Units and Key Constants
1
• Conventional Units
Parameter
English Units
SI Units
–
–
–
–
Feet, Inches
Seconds
Pounds (force), lbf
psf, psi
Meters, M
Seconds, s
4.448 Newton, N
Pascal, Pa (1N/1m2)
bar (105Pa)
2.989 kPa
0.4536 kilogram
Joule, J
0.7457 kWatt
Distance
Time
Force
Pressure
– Mass
– Energy
– Power
1 ft H2O
Pounds (mass), lbm
Btu
1 Hp
2
Equivalent Systems of Units
System
English Eng.
English Gravitational
Metric
Metric
International System (SI)
Force Mass
lbf
lbm
lbf
slug
kgf
kg
dyne
gm
Newton
kg
Length
ft
ft
m
cm
m
time
s
s
s
s
s
3
Important Constants for Air
Variable
pressure
density
Universal gas
constant
Spec. gas constant
(air)
Air
Air
Joule constant
speed of sound
Symbol
p


R=/M
Cp
Cv
J
a
lbm unit
lb/ft2
slug/ft3
4.97+4
ft-lb/slug-mole-R
1716
ft-lb/slug-R
7.73
5.5
778.16 ft-lbf/BTU
1100 ft/s
lbf unit
lb/ft2
lbm/ft3
1545.33
ft-lb/lbm-mole-R
53.35
ft-lb/lbm-R
0.24
0.172
kg u
N/m
N/m
831
J/kg-m
287
J/kg
1.00
0.71
1100 ft/s
440 m
4
Useful Equivalents
Quantity
Original Unit
Flow
Specific Energy
Mass
Rotational speed
Kinematic viscosity
Pressue
1.0
1.0
1.0
1.0
1.0
1.0
cfs [ft3/sec]
ft2/s2
slug
rad/s
ft2/s
in. H2O
Equivalent
448. gal/min
1.0 ft-lbf/slug
32.174 lbm
9.549 rev/min
92,903 centistokes
5.2 lbf/ft2
5
• For Liquid Water :
  62.4lbm / ft 3
• U.S. Standard Atmosphere - 1976
lbf
pressure  14.696 2  101,325 Pa
in
temperature  518.67R  273 K
o
6
Standard Atmosphere
Altitude
Stratosphere
>65,000 ft
36,089 ft
59 F
Temperature
Altitude
36,089 ft
3.202 psia
14.696 psia
Pressure
7
8
9
Thermodynamics Review
10
Thermodynamics Review
• Thermodynamic views
– microscopic: collection of particles in random motion.
Equilibrium refers to maximum state of disorder
– macroscopic: gas as a continuum. Equilibrium is
evidenced by no gradients
• 0th Law of Thermo [thermodynamic definition of
temperature]:
– When any two bodies are in thermal equilibrium with
a third, they are also in thermal equilibrium with each
other.
– Correspondingly, when two bodies are in thermal
equilibrium with one another they are said to be at
the same temperature.
11
Thermodynamics Review
• 1st Law of Thermo [Conservation of energy]: Total work
is same in all adiabatic processes between any two
equilibrium states having same kinetic and potential
energy.
– Introduces idea of stored or internal energy E
– dE = dQ - dW
• dW = Work done by system [+]=dWout= - pdV
• Some books have dE=dQ+dW [where dW is work done ON
system]
• dQ = Heat added to system [+]=dQin
– Heat and work are mutually convertible. Ratio of conversion is
called mechanical equivalent of heat J = joule
12
Review of Thermodynamics
• Stored energy E components
– Internal energy (U), kinetic energy (mV2/2), potential energy,
chemical energy
• Energy definitions
– Introduces e = internal energy = e(T, p)
– e = e(T)  de = Cv(T) dT thermally perfect
– e = Cv T
calorically perfect
• 2nd law of Thermo
– Introduces idea of entropy S
– Production of s must be positive
– Every natural system, if left undisturbed, will change
spontaneously and approach a state of equilibrium or rest. The
property associated with the capability of systems for change is
called entropy.
dS 
 Qrev
T

TdS  dE  dW
13
Review of Thermodynamics
• Extensive variables – depend on total mass of the system, e.g. M, E,
S, V
• Intensive variables – do not depend on total mass of the system, e.g.
p, T, s,  (1/v)
• Equilibrium (state of maximum disorder) – bodies that are at the same
temperature are called in thermal equilibrium.
• Reversible – process from one state to another state during which the
whole process is in equilibrium
• Irreversible – all natural or spontaneous processes are irreversible,
e.g. effects of viscosity, conduction, etc.
14
Thermodynamic Properties
Derived
Primitive
Extensive
Intensive
Extensive
Intensive
Mass – M
Density - 
Energy – Eo
Specific energy – eo
-
Pressure – p
Kinetic energy – Ek
Sp. kin. energy – V2/2
-
Temperature – T
Potential energy – Ep
Sp. pot. energy – gz
Volume - V
Specific volume - 
Internal energy - E
Sp. int. energy - e
E0  E  Ek  E p
 0   T
or
V2
e0  e 
 gz
2
 Total or stagnation state
15
1st Law of Thermodynamics
• For steady flow, defining:
V2 /2
specific kinetic energy
gz
specific potential energy
eu
specific internal energy
h =e+pv  e+
p
specific enthalpy

V2
e0  e 
 gz
2
total specific energy
• We can write:
V2
e0  pv  e 
 gz  pv
2
• and
h  e  pv
and
h0  e0  pv
16
1st Law of Thermodynamics
• Substituting back into 1st law:
E0  Q  W   m  h  V 2 / 2  gz    m  h  V 2 / 2  gz 
out
in
– Height term often negligible (not for hydraulic machines)
• Defining total or stagnation enthalpy:
h0  h  V 2 / 2
• The first law for open systems is:
Q  W   m  h0   m  ho
out
in
17
Equation of State
• The relation between the thermodynamic properties of a pure
substance is referred to as the equation of state for that substance, i.e.
F(p, v, T) = 0
• Ideal (Perfect) Gas
– Intermolecular forces are neglected
– The ratio pV/T in limit as p  0 is known as the universal gas
constant (R).
p  /T  R = 8.3143e3
– At sufficiently low pressures, for all gases
p/T = R
or
p   RT
• Real gas: intermolecular forces are important
18
Real Gas
 1150 R
19
Real Gas
20
1st & 2nd Law of Thermodynamics
• Gibbs Eqn. relates 2nd law properties to 1st law properties:
Tds  pdv  de
h  e  pv
dh  de  pdv  vdp
Tds  dh 
dp

21
Gibbs Equation
• Isentropic form of Gibbs equation:
dh 
dp

• and using specific heat at constant pressure:
RT
c p dT 
dP
P
dT R dP

T
cp P
22
Thermally & Calorically Perfect Gas
• Also, for a thermally perfect gas Cp[T]:
cP  cv  R
kT c p
 =k= =
ks cv
 -1 R


cp
 R 1.4 R
dT   1 dP
cp

 3.5R for air

  1 0.4
T
 P
• Calorically perfect gas - Constant Cp
dT   1 dP
1 T   1 P
2
2
23
Isentropic Flow
• For Isentropic Flow [if dQ=0, Adiabatic Gas Law]:
T2  P2 
 
T1  P1 
also
T0  P0 
 
T P
  1 / 
or
T  CP  1 /
  1 / 
• Precise gas tables available for design work
• Thermally Perfect Gas good flows at moderate
temperature.
24
Common Gases
Gas

Argon
1.67
Helium
1.67
Air
1.40
Hydrogen
1.40
Nitrogen
1.40
Oxygen
1.39
Water vapor
1.33
Carbon dioxide
1.29
Sulfur dioxide
1.29
Butane
1.10
monatomic
diatomic
polyatomic
25
Important Constants for Air
Rair   / M  8314.3 / 28.97  287 m 2 / s 2  K
R
R  53.35 ft  lb / lbm  R c p  air 
 0.24 Btu / lbm  R
 1
R
R  1716 ft  lbf / slug  R c p  air 
 7.73 Btu / lbf  R
 1
R
R  287 J / kg  K
c p  air 
 1004.5 J / kg  K
 1
26
Gibbs Equation
• Rewriting Gibbs Equation:
Tds  dh 
dP

c p dT
1 RT dP
ds 

T
T P P
ds dT   1 dP


cP
T
 P
 T2    1  P2 
s2  s1
 ln   
ln  
cp

 T1 
 P1 
27
Gibbs Equation
• Rewriting Gibbs Equation:
Apply at stagnation state
 T02    1  P02 
s2  s1
 ln 
ln 


cp

 T01 
 P01 
For adiabatic processes, T0  constant
s2  s1
  1  P02 

ln 

cp

P
 01 
 P02 
 s2  s1 
1

  exp  

R 

 P01 
28
Mollier Chart for Air
3,000
P=50Atm
Temperature Deg R
2,500
Isobars are not parallel
20
2,000
10
1,500
5
2
1,000
500
0.00
1
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Entropy - BTU/Lbm/deg R
29
Mollier for Static / Total States
Poout
h02
1,650
h02i
We will soon see
1,450
P out
V2/2
2
V
h0  h 
2
1,250
T 1,050
Real
Ideal
850
Poin
650
h01
P in
450
-0.02
s
-0.01
0.00
0.01
0.02
S
0.03
0.04
0.05
0.06
30