00
−771.53
−793.15
−814.87
−1117.02
−1145.51
−1174.08
−1202.75
−1231.51
−1586.74
−1622.04
−1657.49
−1693.10
−1728.85
−1269.49
−1299.83
−1330.32
−1360.95
−1391.71
−973.02
−1002.97
−1033.01
−1063.14
−1093.36
−758.26
−779.16
−800.12
−821.13
−842.18
−903.85
−930.65
−957.55
−984.54
−1011.61
−536.22
−557.83
−579.48
−601.19
−622.95
−379.81
−396.70
−413.65
−430.64
−447.68
4500
4600
4700
4800
4900
−1133.65
−1162.23
−1190.90
−1219.64
−1248.47
−1120.40
−1148.65
−1176.98
−1205.40
−1233.89
−1195.40
−1225.57
−1255.82
−1286.17
−1316.60
−836.67
−858.57
−880.55
−902.62
−924.77
−1260.35
−1289.27
−1318.28
−1347.37
−1376.54
−1764.75
−1800.78
−1836.96
−1873.28
−1909.72
−1422.60
−1453.63
−1484.78
−1516.06
−1547.46
−1123.66
−1154.05
−1184.52
−1215.07
−1245.71
−863.29
−884.44
−905.63
−926.87
−948.15
−1038.77
−1066.02
−1093.35
−1120.76
−1148.26
−644.75
−666.61
−688.51
−710.45
−732.44
−464.77
−481.91
−499.09
−516.32
−533.59
−18.38
20.51
−37.98
5.62
−57.70 −9.40
−77.55 −24.55
−97.51 −39.82
5000 −1277.38 −1262.46 −1347.12 −947.00 −1405.78 −1946.30 −1578.99 −1276.42 −969.48 −1175.83 −754.48 −550.90
The Gibbs energy G is by definition G ◦ = H − T S ◦ with the superscript ◦ referring to the datum pressure p ◦ of 1 bar. The Gibbs
energy can be evaluated at other pressures in a similar way to the entropy through the use of G = G ◦ + RT ln(p/p ◦ ).
When the Gibbs energy of a mixture is being evaluated, then the properties of the individual constituents are summed, but the
pressure p now refers to the partial pressure of each constituent.
80
ENGINEERING TABLES AND DATA
−1
Absolute molar internal energy (MJ kmol )
T (K)
C6 H6
C7 H8
−82.660 −115.697
−81.302 −115.471
−79.403 −112.489
74.264
72.339
74.628
36.827 −213.096 −252.754 −431.071 100.199 −210.113 −246.415 −84.821
36.032 −211.181 −250.583 −425.839 97.352 −208.484 −245.303 −83.957
39.826 −203.207 −241.458 −406.960 102.575 −206.208 −242.216 −81.323
298.15 −76.999 −107.159
80.451
47.521 −190.129 −226.489 −376.649 114.381 −203.419 −237.429 −77.209
300
400
−76.948 −107.037
−73.932 −99.386
80.592
89.735
47.701 −189.836 −226.154 −375.975 114.660 −203.362 −237.323 −77.118
59.177 −171.683 −205.391 −334.313 132.521 −200.040 −230.788 −71.526
500
600
700
800
900
−70.354
−66.220
−61.544
−56.346
−50.653
−89.785
−78.464
−65.634
−51.487
−36.197
101.603 73.804 −149.318 −179.834 −283.293 155.194 −196.354
115.785 91.161 −123.263 −150.092 −224.124 181.833 −192.433
131.914 110.859 −93.996 −116.720 −157.901 211.717 −188.423
149.664 132.535 −61.949 −80.218 −85.612 244.242 −184.486
−8.131 278.928 −180.803
168.752 155.857 −27.508 −41.031
1000
1100
1200
1300
1400
−44.499
−37.925
−30.977
−23.711
−16.186
−19.915
−2.776
15.106
33.634
52.734
188.938
210.026
231.862
254.333
277.372
1500
−8.471
0
100
200
CH4
C3 H8
C7 H16
C8 H18
C16 H34 C10 H7 CH3 CH3 OH
315.414
353.461
392.951
433.887
476.391
−177.571
−175.004
−173.334
−172.809
−173.695
C2 H5 OH
−222.777
−213.454
−202.981
−191.522
−179.237
CH3 NO2
−64.720
−56.859
−48.092
−38.552
−28.363
−166.289 −17.633
−152.836 −6.461
−139.038
5.068
−125.054
16.883
−111.040
28.924
180.523
206.260
232.824
260.001
287.606
8.988
47.246
87.016
128.099
170.338
0.453
43.898
89.025
135.609
183.482
73.778
159.463
248.381
340.101
434.304
72.348 300.952 315.485
213.622
232.528
530.781 520.709 −176.274
−97.153
41.143
C7 H16
C8 H18
C16 H34 C10 H7 CH3 CH3 OH
C2 H5 OH
CH3 NO2
−1
Absolute molar enthalpy (MJ kmol )
T (K)
0
100
200
CH4
C3 H8
C6 H6
C7 H8
−82.660 −115.697
−80.470 −114.640
−77.740 −110.826
74.264
73.170
76.290
36.827 −213.096 −252.754 −431.071 100.199 −210.113 −246.415 −84.821
36.863 −210.350 −249.752 −425.008 98.184 −207.653 −244.471 −83.125
41.489 −201.545 −239.795 −405.297 104.238 −204.545 −240.553 −79.660
298.15 −74.520 −104.680
82.930
50.000 −187.650 −224.010 −374.170 116.860 −200.940 −234.950 −74.730
300
400
−74.454 −104.543
−70.606 −96.060
83.086
93.061
50.195 −187.342 −223.660 −373.480 117.155 −200.867 −234.829 −74.624
62.502 −168.357 −202.065 −330.987 135.847 −196.714 −227.463 −68.200
500
600
700
800
900
−66.197
−61.231
−55.724
−49.694
−43.170
−85.627
−73.475
−59.814
−44.836
−28.713
105.760 77.961 −145.160 −175.677 −279.136 159.351 −192.197
120.774 96.150 −118.274 −145.103 −219.135 186.822 −187.444
137.734 116.679 −88.176 −110.900 −152.081 217.537 −182.602
156.315 139.186 −55.298 −73.567 −78.961 250.894 −177.834
−0.648 286.411 −173.320
176.235 163.340 −20.025 −33.548
−218.620
−208.465
−197.161
−184.870
−171.754
−60.563
−51.871
−42.272
−31.901
−20.880
1000
1100
1200
1300
1400
−36.185
−28.779
−21.000
−12.902
−4.546
−11.600
6.370
25.083
44.443
64.374
197.253
219.172
241.839
265.142
289.012
1500
4.001
188.838
215.406
242.801
270.810
299.247
17.303
56.392
96.994
138.908
181.978
8.767
53.044
99.002
146.418
195.122
82.093
168.609
258.358
350.910
445.945
−169.256
−165.858
−163.357
−162.000
−162.055
−157.974
−143.690
−129.061
−114.245
−99.399
−9.319
2.685
15.046
27.692
40.564
84.819 313.424 327.957
226.094
245.000
543.253 533.181 −163.803
−84.681
53.614
323.729
362.607
402.929
444.696
488.032
PROPERTIES OF MATTER
81
−1
Absolute molar entropy (kJ kmol
−1
K )
T (K)
CH4
C3 H8
C6 H6
C7 H8
C7 H16
C8 H18
C16 H34
C10 H7 CH3 CH3 OH
C2 H5 OH
CH3 NO2
0
100
200
0.000
154.754
173.463
0.000
220.323
245.721
0.000
222.796
242.855
0.000
255.497
285.836
0.000
313.582
372.338
0.000
293.320
359.722
0.000
524.594
656.095
0.000
288.137
327.134
0.000
204.204
225.502
0.000
231.886
258.289
0.000
232.013
255.365
298.15 186.270
270.200
269.200
319.740
427.980
422.960
781.020
377.440
239.880
280.640
275.010
300
400
186.709
197.739
270.877
295.137
269.941
298.432
320.612
355.785
429.229
483.534
424.350
486.120
783.545
905.102
378.644
432.056
240.342
252.261
281.265
302.348
275.584
293.974
500
600
700
800
900
207.556
216.594
225.073
233.117
240.796
318.341
340.454
361.484
381.467
400.444
326.667
353.982
380.091
404.880
428.328
390.157
423.250
454.853
484.880
513.312
535.134
584.058
630.395
674.259
715.779
544.819
600.453
653.110
702.919
750.026
1020.442
1129.626
1232.860
1330.416
1422.599
484.322
534.308
581.597
626.103
667.914
262.329
270.989
278.452
284.821
290.143
322.022
340.503
357.907
374.305
389.745
310.969
326.790
341.570
355.408
368.382
1000
1100
1200
1300
1400
248.152
255.207
261.974
268.454
274.646
418.466
435.587
451.865
467.358
482.126
450.463
471.349
491.067
509.716
527.403
540.165
565.480
589.313
611.729
632.801
755.090
792.333
827.653
861.196
893.109
794.590
836.776
876.755
914.701
950.789
1509.738
1592.170
1670.243
1744.310
1814.728
707.217
744.261
779.336
812.760
844.868
294.429
297.674
299.857
300.949
300.916
404.258
417.869
430.597
442.455
453.457
380.558
391.996
402.749
412.870
422.409
1500
280.541
496.230
544.243
652.608
923.542
985.196
1881.854
876.010
299.718
463.612
431.412
C8 H18
C16 H34
−1
Absolute molar Gibbs energy (MJ kmol )
T (K)
CH4
C3 H8
C 6 H6
C7 H 8
C7 H16
C10 H7 CH3 CH3 OH
C2 H5 OH CH3 NO2
−82.660 −115.697
−95.946 −136.672
−112.432 −159.970
74.264
36.827 −213.096 −252.754 −431.071
50.891
11.314 −241.708 −279.084 −477.467
27.719 −15.678 −276.012 −311.739 −536.516
100.199 −210.113 −246.415 −84.821
69.370 −228.073 −267.660 −106.326
38.811 −249.645 −292.211 −130.733
298.15 −130.056 −185.240
2.668 −45.330 −315.252 −350.116 −607.031
4.326 −272.460 −318.623 −156.724
0
100
200
300
400
−130.467 −185.806
2.104 −45.988 −316.111 −350.965 −608.544
3.561 −272.970 −319.208 −157.299
−149.702 −214.115 −26.312 −79.811 −361.771 −396.513 −693.028 −36.975 −297.618 −348.402 −185.790
500
600
700
800
900
−169.975
−191.188
−213.275
−236.188
−259.886
−244.798
−277.747
−312.853
−350.009
−389.113
−57.573
−91.615
−128.329
−167.589
−209.260
−117.118
−157.800
−201.718
−248.718
−298.641
−412.727
−468.709
−529.452
−594.705
−664.226
−448.086
−505.375
−568.077
−635.902
−708.571
−789.357
−896.911
−1015.083
−1143.293
−1280.988
−82.810
−133.763
−189.581
−249.989
−314.712
−323.361
−350.037
−377.519
−405.691
−434.448
−379.631
−412.767
−447.696
−484.314
−522.525
−216.047
−247.945
−281.371
−316.227
−352.424
1000
1100
1200
1300
1400
−284.336
−309.507
−335.368
−361.892
−389.049
−430.066
−472.776
−517.155
−563.123
−610.603
−253.211
−299.311
−347.441
−397.489
−449.353
−351.328
−406.623
−464.374
−524.438
−586.675
−737.787
−815.175
−896.190
−980.647
−1068.375
−785.823
−867.410
−953.104
−1042.693
−1135.982
−1427.645
−1582.778
−1745.934
−1916.693
−2094.674
−383.488
−456.080
−532.275
−611.892
−694.783
−463.686
−493.299
−523.185
−553.234
−583.337
−562.232
−603.346
−645.777
−689.437
−734.239
−389.877
−428.511
−468.253
−509.039
−550.808
1500
−416.811 −659.526 −502.941 −650.956 −1159.219 −1232.795 −2279.529 −780.833 −613.379 −780.100 −593.503
82
ENGINEERING TABLES AND DATA
Thermochemical data for equilibrium reactions
Stoichiometric equations
νi Ai = 0 where νi is the stoichiometric coefficient of the substance whose chemical symbol is Ai .
(1)
−2H + H2 = 0
(6)
− 12 H2 − OH + H2 O = 0
(2)
−2N + N2 = 0
(7)
−CO − 12 O2 + CO2 = 0
(3)
−2O + O2 = 0
(8)
−CO − H2 O + CO2 + H2 = 0
(4)
−2NO + N2 + O2 = 0
(9)
− 12 N2 − 32 H2 + NH3 = 0
(5)
−H2 − 12 O2 + H2 O = 0
Standard enthalpy of reaction
HT◦ =
νi [h̄i ]◦T where [h̄i ]◦T = enthalpy per kmol of substance Ai at 1 bar pressure and absolute temperature T .
i
Warning: this table lists absolute temperatures.
Reaction number
1
2
3
4
5
HT◦ MJ
6
7
8
9
Temp
(K)
200
298
400
600
800
1000
−434.7
−436.0
−437.3
−439.7
−442.1
−444.5
−944.1
−945.3
−946.6
−948.9
−951.1
−953.0
−496.9
−498.4
−499.8
−502.1
−503.9
−505.4
−180.4
−180.6
−180.7
−180.7
−180.8
−180.9
−240.9
−241.8
−242.8
−244.8
−246.5
−247.9
−280.2
−281.3
−282.4
−284.1
−285.5
−286.6
−282.1
−283.0
−283.5
−283.6
−283.3
−282.6
−41.21
−41.17
−40.63
−38.88
−36.82
−34.74
−43.71
−45.90
−48.04
−51.39
−53.66
−55.07
Temp
(K)
200
298
400
600
800
1000
1200
1400
1600
1800
2000
−446.7
−448.7
−450.6
−452.3
−453.8
−954.7
−956.1
−957.5
−958.7
−959.9
−506.7
−507.8
−508.9
−509.8
−510.6
−180.9
−181.0
−181.0
−181.0
−181.0
−249.0
−249.9
−250.6
−251.2
−251.7
−287.4
−287.9
−288.4
−288.6
−288.8
−281.8
−280.9
−279.9
−278.9
−277.9
−32.79
−30.98
−29.29
−27.71
−26.22
−55.83
−56.07
−55.99
−55.66
−55.19
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
−455.2
−456.4
−457.6
−458.6
−459.6
−961.0
−962.1
−963.1
−964.1
−965.0
−511.4
−512.0
−512.5
−513.0
−513.4
−180.8
−180.7
−180.4
−180.1
−179.7
−252.1
−252.4
−252.7
−253.0
−253.3
−288.9
−289.0
−289.0
−288.9
−288.9
−276.8
−275.8
−274.8
−273.7
−272.7
−24.79
−23.41
−22.07
−20.77
−19.49
−54.61
−53.92
−53.12
−52.22
−51.20
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
−460.4
−461.2
−461.9
−462.5
−463.0
−966.0
−967.0
−968.1
−969.2
−970.4
−513.8
−514.1
−514.4
−514.6
−514.8
−179.3
−178.7
−178.2
−177.6
−176.9
−253.5
−253.8
−254.1
−254.5
−254.8
−288.8
−288.7
−288.6
−288.5
−288.4
−271.7
−270.7
−269.8
−268.8
−267.8
−18.19
−16.91
−15.62
−14.33
−13.00
−50.10
−48.94
−47.75
−46.49
−45.19
3200
3400
3600
3800
4000
4500
5000
5500
6000
−464.0
−464.6
−464.8
−464.7
−973.8
−977.9
−982.9
−989.0
−515.3
−515.9
−516.5
−517.2
−175.2
−173.2
−171.1
−169.0
−255.9
−257.2
−258.6
−260.3
−288.1
−288.0
−287.9
−287.9
−265.5
−263.1
−260.7
−258.2
−9.57
−5.95
−2.10
2.01
−41.68
−37.79
−33.56
−28.98
4500
5000
5500
6000
PROPERTIES OF MATTER
83
Equilibrium constants
ln Kp =
νi ln pi∗ where the dimensionless quantity pi∗ is numerically equal to the partial pressure of substance Ai in bars.
i
Warning: this table lists absolute temperatures.
Reaction number
1
2
3
4
5
ln Kp
6
7
8
9
Temp
(K)
200
298
400
600
800
1000
250.149
163.986
119.150
75.217
53.126
39.803
554.472
367.479
270.329
175.356
127.753
99.127
285.471
186.975
135.715
85.523
60.319
45.150
105.592
69.865
51.311
33.203
24.145
18.706
139.972
92.207
67.321
42.897
30.592
23.162
161.789
106.228
77.284
48.905
34.634
26.033
159.692
103.762
74.669
46.245
32.036
23.528
19.719
11.554
7.348
3.348
1.444
0.366
15.433
6.593
1.778
−3.191
−5.822
−7.457
Temp
(K)
200
298
400
600
800
1000
1200
1400
1600
1800
2000
30.874
24.463
19.632
15.865
12.835
80.011
66.329
56.055
48.051
41.645
35.005
27.742
22.285
18.030
14.622
15.082
12.489
10.546
9.035
7.824
18.182
14.608
11.921
9.825
8.145
20.281
16.160
13.065
10.657
8.727
17.871
13.841
10.829
8.497
6.634
−0.311
−0.767
−1.091
−1.329
−1.510
−8.570
−9.371
−9.972
−10.439
−10.810
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
10.353
8.276
6.512
5.002
3.685
36.391
32.011
28.304
25.117
22.359
11.827
9.497
7.521
5.286
4.357
6.834
6.010
5.314
4.720
4.205
6.768
5.619
4.647
3.811
3.086
7.148
5.831
4.718
3.763
2.936
5.119
3.859
2.800
1.893
1.110
−1.649
−1.759
−1.847
−1.918
−1.976
−11.109
−11.358
−11.563
−11.738
−11.885
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
2.533
1.516
0.609
−0.207
−0.939
19.936
17.800
15.898
14.198
12.660
3.072
1.935
0.926
0.019
−0.796
3.753
3.357
3.007
2.694
2.413
2.450
1.891
1.391
0.944
0.541
2.211
1.575
1.007
0.500
0.044
0.429
−0.170
−0.702
−1.176
−1.600
−2.022
−2.061
−2.093
−2.121
−2.141
−12.012
−12.122
−12.217
−12.300
−12.373
3200
3400
3600
3800
4000
4500
5000
5500
6000
−2.486
−3.725
−4.743
−5.590
9.414
6.807
4.666
2.865
−2.514
−3.895
−5.024
−5.963
1.828
1.363
0.986
0.677
−0.313
−0.997
−1.561
−2.033
−0.921
−1.690
−2.318
−2.843
−2.491
−3.198
−3.771
−4.246
−2.178
−2.201
−2.210
−2.213
−12.519
−12.624
−12.703
−12.760
4500
5000
5500
6000
Standard free enthalpy of reaction
At a given temperature, the standard free enthalpy of reaction
the listed value of ln Kp by the following equation:
GT◦ = −R0 T ln Kp = −8.3145 T ln Kp kJ kmol−1
GT◦ (or standard Gibbs function change) may be calculated from
–1
0
–10
–5
–5
0
0
0
5
50
5
10
60
10
70
10
15
90
20
15
0.84
20
20
30
%
30
3
%
o
pe
ure
30
tiv
Rela
rat
5%
10%
15%
20
25
0%
tem
%
bu
lb
25
We
t
0
e hu
o
C
ity
40
mid
13
0
50
0.96
0.86
0.82
0.80
0.78
0.76
%
Dry bulb temperature C
25
0
12
0.90
0
10
g
r
ai
10
0.88
20
30
40
E
90%
80%
dr
y
/k
kJ
py
al
nt
h
C
e
o
atu
r
p
80
er
em
%
70
Based on barometric pressure of 101.325 kPa
t
ion
t
ra
Sa
tu
%
60
dry
50
m
Volu
e 0.
3 /kg
92 m
40
11
0
35
60
0.00
0.01
0.02
0.03
84
ENGINEERING TABLES AND DATA
Psychrometric chart
0.98
0.94
air
Specific humidity kg/kg dry air
THERMODYNAMICS AND FLUID MECHANICS
Part 4
Thermodynamics and fluid mechanics
85
86
ENGINEERING TABLES AND DATA
Notation
In the following, T is absolute temperature, q heat input per unit mass, w work output per unit mass, v is volume per unit mass,
V is velocity, μ dynamic viscosity and ν kinematic viscosity. Other symbols are as defined on page 54 or below.
Thermodynamic relations
Basic relations
Process relations
Enthalpy: h = u + pv
Reversible polytropic, closed system
Helmholtz function: f = u − T s
pv n = constant
p2 v2 − p1 v1
w =
1−n
For a perfect gas,
Gibbs function: g = h − T s
du = T ds − p dv
dh = T ds + v dp
R
( T − T1 )
1−n 2
R
q = cv +
( T2 − T1 )
1−n
(n −1)/n
p2
T2
=
p1
T1
w =
df = −p dv − s dT
dg = v dp − s dT
Maxwell’s relations
∂p
∂s v
s
∂T
∂v
=
∂p s
∂s p
∂p
∂s
=
∂T v
∂v T
∂v
∂s
=−
∂T p
∂p T
∂T
∂v
and, if adiabatic (q = 0), n = γ , the ratio of specific heats
cp /cv , or if isothermal, n = 1 and
=−
q = w = RT ln v2 /v1 .
Steady flow
In terms of enthalpy, the energy equation relating any two
sections of a steady flow at which properties are uniform is
q − ws = h2 − h1 +
Specific heats
cp =
cv =
∂h
∂T
∂u
∂T
(n = 1)
1
2
V22 − V12 + g (z2 − z1 )
where g is the acceleration of gravity, ws is shaft work and z is
the height above a horizontal datum.
p
Nozzle flow
v
Coefficients
1 ∂v
v ∂T p
1 ∂v
Compressibility κ = −
v ∂p T
∂T
Joule-Thomson μ =
∂p h
Volume expansion β =
(for a perfect gas μ = 0)
Tv
κ
(for a perfect gas cp − cv = R)
cp − cv = β 2
Equations of State
n
is constant the critical
which gives the mass flux
#
(n +1)/(n −1)
p1
2
ṁ
= n
.
v1
n +1
A2
Pressure and area are related by
1
dA
1
−
= vA
dp
V 2 a2
where a is the sonic velocity, given by
∂p
∂p
=
.
a 2 = −v 2
∂v s
∂ρ s
Entropy change
For unit mass:
Perfect gas pv = RT =
For an expansion in which pv
pressure ratio is
n /(n −1)
p2
2
=
p1
n +1
R0 T
M
for a gas of molecular weight M with gas constant R.
Van der Waals’ gas ( p + a/v 2 )(v − b) = RT
where a, b are constants of the gas.
For any process the specific entropy change of a perfect gas is
p2
v2
s2 − s1 = cv ln
+ cp ln
p1
v1
T2
v2
= cv ln
+ R ln
v1
T1
T2
p2
= cp ln
− R ln
.
p1
T1
THERMODYNAMICS AND FLUID MECHANICS
87
Equations for fluid flow
Continuity
τxy = τy x = μ
∂ρ
= − div (ρV)
∂t
DVx
Momentum
∂
∂t
Dt
+
A
A
∂x
1 ∂p
+ν
ρ ∂x
∂2 Vx
∂x 2
+
∂2 Vx
∂y 2
+
∂2 Vx
∂z 2
1 ∂θ Fx
+
ν
ρ
3 ∂x
F
DV
1
1
= − ∇p + ν ∇2 V + ν ∇θ +
ρ
ρ
3
Dt
where
∂
D
+ V·∇
≡
Dt
∂t
Energy
q̇ + div (k grad T ) = ρ
∂u
+ V · grad u + p div V
∂t
at any point; terms for dissipation and radiation may be
included in q̇ , the rate of heat production per unit volume. The
integrated form for a control volume τ bounded by a surface A
is
∂
q̇ − ẇ =
u + 12 V 2 + gz ρ dτ
∂t
τ
p
V · dA
+ ρ u + 12 V 2 + gz +
ρ
A
where q̇ , ẇ are now the total rates of heat input and work
output for the volume. For steady flow,
∂Vx
∂x
+
∂Vy
∂y
∂Vx
∂x
+
∂Vy
∂y
+
∂Vz
∂z
= ∇·V.
Stream function and velocity potential
For two-dimensional incompressible flow the stream function
is a scalar field ψ such that, in Cartesian coordinates,
Vx = −
∂ψ
∂ψ
, Vy =
.
∂y
∂x
If in addition the flow is irrotational, the velocity potential is a
scalar field φ such that
∂φ
,
∂x
Vy = −
∂φ
.
∂y
The corresponding equations in polar coordinates are
etc., where Fx is the body force per unit volume and
2
− μ
σx = −p + 2 μ
3
∂x
θ=
φ and ψ are both harmonic functions satisfying Laplace’s
equation; the complex potential φ + jψ satisfies the
Cauchy-Riemann conditions. It may also be so defined as to
reverse the signs in all four of the above relations.
In Cartesian coordinates
∂Vx
∂Vx
∂Vx
∂Vx
+ Vx
+ Vy
+ Vz
ρ
∂t
∂x
∂y
∂z
∂τ
∂τz x
∂σx
yx
= Fx +
+
+
∂x
∂y
∂z
∂Vx
and
Vx = −
p
V · dA .
ρ u + 12 V 2 + gz +
ρ
Navier-Stokes equations
etc.
+
etc. or
where F is the net force on a control volume τ bounded by a
surface A. For steady flow
F = ρV (V · dA) .
q̇ − ẇ =
=−
ρV dτ + ρV (V · dA)
τ
∂y
∂Vy
etc. These give, for constant viscosity,
(for an incompressible fluid, div V = 0)
F=
∂Vx
+
∂Vz
∂z
Vr = −
1 ∂ψ
,
r ∂θ
Vr = −
∂φ
,
∂r
Vθ =
Vθ = −
∂ψ
,
∂r
1 ∂φ
.
r ∂θ
88
ENGINEERING TABLES AND DATA
Dimensionless groups
General
h
L
Hydraulic machines
heat transfer coefficient (rate of heat flow per unit area
and per degree of temperature difference T )
characteristic dimension (length or diameter)
Froude number
V
Fr = Lg
Grashof number
Gr =
2
V
Lg
or
2 3
β gρ L
Nusselt number
Prandtl number
Reynolds number
Stanton number
P
ρN 3 D 5
Q
ND3
gH
Power number
T
μ2
Discharge number
V
a
hL
Nu =
k
cp μ
Pr =
k
V Lρ
Re =
μ
h
St =
ρV cp
M =
Mach number
power
speed
diameter
discharge
head
P
N
D
Q
H
Head number
N 2D 2
ρN D
μ
Reynolds number
2
Addison’s shape parameter:
N
Turbine
P /ρ
(gH )5/4
√
N Q
Pump
(gH )3/4
Generalized compressibility chart
The compressibility factor Z as a function of reduced pressure and temperature pR and TR is to a good approximation the same
for all gases. The function is plotted here with pR as independent variable and TR as parameter. Reduced pressure and
temperature are dimensionless numbers, defined as pR = p/pc and TR = T /Tc , which relate the actual values of pressure and
temperature to the critical values pc and Tc .
3.50
1.1
5.00
2.50
1.0
2.00
1.80
0.9
Compressibility factor Z =
pv
RT
1.60
1.50
0.8
1.40
0.7
1.30
0.6
1.20
0.5
1.15
1.10
0.4
1.05
0.3
TR = 1.00
0.2
0
1
2
3
4
Reduced pressure pR
5
6
7
THERMODYNAMICS AND FLUID MECHANICS
89
Nusselt numbers for convective heat transfer
In calculating heat transfer the temperature difference T to be used is that between the wall temperature Tw and either the
undisturbed free-stream temperature Tf (for external convection) or the average temperature Tb of the bulk flow (for convection in
pipes). The fluid properties are to be evaluated at the mean film temperature 12 (Tw + Tf ), for external convection, or at Tb for pipe
flow.
Unless otherwise indicated, the dimension L specified is that to be used in the groups Gr, Nu and Re (see Dimensionless groups
on page 88) and the expressions given are for constant wall temperature Tw .
Natural convection
Forced convection
Over the ranges indicated, Nu can be expressed as C (Gr Pr)n
where C and n are as shown.
Flat plates (streamwise length L)
Gr Pr
C
n
0.56
1
4
1
3
Vertical plates and cylinders (height L)
104 − 109
Laminar flow
9
10 − 10
Turbulent flow
13
0.13
Horizontal cylinders (diameter L)
Laminar flow
104 − 109
Turbulent flow
10 − 10
9
12
0.53
0.126
1
4
1
3
Horizontal plates
Heated facing up, or
7
cooled facing down
Heated facing down, or
cooled facing up
105 − 107
10 − 10
11
5
11
10 − 10
0.14
1
4
1
3
0.27
1
4
0.54
Laminar flow, Re < 5 × 105 :
Nu = 0.664 Re1/2 Pr1/3
For laminar flow with constant
heat flux, the local Nusselt
number at any distance L from
the leading edge is:
Nu = 0.453 Re1/2 Pr1/3
Turbulent flow, Re > 5 × 105 :
Nu = 0.036 Re0.8 Pr1/3
Flow in circular pipes (length L, diameter D ,
Re and Nu based on D )
Laminar flow, Re < 2000:
Nu = 1.615 (Re Pr D /L)1/3
For laminar flow with constant
heat flux:
Nu = 4.36
Turbulent flow, fully developed,
Re > 2000:
Nu = 0.023 Re0.8 Pr 0.4
Cylinder in cross flow (diameter L)
For air the Nusselt number can be approximated as
Nu = C Ren
where C and n depend on Re as follows:
Re
0.0001 – 0.004
0.004 – 0.09
0.09 – 1.0
1 – 35
35 – 5000
5000 – 50000
50000 – 200000
C
n
0.437
0.565
0.800
0.0895
0.136
0.280
0.795
0.583
0.148
0.0208
0.384
0.471
0.633
0.814
For other fluids the following approximations may be used:
Nu = 0.91 Re0.385 Pr 0.31
0.5
Nu = 0.60 Re
Pr
0.31
(0.1 < Re < 50)
(50 < Re < 10000)
90
ENGINEERING TABLES AND DATA
Friction in pipes
Moody diagram
Coefficients of loss for pipe fittings
The head loss for an average flow velocity V in a pipe of
diameter d and length L is given by Darcy’s equation:
The loss of head incurred by fittings, valves or sudden
contractions of area is given by the loss coefficient KL
according to the relation
2
hL = f
4L V
d 2g
hL = KL
in which f is the friction coefficient given by τ / 12 ρV 2 where τ
is the shear stress at the wall. The curves opposite show f as
a function of Re for various values of relative roughness
r = k /d , where k is the effective surface roughness and Re is
based on the diameter d . Typical values of k are given below.
Surface
k mm
Riveted steel
Concrete
Wood stave
Cast iron
Galvanized steel
Asphalted cast iron
Commercial steel or wrought iron
Drawn tubing
1 – 10
0.3 – 3
0.2 – 1
0.25
0.15
0.12
0.045
0.0015
For laminar flow
f =
16
.
Re
The Blasius formula for smooth pipes is
f = 0.079Re−0.25 .
V2
2g
where V is the average flow velocity. Values of KL for fittings,
valves and contractions of area ratio A2 /A1 are given below.
Fitting
Glove valve, fully open
Angle valve, fully open
Swing check valve, fully open
Gate valve, fully open
Three-quarters open
One-half open
One-quarter open
Close return bend
Standard tee
Standard 90◦ elbow
Medium sweep 90◦ elbow
Long sweep 90◦ elbow
45◦ elbow
Rounded inlet
Re-entrant inlet
Sharp-edged inlet
Contraction, A2 /A1 = 0.1
= 0.2
= 0.4
= 0.6
= 0.8
= 0.9
KL
10.0
5.0
2.5
0.19
1.15
5.6
24.0
2.2
1.8
0.9
0.75
0.60
0.42
0.04
0.8
0.5
0.37
0.35
0.27
0.17
0.06
0.02
Coefficient of friction f
10
3
4
5 6 7 8
10
4
2
Smooth pipes r = 0
2
3
4
r = 5e−006
6
r = 1e−006
Complete turbulence, rough pipes
5 6 7 8
10
7
2
3
4
5 6 7 8
10
0.06
0.07
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
0.0055
0.006
0.007
0.008
0.009
0.01
0.012
0.014
3
4
5 6 7 8
10
5 6 7 8
Reynolds number Re
5
Blasius formula
10
2
3
4
8
1e−005
2e−005
5e−005
0.0001
0.0002
0.0004
0.0006
0.001
0.0008
0.0015
0.002
0.003
0.004
0.006
0.008
0.01
0.015
0.02
0.03
0.04
2
Transition zone
0.016
3
Critical
zone
0.05
6 7 8
Laminar
flow
0.018
0.02
0.025
THERMODYNAMICS AND FLUID MECHANICS
91
Relative roughness r
92
ENGINEERING TABLES AND DATA
Boundary-layer friction and drag
In a two-dimensional constant-pressure boundary layer the
local skin-friction coefficient f at distance x from the leading
edge, and the friction drag coefficient CF for unit width of a
plate of length l (one surface) are as follows.
5
Laminar, 0 < Re 10
f =
τ
1
ρV 2
2
CF =
= 0.664
D
f = 0.0576
Vx
ν
= 1.328
−1/2
Vl
ν
−1/2
Vx
ν
(Blasius solution)
1
ρV 2 l
2
Turbulent, Re 105 ( 17 th root velocity profile)
CF = 0.072
Vl
ν
−1/5
−1/5
or
−2.58
Vl
CF = 0.455 log10
ν
which is the Prandtl-Schlichting formula.
In the above τ is the shear stress at the wall, V the velocity
outside the boundary layer and D the drag.
Open channel flow
The velocity V of uniform flow in an open channel of slope S
may be estimated from the Chézy formula
V = C RS
or from the Manning formula
0.82 2/3 1/2
V =
R S
n
in
which C is the Chézy coefficient given by 8g/f or
2g/Cf , and R is the hydraulic radius (ratio of flow section to
wetted perimeter) of the channel. Values of n are given at
right.
Surface
n (m1/6 )
Smooth surface
Neat cement surface
Finished concrete, planed wood, or steel
surface
Mortar, clay, or glazed brick surface
Vitrified clay surface
Brick surface lined with cement mortar
Unfinished cement surface
Rubble masonry or corrugated metal surface
Earth channel with gravel bottom
Earth channel with dense weed
Natural channel with clean bottom, brush
on sides
Flood plain with dense brush
0.008
0.009
0.010
0.011
0.012
0.012
0.014
0.016
0.020
0.030
0.040
0.080
Black-body radiation
The power radiated by a black body in all directions over a
solid angle 2π, per unit surface area and per unit frequency
interval, is
3
Eν =
2πh ν /c
2
λ m T = 0.0029
e h c /λk T − 1
∞
.
∞
Eν dν =
0
Eλ dλ = σ T 4
0
Radiation exchange
The net power exchanged between two black (or diffusely
radiating) surface elements dA1 and dA2 a distance r apart is
q̇ = (E1 − E2 ) cos φ1 cos φ2 dA1 dA2
2πh c 2 /λ 5
The total power per unit area is
E=
mK
e h ν /k T − 1
in the region of the frequency ν , where h is the Planck
constant, k the Boltzmann constant, and c the velocity of
light; alternatively, the power per unit wavelength interval in the
region of the wavelength λ is
Eλ =
in which σ is the Stefan-Boltzmann constant. The wavelength
λ m at which Eλ is a maximum is given by the Wien
displacement law:
where φ1 , φ2 are the angles between r and each normal.
1.0000
1.0000
.9998
.9996
.9992
1.0000
.9999
.9997
.9994
.9989
.9983
.9975
.9966
.9955
.9944
.9930
.9916
.9900
.9883
.9864
.9844
.9823
.9800
.9776
.9751
.9725
.9697
.9668
.9638
.9607
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.9803
.9783
.9762
.9740
.9718
.9888
.9873
.9857
.9840
.9822
.9950
.9940
.9928
.9916
.9903
.9988
.9982
.9976
.9968
.9960
ρ
ρ0
p
p0
M
.9921
.9913
.9904
.9895
.9886
.9055
.9949
.9943
.9936
.9928
.9980
.9976
.9971
.9966
.9961
.9995
.9993
.9990
.9987
.9984
1.0000
1.0000
.9999
.9998
.9997
T
T0
.9960
.9956
.9952
.9948
.9943
.9978
.9974
.0971
.9968
.9964
.9990
.9988
.9986
.9983
.9980
.9998
.9996
.9995
.9994
.9992
1.0000
1.0000
1.0000
.9999
.9998
a
a0
Table 1
Flow parameters versus M for subsonic isentropic flow, γ = 1.4
.3374
.3534
.3693
.3851
.4007
.2557
.2723
.2887
.3051
.3213
.1718
.1887
.2056
.2224
.2391
.08627
.1035
.1206
.1377
.1548
.00000
.01728
.03455
.05181
.06905
A∗
A
.45
.46
.47
.48
.49
.40
.41
.42
.43
.44
.35
.36
.37
.38
.39
.30
.31
.32
.33
.34
.25
.26
.27
.28
.29
M
.8703
.8650
.8596
.8541
.8486
.8956
.8907
.8857
.8807
.8755
.9188
.9143
.9098
.9052
.9004
.9395
.9355
.9315
.9274
.9231
.9575
.9541
.9506
.9470
.9433
p
p0
Table 1 (continued)
.9055
.9016
.8976
.8935
.8894
.9243
.9207
.9170
.9132
.9094
.9413
.9380
.9347
.9313
.9278
.9564
.9535
.9506
.9476
.9445
.9694
.9670
.9645
.9619
.9592
ρ
ρ0
.9611
.9594
.9577
.9560
.9542
.9690
.9675
.9659
.9643
.9627
.9761
.9747
.9733
.9719
.9705
.9823
.9811
.9799
.9787
.9774
.9877
.9867
.9856
.9846
.9835
T
T0
.9803
.9795
.9786
.9777
.9768
.9844
.9836
.9828
.9820
.9812
.9880
.9873
.9866
.9859
.9851
.9911
.9905
.9899
.9893
.9886
.9938
.9933
.9928
.9923
.9917
a
a0
.6903
.7019
.7134
.7246
.7356
.6289
.6416
.6541
.6663
.6784
.5624
.5761
.5896
.6029
.6160
.4914
.5059
.5203
.5345
.5486
.4162
.4315
.4467
.4618
.4767
A∗
A
M
Mach number
A
Cross-sectional area of duct
I
Impulse function
a
Speed of sound
θ
Prandtl-Meyer angle
Lmax Maximum length for choked flow in a duct of diameter D with friction coefficient f
Asterisk (*) denotes reference value when M = 1.
Subscript or superscript 0 denotes stagnation condition; subscripts 1, 2 denote conditions upstream, downstream of shock; subscript n denotes normal to shock.
Tables for compressible flow of a perfect gas
THERMODYNAMICS AND FLUID MECHANICS
93
p
p0
.8430
.8374
.8317
.8259
.8201
.8142
.8082
.8022
.7962
.7901
.7840
.7778
.7716
.7654
.7591
.7528
.7465
.7401
.7338
.7274
.7209
.7145
.7080
.7016
.6951
M
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
Table 1 (continued)
.7916
.7865
.7814
.7763
.7712
.8164
.8115
.8066
.8016
.7966
.8405
.8357
.8310
.8262
.8213
.8634
.8589
.8544
.8498
.8451
.8852
.8809
.8766
.8723
.8679
ρ
ρ0
.9107
.9084
.9061
.9037
.9013
.9221
.9199
.9176
.9153
.9131
.9328
.9307
.9286
.9265
.9243
.9430
.9410
.9390
.9370
.9349
.9524
.9506
.9487
.9468
.9449
T
T0
.9543
.9531
.9519
.9506
.9494
.9603
.9591
.9579
.9567
.9555
.9658
.9647
.9636
.9625
.9614
.9711
.9701
.9690
.9680
.9669
.9759
.9750
.9740
.9730
.9721
a
a0
.9138
.9197
.9254
.9309
.9362
.8806
.8877
.8945
.9012
.9076
.8416
.8499
.8579
.8657
.8732
.7968
.8063
.8155
.8244
.8331
.7464
.7569
.7672
.7773
.7872
A
A
∗
.5595
.5532
.5469
.5407
.5345
.5283
1.00
.5913
.5849
.5785
.5721
.5658
.6235
.6170
.6106
.6041
.5977
.6560
.6495
.6430
.6365
.6300
.6886
.6821
.6756
.6690
.6625
p
p0
.95
.96
.97
.98
.99
.90
.91
.92
.93
.94
.85
.86
.87
.88
.89
.80
.81
.82
.83
.84
.75
.76
.77
.78
.79
M
Table 1 (continued)
.6339
.6604
.6551
.6498
.6445
.6392
.6870
.6817
.6764
.6711
.6658
.7136
.7083
.7030
.6977
.6924
.7400
.7347
.7295
.7242
.7189
.7660
.7609
.7557
.7505
.7452
ρ
ρ0
.8333
.8471
.8444
.8416
.8389
.8361
.8606
.8579
.8552
.8525
.8498
.8737
.8711
.8685
.8659
.8632
.8865
.8840
.8815
.8789
.8763
.8989
.8964
.8940
.8915
.8890
T
T0
.9129
.9204
.9189
.9174
.9159
.9144
.9277
.9262
.9248
.9233
.9218
.9347
.9333
.9319
.9305
.9291
.9416
.9402
.9389
.9375
.9361
.9481
.9468
.9455
.9442
.9429
a
a0
1.0000
.9979
.9986
.9992
.9997
.9999
.9912
.9929
.9944
.9958
.9969
.9797
.9824
.9849
.9872
.9893
.9632
.9669
.9704
.9737
.9769
.9412
.9461
.9507
.9551
.9592
∗
A
A
94
ENGINEERING TABLES AND DATA
p
p0
.5283
.5221
.5160
.5099
.5039
.4979
.4919
.4860
.4800
.4742
.4684
.4626
.4568
.4511
.4455
.4398
.4343
.4287
.4232
.4178
.4124
.4070
.4017
.3964
.3912
.3861
.3809
.3759
.3708
.3658
.3609
.3560
.3512
.3464
.3417
M
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
.4829
.4782
.4736
.4690
.4644
.5067
.5019
.4971
.4923
.4876
.5311
.5262
.5213
.5164
.5115
.5562
.5511
.5461
.5411
.5361
.5817
.5766
.5714
.5663
.5612
.6077
.6024
.5972
.5920
.5869
.6339
.6287
.6234
.6181
.6129
ρ
ρ0
.7474
.7445
.7416
.7387
.7358
.7619
.7590
.7561
.7532
.7503
.7764
.7735
.7706
.7677
.7648
.7908
.7879
.7851
.7822
.7793
.8052
.8023
.7994
.7966
.7937
.8193
.8165
.8137
.8108
.8080
.8333
.8306
.8278
.8250
.8222
T
T0
.8645
.8628
.8611
.8595
.8578
.8729
.8712
.8695
.8679
.8662
.8811
.8795
.8778
.8762
.8745
.8893
.8877
.8860
.8844
.8828
.8973
.8957
.8941
.8925
.8909
.9052
.9036
.9020
.9005
.8989
.9129
.9113
.9098
.9083
.9067
a
a0
.9378
.9341
.9302
.9263
.9223
.9553
.9520
.9486
.9151
.9415
.9705
.9676
.9647
.9617
.9586
.9828
.9806
.9782
.9758
.9732
.9921
.9905
.9888
.9870
.9850
.9980
.9971
.9961
.9949
.9936
1.0000
.9999
.9997
.9993
.9987
A
A
∗
.4270
.4277
.4283
.4289
.4294
.4223
.4233
.4244
.4253
.4262
.4157
.4171
.4185
.4198
.4211
.4072
.4090
.4108
.4125
.4141
.3967
.3990
.4011
.4032
.4052
.3842
.3869
.3895
.3919
.3944
.3698
.3728
.3758
.3787
.3815
p0
1
ρV 2
2
Table 2
Flow parameters versus M for supersonic isentropic flow, γ = 1.4
6.170
6.445
6.721
7.000
7.279
4.830
5.093
5.359
5.627
5.898
3.558
3.806
4.057
4.312
4.569
2.381
2.607
2.839
3.074
3.314
1.336
1.532
1.735
1.944
2.160
.4874
.6367
.7973
.9680
1.148
0
.04473
.1257
.2294
.3510
θ
.2724
.2685
.2646
.2608
.2570
1.50
1.51
1.52
1.53
1.54
1.65
1.66
1.67
1.68
1.69
1.60
1.61
1.62
1.63
1.64
.2184
.2151
.2119
.2088
.2057
.2353
.2318
.2284
.2250
.2217
.2533
.2496
.2459
.2423
.2388
.2927
.2886
.2845
.2804
.2764
1.45
1.46
1.47
1.48
1.49
1.55
1.56
1.57
1.58
1.59
.3142
.3098
.3055
.3012
.2969
.3370
.3323
.3277
.3232
.3187
p
p0
1.40
1.41
1.42
1.43
1.44
1.35
1.36
1.37
1.38
1.39
M
Table 2 (continued)
.3373
.3337
.3302
.3266
.3232
.3557
.3520
.3483
.3446
.3409
.3750
.3710
.3672
.3633
.3595
.3950
.3909
.3869
.3829
.3789
.4158
.4116
.4074
.4032
.3991
.4374
.4330
.4287
.4244
.4201
.4598
.4553
.4508
.4463
.4418
ρ
ρ0
.6475
.6447
.6419
.6392
.6364
.6614
.6586
.6558
.6530
.6502
.6754
.6726
.6698
.6670
.6642
.6897
.6868
.6840
.6811
.6783
.7040
.7011
.6982
.6954
.6925
.7184
.7155
.7126
.7097
.7069
.7329
.7300
.7271
.7242
.7213
T
T0
.8046
.8029
.8012
.7995
.7978
.8133
.8115
.8098
.8081
.8064
.8219
.8201
.8184
.8167
.8150
.8305
.8287
.8270
.8253
.8236
.8390
.8373
.8356
.8339
.8322
.8476
.8459
.8442
.8425
.8407
.8561
.8544
.8527
.8510
.8493
a
a0
.7739
.7686
.7634
.7581
.7529
.7998
.7947
.7895
.7843
.7791
.8254
.8203
.8152
.8101
.8050
.8502
.8453
.8404
.8354
.8304
.8742
.8695
.8647
.8599
.8551
.8969
.8925
.8880
.8834
.8788
.9182
.9141
.9099
.9056
.9013
∗
A
A
.4162
.4150
.4138
.4125
.4112
.4216
.4206
.4196
.4185
.4174
.4259
.4252
.4243
.4235
.4226
.4290
.4285
.4279
.4273
.4266
.4308
.4306
.4303
.4299
.4295
.4311
.4312
.4312
.4311
.4310
.4299
.4303
.4306
.4308
.4310
p0
1
ρV 2
2
16.34
16.63
16.93
17.22
17.52
14.86
15.16
15.45
15.75
16.04
13.38
13.68
13.97
14.27
14.56
11.91
12.20
12.49
12.79
13.09
10.44
10.73
11.02
11.32
11.61
8.987
9.276
9.565
9.855
10.15
7.561
7.844
8.128
8.413
8.699
θ
THERMODYNAMICS AND FLUID MECHANICS
95
p
p0
.2026
.1996
.1966
.1936
.1907
.1878
.1850
.1822
.1794
.1767
.1740
.1714
.1688
.1662
.1637
.1612
.1587
.1563
.1539
.1516
.1492
.1470
.1447
.1425
.1403
.1381
.1360
.1339
.1318
.1298
.1278
.1258
.1239
.1220
.1201
M
1.70
1.71
1.72
1.73
1.74
1.75
1.76
1.77
1.78
1.79
1.80
1.81
1.82
1.83
1.84
1.85
1.86
1.87
1.88
1.89
1.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2.00
2.01
2.02
2.03
2.04
Table 2 (continued)
.2300
.2275
.2250
.2225
.2200
.2432
.2405
.2378
.2352
.2326
.2570
.2542
.2514
.2486
.2459
.2715
.2686
.2656
.2627
.2598
.2868
.2837
.2806
.2776
.2745
.3029
.2996
.2964
.2932
.2900
.3197
.3163
.3129
.3095
.3062
ρ
ρ0
.5556
.5531
.5506
.5482
.5458
.5680
.5655
.5630
.5605
.5580
.5807
.5782
.5756
.5731
.5705
.5936
.5910
.5884
.5859
.5833
.6068
.6041
.6015
.5989
.5963
.6202
.6175
.6148
.6121
.6095
.6337
.6310
.6283
.6256
.6229
T
T0
.7454
.7437
.7420
.7404
.7388
.7537
.7520
.7503
.7487
.7470
.7620
.7604
.7587
.7570
.7553
.7705
.7688
.7671
.7654
.7637
.7790
.7773
.7756
.7739
.7722
.7875
.7858
.7841
.7824
.7807
.7961
.7943
.7926
.7909
.7892
a
a0
.5926
.5877
.5828
.5779
.5730
.6175
.6125
.6075
.6025
.5975
.6430
.6379
.6328
.6277
.6226
.6688
.6636
.6584
.6533
.6481
.6949
.6897
.6845
.6792
.6740
.7212
.7160
.7107
.7054
.7002
.7476
.7423
.7371
.7318
.7265
∗
A
A
.3579
.3559
.3539
.3518
.3498
.3677
.3657
.3638
.3618
.3598
.3771
.3753
.3734
.3715
.3696
.3862
.3844
.3826
.3808
.3790
.3947
.3931
.3914
.3897
.3879
.4026
.4011
.3996
.3980
.3964
.4098
.4086
.4071
.4056
.4041
p0
1
ρV
2
2
26.38
26.66
26.93
27.20
27.48
24.99
25.27
25.55
25.83
26.10
23.59
23.87
24.15
24.43
24.71
22.16
22.45
22.73
23.02
23.30
20.73
21.01
21.30
21.59
21.88
19.27
19.56
19.86
20.15
20.44
17.81
18.10
18.40
18.69
18.98
θ
2.35
2.36
2.37
2.38
2.39
2.30
2.31
2.32
2.33
2.34
2.25
2.26
2.27
2.28
2.29
2.20
2.21
2.22
2.23
2.24
2.15
2.16
2.17
2.18
2.19
2.10
2.11
2.12
2.13
2.14
2.05
2.06
2.07
2.08
2.09
M
.07396
.07281
.07168
.07057
.06948
.07997
.07873
.07751
.07631
.07512
.08648
.08514
.08382
.08252
.08123
.09352
.09207
.09064
.08923
.08785
.1011
.09956
.09802
.09650
.09500
.1094
.1077
.1060
.1043
.1027
.1182
.1164
.1146
.1128
.1111
p
p0
Table 2 (continued)
.1556
.1539
.1522
.1505
.1488
.1646
.1628
.1609
.1592
.1574
.1740
.1721
.1702
.1683
.1664
.1841
.1820
.1800
.1780
.1760
.1946
.1925
.1903
.1882
.1861
.2058
.2035
.2013
.1990
.1968
.2176
.2152
.2128
.2104
.2081
ρ
ρ0
.4752
.4731
.4709
.4688
.4668
.4859
.4837
.4816
.4794
.4773
.4969
.4947
.4925
.4903
.4881
.5081
.5059
.5036
.5014
.4991
.5196
.5173
.5150
.5127
.5104
.5313
.5290
.5266
.5243
.5219
.5433
.5409
.5385
.5361
.5337
T
T0
.6893
.6878
.6863
.6847
.6832
.6971
.6955
.6940
.6924
.6909
.7049
.7033
.7018
.7002
.6986
.7128
.7112
.7097
.7081
.7065
.7208
.7192
.7176
.7160
.7144
.7289
.7273
.7257
.7241
.7225
.7371
.7355
.7338
.7322
.7306
a
a0
.4357
.4317
.4278
.4239
.4200
.4560
.4519
.4478
.4437
.4397
.4770
.4727
.4685
.4643
.4601
.4988
.4944
.4900
.4856
.4813
.5212
.5167
.5122
.5077
.5032
.5444
.5397
.5350
.5304
.5258
.5682
.5634
.5586
.5538
.5491
∗
A
A
.2859
.2839
.2818
.2798
.2778
.2961
.2941
.2920
.2900
.2879
.3065
.3044
.3023
.3003
.2982
.3169
.3148
.3127
.3106
.3085
.3272
.3252
.3231
.3210
.3189
.3376
.3355
.3334
.3314
.3293
.3478
.3458
.3437
.3417
.3396
p0
1
ρV 2
2
35.53
35.77
36.02
36.26
36.50
34.28
34.53
34.78
35.03
35.28
33.02
33.27
33.53
33.78
34.03
31.73
31.99
32.25
32.51
32.76
30.43
30.69
30.95
31.21
31.47
29.10
29.36
29.63
29.90
30.16
27.75
28.02
28.29
28.56
28.83
θ
96
ENGINEERING TABLES AND DATA
p
p0
.06840
.06734
.06630
.06527
.06426
.06327
.06229
.06133
.06038
.05945
.05853
.05762
.05674
.05586
.05500
.05415
.05332
.05250
.05169
.05090
.05012
.04935
.04859
.04784
.04711
.04639
.04568
.04498
.04429
.04362
.04295
.04229
.04165
.04102
.04039
M
2.40
2.41
2.42
2.43
2.44
2.45
2.46
2.47
2.48
2.49
2.50
2.51
2.52
2.53
2.54
2.55
2.56
2.57
2.58
2.59
2.60
2.61
2.62
2.63
2.64
2.65
2.66
2.67
2.68
2.69
2.70
2.71
2.72
2.73
2.74
Table 2 (continued)
.1056
.1044
.1033
.1022
.1010
.1115
.1103
.1091
.1079
.1067
.1179
.1166
.1153
.1140
.1128
.1246
.1232
.1218
.1205
.1192
.1317
.1302
.1288
.1274
.1260
.1392
.1377
.1362
.1347
.1332
.1472
.1456
.1439
.1424
.1408
ρ
ρ0
.4068
.4051
.4033
.4015
.3998
.4159
.4141
.4122
.4104
.4086
.4252
.4233
.4214
.4196
.4177
.4347
.4328
.4309
.4289
.4271
.4444
.4425
.4405
.4386
.4366
.4544
.4524
.4504
.4484
.4464
.4647
.4626
.4606
.4585
.4565
T
T0
.6378
.6364
.6350
.6337
.6323
.6449
.6435
.6421
.6406
.6392
.6521
.6506
.6492
.6477
.6463
.6593
.6579
.6564
.6549
.6535
.6667
.6652
.6637
.6622
.6608
.6741
.6726
.6711
.6696
.6681
.6817
.6802
.6786
.6771
.6756
a
a0
.3142
.3112
.3083
.3054
.3025
.3294
.3263
.3232
.3202
.3172
.3453
.3421
.3389
.3357
.3325
.3619
.3585
.3552
.3519
.3486
.3793
.3757
.3722
.3688
.3653
.3973
.3937
.3900
.3864
.3828
.4161
.4123
.4085
.4048
.4010
∗
A
A
.2192
.2174
.2157
.2140
.2123
.2280
.2262
.2245
.2227
.2209
.2371
.2353
.2335
.2317
.2298
.2465
.2446
.2427
.2409
.2390
.2561
.2541
.2522
.2503
.2484
.2658
.2639
.2619
.2599
.2580
.2758
.2738
.2718
.2698
.2678
p0
1
ρV 2
2
43.62
43.84
44.05
44.27
44.48
42.53
42.75
42.97
43.19
43.40
41.41
41.64
41.86
42.09
42.31
40.28
40.51
40.75
40.96
41.19
39.12
39.36
39.59
39.82
40.05
37.95
38.18
38.42
38.66
38.89
36.75
36.99
37.23
37.47
37.71
θ
3.05
3.06
3.07
3.08
3.09
3.00
3.01
3.02
3.03
3.04
2.95
2.96
2.97
2.98
2.99
2.90
2.91
2.92
2.93
2.94
2.85
2.86
2.87
2.88
2.89
2.80
2.81
2.82
2.83
2.84
2.75
2.76
2.77
2.78
2.79
M
.02526
.02489
.02452
.02416
.02380
.02722
.02682
.02642
.02603
.02564
.02935
.02891
.02848
.02805
.02764
.03165
.03118
.03071
.03025
.02980
.03415
.03363
.03312
.03263
.03213
.03685
.03629
.03574
.03520
.03467
.03978
.03917
.03858
.03799
.03742
p
p0
Table 2 (continued)
.07226
.07149
.07074
.06999
.06925
.07623
.07541
.07461
.07382
.07303
.08043
.07957
.07872
.07788
.97705
.08489
.08398
.08307
.08218
.08130
.08962
.08865
.08769
.08675
.08581
.09463
.09360
.09259
.09158
.09059
.09994
.09885
.09778
.09671
.09566
ρ
ρ0
.3496
.3481
.3466
.3452
.3437
.3571
.3556
.3541
.3526
.3511
.3649
.3633
.3618
.3602
.3587
.3729
.3712
.3696
.3681
.3665
.3810
.3794
.3777
.3761
.3745
.3894
.3877
.3860
.3844
.3827
.3980
.3963
.3945
.3928
.3911
T
T0
.5913
.5900
.5887
.5875
.5862
.5976
.5963
.5951
.5938
.5925
.6041
.6028
.6015
.6002
.5989
.6106
.6093
.6080
.6067
.6054
.6173
.6159
.6146
.6133
.6119
.6240
.6227
.6213
.6200
.6186
.6309
.6295
.6281
.6268
.6254
a
a0
.2252
.2230
.2209
.2188
.2168
.2362
.2339
.2317
.2295
.2273
.2477
.2453
.2430
.2407
.2384
.2598
.2573
.2549
.2524
.2500
.2724
.2698
.2673
.2648
.2622
.2857
.2830
.2803
.2777
.2750
.2996
.2968
.2940
.2912
.2884
∗
A
A
.1645
.1631
.1618
.1604
.1591
.1715
.1701
.1687
.1673
.1659
.1788
.1773
.1758
.1744
.1729
.1863
.1848
.1833
.1818
.1803
.1941
.1926
.1910
.1894
.1879
.2022
.2006
.1990
.1973
.1957
.2106
.2089
.2072
.2055
.2039
p0
1
ρV 2
2
50.71
50.90
51.09
51.28
51.46
49.76
49.95
50.14
50.33
50.52
48.78
48.98
49.18
49.37
49.56
47.79
47.99
48.19
48.39
48.59
46.78
46.98
47.19
47.39
47.59
45.75
45.95
46.16
46.37
46.57
44.69
44.91
45.12
45.33
45.54
θ
THERMODYNAMICS AND FLUID MECHANICS
97
p
p0
.02345
.02310
.02276
.02243
.02210
.02177
.02146
.02114
.02083
.02053
.02023
.01993
.01964
.01936
.01908
.01880
.01853
.01826
.01799
.01773
.01748
.01722
.01698
.01673
.01649
.01625
.01602
.01579
.01557
.01534
.01513
.01491
.01470
.01449
.01428
M
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
3.33
3.34
3.35
3.36
3.37
3.38
3.39
3.40
3.41
3.42
3.43
3.44
Table 2 (continued)
.05009
.04958
.04908
.04858
.04808
.05274
.05220
.05166
.05113
.05061
.05554
.05497
.05440
.05384
.05329
.05851
.05790
.05730
.05671
.05612
.06165
.06101
.06037
.05975
.05912
.06499
.06430
.06363
.06296
.06231
.06852
.06779
.06708
.06637
.06568
ρ
ρ0
.3019
.3007
.2995
.2982
.2970
.3082
.3069
.3057
.3044
.3032
.3147
.3134
.3121
.3108
.3095
.3213
.3199
.3186
.3173
.3160
.3281
.3267
.3253
.3240
.3226
.3351
.3337
.3323
.3309
.3295
.3422
.3408
.3393
.3379
.3365
T
T0
.5495
.5484
.5472
.5461
.5450
.5552
.5540
.5529
.5517
.5506
.5609
.5598
.5586
.5575
.5563
.5668
.5656
.5645
.5633
.5621
.5728
.5716
.5704
.5692
.5680
.5788
.5776
.5764
.5752
.5740
.5850
.5838
.5825
.5813
.5801
a
a0
.1617
.1602
.1587
.1572
.1558
.1695
.1679
.1663
.1648
.1632
.1777
.1760
.1743
.1727
.1711
.1863
.1845
.1828
.1810
.1793
.1953
.1934
.1916
.1898
.1880
.2048
.2028
.2009
.1990
.1971
.2147
.2127
.2107
.2087
.2067
∗
A
A
.1224
.1214
.1203
.1193
.1183
.1277
.1266
.1255
.1245
.1234
.1332
.1321
.1310
.1299
.1288
.1390
.1378
.1367
.1355
.1344
.1450
.1438
.1426
.1414
.1402
.1512
.1500
.1487
.1475
.1462
.1577
.1564
.1551
.1538
.1525
p0
1
ρV
2
2
56.91
57.07
57.24
57.40
57.56
56.07
56.24
56.41
56.58
56.75
55.22
55.39
55.56
55.73
55.90
54.35
54.53
54.71
54.88
55.05
53.47
53.65
53.83
54.00
54.18
52.57
52.75
52.93
53.11
53.29
51.65
51.84
52.02
52.20
52.39
θ
.01408
.01388
.01368
.01349
.01330
−3
.01134
5.194×10−3
2.609×10−3
1.414×10−3
8.150×10−4
−4
1.890×10−3
6.334×10−4
2.416×10−4
1.024×10−4
4.739×10−5
−5
5.00
6.00
7.00
8.00
9.00
.1667
.1220
.09259
.07246
.05814
.1980
.1911
.1846
.1783
.1724
.2381
.2293
.2208
.2129
.2053
.2899
.2784
.2675
.2572
.2474
.2958
.2946
.2934
.2922
.2910
T
T0
.4082
.3492
.3043
.2692
.2411
.4450
.4372
.4296
.4223
.4152
.4880
.4788
.4699
.4614
.4531
.5384
.5276
.5172
.5072
.4974
.5439
.5428
.5417
.5406
.5395
a
a0
.04898
.04521
.04177
.03861
.03572
.07376
.06788
.06251
.05759
.05309
.1124
.1033
.09490
.08722
.08019
.1173
.1163
.1153
.1144
.1134
p0
1
ρV 2
2
−3
−3
.04000
.03308
.01880
.01596
9.602×10−3 8.285×10−3
5.260×10−3 4.589×10−3
3.056×10−3 2.687×10−3
.06038
.05550
.05107
.04703
.04335
.09329
.08536
.07818
.07166
.06575
.1473
.1342
.1224
.1117
.1021
.1543
.1529
.1515
.1501
.1487
∗
A
A
76.92
84.96
90.97
95.62
99.32
71.83
72.92
73.97
74.99
75.97
65.78
67.08
68.33
69.54
70.71
58.53
60.09
61.60
63.04
64.44
57.73
57.89
58.05
58.21
58.37
θ
10.00 2.356×10
4.948×10
.04762
.2182 1.866×10 1.649×10
102.3
100.00 2.790×10−12 5.583×10−9 4.998×10−4 .02236 2.157×10−8 1.953×10−8 127.6
∞
0
0
0
0
0
0
130.5
.01745
.01597
.01464
.01343
.01233
.02766
.02516
.02292
.02090
.01909
3.455×10−3
3.053×10−3
2.701×10−3
2.394×10−3
−3
2.126×10
6.586×10
−3
5.769×10
5.062×10−3
4.449×10−3
3.918×10−3
.04523
.04089
.03702
.03355
.03044
.04759
.04711
.04663
.04616
.04569
ρ
ρ0
4.50
4.60
4.70
4.80
4.90
4.00
4.10
4.20
4.30
4.40
3.50
.01311
3.60
.01138
3.70 9.903×10−3
3.80 8.629×10−3
3.90 7.532×10−3
3.45
3.46
3.47
3.48
3.49
M
p
p0
Table 2 (continued)
98
ENGINEERING TABLES AND DATA
p2
p1
1.000
1.023
1.047
1.071
1.095
1.120
1.144
1.169
1.194
1.219
1.245
1.271
1.297
1.323
1.350
1.376
1.403
1.430
1.458
1.485
1.513
1.541
1.570
1.598
1.627
1.656
1.686
1.715
1.745
1.775
1.805
1.835
1.866
1.897
1.928
M1n
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.516
1.533
1.551
1.568
1.585
1.429
1.446
1.463
1.481
1.498
1.342
1.359
1.376
1.394
1.411
1.255
1.272
1.290
1.307
1.324
1.169
1.186
1.203
1.221
1.238
1.084
1.101
1.118
1.135
1.152
1.000
1.017
1.033
1.050
1.067
ρ2
ρ1
1.191
1.197
1.204
1.210
1.216
1.159
1.166
1.172
1.178
1.185
1.128
1.134
1.141
1.147
1.153
1.097
1.103
1.109
1.115
1.122
1.065
1.071
1.078
1.084
1.090
1.033
1.039
1.046
1.052
1.059
1.000
1.007
1.013
1.020
1.026
T2
T1
Table 3
Parameters for shock flow, γ = 1.4
1.091
1.094
1.097
1.100
1.103
1.077
1.080
1.083
1.085
1.088
1.062
1.065
1.068
1.071
1.074
1.047
1.050
1.053
1.056
1.059
1.032
1.035
1.038
1.041
1.044
1.016
1.019
1.023
1.026
1.029
1.000
1.003
1.007
1.010
1.013
a2
a1
.9794
.9776
.9758
.9738
.9718
.9871
.9857
.9842
.9827
.9811
.9928
.9918
.9907
.9896
.9884
.9967
.9961
.9953
.9946
.9937
.9989
.9986
.9982
.9978
.9973
.9999
.9998
.9996
.9994
.9992
1.0000
1.0000
1.0000
1.0000
.9999
p10
p20
M2
.7860
.7809
.7760
.7712
.7664
.8126
.8071
.8016
.7963
.7911
.8422
.8360
.8300
.8241
.8183
.8750
.8682
.8615
.8549
.8485
.9118
.9041
.8966
.8892
.8820
.9531
.9444
.9360
.9277
.9196
1.0000
.9901
.9805
.9712
.9620
for normal
shocks only
1.65
1.66
1.67
1.68
1.69
1.60
1.61
1.62
1.63
1.64
1.55
1.56
1.57
1.58
1.59
1.50
1.51
1.52
1.53
1.54
1.45
1.46
1.47
1.48
1.49
1.40
1.41
1.42
1.43
1.44
1.35
1.36
1.37
1.38
1.39
M1n
3.010
3.048
3.087
3.126
3.165
2.820
2.857
2.895
2.933
2.971
2.636
2.673
2.709
2.746
2.783
2.458
2.493
2.529
2.564
2.600
2.286
2.320
2.354
2.389
2.423
2.120
2.153
2.186
2.219
2.253
1.960
1.991
2.023
2.055
2.087
p2
p1
Table 3 (continued)
2.115
2.132
2.148
2.165
2.181
2.032
2.049
2.065
2.082
2.099
1.947
1.964
1.981
1.998
2.015
1.862
1.879
1.896
1.913
1.930
1.776
1.793
1.811
1.828
1.845
1.690
1.707
1.724
1.742
1.759
1.603
1.620
1.638
1.655
1.672
ρ2
ρ1
1.423
1.430
1.437
1.444
1.451
1.388
1.395
1.402
1.409
1.416
1.354
1.361
1.367
1.374
1.381
1.320
1.327
1.334
1.340
1.347
1.287
1.294
1.300
1.307
1.314
1.255
1.261
1.268
1.274
1.281
1.223
1.229
1.235
1.242
1.248
T2
T1
1.193
1.196
1.199
1.202
1.205
1.178
1.181
1.184
1.187
1.190
1.164
1.166
1.169
1.172
1.175
1.149
1.152
1.155
1.158
1.161
1.135
1.137
1.140
1.143
1.146
1.120
1.123
1.126
1.129
1.132
1.106
1.109
1.111
1.114
1.117
a2
a1
.8760
.8720
.8680
.8640
.8599
.8952
.8914
.8877
.8838
.8799
.9132
.9097
.9061
.9026
.8989
.9298
.9266
.9233
.9200
.9166
.9448
.9420
.9390
.9360
.9329
.9582
.9557
.9531
.9504
.9476
.9697
.9676
.9653
.9630
.9606
p10
p20
M2
.6540
.6512
.6485
.6458
.6431
.6684
.6655
.6625
.6596
.6568
.6841
.6809
.6777
.6746
.6715
.7011
.6976
.6941
.6907
.6874
.7196
.7157
.7120
.7083
.7047
.7397
.7355
.7314
.7274
.7235
.7618
.7572
.7527
.7483
.7440
for normal
shocks only
THERMODYNAMICS AND FLUID MECHANICS
99
p2
p1
3.205
3.245
3.285
3.325
3.366
3.406
3.447
3.488
3.530
3.571
3.613
3.655
3.698
3.740
3.783
3.826
3.870
3.913
3.957
4.001
4.045
4.089
4.134
4.179
4.224
4.270
4.315
4.361
4.407
4.453
4.500
4.547
4.594
4.641
4.689
M1n
1.70
1.71
1.72
1.73
1.74
1.75
1.76
1.77
1.78
1.79
1.80
1.81
1.82
1.83
1.84
1.85
1.86
1.87
1.88
1.89
1.90
1.91
1.92
1.93
1.94
1.95
1.96
1.97
1.98
1.99
2.00
2.01
2.02
2.03
2.04
Table 3 (continued)
2.667
2.681
2.696
2.711
2.725
2.592
2.607
2.622
2.637
2.652
2.516
2.531
2.546
2.562
2.577
2.438
2.454
2.469
2.485
2.500
2.359
2.375
2.391
2.407
2.422
2.279
2.295
2.311
2.327
2.343
2.198
2.214
2.230
2.247
2.263
ρ2
ρ1
1.688
1.696
1.704
1.712
1.720
1.647
1.655
1.663
1.671
1.679
1.608
1.616
1.624
1.631
1.639
1.569
1.577
1.585
1.592
1.600
1.532
1.539
1.547
1.554
1.562
1.495
1.502
1.509
1.517
1.524
1.458
1.466
1.473
1.480
1.487
T2
T1
1.299
1.302
1.305
1.308
1.312
1.283
1.287
1.290
1.293
1.296
1.268
1.271
1.274
1.277
1.280
1.253
1.256
1.259
1.262
1.265
1.238
1.241
1.244
1.247
1.250
1.223
1.226
1.229
1.232
1.235
1.208
1.211
1.214
1.217
1.220
a2
a1
.7209
.7162
.7115
.7069
.7022
.7442
.7395
.7349
.7302
.7255
.7674
.7628
.7581
.7535
.7488
.7902
.7857
.7811
.7765
.7720
.8127
.8082
.8038
.7993
.7948
.8346
.8302
.8259
.8215
.8171
.8557
.8516
.8474
.8431
.8389
p20
p10
.5773
.5757
.5740
.5723
.5707
.5862
.5844
.5826
.5808
.5791
.5956
.5937
.5918
.5899
.5880
.6057
.6036
.6016
.5996
.5976
.6165
.6143
.6121
.6099
.6078
.6281
.6257
.6234
.6210
.6188
.6405
.6380
.6355
.6330
.6305
for normal
shocks only
M2
2.35
2.36
2.37
2.38
2.39
2.30
2.31
2.32
2.33
2.34
2.25
2.26
2.27
2.28
2.29
2.20
2.21
2.22
2.23
2.24
2.15
2.16
2.17
2.18
2.19
2.10
2.11
2.12
2.13
2.14
2.05
2.06
2.07
2.08
2.09
M1n
6.276
6.331
6.386
6.442
6.497
6.005
6.059
6.113
6.167
6.222
5.740
5.792
5.845
5.808
5.951
5.480
5.531
5.583
5.635
5.687
5.226
5.277
5.327
5.378
5.429
4.978
5.027
5.077
5.126
5.176
4.736
4.784
4.832
4.881
4.929
p2
p1
Table 3 (continued)
3.149
3.162
3.174
3.187
3.199
3.085
3.098
3.110
3.123
3.136
3.019
3.032
3.045
3.058
3.071
2.951
2.965
2.978
2.992
3.005
2.882
2.896
2.910
2.924
2.938
2.812
2.826
2.840
2.854
2.868
2.740
2.755
2.769
2.783
2.798
ρ2
ρ1
1.993
2.002
2.012
2.021
2.031
1.947
1.956
1.965
1.974
1.984
1.901
1.910
1.919
1.929
1.938
1.857
1.866
1.875
1.883
1.892
1.813
1.822
1.831
1.839
1.848
1.770
1.779
1.787
1.796
1.805
1.729
1.737
1.745
1.754
1.762
T2
T1
1.412
1.415
1.418
1.422
1.425
1.395
1.399
1.402
1.405
1.408
1.379
1.382
1.385
1.389
1.392
1.363
1.366
1.369
1.372
1.376
1.347
1.350
1.353
1.356
1.359
1.331
1.334
1.337
1.340
1.343
1.315
1.318
1.321
1.324
1.327
a2
a1
.5615
.5572
.5529
.5486
.5444
.5833
.5789
.5745
.5702
.5658
.6055
.6011
.5966
.5921
.5877
.6281
.6236
.6191
.6145
.6100
.6511
.6464
.6419
.6373
.6327
.6742
.6696
.6649
.6603
.6557
.6975
.6928
.6882
.6835
.6789
p10
p20
M2
.5286
.5275
.5264
.5253
.5242
.5344
.5332
.5321
.5309
.5297
.5406
.5393
.5381
.5368
.5356
.5471
.5457
.5444
.5431
.5418
.5540
.5525
.5511
.5498
.5484
.5613
.5598
.5583
.5568
.5554
.5691
.5675
.5659
.5643
.5628
for normal
shocks only
100
ENGINEERING TABLES AND DATA
p2
p1
6.553
6.609
6.666
6.722
6.779
6.836
6.894
6.951
7.009
7.067
7.125
7.183
7.242
7.301
7.360
7.420
7.479
7.539
7.599
7.659
7.720
7.781
7.842
7.903
7.965
8.026
8.088
8.150
8.213
8.275
8.338
8.401
8.465
8.528
8.592
M1n
2.40
2.41
2.42
2.43
2.44
2.45
2.46
2.47
2.48
2.49
2.50
2.51
2.52
2.53
2.54
2.55
2.56
2.57
2.58
2.59
2.60
2.61
2.62
2.63
2.64
2.65
2.66
2.67
2.68
2.69
2.70
2.71
2.72
2.73
2.74
Table 3 (continued)
3.559
3.570
3.580
3.591
3.601
3.505
3.516
3.527
3.537
3.548
3.449
3.460
3.471
3.483
3.494
3.392
3.403
3.415
3.426
3.438
3.333
3.345
3.357
3.369
3.380
3.273
3.285
3.298
3.310
3.321
3.212
3.224
3.237
3.249
3.261
ρ2
ρ1
2.343
2.354
2.364
2.375
2.386
2.290
2.301
2.311
2.322
2.332
2.238
2.249
2.259
2.269
2.280
2.187
2.198
2.208
2.218
2.228
2.138
2.147
2.157
2.167
2.177
2.088
2.098
2.108
2.118
2.128
2.040
2.050
2.059
2.069
2.079
T2
T1
1.531
1.534
1.538
1.541
1.545
1.513
1.517
1.520
1.524
1.527
1.496
1.500
1.503
1.506
1.510
1.479
1.482
1.486
1.489
1.493
1.462
1.465
1.469
1.472
1.476
1.445
1.449
1.452
1.455
1.459
1.428
1.432
1.435
1.438
1.442
a2
a1
.4236
.4201
.4166
.4131
.4097
.4416
.4379
.4343
.4307
.4271
.4601
.4564
.4526
.4489
.4452
.4793
.4754
.4715
.4677
.4639
.4990
.4950
.4911
.4871
.4832
.5193
.5152
.5111
.5071
.5030
.5401
.5359
.5317
.5276
.5234
p10
p20
M2
.4956
.4949
.4941
.4933
.4926
.4996
.4988
.4980
.4972
.4964
.5039
.5030
.5022
.5013
.5005
.5083
.5074
.5065
.5056
.5047
.5130
.5120
.5111
.5102
.5092
.5179
.5169
.5159
.5149
.5140
.5231
.5221
.5210
.5200
.5189
for normal
shocks only
3.50
3.60
3.70
3.80
3.90
3.00
3.10
3.20
3.30
3.40
2.95
2.96
2.97
2.98
2.99
2.90
2.91
2.92
2.93
2.94
2.85
2.86
2.87
2.88
2.89
2.80
2.81
2.82
2.83
2.84
2.75
2.76
2.77
2.78
2.79
M1n
14.13
14.95
15.80
16.68
17.58
10.33
11.05
11.78
12.54
13.32
9.986
10.06
10.12
10.19
10.26
9.645
9.713
9.781
9.849
9.918
9.310
9.376
9.443
9.510
9.577
8.980
9.045
9.111
9.177
9.243
8.656
8.721
8.785
8.850
8.915
p2
p1
Table 3 (continued)
4.261
4.330
4.395
4.457
4.516
3.857
3.947
4.031
4.112
4.188
3.811
3.820
3.829
3.839
3.848
3.763
3.773
3.782
3.792
3.801
3.714
3.724
3.734
3.743
3.753
3.664
3.674
3.684
3.694
3.704
3.612
3.622
3.633
3.643
3.653
ρ2
ρ1
3.315
3.454
3.596
3.743
3.893
2.679
2.799
2.922
3.049
3.180
2.621
2.632
2.644
2.656
2.667
2.563
2.575
2.586
2.598
2.609
2.507
2.518
2.529
2.540
2.552
2.451
2.462
2.473
2.484
2.496
2.397
2.407
2.418
2.429
2.440
T2
T1
1.821
1.858
1.896
1.935
1.973
1.637
1.673
1.709
1.746
1.783
1.619
1.622
1.626
1.630
1.633
1.601
1.605
1.608
1.612
1.615
1.583
1.587
1.590
1.594
1.597
1.566
1.569
1.573
1.576
1.580
1.548
1.552
1.555
1.559
1.562
a2
a1
.2129
.1953
.1792
.1645
.1510
.3283
.3012
.2762
.2533
.2322
.3428
.3398
.3369
.3340
.3312
.3577
.3547
.3517
.3487
.3457
.3733
.3701
.3670
.3639
.3608
.3895
.3862
.3829
.3797
.3765
.4062
.4028
.3994
.3961
.3928
p10
p20
M2
.4512
.4474
.4439
.4407
.4377
.4752
.4695
.4643
.4596
.4552
.4782
.4776
.4770
.4764
.4758
.4814
.4807
.4801
.4795
.4788
.4847
.4840
.4833
.4827
.4820
.4882
.4875
.4868
.4861
.4854
.4918
.4911
.4903
.4896
.4889
for normal
shocks only
THERMODYNAMICS AND FLUID MECHANICS
101
18.50
29.00
41.83
57.00
74.50
4.00
5.00
6.00
7.00
8.00
4.047
5.800
7.941
10.47
13.39
T2
T1
5.651
16.69
5.714
20.39
5.997 1945.4
6
∞
4.571
5.000
5.268
5.444
5.565
ρ2
ρ1
4.086
4.515
44.11
∞
2.012
2.408
2.818
3.236
3.659
a2
a1
M2
.4350
.4152
.4042
.3974
.3929
.3898
.3876
.3781
.3780
4.964×10−3
3.045×10−3
3.593×10−8
0
for normal
shocks only
.1388
.06172
.02965
.01535
8.488×10−3
p20
p10
∞
109.544
54.770
36.511
27.382
21.903
18.251
15.642
13.684
12.162
10.9435
9.9465
9.1156
8.4123
7.8093
7.2866
6.8291
6.4252
6.0662
5.7448
T
T∗
1.2000
1.2000
1.1999
1.1998
1.1996
1.1994
1.1991
1.1988
1.1985
1.1981
1.1976
1.1971
1.1966
1.1960
1.1953
1.1946
1.1939
1.1931
1.1923
1.1914
M
0
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
p
p∗
3.9103
3.6727
3.4635
3.2779
3.1123
5.8218
5.2992
4.8643
4.4968
4.1824
11.5914
9.6659
8.2915
7.2616
6.4614
∞
57.874
28.942
19.300
14.482
p0∗
p0
.16395
.17482
.18568
.19654
.20739
.10943
.12035
.13126
.14216
.15306
.05476
.06570
.07664
.08758
.09851
0
.01095
.02191
.03286
.04381
V
V∗
3.1317
2.9474
2.7855
2.6422
2.5146
4.6236
4.2146
3.8747
3.5880
3.3432
9.1584
7.6428
6.5620
5.7529
5.1249
∞
45.650
22.834
15.232
11.435
I
I∗
27.932
24.198
21.115
18.543
16.375
66.922
54.688
45.408
38.207
32.511
280.02
193.03
140.66
106.72
83.496
∞
7134.40
1778.45
787.08
440.35
D
f Lmax
Fanno line – one-dimensional, adiabatic, constant-area flow of a perfect gas
(constant specific heat and molecular weight) γ = 1.4
Table 4
9.00
94.33
10.00
116.5
100.00 11666.5
∞
∞
p2
p1
M1n
Table 3 (continued)
2.6958
2.6280
2.5634
2.5017
2.4428
2.3865
2.3326
2.2809
2.2314
2.1838
2.1381
2.0942
2.0519
2.0112
1.9719
1.1628
1.1610
1.1591
1.1572
1.1553
1.1533
1.1513
1.1492
1.1471
1.1450
1.1429
1.1407
1.1384
1.1362
1.1339
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
.51
.52
.53
.54
3.6190
3.5002
3.3888
3.2840
3.1853
1.1788
1.1774
1.1759
1.1744
1.1729
.30
.31
.32
.33
.34
3.0922
3.0042
2.9209
2.8420
2.7671
4.3546
4.1850
4.0280
3.8820
3.7460
1.1852
1.1840
1.1828
1.1815
1.1802
.25
.26
.27
.28
.29
1.1713
1.1697
1.1680
1.1663
1.1646
5.4555
5.1936
4.9554
4.7378
4.5383
1.1905
1.1895
1.1885
1.1874
1.1863
.20
.21
.22
.23
.24
.35
.36
.37
.38
.39
p
p∗
T
T∗
M
Table 4 (continued) γ = 1.4
p0
1.3399
1.3212
1.3034
1.2864
1.2702
1.4486
1.4246
1.4018
1.3801
1.3595
1.5901
1.5587
1.5289
1.5007
1.4739
1.7780
1.7358
1.6961
1.6587
1.6234
2.0351
1.9765
1.9219
1.8708
1.8229
2.4027
2.3173
2.2385
2.1656
2.0979
2.9635
2.8293
2.7076
2.5968
2.4956
p0∗
.53453
.54469
.55482
.56493
.57501
.48326
.49357
.50385
.51410
.52433
.43133
.44177
.45218
.46257
.47293
.37880
.38935
.39988
.41039
.42087
.32572
.33637
.34700
.35762
.36822
.27217
.28291
.29364
.30435
.31504
.21822
.22904
.23984
.25063
.26141
V
V∗
1.2027
1.1903
1.1786
1.1675
1.1571
1.2763
1.2598
1.2443
1.2296
1.2158
1.3749
1.3527
1.3318
1.3122
1.2937
1.5094
1.4789
1.4503
1.4236
1.3985
1.6979
1.6546
1.6144
1.5769
1.5420
1.9732
1.9088
1.8496
1.7950
1.7446
2.4004
2.2976
2.2046
2.1203
2.0434
I
I∗
1.06908
.99042
.91741
.84963
.78662
1.5664
1.4509
1.3442
1.2453
1.1539
2.3085
2.1344
1.9744
1.8272
1.6915
3.4525
3.1801
2.9320
2.7055
2.4983
5.2992
4.8507
4.4468
4.0821
3.7520
8.4834
7.6876
6.9832
6.3572
5.7989
14.533
12.956
11.596
10.416
9.3865
D
f Lmax
102
ENGINEERING TABLES AND DATA
p
p∗
1.9341
1.8976
1.8623
1.8282
1.7952
1.7634
1.7325
1.7026
1.6737
1.6456
1.6183
1.5919
1.5662
1.5413
1.5170
1.4934
1.4705
1.4482
1.4265
1.4054
1.3848
1.3647
1.3451
1.3260
1.3074
1.2892
1.2715
1.2542
1.2373
1.2208
1.2047
1.1889
1.1735
1.1584
1.1436
T
T∗
1.1315
1.1292
1.1268
1.1244
1.1219
1.1194
1.1169
1.1144
1.1118
1.1091
1.10650
1.10383
1.10114
1.09842
1.09567
1.09290
1.09010
1.08727
1.08442
1.08155
1.07865
1.07573
1.07279
1.06982
1.06684
1.06383
1.06080
1.05775
1.05468
1.05160
1.04849
1.04537
1.04223
1.03907
1.03589
M
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
Table 4 (continued) γ = 1.4
p0
1.02067
1.01787
1.01529
1.01294
1.01080
1.03823
1.03422
1.03047
1.02696
1.02370
1.06242
1.05700
1.05188
1.04705
1.04250
1.09436
1.08729
1.08057
1.07419
1.06815
1.1356
1.1265
1.1179
1.1097
1.1018
1.1882
1.1766
1.1656
1.1551
1.1451
1.2549
1.2403
1.2263
1.2130
1.2003
p0∗
.87037
.87929
.88818
.89703
.90583
.82514
.83426
.84334
.85239
.86140
.77893
.78825
.79753
.80677
.81598
.73179
.74129
.75076
.76019
.76958
.68374
.69342
.70306
.71267
.72225
.63481
.64467
.65449
.66427
.67402
.58506
.59507
.60505
.61500
.62492
V
V∗
1.00966
1.00829
1.00704
1.00591
1.00490
1.01853
1.01646
1.01455
1.01278
1.01115
1.03137
1.02844
1.02570
1.02314
1.02075
1.04915
1.04514
1.04137
1.03783
1.03450
1.07314
1.06777
1.06271
1.05792
1.05340
1.10504
1.09793
1.09120
1.08485
1.07883
1.1472
1.1378
1.1289
1.1205
1.1126
I
I∗
.03632
.03097
.02613
.02180
.01793
.07229
.06375
.05593
.04878
.04226
.12728
.11446
.10262
.09167
.08159
.20814
.18949
.17215
.15606
.14113
.32460
.29785
.27295
.24978
.22821
.49081
.45270
.41720
.38411
.35330
.72805
.67357
.62286
.57568
.53174
D
f Lmax
.94899
.94554
.94208
.93862
.93515
.93168
.92820
.92473
.92125
.91777
1.20
1.21
1.22
1.23
1.24
.96618
.96276
.95933
.95589
.95244
.98320
.97982
.97642
.97302
.96960
1.00000
.99666
.99331
.98995
.98658
1.01652
1.01324
1.00995
1.00664
1.00333
1.03270
1.02950
1.02627
1.02304
1.01978
T
T∗
1.15
1.16
1.17
1.18
1.19
1.10
1.11
1.12
1.13
1.14
1.05
1.06
1.07
1.08
1.09
1.00
1.01
1.02
1.03
1.04
.95
.96
.97
.98
.99
.90
.91
.92
.93
.94
M
.80436
.79623
.78822
.78034
.77258
.84710
.83827
.82958
.82104
.81263
.89359
.88397
.87451
.86522
.85608
.94435
.93383
.92350
.91335
.90338
1.00000
.98844
.97711
.96598
.95506
1.06129
1.04854
1.03605
1.02379
1.01178
1.12913
1.11500
1.10114
1.08758
1.07430
p
p∗
Table 4 (continued) γ = 1.4
p0
1.03044
1.03344
1.03657
1.03983
1.04323
1.01746
1.01978
1.02224
1.02484
1.02757
1.00793
1.00955
1.01131
1.01322
1.01527
1.00203
1.00291
1.00394
1.00512
1.00645
1.00000
1.00008
1.00033
1.00073
1.00130
1.00215
1.00137
1.00076
1.00033
1.00008
1.00887
1.00714
1.00560
1.00426
1.00311
p0∗
1.1583
1.1658
1.1732
1.1806
1.1879
1.1203
1.1280
1.1356
1.1432
1.1508
1.08124
1.08913
1.09698
1.10479
1.11256
1.04115
1.04925
1.05731
1.06533
1.07331
1.00000
1.00831
1.01658
1.02481
1.03300
.95782
.96634
.97481
.98324
.99164
.91459
.92332
.93201
.94065
.94925
V
V∗
1.01082
1.01178
1.01278
1.01381
1.01486
1.00646
1.00726
1.00810
1.00897
1.00988
1.00305
1.00365
1.00429
1.00497
1.00569
1.00082
1.00116
1.00155
1.00200
1.00250
1.00000
1.00003
1.00013
1.00030
1.00053
1.00093
1.00059
1.00033
1.00014
1.00003
1.00399
1.00318
1.00248
1.00188
1.00136
I
I∗
.03364
.03650
.03942
.04241
.04547
.02053
.00298
.02552
.02814
.03085
.009933
.011813
.013824
.015949
.018187
.002712
.003837
.005129
.006582
.008185
0
.000114
.000458
.001013
.001771
.003280
.002056
.001135
.000493
.000120
.014513
.011519
.008916
.006694
.004815
D
f Lmax
THERMODYNAMICS AND FLUID MECHANICS
103
T
T∗
.91429
.91080
.90732
.90383
.90035
.89686
.89338
.88989
.88641
.88292
.87944
.87596
.87249
.86901
.86554
.86207
.85860
.85514
.85168
.84822
.84477
.84133
.83788
.83445
.83101
.82759
.82416
.82075
.81734
.81394
.81054
.80715
.80376
.80038
.79701
M
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
.58084
.57591
.57104
.56623
.56148
.60648
.60122
.59602
.59089
.58583
.63387
.62824
.62269
.61722
.61181
.66320
.65717
.65122
.64536
.63958
.69466
.68818
.68180
.67551
.66931
.72848
.72152
.71465
.70789
.70123
.76495
.75743
.75003
.74274
.73556
p
p∗
Table 4 (continued) γ = 1.4
p0
1.2116
1.2190
1.2266
1.2343
1.2422
1.1762
1.1830
1.1899
1.1970
1.2043
1.1440
1.1502
1.1565
1.1629
1.1695
1.1149
1.1205
1.1262
1.1320
1.1379
1.08904
1.09397
1.09902
1.10419
1.10948
1.06630
1.07060
1.07502
1.07957
1.08424
1.04676
1.05041
1.05419
1.05809
1.06213
p0∗
1.3955
1.4015
1.4075
1.4135
1.4195
1.3646
1.3708
1.3770
1.3832
1.3894
1.3327
1.3392
1.3456
1.3520
1.3583
1.2999
1.3065
1.3131
1.3197
1.3262
1.2660
1.2729
1.2797
1.2864
1.2932
1.2311
1.2382
1.2452
1.2522
1.2591
1.1952
1.2025
1.2097
1.2169
1.2240
V
V∗
1.05604
1.05752
1.05900
1.06049
1.06198
1.04870
1.05016
1.05162
1.05309
1.05456
1.04153
1.04295
1.04438
1.04581
1.04725
1.03458
1.03595
1.03733
1.03872
1.04012
1.02794
1.02924
1.03056
1.03189
1.03323
1.02169
1.02291
1.02415
1.02540
1.02666
1.01594
1.01705
1.01818
1.01933
1.02050
I
I∗
.15427
.15790
.16152
.16514
.16876
.13605
.13970
.14335
.14699
.15063
.11782
.12146
.12510
.12875
.13240
.09974
.10333
.10694
.11056
.11419
.08199
.08550
.08904
.09259
.09616
.06483
.06820
.07161
.07504
.07850
.04858
.05174
.05494
.05820
.06150
D
f Lmax
1.90
1.91
1.92
1.93
1.94
1.85
1.86
1.87
1.88
1.89
1.80
1.81
1.82
1.83
1.84
1.75
1.76
1.77
1.78
1.79
1.70
1.71
1.72
1.73
1.74
1.65
1.66
1.67
1.68
1.69
1.60
1.61
1.62
1.63
1.64
M
.69686
.69379
.69074
.68769
.68465
.71238
.70925
.70614
.70304
.69995
.72816
.72498
.72181
.71865
.71551
.74419
.74096
.73774
.73453
.73134
.76046
.75718
.75392
.75067
.74742
.77695
.77363
.77033
.76703
.76374
.79365
.79030
.78695
.78361
.78028
T
T∗
.43936
.43610
.43287
.42967
.42651
.45623
.45278
.44937
.44600
.44266
.47407
.47042
.46681
.46324
.45972
.49295
.48909
.48527
.48149
.47776
.51297
.50887
.50482
.50082
.49686
.53421
.52986
.52556
.52131
.51711
.55679
.55216
.54759
.54308
.53862
p
p∗
Table 4 (continued) γ = 1.4
p0
1.5552
1.5677
1.5804
1.5932
1.6062
1.4952
1.5069
1.5188
1.5308
1.5429
1.4390
1.4499
1.4610
1.4723
1.4837
1.3865
1.3967
1.4070
1.4175
1.4282
1.3376
1.3471
1.3567
1.3665
1.3764
1.2922
1.3010
1.3099
1.3190
1.3282
1.2502
1.2583
1.2666
1.2750
1.2835
p0∗
1.5861
1.5909
1.5957
1.6005
1.6052
1.5614
1.5664
1.5714
1.5763
1.5812
1.5360
1.5412
1.5463
1.5514
1.5564
1.5097
1.5150
1.5203
1.5256
1.5308
1.4825
1.4880
1.4935
1.4989
1.5043
1.4544
1.4601
1.4657
1.4713
1.4769
1.4254
1.4313
1.4371
1.4429
1.4487
V
V∗
1.1083
1.1097
1.1112
1.1126
1.1141
1.1009
1.1024
1.1039
1.1054
1.1068
1.09352
1.09500
1.09649
1.09798
1.09946
1.08603
1.08753
1.08903
1.09053
1.09202
1.07851
1.08002
1.08152
1.08302
1.08453
1.07098
1.07249
1.07399
1.07550
1.07701
1.06348
1.06498
1.06648
1.06798
1.06948
I
I∗
.27433
.27748
.28061
.28372
.28681
.25832
.26156
.26478
.26798
.27116
.24189
.24521
.24851
.25180
.25507
.22504
.22844
.23183
.23520
.23855
.20780
.21128
.21474
.21819
.22162
.19022
.19376
.19729
.20081
.20431
.17236
.17595
.17953
.18311
.18667
D
f Lmax
104
ENGINEERING TABLES AND DATA
T
T∗
.68162
.67861
.67561
.67262
.66964
.66667
.66371
.66076
.65783
.65491
.65200
.64910
.64621
.64333
.64047
.63762
.63478
.63195
.62914
.62633
.62354
.62076
.61799
.61523
.61249
.60976
.60704
.60433
.60163
.59895
.59627
.59361
.59096
.58833
.58570
M
1.95
1.96
1.97
1.98
1.99
2.00
2.01
2.02
2.03
2.04
2.05
2.06
2.07
2.08
2.09
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
.34319
.34091
.33865
.33641
.33420
.35494
.35254
.35017
.34782
.34550
.36728
.36476
.36227
.35980
.35736
.38024
.37760
.37498
.37239
.36982
.39389
.39110
.38834
.38562
.38292
.40825
.40532
.40241
.39954
.39670
.42339
.42030
.41724
.41421
.41121
p
p∗
Table 4 (continued) γ = 1.4
p0
2.0964
2.1154
2.1345
2.1538
2.1733
2.0050
2.0228
2.0409
2.0592
2.0777
1.9185
1.9354
1.9525
1.9698
1.9873
1.8369
1.8528
1.8690
1.8853
1.9018
1.7600
1.7750
1.7902
1.8056
1.8212
1.6875
1.7017
1.7160
1.7305
1.7452
1.6193
1.6326
1.6461
1.6597
1.6735
p0∗
1.7374
1.7412
1.7450
1.7488
1.7526
1.7179
1.7219
1.7258
1.7297
1.7336
1.6977
1.7018
1.7059
1.7099
1.7139
1.6769
1.6811
1.6853
1.6895
1.6936
1.6553
1.6597
1.6640
1.6683
1.6726
1.6330
1.6375
1.6420
1.6465
1.6509
1.6099
1.6146
1.6193
1.6239
1.6284
V
V∗
1.1565
1.1578
1.1590
1.1603
1.1616
1.1500
1.1513
1.1526
1.1539
1.1552
1.1434
1.1447
1.1460
1.1474
1.1487
1.1366
1.1380
1.1393
1.1407
1.1420
1.1297
1.1311
1.1325
1.1339
1.1352
1.1227
1.1241
1.1255
1.1269
1.1283
1.1155
1.1170
1.1184
1.1198
1.1213
I
I∗
.37378
.37630
.37881
.38130
.38377
.36091
.36352
.36611
.36868
.37124
.34760
.35030
.35298
.35564
.35828
.33385
.33664
.33940
.34215
.34488
.31965
.32253
.32538
.32822
.33104
.30499
.30796
.31091
.31384
.31675
.28989
.29295
.29599
.29901
.30201
D
f Lmax
2.60
2.61
2.62
2.63
2.64
2.55
2.56
2.57
2.58
2.59
2.50
2.51
2.52
2.53
2.54
2.45
2.46
2.47
2.48
2.49
2.40
2.41
2.42
2.43
2.44
2.35
2.36
2.37
2.38
2.39
2.30
2.31
2.32
2.33
2.34
M
.51020
.50795
.50571
.50349
.50127
.52163
.51932
.51702
.51474
.51247
.53333
.53097
.52862
.52627
.52394
.54533
.54291
.54050
.53810
.53571
.55762
.55514
.55267
.55021
.54776
.57021
.56767
.56514
.56262
.56011
.58309
.58049
.57790
.57532
.57276
T
T∗
.27473
.27307
.27143
.26980
.26818
.28323
.28150
.27978
.27808
.27640
.29212
.29031
.28852
.28674
.28498
.30141
.29952
.29765
.29579
.29395
.31114
.30916
.30720
.30525
.30332
.32133
.31925
.31720
.31516
.31314
.33200
.32983
.32767
.32554
.32342
p
p∗
Table 4 (continued) γ = 1.4
p0
2.8960
2.9234
2.9511
2.9791
3.0074
2.7630
2.7891
2.8154
2.8420
2.8689
2.6367
2.6615
2.6865
2.7117
2.7372
2.5168
2.5403
2.5640
2.5880
2.6122
2.4031
2.4254
2.4479
2.4706
2.4936
2.2953
2.3164
2.3377
2.3593
2.3811
2.1931
2.2131
2.2333
2.2537
2.2744
p0∗
1.8571
1.8602
1.8632
1.8662
1.8691
1.8417
1.8448
1.8479
1.8510
1.8541
1.8257
1.8290
1.8322
1.8354
1.8386
1.8092
1.8126
1.8159
1.8192
1.8225
1.7922
1.7956
1.7991
1.8025
1.8059
1.7745
1.7781
1.7817
1.7852
1.7887
1.7563
1.7600
1.7637
1.7673
1.7709
V
V∗
1.1978
1.1989
1.2000
1.2011
1.2021
1.1923
1.1934
1.1945
1.1956
1.1967
1.1867
1.1879
1.1890
1.1901
1.1912
1.1810
1.1821
1.1833
1.1844
1.1856
1.1751
1.1763
1.1775
1.1786
1.1798
1.1690
1.1703
1.1715
1.1727
1.1739
1.1629
1.1641
1.1653
1.1666
1.1678
I
I∗
.45259
.45457
.45654
.45850
.46044
.44247
.44452
.44655
.44857
.45059
.43197
.43410
.43621
.43831
.44040
.42113
.42333
.42551
.42768
.42983
.40989
.41216
.41442
.41667
.41891
.39826
.40062
.40296
.40528
.40760
.38623
.38867
.39109
.39350
.39589
D
f Lmax
THERMODYNAMICS AND FLUID MECHANICS
105
T
T∗
.49906
.49687
.49469
.49251
.49035
.48820
.48606
.48393
.48182
.47971
.47761
.47553
.47346
.47139
.46933
.46729
.46526
.46324
.46122
.45922
.45723
.45525
.45328
.45132
.44937
M
2.65
2.66
2.67
2.68
2.69
2.70
2.71
2.72
2.73
2.74
2.75
2.76
2.77
2.78
2.79
2.80
2.81
2.82
2.83
2.84
2.85
2.86
2.87
2.88
2.89
.23726
.23592
.23458
.23326
.23196
.24414
.24274
.24135
.23997
.23861
.25131
.24985
.24840
.24697
.24555
.25878
.25726
.25575
.25426
.25278
.26658
.26499
.26342
.26186
.26032
p
p∗
Table 4 (continued) γ = 1.4
p0
3.6707
3.7058
3.7413
3.7771
3.8133
3.5001
3.5336
3.5674
3.6015
3.6359
3.3376
3.3695
3.4017
3.4342
3.4670
3.1830
3.2133
3.2440
3.2749
3.3061
3.0359
3.0647
3.0938
3.1234
3.1530
p0∗
1.9271
1.9297
1.9322
1.9348
1.9373
1.9140
1.9167
1.9193
1.9220
1.9246
1.9005
1.9032
1.9060
1.9087
1.9114
1.8865
1.8894
1.8922
1.8950
1.8978
1.8721
1.8750
1.8779
1.8808
1.8837
V
V∗
1.2230
1.2240
1.2249
1.2258
1.2268
1.2182
1.2192
1.2202
1.2211
1.2221
1.2133
1.2143
1.2153
1.2163
1.2173
1.2083
1.2093
1.2103
1.2113
1.2123
1.2031
1.2042
1.2052
1.2062
1.2073
I
I∗
.49828
.49995
.50161
.50326
.50489
.48976
.49148
.49321
.49491
.49660
.48095
.48274
.48452
.48628
.48803
.47182
.47367
.47551
.47734
.47915
.46237
.46429
.46619
.46807
.46996
D
f Lmax
∞
6.0
7.0
8.0
9.0
10.0
3.0
3.5
4.0
4.5
5.0
2.95
2.96
2.97
2.98
2.99
2.90
2.91
2.92
2.93
2.94
M
0
.14634
.11111
.08696
.06977
.05714
.42857
.34783
.28571
.23762
.20000
.43788
.43600
.43413
.43226
.43041
.44743
.44550
.44358
.44167
.43977
T
T∗
0
.06376
.04762
.03686
.02935
.02390
.21822
.16850
.13363
.10833
.08944
.22431
.22307
.22185
.22063
.21942
.23066
.22937
.22809
.22682
.22556
p
p∗
Table 4 (continued) γ = 1.4
p0
∞
53.180
104.14
190.11
327.19
535.94
4.2346
6.7896
10.719
16.562
25.000
4.0376
4.0763
4.1153
4.1547
4.1944
3.8498
3.8866
3.9238
3.9614
3.9993
p0∗
2.4495
2.2953
2.3333
2.3591
2.3772
2.3905
1.9640
2.0642
2.1381
2.1936
2.2361
1.9521
1.9545
1.9569
1.9592
1.9616
1.9398
1.9423
1.9448
1.9472
1.9497
V
V∗
1.4289
1.3655
1.3810
1.3915
1.3989
1.4044
1.2366
1.2743
1.3029
1.3247
1.3416
1.2322
1.2331
1.2340
1.2348
1.2357
1.2277
1.2286
1.2295
1.2304
1.2313
I
I∗
.82153
.72987
.75281
.76820
.77898
.78683
.52216
.53643
.63306
.66764
.69381
.51447
.51603
.51758
.51912
.52064
.50651
.50812
.50973
.51133
.51291
D
f Lmax
106
ENGINEERING TABLES AND DATA
p0
1.0714
1.0441
1.0240
1.0104
1.0026
1.333
1.250
1.176
1.111
1.0526
1.0000
.9524
.9091
.8695
.8333
.8000
.7692
.7407
.7143
.6897
.6667
.6452
.6250
.6061
.5882
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.245
1.300
1.363
1.434
1.514
1.0598
1.0862
1.118
1.154
1.196
1.0000
1.0025
1.0097
1.0217
1.0384
1.375
1.283
1.210
1.153
1.107
2.000
1.818
1.667
1.539
1.429
.50
.55
.60
.65
.70
1.000
2.503
2.115
1.842
1.643
1.492
0
.05
.10
.15
.20
4.000
3.333
2.857
2.500
2.222
p0∗
.25
.30
.35
.40
.45
p
p∗
∞
12.146
6.096
4.089
3.094
T
T∗
∞
20.000
10.000
6.667
5.000
M
Table 4 (continued) γ = 1.0
1.500
1.550
1.600
1.650
1.700
1.250
1.300
1.350
1.400
1.450
1.0000
1.0500
1.100
1.150
1.200
.7500
.8000
.8500
.9000
.9500
.5000
.5500
.6000
.6500
.7000
.2500
.3000
.3500
.4000
.4500
0
.0500
.1000
.1500
.2000
V
V∗
1.0833
1.0976
1.112
1.128
1.144
1.0250
1.0346
1.0453
1.0571
1.0698
1.0000
1.0012
1.0045
1.0098
1.0167
1.0417
1.0250
1.0132
1.0056
1.0013
1.250
1.184
1.133
1.0942
1.0643
2.125
1.817
1.604
1.450
1.336
∞
10.025
5.050
3.408
2.600
I
I∗
.2554
.2927
.3306
.3689
.4073
.08629
.1164
.1489
.1831
.2188
0
.00461
.01707
.03567
.05909
.2024
.1162
.05904
.02385
.00545
1.614
1.110
.7561
.5053
.3275
12.227
7.703
5.064
3.417
2.341
∞
393.01
94.39
39.65
20.78
D
f Lmax
.3636
.3571
.3509
.3449
.3390
2.75
2.80
2.85
2.90
2.95
.1667
.1429
.1250
.1111
.1000
0
6.00
7.00
8.00
9.00
10.00
∞
.3333
.2857
.2500
.2222
.2000
.4000
.3922
.3847
.3774
.3704
2.50
2.55
2.60
2.65
2.70
3.00
3.50
4.00
4.50
5.00
.4444
.4348
.4256
.4167
.4082
2.25
2.30
2.35
2.40
2.45
.5714
.5556
.5406
.5263
.5128
p
p∗
.5000
.4878
.4762
.4651
.4545
1.000
T
T∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
Table 4 (continued) γ = 1.0
p0
∞
664×104
378×107
599×1010
262×1014
314×1018
18.20
79.22
452.01
3364
32550
9.676
10.92
12.35
14.02
15.95
5.522
6.142
6.852
7.665
8.600
3.388
3.714
4.083
4.502
4.979
2.241
2.419
2.620
2.846
3.100
1.603
1.703
1.815
1.941
2.082
p0∗
∞
6.000
7.000
8.000
9.000
10.000
3.000
3.500
4.000
4.500
5.000
2.750
2.800
2.850
2.900
2.950
2.500
2.550
2.600
2.650
2.700
2.250
2.300
2.350
2.400
2.450
2.000
2.050
2.100
2.150
2.200
1.750
1.800
1.850
1.900
1.950
V
V∗
∞
3.083
3.571
4.062
4.556
5.050
1.667
1.893
2.125
2.361
2.600
1.557
1.579
1.600
1.622
1.644
1.450
1.471
1.492
1.514
1.535
1.347
1.367
1.388
1.408
1.429
1.250
1.269
1.288
1.308
1.327
1.161
1.178
1.195
1.213
1.231
I
I∗
∞
2.611
2.912
3.174
3.407
3.615
1.308
1.587
1.835
2.058
2.259
1.155
1.187
1.218
1.248
1.279
.9926
1.0260
1.0590
1.0916
1.1237
.8194
.8549
.8900
.9246
.9588
.6363
.6736
.7106
.7472
.7835
.4458
.4842
.5225
.5607
.5986
D
f Lmax
THERMODYNAMICS AND FLUID MECHANICS
107
4.092
3.408
2.919
2.552
2.266
2.037
1.849
1.693
1.560
1.446
1.347
1.261
1.184
1.116
1.0551
1.0500
1.0499
1.0495
1.0488
1.0479
1.0467
1.0453
1.0436
1.0417
1.0395
1.0370
1.0343
1.0314
1.0283
1.0249
1.0213
1.0174
1.0133
1.0091
1.0047
1.0000
.9951
.9901
.9849
.9795
.9739
.9682
.9623
.9563
.9501
.9438
.9374
.9309
.9242
.9174
0
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
.6476
.6246
.6030
.5826
.5634
.7895
.7569
.7266
.6985
.6722
1.0000
.9501
.9046
.8630
.8247
∞
20.493
10.244
6.828
5.118
T
T∗
M
p
p∗
Table 4 (continued) γ = 1.1
p0
1.223
1.272
1.326
1.387
1.454
1.0559
1.0801
1.109
1.142
1.180
1.0000
1.0023
1.0092
1.0204
1.0360
1.0689
1.0425
1.0231
1.0100
1.0024
1.365
1.275
1.204
1.148
1.104
2.476
2.094
1.825
1.628
1.480
∞
11.999
6.023
4.042
3.059
p0∗
1.457
1.501
1.544
1.586
1.628
1.234
1.279
1.324
1.369
1.413
1.0000
1.0474
1.0945
1.1412
1.1876
.7579
.8069
.8557
.9041
.9522
.5092
.5594
.6094
.6591
.7086
.2558
.3067
.3575
.4082
.4588
0
.05123
.1024
.1536
.2047
V
V∗
1.0717
1.0835
1.0958
1.108
1.121
1.0221
1.0304
1.0397
1.0498
1.0605
1.0000
1.0011
1.0041
1.0087
1.0148
1.0386
1.0231
1.0122
1.0051
1.0012
1.237
1.174
1.125
1.0882
1.0599
2.083
1.784
1.577
1.429
1.319
∞
9.785
4.932
3.332
2.545
I
I∗
.2138
.2443
.2749
.3056
.3362
.07350
.09885
.1260
.1544
.1838
0
.00398
.01468
.03058
.05050
.1780
.1019
.05160
.02078
.00472
1.439
.9871
.6705
.4468
.2887
11.03
6.936
4.549
3.062
2.093
∞
357.05
85.65
35.92
18.79
D
f Lmax
.8750
.8677
.8603
.8529
.8454
.8379
.8304
.8228
.8152
.8076
.8000
.7924
.7848
.7771
.7695
.7619
.7543
.7467
.7392
.7316
.7241
.6512
.5833
.5217
.4667
.3750
.3043
.2500
.2079
.1750
0
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.9105
.9036
.8966
.8895
.8823
T
T∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
0
.1021
.07881
.06250
.05067
.04183
.2837
.2305
.1909
.1605
.1366
.3174
.3102
.3032
.2965
.2900
.3578
.3491
.3407
.3327
.3249
.4068
.3962
.3860
.3762
.3668
.4677
.4544
.4417
.4295
.4179
.5453
.5281
.5118
.4964
.4817
p
p∗
Table 4 (continued) γ = 1.1
p0
∞
4911
37919
263×103
4
161×10
889×104
9.880
25.83
71.74
205.7
597.3
6.320
6.895
7.532
8.237
9.016
4.165
4.515
4.902
5.328
5.799
2.846
3.061
3.299
3.560
3.848
2.032
2.165
2.312
2.473
2.651
1.528
1.610
1.701
1.801
1.911
p0∗
4.583
3.674
3.862
4.000
4.104
4.183
2.553
2.824
3.055
3.250
3.416
2.400
2.432
2.463
2.493
2.523
2.236
2.270
2.303
2.336
2.368
2.060
2.096
2.132
2.167
2.202
1.871
1.910
1.948
1.986
2.023
1.670
1.711
1.752
1.792
1.832
V
V∗
2.400
1.973
2.060
2.125
2.174
2.211
1.472
1.589
1.691
1.779
1.854
1.409
1.422
1.434
1.447
1.460
1.342
1.355
1.369
1.382
1.395
1.273
1.286
1.300
1.314
1.328
1.203
1.217
1.231
1.245
1.259
1.134
1.148
1.161
1.175
1.189
I
I∗
1.997
1.601
1.689
1.752
1.798
1.832
.9812
1.147
1.280
1.386
1.472
.8828
.9034
.9235
.9432
.9624
.7726
.7957
.8182
.8402
.8617
.6498
.6754
.7005
.7251
.7491
.5140
.5422
.5698
.5970
.6237
.3667
.3969
.4268
.4563
.4854
D
f Lmax
108
ENGINEERING TABLES AND DATA
4.182
3.480
2.978
2.601
2.307
2.072
1.879
1.717
1.581
1.463
1.361
1.271
1.192
1.121
1.0573
1.1000
1.0997
1.0989
1.0975
1.0956
1.0932
1.0902
1.0867
1.0827
1.0782
1.0732
1.0677
1.0618
1.0554
1.0486
1.0414
1.0338
1.0259
1.0176
1.0089
1.0000
.9908
.9813
.9715
.9615
.9514
.9410
.9304
.9197
.9089
.8980
.8869
.8758
.8646
.8534
0
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
.6317
.6076
.5849
.5635
.5434
.7803
.7462
.7145
.6850
.6575
1.0000
.9480
.9005
.8571
.8172
∞
20.974
10.483
6.984
5.234
T
T∗
M
p
p∗
Table 4 (continued) γ = 1.2
p0
1.205
1.248
1.296
1.349
1.407
1.0525
1.0749
1.101
1.132
1.166
1.0000
1.0022
1.0087
1.0194
1.0340
1.0666
1.0410
1.0222
1.0096
1.0023
1.356
1.268
1.199
1.144
1.100
2.451
2.073
1.809
1.615
1.469
∞
11.857
5.953
3.996
3.026
p0∗
1.421
1.459
1.497
1.534
1.570
1.219
1.261
1.302
1.342
1.382
1.0000
1.0451
1.0896
1.134
1.177
.7654
.8134
.8609
.9078
.9542
.5179
.5683
.6183
.6678
.7168
.2614
.3133
.3649
.4162
.4672
0
.05243
.1048
.1571
.2093
V
V∗
1.0625
1.0724
1.0826
1.0930
1.1036
1.0197
1.0270
1.0351
1.0437
1.0529
1.0000
1.0010
1.0037
1.0079
1.0134
1.0360
1.0214
1.0112
1.0047
1.0011
1.224
1.164
1.118
1.0826
1.0561
2.044
1.753
1.553
1.409
1.304
∞
9.562
4.822
3.260
2.493
I
I∗
.1817
.2069
.2323
.2575
.2825
.06338
.08500
.1080
.1320
.1567
0
.00347
.01277
.02657
.04368
.1579
.09016
.04554
.01829
.00414
1.294
.8855
.5999
.3987
.2570
10.04
6.298
4.121
2.768
1.887
∞
327.09
78.36
32.81
17.13
D
f Lmax
.7857
.7745
.7634
.7523
.7413
.7303
.7194
.7086
.6980
.6874
.6769
.6665
.6563
.6462
.6362
.6263
.6166
.6070
.5975
.5882
.5789
.4944
.4231
.3636
.3143
.2391
.1864
.1486
.1209
.1000
0
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.8421
.8308
.8195
.8082
.7970
T
T∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
0
.08150
.06168
.04819
.03863
.03162
.2536
.2009
.1626
.1340
.1121
.2878
.2804
.2733
.2665
.2600
.3291
.3202
.3116
.3033
.2954
.3798
.3688
.3582
.3481
.3384
.4432
.4293
.4160
.4034
.3913
.5244
.5064
.4894
.4732
.4578
p
p∗
Table 4 (continued) γ = 1.2
p0
∞
434.7
1458
4353
13156
29601
6.735
13.76
28.35
57.96
116.31
4.767
5.103
5.466
5.858
6.280
3.420
3.650
3.898
4.166
4.455
2.504
2.660
2.829
3.011
3.208
1.884
1.989
2.103
2.226
2.359
1.471
1.540
1.615
1.697
1.787
p0∗
3.317
2.934
3.023
3.084
3.129
3.162
2.283
2.461
2.602
2.714
2.803
2.176
2.199
2.220
2.242
2.263
2.057
2.082
2.106
2.130
2.154
1.923
1.951
1.978
2.005
2.031
1.773
1.804
1.835
1.865
1.894
1.606
1.641
1.675
1.708
1.741
V
V∗
1.809
1.637
1.677
1.704
1.724
1.739
1.360
1.434
1.493
1.541
1.580
1.318
1.327
1.335
1.344
1.352
1.272
1.281
1.291
1.300
1.309
1.221
1.232
1.242
1.252
1.262
1.168
1.179
1.190
1.201
1.211
1.114
1.125
1.136
1.147
1.158
I
I∗
1.365
1.163
1.212
1.245
1.268
1.286
.7724
.8857
.9718
1.0380
1.0896
.7026
.7173
.7316
.7456
.7592
.6222
.6392
.6557
.6718
.6874
.5299
.5493
.5683
.5868
.6047
.4247
.4468
.4684
.4894
.5099
.3072
.3316
.3556
.3791
.4021
D
f Lmax
THERMODYNAMICS AND FLUID MECHANICS
109
4.270
3.551
3.036
2.649
2.348
2.106
1.907
1.741
1.600
1.479
1.373
1.280
1.198
1.125
1.0594
1.150
1.149
1.148
1.146
1.143
1.139
1.134
1.129
1.123
1.116
1.1084
1.1001
1.0911
1.0815
1.0713
1.0605
1.0493
1.0376
1.0254
1.0129
1.0000
.9868
.9733
.9596
.9457
.9316
.9174
.9031
.8887
.8743
.8598
.8454
.8309
.8165
.8022
0
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
.6182
.5932
.5697
.5477
.5269
.7722
.7368
.7039
.6734
.6448
1.0000
.9461
.8969
.8518
.8104
∞
21.444
10.716
7.137
5.346
T
T∗
M
p
p∗
Table 4 (continued) γ = 1.3
p0
1.189
1.228
1.271
1.318
1.369
1.0495
1.0704
1.0948
1.1227
1.1543
1.0000
1.0021
1.0083
1.0183
1.0321
1.0644
1.0395
1.0214
1.0092
1.0022
1.348
1.261
1.193
1.140
1.0972
2.426
2.054
1.793
1.602
1.459
∞
11.721
5.885
3.952
2.994
p0∗
1.391
1.425
1.458
1.491
1.523
1.206
1.245
1.283
1.320
1.356
1.0000
1.0430
1.0852
1.1266
1.1670
.7724
.8195
.8658
.9113
.9561
.5264
.5769
.6267
.6759
.7245
.2668
.3195
.3719
.4239
.4754
0
.05361
.1072
.1606
.2138
V
V∗
1.0549
1.0634
1.0721
1.0808
1.0897
1.0177
1.0241
1.0312
1.0388
1.0467
1.0000
1.0009
1.0033
1.0071
1.0120
1.0336
1.0199
1.0104
1.0043
1.0010
1.213
1.155
1.111
1.0777
1.0524
2.007
1.724
1.530
1.391
1.289
∞
9.354
4.720
3.194
2.445
I
I∗
.1564
.1777
.1989
.2200
.2408
.05524
.07388
.09365
.11417
.13513
0
.00305
.01122
.02324
.03820
.14131
.08044
.04053
.01623
.00367
1.172
.8004
.5409
.3586
.2305
9.201
5.759
3.760
2.520
1.714
∞
301.74
72.20
30.18
15.73
D
f Lmax
.5935
.5822
.5711
.5601
.5493
.5388
.5285
.5184
.5085
.4988
.4894
.4053
.3382
.2848
.2421
.1797
.1377
.1085
.08745
.07188
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
0
.6536
.6412
.6290
.6170
.6051
2.25
2.30
2.35
2.40
2.45
∞
.7188
.7054
.6922
.6791
.6662
.7880
.7739
.7599
.7460
.7323
T
T∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
0
.07065
.05302
.04117
.03286
.02769
.2332
.1819
.1454
.1186
.09841
.2669
.2596
.2526
.2459
.2394
.3082
.2992
.2906
.2824
.2745
.3593
.3482
.3375
.3273
.3175
.4239
.4097
.3962
.3833
.3710
.5073
.4887
.4712
.4546
.4388
p
p∗
Table 4 (continued) γ = 1.3
p0
∞
120.1
285.3
625.2
1275
2438
5.160
9.110
15.94
27.39
45.95
3.892
4.116
4.354
4.607
4.875
2.954
3.119
3.295
3.482
3.681
2.268
2.388
2.517
2.654
2.800
1.773
1.859
1.951
2.050
2.156
1.424
1.484
1.549
1.618
1.693
p0∗
2.769
2.543
2.598
2.635
2.662
2.681
2.099
2.228
2.326
2.402
2.460
2.019
2.036
2.052
2.068
2.084
1.926
1.946
1.965
1.983
2.001
1.819
1.842
1.864
1.885
1.906
1.696
1.722
1.747
1.772
1.796
1.554
1.584
1.613
1.641
1.669
V
V∗
1.565
1.468
1.491
1.507
1.519
1.527
1.288
1.338
1.378
1.409
1.433
1.257
1.263
1.270
1.276
1.282
1.223
1.230
1.237
1.244
1.250
1.184
1.192
1.200
1.208
1.215
1.143
1.151
1.160
1.168
1.176
1.0986
1.108
1.116
1.125
1.134
I
I∗
1.0326
.9037
.9355
.9570
.9722
.9832
.6277
.7110
.7726
.8189
.8543
.5752
.5864
.5972
.6077
.6179
.5136
.5267
.5394
.5517
.5636
.4413
.4566
.4715
.4860
.5000
.3573
.3751
.3924
.4092
.4255
.2613
.2814
.3010
.3202
.3390
D
f Lmax
110
ENGINEERING TABLES AND DATA
4.574
3.795
3.235
2.814
2.485
2.220
2.002
1.819
1.664
1.530
1.413
1.311
1.220
1.139
1.0657
1.335
1.334
1.331
1.325
1.317
1.308
1.296
1.282
1.267
1.250
1.232
1.212
1.191
1.169
1.146
1.1233
1.0993
1.0748
1.0501
1.0251
1.0000
.9749
.9499
.9251
.9006
.8763
.8524
.8289
.8059
.7833
.7612
.7397
.7187
.6982
.6783
0
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
.5817
.5549
.5298
.5064
.4845
.7489
.7102
.6744
.6412
.6104
1.0000
.9404
.8860
.8364
.7908
∞
23.099
11.535
7.674
5.739
T
T∗
M
p
p∗
Table 4 (continued) γ = 1.67
p0
1.148
1.176
1.207
1.240
1.275
1.0406
1.0573
1.0765
1.0981
1.1220
1.0000
1.0018
1.0070
1.0154
1.0266
1.0576
1.0351
1.0189
1.0081
1.0019
1.320
1.239
1.176
1.126
1.0874
2.344
1.989
1.741
1.560
1.424
∞
11.265
5.661
3.805
2.887
p0∗
1.309
1.333
1.356
1.378
1.400
1.170
1.200
1.229
1.257
1.284
1.0000
1.0368
1.0721
1.1061
1.1388
.7949
.8388
.8812
.9222
.9618
.5549
.6056
.6548
.7029
.7496
.2859
.3415
.3963
.4502
.5031
0
.05775
.1154
.1727
.2296
V
V∗
1.0364
1.0416
1.0468
1.0520
1.0572
1.0124
1.0167
1.0213
1.0262
1.0313
1.0000
1.0006
1.0024
1.0051
1.0084
1.0265
1.0155
1.0080
1.0033
1.0008
1.178
1.128
1.0909
1.0628
1.0418
1.892
1.635
1.460
1.336
1.245
∞
8.687
4.392
2.982
2.293
I
I∗
.09749
.1101
.1225
.1346
.1465
.03564
.04733
.05957
.07212
.08481
0
.00203
.00740
.01522
.02481
.09870
.05576
.02780
.01106
.00248
.8549
.5787
.3877
.2548
.1625
6.980
4.337
2.810
1.868
1.260
∞
234.36
55.83
23.21
12.11
D
f Lmax
.4315
.4200
.4089
.3982
.3878
.3778
.3681
.3587
.3497
.3410
.3325
.2616
.2099
.1715
.1424
.10222
.07666
.05949
.04745
.03870
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
0
.4952
.4816
.4684
.4557
.4434
2.25
2.30
2.35
2.40
2.45
∞
.5705
.5544
.5388
.5238
.5093
.6590
.6402
.6219
.6042
.5871
T
T∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
0
.05329
.03955
.03049
.02420
.01996
.1922
.1461
.1145
.09203
.07547
.2235
.2167
.2102
.2039
.1979
.2628
.2542
.2460
.2381
.2306
.3128
.3017
.2912
.2813
.2718
.3776
.3632
.3496
.3367
.3244
.4639
.4445
.4263
.4091
.3929
p
p∗
Table 4 (continued) γ = 1.67
p0
∞
15.68
23.85
34.58
48.24
65.18
2.990
4.134
5.608
7.456
9.721
2.529
2.616
2.705
2.797
2.892
2.135
2.209
2.285
2.364
2.445
1.803
1.865
1.929
1.995
2.064
1.530
1.580
1.632
1.687
1.744
1.312
1.351
1.392
1.436
1.482
p0∗
1.996
1.918
1.938
1.951
1.960
1.967
1.730
1.790
1.833
1.864
1.887
1.691
1.699
1.707
1.715
1.723
1.642
1.653
1.663
1.672
1.682
1.583
1.596
1.608
1.620
1.631
1.510
1.526
1.541
1.556
1.570
1.421
1.440
1.459
1.477
1.494
V
V∗
1.249
1.220
1.227
1.232
1.235
1.238
1.154
1.174
1.189
1.200
1.208
1.141
1.144
1.146
1.149
1.152
1.126
1.129
1.132
1.135
1.138
1.107
1.111
1.115
1.119
1.122
1.0863
1.0908
1.0952
1.0994
1.1035
1.0623
1.0673
1.0722
1.0770
1.0817
I
I∗
.5064
.4594
.4714
.4793
.4849
.4889
.3440
.3810
.4071
.4261
.4402
.3196
.3248
.3299
.3348
.3395
.2901
.2965
.3026
.3085
.3141
.2542
.2620
.2694
.2766
.2835
.2105
.2199
.2290
.2377
.2461
.1580
.1692
.1800
.1905
.2007
D
f Lmax
THERMODYNAMICS AND FLUID MECHANICS
111
0
.000480
.00192
.00431
.00765
.01192
.01712
.02322
.03021
.03807
.04678
.05630
.06661
.07768
.08947
.10196
.11511
.12888
.14324
.15814
.17355
.18943
.20574
.22244
.23948
.25684
.27446
.29231
.31035
.32855
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
T0∗
0
.01
.02
.03
.04
M
T0
.30440
.32496
.34573
.36667
.38773
.20661
.22533
.24452
.26413
.28411
.12181
.13743
.15377
.17078
.18841
.05602
.06739
.07970
.09290
.10695
.01430
.02053
.02784
.03621
.04562
0
.000576
.00230
.00516
.00917
T
T∗
1.2679
1.2678
1.2675
1.2671
1.2665
1.2657
1.2647
1.2636
1.2623
1.2608
1.2591
1.2573
1.2554
1.2533
1.2510
1.2486
1.2461
1.2434
1.2406
1.2377
1.2346
1.2314
1.2281
1.2248
1.2213
1.2177
1.2140
1.2102
1.2064
1.2025
2.3916
2.3880
2.3837
2.3787
2.3731
2.3669
2.3600
2.3526
2.3445
2.3359
2.3267
2.3170
2.3067
2.2959
2.2845
2.2727
2.2604
2.2477
2.2345
2.2209
2.2069
2.1925
2.1777
2.1626
2.1472
p0∗
p0
2.4000
2.3997
2.3987
2.3970
2.3946
p
p∗
.13793
.14821
.15876
.16955
.18058
.09091
.09969
.10879
.11820
.12792
.05235
.05931
.06666
.07438
.08247
.02367
.02856
.03388
.03962
.04578
.00598
.00860
.01168
.01522
.01922
0
.000240
.000959
.00216
.00383
V
V∗
Table 5
Rayleigh Line – one-dimensional, frictionless, constant-area
flow with stagnation temperature change for a perfect gas, γ = 1.4
isoth
.88594
.88683
.88776
.88872
.88972
.88200
.88272
.88347
.88426
.88508
.87894
.87948
.88006
.88067
.88132
.87675
.87712
.87752
.87796
.87843
.87544
.87563
.87586
.87612
.87642
.87500
.87502
.87507
.87516
.87528
T0∗
T0
.55
.56
.57
.58
.59
.50
.51
.52
.53
.54
.45
.46
.47
.48
.49
.40
.41
.42
.43
.44
.35
.36
.37
.38
.39
.30
.31
.32
.33
.34
M
.75991
.77248
.78467
.79647
.80789
.69136
.70581
.71990
.73361
.74695
.61393
.63007
.64589
.66139
.67655
.52903
.54651
.56376
.58075
.59748
.43894
.45723
.47541
.49346
.51134
.34686
.36525
.38369
.40214
.42057
T0∗
T0
.85987
.87227
.88415
.89552
.90637
.79012
.80509
.81955
.83351
.84695
.70803
.72538
.74228
.75871
.77466
.61515
.63448
.65345
.67205
.69025
.51413
.53482
.55530
.57553
.59549
.40887
.43004
.45119
.47228
.49327
T
T∗
Table 5 (continued) γ = 1.4
1.1140
1.1099
1.1059
1.1019
1.0979
1.09397
1.09010
1.08630
1.08255
1.07887
1.6860
1.6678
1.6496
1.6316
1.6136
1.1351
1.1308
1.1266
1.1224
1.1182
1.1566
1.1523
1.1480
1.1437
1.1394
1.1779
1.1737
1.1695
1.1652
1.1609
1.1985
1.1945
1.1904
1.1863
1.1821
p0∗
p0
1.7778
1.7594
1.7410
1.7226
1.7043
1.8699
1.8515
1.8331
1.8147
1.7962
1.9608
1.9428
1.9247
1.9065
1.8882
2.0487
2.0314
2.0140
1.9964
1.9787
2.1314
2.1154
2.0991
2.0825
2.0657
p
p∗
.51001
.52302
.53597
.54887
.56170
.44444
.45761
.47075
.48387
.49696
.37865
.39178
.40493
.41810
.43127
.31372
.32658
.33951
.35251
.36556
.25096
.26327
.27572
.28828
.30095
.19183
.20329
.21494
.22678
.23879
V
V∗
isoth
.92794
.92988
.93186
.93387
.93592
.91875
.92052
.92232
.92416
.92603
.91044
.91203
.91366
.91532
.91702
.90300
.90442
.90587
.90736
.90888
.89644
.89768
.89896
.90027
.90162
.89075
.89182
.89292
.89406
.89523
T0∗
T0
112
ENGINEERING TABLES AND DATA
.81892
.82956
.83982
.84970
.85920
.86833
.87709
.88548
.89350
.90117
.90850
.91548
.92212
.92843
.93442
.94009
.94546
.95052
.95528
.95975
.96394
.96786
.97152
.97492
.97807
.98097
.98363
.98607
.98828
.99028
.99207
.99366
.99506
.99627
.99729
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
T0∗
.60
.61
.62
.63
.64
M
T0
1.04310
1.04033
1.03764
1.03504
1.03253
1.4235
1.4070
1.3907
1.3745
1.3585
1.3427
1.3270
1.3115
1.2961
1.2809
1.2658
1.2509
1.2362
1.2217
1.2073
1.1931
1.1791
1.1652
1.1515
1.1380
1.1246
1.1114
1.09842
1.08555
1.07285
.99289
.99796
1.00260
1.00682
1.01062
1.01403
1.01706
1.01971
1.02198
1.02390
1.02548
1.02672
1.02763
1.02823
1.02853
1.02854
1.02826
1.02771
1.02690
1.02583
1.02451
1.02297
1.02120
1.01921
1.01702
1.00485
1.00393
1.00310
1.00237
1.00174
1.01091
1.00951
1.00819
1.00698
1.00587
1.01934
1.01746
1.01569
1.01399
1.01240
1.03010
1.02776
1.02552
1.02337
1.02131
1.05820
1.05502
1.05192
1.04890
1.04596
.96081
.96816
.97503
.98144
.98739
.91670
.92653
.93585
.94466
.95298
1.5080
1.4908
1.4738
1.4569
1.4401
p0∗
p0
1.07525
1.07170
1.06821
1.06480
1.06146
p
p∗
1.5957
1.5780
1.5603
1.5427
1.5253
T
T∗
Table 5 (continued) γ = 1.4
.91097
.92039
.92970
.93889
.94796
.86204
.87206
.88196
.89175
.90142
.81012
.82075
.83126
.84164
.85190
.75525
.76646
.77755
.78852
.79938
.69751
.70927
.72093
.73248
.74392
.63713
.64941
.66159
.67367
.68564
.57447
.58716
.59978
.61232
.62477
V
V∗
isoth
.98700
.98982
.99267
.99556
.99848
.97344
.97608
.97876
.98147
.98422
.96075
.96322
.96572
.96826
.97083
.94894
.95123
.95356
.95592
.95832
.93800
.94012
.94227
.94446
.94668
T0∗
T0
1.01675
1.01992
1.02312
1.02636
1.02963
1.00144
1.00443
1.00746
1.01052
1.01362
.99838
.99769
.99690
.99600
.99501
.99392
.99274
.99148
.99013
.98871
.98721
.98564
.98400
.98230
.98054
.97872
.97685
.97492
.97294
.97092
.96886
.96675
.96461
.96243
.96022
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.00000
.99993
.99973
.99940
.99895
.99814
.99883
.99935
.99972
.99993
T0∗
1.05
1.06
1.07
1.08
1.09
1.00
1.01
1.02
1.03
1.04
.95
.96
.97
.98
.99
M
T0
.88581
.88052
.87521
.86988
.86453
.91185
.90671
.90153
.89632
.89108
.93685
.93195
.92700
.92200
.91695
.96031
.95577
.95115
.94646
.94169
.98161
.97755
.97339
.96913
.96477
1.00000
.99659
.99304
.98936
.98553
1.01463
1.01205
1.00929
1.00636
1.00326
T
T∗
Table 5 (continued) γ = 1.4
.75294
.74473
.73663
.72865
.72078
.79576
.78695
.77827
.76971
.76127
.84166
.83222
.82292
.81374
.80468
.89086
.88075
.87078
.86094
.85123
.94358
.93275
.92206
.91152
.90112
1.00000
.98841
.97697
.96569
.95456
1.06030
1.04792
1.03570
1.02364
1.01174
p
p∗
1.03032
1.03280
1.03536
1.03803
1.04080
1.01941
1.02140
1.02348
1.02566
1.02794
1.01092
1.01243
1.01403
1.01572
1.01752
1.00486
1.00588
1.00699
1.00820
1.00951
1.00121
1.00175
1.00238
1.00311
1.00394
1.00000
1.00004
1.00019
1.00043
1.00077
1.00121
1.00077
1.00043
1.00019
1.00004
p0∗
p0
1.1764
1.1823
1.1881
1.1938
1.1994
1.1459
1.1522
1.1584
1.1645
1.1705
1.1131
1.1198
1.1264
1.1330
1.1395
1.07795
1.08518
1.09230
1.09933
1.10626
1.04030
1.04804
1.05567
1.06320
1.07062
1.00000
1.00828
1.01644
1.02450
1.03246
.95692
.96576
.97449
.98311
.99161
V
V∗
T0∗
T0
isoth
1.14844
1.15283
1.15726
1.16172
1.16622
1.12700
1.13122
1.13547
1.13976
1.14408
1.10644
1.11048
1.11456
1.11867
1.12282
1.08675
1.09062
1.09452
1.09846
1.10243
1.06794
1.07163
1.07536
1.07912
1.08292
1.05000
1.05352
1.05707
1.06066
1.06428
1.03294
1.03628
1.03966
1.04307
1.04652
THERMODYNAMICS AND FLUID MECHANICS
113
.95798
.95571
.95341
.95108
.94873
.94636
.94397
.94157
.93915
.93671
.93425
.93178
.92931
.92683
.92434
.92184
.91933
.91682
.91431
.91179
.90928
.90676
.90424
.90172
.89920
.89669
.89418
.89167
.88917
.88668
.88419
.88170
.87922
.87675
.87429
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
1.51
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
1.61
1.62
1.63
1.64
T0∗
1.30
1.31
1.32
1.33
1.34
M
T0
.70173
.69680
.69190
.68703
.68219
.72680
.72173
.71669
.71168
.70669
.75250
.74731
.74215
.73701
.73189
.77875
.77346
.76819
.76294
.75771
.80540
.80004
.79469
.78936
.78405
.83227
.82698
.82151
.81613
.81076
.85917
.85380
.84843
.84305
.83766
T
T∗
Table 5 (continued) γ = 1.4
.52356
.51848
.51346
.50851
.50363
.55002
.54458
.53922
.53393
.52871
.57831
.57250
.56677
.56111
.55553
.60860
.60237
.59623
.59018
.58421
.64102
.63436
.62779
.62131
.61491
.67577
.66863
.66159
.65464
.64778
.71301
.70535
.69780
.69035
.68301
p
p∗
1.1756
1.1816
1.1877
1.1939
1.2002
1.1473
1.1527
1.1582
1.1639
1.1697
1.1215
1.1264
1.1315
1.1367
1.1420
1.0983
1.1028
1.1073
1.1120
1.1167
1.07765
1.08159
1.08563
1.08977
1.09400
1.05943
1.06288
1.06642
1.07006
1.07380
1.04365
1.04661
1.04967
1.05283
1.05608
p0∗
p0
1.3403
1.3439
1.3475
1.3511
1.3546
1.3214
1.3253
1.3291
1.3329
1.3366
1.3012
1.3054
1.3095
1.3135
1.3175
1.2796
1.2840
1.2884
1.2927
1.2970
1.2564
1.2612
1.2659
1.2705
1.2751
1.2316
1.2367
1.2417
1.2467
1.2516
1.2050
1.2105
1.2159
1.2212
1.2264
V
V∗
T0∗
T0
isoth
1.32300
1.32862
1.33427
1.33996
1.34568
1.29544
1.30088
1.30636
1.31187
1.31742
1.26875
1.27402
1.27932
1.28466
1.29003
1.24294
1.24803
1.25316
1.25832
1.26352
1.21800
1.22292
1.22787
1.23286
1.23788
1.19394
1.19868
1.20346
1.20827
1.21312
1.17075
1.17532
1.17992
1.18456
1.18923
1.95
1.96
1.97
1.98
1.99
1.90
1.91
1.92
1.93
1.94
1.85
1.86
1.87
1.88
1.89
1.80
1.81
1.82
1.83
1.84
1.75
1.76
1.77
1.78
1.79
1.70
1.71
1.72
1.73
1.74
1.65
1.66
1.67
1.68
1.69
M
.80359
.80152
.79946
.79742
.79540
.81414
.81200
.80987
.80776
.80567
.82504
.82283
.82064
.81846
.81629
.83628
.83400
.83174
.82949
.82726
.84785
.84551
.84318
.84087
.83857
.85970
.85731
.85493
.85256
.85020
.87184
.86940
.86696
.86453
.86211
T0∗
T0
.54774
.54391
.54012
.53636
.53263
.56734
.56336
.55941
.55549
.55160
.58773
.58359
.57948
.57540
.57135
.60894
.60463
.60036
.59612
.59191
.63096
.62649
.62205
.61765
.61328
.65377
.64914
.64455
.63999
.63546
.67738
.67259
.66784
.66312
.65843
T
T∗
Table 5 (continued) γ = 1.4
.37954
.37628
.37306
.36988
.36674
.39643
.39297
.38955
.38617
.38283
.41440
.41072
.40708
.40349
.39994
.43353
.42960
.42573
.42191
.41813
.45390
.44972
.44559
.44152
.43750
.47563
.47117
.46677
.46242
.45813
.49881
.49405
.48935
.48471
.48014
p
p∗
1.4516
1.4616
1.4718
1.4821
1.4925
1.4033
1.4127
1.4222
1.4319
1.4417
1.3581
1.3669
1.3758
1.3848
1.3940
1.3159
1.3241
1.3324
1.3408
1.3494
1.2767
1.2843
1.2920
1.2998
1.3078
1.2402
1.2473
1.2545
1.2618
1.2692
1.2066
1.2131
1.2197
1.2264
1.2332
p0∗
p0
1.4432
1.4455
1.4478
1.4501
1.4523
1.4311
1.4336
1.4360
1.4384
1.4408
1.4183
1.4209
1.4235
1.4261
1.4286
1.4046
1.4074
1.4102
1.4129
1.4156
1.3901
1.3931
1.3960
1.3989
1.4018
1.3745
1.3777
1.3809
1.3840
1.3871
1.3580
1.3614
1.3648
1.3681
1.3713
V
V∗
T0∗
T0
isoth
1.54044
1.54728
1.55416
1.56107
1.56802
1.50675
1.51342
1.52012
1.52686
1.53363
1.47394
1.48043
1.48696
1.49352
1.50012
1.44200
1.44832
1.45467
1.46106
1.46748
1.41094
1.41708
1.42326
1.42947
1.43572
1.38075
1.38672
1.39272
1.39876
1.40483
1.35144
1.35723
1.36306
1.36892
1.37482
114
ENGINEERING TABLES AND DATA
.79339
.79139
.78941
.78744
.78549
.78355
.78162
.77971
.77781
.77593
.77406
.77221
.77037
.76854
.76673
.76493
.76314
.76137
.75961
.75787
.75614
.75442
.75271
.75102
.74934
.74767
.74602
.74438
.74275
.74114
.73954
.73795
.73638
.73482
.73327
2.05
2.06
2.07
2.08
2.09
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.30
2.31
2.32
2.33
2.34
T0∗
2.00
2.01
2.02
2.03
2.04
M
T0
.43122
.42837
.42555
.42276
.41999
.44582
.44285
.43990
.43698
.43409
.46106
.45796
.45489
.45184
.44882
.47696
.47373
.47052
.46734
.46419
.49356
.49018
.48683
.48351
.48022
.51087
.50735
.50386
.50040
.49697
.52893
.52526
.52161
.51800
.51442
T
T∗
Table 5 (continued) γ = 1.4
.28551
.28333
.28118
.27905
.27695
.29675
.29445
.29218
.28993
.28771
.30864
.30621
.30381
.30143
.29908
.32122
.31864
.31610
.31359
.31110
.33454
.33181
.32912
.32646
.32383
.34866
.34577
.34291
.34009
.33730
.36364
.36057
.35754
.35454
.35158
p
p∗
1.8860
1.9012
1.9165
1.9320
1.9476
1.8128
1.8271
1.8416
1.8562
1.8710
1.7434
1.7570
1.7707
1.7846
1.7986
1.6780
1.6908
1.7037
1.7168
1.7300
1.6161
1.6282
1.6404
1.6528
1.6653
1.5579
1.5693
1.5808
1.5924
1.6042
1.5031
1.5138
1.5246
1.5356
1.5467
p0∗
p0
1.5104
1.5119
1.5134
1.5150
1.5165
1.5024
1.5040
1.5056
1.5072
1.5088
1.4939
1.4956
1.4973
1.4990
1.5007
1.4849
1.4867
1.4885
1.4903
1.4921
1.4753
1.4773
1.4792
1.4811
1.4830
1.4652
1.4673
1.4694
1.4714
1.4734
1.4545
1.4567
1.4589
1.4610
1.4631
V
V∗
T0∗
T0
isoth
1.80075
1.80882
1.81692
1.82506
1.83323
1.76094
1.76883
1.77676
1.78472
1.79272
1.72200
1.72972
1.73747
1.74526
1.75308
1.68394
1.69148
1.69906
1.70667
1.71432
1.64675
1.65412
1.66152
1.66896
1.67643
1.61044
1.61763
1.62486
1.63212
1.63942
1.57500
1.58202
1.58907
1.59616
1.60328
2.65
2.66
2.67
2.68
2.69
2.60
2.61
2.62
2.63
2.64
2.55
2.56
2.57
2.58
2.59
2.50
2.51
2.52
2.53
2.54
2.45
2.46
2.47
2.48
2.49
2.40
2.41
2.42
2.43
2.44
2.35
2.36
2.37
2.38
2.39
M
.69084
.68964
.68845
.68727
.68610
.69699
.69574
.69450
.69327
.69205
.70340
.70210
.70081
.69953
.69825
.71005
.70870
.70736
.70603
.70471
.71700
.71559
.71419
.71280
.71142
.72421
.72274
.72129
.71985
.71842
.73173
.73020
.72868
.72718
.72569
T0∗
T0
.34478
.34267
.34057
.33849
.33643
.35561
.35341
.35123
.34906
.34691
.36691
.36461
.36233
.36007
.35783
.37870
.37630
.37392
.37157
.36923
.39100
.38850
.38602
.38356
.38112
.40383
.40122
.39863
.39606
.39352
.41724
.41451
.41181
.40913
.40647
T
T∗
Table 5 (continued) γ = 1.4
.22158
.22007
.21857
.21709
.21562
.22936
.22777
.22620
.22464
.22310
.23754
.23587
.23422
.23258
.23096
.24616
.24440
.24266
.24094
.23923
.25523
.25337
.25153
.24972
.24793
.26478
.26283
.26090
.25899
.25710
.27487
.27281
.27077
.26875
.26675
p
p∗
2.5233
2.5451
2.5671
2.5892
2.6116
2.4177
2.4384
2.4593
2.4804
2.5017
2.3173
2.3370
2.3569
2.3770
2.3972
2.2218
2.2405
2.2594
2.2785
2.2978
2.1311
2.1489
2.1669
2.1850
2.2033
2.0450
2.0619
2.0789
2.0961
2.1135
1.9634
1.9794
1.9955
2.0118
2.0283
p0∗
p0
1.5560
1.5571
1.5582
1.5593
1.5603
1.5505
1.5516
1.5527
1.5538
1.5549
1.5446
1.5458
1.5470
1.5482
1.5494
1.5385
1.5398
1.5410
1.5422
1.5434
1.5320
1.5333
1.5346
1.5359
1.5372
1.5252
1.5266
1.5279
1.5293
1.5306
1.5180
1.5195
1.5209
1.5223
1.5237
V
V∗
T0∗
T0
isoth
2.10394
2.11323
2.12256
2.13192
2.14132
2.05800
2.06711
2.07627
2.08546
2.09468
2.01294
2.02188
2.03086
2.03987
2.04892
1.96875
1.97752
1.98632
1.99515
2.00403
1.92544
1.93403
1.94266
1.95132
1.96002
1.88300
1.89142
1.89987
1.90836
1.91688
1.84144
1.84968
1.85796
1.86627
1.87462
THERMODYNAMICS AND FLUID MECHANICS
115
.68494
.68378
.68263
.68150
.68038
.67926
.67815
.67704
.67595
.67487
.67380
.67273
.67167
.67062
.66958
.66855
.66752
.66650
.66549
.66449
2.75
2.76
2.77
2.78
2.79
2.80
2.81
2.82
2.83
2.84
2.85
2.86
2.87
2.88
2.89
T0∗
2.70
2.71
2.72
2.73
2.74
M
T0
.30568
.30389
.30211
.30035
.29860
.31486
.31299
.31114
.30931
.30749
.32442
.32248
.32055
.31864
.31674
.33439
.33236
.33035
.32836
.32638
T
T∗
Table 5 (continued) γ = 1.4
.19399
.19274
.19151
.19029
.18908
.20040
.19909
.19780
.19652
.19525
.20712
.20575
.20439
.20305
.20172
.21417
.21273
.21131
.20990
.20850
p
p∗
3.0013
3.0277
3.0544
3.0813
3.1084
2.8731
2.8982
2.9236
2.9493
2.9752
2.7508
2.7748
2.7990
2.8235
2.8482
2.6342
2.6571
2.6802
2.7035
2.7270
p0∗
p0
1.5757
1.5766
1.5775
1.5784
1.5792
1.5711
1.5721
1.5730
1.5739
1.5748
1.5663
1.5673
1.5683
1.5692
1.5702
1.5613
1.5623
1.5633
1.5644
1.5654
V
V∗
T0∗
T0
isoth
2.29644
2.30643
2.31646
2.32652
2.33662
2.24700
2.25682
2.26667
2.27655
2.28648
2.19844
2.20808
2.21776
2.22747
2.23722
2.15075
2.16022
2.16972
2.17925
2.18883
.53633
.52437
.51646
.51098
.50702
.48980
∞
.65398
.61580
.58909
.56983
.55555
.65865
.65770
.65676
.65583
.65490
.66350
.66252
.66154
.66057
.65961
T0∗
6.00
7.00
8.00
9.00
10.00
3.00
3.50
4.00
4.50
5.00
2.95
2.96
2.97
2.98
2.99
2.90
2.91
2.92
2.93
2.94
M
T0
0
.07849
.05826
.04491
.03565
.02897
.28028
.21419
.16831
.13540
.11111
.28841
.28676
.28512
.28349
.28188
.29687
.29515
.29344
.29175
.29007
T
T∗
Table 5 (continued) γ = 1.4
0
.04669
.03448
.02649
.02098
.01702
.17647
.13223
.10256
.08177
.06667
.18205
.18091
.17978
.17867
.17757
.18788
.18669
.18551
.18435
.18320
p
p∗
∞
38.946
75.414
136.62
233.88
381.62
3.4244
5.3280
8.2268
12.502
18.634
3.2768
3.3058
3.3351
3.3646
3.3944
3.1358
3.1635
3.1914
3.2196
3.2481
p0∗
p0
1.7143
1.6809
1.6896
1.6954
1.6993
1.7021
1.5882
1.6198
1.6410
1.6559
1.6667
1.5843
1.5851
1.5859
1.5867
1.5875
1.5801
1.5809
1.5818
1.5826
1.5834
V
V∗
T0∗
T0
isoth
∞
7.17499
9.45000
12.07500
15.05003
18.37500
2.45000
3.01875
3.67500
4.41874
5.25000
2.39794
2.40828
2.41865
2.42907
2.43952
2.34675
2.35692
2.36712
2.37735
2.38763
116
ENGINEERING TABLES AND DATA
.2215
.3030
.3889
.4756
.5602
.6400
.7132
.7785
.8352
.8828
.9216
.9518
.9740
.9890
.9974
1.0000
.9976
.9910
.9807
.9675
.9518
.9342
.9151
.8948
.8737
.8521
.8301
.8080
.7859
.7639
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
T∗
T
.25
.30
.35
.40
.45
=
0
.00995
.03921
.08608
.14793
T0∗
T0
0
.05
.10
.15
.20
M
1.0997
1.0834
1.0679
1.0534
1.0402
1.0285
1.0186
1.0107
1.0048
1.0012
1.600
1.536
1.471
1.406
1.342
1.280
1.220
1.161
1.105
1.0512
.6154
.5878
.5618
.5373
.5141
.7805
.7435
.7086
.6757
.6447
1.150
1.186
1.226
1.271
1.323
1.0340
1.0498
1.0690
1.0919
1.1187
1.0000
1.0013
1.0052
1.0118
1.0214
1.178
1.164
1.149
1.133
1.116
1.882
1.835
1.782
1.724
1.663
1.0000
.9512
.9049
.8611
.8197
1.213
1.212
1.207
1.200
1.190
p0∗
p0
2.000
1.995
1.980
1.956
1.923
p
p∗
Table 5 (continued) γ = 1.0
1.384
1.412
1.438
1.463
1.486
1.220
1.257
1.291
1.324
1.355
1.0000
1.0488
1.0951
1.1389
1.1802
.7200
.7805
.8389
.8950
.9488
.4000
.4645
.5294
.5940
.6577
.1176
.1651
.2183
.2758
.3368
0
.00499
.01980
.04401
.07692
V
V∗
=
T∗
T
.10519
.07840
.06059
.04818
.03921
0
∞
.3600
.2791
.2215
.1794
.1479
6.00
7.00
8.00
9.00
10.00
3.00
3.50
4.00
4.50
5.00
.4126
.4013
.3904
.3799
.3698
.4757
.4621
.4490
.4364
.4243
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
.5510
.5348
.5192
.5042
.4897
.6400
.6211
.6027
.5849
.5677
.7422
.7209
.7000
.6795
.6595
T0∗
T0
2.25
2.30
2.35
2.40
2.45
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
0
.05405
.04000
.03077
.02439
.01980
.2000
.1509
.1176
.09412
.07692
.2336
.2262
.2192
.2125
.2061
.2759
.2666
.2577
.2493
.2413
.3299
.3179
.3066
.2959
.2857
.4000
.3844
.3697
.3557
.3425
.4923
.4717
.4522
.4338
.4164
p
p∗
Table 5 (continued) γ = 1.0
∞
215×104
106×107
147×1010
574×1013
623×1017
10.92
41.85
212.71
1425
12519
6.215
6.916
7.719
8.640
9.699
3.808
4.175
4.591
5.064
5.602
2.515
2.716
2.942
3.197
3.484
1.793
1.907
2.034
2.176
2.336
1.381
1.446
1.519
1.600
1.691
p0∗
p0
2.000
1.946
1.960
1.969
1.976
1.980
1.800
1.849
1.882
1.906
1.923
1.766
1.774
1.781
1.787
1.794
1.724
1.733
1.742
1.751
1.759
1.670
1.682
1.693
1.704
1.714
1.601
1.616
1.630
1.644
1.657
1.508
1.528
1.547
1.566
1.584
V
V∗
THERMODYNAMICS AND FLUID MECHANICS
117
0
.01044
.04111
.09009
.15444
.2305
.3144
.4020
.4898
.5746
.6540
.7261
.7898
.8446
.8902
.9270
.9554
.9761
.9899
.9976
1.0000
.9979
.9919
.9827
.9710
.9572
.9418
.9251
.9074
.8892
.8706
.8518
.8329
.8141
.7955
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
T0∗
0
.05
.10
.15
.20
M
T0
.8217
.7984
.7753
.7524
.7298
.9322
.9118
.8902
.8678
.8449
1.0000
.9930
.9821
.9679
.9511
.9467
.9720
.9892
.9989
1.0023
.6782
.7510
.8147
.8684
.9123
.2413
.3286
.4195
.5102
.5973
0
.01097
.04315
.09449
.16184
T
T∗
Table 5 (continued) γ = 1.1
1.1040
1.0867
1.0702
1.0550
1.0412
1.0291
1.0189
1.0109
1.0050
1.0013
1.647
1.576
1.504
1.434
1.365
1.297
1.232
1.170
1.111
1.0538
.6043
.5765
.5503
.5257
.5025
.7724
.7345
.6989
.6654
.6339
1.141
1.173
1.210
1.251
1.297
1.0331
1.0483
1.0665
1.0880
1.1130
1.0000
1.0013
1.0051
1.0116
1.0209
1.190
1.174
1.157
1.140
1.122
1.965
1.911
1.851
1.786
1.717
1.0000
.9490
.9009
.8555
.8127
1.228
1.226
1.221
1.213
1.203
p0∗
p0
2.100
2.094
2.077
2.049
2.011
p
p∗
1.360
1.385
1.409
1.431
1.452
1.207
1.241
1.273
1.304
1.333
1.0000
1.0463
1.0901
1.1314
1.1703
.7297
.7887
.8453
.8995
.9511
.4118
.4766
.5416
.6057
.6686
.1228
.1720
.2267
.2857
.3478
0
.00524
.02077
.04611
.08046
V
V∗
isoth
.98342
.98713
.99108
.99526
.99968
.96848
.97099
.97374
.97673
.97996
.95951
.96083
.96238
.96417
.96621
.95652
.95664
.95700
.95760
.95843
T0∗
T0
1.06413
1.07142
1.07896
1.08673
1.09474
1.03125
1.03735
1.04368
1.05026
1.05708
1.00435
1.00925
1.01439
1.01977
1.02539
.6914
.6756
.6603
.6456
.6313
.6175
.6042
.5914
.5790
.5671
.5556
.5445
.5338
.5235
.5136
.5041
.4949
.4860
.4775
.4693
.4613
.3960
.3496
.3160
.2909
.2568
.2356
.2215
.2116
.2045
.1736
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.7771
.7591
.7415
.7243
.7076
T0∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
T0
0
.09631
.07169
.05536
.04400
.03579
.3341
.2578
.2040
.1648
.1357
.3840
.3733
.3629
.3529
.3433
.4444
.4314
.4189
.4068
.3952
.5174
.5017
.4866
.4720
.4580
.6049
.5862
.5681
.5506
.5337
.7076
.6859
.6648
.6443
.6243
T
T∗
Table 5 (continued) γ = 1.1
0
.05172
.03825
.02941
.02331
.01892
.1927
.1451
.1129
.0902
.0737
.2253
.2182
.2114
.2049
.1986
.2667
.2576
.2489
.2406
.2328
.3197
.3079
.2968
.2863
.2763
.3889
.3735
.3589
.3451
.3321
.4807
.4601
.4407
.4224
.4052
p
p∗
∞
2508
18430
123×103
743×103
4
401×10
6.710
16.26
42.42
115.70
322.33
4.487
4.851
5.251
5.692
6.176
3.104
3.332
3.581
3.855
4.156
2.237
2.380
2.537
2.709
2.897
1.689
1.780
1.879
1.987
2.106
1.347
1.403
1.465
1.532
1.607
p0∗
p0
1.909
1.862
1.874
1.882
1.888
1.892
1.734
1.777
1.806
1.827
1.842
1.704
1.711
1.717
1.723
1.729
1.667
1.675
1.683
1.690
1.697
1.618
1.629
1.639
1.649
1.658
1.556
1.570
1.583
1.595
1.607
1.472
1.491
1.508
1.525
1.541
V
V∗
T0∗
T0
isoth
∞
2.67826
3.30000
4.01739
4.83044
5.73913
1.38696
1.54239
1.72174
1.92500
2.15217
1.31821
1.33148
1.34499
1.35874
1.37273
1.25543
1.26751
1.27983
1.29238
1.30517
1.19864
1.20952
1.22064
1.23200
1.24360
1.14783
1.15751
1.16743
1.17760
1.18800
1.10299
1.11148
1.12021
1.21917
1.13838
118
ENGINEERING TABLES AND DATA
0
.01094
.04301
.09408
.16089
.2395
.3255
.4147
.5034
.5884
.6672
.7381
.8003
.8531
.8969
.9318
.9585
.9779
.9907
.9978
1.0000
.9981
.9927
.9845
.9740
.9617
.9481
.9334
.9180
.9021
.8859
.8695
.8532
.8370
.8211
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
T0∗
0
.05
.10
.15
.20
M
T0
.7955
.7712
.7473
.7237
.7007
.9149
.8921
.8685
.8443
.8199
1.0000
.9888
.9741
.9564
.9365
.9704
.9910
1.0032
1.0081
1.0067
.7160
.7881
.8497
.9004
.9405
.2618
.3548
.4507
.5450
.6343
0
.01203
.04726
.10325
.17627
T
T∗
Table 5 (continued) γ = 1.2
1.1078
1.0895
1.0722
1.0563
1.0420
1.0296
1.0191
1.0109
1.0049
1.0012
1.692
1.614
1.536
1.460
1.385
1.313
1.244
1.178
1.115
1.0562
.5946
.5666
.5403
.5156
.4924
.7653
.7266
.6903
.6563
.6245
1.134
1.164
1.197
1.234
1.275
1.0322
1.0467
1.0640
1.0843
1.1077
1.0000
1.0013
1.0050
1.0114
1.0204
1.199
1.183
1.165
1.146
1.127
2.047
1.986
1.918
1.846
1.770
1.0000
.9471
.8972
.8504
.8065
1.242
1.239
1.234
1.226
1.214
p0∗
p0
2.200
2.193
2.173
2.141
2.099
p
p∗
1.338
1.361
1.383
1.404
1.423
1.196
1.228
1.258
1.286
1.313
1.0000
1.0441
1.0856
1.1247
1.1613
.7388
.7964
.8514
.9037
.9532
.4231
.4884
.5531
.6168
.6788
.1279
.1787
.2350
.2953
.3584
0
.00548
.02174
.04820
.08397
V
V∗
isoth
.94615
.95100
.95631
.96208
.96831
.92885
.93138
.93438
.93785
.94177
.92308
.92331
.92400
.92515
.92677
T0∗
T0
1.13077
1.14485
1.15938
1.17438
1.18985
1.06731
1.07908
1.09131
1.10400
1.11715
1.01538
1.02485
1.03477
1.04515
1.05600
.97500
.98215
.98977
.99785
1.00638
.7325
.7192
.7063
.6939
.6819
.6703
.6591
.6484
.6381
.6281
.6185
.6093
.6004
.5918
.5836
.5757
.5681
.5608
.5537
.5469
.5404
.4865
.4486
.4211
.4006
.3730
.3557
.3443
.3363
.3306
.3056
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.8054
.7900
.7750
.7604
.7462
T0∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
T0
0
.08919
.06632
.05118
.04065
.03306
.3128
.2405
.1898
.1531
.1259
.3606
.3503
.3404
.3309
.3217
.4187
.4062
.3941
.3825
.3713
.4895
.4742
.4595
.4453
.4317
.5755
.5570
.5391
.5219
.5054
.6782
.6563
.6351
.6146
.5947
T
T∗
Table 5 (continued) γ = 1.2
0
.04977
.03679
.02828
.02240
.01818
.1864
.1401
.1089
.08696
.07097
.2184
.2114
.2047
.1983
.1923
.2588
.2499
.2414
.2334
.2257
.3109
.2994
.2884
.2780
.2682
.3793
.3641
.3497
.3360
.3231
.4706
.4501
.4308
.4126
.3955
p
p∗
∞
266.2
875.9
2621
7181
18182
4.951
9.597
18.99
37.61
73.64
3.617
3.847
4.094
4.359
4.644
2.690
2.849
3.021
3.205
3.403
2.050
2.159
2.277
2.405
2.542
1.612
1.687
1.767
1.854
1.948
1.320
1.369
1.422
1.480
1.543
p0∗
p0
1.833
1.792
1.803
1.810
1.815
1.818
1.678
1.717
1.743
1.761
1.774
1.651
1.657
1.663
1.668
1.673
1.618
1.625
1.632
1.639
1.645
1.574
1.584
1.593
1.602
1.610
1.517
1.530
1.542
1.553
1.564
1.441
1.458
1.474
1.490
1.504
V
V∗
T0∗
T0
isoth
∞
4.24615
5.44615
6.83077
8.40002
10.15385
1.75385
2.05385
2.40000
2.79230
3.23077
1.62115
1.64677
1.67285
1.69938
1.72638
1.50000
1.52331
1.54708
1.57131
1.59600
1.39039
1.41139
1.43285
1.45477
1.47715
1.29231
1.31100
1.33015
1.34977
1.36985
1.20577
1.22215
1.23900
1.25631
1.27408
THERMODYNAMICS AND FLUID MECHANICS
119
0
.01143
.04489
.09803
.16726
.2482
.3363
.4270
.5165
.6015
.6796
.7494
.8099
.8611
.9029
.9361
.9614
.9795
.9914
.9980
1.0000
.9982
.9933
.9859
.9765
.9656
.9534
.9404
.9268
.9128
.8986
.8843
.8701
.8560
.8421
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
T0∗
0
.05
.10
.15
.20
M
T0
.7726
.7475
.7230
.6990
.6756
.8996
.8747
.8493
.8237
.7980
1.0000
.9851
.9669
.9461
.9235
.9928
1.0088
1.0163
1.0166
1.0108
.7533
.8244
.8837
.9312
.9673
.2828
.3816
.4822
.5800
.6713
0
.01314
.05155
.11236
.19120
T
T∗
Table 5 (continued) γ = 1.3
1.1112
1.0919
1.0739
1.0574
1.0426
1.0299
1.0193
1.0109
1.0049
1.0012
1.736
1.651
1.567
1.485
1.405
1.328
1.255
1.186
1.120
1.0583
.5860
.5578
.5314
.5067
.4835
.7588
.7194
.6826
.6483
.6161
1.128
1.155
1.185
1.219
1.256
1.0312
1.0451
1.0617
1.0809
1.1028
1.0000
1.0012
1.0049
1.0111
1.0199
1.209
1.191
1.172
1.152
1.131
2.127
2.059
1.984
1.904
1.821
1.0000
.9452
.8939
.8458
.8008
1.255
1.253
1.247
1.237
1.224
p0∗
p0
2.300
2.293
2.270
2.234
2.186
p
p∗
1.318
1.340
1.360
1.379
1.397
1.186
1.216
1.244
1.270
1.295
1.0000
1.0421
1.0816
1.1186
1.1532
.7473
.8035
.8569
.9075
.9552
.4340
.4994
.5640
.6272
.6885
.1329
.1853
.2430
.3046
.3687
0
.00573
.02270
.05028
.08745
V
V∗
isoth
.93017
.93723
.94497
.95337
.96245
.90496
.90866
.91303
.91807
.92378
.89655
.89689
.89790
.89958
.90193
T0∗
T0
1.19914
1.21965
1.24083
1.26268
1.28521
1.10668
1.12383
1.14165
1.16014
1.17930
1.03103
1.04482
1.05928
1.07441
1.09021
.97220
.98262
.99372
1.00548
1.01792
.7659
.7545
.7435
.7329
.7227
.7129
.7034
.6943
.6855
.6771
.6690
.6612
.6537
.6465
.6396
.6329
.6265
.6203
.6144
.6087
.6032
.5582
.5265
.5037
.4867
.4639
.4496
.4402
.4336
.4289
.4083
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.8285
.8153
.8024
.7898
.7776
T0∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
T0
0
.08335
.06192
.04775
.03792
.03082
.2952
.2262
.1781
.1435
.1178
.3410
.3311
.3216
.3124
.3036
.3971
.3850
.3733
.3621
.3513
.4659
.4510
.4367
.4229
.4097
.5505
.5322
.5146
.4977
.4815
.6529
.6309
.6097
.5892
.5695
T
T∗
Table 5 (continued) γ = 1.3
0
.04812
.03555
.02732
.02164
.01756
.1811
.1359
.1055
.08417
.06866
.2123
.2055
.1990
.1928
.1868
.2521
.2433
.2350
.2271
.2195
.3034
.2920
.2812
.2710
.2613
.3710
.3559
.3416
.3281
.3154
.4617
.4413
.4221
.4040
.3870
p
p∗
∞
76.97
191.3
413.4
833.4
1582
4.007
6.806
11.57
19.44
32.06
3.096
3.258
3.429
3.611
3.804
2.416
2.536
2.664
2.800
2.944
1.915
2.003
2.097
2.197
2.303
1.552
1.615
1.683
1.755
1.832
1.296
1.340
1.387
1.438
1.493
p0∗
p0
1.769
1.732
1.742
1.748
1.753
1.756
1.630
1.665
1.688
1.704
1.716
1.606
1.611
1.616
1.621
1.626
1.575
1.582
1.588
1.594
1.600
1.536
1.545
1.553
1.561
1.568
1.484
1.495
1.506
1.517
1.527
1.414
1.430
1.445
1.459
1.472
V
V∗
T0∗
T0
isoth
∞
5.73793
7.48621
9.50345
11.78968
14.34483
2.10690
2.54397
3.04828
3.61982
4.25862
1.91358
1.95090
1.98889
2.02755
2.06689
1.73707
1.77103
1.80565
1.84096
1.87693
1.57737
1.60797
1.63923
1.67117
1.70378
1.43448
1.46172
1.48962
1.51820
1.54745
1.30841
1.33227
1.35682
1.38204
1.40792
120
ENGINEERING TABLES AND DATA
0
.01325
.05183
.11243
.19020
.2794
.3742
.4693
.5606
.6448
.7201
.7853
.8402
.8853
.9213
.9491
.9697
.9842
.9935
.9985
1.0000
.9987
.9952
.9899
.9833
.9757
.9674
.9586
.9495
.9403
.9310
.9217
.9125
.9035
.8947
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
T0∗
0
.05
.10
.15
.20
M
T0
1.377
1.291
1.210
1.135
1.0649
1.0662
1.0660
1.0578
1.0432
1.0235
.7087
.6818
.6559
.6309
.6069
.8550
.8246
.7946
.7652
.7365
1.0000
.9736
.9454
.9158
.8855
.5612
.5327
.5062
.4814
.4583
.7397
.6985
.6603
.6249
.5919
1.0000
.9398
.8839
.8321
.7842
1.1202
1.0981
1.0778
1.0597
1.0438
1.884
1.774
1.667
1.565
1.468
.8870
.9519
1.0010
1.0354
1.0565
1.108
1.130
1.154
1.179
1.206
1.0280
1.0400
1.0540
1.0700
1.0880
1.0000
1.0012
1.0046
1.0103
1.0181
1.0303
1.0193
1.0108
1.0048
1.0012
1.239
1.216
1.192
1.168
1.144
.3653
.4849
.6018
.7103
.8062
2.418
2.321
2.216
2.107
1.995
p0∗
p0
1.299
1.297
1.289
1.276
1.259
p
p∗
2.670
2.659
2.626
2.573
2.503
0
.01767
.06896
.1490
.2506
T
T∗
Table 5 (continued) γ = 1.67
1.263
1.280
1.296
1.311
1.324
1.156
1.181
1.204
1.225
1.245
1.0000
1.0361
1.0695
1.1005
1.1292
.7744
.8260
.8742
.9192
.9611
.4709
.5366
.6003
.6614
.7195
.1511
.2089
.2715
.3371
.4040
0
.00665
.02626
.05790
.10011
V
V∗
isoth
.90267
.91732
.93337
.95081
.96964
.85036
.85803
.86710
.87756
.88942
.83292
.83362
.83571
.83920
.84408
T0∗
T0
1.46073
1.50328
1.54723
1.59257
1.63931
1.26890
1.30447
1.34145
1.37981
1.41957
1.11195
1.14055
1.17054
1.20193
1.23472
.98987
1.01150
1.03452
1.05893
1.08474
.8474
.8405
.8338
.8274
.8213
.8154
.8097
.8043
.7991
.7941
.7893
.7847
.7803
.7761
.7721
.7682
.7644
.7608
.7574
.7541
.7509
.7251
.7072
.6943
.6848
.6721
.6642
.6590
.6553
.6528
.6414
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.50
4.00
4.50
5.00
6.00
7.00
8.00
9.00
10.00
∞
.8862
.8779
.8699
.8621
.8546
T0∗
2.00
2.05
2.10
2.15
2.20
1.75
1.80
1.85
1.90
1.95
M
T0
0
.06870
.05092
.03920
.03110
.02526
.2497
.1897
.1484
.1191
.0975
.2902
.2814
.2730
.2649
.2571
.3406
.3296
.3191
.3090
.2994
.4038
.3899
.3767
.3641
.3521
.4835
.4660
.4493
.4334
.4183
.5840
.5620
.5410
.5209
.5018
T
T∗
Table 5 (continued) γ = 1.67
0
.04368
.03224
.02475
.01959
.01589
.1666
.1244
.09632
.07669
.06246
.1959
.1895
.1834
.1775
.1719
.2334
.2251
.2173
.2098
.2027
.2824
.2715
.2612
.2514
.2422
.3477
.3330
.3192
.3062
.2940
.4367
.4165
.3976
.3799
.3633
p
p∗
∞
12.86
19.44
28.07
39.05
52.66
2.587
3.521
4.716
6.213
8.044
2.216
2.287
2.360
2.435
2.512
1.897
1.956
2.018
2.082
2.148
1.628
1.678
1.729
1.783
1.839
1.408
1.448
1.490
1.534
1.580
1.235
1.266
1.299
1.334
1.370
p0∗
p0
1.599
1.573
1.580
1.584
1.587
1.589
1.499
1.524
1.541
1.553
1.561
1.481
1.485
1.489
1.493
1.496
1.459
1.464
1.469
1.473
1.477
1.430
1.436
1.442
1.448
1.454
1.391
1.400
1.408
1.415
1.423
1.337
1.349
1.360
1.371
1.381
V
V∗
T0∗
T0
isoth
∞
10.87789
14.50526
18.69067
23.43419
28.73566
3.34416
4.25100
5.29736
6.48321
7.80860
2.94307
3.02049
3.09932
3.17954
3.26115
2.57684
2.64729
2.71914
2.79239
2.86703
2.24550
2.30897
2.37385
2.44011
2.50778
1.94903
2.00553
2.06343
2.12273
2.18341
1.68744
1.73696
1.78789
1.84021
1.89392
THERMODYNAMICS AND FLUID MECHANICS
121
Shock-wave angle β degrees
0
5
1.3
1.35
10
1.4
1.45 1.5
15
1.6
1.7
20
1.8
1.9
M1= 2
25
M2 = 1
2.2
2.4
30
2.6
2.8
3
35
M1
3.2 3.4 3.6
4
Shock wave
4.5 5
6
20
β
45
8 10
δ
60
70
Deflection angle δ degrees
0
50
0
40
10
Streamline
20
10
20
30
1.25
30
1.2
40
1.15
80
40
1.1
50
1.05
90
50
60
70
80
Perfect gas γ = 1.4
8
90
122
ENGINEERING TABLES AND DATA
Oblique shocks: shock-wave angle versus flow deflection angle
p2 / p1
1
2
3
4
5
6
7
8
9
1
1.2
1.4
1.6
5
Shock wave
Perfect gas γ = 1.4
Streamline
M1 p1
rm
10
No
12
ck
14
ho
16
al
s
18
10
15
2
δ = 20 deg
Min M1 for const δ
1.8
δ
M2 p2
2.2
2
M = 1.0
25
2.4
M1
=
2
2.6
M
5
1.
30
2.8
M
=
2
0
2.
3
M
2
=
3.2
5
2.
35
2
3.4
M
=
0
3.
3.6
M
2
=
3.8
5
3.
4
1
2
3
4
5
6
7
8
9
10
12
14
16
18
THERMODYNAMICS AND FLUID MECHANICS
123
Oblique shocks: pressure ratio and downstream Mach number
124
Part 5
Solid mechanics and structures
ENGINEERING TABLES AND DATA
SOLID MECHANICS AND STRUCTURES
125
Notation
In the following, u, v , w represent small displacements in the x, y , z directions (or as stated); σ , represent direct stress and
strain (positive for tension), and τ , γ shear stress and strain; E , G , K , ν are Young’s modulus, shear modulus, bulk modulus and
Poisson’s ratio; l , m , n are direction cosines (components of a unit vector); ψ is a stress function; ρ is mass density. M , T are
moment and torque; I is second moment or product moment of area; ω is a rotation unless otherwise stated.
Two-dimensional stress and strain
Cartesian coordinates
Transformation of stress
Equilibrium equations
For axes Ox and Oy inclined at θ, anticlockwise, to axes Ox
and Oy
σx x =
σy
y
=
τx y =
σxx + σy y
σxx
2
+ σy y
σy y
2
− σxx
2
σxx − σy y
+
σxx
−
cos 2θ + τxy sin 2θ
2
− σy y
cos 2θ − τxy sin 2θ
2
sin 2θ + τxy cos 2θ .
σ
xx
− σy y 2
2
1
2
+
+
∂x
∂τy x
+
∂x
∂τxy
+ Fx = 0
∂y
∂σy y
+ Fy = 0
∂y
where Fx , Fy are body forces per unit volume.
Boundary conditions
) '
'
)'
)
Tx
σxx τxy
l
=
m
τy x σy y
Ty
Principal stresses
σxx + σy y
σmax
±
=
σmin
2
∂σxx
where l , m are the direction cosines of the outward normal,
and Tx , Ty are the surface tractions (forces per unit area) at
the boundary.
2
τxy
The principal directions are given by
tan 2θ =
Relations between strains and small displacements
2τxy
σxx − σy y
.
xx
Mohr’s circle of stress
∂u
;
∂x
=
2ωz =
Shear stress is taken as clockwise positive.
σyy
Y
τy' x'
τxy
X
τ
Y'
σxx
∂v
;
∂y
γxy =
∂u
∂v
+
∂x
∂y
∂u
∂v
−
∂x
∂y
Transformation of strain
σy' y '
τyx
=
yy
τx' y'
σx' x'
For axes Ox and Oy inclined at θ, anticlockwise, to axes Ox
and Oy
θ
X'
=
xx
+
xx
xx
y y
=
2
+
yy
2
γx y = (
Y (σyy ,τyx )
yy
yy
−
+
xx
−
−
xx
2
−
yy
yy
2
xx ) sin 2θ
cos 2θ +
cos 2θ −
γxy
2
γxy
2
sin 2θ
sin 2θ
+ γxy cos 2θ .
Principal strains
max
Y' (σy' y' ,τy' x' )
=
+
xx
yy
2
min
±
xx
−
yy
2
σ
0
2θ
X' (σx' x' ,-τx' y' )
The principal directions are given by
γxy
tan 2θ =
xx
−
.
yy
X (σxx ,-τxy )
Compatibility of strains
∂2
xx
∂y 2
+
∂2
yy
∂x 2
=
∂2 γxy
∂x∂y
2
+
γ
2 2
1
xy
2
126
ENGINEERING TABLES AND DATA
Hooke’s law
Stress function
Plane stress, σz z = 0
If gravity is the only body force, the stresses are
xx
yy
zz
σxx
σy y
τxy
1
σ − ν σy y
E xx
1
σ − ν σxx
=
E yy
ν
σ + σy y
=−
E xx
E
=
xx + ν
1 − ν2
E
=
yy + ν
1 − ν2
= G γxy
=
=
yy
=
σxx =
σy y =
σy y =
τxy
yy
∂y 2
− ρgy
2
∂ ψ
− ρgy
∂x 2
2
∂ ψ
=−
∂x∂y
and the compatibility equation is
4
∂ ψ
∂x 4
zz
σz z = ν σxx + σy y =
2
∂ ψ
xx
=0
1 − ν2
ν
σxx −
σy y
1−ν
E
2 1−ν
ν
σy y −
σxx
1−ν
E
/
E
(1 − ν ) xx + ν
(1 + ν ) (1 − 2ν )
/
E
(1 − ν ) y y + ν
(1 + ν ) (1 − 2ν )
Plane strain,
xx
σxx =
Eν
xx
+
+
4
2∂ ψ
∂x 2 ∂y 2
+
4
∂ ψ
∂y 4
=0
or
∇4 ψ = 0 .
0
yy
0
xx
yy
(1 + ν ) (1 − 2ν )
τxy = G γxy
Polar coordinates
Equilibrium equations
∂σr r
∂r
+
and the compatibility equation is
σr r − σθθ
1 ∂τr θ
+
+ Fr = 0
r ∂θ
r
∂τr θ
2τr θ
1 ∂σθθ
+
+
+ Fθ = 0
r ∂θ
r
∂r
2
1 ∂2
1 ∂
∂
+
+
∂r 2 r ∂r
r 2 ∂θ 2
2
∂ ψ
∂r 2
2
1 ∂ ψ
1 ∂ψ
+
+
r ∂r
r 2 ∂θ 2
= 0.
Rotating discs and cylinders
where Fr , Fθ are the body forces per unit volume.
Thick cylinder under uniform pressure
B
B
;
σθθ = A −
2
r
r2
where A, B are constants.
σr r = A +
If the angular velocity is ω, the body force is
Fr = ρω 2 r
and the stresses are given by
r σr r = ψ ;
σθθ =
∂ψ
+ ρω 2 r 2 .
∂r
Relations between strains and small displacements
rr
∂u
;
=
∂r
θθ
u 1 ∂v
;
= +
r
r ∂θ
γr θ
v
∂v
1 ∂u
− +
=
r
r ∂θ
∂r
where u, v are displacements in the radial and tangential
directions.
If the body forces are zero, the stresses are
2
τr θ
For symmetrical loading with an outward normal component p
per unit area, the hoop stress σθθ and meridional stress σφ φ
are related by
σφ φ
σθθ
p
+
=
rθ
rφ
t
Stress function
1 ∂ ψ
1 ∂ψ
+
;
r ∂r
r 2 ∂θ 2
∂ 1 ∂ψ
=−
∂r r ∂θ
σr r =
Thin shell of revolution
2
σθθ =
∂ ψ
∂r 2
where rθ and rφ are the corresponding radii of curvature and t
is the thickness.
SOLID MECHANICS AND STRUCTURES
127
Three-dimensional stress and strain
Cartesian coordinates
Transformation of stress
Transformation of strain
x
Original axes x, y , z , new
axes x , y , z with direction
cosines as shown.
x
y
z
l1
l2
l3
y
m1
m2
m3
z
n1
n2
n3
σx x = σxx l12 + σy y m12 + σz z n12
Original axes x, y , z , new
axes x , y , z with direction
cosines as shown.
=
xx
+ 2τxy l1 m1 + 2τy z m1 n1 + 2τz x n1 l1 etc.
τx y = σxx l1 l2 + σy y m1 m2 + σz z n1 n2
2
xx l1
+
yy
m12 +
x
y
z
zz
x
y
z
l1
l2
l3
m1
m2
m3
n1
n2
n3
n12
+ γxy l1 m1 + γy z m1 n1 + γz x n1 l1 etc.
γx y = 2
xx l1 l2
+2
yy
m1 m2 + 2
zz
n1 n2
+ τxy (l1 m2 + m1 l2 ) + τy z (m1 n2 + n1 m2 )
+ γxy (l1 m2 + m1 l2 ) + γy z (m1 n2 + n1 m2 )
+ τz x (n1 l2 + l1 n2 ) etc.
+ γz x (n1 l2 + l1 n2 ) etc.
Principal stresses
Principal strains
The direction cosines of the normal to a principal plane
subject to direct stress σ satisfy the equations
The direction cosines of the normal to a principal plane
subject to direct strain satisfy the equations
⎡
σxx − σ
⎣
τy x
τz x
τxy
σy y − σ
τz y
⎤⎡
⎤ ⎡
⎤
τxz
l
0
⎦⎣ m ⎦ = ⎣ 0 ⎦.
τy z
σz z − σ
n
0
By setting the determinant of the coefficients to zero a cubic in
σ is obtained, the roots of which are the principal stresses.
The corresponding direction cosines can be found from the
above and
l 2 + m2 + n2 = 1 .
∂x
∂τy x
+
∂x
∂τz x
∂x
+
+
∂τxy
∂y
∂σy y
∂y
∂τz y
∂y
+
+
+
∂τxz
∂z
∂τy z
∂z
∂σz z
∂z
−
⎣ γy x /2
γz x /2
⎤⎡
⎤ ⎡
⎤
γxz /2
l
0
γy z /2 ⎦ ⎣ m ⎦ = ⎣ 0 ⎦ .
n
0
zz −
γxy /2
yy −
γz y /2
xx
By setting the determinant of the coefficients to zero a cubic in
is obtained, the roots of which are the principal strains. The
corresponding direction cosines can be found from the above
and
l 2 + m2 + n2 = 1 .
Equilibrium equations
∂σxx
⎡
Compatibility of strains
+ Fx = 0
+ Fy = 0
∂2
xx
∂y 2
+
∂2
yy
∂x 2
∂2 γxy
∂x∂y
and two similar equations.
2
+ Fz = 0
=
∂
∂
2
=
∂y ∂z
∂x
xx
−
∂γy z
∂x
where Fx , Fy , Fz are body forces per unit volume.
and two similar equations.
Boundary conditions
⎤⎡
⎤ ⎡
⎤
⎡
l
Tx
σxx τxy τxz
⎣ τy x σy y τy z ⎦ ⎣ m ⎦ = ⎣ Ty ⎦
n
τ z x τ z y σz z
Tz
Hooke’s law
where l , m , n are the direction cosines of the outward normal,
and Tx , Ty , Tz are the surface tractions (forces per unit area)
at the boundary.
xx
+
∂γxz
∂y
+
∂γxy
∂z
2
1 1
σxx − ν σy y + σz z
etc.
E
2
1
E
νE
=
xx + y y + z z +
1+ν
(1 + ν ) (1 − 2ν )
=
σxx
xx
etc.
τxy = G γxy etc.
Relations between elastic constants
Relations between strains and small displacements
∂u
∂v
∂w
;
;
yy =
zz =
∂x
∂y
∂z
∂u
∂v
∂v
∂w
∂u ∂w
+
; γy z =
+
; γz x =
+
γxy =
∂x
∂y
∂y
∂z
∂z
∂x
∂v
∂u
∂w
∂u ∂w
∂v
−
; 2ωy =
−
; 2ωz =
−
2ωx =
∂y
∂z
∂z
∂x
∂x
∂y
xx
G=
=
E
;
2 (1 + ν )
K =
E
3 (1 − 2ν )
The Lamé constants are
λ=
2
νE
=K − G;
3
(1 + ν )(1 − 2ν )
μ=
E
=G.
2(1 + ν )
128
ENGINEERING TABLES AND DATA
Cylindrical coordinates
Spherical coordinates
Relations between strains and small displacements
Relations between strain and small displacements
rr
γr θ
γz r
∂u
u 1 ∂v
∂w
;
;
+
θθ =
zz =
r
r
∂r
∂θ
∂z
v
∂v
∂v
1 ∂w
1 ∂u
− +
;
γθz =
+
=
r
r ∂θ
r ∂θ
∂r
∂z
∂u
∂w
+
=
∂r
∂z
=
where u, v , w are displacements in the radial, tangential and
axial directions.
For displacements ur , uθ , uφ
rr
φφ
γr θ
γθφ
γφ r
ur
1 ∂uθ
;
+
θθ =
r ∂θ
r
∂r
uθ
ur
1 ∂uφ
+
=
cot θ +
r
r
r sin θ ∂φ
uθ
∂uθ
1 ∂ur
−
=
+
r
r ∂θ
∂r
uφ
1 ∂uφ
1 ∂uθ
−
=
cot θ +
r ∂θ
r
r sin θ ∂φ
∂uφ
uφ
1 ∂ur
+
−
=
.
r
r sin θ ∂φ
∂r
=
∂ur
z
P r,θ,φ
r
θ
O
φ
y
x
Bending of laterally loaded plates
For a plate in the x-y plane, on the assumption that σz z = 0,
σxx = −
σy y
Ez
1 − ν2
Ez
=−
1 − ν2
∂2 w
∂2 w
+ν
2
∂x
∂y 2
∂2 w
∂2 w
+ν
∂y 2
∂x 2
E z ∂2 w
τxy = −
1 + ν ∂x∂y
∂2 w
∂2 w
+ν
Mx = −D
∂x 2
∂y 2
∂2 w
∂2 w
My = −D
+ν
∂y 2
∂x 2
Mxy = −D (1 − ν )
∂2 w
∂x∂y
where for plate thickness t,
D=
Et
3
12 1 − ν 2
The differential equation for deflection is
4
p
∂4 w
∂4 w
∂ w
+2
+
=
4
2
2
4
D
∂x
∂x ∂y
∂y
where p is the load per unit area in the z direction.
Circular plates
,
2
1 ∂ w
1 ∂w
+
r ∂r
r 2 ∂θ 2
,
+
1 ∂2 w
1 ∂w
∂2 w
Mθ = −D
+
+ν
r ∂r
r 2 ∂θ 2
∂r 2
1 ∂w
1 ∂2 w
Mr θ = − (1 − ν ) D
−
r ∂r ∂θ r 2 ∂θ
+
Mr = −D
∂2 w
+ν
∂r 2
The differential equation for deflection is
∂2
1 ∂2
1 ∂
+
+
∂r 2 r ∂r
r 2 ∂θ 2
2
w =
p
D
.
which for radial symmetry becomes
1 d
r dr
'
)
p
d 1 d
dw
r
r
=
.
D
dr r dr
dr
SOLID MECHANICS AND STRUCTURES
129
Yield and failure criteria
Yield criteria
In the following σ1 , σ2 and σ3 are principal stresses and σY the
yield stress in uniaxial tension.
Von Mises
where γ is the free surface energy per unit area, or
3
σ =
2E ( γ + p)
πc
where p is an additional energy per unit area due to plastic
deformation at the crack.
(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 = 2σY2
For plane stress with σ3 = 0,
σ12 − σ1 σ2 + σ22 = σY2
Fatigue
In the following Nf is the number of cycles to failure and a, b
are constants. The relations are empirical and of varying
accuracy.
and in pure shear yielding occurs at the shear stress
τ = σY / 3 .
Tresca
If σ1 σ2 σ3 , then yielding occurs when
σ1 − σ3 = σY
Basquin’s law
If stress is always less than the yield value and has a
peak-to-peak variation σ about a zero mean, then
σ Nfa = constant .
and the shear stress is then σY /2.
If the mean stress is σm , the value of σ for a given Nf is
reduced by the factor 1 − |σm | /σUTS where σUTS is the
ultimate tensile strength of the material (Goodman’s rule).
Ultimate tensile strength
Coffin-Manson law
On the assumption that necking occurs at constant volume,
the point of instability (i.e. maximum load) is that point on the
curve of true stress against nominal strain n touched by the
tangent which intersects the axis at n = −1; this is
Considère’s construction. If true stress varies as the n th power
of true strain t this point occurs at t = n . True strain is given
by ln l /l0 when the length l has the unstrained value l0 .
If the peak stress exceeds the yield value and the plastic
strain has a peak-to-peak range
p , then
Griffith theory for cracks
A crack of half-length c in a material of Young’s modulus E will
lead to tensile failure at a stress
3
2γ E
σ =
πc
p
Nfb = constant .
Miner’s rule
If stress ranges σi which would cause failure in Ni cycles are
applied for ni cycles, then failure occurs when
n2
n1
+
+ ··· = 1.
N 1 N2
130
ENGINEERING TABLES AND DATA
Elastic behaviour of structural members
In this section the origin of right-handed axes x, y , z is at the centroid of a cross-section of area A and the member has length
L, aligned with the x-axis; unless otherwise stated, y and z are principal axes. Ixx is denoted Ix , etc. and I without subscript
refers to a principal axis about which the moment M acts. A bending moment at any cross-section is taken to be positive if a
positive couple acts on the positive x-face. A positive couple is one which acts in a right-handed sense about its axis. For
sections without circular symmetry Ix denotes an effective polar moment such that G Ix is the torsional rigidity.
Bending stress for straight beams
For any orientation of the y, z axes the stress due to elastic
bending is
My Iz + Mz Iy z z − Mz Iy + My Iy z y
σxx =
Iy Iz − Iy2z
δe
P L2
2E I
PL
3E I
5P L
48E I
pL3
6E I
pL4
8E I
17pL4
384E I
P L2
16E I
0
P L3
48E I
pL3
24E I
0
5pL4
384E I
0
0
P L3
192E I
0
0
pL
384E I
ML
EI
ML
2E I
P
δc
3
3
.
p
For bending about a principal axis z this becomes σ = −
My
.
I
P
y
p
z
θe
x
M
σ
P
Winkler theory for curved beams
The direct stress due to a moment M about the z -axis is
+
,
2
R0 y
M
1+
σxx = −
AR0
h 2 ( R0 + y )
p
4
where
h2 =
R0
A
M
y2
dA
R0 + y
and R0 is the original radius of curvature.
The curvature due to a displacement v (x) is
1
= R
d v /dx
1 + dv /dx
2
Shear stress in bending
Fy = −
2
2 3/2
≈
d v
dx 2
For any orientation of the y, z axes the curvatures due to
bending are
My Iz + Mz Iy z
1
=
.
Ry
E Iy Iz − Iy2z
dMz
dx
;
Fz =
dMy
dx
.
In a beam of open thin-walled section, for any orientation of
the y, z axes a shear force Fy produces at any point P a shear
stress given by
for small v .
Mz Iy + My Iy z
1
=
;
Rz
E Iy Iz − Iy2z
2
ML
8E I
On any cross-section the shear force is given by
Deflection of beams
2
2
τ =
Fy AP ( Iy ȳ − Iy z z̄ )
t( Iy Iz − Iy2z )
where AP is the area of the section beyond the point P, ȳ and
z̄ define the centroid of AP , and t is the thickness at point P.
When y is a principal axis this becomes
For bending about a principal axis the curvature is
M
1
=
.
R
EI
The magnitudes of the end slope θe , end deflection δe and
central deflection δc are given below for the loadings shown.
In each case the length is L.
τ =
Fy AP ȳ
tIz
and this expression applies also to a section of any shape if t
signifies the breadth at any y and if τ is assumed to be
constant over the breadth.
SOLID MECHANICS AND STRUCTURES
131
Torsion
In the following, θ is the angle of twist per unit length about
the x-axis.
and the torsional rigidity is
T
= G Ix
θ
General
Closed thin-walled sections
The displacements in a cross-section rotated by θx are
u = θφ ( y, z ) ;
v = − θxz ;
For any closed thin-walled section in torsion the shear flow is
constant and given by
w = θxy
where φ is a function which satisfies the equation
2
∂ φ
∂y 2
τt =
2
+
∂ φ
∂z 2
= 0.
∂ψ
;
∂z
T
2Ae
where Ae is the area enclosed and t the wall thickness.
The torsional rigidity is
The shear stresses are given by
τxy =
(or G J ) .
4G A2e
T
= θ
ds /t
∂ψ
∂y
τxz = −
where ψ is a stress function which satisfies the equation
2
∂ ψ
∂y 2
+
2
∂ ψ
∂z 2
where ds is an element of perimeter. The quantity
2 4Ae / ds /t is the effective polar moment of area, and except
for circular tubes differs from the actual polar moment Ix (or J ).
= −2G θ .
The torque is given by
T = 2 ψ dy dz .
Open thin sections
The torsional rigidity of a thin rectangular section of thickness
t and breadth b is
Circular sections
1
T
= bt 3 G .
3
θ
The shear stress at radius r is given by
The same expression can be applied to any thin open section
if b is taken to be the edge to edge perimeter.
τ
T
= Gθ
=
r
Ix
Flexibility matrix: one-dimensional member
For a cantilever the deflections which result from the loading W = Px · · · Mz shown are given by [F ] {W } where
[F ] =
⎡
Px
Py
Pz
Mx
My
Mz
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
L
EA
0
0
0
0
0
3
0
0
0
2
L
3E Iz
0
0
0
L3
3E Iy
0
L2
−
2E Iy
0
0
L
G Ix
0
0
0
L
E Iy
0
0
0
L
E Iz
0
2
0
0
0
L2
2E Iz
L
2E Iz
0
−
L
2E Iy
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
y
Py
My
x
Mz
z
Pz
Mx Px
132
ENGINEERING TABLES AND DATA
Stiffness matrix: one-dimensional member
y
v1
θy1
The loading to produce deflection δ = u1 · · · θz 2 as shown is given by
[K ] {δ} where
1
z
u1
v1
EA
L
0
w1
θx1
θy 1
θz 1
−
⎢
⎢
⎢
6E Iz
12E Iz
⎢
⎢
0
0
0
0
3
⎢
L
L2
⎢
12E Iy
6E Iy
⎢
⎢
0
0
0 −
0
⎢
L3
L2
⎢
⎢
G Ix
⎢
0
0
0
0
0
⎢
L
⎢
⎢
4E Iy
6E Iy
⎢
⎢
0
0
0
0
−
⎢
L
L2
⎢
⎢
4E Iz
6E Iz
0
0
0
0
[K ] = ⎢
⎢
2
L
L
⎢
⎢
⎢
⎢ − EA
0
0
0
0
0
⎢
L
⎢
⎢
6E Iz
12E Iz
⎢
0
0
0
−
0 −
⎢
⎢
L3
L2
⎢
12E Iy
6E Iy
⎢
⎢
0
0
−
0
0
⎢
L3
L2
⎢
⎢
G Ix
⎢
0
0
0
−
0
0
⎢
L
⎢
⎢
6E Iy
2E Iy
⎢
0
0
0
−
0
⎢
L
⎢
L2
⎢
6E Iz
2E Iz
⎣
0
0
0
0
L
L2
0
0
0
0
θy 2
u1
w2
θx2
θy 2
θz 2
EA
L
0
0
0
0
0
0
0
0
12E Iz
L3
0
0
0
0
0
0
0
−
EA
L
0
6E Iz
L2
0
12E Iz
L3
0
0
0
0
0
0
0
−
6E Iz
L2
x
w2
v2
−
θx2
θz2
w1
u2
0
u2
2
θx1
θz1
⎡
v2
−
12E Iy
0
L3
−
0
6E Iy
L2
G Ix
L
0
−
⎤
6E Iz
L2
6E Iy
0
L2
0
0
2E Iy
0
L
0
0
0
2E Iz
L
0
0
0
0
0
0
0
12E Iy
L3
0
G Ix
L
0
6E Iy
L2
0
0
0
6E Iy
L2
0
4E Iy
L
0
−
6E Iz
L2
0
0
0
4E Iz
L
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
For deflection in the x, y plane only, this becomes
⎡
u1
EA
L
v1
θz 1
⎢
0
0
⎢
⎢
⎢
⎢
6E Iz
12E Iz
⎢
0
⎢
L3
L2
⎢
⎢
⎢
6E Iz
4E Iz
⎢
[K ] = ⎢
0
2
⎢
L
L
⎢
⎢
⎢
⎢ − EA
0
0
⎢
L
⎢
⎢
⎢
6E Iz
12E Iz
⎢
−
0
−
⎢
3
⎢
L
L2
⎢
⎢
6E Iz
2E Iz
⎣
0
L
L2
u2
v2
θz 2
EA
−
L
0
0
0
−
0
−
EA
L
0
0
12E Iz
6E Iz
L3
L2
6E Iz
2E Iz
L
L2
0
12E Iz
L3
−
6E Iz
L2
0
−
6E Iz
L2
4E Iz
L
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
y
v1
1
θz 1
z
v2
u1
2
θz 2
u2
x
SOLID MECHANICS AND STRUCTURES
133
Slope-deflection relations
In the slope-deflection method, moments and deflections are taken to be positive if clockwise.
The slope-deflection equations for a uniform straight member AB of length L are
MAB =
2E I
L
3δ
2θA + θB −
+ MFA
L
MBA =
2E I
L
3δ
2θB + θA −
+ MFB
L
or
MAB
MBA
EI
=
L
'
4
2
2 −6
4 −6
⎧
⎫
) ⎨ θA ⎬ MFA
+
θB
⎭
⎩
MFB
δ/L
where δ is the deflection of B relative to A (positive if it produces a clockwise rotation of the chord AB) and MFA , MFB are the
fixed-end moments due to transverse loading, if any (see next section). The equations give the following end moments for the
cases shown with MFA = MFB = 0.
End A fixed
End A pinned
Symmetry
Skew symmetry
Sidesway
θ
MAB
MBA
-θ/2
θ
MAB
MBA
-θ
θ
MAB
MAB
MAB
MBA
θ
θ
MBA
δ
MBA
θA
θB
δ
MAB
MBA
0
θ
0
2E I θ
L
4E I θ
L
θ
2
θ
0
0
3E I θ
L
−θ
θ
0
θ
θ
0
0
0
δ
−
−
2E I θ
L
6E I θ
L
−
6E I δ
L2
2E I θ
L
6E I θ
L
−
6E I δ
L2
134
ENGINEERING TABLES AND DATA
Fixed end moments
For a clamped member, the moment reactions due to transverse loading are as shown.
If these diagrams are used in conjunction with the slope-deflection method (see previous section), note that MFA = −MA and
MFB = +MB .
Concentrated load P
MA =
P ab 2
;
L2
Uniformly distributed load p
MB =
A
P a2b
L2
P
2
MA = MB =
A
B
MA
B
p
MA
MB
a
pL
12
MB
L
b
General loading
If the loading would give, when applied to the same span with free ends, a bending-moment diagram of area X with centroid
distant a from A and b from B, then it produces moment reactions
MA =
2X (2b − a)
L2
MB =
;
2X (2a − b)
L2
.
Stability of struts
Euler critical loads
L
π 2E I
π 2E I
= 0.25
(2L) 2
L2
Pc =
π 2E I
L2
π 2E I
π 2E I
= 2.05
(0.70L) 2
L2
Energy methods
Rayleigh
L
EI
Pc =
2
d y
Timoshenko
2
dx 2
L dx
0
L 0
dy
dx
Pc =
2
dy
dx
2
dx
0
L
dx
0
2
y
dx
EI
π 2E I
π 2E I
=4
(L/2) 2
L2
SOLID MECHANICS AND STRUCTURES
135
Dimensions and properties of British Standard sections
In the following tables the symbols are not always consistent with those defined earlier. The centroid axes of a cross-section
which are parallel to the flange and the web are denoted by x and y respectively, and the principal pair, if different, by u and v .
The elastic modulus of a section is the ratio of the moment at which yielding first occurs, in pure bending, to the yield stress; the
plastic modulus is the ratio of the fully plastic moment, in pure bending to the yield stress. The buckling parameter u and torsion
index x govern the load for the onset of lateral-torsional buckling in bending (see BS5950 Part 1:2000). The warping constant H
governs the effect of warping on torsion; the torque required for a twist θ per unit length z is
T = GJ θ − EH
d2θ
dz 2
where G J is the torsional rigidity and J is here referred to as the torsional constant (page 131). For beams, columns and
channels the warping constant is given approximately by
2
H =
3
(D − T ) B T
24
in which B , D and T are dimensions defined in the tables.
The section property data have been provided by the Steel Construction Institute on behalf of Corus Group plc. Care has been
taken to ensure that this information is accurate, but Corus Group plc and its subsidiaries does not accept responsibility or
liability for errors or information which is found to be misleading. An extended version of the tables can be found and
downloaded from the Corus web site.
Indicative values of the strength of sections are given in the tables for beam, column and channel sections. The figures quoted
have been calculated in the Department of Engineering Science according to BS5950 Part 1:2000 for grade S275 steel. The
column headed Mcx is the moment capacity of the section about the x-axis in pure bending and excluding the effects of
lateral-torsional buckling. The column headed Pcy is the axial capacity of the section in compression for buckling about the y -axis
for a member with an effective length LE of 3.5 m in the absence of other loads.
136
ENGINEERING TABLES AND DATA
y
B
Universal Beams – Dimensions and Properties
To BS 4–1: 2005
D
d x
x
t
r
T
b
y
Section
designation
Serial size
Second moment
of area
Axis
Axis
x-x
y-y
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
799.6
799.6
719635
625780
45438
39156
38.2
37.8
9.59
9.46
19.1
19.1
19.1
19.1
824.4
824.4
824.4
824.4
504187
436305
376414
325254
15597
13301
11236
9423
37.0
36.8
36.3
35.7
6.51
6.42
6.27
6.07
26.8
21.7
18.8
17.8
17.8
17.8
761.7
761.7
761.7
339704
279175
246021
11360
9066
7799
34.3
33.6
33.1
6.27
6.06
5.90
15.6
14.3
12.8
12.0
25.4
21.6
17.5
15.5
16.5
16.5
16.5
16.5
686.0
686.0
686.0
686.0
239957
205282
168502
150692
8175
6850
5455
4788
30.9
30.5
30.0
29.7
5.71
5.58
5.40
5.30
255.8
254.5
253.7
253.0
14.5
13.2
12.4
11.7
23.7
21.0
19.0
16.2
15.2
15.2
15.2
15.2
615.1
615.1
615.1
615.1
170326
150355
136267
117992
6630
5784
5183
4383
28.0
27.8
27.6
27.2
5.53
5.46
5.39
5.24
635.8
620.2
612.4
311.4
307.1
304.8
18.4
14.1
11.8
31.4
23.6
19.7
16.5
16.5
16.5
540.0
540.0
540.0
209471
153024
125876
15837
11408
9308
26.3
25.9
25.7
7.23
7.07
7.00
139.9
125.1
113.0
101.2
617.2
612.2
607.6
602.6
230.2
229.0
228.2
227.6
13.1
11.9
11.1
10.5
22.1
19.6
17.3
14.8
12.7
12.7
12.7
12.7
547.6
547.6
547.6
547.6
111777
98610
87318
75780
4505
3932
3434
2915
25.0
24.9
24.6
24.2
5.03
4.97
4.88
4.75
533×210×122
533×210×109
533×210×101
533×210×92
533×210×82
122.0
109.0
101.0
92.1
82.2
544.5
539.5
536.7
533.1
528.3
211.9
210.8
210.0
209.3
208.8
12.7
11.6
10.8
10.1
9.6
21.3
18.8
17.4
15.6
13.2
12.7
12.7
12.7
12.7
12.7
476.5
476.5
476.5
476.5
476.5
76043
66822
61519
55227
47539
3388
2943
2692
2389
2007
22.1
21.9
21.9
21.7
21.3
4.67
4.60
4.57
4.51
4.38
457×191×98
457×191×89
457×191×82
457×191×74
457×191×67
98.3
89.3
82.0
74.3
67.1
467.2
463.4
460.0
457.0
453.4
192.8
191.9
191.3
190.4
189.9
11.4
10.5
9.9
9.0
8.5
19.6
17.7
16.0
14.5
12.7
10.2
10.2
10.2
10.2
10.2
407.6
407.6
407.6
407.6
407.6
45727
41015
37051
33319
29380
2347
2089
1871
1671
1452
19.1
19.0
18.8
18.8
18.5
4.33
4.29
4.23
4.20
4.12
Mass
per
metre
kg/m
Depth
of
section
D
mm
Width
of
section
B
mm
Thickness
Root
radius
Web
t
mm
Flange
T
mm
r
mm
Depth
between
fillets
d
mm
914×419×388
914×419×343
388.0
343.3
921.0
911.8
420.5
418.5
21.4
19.4
36.6
32.0
24.1
24.1
914×305×289
914×305×253
914×305×224
914×305×201
289.1
253.4
224.2
200.9
926.6
918.4
910.4
903.0
307.7
305.5
304.1
303.3
19.5
17.3
15.9
15.1
32.0
27.9
23.9
20.2
838×292×226
838×292×194
838×292×176
226.5
193.8
175.9
850.9
840.7
834.9
293.8
292.4
291.7
16.1
14.7
14.0
762×267×197
762×267×173
762×267×147
762×267×134
196.8
173.0
146.9
133.9
769.8
762.2
754.0
750.0
268.0
266.7
265.2
264.4
686×254×170
686×254×152
686×254×140
686×254×125
170.2
152.4
140.1
125.2
692.9
687.5
683.5
677.9
610×305×238
610×305×179
610×305×149
238.1
179.0
149.2
610×229×140
610×229×125
610×229×113
610×229×101
SOLID MECHANICS AND STRUCTURES
137
y
B
Universal Beams – Dimensions and Properties
To BS 4–1: 2005
D
d x
x
t
r
T
b
y
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Buckling Torsional Warping Torsional
parameter index constant constant
u
x
H
dm6
J
cm4
Area
of
section
A
cm2
Indicative values
Mcx
kN m
Pcy
LE = 3.5 m
kN
Section
designation
Serial size
15627
13726
2161
1871
17666
15478
3340
2889
0.885
0.883
26.7
30.1
88.9
75.8
1734
1193
494
437
4690
4110
12100
10700
914×419×388
914×419×343
10883
9501
8269
7204
1014
871
739
621
12570
10942
9535
8352
1601
1370
1163
982
0.867
0.866
0.861
0.854
31.9
36.2
41.3
46.8
31.2
26.4
22.1
18.4
926
626
422
291
368
323
286
256
3340
2890
2530
2210
8220
7180
6300
5570
914×305×289
914×305×253
914×305×224
914×305×201
7985
6641
5893
773
620
535
9155
7640
6808
1211
974
842
0.870
0.862
0.856
35.0
41.6
46.5
19.3
15.2
13.0
514
306
221
289
247
224
2430
2020
1800
6370
5370
4810
838×292×226
838×292×194
838×292×176
6234
5387
4470
4018
610
514
411
362
7167
6198
5156
4644
958
807
647
570
0.869
0.864
0.858
0.854
33.2
38.1
45.2
49.8
11.3
9.39
7.40
6.46
404
267
159
119
251
220
187
171
1900
1640
1370
1280
5310
4600
3840
3580
762×267×197
762×267×173
762×267×147
762×267×134
4916
4374
3987
3481
518
455
409
346
5631
5001
4558
3994
811
710
638
542
0.872
0.871
0.868
0.862
31.8
35.5
38.7
43.9
7.42
6.42
5.72
4.80
308
220
169
116
217
194
178
159
1490
1330
1210
1060
4520
4010
3660
3210
686×254×170
686×254×152
686×254×140
686×254×125
6589
4935
4111
1017
743
611
7486
5548
4594
1574
1144
937
0.886
0.886
0.886
21.3
27.7
32.7
14.5
10.2
8.17
785
340
200
303
228
190
1980
1470
1220
7000
5230
4350
610×305×238
610×305×179
610×305×149
3622
3221
2874
2515
391
343
301
256
4142
3676
3281
2881
611
535
469
400
0.875
0.873
0.870
0.864
30.6
34.1
38.0
43.1
3.99
3.45
2.99
2.52
216
154
111
77.0
178
159
144
129
1100
975
869
792
3510
3110
2780
2510
610×229×140
610×229×125
610×229×113
610×229×101
2793
2477
2292
2072
1800
320
279
256
228
192
3196
2829
2612
2360
2059
500
436
399
355
300
0.877
0.875
0.874
0.872
0.864
27.6
30.9
33.2
36.5
41.6
2.32
1.99
1.81
1.60
1.33
178
126
101
75.7
51.5
155
139
129
117
105
848
750
692
649
567
2900
2570
2370
2180
1910
533×210×122
533×210×109
533×210×101
533×210×92
533×210×82
1957
1770
1611
1458
1296
243
218
196
176
153
2232
2014
1831
1653
1471
379
338
304
272
237
0.881
0.880
0.877
0.877
0.872
25.7
28.3
30.9
33.9
37.9
1.18
1.04
0.922
0.818
0.705
121
90.7
69.2
51.8
37.1
125
114
104
94.6
85.5
591
533
503
454
404
2190
1980
1830
1650
1460
457×191×98
457×191×89
457×191×82
457×191×74
457×191×67
138
ENGINEERING TABLES AND DATA
y
B
Universal Beams – Dimensions and Properties
To BS 4–1: 2005
D
d x
x
t
r
T
b
y
Section
designation
Serial size
Second moment
of area
Axis
Axis
x-x
y-y
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
407.6
407.6
407.6
407.6
407.6
36589
32674
28927
25500
21369
1185
1047
913
795
645
18.7
18.6
18.4
18.3
17.9
3.37
3.33
3.27
3.23
3.11
10.2
10.2
10.2
10.2
360.4
360.4
360.4
360.4
27310
24331
21596
18722
1545
1365
1203
1021
17.0
16.9
16.8
16.5
4.04
3.99
3.97
3.85
11.2
8.6
10.2
10.2
360.4
360.4
15685
12508
538
410
16.4
15.9
3.03
2.87
9.1
8.1
7.4
7.0
15.7
13.0
11.5
9.7
10.2
10.2
10.2
10.2
311.6
311.6
311.6
311.6
19463
16038
14136
12066
1362
1108
968
811
15.1
14.9
14.8
14.5
3.99
3.91
3.86
3.76
126.0
125.4
6.6
6.0
10.7
8.5
10.2
10.2
311.6
311.6
10172
8249
358
280
14.3
14.0
2.68
2.58
310.4
306.6
303.4
166.9
165.7
165.0
7.9
6.7
6.0
13.7
11.8
10.2
8.9
8.9
8.9
265.2
265.2
265.2
11696
9899
8503
1063
896
764
13.0
13.0
12.9
3.93
3.90
3.86
48.1
41.9
37.0
311.0
307.2
304.4
125.3
124.3
123.4
9.0
8.0
7.1
14.0
12.1
10.7
8.9
8.9
8.9
265.2
265.2
265.2
9575
8196
7171
461
389
336
12.5
12.4
12.3
2.74
2.70
2.67
305×102×33
305×102×28
305×102×25
32.8
28.2
24.8
312.7
308.7
305.1
102.4
101.8
101.6
6.6
6.0
5.8
10.8
8.8
7.0
7.6
7.6
7.6
275.9
275.9
275.9
6501
5366
4455
194
155
123
12.5
12.2
11.9
2.15
2.08
1.97
254×146×43
254×146×37
254×146×31
43.0
37.0
31.1
259.6
256.0
251.4
147.3
146.4
146.1
7.2
6.3
6.0
12.7
10.9
8.6
7.6
7.6
7.6
219.0
219.0
219.0
6544
5537
4413
677
571
448
10.9
10.8
10.5
3.52
3.48
3.36
254×102×28
254×102×25
254×102×22
28.3
25.2
22.0
260.4
257.2
254.0
102.2
101.9
101.6
6.3
6.0
5.7
10.0
8.4
6.8
7.6
7.6
7.6
225.2
225.2
225.2
4005
3415
2841
179
149
119
10.5
10.3
10.1
2.22
2.15
2.06
203×133×30
203×133×25
30.0
25.1
206.8
203.2
133.9
133.2
6.4
5.7
9.6
7.8
7.6
7.6
172.4
172.4
2896
2340
385
308
8.71
8.56
3.17
3.10
203×102×23
23.1
203.2
101.8
5.4
9.3
7.6
169.4
2105
164
8.46
2.36
178×102×19
19.0
177.8
101.2
4.8
7.9
7.6
146.8
1356
137
7.48
2.37
152×89×16
16.0
152.4
88.7
4.5
7.7
7.6
121.8
834
89.8
6.41
2.10
127×76×13
13.0
127.0
76.0
4.0
7.6
7.6
96.6
473
55.7
5.35
1.84
Mass
per
metre
kg/m
Depth
of
section
D
mm
Width
of
section
B
mm
Thickness
Root
radius
Web
t
mm
Flange
T
mm
r
mm
Depth
between
fillets
d
mm
457×152×82
457×152×74
457×152×67
457×152×60
457×152×52
82.1
74.2
67.2
59.8
52.3
465.8
462.0
458.0
454.6
449.8
155.3
154.4
153.8
152.9
152.4
10.5
9.6
9.0
8.1
7.6
18.9
17.0
15.0
13.3
10.9
10.2
10.2
10.2
10.2
10.2
406×178×74
406×178×67
406×178×60
406×178×54
74.2
67.1
60.1
54.1
412.8
409.4
406.4
402.6
179.5
178.8
177.9
177.7
9.5
8.8
7.9
7.7
16.0
14.3
12.8
10.9
406×140×46
406×140×39
46.0
39.0
403.2
398.0
142.2
141.8
6.8
6.4
356×171×67
356×171×57
356×171×51
356×171×45
67.1
57.0
51.0
45.0
363.4
358.0
355.0
351.4
173.2
172.2
171.5
171.1
356×127×39
356×127×33
39.1
33.1
353.4
349.0
305×165×54
305×165×46
305×165×40
54.0
46.1
40.3
305×127×48
305×127×42
305×127×37
SOLID MECHANICS AND STRUCTURES
139
y
B
Universal Beams – Dimensions and Properties
To BS 4–1: 2005
D
d x
x
t
r
T
b
y
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
1571
1414
1263
1122
950
153
136
119
104
84.6
1812
1627
1453
1287
1096
240
213
187
163
133
0.873
0.873
0.869
0.868
0.859
1323
1189
1063
930
172
153
135
115
1501
1346
1200
1055
267
237
209
178
778
629
75.7
57.8
888
724
1071
896
796
687
157
129
113
94.8
576
473
Buckling Torsional Warping Torsional
parameter index constant constant
H
dm6
J
cm4
Area
of
section
A
cm2
27.4
30.1
33.6
37.5
43.9
0.591
0.518
0.448
0.387
0.311
89.2
65.9
47.7
33.8
21.4
0.882
0.880
0.880
0.871
27.6
30.5
33.8
38.3
0.608
0.533
0.466
0.392
118
90.8
0.871
0.858
38.9
47.5
1211
1010
896
775
243
199
174
147
0.886
0.882
0.881
0.874
56.8
44.7
659
543
89.0
70.2
754
646
560
127
108
92.6
846
720
623
616
534
471
73.6
62.6
54.5
416
348
292
Indicative values
kN m
Pcy
LE = 3.5 m
kN
105
94.5
85.6
76.2
66.6
480
432
399
355
303
1380
1220
1100
959
793
457×152×82
457×152×74
457×152×67
457×152×60
457×152×52
62.8
46.1
33.3
23.1
94.5
85.5
76.5
69.0
413
371
330
289
1580
1410
1260
1090
406×178×74
406×178×67
406×178×60
406×178×54
0.207
0.155
19.0
10.7
58.6
49.7
244
199
671
523
406×140×46
406×140×39
24.4
28.8
32.1
36.8
0.412
0.330
0.286
0.237
55.7
33.4
23.8
15.8
85.5
72.6
64.9
57.3
333
278
246
213
1410
1170
1030
884
356×171×67
356×171×57
356×171×51
356×171×45
0.871
0.863
35.2
42.2
0.105
0.081
15.1
8.79
49.8
42.1
181
149
469
372
356×127×39
356×127×33
196
166
142
0.889
0.891
0.889
23.6
27.1
31.0
0.234
0.195
0.164
34.8
22.2
14.7
68.8
58.7
51.3
233
198
171
1120
945
816
305×165×54
305×165×46
305×165×40
711
614
539
116
98.4
85.4
0.873
0.872
0.872
23.3
26.5
29.7
0.102
0.0846
0.0725
31.8
21.1
14.8
61.2
53.4
47.2
195
169
148
597
509
442
305×127×48
305×127×42
305×127×37
37.9
30.5
24.2
481
403
342
60.0
48.4
38.8
0.866
0.859
0.846
31.6
37.4
43.4
0.0442
0.0349
0.0273
12.2
7.40
4.77
41.8
35.9
31.6
132
111
94.1
269
218
174
305×102×33
305×102×28
305×102×25
504
433
351
92.0
78.0
61.3
566
483
393
141
119
94.1
0.891
0.890
0.880
21.2
24.3
29.6
0.1030
0.0857
0.0660
23.9
15.3
8.55
54.8
47.2
39.7
156
133
108
777
659
529
254×146×43
254×146×37
254×146×31
308
266
224
34.9
29.2
23.5
353
306
259
54.8
46.0
37.3
0.874
0.866
0.856
27.5
31.5
36.4
0.0280
0.0230
0.0182
9.57
6.42
4.15
36.1
32.0
28.0
97.0
84.0
71.2
246
206
167
254×102×28
254×102×25
254×102×22
280
230
57.5
46.2
314
258
88.2
70.9
0.881
0.877
21.5
25.6
0.0374
0.0294
10.3
5.96
38.2
32.0
86.5
70.9
468
379
203×133×30
203×133×25
207
32.2
234
49.7
0.888
22.5
0.0154
7.02
29.4
64.4
223
203×102×23
153
27.0
171
41.6
0.888
22.6
0.0099
4.41
24.3
47.1
186
178×102×19
109
20.2
123
31.2
0.890
19.6
0.0047
3.56
20.3
33.9
125
152×89×16
74.6
14.7
84.2
22.6
0.895
16.3
0.0020
2.85
16.5
23.1
80.3
127×76×13
u
x
Mcx
Section
designation
Serial size
140
ENGINEERING TABLES AND DATA
y
B
Universal Columns – Dimensions and Properties
To BS 4–1: 2005
D
d
x
x
t
r
T
b
y
Section
designation
Serial size
Mass
per
metre
Second moment
of area
Axis
Axis
x-x
y-y
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
290.2
290.2
290.2
290.2
290.2
290.2
290.2
274845
226938
183003
146618
122543
99875
79085
98125
82671
67834
55367
46853
38677
30993
18.4
18.0
17.5
17.1
16.8
16.5
16.3
11.0
10.9
10.7
10.5
10.4
10.3
10.2
15.2
15.2
15.2
15.2
290.2
290.2
290.2
290.2
66261
57118
48589
40246
23688
20529
17553
14611
16.1
15.9
15.8
15.6
9.60
9.54
9.50
9.43
44.1
37.7
31.4
25.0
21.7
18.7
15.4
15.2
15.2
15.2
15.2
15.2
15.2
15.2
246.7
246.7
246.7
246.7
246.7
246.7
246.7
78872
64203
50904
38747
32814
27672
22249
24635
20315
16299
12569
10700
9059
7308
14.8
14.5
14.2
13.9
13.7
13.6
13.4
8.30
8.15
8.00
7.90
7.80
7.77
7.70
19.2
15.3
12.8
10.3
8.6
31.7
25.3
20.5
17.3
14.2
12.7
12.7
12.7
12.7
12.7
200.3
200.3
200.3
200.3
200.3
29998
22529
17510
14268
11407
9870
7531
5928
4857
3908
11.9
11.6
11.3
11.2
11.1
6.81
6.70
6.59
6.60
6.48
209.1
206.4
205.8
204.3
203.6
12.7
10.0
9.4
7.9
7.2
20.5
17.3
14.2
12.5
11.0
10.2
10.2
10.2
10.2
10.2
160.8
160.8
160.8
160.8
160.8
9449
7618
6125
5259
4568
3127
2537
2065
1778
1548
9.28
9.20
8.96
8.90
8.82
5.34
5.30
5.20
5.20
5.13
154.4
152.9
152.2
8.0
6.5
5.8
11.5
9.4
6.8
7.6
7.6
7.6
123.6
123.6
123.6
2210
1748
1250
706
560
400
6.90
6.76
6.50
3.90
3.83
3.70
kg/m
Depth
of
section
D
mm
Width
of
section
B
mm
Thickness
Root
radius
Web
t
mm
Flange
T
mm
r
mm
Depth
between
fillets
d
mm
356×406×634
356×406×551
356×406×467
356×406×393
356×406×340
356×406×287
356×406×235
633.9
551.0
467.0
393.0
339.9
287.1
235.1
474.6
455.6
436.6
419.0
406.4
393.6
381.0
424.0
418.5
412.2
407.0
403.0
399.0
394.8
47.6
42.1
35.8
30.6
26.6
22.6
18.4
77.0
67.5
58.0
49.2
42.9
36.5
30.2
15.2
15.2
15.2
15.2
15.2
15.2
15.2
356×368×202
356×368×177
356×368×153
356×368×129
201.9
177.0
152.9
129.0
374.6
368.2
362.0
355.6
374.7
372.6
370.5
368.6
16.5
14.4
12.3
10.4
27.0
23.8
20.7
17.5
305×305×283
305×305×240
305×305×198
305×305×158
305×305×137
305×305×118
305×305×97
282.9
240.0
198.1
158.1
136.9
117.9
96.9
365.3
352.5
339.9
327.1
320.5
314.5
307.9
322.2
318.4
314.5
311.2
309.2
307.4
305.3
26.8
23.0
19.1
15.8
13.8
12.0
9.9
254×254×167
254×254×132
254×254×107
254×254×89
254×254×73
167.1
132.0
107.1
88.9
73.1
289.1
276.3
266.7
260.3
254.1
265.2
261.3
258.8
256.3
254.6
203×203×86
203×203×71
203×203×60
203×203×52
203×203×46
86.1
71.0
60.0
52.0
46.1
222.2
215.8
209.6
206.2
203.2
152×152×37
152×152×30
152×152×23
37.0
30.0
23.0
161.8
157.6
152.4
SOLID MECHANICS AND STRUCTURES
141
y
B
Universal Columns – Dimensions and Properties
To BS 4–1: 2005
D
d
x
x
t
r
T
b
y
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Buckling Torsional Warping Torsional
parameter index constant constant
u
x
H
dm6
J
cm4
Area
of
section
A
cm2
Indicative values
Mcx
kN m
Pcy
LE = 3.5 m
kN
Section
designation
Serial size
11582
9962
8383
6998
6031
5075
4151
4629
3951
3291
2721
2325
1939
1570
14235
12076
10003
8223
6999
5813
4687
7108
6058
5034
4154
3544
2949
2383
0.843
0.841
0.839
0.837
0.836
0.835
0.834
5.46
6.05
6.86
7.86
8.85
10.2
12.1
38.8
31.1
24.3
18.9
15.5
12.3
9.54
13720
9240
5809
3545
2343
1441
812
808
702
595
501
433
366
299
3410
2930
2550
2100
1790
1540
1240
17600
15300
13400
11200
9650
8780
7150
356×406×634
356×406×551
356×406×467
356×406×393
356×406×340
356×406×287
356×406×235
3538
3103
2684
2264
1264
1102
948
793
3972
3455
2965
2479
1919
1671
1435
1199
0.844
0.844
0.844
0.844
13.4
15.0
17.0
19.9
7.20
6.09
5.10
4.18
558
381
251
153
257
226
195
164
1050
917
784
652
6060
5320
4590
3850
356×368×202
356×368×177
356×368×153
356×368×129
4318
3643
2995
2369
2048
1760
1445
1529
1276
1037
808
692
589
479
5105
4247
3440
2681
2297
1958
1592
2342
1950
1581
1230
1052
895
726
0.855
0.854
0.854
0.851
0.851
0.850
0.850
7.65
8.74
10.2
12.5
14.2
16.2
19.3
6.40
5.03
3.90
2.87
2.40
1.98
1.60
2034
1271
734
378
249
161
91.2
360
306
252
201
174
150
123
1300
1130
912
710
610
519
437
7410
6890
5650
4480
3870
3330
2810
305×305×283
305×305×240
305×305×198
305×305×158
305×305×137
305×305×118
305×305×97
2075
1631
1313
1096
898
744
576
458
379
307
2424
1869
1485
1224
992
1137
878
697
575
465
0.851
0.850
0.848
0.850
0.849
8.49
10.3
12.4
14.5
17.3
1.63
1.20
0.898
0.700
0.562
626
319
172
102
57.6
213
168
136
113
93.1
641
496
392
323
273
4490
3510
2820
2340
1980
254×254×167
254×254×132
254×254×107
254×254×89
254×254×73
850
706
584
510
450
299
246
201
174
152
977
799
656
567
497
456
374
305
264
231
0.850
0.853
0.846
0.848
0.847
10.2
11.9
14.1
15.8
17.7
0.318
0.300
0.197
0.200
0.143
137
80.2
47.2
31.8
22.2
110
90.0
76.4
66.0
58.7
259
212
180
156
137
2030
1660
1430
1230
1080
203×203×86
203×203×71
203×203×60
203×203×52
203×203×46
273
222
164
91.0
73.3
53.0
309
248
182
140
112
80.0
0.848
0.849
0.840
13.3
16.0
20.7
0.0
0.0308
0.0
19.2
10.5
4.63
47.0
38.3
29.0
84.9
68.1
48.5
667
536
392
152×152×37
152×152×30
152×152×23
142
ENGINEERING TABLES AND DATA
B
Channels – Dimensions and Properties
r
To BS 4–1: 2005
t
D
d
b
T
Section
designation
Serial size
Mass
per
metre
kg/m
Depth Width
of
of
section section
D
B
mm
mm
Thickness
Distance
Web
t
mm
Flange
T
mm
Cy
cm
Root
Depth Second moment
of area
radius between
Axis
Axis
fillets
r
d
x-x
y-y
mm
mm
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
430×100×64
64.4
430
100
11.0
19.0
2.6
15
362
21900
722
16.3
2.97
380×100×54
54.0
380
100
9.5
17.5
2.8
15
315
15000
643
14.8
3.06
300×100×46
300×90×41
45.5
41.4
300
300
100
90
9.0
9.0
16.5
15.5
3.1
2.6
15
12
237
245
8230
7220
568
404
11.9
11.7
3.13
2.77
260×90×35
260×75×28
34.8
27.6
260
260
90
75
8.0
7.0
14.0
12.0
2.7
2.1
12
12
208
212
4730
3620
353
185
10.3
10.1
2.82
2.30
230×90×32
230×75×26
32.2
25.7
230
230
90
75
7.5
6.5
14.0
12.5
2.9
2.3
12
12
178
181
3520
2750
334
181
9.27
9.17
2.86
2.35
200×90×30
200×75×23
29.7
23.4
200
200
90
75
7.0
6.0
14.0
12.5
3.1
2.5
12
12
148
151
2520
1960
314
170
8.16
8.11
2.88
2.39
180×90×26
180×75×20
26.1
20.3
180
180
90
75
6.5
6.0
12.5
10.5
3.2
2.4
12
12
131
135
1820
1370
277
146
7.40
7.27
2.89
2.38
150×90×24
150×75×18
23.9
17.9
150
150
90
75
6.5
5.5
12.0
10.0
3.3
2.6
12
12
102
106
1160
861
253
131
6.18
6.15
2.89
2.40
125×65×15
14.8
125
65
5.5
9.5
2.3
12
82.0
483
80.0
5.07
2.06
100×50×10
10.2
100
50
5.0
8.5
1.7
9
65.0
208
32.3
4.00
1.58
SOLID MECHANICS AND STRUCTURES
143
B
Channels – Dimensions and Properties
r
To BS 4–1: 2005
t
D
d
b
T
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
1020
97.9
1220
176
0.917
791
89.2
933
161
0.933
549
481
81.7
63.1
641
568
148
114
364
278
56.3
34.4
425
328
306
239
55.0
34.8
252
196
Buckling Torsional Warping Torsional
parameter index constant constant
H
dm6
J
cm4
Area
of
section
A
cm2
22.5
0.219
63.0
21.2
0.150
45.7
0.944
0.934
17.0
18.4
0.0813
0.0581
102
62.0
0.943
0.932
17.2
20.5
355
278
98.9
63.2
0.949
0.945
53.4
33.8
291
227
94.5
60.6
202
152
47.4
28.8
232
176
155
115
44.4
26.6
77.3
41.5
Indicative values
kN m
Pcy
LE = 3.5 m
kN
82.1
336
819
430×100×64
68.7
257
716
380×100×54
36.8
28.8
58.0
52.7
176
156
624
473
300×100×46
300×90×41
0.0379
0.0203
20.6
11.7
44.4
35.1
117
90.2
410
234
260×90×35
260×75×28
15.1
17.3
0.0279
0.0153
19.3
11.8
41.0
32.7
97.6
76.5
387
226
230×90×32
230×75×26
0.952
0.956
12.9
14.7
0.0197
0.0107
18.3
11.1
37.9
29.9
80.0
62.4
361
212
200×90×30
200×75×23
83.5
51.8
0.950
0.945
12.8
15.3
0.0141
0.0075
13.3
7.34
33.2
25.9
63.8
48.4
318
183
180×90×26
180×75×20
179
132
76.9
47.2
0.937
0.945
10.8
13.1
0.0089
0.0047
11.8
6.10
30.4
22.8
49.2
36.3
291
163
150×90×24
150×75×18
18.8
89.9
33.2
0.942
11.1
0.0019
4.72
18.8
24.7
104
125×65×15
9.89
48.9
17.5
0.942
10.0
0.0005
2.53
13.0
13.4
44.9
100×50×10
u
x
Mcx
Section
designation
Serial size
144
ENGINEERING TABLES AND DATA
y
v
Equal Angles – Dimensions and Properties
u
t
To BS EN 10056-1:1999
90o
A
r1
x
x
c
r2
t
c
u
Section
Second moment
Radius
Mass
Radius
Area Distance
designation
of
area
of
gyration
per
of
to
Size
Thickness metre Root Toe section centroid Axis
Axis
Axis
Axis
Axis
Axis
A×A
t
r1
r2
c
x-x,y-y u-u
v-v x-x,y-y u-u
v-v
mm
mm
kg/m mm mm
cm2
cm
cm4
cm4
cm4
cm
cm
cm
y
A
v
Elastic Torsional
modulus constant
Axis
x-x,y-y
J
cm3
cm4
200×200
24
20
18
16
71.1
59.9
54.2
48.5
18.0
18.0
18.0
18.0
9.00
9.00
9.00
9.00
90.6
76.3
69.1
61.8
5.84
5.68
5.60
5.52
3331
2851
2600
2342
5280
4530
4150
3720
1380
1170
1050
960
6.06
6.11
6.13
6.16
7.64
7.70
7.75
7.76
3.90
3.92
3.90
3.94
235
199
181
162
182
107
78.9
56.1
150×150
15
12
10
33.8
27.3
23.0
16.0 8.00
16.0 8.00
16.0 8.00
43.0
34.8
29.3
4.25
4.12
4.03
898
737
624
1430
1170
990
370
303
258
4.57
4.60
4.62
5.76
5.80
5.82
2.93
2.95
2.97
83.5
67.8
56.9
34.6
18.2
10.8
120×120
12
10
21.6
18.2
13.0 6.50
13.0 6.50
27.5
23.2
3.40
3.31
368
313
584
497
152
129
3.65
3.67
4.60
4.63
2.35
2.36
42.7
36.0
14.2
8.41
100×100
12
10
8
17.8
15.0
12.2
12.0 6.00
12.0 6.00
12.0 6.00
22.7
19.2
15.5
2.90
2.82
2.74
207
177
145
328
280
230
85.7
73.0
59.9
3.02
3.04
3.06
3.80
3.83
3.85
1.94
1.95
1.96
29.1
24.6
20.0
11.8
6.97
3.68
90×90
10
8
7
13.4
10.9
9.61
11.0 5.50
11.0 5.50
11.0 5.50
17.1
13.9
12.2
2.58
2.50
2.45
127
104
92.6
201
166
147
52.6
43.1
38.3
2.72
2.74
2.75
3.42
3.45
3.46
1.75
1.76
1.77
19.8
16.1
14.1
6.20
3.28
2.24
SOLID MECHANICS AND STRUCTURES
145
y
Rectangular Hollow Sections – Dimensions and Properties
t
D
To BS EN 10210-2: 2006
x
x
y
B
Section
designation
Size
Thickness
D×B
t
mm
mm
Mass
per
metre
kg/m
Area Second moment
of area
of
Axis
section Axis
A
x-x
y-y
cm2
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Torsional
constant
J
cm4
50 × 30
3.2
3.61
4.60
14.2
6.20
1.76
1.16
5.68
4.13
7.25
5.00
14.2
60 × 40
3.0
4.0
5.0
4.35
5.64
6.85
5.54
7.19
8.73
26.5
32.8
38.1
13.9
17.0
19.5
2.18
2.14
2.09
1.58
1.54
1.50
8.82
10.9
12.7
6.95
8.52
9.77
10.9
13.8
16.4
8.19
10.3
12.2
29.2
36.7
43.0
80 × 40
3.2
4.0
5.0
6.3
8.0
5.62
6.90
8.42
10.3
12.5
7.16
8.79
10.7
13.1
16.0
57.2
68.2
80.3
93.3
106
18.9
22.2
25.7
29.2
32.1
2.83
2.79
2.74
2.67
2.58
1.63
1.59
1.55
1.49
1.42
14.3
17.1
20.1
23.3
26.5
9.46
11.1
12.9
14.6
16.1
18.0
21.8
26.1
31.1
36.5
11.0
13.2
15.7
18.4
21.2
46.2
55.2
65.1
75.6
85.8
90 × 50
3.6
5.0
6.3
7.40
9.99
12.3
9.42
12.7
15.6
98.3
127
150
38.7
49.2
57.0
3.23
3.16
3.10
2.03
1.97
1.91
21.8
28.3
33.3
15.5
19.7
22.8
27.2
36.0
43.2
18.0
23.5
28.0
89.4
116
138
100 × 50
3.0
3.2
4.0
5.0
6.3
8.0
6.71
7.13
8.78
10.8
13.3
16.3
8.54
9.08
11.2
13.7
16.9
20.8
110
116
140
167
197
230
36.8
38.8
46.2
54.3
63.0
71.7
3.58
3.57
3.53
3.48
3.42
3.33
2.08
2.07
2.03
1.99
1.93
1.86
21.9
23.2
27.9
33.3
39.4
46.0
14.7
15.5
18.5
21.7
25.2
28.7
27.3
28.9
35.2
42.6
51.3
61.4
16.8
17.7
21.5
25.8
30.8
36.3
88.4
93.4
113
135
160
186
100 × 60
3.6
5.0
6.3
8.0
8.53
11.6
14.2
17.5
10.9
14.7
18.1
22.4
145
189
225
264
64.8
83.6
98.1
113
3.65
3.58
3.52
3.44
2.44
2.38
2.33
2.25
28.9
37.8
45.0
52.8
21.6
27.9
32.7
37.8
35.6
47.4
57.3
68.7
24.9
32.9
39.5
47.1
142
188
224
265
120 × 60
3.6
5.0
6.3
8.0
9.66
13.1
16.2
20.1
12.3
16.7
20.7
25.6
227
299
358
425
76.3
98.8
116
135
4.30
4.23
4.16
4.08
2.49
2.43
2.37
2.30
37.9
49.9
59.7
70.8
25.4
32.9
38.8
45.0
47.2
63.1
76.7
92.7
28.9
38.4
46.3
55.4
183
242
290
344
120 × 80
5.0
6.3
8.0
10.0
14.7
18.2
22.6
27.4
18.7
23.2
28.8
34.9
365
440
525
609
193
230
273
313
4.42
4.36
4.27
4.18
3.21
3.15
3.08
2.99
60.9
73.3
87.5
102
48.2
57.6
68.1
78.1
74.6
91.0
111
131
56.1
68.2
82.6
97.3
401
487
587
688
150 × 100
5.0
6.3
8.0
10.0
12.5
18.6
23.1
28.9
35.3
42.8
23.7
29.5
36.8
44.9
54.6
739
898
1090
1280
1490
392
474
569
665
763
5.58
5.52
5.44
5.34
5.22
4.07
4.01
3.94
3.85
3.74
98.5
120
145
171
198
78.5
94.8
114
133
153
119
147
180
216
256
90.1
110
135
161
190
807
986
1200
1430
1680
150 × 125
4.0
5.0
6.3
8.0
10.0
12.5
16.6
20.6
25.6
32.0
39.2
47.7
21.2
26.2
32.6
40.8
49.9
60.8
714
870
1060
1290
1530
1780
539
656
798
966
1140
1330
5.80
5.76
5.70
5.62
5.53
5.42
5.04
5.00
4.94
4.87
4.78
4.67
95.2
116
141
172
204
238
86.3
105
128
155
183
212
112
138
169
208
251
299
98.9
121
149
183
221
262
949
1160
1430
1750
2100
2490
146
ENGINEERING TABLES AND DATA
y
Rectangular Hollow Sections – Dimensions and Properties
t
D
To BS EN 10210-2: 2006
x
x
y
B
Section
designation
Size
Thickness
D×B
t
mm
mm
Mass
per
metre
kg/m
Area Second moment
of area
of
Axis
section Axis
A
x-x
y-y
cm2
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Torsional
constant
J
cm4
160 × 80
4.0
5.0
6.3
8.0
10.0
14.4
17.8
22.2
27.6
33.7
18.4
22.7
28.2
35.2
42.9
612
744
903
1090
1280
207
249
299
356
411
5.77
5.72
5.66
5.57
5.47
3.35
3.31
3.26
3.18
3.10
76.5
93.0
113
136
161
51.7
62.3
74.8
89.0
103
94.7
116
142
175
209
58.3
71.1
86.8
106
125
493
600
730
883
1040
200 × 100
5.0
6.3
8.0
10.0
12.5
22.6
28.1
35.1
43.1
52.7
28.7
35.8
44.8
54.9
67.1
1500
1830
2230
2660
3140
505
613
739
869
1000
7.21
7.15
7.06
6.96
6.84
4.19
4.14
4.06
3.98
3.87
149
183
223
266
314
101
123
148
174
201
185
228
282
341
408
114
140
172
206
245
1200
1480
1800
2160
2540
200 × 120
5.0
6.3
8.0
10.0
14.2
24.1
30.1
37.6
46.3
63.3
30.7
38.3
48.0
58.9
80.7
1690
2070
2530
3030
3910
762
929
1130
1340
1690
7.40
7.34
7.26
7.17
6.96
4.98
4.92
4.85
4.76
4.58
168
207
253
303
391
127
155
188
223
282
205
253
313
379
503
144
177
218
263
346
1650
2030
2500
3000
3920
200 × 150
8.0
10.0
41.4
51.0
52.8
64.9
2970
3570
1890
2260
7.50
7.41
5.99
5.91
297
357
253
302
359
436
294
356
3640
4410
250 × 120
10.0
12.5
14.2
54.1
66.4
74.5
68.9
84.6
94.9
5310
6330
6960
1640
1930
2090
8.78
8.65
8.56
4.88
4.77
4.70
425
506
556
273
321
349
539
651
722
318
381
421
4090
4880
5360
250 × 150
5.0
6.3
8.0
10.0
12.5
14.2
16.0
30.4
38.0
47.7
58.8
72.3
81.1
90.3
38.7
48.4
60.8
74.9
92.1
103
115
3360
4140
5110
6170
7390
8140
8880
1530
1870
2300
2760
3270
3580
3870
9.31
9.25
9.17
9.08
8.96
8.87
8.79
6.28
6.22
6.15
6.06
5.96
5.88
5.80
269
331
409
494
591
651
710
204
250
306
367
435
477
516
324
402
501
611
740
823
906
228
283
350
426
514
570
625
3280
4050
5020
6090
7330
8100
8870
250 × 200
10.0
12.5
14.2
66.7
82.1
92.3
84.9
105
118
7610
9150
10100
5370
6440
7100
9.47
9.35
9.28
7.95
7.85
7.77
609
732
809
537
644
710
731
888
990
626
760
846
9890
12000
13300
260 × 140
5.0
6.3
8.0
10.0
12.5
14.2
16.0
30.4
38.0
47.7
58.8
72.3
81.1
90.3
38.7
48.4
60.8
74.9
92.1
103
115
3530
4360
5370
6490
7770
8560
9340
1350
1660
2030
2430
2880
3140
3400
9.55
9.49
9.40
9.31
9.18
9.10
9.01
5.91
5.86
5.78
5.70
5.59
5.52
5.44
272
335
413
499
597
658
718
193
237
290
347
411
449
486
331
411
511
624
756
840
925
216
267
331
402
485
537
588
3080
3800
4700
5700
6840
7560
8260
300 × 100
8.0
10.0
14.2
47.7
58.8
81.1
60.8
74.9
103
6310
7610
10000
1080
1280
1610
10.2
10.1
9.85
4.21
4.13
3.94
420
508
669
216
255
321
546
666
896
245
296
390
3070
3680
4760
SOLID MECHANICS AND STRUCTURES
147
y
Rectangular Hollow Sections – Dimensions and Properties
t
D
To BS EN 10210-2: 2006
x
x
y
B
Section
designation
Size
Thickness
D×B
t
mm
mm
Mass
per
metre
kg/m
Area Second moment
of area
of
Axis
section Axis
A
x-x
y-y
cm2
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Torsional
constant
J
cm4
300 × 150
8.0
10.0
12.5
14.2
16.0
54.0
66.7
82.1
92.3
103
68.8
84.9
105
118
131
8010
9720
11700
12900
14200
2700
3250
3860
4230
4600
10.8
10.7
10.6
10.5
10.4
6.27
6.18
6.07
6.00
5.92
534
648
779
862
944
360
433
514
564
613
663
811
986
1100
1210
407
496
600
666
732
6450
7840
9450
10500
11500
300 × 200
6.3
8.0
10.0
12.5
14.2
16.0
47.9
60.3
74.5
91.9
103
115
61.0
76.8
94.9
117
132
147
7830
9720
11800
14300
15800
17400
4190
5180
6280
7540
8330
9110
11.3
11.3
11.2
11.0
11.0
10.9
8.29
8.22
8.13
8.02
7.95
7.87
522
648
788
952
1060
1160
419
518
628
754
833
911
624
779
956
1170
1300
1440
472
589
721
877
978
1080
8480
10600
12900
15700
17500
19300
300 × 250
5.0
6.3
8.0
10.0
12.5
14.2
16.0
42.2
52.8
66.5
82.4
102
115
128
53.7
67.3
84.8
105
130
146
163
7410
9190
11400
13900
16900
18700
20600
5610
6950
8630
10500
12700
14100
15500
11.7
11.7
11.6
11.5
11.4
11.3
11.2
10.2
10.2
10.1
10.0
9.89
9.82
9.74
494
613
761
928
1120
1250
1380
449
556
690
840
1010
1130
1240
575
716
896
1100
1350
1510
1670
508
633
791
971
1190
1330
1470
9770
12200
15200
18600
22700
25400
28100
350 × 150
5.0
6.3
8.0
10.0
12.5
14.2
16.0
38.3
47.9
60.3
74.5
91.9
103
115
48.7
61.0
76.8
94.9
117
132
147
7660
9480
11800
14300
17300
19200
21100
2050
2530
3110
3740
4450
4890
5320
12.5
12.5
12.4
12.3
12.2
12.1
12.0
6.49
6.43
6.36
6.27
6.17
6.09
6.01
437
542
673
818
988
1100
1210
274
337
414
498
593
652
709
543
676
844
1040
1260
1410
1560
301
373
464
566
686
763
840
5160
6390
7930
9630
11600
12900
14100
350 × 250
5.0
6.3
8.0
10.0
12.5
14.2
16.0
46.1
57.8
72.8
90.2
112
126
141
58.7
73.6
92.8
115
142
160
179
10600
13200
16400
20100
24400
27200
30000
6360
7890
9800
11900
14400
16000
17700
13.5
13.4
13.3
13.2
13.1
13.0
12.9
10.4
10.4
10.3
10.2
10.1
10.0
9.93
607
754
940
1150
1400
1550
1720
509
631
784
955
1160
1280
1410
716
892
1120
1380
1690
1890
2100
569
709
888
1090
1330
1490
1660
12200
15200
19000
23400
28500
31900
35300
400 × 120
5.0
6.3
8.0
10.0
12.5
14.2
16.0
39.8
49.9
62.8
77.7
95.8
108
120
50.7
63.5
80.0
98.9
122
137
153
9520
11800
14600
17800
21600
23900
26300
1420
1740
2130
2550
3010
3290
3560
13.7
13.6
13.5
13.4
13.3
13.2
13.1
5.30
5.24
5.17
5.08
4.97
4.89
4.82
476
590
732
891
1080
1200
1320
237
291
356
425
502
549
593
612
762
952
1170
1430
1590
1760
259
320
397
483
583
646
709
4090
5040
6220
7510
8980
9890
10800
148
ENGINEERING TABLES AND DATA
y
Rectangular Hollow Sections – Dimensions and Properties
t
D
To BS EN 10210-2: 2006
x
x
y
B
Section
designation
Size
Thickness
D×B
t
mm
mm
Mass
per
metre
kg/m
Area Second moment
of area
of
Axis
section Axis
A
x-x
y-y
cm2
cm4
cm4
Radius
of gyration
Axis
Axis
x-x
y-y
cm
cm
Elastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Plastic
modulus
Axis
Axis
x-x
y-y
cm3
cm3
Torsional
constant
J
cm4
400 × 150
5.0
6.3
8.0
10.0
12.5
14.2
16.0
42.2
52.8
66.5
82.4
102
115
128
53.7
67.3
84.8
105
130
146
163
10700
13300
16500
20100
24400
27100
29800
2320
2850
3510
4230
5040
5550
6040
14.1
14.0
13.9
13.8
13.7
13.6
13.5
6.57
6.51
6.43
6.35
6.24
6.16
6.09
534
663
824
1010
1220
1360
1490
309
380
468
564
672
740
805
671
836
1050
1290
1570
1760
1950
337
418
521
636
772
859
947
6130
7600
9420
11500
13800
15300
16800
400 × 200
8.0
10.0
12.5
14.2
16.0
72.8
90.2
112
126
141
92.8
115
142
160
179
19600
23900
29100
32400
35700
6660
8080
9740
10800
11800
14.5
14.4
14.3
14.2
14.1
8.47
8.39
8.28
8.21
8.13
978
1200
1450
1620
1790
666
808
974
1080
1180
1200
1480
1810
2030
2260
743
911
1110
1240
1370
15700
19300
23400
26100
28900
400 × 300
8.0
10.0
12.5
14.2
16.0
85.4
106
131
148
166
109
135
167
189
211
25700
31500
38500
43000
47500
16500
20200
24600
27400
30300
15.4
15.3
15.2
15.1
15.0
12.3
12.2
12.1
12.1
12.0
1290
1580
1920
2150
2380
1100
1350
1640
1830
2020
1520
1870
2300
2580
2870
1250
1540
1880
2110
2350
31000
38200
46800
52500
58300
450 × 250
8.0
10.0
12.5
14.2
16.0
85.4
106
131
148
166
109
135
167
189
211
30100
36900
45000
50300
55700
12100
14800
18000
20000
22000
16.6
16.5
16.4
16.3
16.2
10.6
10.5
10.4
10.3
10.2
1340
1640
2000
2240
2480
971
1190
1440
1600
1760
1620
2000
2460
2760
3070
1080
1330
1630
1830
2030
27100
33300
40700
45600
50500
500 × 200
8.0
10.0
12.5
14.2
16.0
85.4
106
131
148
166
109
135
167
189
211
34000
41800
51000
56900
63000
8140
9890
11900
13200
14500
17.7
17.6
17.5
17.4
17.3
8.65
8.56
8.45
8.38
8.30
1360
1670
2040
2280
2520
814
989
1190
1320
1450
1710
2110
2590
2900
3230
896
1100
1350
1510
1670
21100
25900
31500
35200
38900
500 × 300
8.0
10.0
12.5
14.2
16.0
20.0
97.9
122
151
170
191
235
125
155
192
217
243
300
43700
53800
65800
73700
81800
98800
20000
24400
29800
33200
36800
44100
18.7
18.6
18.5
18.4
18.3
18.2
12.6
12.6
12.5
12.4
12.3
12.1
1750
2150
2630
2950
3270
3950
1330
1630
1990
2220
2450
2940
2100
2600
3200
3590
4010
4890
1480
1830
2240
2520
2800
3410
42600
52500
64400
72200
80300
97400
SOLID MECHANICS AND STRUCTURES
149
y
Square Hollow Sections – Dimensions and Properties
t
To BS EN 10210-2: 2006
D
x
x
y
D
Section
designation
Size
Thickness
D×D
t
mm
mm
Mass
per
metre
kg/m
Area
of
section
A
cm2
Second Radius
Elastic
Plastic Torsional
moment
of
modulus modulus constant
of area gyration
I
r
Z
S
J
cm4
cm
cm3
cm3
cm4
40 × 40
3.0
3.2
4.0
5.0
3.41
3.61
4.39
5.28
4.34
4.60
5.59
6.73
9.78
10.2
11.8
13.4
1.50
1.49
1.45
1.41
4.89
5.11
5.91
6.68
5.97
6.28
7.44
8.66
15.7
16.5
19.5
22.5
50 × 50
3.0
3.2
4.0
5.0
6.3
4.35
4.62
5.64
6.85
8.31
5.54
5.88
7.19
8.73
10.6
20.2
21.2
25.0
28.9
32.8
1.91
1.90
1.86
1.82
1.76
8.08
8.49
9.99
11.6
13.1
9.70
10.2
12.3
14.5
17.0
32.1
33.8
40.4
47.6
55.2
60 × 60
3.0
3.2
4.0
5.0
6.3
8.0
5.29
5.62
6.90
8.42
10.3
12.5
6.74
7.16
8.79
10.7
13.1
16.0
36.2
38.2
45.4
53.3
61.6
69.7
2.32
2.31
2.27
2.23
2.17
2.09
12.1
12.7
15.1
17.8
20.5
23.2
14.3
15.2
18.3
21.9
26.0
30.4
56.9
60.2
72.5
86.4
102
118
70 × 70
3.6
5.0
6.3
8.0
7.40
9.99
12.3
15.0
9.42
12.7
15.6
19.2
68.6
88.5
104
120
2.70
2.64
2.58
2.50
19.6
25.3
29.7
34.2
23.3
30.8
36.9
43.8
108
142
169
200
80 × 80
3.6
4.0
5.0
6.3
8.0
8.53
9.41
11.6
14.2
17.5
10.9
12.0
14.7
18.1
22.4
105
114
137
162
189
3.11
3.09
3.05
2.99
2.91
26.2
28.6
34.2
40.5
47.3
31.0
34.0
41.1
49.7
59.5
164
180
217
262
312
90 × 90
3.6
4.0
5.0
6.3
8.0
9.66
10.7
13.1
16.2
20.1
12.3
13.6
16.7
20.7
25.6
152
166
200
238
281
3.52
3.50
3.45
3.40
3.32
33.8
37.0
44.4
53.0
62.6
39.7
43.6
53.0
64.3
77.6
237
260
316
382
459
100 × 100
4.0
5.0
6.3
8.0
10.0
11.9
14.7
18.2
22.6
27.4
15.2
18.7
23.2
28.8
34.9
232
279
336
400
462
3.91
3.86
3.80
3.73
3.64
46.4
55.9
67.1
79.9
92.4
54.4
66.4
80.9
98.2
116
361
439
534
646
761
150
ENGINEERING TABLES AND DATA
y
Square Hollow Sections – Dimensions and Properties
t
To BS EN 10210-2: 2006
D
x
x
y
D
Section
designation
Size
Thickness
D×D
t
mm
mm
Mass
per
metre
kg/m
Area
of
section
A
cm2
Second Radius
Elastic
Plastic Torsional
moment
of
modulus modulus constant
of area gyration
I
r
Z
S
J
cm4
cm
cm3
cm3
cm4
120 × 120
5.0
6.3
8.0
10.0
12.5
17.8
22.2
27.6
33.7
40.9
22.7
28.2
35.2
42.9
52.1
498
603
726
852
982
4.68
4.62
4.55
4.46
4.34
83.0
100
121
142
164
97.6
120
146
175
207
777
950
1160
1380
1620
140 × 140
5.0
6.3
8.0
10.0
12.5
21.0
26.1
32.6
40.0
48.7
26.7
33.3
41.6
50.9
62.1
807
984
1200
1420
1650
5.50
5.44
5.36
5.27
5.16
115
141
171
202
236
135
166
204
246
293
1250
1540
1890
2270
2700
150 × 150
5.0
6.3
8.0
10.0
12.5
22.6
28.1
35.1
43.1
52.7
28.7
35.8
44.8
54.9
67.1
1000
1220
1490
1770
2080
5.90
5.85
5.77
5.68
5.57
134
163
199
236
277
156
192
237
286
342
1550
1910
2350
2830
3380
160 × 160
5.0
6.3
8.0
10.0
12.5
14.2
24.1
30.1
37.6
46.3
56.6
63.3
30.7
38.3
48.0
58.9
72.1
80.7
1230
1500
1830
2190
2580
2810
6.31
6.26
6.18
6.09
5.98
5.90
153
187
229
273
322
351
178
220
272
329
395
436
1890
2330
2880
3480
4160
4580
180 × 180
6.3
8.0
10.0
12.5
14.2
16.0
34.0
42.7
52.5
64.4
72.2
80.2
43.3
54.4
66.9
82.1
92.0
102
2170
2660
3190
3790
4150
4500
7.07
7.00
6.91
6.80
6.72
6.64
241
296
355
421
462
500
281
349
424
511
566
621
3360
4160
5050
6070
6710
7340
200 × 200
5.0
6.3
8.0
10.0
12.5
14.2
16.0
30.4
38.0
47.7
58.8
72.3
81.1
90.3
38.7
48.4
60.8
74.9
92.1
103
115
2450
3010
3710
4470
5340
5870
6390
7.95
7.89
7.81
7.72
7.61
7.54
7.46
245
301
371
447
534
587
639
283
350
436
531
643
714
785
3760
4650
5780
7030
8490
9420
10300
SOLID MECHANICS AND STRUCTURES
151
y
Square Hollow Sections – Dimensions and Properties
t
To BS EN 10210-2: 2006
D
x
x
y
D
Section
designation
Size
Thickness
D×D
t
mm
mm
Mass
per
metre
kg/m
Area
of
section
A
cm2
Second Radius
Elastic
Plastic Torsional
moment
of
modulus modulus constant
of area gyration
I
r
Z
S
J
cm4
cm
cm3
cm3
cm4
250 × 250
6.3
8.0
10.0
12.5
14.2
16.0
47.9
60.3
74.5
91.9
103
115
61.0
76.8
94.9
117
132
147
6010
7460
9060
10900
12100
13300
9.93
9.86
9.77
9.66
9.58
9.50
481
596
724
873
967
1060
556
694
851
1040
1160
1280
9240
11500
14100
17200
19100
21100
260 × 260
6.3
8.0
10.0
12.5
14.2
16.0
49.9
62.8
77.7
95.8
108
120
63.5
80.0
98.9
122
137
153
6790
8420
10200
12400
13700
15100
10.3
10.3
10.2
10.1
9.99
9.91
522
648
788
951
1060
1160
603
753
924
1130
1260
1390
10400
13000
15900
19400
21700
23900
300 × 300
6.3
8.0
10.0
12.5
14.2
16.0
57.8
72.8
90.2
112
126
141
73.6
92.8
115
142
160
179
10500
13100
16000
19400
21600
23900
12.0
11.9
11.8
11.7
11.6
11.5
703
875
1070
1300
1440
1590
809
1010
1250
1530
1710
1900
16100
20200
24800
30300
33900
37600
350 × 350
8.0
10.0
12.5
14.2
16.0
85.4
106
131
148
166
109
135
167
189
211
21100
25900
31500
35200
38900
13.9
13.9
13.7
13.7
13.6
1210
1480
1800
2010
2230
1390
1720
2110
2360
2630
32400
39900
48900
54900
61000
400 × 400
10.0
12.5
14.2
16.0
20.0
122
151
170
191
235
155
192
217
243
300
39100
47800
53500
59300
71500
15.9
15.8
15.7
15.6
15.4
1960
2390
2680
2970
3580
2260
2780
3130
3480
4250
60100
73900
83000
92400
112000
152
ENGINEERING TABLES AND DATA
y
Circular Hollow Sections – Dimensions and Properties
D
To BS EN 10210-2: 2006
x
x
t
y
Section
designation
Outside Thickness
diameter
D×D
t
mm
mm
Mass
per
metre
Area
of
section
Second Radius
Elastic
Plastic Torsional
moment
of
modulus modulus constant
of area gyration
kg/m
A
cm2
I
cm4
r
cm
Z
cm3
S
cm3
J
cm4
26.9
3.2
1.87
2.38
1.70
0.846
1.27
1.81
3.41
33.7
2.6
3.2
4.0
1.99
2.41
2.93
2.54
3.07
3.73
3.09
3.60
4.19
1.10
1.08
1.06
1.84
2.14
2.49
2.52
2.99
3.55
6.19
7.21
8.38
42.4
2.6
3.2
4.0
5.0
2.55
3.09
3.79
4.61
3.25
3.94
4.83
5.87
6.46
7.62
8.99
10.5
1.41
1.39
1.36
1.33
3.05
3.59
4.24
4.93
4.12
4.93
5.92
7.04
12.9
15.2
18.0
20.9
48.3
3.2
4.0
5.0
3.56
4.37
5.34
4.53
5.57
6.80
11.6
13.8
16.2
1.60
1.57
1.54
4.80
5.70
6.69
6.52
7.87
9.42
23.2
27.5
32.3
60.3
3.2
4.0
5.0
4.51
5.55
6.82
5.74
7.07
8.69
23.5
28.2
33.5
2.02
2.00
1.96
7.78
9.34
11.1
10.4
12.7
15.3
46.9
56.3
67.0
76.1
2.9
3.2
4.0
5.0
5.24
5.75
7.11
8.77
6.67
7.33
9.06
11.2
44.7
48.8
59.1
70.9
2.59
2.58
2.55
2.52
11.8
12.8
15.5
18.6
15.5
17.0
20.8
25.3
89.5
97.6
118
142
88.9
3.2
4.0
5.0
6.3
6.76
8.38
10.3
12.8
8.62
10.7
13.2
16.3
79.2
96.3
116
140
3.03
3.00
2.97
2.93
17.8
21.7
26.2
31.5
23.5
28.9
35.2
43.1
158
193
233
280
114.3
3.2
3.6
4.0
5.0
6.3
8.77
9.83
10.9
13.5
16.8
11.2
12.5
13.9
17.2
21.4
172
192
211
257
313
3.93
3.92
3.90
3.87
3.82
30.2
33.6
36.9
45.0
54.7
39.5
44.1
48.7
59.8
73.6
345
384
422
514
625
139.7
5.0
6.3
8.0
10.0
16.6
20.7
26.0
32.0
21.2
26.4
33.1
40.7
481
589
720
862
4.77
4.72
4.66
4.60
68.8
84.3
103
123
90.8
112
139
169
961
1177
1441
1724
168.3
5.0
6.3
8.0
10.0
12.5
20.1
25.2
31.6
39.0
48.0
25.7
32.1
40.3
49.7
61.2
856
1050
1300
1560
1870
5.78
5.73
5.67
5.61
5.53
102
125
154
186
222
133
165
206
251
304
1710
2110
2600
3130
3740
193.7
5.0
6.3
8.0
10.0
12.5
23.3
29.1
36.6
45.3
55.9
29.6
37.1
46.7
57.7
71.2
1320
1630
2020
2440
2930
6.67
6.63
6.57
6.50
6.42
136
168
208
252
303
178
221
276
338
411
2640
3260
4030
4880
5870
SOLID MECHANICS AND STRUCTURES
153
y
Circular Hollow Sections – Dimensions and Properties
D
To BS EN 10210-2: 2006
x
x
t
y
Section
designation
Outside Thickness
diameter
D×D
t
mm
mm
Mass
per
metre
Area
of
section
Second Radius
Elastic
Plastic Torsional
moment
of
modulus modulus constant
of area gyration
kg/m
A
cm2
I
cm4
r
cm
Z
cm3
S
cm3
J
cm4
219.1
5.0
6.3
8.0
10.0
12.5
14.2
16.0
26.4
33.1
41.6
51.6
63.7
71.8
80.1
33.6
42.1
53.1
65.7
81.1
91.4
102
1930
2390
2960
3600
4350
4820
5300
7.57
7.53
7.47
7.40
7.32
7.26
7.20
176
218
270
328
397
440
483
229
285
357
438
534
597
661
3860
4770
5920
7200
8690
9640
10600
244.5
8.0
10.0
12.5
14.2
16.0
46.7
57.8
71.5
80.6
90.2
59.4
73.7
91.1
103
115
4160
5070
6150
6840
7530
8.37
8.30
8.21
8.16
8.10
340
415
503
559
616
448
550
673
754
837
8320
10100
12300
13700
15100
273.0
6.3
8.0
10.0
12.5
14.2
16.0
41.4
52.3
64.9
80.3
90.6
101
52.8
66.6
82.6
102
115
129
4700
5850
7150
8700
9700
10700
9.43
9.37
9.31
9.22
9.16
9.10
344
429
524
637
710
784
448
562
692
849
952
1060
9390
11700
14300
17400
19400
21400
323.9
6.3
8.0
10.0
12.5
14.2
16.0
49.3
62.3
77.4
96.0
108
121
62.9
79.4
98.6
122
138
155
7930
9910
12200
14800
16600
18400
11.2
11.2
11.1
11.0
11.0
10.9
490
612
751
917
1030
1140
636
799
986
1210
1360
1520
15900
19800
24300
29700
33200
36800
355.6
14.2
16.0
120
134
152
171
22200
24700
12.1
12.0
1250
1390
1660
1850
44500
49300
406.4
6.3
8.0
10.0
12.5
14.2
16.0
62.2
78.6
97.8
121
137
154
79.2
100
125
155
175
196
15800
19900
24500
30000
33700
37400
14.1
14.1
14.0
13.9
13.9
13.8
780
978
1210
1480
1660
1840
1010
1270
1570
1940
2190
2440
31700
39700
49000
60100
67400
74900
457.0
8.0
10.0
12.5
14.2
16.0
88.6
110
137
155
174
113
140
175
198
222
28400
35100
43100
48500
54000
15.9
15.8
15.7
15.7
15.6
1250
1540
1890
2120
2360
1610
2000
2470
2790
3110
56900
70200
86300
96900
108000
508.0
10.0
12.5
14.2
16.0
123
153
173
194
156
195
220
247
48500
59800
67200
74900
17.6
17.5
17.5
17.4
1910
2350
2650
2950
2480
3070
3460
3870
97000
120000
134000
150000
154
Part 6
Mechanics
ENGINEERING TABLES AND DATA
MECHANICS
155
Statics
Moment of a force
Belt friction
The moment of a force F about any axis is (r × F · e) where r is
the vector from any point on the axis to any point on the line of
F, and e is the unit vector along the axis.
The limiting ratio of tensions at the ends of an arc of flat belt
subtending angle θ is
Laws of Coulomb friction
1. The friction force developed is independent of the
magnitude of the area of contact.
2. The limiting surface friction force is proportional to the
normal force.
3. At low velocity of sliding the kinetic friction force is
independent of the velocity and proportional to the
normal force.
T1
= eμθ
T2
where μ is the coefficient of friction. For a belt in a V-groove of
semiangle α ,
T1
= e μ θ cosec α .
T2
Funicular curve
The funicular curve for a load p per unit length is given by
2
d y
dx 2
=
p
H
where H is the polar distance. This is also the equation for the
shape of a flexible cable carrying a vertical load p per unit of
x, when H is the horizontal reaction.
Kinematics
In the following, v and a are velocity and acceleration, s is arc length and ω is angular velocity. Unit vectors are i , j , k for
Cartesian, and e with the relevant subscript for other coordinates.
Rectangular coordinates
v = vx i + vy j + vz k
a = ax i + ay j + az k
Normal and tangential components
v=
ds
e
dt t
a=
(ds /dt)
d2s
et −
en
2
R
dt
Spherical polar coordinates
v = r˙ er + r θ̇ eθ + r φ̇ sin θ eφ
.
a = r¨ − r θ̇ 2 − r φ̇ 2 sin2 θ er
.
+ r θ̈ + 2r˙ θ̇ − r φ̇ 2 sin θ cos θ eθ
.
+ r φ̈ + 2r˙ φ̇ sin θ + 2r φ̇ θ̇ cos θ eφ
2
z
r
Cylindrical coordinates
θ
v = r˙ er + r θ̇ eθ + ż ez
a = r¨ − r θ̇ 2 er + r θ̈ + 2r˙ θ̇ eθ + z̈ ez
O
φ
x
y
156
ENGINEERING TABLES AND DATA
Motion referred to a moving coordinate system
Y
r =R+ρ
P
y
ṙ = Ṙ + ρ̇ + ω × ρ
ρ
r̈ = R̈ + ω × (ω × ρ) + ω̇ × ρ + ρ̈ + 2ω × ρ̇
O'
r
ρ̇ is the velocity of P measured relative to O xyz, which has
angular velocity ω relative to OXYZ . The term 2ω × ρ̇ is
known as the Coriolis acceleration.
R
x
ω
z
O
In matrix form (see page 22),
X
Z
{r } = {R} + {ρ}
{r˙ } = {Ṙ} + { ρ̇} + [ω] {ρ}
{r¨ } = {R̈} + [ω]2 {ρ} + [ω̇] {ρ} + { ρ̈} + 2 [ω] { ρ̇}
Dynamics
In the following, m is mass, F force, H angular momentum
and M moment except where specified; V , T and E are,
respectively, potential, kinetic and total energy.
Potential energy
If F = grad φ , F is conservative. Then
r2
F · dr = φ2 − φ1
Newton’s laws
1. Every body stays in a state of rest or uniform motion in
a straight line unless it is acted on by a force which may
change that state.
2. The rate of change of momentum with respect to time is
equal to the force producing it. The change takes place
in the direction of the force.
r1
and the change in potential energy is
r2
−
F · dr = V2 − V1 .
r1
For an inverse square law
3. To every action there is an equal and opposite reaction.
F=−
Particle dynamics
Impulse and momentum
F dt = m v2 − m v1
t1
Moment of momentum
r ×F=
dH0
er
μm
r
if V = 0 at r → ∞.
Conservation of energy
For a conservative system V + T = constant.
dt
where H0 = r × m v.
Conservation of momentum
If in a system of particles only mutual interactions are involved,
the momentum of the system is constant.
Kinetic energy
r2
F · dr = 12 m v22 − 12 m v12 = T2 − T1
r1
r2
where μ is a constant of the field of force acting on m , and
V =
t2
μm
Central force motion
Kepler’s laws
1. Each planet has an elliptical orbit with the sun at a
focus.
2. The radius vector drawn from the sun to the planet
sweeps out equal areas in equal times.
3. The squares of the periods of the planets are
proportional to the cubes of the semi-major axes of the
elliptical orbit.
MECHANICS
157
The attractive force on a particle of mass m and distance r
from a fixed source of attraction may be written
F =
Euler’s equations
If x, y , z are principal axes,
μm
Mx = Ixx ω̇x + Iz z − Iy y ωy ωz
r2
where μ is the intensity of the source. The orbit of the particle
is a conic section (see page 23), with the source as focus,
having eccentricity.
= 1 + 2h 2 E/μ 2
My = Iy y ω̇y + ( Ixx − Iz z ) ωz ωx
Mz = Iz z ω̇z + Iy y − Ixx ωx ωy .
Kinetic energy
1/2
where h is the angular momentum of the particle per unit
mass, given by r 2 ω, and E the total energy per unit mass
given by
T = 12 m vC2 +
=
1
2
ω · HC
1
m {vC }T {vC }
2
+ 12 {ω}T [I ]{ω}
Gyroscopic motion
E = 12 v 2 − μ /r .
In Cartesian coordinates, a particle having the velocity
components vx0 , vy 0 at the point x0 , 0 has h = x0 vy 0 and its
orbit has the equation
2
h
h
h
r =
+ x 1 − vy 0 + y
v .
μ
μ
μ x0
An elliptical orbit (E < 0,
{H } = [I ] {ω} + [I ] {Ω} .
For principal axes
{M } = {Ḣ } + [ω]{H }
= [I ]{ω̇} + [ω] [I ] {ω} + [I ]{Ω̇} + [ω] [I ] {Ω} .
< 1) of area A has period
τ = 2A/h .
Rigid-body dynamics
In the following, x, y , z are Cartesian axes fixed on a rotating
body with inertia matrix [I ] defined on page 29; for matrix
notation see page 22.
Moment of momentum
Hx =
If ω is the angular velocity of the housing and Ω is that of the
rotor relative to the housing,
Ixx ωx − Ixy ωy − Ixz ωz
Hy = −Iy x ωx + Iy y ωy − Iy z ωz
Hz = −Iz x ωx − Iz y ωy + Iz z ωz
or
Lagrange’s equations
d
dt
∂T
∂q̇j
∂T
= Qj
∂qj
−
where qj is a generalized coordinate and Qj is a generalized
force.
For a conservative system
d
dt
∂L
∂q̇j
∂L
=0
∂qj
−
where L = T − V .
Euler’s differential equation
{H } = [I ] {ω}
General equations of motion
d
dx
My = Ḣy − ωx Hz + ωz Hx
Mz = Ḣz − ωy Hx + ωx Hy
or
{M } = {Ḣ } + [ω]{H }
= [I ]{ω̇} + [ω][I ]{ω} .
−
∂f
=0
∂y
where
If the origin is either fixed or at the centre of mass then
Mx = Ḣx − ωz Hy + ωy Hz
∂f
∂y
y =
dy
.
dx
Hamilton’s principle
t2
δ (T − V ) dt = 0
t1
158
ENGINEERING TABLES AND DATA
Vibrations
In the following, k is spring stiffness, c a viscous damping constant, ω0 an undamped natural or resonant frequency and m , M
masses.
Free vibration with viscous damping
the damping ratio is ζ = c /cc , and the logarithmic decrement
is
6
ln d = 2πn ζ / 1 − ζ 2 ≈ 2πn ζ
For a mass m the undamped natural frequency is
6
ω0 = k /m ,
the critical damping constant is
for small ζ , where d is the ratio of the amplitude at any instant
to that n periods later.
cc = 2 k m ,
Steady-state vibration with viscous damping
The governing equation is
ẍ + 2ζ ω0 ẋ + ω02 x =
and the phase angle φ is given by
P sin ωt
m
tan φ =
2ζ ω/ω0
1 − ω/ω0
2
.
for an applied force P sin ωt.
These relations yield the curves given below. Except for the
linear scales
they are identical to those on page
167. For
√
ζ < 1/ 2 the amplitude has a peak at ω/ω0 = 1 − 2ζ 2 .
The ratio of peak amplitude X to the steady displacement
X 0 = P /k is
1
)1/2
2
2
2
1 − ω/ω0
+ 2ζ ω/ω0
X
= '
X0
o
0
3.0
c
k
m
x
Magnifcation factor
X
X0
0.05
0.10
0.15
c
ζ=c
Phase angle φ
180
c
2.0
ζ = 1.0
o
0
0.25
1.0
90
0.05
0.15
0.375
o
0
1.0
2.0
3.0
4.0
ω
Frequency ratio
ω0
5.0
0.375
0.50
1.0
P sin!t
0
1.0
2.0
3.0
ω
Frequency ratio ω
0
4.0
5.0
MECHANICS
159
Rotating unbalance
The governing equation is
ẍ + 2ζ ω0 ẋ + ω02 x =
and
m ω 2e
sin ωt .
M
tan φ =
The peak amplitude X and the phase φ are given by
ω/ω0
MX
= '
me
1 − ω/ω0
2 2
2ζ ω/ω0
1 − ω/ω0
2
.
The curves below show the variation of X and φ with ω.
2
+ 2ζ ω/ω0
2
)1/2
o
0.05
0.05
3.0
0.10
!
0.15
0.15
m
e
x
k
2
c
k
2
90
0.05
0.25
0.50
1.0
ζz==1.0
o
0
2.0
o
0.25
0.25
MX
me
M
Phase angle φ
180
0
1.0
2.0
3.0
4.0
ω
Frequency ratio
ω0
5.0
0.375
0.375
0.50
0.50
1.0
c
ζ = c = 1.0
c
0
1.0
2.0
3.0
ω
Frequency ratio ω
0
4.0
5.0
160
ENGINEERING TABLES AND DATA
Displacement excitation
The governing equation is
and their phase difference is given by
ẍ + 2ζ ω0 ẋ + ω02 x = 2ζ ω0 ẏ + ω02 y .
tan φ =
The ratio of the peak amplitudes of the mass and the base is
⎡
⎤1/2
2
1 + 2ζ ω/ω0
2 2
+ 2ζ ω/ω0
1 − ω/ω0
X
⎢
= ⎣
Y
2
2ζ ω/ω0
1 − ω/ω0
2
3
+ 2ζ ω/ω0
2
.
These results are shown in the curves below. The amplitude
ratio is also the ratio of the transmitted force to the exciting
force in the vibration isolator with the constants k and c
separating mass m from ground.
⎥
⎦
o
180
3.0
0.10
0.15
m
0.25
x
Phase angle φ
0.05
X
Y
0.375
0.25
0.375
ζ = 0.50
o
0
1.0
5.0
2.0
3.0
4.0
ω
Frequency ratio
ω0
0.50
k
2
c
o
0
2.0
k
2
90
0.05
0.10
0.15
1.0
1.0
c
ζ=c
c
y = Y sin ! t
0
1.0
Ö2
2.0
3.0
4.0
5.0
ω
Frequency ratio ω
0
Vibration of beams of uniform section with uniformly distributed load
The natural frequencies of a beam of length L are given by
#
ω0 = k
Support
EI
m L4
1
2
3
3.52
22.0
61.7
9.87
39.5
88.8
15.4
50.0
104
22.4
61.7
121
where m is the mass per unit length.
The table gives the value of k for the fundamental, 2nd and
3rd harmonics with different support conditions.
(also Free - Free)
ELECTRICITY
Part 7
Electricity
161
162
ENGINEERING TABLES AND DATA
Electromagnetism
In the following, E, D are electric field strength and flux density; H, B are magnetic field strength and flux density; J, i are current
density and current; q is electric charge; ρ, ρs are volume and surface charge density; A, φ are vector and scalar potentials; V ,
S, l and r are volume, area, length and distance; ar , n are radial and normal unit vectors; subscripts t and n denote tangential
and normal; brackets represent retarded values; , μ and σ are permittivity, permeability and conductivity, and v is velocity.
Lorentz force
Coulomb law
The force between point charges q1 and q2 is
q1 q2 ar
4π r 2
.
Magnetic force
Biot-Savart law
dH =
The force on a unit positive charge is E + v × B .
i dl × ar
The force on a current element i dl is i dl × B and the force
on a distributed current is J × B per unit volume.
4πr 2
Maxwell’s equations
Integral form
D · dS = ρ dV
Gauss law
Faraday law
B · dS = 0
div B = 0
E · dl = − Ḃ · dS
curl E = −Ḃ
C
⎫
⎬
S
H · dl = (J + Ḋ) · dS
Ampère law
Work law
⎭
Magnetic circuit law
C
Equation of continuity
J · dS = − ρ̇ dV
div J = −ρ̇
V
Constitutive equations
B = μH ;
curl H = J + Ḋ
S
S
D = E;
div D = ρ
V
S
S
Differential form
Potential functions
J = σE
Boundary conditions
Et1 = Et2 ;
Ht1 = Ht2 (for no surface current)
Dn 1 − Dn 2 = ρs ;
B n 1 = Bn 2
and
A=
μ
4π
[ ρ]
dV
r
V
[J]
dV
r
or
A=
μ
4π
V
[i]
dl
r
C
B = curl A
E = −grad φ − Ȧ
At the surface of a perfect conductor
D = ρs n
1
φ=
4π
If div A = −μ φ̇ then ∇2 φ = μ φ̈ − ρ/ .
n × H = Js .
∇2 A = μ Ä − μ J
Energy in a field
1
2
The energy stored in an electric field is
V
1
2
energy stored in a magnetic field is
V
D · E dV and the
B · H dV .
2
Poisson’s equation:
∇ φ =−
Laplace’s equation:
2
∇ φ =0
ρ
ELECTRICITY
163
Analysis of circuits
In the following Z is impedance and Y is admittance. Unless otherwise stated, V and I are r.m.s. (or d.c.) values of voltage and
current.
Star-delta and delta-star transformation
Thévenin’s theorem and equivalent circuit
Star to delta:
Z
+
ZR ,YR
YY YB
YR =
YR + YY + YB
Delta to star:
Z'Y ,Y'Y
Z'B ,Y'B
ET
−
A two-terminal network containing sources and impedances
can always be replaced, as far as any load is concerned, by a
voltage source and impedance as shown. The value of ET is
the voltage which is measured at the terminals when
open-circuited. The value of Z is the impedance presented at
the open-circuited terminals when all independent sources are
de-activated, given by
ZY ,YY
ZB ,YB
Network of Sources
and Impedances
ZR =
ZY ZB
ZR + Z Y + Z B
Z'R ,Y'R
Self-inductance of two coils
If the self-inductances of the individual coils are L1 and L2 and
they are placed so that their mutual inductance is M , then the
self-inductance of the combination, when connected in series,
is
L1 + L2 ± 2M
dependent upon the relative directions of current flow.
When connected in parallel the self-inductance of the
combination is
L1 L2 − M 2
.
L1 + L2 ∓ 2M
Reciprocity theorem
Z =
VOC
ISC
.
Norton’s theorem and equivalent circuit
This is the equivalent to Thévenin’s theorem in terms of a
current source.
+
Network of Sources
and Impedances
IN
Z
−
IN is the current which flows in a short-circuit on the terminals
and Z is as defined for the Thévenin equivalent circuit. It
follows that the two equivalent circuits are related by
E T = IN Z .
Maximum power transfer from source to load
Z=R+jX
+
ZL = RL + j XL
E
_
If a current is produced at a point a in a linear passive network
by a voltage source acting at a point b, then the same current
would be produced at point b by the source acting at point a.
The principle also holds for voltages produced by current
sources.
The power in the load is a maximum when the load is matched
to the source, that is when
Superposition principle
and the load power is then E 2 /4R. If the load is restricted to
resistance RL only, then maximum power is obtained when
The response of a linear system to a number of
simultaneously applied excitations is equal to the sum of the
responses taken one at a time. When any one source is being
considered all the others are de-activated; de-activation
means that independent voltage sources are replaced by short
circuits and independent current sources by open circuits.
ZL = Z ∗
RL = |Z | .
or
RL = R,
X L = −X
164
ENGINEERING TABLES AND DATA
Power in a.c. circuits
Symmetrical components
If r.m.s. voltage V and current I are expressed as complex
numbers the mean power is given by
The following matrix relation gives the various sequence
components in terms of the unbalanced quantities.
⎡
⎤
V1
1
1
⎣ V2 ⎦ = ⎢
⎣ 1
3
V0
1
1 2
1 2
P = Re VI∗ = Re V∗ I = |V| | I | cos φ
⎡
where φ is the angle of I relative to V; also the reactive
component of volt-amperes is given by
1 2
Q = I m V∗ I = |V| | I | sin φ
where a is the operator
if reactive VA is defined as +ve for leading current, or
if reactive VA is defined as +ve for lagging current. Reactive
VA is sometimes referred to as reactive (or imaginary) power
and its units designated VAR.
Power measurement in three-phase circuits
It is normal to quote line voltage V and line current I rather
than phase values, but the phase angle φ relates to the
conditions in a phase. The total power is
3 V I cos φ
for balanced star or delta connections.
In a system fed through n wires, it is necessary to take n − 1
power readings for the sum to give the true total power.
Therefore for a delta-connected load and a three-wire
star-connected load, two wattmeter readings are sufficient. If
the loads are balanced then
W − W 1
2
tan φ = 3
.
W1 + W2
Parameter set
Impedance, z
Admittance, y
Hybrid, h
Inverse hybrid, g
Transmission, a
Inverse transmission, b
V1
V2
I1
I2
V1
I2
I1
V2
V1
I1
V2
I2
1
2
⎤⎡
⎤
a2
VR
⎥⎣
a ⎦ VY ⎦
VB
1
√ −1 + j 3 .
The matrix can be inverted to give
⎤ ⎡
1
VR
2
⎣ VY ⎦ = ⎢
⎣ a
VB
a
⎡
1 2
Q = I m VI∗ = −|V| | I | sin φ
P =
a
a2
1
1
a
a2
⎤⎡
⎤
1
V1
⎥⎣
1 ⎦ V2 ⎦ .
V0
1
√
Other forms give 012 sequence rather than 120, or 1/ 3
multipliers in both equations.
Two-port or four-terminal networks
Each of the six possible pairs of equations which define the
behaviour of a two-port can be written in terms of a matrix of
parameters as shown in the table below. The requirement for
reciprocity and the additional requirement for symmetry are
also shown for each. A two-port containing only passive linear
elements, and no independent or controlled source, satisfies
the reciprocity condition.
I1
I2
V1
Equation
'
)
z11 z12
I1
=
z21 z22
I2
'
)
y11 y12
V1
=
y21 y22
V2
'
)
h11 h12
I1
=
h21 h22
V2
'
)
g11 g12
V1
=
g21 g22
I2
'
)
a11 a12
V2
=
a21 a22
I2
'
)
b11 b12
V1
=
b21 b22
I1
V2
Reciprocity condition
Symmetry condition
z12 = z21
z11 = z22
y12 = y21
y11 = y22
h12 = −h21
det H = 1
g12 = −g21
det G = 1
det A = −1
a11 = −a22
det B = −1
b11 = −b22
ELECTRICITY
165
Sometimes, especially in power systems, I2 is defined in the
opposite sense and two parameters in each set change sign
accordingly. In that case it is common to use A, B , C , D for
the transmission parameters, thus
V1
I1
'
=
A
C
B
D
)
V2
I2
I1
The image impedances ZM1 and ZM2 are functions of the
parameters such that when connected as shown the
impedance to the right at terminals 1 is ZM1 and the
impedance to the left at terminals 2 is ZM2 . Then:
3
.
ZM1 =
#
AB
=
CD
a11 a12
; ZM2 =
a21 a22
3
#
BD
=
AC
a12 a22
a11 a21
I2
V1
V2
The iterative impedance Z0 of a two-port is a function of its
parameters such that when Z0 is connected to the output
terminals the impedance at the input terminals is also Z0 .
1
ZM1
2
ZM 2
For a symmetrical circuit ZM1 = ZM2 = Z0 .
Resonance and response
Resonant frequency and quality factor
Half power bandwidth; half power or 3 dB frequencies
The undamped natural frequency ω0 of a series or parallel
LCR circuit is that for which the impedance Z or admittance Y
is real. This is also the resonant frequency of these circuits.
For serial and parallel resonant circuits, the value of δω, Δω
corresponding to 2 Q = 1 is
The quality factor Q may be defined generally as
Q = 2π ×
maximum energy stored in L or C
.
energy dissipated per cycle
Δω = ±
ω0
2Q
.
The corresponding frequencies ω0 ± Δω are known as the
half-power or 3 dB frequencies or points. The interval between
them, 2Δω, is the half-power bandwidth. Hence an alternative
definition of Q is
Q=
ω0
2Δω
.
Series resonant circuit
R
L
C
The impedance just off resonance, at a frequency
ω = ω0 ± δω = ω0 (1 ± )
At the resonant frequency the impedance is minimum and
ω0 = √
1
.
LC
The quality factor Q is the voltage magnification at resonance,
that is
ω0 L
voltage across L or C
1
.
Q=
=
=
R
voltage across whole circuit
ω0 C R
where
= δω/ω0 1, is given by
Z = R (1 ± j2 Q ) .
If 2 Q = 1, then |Z | =
I is then
√
2R. For constant voltage, the current
1
I = √ I0
2
where I0 is its maximum value. The power is halved (or 3 dB
less than the maximum).
166
ENGINEERING TABLES AND DATA
Parallel resonant circuit
L
R'
r
º
L'
C'
C
The quality factor Q is the current magnification at resonance,
that is
Q=
current through L or C
R
= ω0 C R .
=
current through whole circuit
ω0 L
The admittance just off resonance is
1
(1 ± j2 Q ) .
R
√
2
. For constant current, the voltage
If 2 Q = 1, then |Y | =
R
V is then
Y =
The circuits above are equivalent with L = L, C = C and
2
R = Q r , if Q = ω0 L/r 1. The right-hand version gives
results complementary to those for the series resonant circuit.
Its resonant frequency, at which the admittance Y is a
minimum, is
ω0 = √
1
.
LC
1
V = √ V0
2
where V0 is its maximum value, and the power is then half (or
3 dB less than) the maximum.
Bode diagrams
The complete diagram for any complex function of frequency H ( jω) has two parts, one showing magnitude and the other
argument or phase, as functions of frequency. The frequency is plotted logarithmically and the magnitude is expressed in
decibels, i.e. as 20 log10 |H | dB. In many cases the curves are closely approximated by their asymptotes.
First-order system
Vin
=
j ωT
1 + j ωT
T = CR
C
(dB)
Maximum
error = 3 dB
Slope of asymptote
= 20 dB/decade
≈ 6 dB/octave
−10
Vout
Vin
Vout
0
20 log10
A first-order system with one reactive element gives rise to
functions with one time-constant T and a Bode diagram with a
breakpoint or corner frequency at ω = 1/T . The example at
right gives the diagram for the voltage transfer function of the
circuit shown.
−20
−30
Vout
−40
0.01
0.1
1
10
100
ωT
90
Vout
(degrees)
Vin
R
Straight line
approximation,
o
slope = 45 /decade
60
30
Maximum
o
error = 5.7
arg
Vin
0
0.01
0.1
1
ωT
10
100
ELECTRICITY
167
Second-order system
A second-order system with two different reactive
(energy-storing) elements gives rise to functions containing a
factor of the form
For small ζ the system shows resonant behaviour, as in the
example of the voltage transfer function and its Bode diagram
given below. In this case the peak response occurs at the
resonant frequency
( jω)2 + 2jζ ωω0 + ω02
ωr = ω0
or
jω/ω0
2
1 − 2ζ 2 .
There is therefore no resonant peak when
+ 2jζ ω/ω0 + 1
1
ζ > √
2
in which ω0 is the undamped natural frequency and ζ is the
damping ratio, defined generally as
ζ =
6
1
Q< √
2
or
and the maximum response then occurs for d.c., ω = 0, and
has magnitude 0 dB.
actual damping
critical damping
(in reference to the transient response: see next section) and
related to the Q -factor by
ζ =
1
.
2Q
20
R
(dB)
10
L
0
C
Vout
20 log10
Vout
Vin
Vin
ζ = 0.05
0.10
0.15
0.20
0.25
0.3
−10
0.4
0.5
0.6
0.8
1.0
−20
−30
Vout
Vin
=
1
−40
0.1
2
( jω) LC + jωRC + 1
0.2
0.3
0.4 0.5 0.6
0.8 1.0
2
3
4
5
6
8
10
2
3
4
5
6
8
10
ω/ω0
6
1
ω0 = 1/ LC , ζ = R C/L
2
0
ζ = 0.05
0.10
0.15
0.20
0.25
−20
−40
arg
Vout
(degrees)
Vin
0.3
−60
0.4
0.5
−80
0.6
0.8
1.0
−100
−120
−140
−160
−180
0.1
0.2
0.3
0.4 0.5 0.6
0.8 1.0
ω/ω0
168
ENGINEERING TABLES AND DATA
Transient response
First-order system
1.0
The transient response of a first-order system shows an
exponential approach to the steady-state output. For the
first-order RC circuit shown on page 166 the response Vout to
a unit step Vin is as shown.
0.8
Vout
0.6
0.4
0.2
0.0
0
1
2
3
4
t/ CR
5
Second-order system
The frequency of oscillation is the damped natural frequency
given by
The transient response of a second-order system is oscillatory
if the damping is less than critical, i.e. for underdamping, when
ζ < 1. For the series RLC circuit ζ = 12 R C/L and oscillation
occurs for R less than the critical value
3
L
.
Rc = 2
C
For the parallel R L C circuit ζ = 1/2R
L /C and
oscillation occurs when R exceeds the critical value
#
1 L
.
Rc =
2 C
1.8
#
6
2
ωn = ω0 1 − ζ = ω0 1 −
1
(2Q )2
and the amplitude decays as e −α t where α = ζ ω0 .
The response Vout to a unit step Vin for the series RLC circuit
on page 165 is shown below for various values of ζ .
ζ = 0.1
1.6
0.2
1.4
0.4
1.2
0.7
1.0
Vout
0.8
1.0
0.6
2.0
0.4
0.2
0.0
0
1
2
3
4
5
6
7
ω0 t
8
9
10
11
12
13
ELECTRICITY
169
Poles and zeros
Both the response to a sinusoidal input and the transient
response of a system can be deduced from the position of the
poles and zeros of the appropriate function H (s ). The transfer
function is written in the form
K (s − z1 ) (s − z2 ) · · ·
.
(s − p1 ) (s − p2 ) · · ·
H (s ) =
where z1 , z2 , . . . , p1 , p2 , . . . are its zeros and poles and can be
plotted as points on an Argand diagram.
For the examples given earlier, typical diagrams are shown
below (pole ×, zero ◦)
Second-order ζ < 1
First-order
Im(s)
Im(s)
x
x
-1
sin ζ
ω0
Re(s)
-1/T
ωn
-α
H (s ) =
s
s + 1/T
Re(s)
-ωn
x
T = CR
2
H (s ) =
ω0
s 2 + 2ζ ω0 s + ω02
ω0 = √
1
LC
ωn = ω0 1 − ζ 2
1
ζ = R
2
3
C
L
α = ζ ω0
Electrical machines
In the following Φ is the flux per pole, P the number of poles, n the number of turns in series per phase, N the speed (rev s−1 ), ω
the speed (rad s−1 ) and f the frequency (Hz).
DC machines
EMF =
ΦZ N P
A
For ordinary windings A = P if lap-wound, and A = 2 if
wave-wound (whatever the value of P ).
where Z is the total number of conductors and A the number
of parallel paths in which they are arranged.
Torque =
ΦZ P I
2πA
If flux is constant, ΦZ P /2πA is the emf or torque constant K
of the machine such that
EMF = K ω
and
Torque = K I .
where I is the total armature current.
AC machines
EMF = 4.44Φn f
Volts (rms)
−1
Synchronous speed = 120 f /P rev min
= 60 f /p rev min−1
= 2π f /p rad s−1
where p is the number of pole-pairs.
Regulation =
change in voltage (or speed)
between no load and full load
rated voltage (or speed)
170
ENGINEERING TABLES AND DATA
where X = X 1 + X 2 , and has a maximum
Transformers
Equivalent circuits:
I1
R1
X1
R2
X2
I2
Tmax =
V2
2X ωs
at the slip
V1
Rm
n1
Xm
n2
V2
σ =
R2
X
provided that R1 X .
n1 2
R1+ n R2
2
n1 2
X1+ n X2
2
Synchronous machines
Equivalent circuit and phasor diagram for one phase
(cylindrical rotor, alternator operation):
V1
Rm
n1
Xm
n2
V2
R
Xs
E
I
Xs I
+
An impedance Z2 in the secondary may be considered to be
replaced by an equivalent impedance Z2 in the primary, where
Z2 = Z2
n1
n2
V
−
RI
I
2
V = E − (R + jX s ) I
.
For an ideal transformer (Rm , X m = ∞ ; R1 , X 1 , R2 , X 2 = 0)
n1
V1
=
,
n2
V2
E
φ V
I2 n2 = −I1 n1 .
where R is armature resistance and X s is synchronous
reactance (including leakage and armature reaction). Usually
R Xs.
For operation as a motor the same equivalent circuit and sign
convention for I can be used; the phasor diagram is then:
Asynchronous or induction motor
Equivalent circuit:
R 1 + R2
I
X1 + X2
E
Rm
V
Xm
R2
1
−1
σ
σ =
ωs − ω
ωs
where ωs and ω are the synchronous and actual speeds
respectively.
Xs I
Alternatively the positive direction of I can be redefined for I ,
to give
R
R2 and X 2 are values referred to the stator as determined by a
locked-rotor test. The slip is defined as
Xs
The torque per phase is
T =
2
V R2 σ /ωs
(R1 σ + R2 )2 + σ 2 X 2
RI'
I'
V
+
E
V
−
Xs I'
I'
E
in which case E = V − (R + jX s ) I .
The power per phase if losses are neglected is
The slip is also given by
rotor copper loss
σ =
.
gross power input to rotor
RI
V
V E sin δ
.
Xs
The synchronous reactance X s can be determined from openand short-circuit tests of the machine, when driven as a
generator, as
Xs =
open-circuit voltage
short-circuit current
at a given excitation current.
ELECTRICITY
171
Wave propagation
The propagation constant of a medium with permeability μ ,
permittivity and conductivity σ is
1/2
γ = {jωμ (σ + jω )}
1
= jω μ
− jσ /ω
21/2
which may be written as α + jβ or as k + jk (see page 173).
If μ = μ0 it may be written as
γ =j
ω
c
r
=j
ω
(n − jk )
c
Reflection and transmission
For normal incidence from medium 1 to medium 2, the
coefficients of reflection R and transmission T for the E-field
are
R=
Z2 − Z1
Z2 + Z1
#
Z1 =
The ratio of E to H is the characteristic impedance of the
medium given by
#
=
.
2
n2
n1
θr = θi .
θi
3
Z =
μ
For a good conductor σ ω
ωμ σ
,
2
θr
Medium 1
Medium 2
.
θt
and
3
Z = (1 + j )
ωμ
.
2σ
Poynting vector
Reflection and transmission coefficients
For a wave with its E-field parallel to the boundary (TE case)
the coefficients of reflection and transmission are
RTE =
The Poynting vector is
S=E×H
Z2 cos θi − Z1 cos θt
Z2 cos θi + Z1 cos θt
and
and for orthogonal fields in an isotropic non-conducting
medium of permeability μ and permittivity , it has the value
S = EH = E2
μ2
.
γ = jω μ ,
γ = (1 + j )
Z2 =
For the reflected wave
For a non-conducting lossless medium
3
#
A wave incident at angle θi to the normal is refracted to an
angle θt by Snell’s law:
sin θt
Characteristic impedance
μ1
1
sin θi
jωμ
σ + jω
2Z2
Z2 + Z1
where
where c is the velocity of light in vacuum, r is the complex
dielectric constant
− jσ /ω / 0 and n − jk is the refractive
index in which k now represents absorption and is known as
the extinction coefficient (values of n and k for various
materials are on page 52).
Z =
T =
6
/μ = H 2
6
μ/ .
For sinusoidally time-varying fields the time-averaged
Poynting vector is
1
2
S = Re E × H∗
where E and H are r.m.s. values. In a plane wave the average
power density is 12 E H where E and H are peak values.
TTE =
2Z2 cos θi
Z2 cos θi + Z1 cos θt
.
For a wave with its H-field parallel to the boundary (TM case)
RTM =
Z2 cos θt − Z1 cos θi
Z1 cos θi + Z2 cos θt
and
TTM =
2Z2 cos θi
Z1 cos θi + Z2 cos θt
.
In general,
Power reflectivity = |R|2
Power transmittivity = 1 − |R|2 .
172
ENGINEERING TABLES AND DATA
Antennas
Isotropic radiator
Non-isotropic radiators
An isotropic radiator emitting a mean power P produces a
mean S of P /4πr 2 at a distance r , and the r.m.s. electric field
in free space is then
Dipole
√
E=
Hertzian
30P
.
r
Half-wave
Current
distribution
Radiation
resistance
Maximum
gain
Beam
width
Aperture
constant
80π 2 L2
λ2
1.5
90
◦
3λ 2
8π
half-cosine
73.1 Ω
1.64
78
◦
30λ 2
73π
Antenna gain and directivity
Here L is the total length of the antenna and λ the wavelength.
The gain G of an antenna is the ratio of the power it emits per
steradian in a given direction to the power per steradian
emitted by a reference antenna of the same total power.
Usually, the direction chosen is that of maximum power
density and the reference antenna is an isotropic radiator:
4π × power/solid angle in direction of maximum
G=
total power received
The directivity may be measured either by the maximum gain
or by the beam width, the angle contained between points at
which the power density is half of the maximum.
Radiation resistance
The radiation resistance Rr of an antenna is such that the
antenna radiates power I 2 Rr when fed with r.m.s. current I .
Effective aperture
The aperture of a receiving antenna is the ratio of the power
received to the power density of the incident field. The
effective aperture of an antenna is greatest when it is
matched; for a lossless antenna of gain G it is then given by
λ 2 G /4π, where λ is the wavelength. The power received by a
matched antenna is
P =
V2
4Rr
where V is the integral of the induced electric field along its
length. The gain G and effective aperture A are related by the
expression
G=
4πA
λ2
for radiation with wavelength λ.
Parabolic antenna
Gain
Aperture
Beam width
to 3 dB
ηπ 2 D 2
ηπD 2
4
70λ ◦
D
λ2
Here η is the transmission efficiency, D the dish diameter, and
λ the wavelength.
ELECTRICITY
173
Transmission lines, optical fibres and waveguides
Transmission lines
IS
IR
Attenuation and wavelength
The voltage at distance x can be written in the form
VS
VR
l
V = Ae +γ x + B e −γ x
= Ae α x e jβ x + B e −α x e −jβ x
In the following, R, G , L, C are the resistance, conductance,
inductance and capacitance per unit length.
The propagation constant γ is
γ = α + jβ =
6
(R + jωL) (G + jωC ) .
The characteristic impedance Z0 is
#
Z0 =
Phase, group velocities and dispersion
The phase velocity is vp = ω/β and the group velocity is
vg = dω/dβ .
R + jωL
.
G + jωC
The voltage and current at a point distant x from the sending
end are given by
'
Vx
Ix
⎡
)
cosh γ x
1
=⎣
sinh γ x
−
Z0
−Z0 sinh γ x
cosh γ x
⎤
⎦
'
VS
IS
)
.
VS
IS
⎡
)
cosh γ l
=⎣ 1
sinh γ l
Z0
Z0 sinh γ l
cosh γ l
⎤
⎦
'
On a dispersive line waves of different frequencies travel at
different velocities and the relation between ω and β is known
as the dispersion relation. In these cases the envelope of a
wave containing different frequencies travels with the group
velocity vg .
Complex propagation constant
The matrix can be inverted with x = l to yield
'
where A, B are determined by the end conditions, and the two
terms represent travelling waves in the negative and positive
x-directions respectively; α represents an attenuation and β a
change of phase with distance, and β λ = 2π where λ is
wavelength.
VR
IR
)
.
Instead of β a propagation constant or wave number k may be
used; the attenuation constant α may be represented as the
imaginary part of k . Introducing a complex k as k − j k gives
j (k − j k ) = k + j k ≡ α + j β .
Input impedance and reflection coefficient
If the line is terminated with an impedance ZL the input
impedance is
ZS = Z0
ZL + Z0 tanh γ l
Z0 + ZL tanh γ l
= Z0
1 + ρ e −2γ l
1 − ρ e −2γ l
where ρ is the reflection coefficient for voltage, defined as
ZL − Z0
ZL + Z 0
=ρ=
voltage of reflected wave
.
voltage of incident wave
For a matched line (ZL = Z0 )
ρ=0
and ZS = Z0 .
For a short-circuited line (ZL = 0)
and ZS = Z0 tanh γ l .
ρ = −1
For an open-circuited line (ZL → ∞)
ρ=1
and ZS = Z0 coth γ l .
Voltage standing-wave ratio
The voltage standing-wave ratio is
VSWR =
1 + | ρ|
1 − | ρ|
.
Special cases
For a lossless line
R = G = 0;
3
Z0 =
α = 0;
L
;
C
vp = vg = √
γ = j β = j ω LC
1
.
LC
For a low-loss line
R ωL ;
G ωC
3
3 1
C
L
;
α=
R
+G
2
L
C
β ≈ ω LC .
If L/R = C/G the line is distortionless since both the
attenuation and the velocity of propagation are then
independent of frequency, and
γ = α + jβ =
3
Z0 =
R
=
G
RG + j ω LC
3
L
.
C
174
ENGINEERING TABLES AND DATA
Parameters for transmission lines
Characteristic
impedance Z0
Configuration
Parallel wires
(air)
d
1/2
μ/
d
Capacitance
per unit length
cosh
π
−1
D
Wire over
ground
1/2
μ/
d
D
2π
W
1/2
μ/
d
Microstrip
cosh−1
D
d
Stripline
μ/
1/2
D
2W
1/2
D
W
D
W
μ/
d
D
2D
d
μ
cosh−1
2π
2π
cosh−1 2D /d
2πD
W +d
D
d
2D
d
ln 2πD /(W + d )
ln
π
1/2
μ/
d
D
ln
2π
W
D
2
πD
W +d
D
d
π
ln πD /(W + d )
2π
ln D /d
2D W
d 2D
μ
2πD
ln
2π
W +d
2π
Conditions
D
W
μ
W
D
1/2
μ/
d
Coaxial
cosh−1 D /d
W
D
Parallel strip
μ
−1
cosh
π
π
W
D
ln
2π
D
W
1/2
μ/
D
d
Inductance
per unit length
2D W
μ
D
2W
D W
d D
μ
D
W
D W
d D
μ
πD
ln
π
W +d
D W
μ
D
ln
2π
d
W d
Skin depth
&6
The skin depth for plane conductors at frequency f (hertz) is given by δ = 1
conductivity. The variation of δ with f for several metals is shown below.
πf μ σ where μ is the permeability and σ the
10
Skin depth (mm)
Solder
1
Brass
Aluminium
-1
10
10
10
10
Copper, Silver
-2
-3
-4
10
2
10
10
3
10
4
10
5
10
6
Frequency (Hz)
10
7
10
8
10
9
10
10
ELECTRICITY
175
Optical fibres
For a symmetric-slab dielectric waveguide the single mode
guidance conditions are the following:
x = d /2
Medium 1
ε1
Medium 2
ε2
x
z
x = -d /2
ε1
Medium 1
2
Brewster angle
The Brewster angle θB is given by
tan θB =
n2
n1
When n1 > n2 the critical value of θi for total reflection is given
by
− kz2
θc = sin−1
and
6
kz2 − ω 2 μ
=
and is the value of θi at which R = 0 for a wave with the
electric field in the plane of incidence.
= propagation constant in x direction in medium 2
αx =
2
1
where the dispersion relation gives
6
ω 2μ
The acceptance angle for optical fibre is the angle to the axis
over which light entering the fibre will be guided.
#
Even TE solutions:
k2x d
αx
tan
=
2
k2x
k2x =
Acceptance angle
n2
.
n1
Numerical aperture
1
= propagation constant in x direction in medium 1.
Odd TE solutions:
αx
k2x d
=−
.
cot
2
k2x
The numerical aperture NA of a lens defines the angle to the
axis over which light can enter or exit the lens, and is given by
NA = n sin θ
where n is the refractive index of the medium surrounding the
lens (normally air, n = 1).
Rectangular waveguides
For any waveguide the phase shift (or wave number) is
β = β02 − βc2
1/2
where
β02 = ω 2 μ
b
y
and for a rectangular waveguide in a TE or TM mode
βc2 =
m π 2
a
+
n π 2
b
.
In this m and n denote the number of half-cycles along the x
and y coordinates, for which the waveguide internal
dimensions are a and b. At cut-off β0 = βc , and for
evanescence β0 < βc . The waveguide wavelength λ g is given
by 2π/β .
a
z
x
Electric (—) and magnetic (- - -) fields at a particular instant for
a rectangular waveguide in the TE10 mode (m = 1, n = 0).
176
ENGINEERING TABLES AND DATA
Resonant cavities
In the table, λ is the resonant wavelength and δ the skin depth, given by
permeability μ at angular frequency ω.
Resonator type
2/ωμ σ for material of conductivity σ and
λ
Q
√
2 2a
1
0.353λ
δ
1 + 0.177λ /h
2h
E
TE101
2a
2a
a
2h
Circular cylinder TM010
E
a
Sphere
2.61a
0.383λ
1
δ
1 + 0.192λ /h
2.28a
0.318λ
δ
b
E
Co-axial TEM
λ
4δ + 7.2h δ/b
2h
4h
For optimum Q
b/a = 3.6 and Z0 = 77 Ω
a
Communication systems
Decibel
A decibel is the ratio of two powers: 10 log10 P2 /P1 .
Noise
Thermal noise power delivered to a matched load is
P = k T0 B
where k is Boltzmann’s constant, T0 is the temperature
(usually assumed to be 290 K unless given otherwise) and B
is the bandwidth in hertz.
The shot noise power delivered to a load R from a device
carrying current I is
P = 2eI B R
where e is the charge on an electron and B is the bandwidth
in hertz.
Noise figure and noise temperature
The noise figure F of a system is the ratio of the signal to
noise powers at the output and input of a matched system,
referred to the system input, given by
F =
SNR in
SNR out
.
The noise temperature T represents the additional noise
power k T B from a system, again referred to the input.
ELECTRICITY
177
Modulation
Amplitude modulation (DSB-AM)
If the baseband signal v (t) is modulated onto a carrier
vc (t) = Vc cos ωc t, the modulated signal vm (t) takes the form
vm (t) = (Vc + v (t)) cos ωt .
vm (t) = Vc cos ωc t + k v (t) dt .
For the special case of a sinusoidal signal v (t) = Vs cos ωs t
then
The modulation index m is defined as
m=
where dφ /dt is the instantaneous frequency of the carrier
wave and k is the frequency deviation constant. Hence
rms value of v (t)
.
rms value of carrier wave
vm (t) = Vc cos (ωc t + β sin ωs t)
For the special case of a sinusoidal signal v (t) = Vs cos ωs t
then
where the modulation index β is defined by
β=
vm (t) = Vc (1 + m cos ωs t) cos ωt .
k Vs
.
ωs
Carson’s rule
Some other amplitude modulation schemes
DSB-SC-AM (DSBSC)
SSB-AM (SSBSC)
VSB-AM (SSBVC)
QAM
Double sideband suppressed carrier
Single sideband suppressed carrier
Single sideband vestigial carrier
Quadrature amplitude modulation
Frequency modulation (FM)
The phase φ of the carrier wave varies as
dφ
= ωc + k v (t)
dt
The bandwidth required for transmission of an FM signal is
approximately equal to 2 ( β + 1) ωs .
Phase modulation (PM)
The phase φ of the carrier wave is given by
φ = k v (t)
and hence
vm (t) = Vc cos (ωc t + k v (t)) .
Digital systems
Channel capacity
−1
The capacity of an ideal channel is 2B log2 Q binits s , where B is the bandwidth and each signal has Q equiprobable levels.
The capacity of a real channel is B log2 (1 + SNR ) (Shannon-Hartley theorem) where SNR is the signal-to-noise ratio.
Pulse Code Modulation Schemes
Scheme
ASK
FSK
PSK
QAM
QPSK
CDMA
Signal information conveyed by
Amplitude shift keying
Frequency shift keying
Phase shift keying
Quadrature amplitude modulation
Quadrature phase shift keying
Code division multiple access
Amplitude of the pulses
Frequency of the pulses
Phase of the pulses
Combination of phase and frequency
Combination of phase and frequency
Frequency or pulse sequence
178
ENGINEERING TABLES AND DATA
Components
Resistors
Colour
code
Letter code for multipliers for resistors
Figure Multiplier Tolerance Temperature
coefficient
◦ −1
(ppm C )
Silver
Gold
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Grey
White
0
1
2
3
4
5
6
7
8
9
These are inserted in place of the decimal point.
Preferred values (BS 2488 E24 series)
−2
10
−1
10
1
1
10
2
10
3
10
104
105
6
10
107
108
9
10
R = 1, K = 103 , M = 106 , G = 109 , T = 1012
10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 43,
47, 51, 56, 62, 68, 75, 82, 91.
±1%
±2%
±5%
±10%
100
50
25
15
Capacitors
Letter code for multipliers for capacitors
−12
10
5
p = 10
, n = 10−9 , μ = 10−6 , m = 10−3
1
4 bands: 2 significant figures, multiplier, tolerance
5 bands: 3 significant figures, multiplier, tolerance
6 bands: 3 significant figures, multiplier, tolerance,
temperature coefficient
If 4 or 5 bands, read from the band nearest to an end, left to
right; if 6 bands, that indicating the temperature coefficient
should be wider and indicates the right-hand end.
Semiconductor devices
Graphical symbols
Diode
or
Photodiode
Schottky diode
LED
Zener diode
ELECTRICITY
179
BJT (Bipolar Junction Transistor)
Typical characteristics and the polarities are shown below.
n-type
iC (mA)
C+
iB = 300 μA
30
vCE
iB
0.5 mA
250
5V
20 V
200
B
20
150
10
electron
flow
Enpn
0
0
2
4
6
100
50
0
8 10 12
vCE
vBE
iB
+
+
+
vCE
vBE
iB
−
−
−
vBE
vCE
0
1V
p-type
−12 −10 −8 −6 −4 −2 0
0
−50
−100
−10
CvCE
B
−1 V
0
vBE
−150
−200
−20
−250
hole
flow
−20 V
−5 V
−30
vCE
iB = −300 μA
E+
pnp
−0.5 mA
iB
iC (mA)
JFET (Junction Field-Effect Transistor)
Typical characteristics and the polarities are shown below.
n-channel
D
iD
iD
vGS
−2
vDG
G
iD
0
−1
I DSS
−3
vGS
Vth
vDS
vGS
vDS
vGS
iD
Vth
IDSS
+
−
+
−
+
vDS
vGS
iD
Vth
IDSS
−
+
−
+
−
S
p-channel
D
iD
iD
vDS
vDG
G
vGS
S
iD
+3
vGS +2
+1
0
Vth
vGS
I DSS
IDSS = drain current at vGS = 0
Vth = pinch-off voltage, i.e. vGS for iD = 0. The symbol Vp is frequently used.
180
ENGINEERING TABLES AND DATA
MOSFET (Metal-Oxide-Silicon FET)
This device is also known as MOST or IGFET, Insulated-Gate FET. The characteristics and polarities in each case are as shown.
The numbers are typical values of vGS in volts.
n-channel enhancement
D+
+9
iD
Drain
Gate
iD
+8
Substrate
G
+7
Source
+6
vDS
S-
Vth
vGS
vDS
vGS
iD
Vth
IDSS
+
+
+
+
0
vDS
vGS
iD
Vth
IDSS
+
− (+)
+
−
+
vDS
vGS
iD
Vth
IDSS
−
−
−
−
0
vDS
vGS
iD
Vth
IDSS
−
+(−)
−
+
−
n-channel depletion
D+
iD
+1
I DSS
−1
−2
−3
vDS
Substrate
G
iD
0
S-
vGS
Vth
p-channel enhancement
DGate
Source
iD
vGS
vDS
−5
−6
−7
−8
Substrate
G
Vth
iD
Drain
−9
S+
p-channel depletion
D-
iD
Substrate
G
S+
+3
+2
+1
0
−1
iD
Vth
vDS
vGS
I DSS
Small-signal models
Notation
When considering time-varying quantities, upper-case
symbols (V, I ) with appropriate subscripts are used for d.c.,
average, or root-mean-square values. Lower-case symbols
are used for instantaneous value and for a.c. components.
Thus, in the diagram,
IB = average (d.c.) value
Upper-case subscripts refer to large-signal (or d.c.) values,
e.g.
hFE =
IC
IB
.
hFE is often referred to as β .
Ib = r.m.s. value of a.c. component
iB = total instantaneous value
ib = instantaneous value of a.c. component.
ib
For circuit parameters, lower-case subscripts refer to
small-signal values, e.g.
hfe =
∂ic
∂ib
iB
.
2 Ib
IB
time
ELECTRICITY
181
BJT models
Hybrid π parameters
Simplified T-model
b
β ib
hie
ib
The above equivalent circuits contain no provision for the
representation of high-frequency effects. The hybrid π
equivalent circuit includes capacitors and is appropriate for
BJTs at high frequencies.
c
e
β=
ic
ib
rπ
hie =
;
b
kT
eIB
rπ
where k is Boltzmann’s constant, T the temperature, e the
charge on the electron and IB is the base average (d.c.)
current. Strictly, β = hfe , but it is assumed here that hFE = hfe .
Cμ
b'
c
vπ
gmvπ
Cπ
e
h-parameters
b
ic
+
hrevce
1
hoe
h fe ib
−
At high frequency
c
hfe (or β ) = hfe0 /(1 + jωCπ rπ )
vce
e
e
hie = input impedance with output short-circuited to a.c.
hre = reverse voltage transfer ratio with input open-circuited
to a.c.
hfe = forward current ratio with output short-circuited to
a.c. (current gain)
hoe = output admittance with input open-circuited to a.c.
Equivalent models can be set up for common base and
common collector circuits. The three h -parameter sets are
then related as follows:
Common
base
Common
emitter
Common
collector
hib = hie (1 − α )
hie = hib (1 + β )
hic = hie
hrb = hie hoc (1 − α )−hre
hre = hib (1 + β )−hrb
hrc = 1
hfb = −α
hfe = β
hfc = −(1 + β )
hob = hoe (1 − α )
hoe = hob (1 + β )
hoc = hoe
α and β satisfy the relationships
α=
β
;
1+β
e
gm is the transconductance ∂iC /∂vπ .
ib h ie ib
vbe
r0
β=
α
;
1−α
1−α =
1
.
1+β
Note that it is assumed that hFE = hfe and hFB = hfb .
where hfe0 is the d.c. value of hfe and Cμ is neglected because
it is√very much less than Cπ . The value of hfe will fall to
(1/ 2 )hfe0 when ωCπ rπ = 1. The cut-off frequency is defined
as
ωc =
1
C π rπ
or
fc =
1
.
2πCπ rπ
The frequency at which the value of hfe falls to unity is given by
ωT =
hfe0
Cπ rπ
or
fT =
hfe0
2πCπ rπ
.
182
ENGINEERING TABLES AND DATA
FET models
Low frequencies
Higher frequencies
Drain
Drain
Cgd
Gate
Gate
gmvgs
gmvgs
rds
vgs
vgs
rds
Cds
Cgs
Source
Source
In these gm is the transconductance ∂iD /∂vGS and rds is the drain resistance ∂vDS /∂iD .
Ebers-Moll large-signal model
For a BJT operating normally in the active region, BE is
forward biased and BC reverse biased. The above circuit then
reduces to the following alternative forms:
The generalized model for an n-p-n BJT is:
αR iC
αF i E
ICO
iE
iC
E
C
IEO
ICO
iE
iC
E
C
αF i E
B
ICO (1+β F)
For the current directions shown, the Ebers-Moll equations are
iC = −αF iE − ICO e vBC /VT − 1
iE = −αR iC − IEO e vBE /VT − 1 .
In these αF and αR are the forward and reverse common-base
current ratios, IEO and ICO are the reverse leakage currents of
the diodes, and VT is the voltage k T /e (approximately 25 mV
at room temperature).
B
B
iB
iC
β Fi B
E
Here βF is the forward common-emitter current gain. The
leakage currents ICO and ICO (1 + βF ) are often neglected.
C
ELECTRICITY
183
Instrumentation
Differential amplifiers
For a general differential amplifier
vout = Ad (v1 − v2 ) + Acm
Example of an instrumentation amplifier:
v1 + v2
2
v1
+
A1
-
where Ad is the differential gain and Acm is the common mode
gain.
R3
vj
R4
R2
v1
A diff
vout
v2
A3
+
R1
The common mode rejection ratio CMRR is given by
CMRR =
Ad
R2
Acm
and is usually expressed in decibels.
A2
+
v2
vo
R3
R4
vj
General second order filter
The circuit shown implements a second order filter using only
resistors and capacitors. It may be designed to be low pass,
bandpass or high pass depending on the choice of
components.
−Y1 Y3
vout
=
vin
Y5 (Y1 + Y2 + Y3 + Y4 ) + Y3 Y4
Y4
I4
Y5
I3
vin
I1
Y3
Y1
vout
I2
Y2
+
184
ENGINEERING TABLES AND DATA
Digital logic
Boolean Algebra
A·0=0
A·1=A
A·A=A
A·A=0
A+0=A
A+1=1
A+A=A
A+A·B=A
A · (A + B) = A
Absorption law
A·B=A+B
Distributive law
(redundancy theorem)
de Morgan’s theorem
A+B=A·B
A+A=1
A·B+B·C+A·C=A·B+A·C
A=A
A·B=B·A
A+B=B+A
A · (B + C) = A · B + A · C
A + B · C = (A + B) · (A + C)
2
Race hazard theorem
Alternative symbols
Commutative law
OR
AND
A · (B · C) = (A · B) · C
Associative law
A + (B + C) = (A + B) + C
+
·
∨
∧
The symbol for exclusive OR is ⊕ .
Minterms and maxterms
Boolean expressions of several variables can be expressed in
terms of the sum of minterms or the product of maxterms.
A minterm is the product of variables, and a maxterm is the
sum of variables. For three independent variables A,B,C, the
sets of minterms and maxterms are shown at right.
A
B
C
Minterm
Maxterm
0
0
0
A·B·C
A+B+C
0
0
1
A·B·C
A+B+C
0
1
0
A·B·C
A+B+C
0
1
1
A·B·C
A+B+C
1
0
0
A·B·C
A+B+C
1
0
1
A·B·C
A+B+C
1
1
0
A·B·C
A+B+C
1
1
1
A·B·C
A+B+C
ELECTRICITY
185
Combinatorial logic gates
Logic gates may be represented either by rectangular symbols (left column) or by distinctive shape symbols (right column). The
symbols below conform to IEEE 91A-1991.
AND
A
B
C
&
f
A
B
C
f
f=A·B·C
OR
A
B
C
≥1
f
A
B
C
f
f=A+B+C
NAND
A
B
C
&
&
f
A
B
C
f
f=A·B·C
NOR
A
B
C
≥1
f
A
B
C
f
f=A+B+C
Buffer
A
f
A
f
f=A
Inverter
A
f
A
f
f=A
f
f=A⊕B
f
f=A⊕B
Exclusive-OR
Exclusive-NOR
A
B
A
B
=1
f
=1
f
A
B
A
B
PLDs : Programmable logic devices
The following table summarises some standard programmable logic devices (control lines are omitted).
Mnemonic
Full name
Brief description
PROM
Programmable read-only memory
Memory chip
inputs: memory address; outputs: data at the address
PLA
Programmable logic array
Implements combinatorial logic
programmable AND and OR arrays
PAL
Programmable array logic
Combinatorial logic
programmable AND arrays
PLD
Programmable logic device
Realise complex logic functions
CPLD
Complex programmable logic device
Realise complex logic functions
FPGA
Field programmable gate array
Realise complex logic functions
186
ENGINEERING TABLES AND DATA
Some useful logic components
A
B
Multiplexer
} G 03
Dout
D0
D1
D2
D3
A
B
} G 03
D0
D1
D2
D3
Decoder
EN
Tristate
IN
OUT
EN
A
Half adder
B
0
0
1
1
A
0
1
0
1
Dout
D0
D1
D2
D3
EN
0
1
1
1
1
B
X
0
0
1
1
EN
0
1
1
IN
X
0
1
A
X
0
1
0
1
D0
0
1
0
0
0
OUT
X
0
1
D1
0
0
1
0
0
D2
0
0
0
1
0
D3
0
0
0
0
1
comment
high impedance state
enabled
enabled
S
B
Cout
A
Full adder
S
B
Cin
Cout
For Verilog example of a multiplexer see page 190.
Status bits resulting from arithmetic operations
N
Z
C
V
set when result is negative
set when result is zero
set when carry
set when overflow
ELECTRICITY
187
Flip-flops
D edge triggered flip-flop
D
D
Ù
Ù
C
C
Positive-edge clock
Negative-edge clock
Conventions: the symbol > represents a clock edge, and a small circle on a line indicates a complement.
For Verilog example of a D flip-flop with negative-edge clock and asynchronous clear see page 190.
Flip-flop technologies
Type
S-R
D
Type
Truth table
S
0
0
1
1
R
0
1
0
1
D
0
1
Qn +1
0
1
Qn +1
Qn
0
1
X
J-K
T
Truth table
J
0
0
1
K
0
1
0
Qn +1
Qn
0
1
1
1
Qn
T
0
Qn +1
Qn
1
Qn
Registers and memory
R
Ù
Register R
Clear
Register
D0
Q0
D1
Q1
D2
Q2
D3
Q3
7 6 5 4 3 2 1 0
Bit and byte allocations
Bit numbering in 8-bit register
0
15
High byte Low byte
Two-byte register R
Address
Read only memory (ROM)
CS
Input data
Address
Random access memory (RAM)
Output data
CS
R/W
Output data
188
ENGINEERING TABLES AND DATA
Verilog: HDL summary of common features
Verilog is used to define a digital system, both for device synthesis and functional simulation. It is one of the two major Hardware
Description Languages – the other being VHDL.
Language structure, hierarchy and syntax
Comment lines are designated by / / or by / ∗ to ∗ / across several lines. Statements end with a semicolon ( ; ) character. Verilog
is case sensitive. Leading spaces are ignored and circuit behaviour is described through module definitions.
Previously defined modules can be used in the hierarchy, for example:
module OR(inA, inB, out);
/ / An OR gate from three NANDs
input inA, inB;
output out;
wire w1, w2;
/ / Instantiation of NAND
NAND NAND1(inA, inA, w1),
NAND2(inB, inB, w2),
NAND3(w1, w2, out);
endmodule
module NAND(inA, inB, out);
input inA, inB;
output out;
assign out=∼(inA & inB);
endmodule
Net types and declarations
reg [3:0] B;
//
//
reg [0:7] B,C;
//
//
wire [3:0] Dataout; / /
integer Count;
//
Declare 4-bit register for storage
MSB = b3
Declare two 8-bit registers for storage
MSB = b0
Declare 4-bit net or signal Dataout
Simple signed 32-bit integer
285
h8FF
o765
3 b10x
5 d3
−4 b11
//
//
//
//
//
//
Decimal number
Hex number
Octal number
3-bit binary number with LSB unknown
5-digit decimal number
4-bit two’s complement of 0011
Unary and binary operators
+
−
∗
/
%
−
>
>=
<
<=
Addition
Subtraction
Multiplication
Division
Modulus
Unary minus
Greater than
Greater than or equal to
Less than
Less than or equal to
==
!=
!
&&
||
∼
&
|
∧
∼&
∼|
∼∧
Logical equality
Logical inequality
Logical negation
Logical AND
Logical OR
Bitwise negation
Bitwise AND
Bitwise OR
Bitwise XOR
Bitwise NAND
Bitwise NOR
Bitwise NOT XOR
&
|
∧
∼&
∼|
∼∧
===
! ==
{ , }
<<
>>
?:
AND reduction
OR reduction
XOR reduction
NAND reduction
NOR reduction
XNOR reduction
Case equality
Case inequality
Concatenation of bits
Shift left
Shift right
Conditional
ELECTRICITY
189
Parallel and sequential statements
Parallel statements (concurrency)
assign [ (strength1,strength0) ] WIRE NET NAME = expression;
initial sequential statements
always sequential statements
Sequential statements
begin [ {declarations} ] , [ {sequential statements} ] , end
if (expression) sequential statement , [else sequential statement ]
case (expression) , [ {sequential statement} ] , [ default : sequential statement ] , endcase
forever sequential statement
repeat (expression) sequential statement
while (expression) sequential statement
wait (expression) sequential statement
for (logical value = expression) sequential statements
assign logical value = expression;
deassign logical value;
# (number) sequential statement
event can be negedge or posedge
@(event) sequential statement
Simulation and compiler directives
timescale 1ns / 100ps
#2 sequential statement
$display (expression);
$write (expression);
$strobe (expression);
$monitor (expression);
$time
$stime
$realtime
$timeformat (unit#, prec#, “unit”, minwidth)
$stop
$finish
$dumpfile (“fn”)
$dumplimit (size)
$dumpvars (levels,[ {module or variable name} ] )
$dumpall
//
//
//
//
//
//
//
//
//
//
//
//
//
//
//
//
Simulation time unit of 1 ns, 100 ps data sampling
Simulates a 2 unit delay
Simulator displays values or text
Similar to $display but no newline
Similar to $display but after all nets have changed
Displays values whenever they change
“now” as TIME
“now” as INTEGER
“now” as REAL
Set time format as specified
Interrupt simulation
Terminate simulation
Name of file for value changed dump
Maximum size of dump file
Specifies output to VCD
Force a dump now
Blocking and nonblocking statements
An important feature of Verilog is the way register transfers
are handled. The example here first shows blocking
assignments, where each statement in the block is executed in
sequence, and is then followed by code using nonblocking
assignments. In the latter case all the statements are
executed at the same time. Nonblocking statements resemble
register hardware more closely than blocking assignments.
module blocks( );
reg a,b,c,d,e,f;
/ / Blocking assignments
initial begin
a = #20 1 b1
/ / At time 20 a = 1
b = #30 1 b0
/ / At time 50 b = 0
c = #40 1 b1
/ / At time 90 c = 1
end
/ / Nonblocking assignments
initial begin
d <= #20 1 b1 / / At time 20 d = 1
e <= #30 1 b0 / / At time 30 e = 0
f <= #40 1 b1 / / At time 40 f = 1
end
endmodule
190
ENGINEERING TABLES AND DATA
Verilog examples
4-to-1 MUX using CASE statement
D flip-flop with negative-edge clock and asynchronous clear
module mux ( D0,D1,D2,D3,A,B,Dout );
input D0,D1,D2,D3,A,B;
output Dout;
reg Dout;
module flop ( C,D,CLR,Q );
input C,D,CLR;
output Q;
reg Q;
always @( D0 or D1 or D2 or D3 or A or B )
begin
case ({ A,B })
2 b00 : Dout = D0;
2 b01 : Dout = D1;
2 b10 : Dout = D2;
default : Dout = D3;
endcase
end
endmodule
always @ ( negedge C or posedge CLR )
begin
if ( CLR ) Q = 1 b0;
else Q = D;
end
endmodule
State machines and computer architecture
ASM chart symbols
Entry
State box
Register operation
or output
Entry
Exit
Entry 1
Entry 2
Scalar decision
box
1-bit condition
Exit 0
Exit 1
Junction
Entry
Exit
Entry
Vector decision
box
Exit 0
n-bit condition
n
Exit 2 -1
Conditional
output box
Register operation
or output
Exit
ELECTRICITY
191
Central processor unit
Addressing modes : examples using LD instruction and register R1
LD addr
LD #addr
LD [addr]
LD $addr
LD addr(R1)
LD R1
LD (R1)
direct
immediate
indirect
relative to PC
index
register
register indirect
Typical data transfer instructions
Load
Store
Move
Push
Pop
Input
Output
LD, LOAD
ST, STORE
MOVE
PUSH
POP
IN
OUT
Typical program control instructions
Branch
Jump
Skip next instruction
Call procedure
Return from procedure
Return from interrupt
Compare (by subtraction)
Test (by ANDing)
BR
JMP
SKP
CALL
RET
RTI
CMP
TEST
Typical Arithmetic and Shift Instructions
Increment
Decrement
Add
Subtract
Multiply
Divide
Add with carry
Negate
Shift right
Shift left
Rotate right
Rotate left
INC
DEC
ADD
SUB
MUL
DIV
ADDC
NEG
SHR
SHL
ROR
ROL
Typical logical and bit manipulation instructions
And
Or
Exclusive or
Complement
Clear
Set
Clear carry
Set carry
Complement carry
AND
OR
XOR
NOT
CLR
SET
CLRC
SETC
COMC
Conditional branch instructions on status bits
Branch if zero
Branch if not zero
Branch if carry
Branch if no carry
Branch if minus
Branch if plus
Branch if overflow
Branch if no overflow
BZ
BNZ
BC
BNC
BN
BNN
BV
BNV
Conditional branch instructions for unsigned numbers
Branch if higher
Branch if higher or equal
Branch if lower
Branch if lower or equal
Branch if equal
Branch if not equal
BH
BHE
BL
BLE
BE
BNE
Conditional branch instructions for signed numbers
Branch if greater
Branch if greater or equal
Branch if less
Branch if less or equal
BG
BGE
BL
BLE
192
ENGINEERING TABLES AND DATA
Representation of numbers
Powers of two
Power of 2
Number systems
Decimal
Dec
Hex
Oct
Bin
Gray
20
1
0
0
0
21
2
1
1
1
1
1
2
4
2
2
2
10
11
23
8
3
3
3
11
10
2
4
16
4
4
4
100
110
25
32
5
5
5
101
111
26
64
6
6
6
110
101
27
128
7
7
7
111
100
28
256
8
8
10
1000
1100
29
512
9
9
11
1001
1101
10
1024
10
A
12
1010
1111
2
11
2048
11
B
13
1011
1110
212
4096
12
C
14
1100
1010
213
8192
13
D
15
1101
1011
214
16384
14
E
16
1110
1001
215
32768
15
F
17
1111
1000
2
2
0
0
Dec = decimal, Hex = hexadecimal, Oct = octal, Bin = binary,
Gray = Gray code
Computer representation of binary numbers
Unsigned offset binary
Signed
n bits representing binary number x
n-1 bits representing binary number x
sign bit s
Fixed point
(signed or unsigned)
binary number x
binary number f
m bits
n-m-1 bits
binary number, x + c
where c is offset
(normally c = −2n−1 ).
positive (s = 0): binary number, x
one’s complement (s = 1): x1 = x̃
two’s complement (s = 1): x2 = x̃ + 1
(note that x̃ inverts every bit of x)
x.f
where x and f have fixed number of bits
sign bit s
IEEE floating point standard
exponent e
binary number f
m bits
n-m-1 bits
sign bit s
−1(s ) ∗ 2 e ∗ 1.f
where e = e − (2m−1 − 1)
and e and f are binary numbers
ELECTRICITY
193
ASCII (American Standard Code for Information Interchange)
or ISO code, also known as CCITT Alphabet No 5.
Bits
6
0
0
0
0
1
1
1
1
5
0
0
1
1
0
0
1
1
4
0
1
0
1
0
1
0
1
0
0
0
0
NULL
DLE
SPACE
0
@
P
0
0
0
1
SOH
DC1
!
1
A
Q
a
q
0
0
1
0
STX
DC2
2
B
R
b
r
0
0
1
1
ETX
DC3
#
3
C
S
c
s
0
1
0
0
EOT
DC4
$
4
D
T
d
t
0
1
0
1
ENQ
NACK
%
5
E
U
e
u
0
1
1
0
ACK
SYN
&
6
F
V
f
v
0
1
1
1
BELL
ETB
7
G
W
g
w
1
0
0
0
BS
CAN
(
8
H
X
h
x
1
0
0
1
TAB
EM
)
9
I
Y
i
y
1
0
1
0
LF
SUB
*
:
J
Z
j
z
1
0
1
1
VT
ESC
+
;
K
[
k
{
1
1
0
0
FF
FS
,
<
L
\
l
|
1
1
0
1
CR
GS
−
=
M
]
m
}
1
1
1
0
SO
RS
.
>
N
∧
n
∼
1
1
1
1
SI
US
/
?
O
o
DELETE
3
2
1
0
Bits
An eighth bit is available for parity purposes.
p
194
ENGINEERING TABLES AND DATA
Electrical properties of materials
In this section unless otherwise stated k denotes a propagation constant or wave number and kB the Boltzmann constant; e is
the magnitude of the electronic charge.
Free charges
Z (E ) dE = 4π
The mobility of charge carriers of mass m and charge e in a
medium is
e
μ=
τ
m
where τ is their average collision time or relaxation time for
momentum transfer. If their number density is n , they produce
a conductivity
2
σ = n eμ =
ne τ
m
at frequencies ω such that ωτ 1. If ωτ 1 the conductivity
becomes σ /(1 + jωτ ).
2m
h2
3/2
E 1/2 dE
in which h is Planck’s constant. The probability of an electron
occupying an available state of energy E is given by the
Fermi-Dirac function
−1
F (E ) = e (E −EF )/kB T + 1
.
In this T is absolute temperature and EF the Fermi energy,
defined in general as that value of E for which F = 12 (or, if
T = 0, above which F = 0). For free electrons the Fermi
energy is
EF =
h2
2m
3n
8π
2/3
.
Propagation constant
If the medium has permittivity , its propagation constant for a
wave at frequency ω is given by
k 2 = ω 2μ
1+
σ
jω (1 + jωτ )
Maxwell-Boltzmann approximation
In the classical (Maxwell-Boltzmann) approximation the
probability of a particle having energy E is proportional to the
Boltzmann factor exp −E/kB T .
in which μ now signifies the magnetic permeability.
Thermionic emission
Plasma frequency
The Richardson-Dushman equation for the current density
due to thermionic emission of electrons from a surface of work
function φ at absolute temperature T is
For ωτ 1 this becomes
k2 = μ
J = AT 2 e −φ /kB T
ω 2 − ωp2
where ωp is the plasma frequency given by
ωp2 =
σ
n e2
=
.
τ
m
in which A is a constant of theoretical value 4πem kB2 /h 3 or
1.2 MA m−2 K−2 .
Semiconductors
Evanescent waves
The wave is evanescent, and the medium reflective, for
ω < ωp .
In the following m ∗ denotes effective mass, μ mobility and D
diffusion coefficient; subscripts e and h denote electrons and
holes respectively.
Effective mass of an electron
Cyclotron frequency
In the presence of an applied magnetic field B the propagation
constant is given by
'
k 2 = ω 2μ
1+
σ
jω {1 + j (ω ± ωc ) τ }
)
where ωc is the cyclotron frequency eB /m .
Density of states and Fermi level
Free electrons confined by an infinite potential barrier have
density of energy states per unit volume in the energy interval
E to E + dE given by
If the E – k relation is assumed to be independent of direction,
that is if the surfaces of constant energy are spherical in
k -space, then the effective mass of an electron within an
energy band is the scalar quantity
me∗
h2
=
4π 2
∂2 E
∂k 2
−1
and the same expression gives the effective mass of a hole
mh∗ .
ELECTRICITY
195
Density of states
The density of states for electrons at the bottom of the
conduction band is then
Ze (E ) = 4π
∗ 3/2
2me
E − Eg
h2
1/2
The diffusion coefficient for a charge carrier is related to its
mobility by the Einstein relation
kB T
D
=
.
μ
e
Continuity equation
and that for holes at the top of the valence band is
Zh (E ) = 4π
2mh∗
The continuity equation for electrons is
3/2
∇ · Je
∂n
= −re +
e
∂t
(−E )1/2 .
h2
In these E is measured from the top of the valence band and
Eg is the energy gap.
∇ · Jh
∂p
= −rh −
.
e
∂t
Effective number density
The approximate effective number density of electrons in the
conduction band is
n =2
∗
2πme kB T
3/2
h2
p =2
2πmh∗ kB T
e −(Eg −EF )/kB T
3/2
h2
The net recombination rates can be expressed as
Δn
;
τe
re =
and the corresponding density of holes in the valence band is
where re is the net loss rate per unit volume by recombination;
for holes
e −EF /kB T .
rh =
Δp
τh
where τe , τh are the lifetimes and Δn , Δp the excess
populations, over the equilibrium values, of electrons and
holes respectively.
The diffusion length for a species with lifetime τ is
L=
Dτ .
Intrinsic semiconductor
An intrinsic semiconductor has n = p and its Fermi energy is
EF =
mh∗
1
3
Eg + kB T ln ∗ .
2
4
me
The rectifier equation for the forward current across a junction
with forward bias voltage V is
I = I0 e eV/kB T − 1
Extrinsic semiconductor
where I0 is the reverse saturation current.
An extrinsic semiconductor has
The contact potential of an abrupt junction has the magnitude
n p = ni2
for any impurity level, where ni is the value of n and p in the
intrinsic material at the same temperature.
V0 =
kB T
NA ND
ln
e
ni2
where NA , ND are the densities of ionized acceptor and donor
atoms on the p and n sides respectively.
Conductivity
The conductivity of a semiconductor is the sum of electron
and hole contributions, that is
σ = n eμe + peμh .
Diffusion current
In a region where n and p vary with distance x a diffusion
current is superimposed on that due to conductivity. The total
current density in an electric field E is then
J = Je + Jh = (n eμe + peμh ) E + eDe
p-n junction
∂p
∂n
− eDh
.
∂x
∂x
If the densities of mobile carriers are assumed to be negligible,
and NA , ND uniform, for distances xp and xn on the respective
sides, then the width of the depletion layer for a reverse bias
voltage V is
⎧#
⎫
#
2 (V0 + V ) 1/2 ⎨ ND
NA ⎬
w = xp + xn =
+
⎩ NA
ND ⎭
e (NA + ND )
and the junction capacitance per unit area is
C=
e NA ND
2 (V0 + V ) (NA + ND )
1/2
.
196
ENGINEERING TABLES AND DATA
Hall effect
The Hall coefficient RH of a material is the ratio of the
transverse electric field induced to the product J B , where J is
the current density flowing through a magnetic flux density B
perpendicular to it. When the charge carriers are electrons of
density n
RH =
1
ne
When both electrons and holes are present, with mobilities μe
and μh ,
n μe2 − pμh2
e (n μe + pμh )
2
−1 3 0
r+2 N
r
√
tan δ is
where c is the speed of light in vacuum and k or 12
the extinction coefficient (the imaginary part of the refractive
index). The attenuation in decibels per kilometre is then
Orientational polarization
In an applied field E molecules having a permanent dipole
moment μd each contribute to the resultant total moment a
fraction of μd given by the Langevin function
1
a
− 1)
If the dipoles have a relaxation time τ for alignment in an a.c.
field, then the effect of frequency ω on orientational
polarization is given by the Debye equations for the dielectric
constant r . If r is written − j , then these are
=
where r is the dielectric constant of the material. The
polarization, or total dipole moment per unit volume, in an
applied field E is
r
ωk
c
when a is μd E/kB T . For small a, L ≈ a/3.
The Clausius-Mossotti equation for the polarizability of
non-polar molecules of number density N in a dielectrically
isotropic material is
P =(
tan δ =
L(a) = coth a −
.
Dielectrics
α=
1ω√
2c
where f is the frequency in hertz.
1
pe
the direction of J × B being taken as positive.
RH =
α=
20 log10 e 1000α = 8686 α = 1.820 f k × 10−4
and if holes of density p
RH = −
due to conductivity σ , where μ is the permeability; or, if
μ = μ0 ,
0E
=χ
H
+
L
−
H
1 + ω 2τ 2
ωτ
=
( L−
1 + ω 2τ 2
H)
in which L is the value of r at frequencies for which ωτ 1,
and H its value when ωτ 1.
0E
Resonant polarization
where χ is the electrical susceptibility.
Dielectric loss
In an electric field of r.m.s. value E the loss per unit volume
2
due to conductivity σ is σ E . In terms of a complex dielectric
constant this becomes
w =ω
0E
2
= ω E 2 tan δ
where is the permittivity
/ .
0
and tan δ, the loss tangent, is
If σ ω , or tan δ 1, the attenuation of a wave is given by
the attenuation constant
3
μ
1
α= σ
2
If bound charges resonate at an undamped frequency ω0 with
a damping ratio ζ , the effect of frequency ω on their induced
polarization is given by the following equations for their
contribution to r :
⎧
⎫
⎨
⎬
1 − r2
= 1 + ( L − 1)
⎩ 1 − r 2 2 + 4ζ 2 r 2 ⎭
⎧
⎫
⎨
⎬
2ζ r
= ( L − 1)
⎩ 1 − r 2 2 + 4ζ 2 r 2 ⎭
In these r is the frequency ratio ω/ω0 and
for r 1.
L
the value of
MISCELLANEOUS
Part 8
Miscellaneous
197
198
ENGINEERING TABLES AND DATA
Screw threads
ISO metric thread – coarse series
Thread
designation
Nominal OD
Pitch
mm
1.6
2
2.5
3
4
5
6
8
10
12
16
20
24
30
36
M1.6
M2
M2.5
M3
M4
M5
M6
M8
M10
M12
M16
M20
M24
M30
M36
mm
Tapping
drill dia.
mm
Clearance hole
dia. (medium fit)
mm
Bolt tensile
stress area
mm2
0.35
0.4
0.45
0.5
0.7
0.8
1
1.25
1.5
1.75
2
2.5
3
3.5
4
1.25
1.6
2.05
2.5
3.3
4.2
5
6.8
8.5
10.2
14
17.5
21
26.5
32
1.8
2.4
2.9
3.4
4.5
5.5
6.6
9
11
14
18
22
26
33
39
1.27
2.07
3.39
5.03
8.78
14.2
20.1
36.6
58
84.3
157
245
353
561
817
Typical notation is M5 × 0.8 where M = metric thread, 5 = nominal thread OD, 0.8 = thread pitch (travel per revolution).
Engineering drawing
Abbreviations and notation
A/F
Across flats (e.g. the separation of faces on a
hexagonal nut)
ASSY
Assembly
BOM
Bill of materials
CBORE
Counterbore
CHAM
Chamfer
CH HD
Cheese head (head of screw)
CRS
Centres
CSK
Countersunk
CSK HD
Countersunk head (of a screw)
CYL
Cylinder or cylindrical
DIA
Diameter
EQUI SP
Equally spaced (e.g. holes equally spaced
around a circle)
ID
Inside diameter
OD
Outside diameter
PCD
Pitched circle diameter (e.g. “6 holes Equi Sp
on 65 PCD”: the centres of the 6 holes lie on a
circle of diameter 65)
R, RAD
Radius
REF
Reference (used to denote a reference
dimension: one not needed for manufacture,
but useful to know)
SFACE
Spotface (an area around a hole machined to
give a good seating surface)
TYP
Typically
UCUT
Undercut (often at the end of thread)
φ
Diameter
MISCELLANEOUS
199
Arrangement of views
LAYOUT OF VIEWS
THIRD ANGLE PROJECTION
(USED IN THE USA AND THE UK)
SYMBOL ON DRAWING
Sections and auxiliary views – third angle
VIEW A
SECTION B-B
B
B
A
200
ENGINEERING TABLES AND DATA
Index
2
χ -distribution, 39–40
t-distribution, 36
z -transform, 17
Abbreviations, engineering drawing, 198
AC machines, 169
Acceptance angle, 175
Adams-Bashforth integration, 34
Adder, 186
Addison’s shape parameter, 88
Adjoint, 22
Admittance, 163, 164
Aerials, see Antennas
Air
composition, 74
International Standard Atmosphere, 74
properties, 75
Algebraic equations, 18, 34
Alloys
mechanical properties, 47
physical properties, 45
Alphabet, Greek, 2
Ammonia, properties, 72
Ampère law, 162
Amplifiers
differential, 183
instrumentation, 183
Amplitude modulation, 177
Analysis of circuits, 163
Angle
acceptance, 175
Brewster, 175
phase, 158, 159
sections, 144
Antennas, 172
Aperture
numerical, 175
of antenna, 172
Archimedean spiral, 25
Area
centroid, 135
conversion factors, 4
of plane figures, 26–28
second moment of, 25, 135
surface, of rigid bodies, 30–33
Arithmetic series, 7
Arrangement of views, 199
ASCII code, 193
ASM chart symbols, 190
Asynchronous machines, 170
Atmosphere, standard, 74
Atomic properties, 43–44
Attenuation
constant, 173, 196
optical fibre, 53
Autocorrelation, 13
Auxiliary views, 199
Avogadro number, 42
Axes
parallel, 25, 29
principal, 25, 29
rotation of, 25, 29
Bands, in semiconductors, 194
Bandwidth, 165
Basquin’s law, 129
Beam width, of antenna, 172
Beams
bending, 130–134
curvature, 130
deflection, 130–134
universal, sections, 136–139
vibration, 160
Belt friction, 155
Bending
beams, 130–134
moment, 130
plates, 128
Bessel functions, 19
Beta function, 12
Binomial
coefficients, 6
distribution, 35
series, 7
Bio-Savart law, 162
Bipolar junction transistor, see BJT
BJT, 179, 181, 182
Black-body radiation, 92
Blasius
formula, 90
solution, 92
Bode diagrams, 166–167
Body-centred cubic, 45
Bohr magneton, 42
Boiling points
gases, 75
liquids, 53
metallic elements, 45
Boltzmann
constant, 42
factor, 194
Boolean algebra, 184
Boundary
conditions
electromagnetic, 162
for stress, 125, 127
INDEX
layers, 92
Brewster angle, 175
Brittle materials, 46, 48
Cantilever, 130, 131, 133
Capacitance
junction, 195
transmission lines, 174
Capacitors, 178
Carbon dioxide, properties, 73, 75
Cardioid, 24
Carson’s rule, 177
Catenary, 24
Cauchy’s
equation, 19
integral, 17
theorem, 17
Cauchy-Riemann relations, 17
Cavities, 176
Central force motion, 156–157
Centroid, 23–25
Ceramics, properties, 46, 48
Chézy formula, 92
Channel
capacity, 177
sections, 142–143
Characteristic impedance, 171, 173–174
Chebyshev polynomials, 19
Chebyshev’s equation, 19
Circle, 23, 26
Circular sections, 152–153
Clausius-Mossotti equation, 196
Close-packed, hexagonal, 45
CMRR, 183
CO2 , properties, 73
Codes
ASCII, 193
for components, 178
Coefficients
binomial, 6
Chézy, 92
compressibility, 86
diffusion, 194
extinction, 52, 171
friction
Coulomb, 155
fluid, 90, 92
Hall, 196
head loss, in pipes, 90
heat transfer, 88
Joule-Thomson, 86
perfect gas, 86
reflection, 173
stoichiometric, 82
temperature, resistance, 45
thermal expansion
fluids, 86
liquids, 53
solids, 45–46
volume expansion, 86
Coercive force, 51
Coffin-Manson law, 129
201
Coils, self-inductance, 163
Colour codes, 178
Column sections, universal, 140–141
Combinatorial logic gates, 185
Common mode rejection ratio, 183
Communication systems, 176
Compatibility of strains, 125, 127
Complex variable, 17
Components, electrical, 178
Composite materials, 46, 48
Compressibility
chart, 88
coefficient, 86
factor, 88
Compressible flow, 93–121
Computer architecture, 190
Conductivity
electrical, 162, 194–196
thermal
gases, 74, 75
liquids, 53
solids, 45–46
water and steam, 72
Conic sections, 23
Conservative field, 21, 156
Considère’s construction, 129
Constants
attenuation, 173, 196
damping, viscous, 158
dielectric, see Dielectric
elastic
relations, 127
values, 47–48
Euler, 6, 12
Lamé, 127
lattice, 50
mathematical, 6
physical, 42
propagation, 171, 173, 175, 194
warping, 135
Constitutive equations, 162
Contact potential, 195
Continuity equations
electrical, 162
electrons, 195
fluid flow, 87
Convection, 89
Conversion factors, 4
Convolution, 13, 16, 17
Coriolis acceleration, 156
Coulomb
friction, 155
law, 162
CPLD, 185
Cracks, Griffith theory, 129
Cramer’s rule, 18
Critical damping, 158, 167, 168
Crystal structure, 45
Cubic
body-centred, 45
equation, 18
face-centred, 45
202
Curvature, 130
Cycloids, 24
Cyclotron frequency, 194
Damping
constant, 158
critical, 158, 167, 168
ratio, 158, 167
viscous, 158
Darcy’s equation, 90
DC machines, 169
De Moivre’s theorem, 9
Debye equations, 196
Decoder, 186
Decrement, logarithmic, 158
Definite integrals, 12
Deflection of beams, 130–133
Delta-star transformation, 163
Density
gases, 75
liquids, 53
of states, 50, 194
power spectral, 13
semiconductors, 50
solids, 45–46
Determinant, 22
Dielectric
constant
gases, 75
liquids, 53
semiconductors, 50
solids, 45–46
loss, 196
polarization, 196
waveguides, 175
Differential
amplifier, 183
equations, 19, 34
Differentials, 10, 34
Diffusion
coefficient, 194
equation, 20
length, 195
Dimensionless groups, 88
Diodes, 178
Dipole antennas, 172
Directivity, of antenna, 172
Discharge number, 88
Discrete Fourier transform, 13
Dispersion, 173, 175
Displacement
and strain, 125–127
excitation, 160
Distortionless line, 173
Distributions, 35–40
Drawing, engineering, 198
Duality, 13
Dynamics, 156–157
Earth, properties, 42
Ebers-Moll model, 182
Effective
ENGINEERING TABLES AND DATA
aperture, of antenna, 172
mass, 50, 194
Einstein relation, 195
Elastic modulus, 47–49, 125, 127, 135
Electric field, 162
Electrical
conductivity, see Conductivity
machines, 169–170
properties of materials, 45–46, 50–51, 194–196
Electromagnetism, 162
Electron
charge and rest mass, 42
continuity equation, 195
density of states, 194
effective mass, 50, 194
emission, 194
free, 194
Elements
atomic properties, 42
metallic
mechanical properties, 47
physical properties, 45
Ellipse, 23, 27
Emission, thermionic, 194
Energy
conservation of, 156
equation, 86, 87
Fermi, 194–195
gap, 50
Gibbs, 79, 81, 83
kinetic, 156, 157
potential, 156
specific internal, steam, 54–70
Engineering drawing, 198
Enthalpy, 86
gases, 77, 80
of reaction, 82–83
specific
refrigerants, 72–73
steam, 54–71
Entropy, 86
gases, 78, 81
specific
refrigerants, 72–73
steam, 54–71
Epicycloids, 24
Equations
algebraic, 18
Bessel, 19
Cauchy, 19
Chebyshev, 19
Clausius-Mossotti, 196
constitutive, 162
continuity
electrical, 162
electrons, 195
fluid, 87
cubic, 18
Darcy’s, 90
Debye, 196
differential, 19, 34
diffusion, 20
INDEX
energy, 86, 87
equilibrium, 125–127
Euler, 157
fluid flow, 87
heat conduction, 20
Lagrange, 157
Laguerre, 19
Laplace, 20, 162
Legendre, 19
Mathieu, 19
Maxwell, 162
Navier-Stokes, 87
of motion, 157
of state, 86
perfect gas, 86
Poisson, 20, 162
polytropic, 86
quadratic, 18
rectifier, 195
Riccati, 19
Richardson-Dushman, 194
simultaneous, 18
steady flow, 86
Van der Waals, 86
wave, 20
Equilibrium
constants, 83
equations, 125–127
reactions, 82–83
Error function, 12
Euler
constant, 6, 12
critical loads, 134
differential equation, 157
equations, 157
Everett’s formula, 34
Expansion, thermal
liquids, 53
solids, 45–46
Exponential
functions, 9
series, 7
Extinction coefficient, 52, 171
Face-centred cubic, 45
Factorials, large, 6
Failure
criteria, 129
strain, 47
stress, 47–49
Fanno line, 102–111
Faraday
constant, 42
law, 162
Fatigue, 129
Fermi energy, 194–195
Fermi-Dirac function, 194
Ferromagnetic materials, 51
FET, 179–180, 182
Fibre
optical, 53, 175
reinforcing, 49
203
single mode, 175
Field
electromagnetic, 162
programmable gate array, 185
vector, 21
Field-effect transistor, see FET
Final-value theorem, 16, 17
Fixed-end moments, 133–134
Flexibility matrix, 131
Flip flops, 187
Fluid flow equations, 87
Forced convection, 89
Four-terminal networks, 164–165
Fourier
series, 7, 8
transform, 13–15
discrete, 13
inverse, 13
FPGA, 185
Fracture toughness, 47–48
Free
convection, 89
electrons, 194
space, properties, 42
Freezing points
gases, 75
liquids, 53
Frequency
cyclotron, 194
modulation, 177
natural
of beam, 160
of circuit, 168
plasma, 194
undamped natural, 158
Friction
belt, 155
boundary-layer, 92
Coulomb, 155
pipe, 90
Froude number, 88
Full adder, 186
Functions
Bessel, 19
beta, 12
error, 12
exponential, 9
Fermi-Dirac, 194
gamma, 12
Gibbs, 79, 81, 83, 86
Hankel, 19
Helmholtz, 86
hyperbolic, 9
impulse, 14
Langevin, 196
potential, 162
probability, 35
sign, 14
step, 14
stream, 87
stress, 126, 131
trigonometric, 9
204
ENGINEERING TABLES AND DATA
work, 49
Funicular curve, 155
Gain, of antenna, 172
Gamma function, 12
Gas constants
specific, 75
universal, 42
Gases
enthalpy, 77, 80
entropy, 78, 81
Gibbs energy, 79, 81
internal energy, 76, 80
perfect, 86
properties, 75
Van der Waals, 86
Gates, logic, 185–187
Gauss law, 162
Gauss’s divergence theorem, 21
Gaussian
distribution, 35
integration, 34
pulse, 15
Geometric series, 7
Gibbs
energy, gases, 79, 81
function, 79, 81, 86
Goodman’s rule, 129
Grashof number, 88
Greek alphabet, 2
Griffith theory, 129
Group velocity, 173
Gyration, radius of, 25
Gyroscopic motion, 157
Half adder, 186
Hall effect, 196
Hamilton’s principle, 157
Hankel functions, 19
Head
loss, 90
number, 88
Heat transfer
coefficient, 88
convection, 89
radiation, 92
Helmholtz function, 86
Hexagonal close-packed, 45
Hollow sections
circular, 152–153
rectangular, 145–148
square, 149–151
Hooke’s law, 126, 127
Hybrid parameters, 164, 181
Hydraulic
machines, 88
radius, 92
Hyperbola, 23
Hyperbolic relations, 9
Hysteresis loss, 51
Ideal
gas, see Perfect gas
transformer, 170
Image impedance, 165
Impedance
characteristic, 171, 173–174
free space, 42
image, 165
input, 173
iterative, 165
parameters, 165
Impulse, 156
function, 14
Indefinite integrals, 11
Indicative values, 135
Inductance
transmission lines, 174
two coils, 163
Induction motor, 170
Inertia
matrix, 29, 157
moment of, 29–33
product of, 29
Initial-value theorem, 16, 17
Input impedance, 173
Instrumentation amplifier, 183
Integrals
definite, 12
elliptic, 12
indefinite, 11
Legendre, 12
Integration
Adams-Bashforth, 34
Gaussian, 34
numerical, 34
Runge-Kutta, 34
Internal energy
gases, 76, 80
specific, steam, 54–70
International Standard Atmosphere, 74
Inverse parameters, two-port, 164
Ionization potential, 43–44
Isentropic flow, 93–98
ISO screw threads, 198
Isotropic radiator, 172
Iterative impedance, 165
Jacobian, 22
JFET, 179
Joule-Thomson coefficient, 86
Junction
capacitance, 195
p-n, 195
Kepler’s laws, 156
Kilomole, 42
Kinematics, 155
Kinetic energy, 156, 157
Lagrange’s equations, 157
Laguerre’s equation, 19
Lamé constants, 127
Laminar flow, 89
Langevin function, 196
Laplace transform, 16
INDEX
Laplace’s equation, 20, 162
Latent heat, 45, 53
Lattice constant, 50
Legendre’s equation, 19
Logarithmic
decrement, 158
series, 7
spiral, 24
Logic, 184–187
Lorentz force, 162
Loschmidt number, 42
Loss factor, dielectric, 45, 46, 196
Mach number, 88
Machines
electrical, 169–170
hydraulic, 88
Maclaurin series, 7
Magnetic
circuit law, 162
field, 162
force, 162
Manning formula, 92
Matching, 163, 172, 173
Mathieu’s equation, 19
Matrices, 22
flexibility, 131
inertia, 157
rotation, 22, 29
stiffness, 132
Maxterms and minterms, 184
Maxwell
equations, 162
relations, 86
Mean free path, 75
Mechanical properties, 47–48
Melting points, 45–46
Metals
mechanical properties, 47
physical properties, 45
Miner’s rule, 129
Minterms and maxterms, 184
Mobility
charge carriers, 194
in semiconductors, 50
Modulation
amplitude, 177
frequency, 177
phase, 177
pulse code, 177
Modulus
bulk, 47, 53, 125, 127
elastic, 135
of sections, 135
plastic, 135
shear, 47, 125, 127
Young’s, 47–49, 125
Mohr’s circle of stress, 125
Molar
enthalpy, 77, 80
entropy, 78, 81
internal energy, 76, 80
205
Moment
bending, 130
fixed-end, 133–134
of area, second, 135
of force, 155
of inertia, 29–33
of momentum, 156, 157
polar, 25
principal, 29
product, 25
second, of area, 25
Momentum, 156
conservation, 156
equation, of fluid, 87
moment of, 156, 157
Moody diagram, 90–91
MOSFET, 180
Moving coordinates, 156
Multiplexer, 186
Natural
convection, 89
frequency, 158, 160, 168
Navier-Stokes equations, 87
Newton’s
laws, 156
method, 34
NH3 , properties, 72
Noise
figure, 176
shot, 176
temperature, 176
thermal, 176
Nominal strain, 129
Non-isotropic radiators, 172
Non-metals
mechanical properties, 48
physical properties, 46
Normal distribution, 35, 37–38
Norton’s theorem, 163
Notation, engineering drawing, 198
Nozzle flow, 86
Number systems, 192
Numerical
analysis, 34
aperture, 175
integration, 34
Nusselt number, 88, 89
Nyquist criterion, 17
Oblique shocks, 122–123
Open channel flow, 92
Optical
fibre, 53
properties, 52–53
waveguides, 175
Orbit, planetary, 156–157
Orientational polarization, 196
p-n junction, 195
PAL, 185
Pappus’s theorems, 23
Parabola, 23, 28
206
Parabolic antenna, 172
Parallel-axis theorem, 25, 29
Parallelogram, 26
Parameter
Addison’s shape, 88
sets
transistor, 180–181
two-port, 164–165
Parseval-Rayleigh theorem, 13
Perfect gas
coefficients, 86
compressible flow, 93–121
equation, 86
specific heats, 86
Periodic table, 42
Permeability, 162
free space, 42
relative, 51
Permittivity, 162
free space, 42
relative, 53
Phase
angle, 158, 159
modulation, 177
velocity, 173
Physical properties
gases, 75
liquids, 53
solids, 45–46
Pipe flow, 89, 90
PLA, 185
Planck constant, 42
Plane
figures, 26–28
strain, 126
stress, 126
Plasma frequency, 194
Plastic modulus, 135
PLD, 185
Poisson distribution, 35
Poisson’s
equation, 20, 162
ratio, 47–48, 125
Polar moment, 25, 131
Polarization, 196
Poles and zeros, 169
Polymers, 46, 48
Polytropic process, 86
Potential
energy, 156
functions, 162
velocity, 87
Power
in a.c. circuits, 164
number, 88
radiated, 172
spectral density, 13
transfer, 163–164
Powers of two, 192
Poynting vector, 171
Prandtl number, 88
air and constituents, 75
ENGINEERING TABLES AND DATA
water and steam, 72
Prandtl-Schlichting formula, 92
Principal
axes, 25, 29
moments of inertia, 29
strains, 125, 127
stresses, 125, 127
Probability function, 35
Product
moment, 29
of inertia, 29
scalar, 20
triple, 20
vector, 20
Programmable
array logic, 185
complex logic device, 185
gate array, field, 185
logic array, 185
logic device, 185
read-only memory, 185
Projection, third angle, 199
PROM, 185
Propagation constant, 171, 173, 175, 194
Proton mass, 42
Psychrometric chart, 84
Pulse code modulation, 177
Pulses, Fourier transforms, 14–15
Quadratic equation, 18
Quality factor, 165–166, 176
Quantum numbers, 43–44
R134a, properties, 73
Radiated power, 172
Radiation
black body, 92
resistance, 172
Radius of gyration, 25
Rayleigh
line, 112–121
strut formula, 134
Reactive volt-amperes, 164
Reciprocity
condition, 164–165
theorem, 163
Recombination, 195
Rectangle, 26
Rectangular sections, 145–148
Rectifier equation, 195
Reduced pressure and temperature, 88
Reflection coefficient, 171, 173
Refractive index, 52, 171
Refrigerant 134a, properties, 73
Refrigerants, 72–73
Remanent flux density, 51
Residue theorem, 17
Resistance, temperature coefficient, 45
Resistivity
ferromagnetic materials, 51
liquids, 53
solids, 45–46
INDEX
Resistors, 178
Resonance, electrical, 165–168
Resonant
cavities, 176
polarization, 196
Response, 165–169
Reynolds number, 88
Riccati’s equation, 19
Richardson-Dushman equation, 194
Rigid bodies, 29, 157
Rigidity, torsional, 131, 135
Rotating
disc, stresses in, 126
unbalance, 159
Rotation
matrix, 22, 29
of axes, 22, 25, 29, 156
Routh-Hurwitz criterion, 18
Runge-Kutta integration, 34
Screw threads, 198
Secant method, 34
Second order filter, 183
Section views, 199
Sections
channel, 142–143
circular, 131, 152–153
closed thin-walled, 131
equal angle, 144
open thin, 131
rectangular, 145–148
square, 149–151
Self-inductance
transmission lines, 174
two coils, 163
Semicircle, 26
Semiconductor
devices, 178–182
properties, 50, 194–195
Series, 7
Shannon-Hartley theorem, 177
Shape parameter, Addison’s, 88
Shear
modulus, 125
stress, 125–128, 130
Shells
moments of inertia, 30–33
stresses in, 126
Shock
flow, 99–102
oblique, 122–123
Shot noise, 176
SI units, 2
Sign function, 14
Simpson’s rule, 34
Simultaneous equations, 18
Skin depth, 174, 176
Slope-deflection relations, 133
Small-signal models, 180–182
Smoothing, 34
Snell’s law, 171
Solenoidal field, 21
207
Specific
heat, 86
gases, 75
liquids, 53
solids, 45–46
volume
air, 75
refrigerants, 72–73
steam, 54–71
Spectral density, 13
Speed of sound, 74, 93–102
Spirals
Archimedean, 25
logarithmic, 24
Square sections, 149–151
Stability of struts, 134
Standard
atmosphere, 74
deviation, 35
temperature and pressure, 75
Standing-wave ratio, 173
Stanton number, 88
Star-delta transformation, 163
State machines, 190
Statistics, 35–40
χ 2 -distribution, 39–40
binomial distribution, 35
Gaussian distribution, 35
normal distribution, 35, 37–38
Poisson distribution, 35
t-distribution, 36
Steady-flow process, 86
Steam
properties, 54–72
saturated, 54–65, 72
supercritical, 71
superheated, 66–70
Stefan-Boltzmann constant, 42
Step function, 14
Stiffness matrix, 132
Stirling’s formula, 6, 12
Stoichiometric coefficient, 82
Stokes’s theorem, 21
Strain
and displacements, 125–127
compatibility, 125, 127
plane, 126
principal, 125, 127
three-dimensional, 127
transformation, 125, 127
true, 129
two-dimensional, 125
Stream function, 87
Stress
bending, 128, 130
failure, 47–49, 129
function, 126, 131
Mohr’s circle, 125
plane, 126
principal, 125, 127
proof, 47
range, 129
208
rotating disc, 126
shear, 125–128, 131
thick cylinder, 126
three-dimensional, 127
transformation, 125, 127
true, 129
two-dimensional, 125
ultimate, 47–48, 129
yield, 47, 129
Struts, 134
Superconductors, 51
Superposition principle, 163
Symbols
ASM chart, 190
Greek, 2
logic, 185–187
semiconductor devices, 178–180
Symmetrical components, 164
Symmetry
half-wave, 7
in two-ports, 164
Synchronous machines, 170
t-distribution, 36
Taylor series, 7
Tensile strength, 47–49, 129
Tetrafluoroethane, properties, 73
Thévenin’s theorem, 163
Thermal
conductivity, see Conductivity
noise, 176
Thermionic emission, 194
Thermochemical data, 82–83
Thermodynamic
properties
fluids, 54–71, 74–81
gases, 76–81
liquids, 53
solids, 45–46
relations, 86
Thick cylinder, stresses in, 126
Third angle projection, 199
Timoshenko strut formula, 134
Torsion, 131
Torsional rigidity, 131, 135
Toughness, fracture, 47–48
Transconductance, of FET, 182
Transformers, 170
Transforms
discrete Fourier, 13
Fourier, 13–15
Laplace, 16
z, 17
Transistors, 178–182
Transmission
coefficient, 171
lines, 173–174
parameters, 165
Trapezium, 26
Trapezoidal rule, 34
Tresca criterion, 129
Triangle, 26
ENGINEERING TABLES AND DATA
Trigonometric
relations, 9
series, 7
Tristate, 186
True stress and strain, 129
Turbulent flow, 89
Two-port networks, 164–165
Ultimate tensile strength, 47–48, 129
Unbalance, rotating, 159
Units
atomic mass, 42
conversion, 4
metric, 2–3
multiples, 3
SI, 2
Universal beams and columns, 136–141
Van der Waals’ gas, 86
Variance, 35
Vectors, 20–21
matrix representation, 22
Poynting, 171
Velocity
group, 173
of light, 42
phase, 173
potential, 87
Verilog, 188
Vibrations, 158–160
Views, engineering drawing, 199
Viscosity
dynamic
gases, 75
liquids, 53
kinematic, of atmosphere, 74
Viscous damping, 158
Voltage standing-wave ratio, 173
Von Mises criterion, 129
Warping constant, 135
Wave
equation, 20
propagation, 171
Waveforms, Fourier series, 8
Waveguides
dielectric, 175
optical, 175
rectangular, 175
White noise, 13
Wien displacement law, 92
Winkler theory, 130
Work
functions, 49
law, 162
shaft, 86
Yield
criteria, 129
stress, 47
Young’s modulus, 47–49, 125
z-transform, 17
Zeros and poles, 169
Errata
Please note the following corrections to the April 2009 edition of Engineering Tables and Data:
Page 7
• Replace odd, bn = 0 by odd, an = 0 in the third row from the bottom of the right-hand column.
Page 75
• Replace 1.178 by 0.178 in the third row from the bottom of the second table.
P.D.M.
January 2010
209