Mechanical Engineering
Equation book
Sagar Wilcy Tom
eggsam
Contents
1 Basics and Engineering
1.1 Equations of motion
1.2 Momentum . . . . .
1.3 Collision . . . . . . .
1.4 Force . . . . . . . . .
1.5 Truss . . . . . . . . .
1.6 Friction . . . . . . .
1.7 Work and energy . .
1.8 Constants . . . . . .
1.9 Units . . . . . . . . .
1.10 Vector . . . . . . . .
1.11 Misc . . . . . . . . .
Mechanics
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2 Strength of Materials
2.1 Stress and strain . . . . .
2.2 Complex stress and strain
2.3 Slopes and deflections . . .
2.4 Shear stress distribution in
2.5 Torsion . . . . . . . . . . .
2.6 Strain energy . . . . . . .
2.7 Applications . . . . . . . .
2.8 Misc . . . . . . . . . . . .
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beams
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3 Theory of Machines
3.1 Analysis of Planar mechanisms
3.2 Gears . . . . . . . . . . . . . .
3.3 Fly wheels . . . . . . . . . . . .
3.4 Governors . . . . . . . . . . . .
3.5 Balancing . . . . . . . . . . . .
3.6 Cam and Follower . . . . . . . .
3.7 Gyroscope . . . . . . . . . . . .
3.8 Vibration . . . . . . . . . . . .
4 Machine design
4.1 Theories of failure
4.2 Fatigue failure . .
4.3 Keys . . . . . . .
4.4 Rivets . . . . . .
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www.eggsam.com
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
CONTENTS
Threaded joints . . . . .
Welded joints . . . . . .
Sliding contact bearings
Rolling contact bearing .
Clutch . . . . . . . . . .
Brakes . . . . . . . . . .
Gear . . . . . . . . . . .
Power screw . . . . . . .
Misc . . . . . . . . . . .
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5 Production Engineering
5.1 Metrology . . . . . . . . .
5.2 Casting . . . . . . . . . .
5.3 Metal Cutting . . . . . . .
5.4 Machining . . . . . . . . .
5.5 Metal forming . . . . . . .
5.6 Welding . . . . . . . . . .
5.7 Non-traditional Machining
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6 IM & OR
6.1 CPM / PERT . . . . . . . . . .
6.2 Inventory control . . . . . . . .
6.3 Forecasting . . . . . . . . . . .
6.4 Break even analysis . . . . . . .
6.5 Queueing theory . . . . . . . .
6.6 Linear Programming . . . . . .
6.7 Transportation . . . . . . . . .
6.8 Assignment . . . . . . . . . . .
6.9 Work study . . . . . . . . . . .
6.10 Scheduling and loading . . . . .
6.11 Line Balancing . . . . . . . . .
6.12 Material Requirement planning
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7 Material Science
7.1 Tests . . . . . . . . .
7.2 Plastics . . . . . . .
7.3 Ceramics . . . . . . .
7.4 Crystal structure and
7.5 Alloys . . . . . . . .
7.6 Phase diagrams . . .
7.7 Heat treatment . . .
7.8 Nanomaterials . . . .
7.9 Misc . . . . . . . . .
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defects .
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8 Mechatronics and robotics
8.1 Microprocessors . . . . .
8.2 Microcontroller . . . . .
8.3 Stepper motor . . . . . .
8.4 Optical encoder . . . . .
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8.5
8.6
8.7
8.8
8.9
Hall sensor . . .
Electromagnetic
Actuators . . .
Robotics . . . .
Transducer . . .
CONTENTS
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induction
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. . . . . .
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9 Fluid Mechanics
9.1 Hydrostatics . . . . . . .
9.2 Viscosity . . . . . . . . .
9.3 Kinematics of fluid flow
9.4 Flow Dynamics . . . . .
9.5 Flow measurement . . .
9.6 Laminar flow . . . . . .
9.7 Turbulent flow . . . . . .
9.8 Losses . . . . . . . . . .
9.9 Boundary Layer Flow . .
9.10 Misc . . . . . . . . . . .
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120
121
121
123
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124
124
126
127
130
131
132
134
135
136
138
10 Fluid Machinery
139
10.1 Impact of jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.2 Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.3 Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11 Heat Transfer
11.1 Named numbers . . . . . . . . .
11.2 Conduction . . . . . . . . . . .
11.3 Convection . . . . . . . . . . . .
11.4 Boiling . . . . . . . . . . . . . .
11.5 Fins and Transient heat transfer
11.6 Heat exchanger . . . . . . . . .
11.7 Radiation . . . . . . . . . . . .
11.8 Misc . . . . . . . . . . . . . . .
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146
147
148
149
151
151
152
153
154
12 Thermodynamics
12.1 Basic Principles . . . . . . .
12.2 Work and Heat . . . . . . .
12.3 Laws of thermodynamics . .
12.4 Entropy . . . . . . . . . . .
12.5 Availability . . . . . . . . .
12.6 Pure Substances . . . . . . .
12.7 Air cycles . . . . . . . . . .
12.8 Psychrometry . . . . . . . .
12.9 Rankine cycle . . . . . . . .
12.10Steam Turbine . . . . . . .
12.11Brayton cycle, Gas Turbines
12.12Nozzle . . . . . . . . . . . .
12.13Jet Propulsion . . . . . . . .
12.14Reciprocating compressor .
12.15Centrifugal compressor . . .
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155
155
157
158
160
161
162
164
165
166
167
168
170
170
171
172
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www.eggsam.com
CONTENTS
12.16Axial flow compressors . . . . . .
12.17Fuels and combustion . . . . . . .
12.18IC Engines . . . . . . . . . . . . .
12.19Refrigeration & Air Conditioning
12.20Boiler . . . . . . . . . . . . . . .
13 Renewable sources
13.1 Solar energy . .
13.2 Wind energy . .
13.3 Biomass energy
13.4 Tidal power . .
13.5 Fuel cells . . . .
of Energy
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
14 Engineering Mathematics
14.1 Linear Algebra . . . . . .
14.2 Calculus . . . . . . . . . .
14.3 Vector Calculus . . . . . .
14.4 Probability and Statistics
14.5 Differential Equations . . .
14.6 Laplace transform . . . . .
14.7 Complex variables . . . . .
14.8 Numerical methods . . . .
14.9 Misc . . . . . . . . . . . .
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172
172
174
177
180
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182
182
186
186
187
187
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189
189
197
202
203
205
209
211
215
218
www.eggsam.com
CONTENTS
[5]
Important
This equation book is a compilation of equations I collected during my studies. The aim
of this book is to make revision easier. If you are studying the subjects for the first
time, then please do not use this book. This book is not for first time study.
Version 3
Chapter 1
Basics and Engineering Mechanics
1.1
Equations of motion
Straight line motion
v = u + at
1
s = ut + at2
2
v 2 − u2 = 2as
1
xn = u + a(2n − 1)
2
dx
v=
dt
dv
dv dx
dv
d2 x
a=
=
=v
= 2
dt
dx dt
dx
dt
Linear momentum p~ = m~v
u:
v:
a:
t:
xn :
θn :
ω0 :
ω:
θ:
r:
Rotational motion
ω = ω0 + αt
1
θ = ω0 t + αt2
2
ω 2 − ω02 = 2αθ
1
θn = ω0 + α(2n − 1)
2
dθ
ω=
dt
dω
dω dθ
dω
d2 θ
α=
=
=ω
= 2
dt
dθ dt
dθ
dt
Angular momentum = Iω
Tangential velocity ~v = ω
~ × ~r
Tangential speed v = rω
Tangential acceleration aT = rα
v2
Radial acceleration=
= rω 2
r
=centrifugal acceleration
=centripetal acceleration
initial velocity
final velocity
acceleration
time
Displacement in the nth second
Angular displacement in the nth second
Initial angular velocity
Final angular velocity
Angular displacement
Radius
Projectile motion
v0 =Launching speed
θ= Launching angle w.r.t horizontal plane
6
www.eggsam.com
Horizontal acceleration
Vertical acceleration
Horizontal velocity
Vertical velocity
Horizontal displacement at t
Vertical displacement at t
Time of flight
Maximum height
Range
For maximum range
Path
1.2. MOMENTUM
[7]
ax = 0
ay = −g
vx = v0 cos θ
vy = v0 sin θ − gt
x = v0 t cos θ
1
y = v0 t sin θ − gt2
2
2v0 sin θ
t=
g
v02 sin2 θ
h=
2g
v02 sin 2θ
R=
g
o
θ = 45
gx2
y = x tan θ − 2
2v0 cos2 θ
Projectile motion on an inclined plane
β: Inclination of plane
θ: launching angle wrt horizontal plane
2v0 sin(θ − β)
Time of flight t =
g cos β
2v02 sin(θ + β) cos θ
Range R =
g cos2 β
Coefficient of restitution
Relative velocity after impact
e=−
Relative velocity before impact
e=0 for perfectly plastic collision
e=1 for perfectly elastic collision
Cylindrical coordinates
Position P~ = rr̂ + θθ̂
Velocity V~ = ṙr̂ + rθ̇θ̂
Acceleration ~a = (r̈ − rθ̇2 )r̂ + (rθ̈ + 2ṙθ̇)θ̂
3/2
[1 + y 02 ]3/2
[1 + (dy/dx)2 ]
=
Radius of curvature r =
d2 y/d2 x
y 00
1.2
Momentum
Momentum P~ = mV~
-Vector
Conservation of momentum: In the absence of external forces, the total momentum
of a system is constant
www.eggsam.com
1.3
1.3. COLLISION
[8]
Collision
Elastic collision
Linear momentum and kinetic energy are conserved.
m1 with velocity u1 collides with m2 with velocity
u2
m1 − m2
2m1
Velocity of m1 after collision v1 =
u1 +
u2
m1 + m2 m1 + m2 m2 − m1
2m2
Velocity of m2 after collision, v2 =
u2 +
u1
m1 + m2
m1 + m2
1.4
Force
Newton’s laws of motion
First law: In an inertial frame of reference, an object either remains at rest or continues
to move at a constant velocity, unless acted upon by a force.
Second law: In an inertial frame of reference, the vector sum of the forces F on an
object is equal to the mass m of that object multiplied by the acceleration a of the object.
F = ma.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
d~v
= m~a
Force F~ = m
dt
Torque ~τ = I~
α
mv 2 r̂
Centripetal force F~c = −
r
Force on a spring F = kx
x: extension/compression of the spring
k: spring constant or stiffness of the spring
Gm1 m2
Gravitational force F~ = −
r̂
r2
G: 6.67408 × 10−11 m3 kg−1 s−2
r: Distance between the bodies
Lami’s Theorem
P
Q
R
=
=
sin α
sin β
sin γ
Impulse
Integral of a force over the time interval for which it acts.
Change in momentum
www.eggsam.com
1.5
1.5. TRUSS
[9]
Truss
Perfect truss, number of members m=2j-3
m>2j-3: Redundant truss
m<2j-3: Deficient truss
j: number of joints
m: number of members
1.6
Friction
For a stationary body, F~f riction = F
For a moving body, F~f riction = µmg
Angle of friction φ = tan−1 µ
Angle of Repose α = tan−1 µ
α=φ
F: horizontal force acting on the body.
Limiting friction: The maximum friction that can be generated between two static
surfaces in contact with each other. Once a force applied to the two surfaces exceeds the
limiting friction, motion will occur.
Stiction is the static friction that needs to be overcome to enable relative motion of
stationary objects in contact.
Amontons’ First Law: The force of friction is directly proportional to the applied load.
Amontons’ Second Law: The force of friction is independent of the apparent area
of contact. [As long as there is contact]
Coulomb’s Law of Friction: Kinetic friction is independent of the sliding velocity.
1.7
Work=
Work and energy
~
x
R2
F~ · d~x
~
x1
x: displacement
Work done by torque on a rotating body W = T × θ
1
Kinetic energy = mv 2
2
1
Kinetic energy = Iω 2
2
Potential energy due to gravity =mgh
1
Potential energy in a spring = kx2
2
1 2
Work done by a spring W = kx
2
www.eggsam.com
1.8
1.8. CONSTANTS
Constants
π
Euler’s constant e
Plank’s constant h
Universal gas constant R
Boltzmann constant k
Stefan-Boltzmann constant σ
Acceleration due to gravity g
Atmospheric pressure at MSL
Density of mercury
Faraday constant F
Water
Density
Molecular mass
Gas constant
Sp. heat of ice
Sp. heat of water
Sp. heat of water vapor
Latent heat of fusion
Latent heat of vaporization
Air
Density
Mean molecular mass
Gas constant
Sp. heat at const volume
Sp. heat at const pressure
Atomic mass
H
He
C
N
O
S
Al
Fe
3.14
2.718
6.625 × 10−34 Js
8.314 J/molK
1.38×10−23 J/K
5.67 × 10−8 Wm−2 K−4
9.81 m/s
101320 Pa
13600 kg/m3
96485 C/mol
1000 kg/m3
18.015 g/mol
461 J/kgK
2.108 kJ/kgK
4.187 kJ/kgK
1.996 kJ/kgK
336k J/kg
2264.7 kJ/kg
(at STP)
1.223 kg/m3
28.9 g/mol
287 J/KgK
718 J/kgK
1005 J/kgK
(g)
1
4
12
14
16
32
27
56
[10]
www.eggsam.com
1.9
101
102
103
106
109
1012
1015
1018
1021
1024
1.9. UNITS
[11]
Units
Deca
Hecto
kilo
Mega
Giga
Tera
Peta
Exa
Zetta
Yotta
(da)
(h)
(k)
(M)
(G)
(T)
(P)
(E)
10−1
10−2
10−3
10−6
10−9
10−12
10−15
10−18
10−21
10−24
Length, Area, Volume
1 in =2.54cm
1 foot=30.48cm
1 foot=12in
1 mile=1.6km
1 km2 =147 acres
1 Liter=1000 cm3
1000 Liter=1 m3
1 gal=3.786L
Mass
1 pound=0.453 kg
1 tonne=1000 kg*
1 US tonne=907 kg
1 long tonne=1016 kg
Temperature
C ×9
+ 32
F =
5o
Kelvin= C + 273.15
Tripple point
1 K=1o C =
273.16
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
(d)
(c)
(m)
(µ)
(n)
(p)
(f)
(a)
Force and pressure
1 kgf=9.8N
1 lbf=4.44N
1 bar=100kPa
1 atm=1.0132bar
1 lbf/in2 =6.89kPa
1 torr=1mmHg
1 kgf/cm2 =0.98 bar
1 N = 105 Dyne
Energy and Power
1 Btu=1054 J
1 cal=4.18 J
1 eV=1.6 × 10−19 J
1 hp=746W* (Mechanical hp)
1 hp=735.5W (Metric hp)
1 kWh= 3.6 MJ
1 poise=0.1 Ns/m2
1 stoke = 10−4 m2 /s
π rad=180o
1 Gauss= 10−4 Tesla
Refrigeration
1TR= 12000BTU/hr
1TR= 3000kcal/hr
1TR= 3.517kW*
1TR= 3.88kW
* Commonly used
Light year
Chandrasekhar limit
Astronomical unit
Parsec
: Distance traveled by light in one year = 9.4607×1012 km
: 1.4 times the mass of sun = 2.8×1034 kg
: Average distance between earth and sun = 149.59×106 km
: 3.26156 light year, distance at which one astronomical unit
subtends an angle of one arcsecond, which corresponds to
astronomical units.
www.eggsam.com
7 Basic units
Quantity
Mass
Length
Time
Electric current
Temperature
Luminous intensity
Amount of substance
1.9. UNITS
Name
kilo gram
meter
second
Ampere
Kelvin
candela
mole
[12]
SI Unit
kg M1 L0 T0
m M0 L1 T0
s M0 L0 T1
A
K
cd
mol
Derived units
Quantity
plane angle
solid angle
frequency
force, weight
pressure, stress
energy, work, heat
power, radiant flux
electric charge
voltage (electrical potential), emf
capacitance
resistance, impedance, reactance
electrical conductance
magnetic flux
magnetic flux density
inductance
luminous flux
illuminance
radioactivity (decays per unit time)
absorbed dose (of ionising radiation)
equivalent dose (of ionising radiation)
catalytic activity
area
volume
speed, velocity
acceleration
wavenumber
density
surface density
specific volume
current density
magnetic field strength
concentration
mass concentration
luminance
refractive index
relative permeability
Name
radian
steradian
hertz
newton
pascal
joule
watt
coulomb
volt
farad
ohm
siemens
weber
tesla
henry
lumen
lux
becquerel
gray
sievert
katal
Symbol
rad
sr
Hz
N
Pa
J
W
C
V
F
Ω
S
Wb
T
H
lm
lx
Bq
Gy
Sv
kat
In SI base units
(mm−1 )
(m2 m −2 )
s−1
kgms−2
kgm−1 s−2
kgm2 s−2
kgm2 s−3
As
kgm2 s−3 A−1
kg−1 m−2 s4 A2
kgm2 s−3 A−2
kg−1 m−2 s3 A2
kgm2 s−2 A−1
kgs−2 A−1
kgm2 s−2 A−2
cd
m−2 cd
s−1
m2 s−2
m2 s−2
mol s−1
m2
m3
ms−1
ms−2
m−1
kgm−3
kgm−2
m3 kg−1
Am−2
Am−1
molm−3
kgm−3
cdm−2
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1.10
1.10. VECTOR
[13]
Vector
P~ = p1 î + p2 ĵ + p3 k̂
~ = q1 î + q2 ĵ + q3 k̂
Q
θ: Angle between the vectors
p
Magnitude P = |P~ |= p21 + p22 + p23
~
~ ~
Vector
p sum R = P + Q = (p1 + q1 )î + (p2 + q2 )ĵ + (p3 + q3 )k̂
R = P 2 + Q2 + 2P Q cos θ
~ = (p1 q1 ) + (p2 q2 ) + (p3 q3 ) = P Q cos θ
Dot product P~ · Q
î ĵ k̂
~ = p1 p2 p 3
Cross product P~ × Q
q1 q2 q3
~
~
|P × Q| = P Q sin θ
Unit vector: A vector whose magnitude is 1
Divergence = ∇ · P
∂
∂
∂
∇=
î +
ĵ + k̂
∂x
∂y
∂z
1.11
Misc
R
x̄dA
Centroid XG = R
dA
m1 m2
Reduced mass µ =
m1 + m2
Parallel axis theorem Id = ICM + md2
Perpendicular axis theorem Izz = Ixx + Iyy
Center of percussion
The point on an extended massive object attached to a pivot where a perpendicular
impact will produce no reactive shock at the pivot. Translational and rotational motions
cancel at the pivot when an impulsive blow is struck at the center of percussion.
The same point is called the center of oscillation for the object suspended from the pivot
as a pendulum, meaning that a simple pendulum with all its mass concentrated at that
point will have the same period of oscillation as the compound pendulum.
Varignon’s Theorem
If many concurrent forces are acting on a body, then the algebraic sum of torques of all
the forces about a point in the plane of the forces is equal to the torque of their resultant
about the same point.
Radial run-out
Result of rotating component running off centre, such as a ball bearing with an offset
centre.
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1.11. MISC
[14]
The rotating tool or shaft, instead of being centrally aligned, will rotate about a secondary
axis.
Simple pendulum
r
l
T = 2π
g
Compound pendulum
r
I
T = 2π
mgl
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1.11. MISC
[15]
Moment of inertia
Shape
Rectangle
Triangle
Circle
Moment of inertia
bd3
Ixx =
12
b3 d
Iyy =
12
bh3
Ixx =
36
πd4
Ixx = Iyy =
64
πd4
Izz =
32
a2 + 4ab + b2 3
h
36(a + b)
Trapezium
I=
Semicircle
Ixx =
Diamond (side a)
Solid sphere
Hollow sphere
Slender rod about
midpoint
Slender rod about end
Circular ring
Solid disc
Cylinder
1 πd4
2 4 64
a
Ixx =
12
2 2
I = mr
5
2
I = mr2
3
1
I = ml2
12
1
I = ml2
3
Izz = mr2
Ixx = Iyy =
1
Izz = mr2
2
Ixx = Iyy =
Center
h/3 from base
Center
2a + b h
a+b 3
2d
3π
Center
Center
Center
Center
Center
1 2
mr
2
Center
1 2
mr
4
Center
I = m(r12 + r22 )
Hemisphere
Cone
CG
1
Volume V= πr2 h
3
Center
3R
8
h
4
Chapter 2
Strength of Materials
2.1
Stress and strain

σxx σxy σxz
σ =  σyx σyy σyz
 σzx νσzy σνzz
xy
xz
xx
 ν
2 ν2
yz
 yx
yy
=
2
 ν2 ν
zy
zx
zz
2
2







Normal stress
Hooke’s law: E = σ/ε
Hooke’s law is valid up to proportionality limit
P
Stress σ =
A
δl
Strain ε =
L
PL
Change in length δl =
AE
Poisson’s ratio
Lateral strain
µ=−
Linear strain
Lies in the range -1.0 to 0.5 (For an isotropic linear elastic materials)
For perfectly isotropic elastic material µ = 0.25
For perfectly incompressible isotropic, µ = 0.5
For rubber µ ≈ 0.5
For cork µ ≈ 0.0
Auxetics: Structures with negative Poisson’s ratio
True stress and strain
Force
σT =
Area at the instant
σT = σe (1 + e )
16
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2.1. STRESS AND STRAIN
L
L0
T = ln(e + 1)
σT = KnT
n: Strain hardening exponent
T = ln
Shear stress
F
τ=
A
= P/G
Volumetric stress
ev = ex + ey + ez
4V
ev =
V
v = P/K
1
= (σx + σy + σz ) (1 − 2µ)
E
Bar of uniform strength
h ρg i
Ax = A exp
x
P
Tapered rod
δl =
PL
πd1 d2
E
4
Uniformly tapered rectangular bar with constant thickness t
δl =
PL
(a − b)
Et
ln(a/b)
Rotating bar about one end
1
σx = ρω 2 [l2 − x2 ] (From axis)
2
1 23
δl = ρω l
3
Strain due to self weight
δl =
PL
ρgL2
=
2AE
2E
Thermal stress, at least one end free
σ=0
δl = Lα∆T
[17]
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2.1. STRESS AND STRAIN
Thermal stress, both ends fixed
δl = 0
σ = Eα∆T
Thermal stress, one support yields by l0
δl = l0
σ = (α∆T − l0 /L)E
Impact load
r
Impact factor IF =1 +
1+
2h
δstatic
σ = IF × σstatic
δ = IF × δstatic
PL
δstatic =
AE
h: Height from where the weight P is dropped.
Sudden load: σ = 2
P
A
Relation between elastic constants
E = 2G (1 + µ)
E = 3K (1 − 2µ)
9KG
E=
3K + G
Lame’s constant
Eµ
λ=
(1 + µ) (1 − 2µ)
Number of independent elastic constants
Isotropic
: 2
Orthotropic : 9
Anisotropic : 21
Equivalent Young’s modulus
E1 A1 + E2 A2
E=
A1 + A2
Angle of failure
Ductile
Tension
45o
Compression 90o
Torsion
90o
Brittle
90o
45o
45o
[18]
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2.2. COMPLEX STRESS AND STRAIN
2.2 Complex stress and
strain
Mohr’s Circle s
2
σx − σy
σx + σy
2
σ1 =
+
+ τxy
2
2
s
2
σx + σy
σx − σy
2
−
+ τxy
σ2 =
2
2
s
2
σx − σy
2
τ=
+ τxy
2
σ1 − σ2
τ=
2
σx + σy σx − σy
+
cos 2θ + τxy sin 2θ
σθ =
2
2
σx − σy
sin 2θ − τxy cos 2θ
τθ =
2
2τxy
tan 2θ =
σx − σy
τθ
Obliquity φ = tan−1
σθ
Obliquity: Angle made by the line connecting a
point on the Mohr circle and origin with the xaxis
Mohr’s circle for strain
σ→
τ → φ2
x + y x − y
φxy
θ =
+
cos 2θ +
sin 2θ
2
2
2
2
2
θ = x cos θ + y sin θ + φxy cos θ sin θ
x − y
φxy
φθ
=
sin 2θ −
cos 2θ
2
2
2
s
2 2
x + y
x − y
φxy
1 , 2 =
±
+
2
2
2
E(1 + µ2 )
σ1 =
1 − µ2
E(2 + µ1 )
σ2 =
1 − µ2
Rectangle strain rosette
φxy = 245 − x − y
[19]
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2.3. SLOPES AND DEFLECTIONS
[20]
Sign Convention
2.3
Slopes and deflections
Counter clockwise M negative
Positive bending moment =⇒ sagging
E
M
σb
=
=
R
I
y
d2 y
EI 2 = M (x) Bending moment
dx
d3 y
EI 3 = S(x) Shear force
dx
d4 y
EI 4 = −w(x) Rate of loading
dx
Beam of uniform strength σb = Constant
Conjugate beam method
Beam
Conjugate beam
Slope
Shear force
Deflection
Bending moment
End pin support
Pin support
Internal Pin support Hinged joint
Hinged joint
Internal pin support
Fixed end
Free end
Free end
Fixed end
Maxwell’s Reciprocal Theorem
In any beam or truss, the deflection at any point D due to load W at any other point C
is the same as the deflection at C due to the same load at D.
Point of contra flexure
Curvature changes from sagging to hogging
Point of inflection
Virtual hinge
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2.3. SLOPES AND DEFLECTIONS
[21]
www.eggsam.com2.4. SHEAR STRESS DISTRIBUTION IN BEAMS
2.4
Shear stress distribution in beams
A: Area above PQ
Ȳ : Centroid of the area above the PQ
S: Shear force at the section
I: Moment of inertia of the full section about Neutral axis (NA)
b: Length of PQ
SAȲ
Shear stress at section PQ, τ =
bI
Section type
Image
τave
3
Rectangular section
τmax = τave
2
4
Solid circular section
τmax = τave
3
4
Triangular section
4
τmax = τave
3
9
Diamond section
τmax = τave
8
2.5
Torsion
τ
Gθ
T
=
=
J
R
L
J
Torsional section modulus Z =
R
GJ
T
Torsional stiffness = =
θ
L
Equivalent Moment and torsion
√
1
Me =
M + M2 + T 2
√2
Te = M 2 + T 2
Pure torsion assumptions
Uniform material
Uniform twist
Shaft is of uniform circular cross section
Cross sections which are plane remains plane
All radii remain straight after twist
[22]
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2.6. STRAIN ENERGY
[23]
Compound shaft
2.6
Shafts in series
Parallel shafts
θ = θ1 + θ2
T = T1 = T2
θ = θ1 = θ2
T = T1 + T2
Strain energy
Sudden or impact loading
U = Pδ
1
P 2L
E
Due to direct stress
U = Pδ =
=
Volume
22
2AE
2
τ
Due to shear stress
U=
Volume
2G2
σ
Volume
Due to volumetric stress
U=
2K
2
T L
1
τ2
Due to torsion
U=
= T θ = max Volume
2GJ
22
4G
2
τmax
R1 + R22
Due to torsion, hollow shaft U =
Volume
4G
R22
RL 1 Mx2
For a beam
U=
dx
0 2 EI
Where Mx is the bending moment in the beam as a function of distance x.
Castigliano’s theorem
∂U
yr =
∂Pr
If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Pr then the partial derivative of the strain energy with respect to Pr gives
the generalized displacement yr in the direction of Pr .
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2.7
2.7. APPLICATIONS
Applications
Thin Cylinder
Hoop stress
Longitudinal stress
Longitudinal strain
Hoop strain
Volumetric strain
Pd
2t
Pd
σl =
4t
Pd
el =
(1 − 2µ)
4tE
Pd
eh =
(2 − µ)
4tE
Pd
ev =
(5 − 4µ)
4tE
σh =
Thin Sphere
Hoop stress
Hoop strain
Volumetric strain
Pd
4t
Pd
(1 − µ)
eh =
4tE
Pd
ev = 3
(1 − µ)
4tE
σh =
Thick Cylinder
Radial stress at radius r
Hoop stress at radius r
Longitudinal stress
Circumferential strain
B
r2
B
σh = A + 2
r
Pi ri2 − Po ro2
σL =
ro2 − ri2
1
h = (σh − µσr − µσL )
E
σr = A −
Thick Spherical shell
Radial stress at radius r
Hoop stress at radius r
Springs
D
Spring index C =
d
Gd4
k=
64R3 n
WR
τmax =
πd3
16
2B
r3
B
σh = A + 3
r
σr = A −
64W R3 n
Maximum deflection δ =
Gd4
4C − 1 0.615
Wahl’s factor =
+
4C − 4
C
Columns and struts
π 2 EI
Euler’s formula PE = 2
le
[24]
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2.8. MISC
le
Effective length
=
K
Least radius of gyration
Type
le
Both ends fixed
L/2√
One end fixed other end hinged L/ 2
Both ends hinged
L
One end fixed, other end free
2L
Slenderness ratio λ =
Rankine’s formula
1
1
1
=
+
PR
PC PE
σc A
PR =
2
Le
1+a
k
σc
a = 2 : Rankine’s constant
π E
PC = Crushing load = σc A
PE = Euler’s load
PR = Rankine’s load
2.8
Misc
Axial rigidity = AE
Torsional rigidity = GJ
I
Section modulus Z =
Y
J
Torsional section modulus Z =
R
GJ
Torsional stiffness =
l
le
Effective length
Slenderness ratio λ =
=
K
Least radius of gyration
Beam
Transverse load
Bending stress on the beam
Column
Axial load
Vertical member
Compressive stress on the column
Moderate load
Fails by buckling if it is slender
Strut
Vertical, horizontal or inclined Heavy load
Fails by buckling if it is slender
[25]
Chapter 3
Theory of Machines
3.1
Analysis of Planar mechanisms
Kinematics: Deals with motion ignoring forces.
Dynamics: Deals with forces and its effects on motion
Link: A material body which is common to two or more kinematic pairs
Fluid can also act as a link
Type
Contact
Higher pair
Point or line contact
Lower pairs
Surface contact
Wrapping pair
One link is wrapped
in another
Example
Gear pairs
Cam and follower
Piston cylinder
Prismatic
Revolute pair
Screw pair
Spherical joint
Belt drive
Degree of freedom (Grubler’s criterion)
DOF = 6(m − 1) − 5J1 − 4J2 − 3J3 − 2J4 − J5
For planar mechanism, DOF = 3(m − 1) − 2J1 − J2
Jn : Number of joints with n degree of freedom restricted
m: Number of links
Completely constrained motion : 1 DOF
Incompletely constrained motion : More than one DOF
Successfully constrained motion : 1 DOF, (The mechanism normally has more than 1
DOF. But the extra DOFs are restricted by some other means)
Structure: DOF=0
Super structure/Preloaded structure: DOF<0
Mechanism: DOF≥1
Statically indeterminate system, DOF ≤ −1
Closed pair : Permanent contact between the links
26
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3.1. ANALYSIS OF PLANAR MECHANISMS
[27]
Force closed/open pairs : Need to apply force to keep links in contact
Kinematic chain : Last link is connected to first link and no link is fixed
Vin
Fout
Torque output
1
=
=
=
Mechanical advantage =
velocity ratio
Vout
Fin
Torque input
At Toggle position : infinite mechanical advantage
4 bar mechanism
dµ
ls sin θ
=
dθ
pq sin µ
µ: Transmission angle
b : Coupler link
c : Output link
Grashoff ’s condition
l+s<p+q Grashoff’s I
l+s>p+q Non-Grashoff’s or Grashoff’s II
l+s=p+q Special Grashoff’s or Grashoff’s III
s: Shortest link
l: longest link
Grashoff ’s I
Shortest link fixed
: crank-crank mechanism.
Adjacent link to shortest link fixed : crank-rocker mechanism.
Link opposite to shortest link fixed : rocker-rocker mechanism.
Coriolis Acceleration =2ωV
Motion
Outwards
Outwards
Inwards
Inwards
Rotation
Clockwise
Counterclockwise
Clockwise
Counterclockwise
Instantaneous center
Coriolis acceleration
Positive
Negative
Negative
Positive
n(n − 1)
Number of instantaneous centers =
2
Where n is the number of links in relative motion.
Kennedy’s Theorem: When three rigid bodies are in relative motion, the three instantaneous centers shared by three bodies all lie on the same straight line.
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3.1. ANALYSIS OF PLANAR MECHANISMS
[28]
Fixed instantaneous centers: will not move (Primary instantaneous center)
Permanent instantaneous centers: will move, but always on a point on the link (Primary
instantaneous center)
Neither fixed nor permanent instantaneous centers: Also called secondary instantaneous
centers
Body centrode
Also called moving centrode
Locus of the instantaneous centre of the fixed body relative to the movable body.
Space centrode
Also called fixed centrode
Locus of the instantaneous centre of the moving body with respect to the fixed body.
Universal joint - Hooke’s joint
-used to connect two coplanar intersecting shafts
θ1 , ω1 : Input angle/velocity
θ2 , ω2 : Output angle/velocity
α: Inclination
tan θ1 = tan θ2 cos α
cos α
ω2
=
ω1
1 − sin2 α cos2 θ1
N1
Maximum speed of driven shaft N2 =
cos α
Mechanisms
Hart mechanism
Scott-Russel mechanism
Tusi couple
Sarrus linkage
PeaucellierLipkin linkage
Quadruplanar-Inversor
Grass hopper mechanism
Robert’s mechanism
Chebyshev Linkage
Watt’s indicator mech
Hoeckens linkage
Drag-link mechanism
Pantograph mechanism
Geneva mechanism
Ratchet mechanism
3.1.1
Exact straight line motion
Exact straight line motion
Exact straight line motion
Exact straight line motion
Exact straight line motion
Exact straight line motion
Approximate straight line mechanism
Approximate straight line mechanism
Approximate straight line mechanism
Approximate straight line mechanism
Approximate straight line mechanism
Quick return mechanism
Copying mechanism
Intermittent motion
Intermittent motion
Steering
w
Equation of correct steering cot α − cot β =
L
w
Inclination angle of track arms tan θ =
2L
α: Angle of outer wheel
β: Angle of inner wheel
6 bar, 7R joints
2R 1S
2 Circle
6R, 3-Dimensional
6R
4 bar
4 bar
4 bar
Single slider
4 bar
4 bar
Indexing milling machine
Clocks
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3.1. ANALYSIS OF PLANAR MECHANISMS
w: Pivot distance of the front stub axles
L: Wheel base
Only turning pairs
Ackermann steering
Correct inner turning angle for all positions.
Davis steering mechanism
Turning pairs and spherical joints
Crossed slider mechanism
Rapson’s slide
Used in ships
Bell-Crank
Uses Rack and pinion for steering
Rack and Pinion
Click here for more information
3.1.2
Slider crank mechanism
Inversions
1st inversion
Ground is fixed
2nd inversion
Crank is fixed
3rd inversion
Rocker is fixed
4th inversion
Slider is fixed
Slider crank mechanism
Whit-worth Quick return
mechanism
Rotary Engine
Gnome engine
Shaping machine
Planning machine
Crank and slotted lever
quick return mechanism,
Oscillating cylinder engine
Hand pump, Bull engine,
pendulum pump
[29]
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3.1. ANALYSIS OF PLANAR MECHANISMS
[30]
n = l/r
h
i
p
Position x = r (1 − cos θ) + n − n2 − sin2 θ
#
"
sin 2θ
Velocity v = rω sin θ + p
2 n2 − sin2 θ
"
#
2
cos
2θ
sin
2θ
Acceleration a = rω 2 cos θ + p
−
3/2
n2 − sin2 θ 4 n2 − sin2 θ
ω cos θ
Angular velocity of connecting rod = p
n2 − sin2 θ
−ω 2 sin θ(n2 − 1)
Angular acceleration of connecting rod =
3/2
n2 − sin2 θ
When n>>1
sin 2θ
Velocity v = rω sin θ +
2n cos
2θ
Acceleration a = rω 2 cos θ +
n
ω
cos θ
Angular velocity of connecting rod =
n 2
ω
Angular acceleration of connecting rod = − sin θ
n
Pressure force FP = Pressure × Area
Force of reciprocating mass = −ma
FP
Crank effort FCR =
cos φ
Force on cylinder wall FW = FCR sin φ
Force on crank bearing FB = FCR cos(φ + θ)
Tangential Force on crank Ft = FCR sin(φ + θ)
Turning moment T = Ft × r
sin(θ + φ)
Turning moment on crank shaft = F
cos φ
Quick return mechanism
time of cutting
180 + 2α
=
time of return
180 − 2α
Klien’s construction
Angular acceleration of crank is zero
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3.2. GEARS
Green: Velocity
Red: Acceleration
3.1.3
Double slider crank mechanism
1st inversion
2nd inversion
3rd inversion
Ground is fixed
One slider is fixed
Connecting rod is fixed
Elliptical trammel
Scotch yoke mechanism, SHM
Oldham coupling
Oldham coupling
For maximum speed ratio,
3.2
ω1
1
=
ω
cos α
Gears
Gears are Positive drive (No slip is possible)
(In negative drive, slip is possible)
T: Number of teeth on gear t: Number of teeth on pinion
D,d: Pitch diameter
Ra , ra : Addendum radius
R,r: Pitch radius
φ: Pressure angle
Pitch circle diameter = d
Base circle diameter= d × cos φ (Constant for a gear)
d
module m =
T
πd
Circular pitch =
T
T
Diametral pitch =
d
Addendum = Ra - R
N1
ω1
T2
Gear ratio G=
=
=
N2
ω2
T1
input speed
Velocity ratio =
output speed
p
Gear path of approach = pRa2 − R2 cos2 φ − R sin φ
sin φ
Gear path of recess = pra2 − r2 cos2 φ − r p
2
2
2
Gear path of contact =
Ra − R cos φ + ra2 − r2 cos2 φ − (R + r) sin φ
Path of contact
Arc of contact =
cos φ
Lenght of action
arc of contact
Angle of action
Contact ratio =
=
=
mπ cos φ
πm
Pitch angle
Working depth = Addendum + Dedendum - clearance
Working depth = Sum of addendum of both the gears
To avoid interference
[31]
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3.3. FLY WHEELS
2AG
T ≥s
1+
G=
1 1
+ 2 sin2 φ − 1
G G
2AP
[32]
T
t
t≥ p
1 +p
G (G + 2) sin2 φ − 1
Ra,max = (R cos φ)2 + (R sin φ + r sin φ)2
In rack and pinion,
2AR
T ≥
sin2 φ
-add undercutting
-increase pressure angle
-tooth stubbing
-increase number of teeth in gear
-slightly increase centre-centre distance
-decrease gear addendum and increase pinion addendum
In an epicyclic gear train
N2 − Na
T1
=−
T2
N1 − Na
Helical or spiral gears
ψ: Spiral angle or helix angle
φ: Pressure angle
θ: angle between axes
θ = ψ1 + ψ2 , same hand are in contact
θ = ψ1 − ψ2 , opposite hands are in contact
m1 cos ψ1 = m2 cos ψ2
Radius R =
m T
2 cos ψ
Worm Gear
1 − sin φ
Maximum efficiency ηmax =
1 + sin φ
φ = tan−1 µ
Fundamental law of gearing
The angular velocity ratio of all gears of a meshed gear system must remain constant
The common normal at the point of contact must pass through the fixed pitch point.
3.3
Fly wheels
Coefficient of fluctuation of speed Cs =
Coefficient of steadiness =
1
CS
∆ω
ωave
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3.4. GOVERNORS
[33]
∆E
Coefficient of fluctuation of energy CE =
E
∆E = 2ECs
∆KE = Iω 2 Cs
CE = 2Cs
r
σ
Maximum velocity of flywheel=
ρ
3.4
Governors
Centrifugal Governor
1
h∝ 2
N
N2 − N1
Sensitiveness=
N
Isochronous governor: Speed is constant
Effort of governor: Mean force acting on the governor
Hunting: Speed of engine fluctuates above and below the mean speed.
Spring controlled governors
F=Ar+B
B<0, stable
B>0, unstable
B=0, isochronous
Watt Governor
Gravity controlled
Pendulum type
Porter Governor
Gravity controlled
Watt Governor + Mass
(M) at the sleeve.
Porter governor can not be
isochronous.
Proell Governor
Hartnell Governor
Hartung Governor
Wilson Hartnell
Pickering Governor
Gravity controlled
Porter governor + arms
Spring controlled
Spring controlled
Hartnell + change in spring
position
Hartnell governor with
extra spring
3 leaf springs
Used in gramophone
rω 2
g
g
895
h= 2 = 2
ω
N
Frictional force f is also present at the
sleeve
β
2mg + (M g ± f )(1 + tan
)
tan θ
2
ω =
2mh
895
Mg ± f
tan β
h= 2 1+
(1 + tan θ )
N
2mg
895 a
Mg ± f
tan β
h= 2
(1 + tan θ )
1+
N e
2mg
tan θ =
mrω 2 a = k(r − r0 )a +
Mg
b
2
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3.5
3.5. BALANCING
[34]
Balancing
Static balancing
Centre of mass of the system must lie on the rotational axis
Σmi ri sin θi
tan θc =
Σmi ri cos θi
p
mc rc = (Σmi ri sin θi )2 + (Σmi ri cos θi )2
Dynamic Balancing
-If there is a couple
Σmrl + mc rc lc = 0
Slider-crank mass balancing
n=
connecting rod length
crank length
Acceleration a = rω
2
cos 2θ
cos θ +
n
F = ma
Primary force balance=cmrω 2 cos θ
Primary force unbalance=(1 − c)mrω 2 cos θ
Primary vertical unbalanced force=cmrω 2 sin θ
Hammer blow: Maximum vertical unbalanced force by the mass used to balance the
reciprocating masses. = cmrω 2
p
Resultant primary unbalanced force = ((1 − c)mrω 2 cos θ)2 + (cmrω 2 sin θ)2
Secondary unbalanced force =
mrω 2 cos 2θ
n
If primary direct is θ clockwise,
Secondary direct is 2θ clockwise,
Secondary reverse is 2θ anticlockwise
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3.6
3.6. CAM AND FOLLOWER
[35]
Cam and Follower
v
dx
=
dθ
ω
d2 x
a
= 2
2
dθ
ω
d3 x
J
= 3
3
dθ
ω
Pitch point: The point on the pitch curve having the maximum pressure angle.
Pitch curve: Path of tracing point
Tracing point: The point of the follower from which the profile of a cam is determined.
Base circle: Smallest circle drawn to cam profile
Prime circle: Smallest circle drawn tangent to the pitch curve.
Pressure angle: Angle between direction of motion of follower and normal to the pitch
curve. Measure of how steepness of the cam profile at the given point.
Constraints on follower: Gravity, Spring, and positive mechanical constraints
Motion
Velocity
h πω
πθ
Vθ =
sin
2 φ
φ
h πω
Vmax =
2 φ
ω
V =h
φ
Acceleration
2
h πω
πθ
aθ =
cos
2 φ
φ
2
h πω
amax =
2 φ
Displacement
0
s=h
ω
Vθ = 2h 2 θ
φ
2
ω
a = 4h
φ
2
θ
s = 2h
φ
hω
2πθ
Cycloidal
Vθ =
1 − cos
φ
φ
φ: Angle of ascent/descent
2hπω 2
a=
sin
φ2
SHM
Uniform
velocity
Const. acceleration
Parabolic
motion
(θ ≤ φ/2)
3.7
2πθ
φ
h
s=
2
πθ
1 − cos
φ
θ
φ
h
s=
π
πθ 1
− sin
φ
2
Gyroscope
Gyroscopic torque = Iω × ωp
Active gyroscopic couple: Rotate the spin vector through 90o about precession axis to
get the direction of gyroscopic torque vector.
Reactive gyroscopic couple: Opposite direction of active gyroscopic couple
(Left turn -ve × CCW from behind -ve = +ve: Dip the nose)
Bow: Front of the boat (Also called Fore)
Stern: Back of the boat (Also called Aft)
Port: Left side
Starboard: Right side
2πθ
φ
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3.8
3.8. VIBRATION
Vibration
Reileigh’s
method
r
g
ωn =
deflection
Energy method
d(KE + P E)
=0
dt
Lagrange’s equation
L=KE-PE
d ∂L
∂L
−
=0
dt ∂ q̇i
∂qi
Damped vibrations
mẍ +c
rẋ + kx = 0
k
ωn =
m
r
c
c 2
k
s1,2 = −
±
−
2m
h
p 2m i m
s1,2 = ωn −ξ ± ξ 2 − 1
x = Xe−ξωn t [sin(ωd t + φ)]
√
Critical damping Cc = 2 mk = 2mωn
C
Damping ratio/factor ξ =
Cc
Degree of damping = ξ 2 p
Damped frequency ωd = ωn 1 − ξ 2
2π
1
p
Damped time period Td =
ωn 1 − ξ 2
x1
2πξ
Logarithmic decrement δ = ln = ξωn Td = p
x2
1 − ξ2
Forced oscillation
(F0 /k)
Steady state Amplitude X = v(
u
2 )2 2
u
ω
t 1− ω
+ 2ξ
ωn
ωn
ω
−2ξ
−cω
ωn
tan φ =
=
2
2
k − mω
ω
1−
ωn
X
1
Magnification factor
= v(
u
Xst
2 )2 2
u
ω
t 1− ω
+ 2ξ
ωn
ωn
Transmissivity ratio
[36]
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3.8. VIBRATION
s
2
ω
1 + 2ξ
ωn
FT
TR =
= v(
u
F
2 )2 2
u
ω
ω
t 1−
+ 2ξ
ωn
ωn
Under damped,
√
ω
< 2
TR increases if
ωn
√
ω
> 2
TR decreases if
ωn
Force transmitted =F × T R
Vibration isolation
0 < TR
√< 1
ω
> 2
ωn
Rotating unbalanced mass
2
mr ω
M ωn
X = v(
u
2 )2 2
u
ω
t 1− ω
+ 2ξ
ωn
ωn
Whirling of shafts
2
ω
e
ωn
r = v(
u
2 )2 2
u
ω
ω
t 1−
+ 2ξ
ωn
ωn
Dunkerly’s method for multiple rotor system
1
1
1
= 2 + 2 + ...
2
ωn
ω1 ω2
Torsional vibration
r
r
Torsional stiffness
GJ
ω=
=
I
Il
[37]
Chapter 4
Machine design
4.1
Theories of failure
4.2
Fatigue failure
Endurance limit: Maximum amplitude of a fully reversed stress that the standard
specimen can sustain for infinite number of cycles without fatigue failure.
38
www.eggsam.com
4.2. FATIGUE FAILURE
Maximum stress σm = σB
2a
1+
b
Corrected endurance limit σe0 = σe Ka Kb Kc Kd Ke
Name
1
Kf
Equations
Ka = 1.0 d ≤ 7.6mm
Size factor
Ka = 0.85 7.6 ≤ d ≤ 50mm
Ka = 0.75 d ≥ 50mm
Kb = 1.0
For reversed bending load
Loading factor
Kb = 0.8
Reversed axial load for steel
Kb = 0.577 Reversed torsional load
Surface factor
Kc = 1, for polished surface
Temperature factor
Kd
Ke = 1.0 50% Reliability
Reliability factor
Ke = 0.89 90% Reliability
Ke = 0.7 99.99% Reliability
Endurance limit for notch free speciment
Fatigue strength
Kf =
reduction factor
Endurance limit for notched specimen
Kf − 1
q=
Kt − 1
max stress
Stress concentration factor Kt =
avg stress
√
1
Notch sensitivity q =
a = Neuber’s constant
,
1 + (a/r)2
Endurance limit σe = 0.5σut for steels
Endurance limit σe = 0.4σut for Cast Iron, Cast steels, Al alloys
Endurance limit σe = 0.3σut for Cast Al alloys
Low and high cycle fatigue
Miners approach
n1
n2
n3
+
+
+ ... = 1
L1 L2 L3
[39]
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4.3
4.3. KEYS
Keys
b: width of the key
t: thickness
l: length
Usually b=D/4
Shear design
F
τ=
bl
Crushing design of key
F
σ=
lt/2
Loose fit
Feather Key
Permits axial movement
Self aligning
Used in tapered
Woodruf Key
Curved
Square/Rectangular Normal
Tapered
Taper sunk key
Edges adjusted
Barth key
Diamond
Kennedy Key
Two keys
Tangent Keys
One key can transmit power in one direction
Hollow saddle key transmits power by
Saddle key
friction
4.4
Rivets
√
Unwin’s formula d = 6.04 t
Bearing/crushing failure Fb = σb dt
Least Strength
Efficiency η =
Strength of plate without rivet
Rivet value R=min(Shear strength, Bearing strength, plate strength,... )
[40]
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4.5. THREADED JOINTS
No of rivets required=F/R
For diamond riveting, η =
b−d
b
Eccentric loading
F A1
ΣAi
F eA1 r1
0
F1 =
ΣAi ri2
p
Resultant force F = F12 + F102 + 2F1 F10 cos θ
F1 : Primary shear stress on rivet 1
F10 : Secondary shear stress on rivet 1
F: Force applied
ri : Distance between ith rivet and CG
Ai : Area of ith rivet
e: Distance between CG of rivets and load
F1 =
4.5
Threaded joints
British association
BSW
British Standard
Whitworth
ISO metric
Square thread
ACME
Trapezoidal
Buttress
ANS
American National
Standard
-Precision threads
-Rounded corners
-47.5o
-Used in micrometer
-For fatigue loads
-Automobiles
-rounded corners
-55o
-VVVVVV
- 60o
-Clearance given
-Very tight
-For power transmission
-Screw jack mechanism
-Modified square thread
-Bidirectional power transfer
-Lathe lead screw
-used in vice
-Unidirectional
-Maximum power transmission
-For rough usage
-Railway carriages
-Water bottle
Knuckle
-Half round
-30o
F
Shear stress in nut/bolt τ =
πdc tn
[41]
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Crushing stress σc =
4.6. WELDED JOINTS
4Fi
− d2c )n
π(d2
t: pitch
Eccentric loading, Load in plane of bolted joints
F A1
ΣAi
F
eA1 r1
F10 =
ΣAi ri2
F1 =
Eccentric loading, load perpendicular to bolt axis
Take moment about bottom point
F A1
A1 + A2 + ...
F el1
= 2
l1 + l22 + ...
Shear force on bolt 1 F11 =
Tensile force on bolt 1 F12
Screw thread designation
Md×pC
M: Metric
d: Normal diameter
p: pitch
C: Coarse diameter
Bolt of uniform strength
Core diameter of the thread = diameter of unthreaded portion
Increases shock absorbing capacity
4.6
Welded joints
Butt weld
P
σt =
hl
Where h is the plate thickness
Parallel Fillet weld
h: weld thickness
t: weld thickness at throat
h
t=
sin θ + cos θ
F
τ = (cos θ + sin θ)
hl
For maximum θ = 45o
F
τmax =
0.707hl
Transverse fillet weld
F
τ=
sin θ(cos θ + sin θ)
hl
For maximum θ = 67.5o
[42]
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4.7. SLIDING CONTACT BEARINGS
F
0.828hl
For both parallel and transverse welds use the formula
τmax =
F
0.707hl
Unsymmetrically welded and Axially loaded
F = F1 + F2
Fb
F1 =
a+b
Fa
F2 =
a+b
Eccentric loading
F
τ1 =
Throat Area
My
σbending =
I
bt3 btd2
d2
I=
+
≈ bt
12
4
4
Circular weld Subjected to torsion
Mr
τ=
J
J = 2πtr3
4.7
Sliding contact bearings
Teflon bearing
Carbon bearing
Hydrodynamic bearing
Sleeve bearing
Self lubricating
High temperature
High loads at high speed
Bushings are used to reduce friction and wear
e
Eccentricity ratio or attitude =
c
e: eccentricity
c: radial clearance
Petroff ’s equation
Coefficient of friction, f = 2π
2
r µN s
c
P
Ns in rps
Bearing load W = Bearing pressure ×LD
Power loss = f W rω
µN
Bearing characteristic number BCN =
P
N: rpm
min(BCN)=k=Bearing modulus
Sommerfield number
-decides
thickness
r film
2
µNs
S=
c
P
Ns in rps
[43]
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4.8. ROLLING CONTACT BEARING
Film thickness depends on Sommerfeld number
Hydrodynamic Journal bearing Radial load only
Rectangular hydrostatic bearing Thrust load only
Friction circle radius =f r
r: radius of the journal
4.8
Rolling contact bearing
Also called anti-friction bearing
b
Reliability R = e−(L/a)
1/b
1
ln
L

R 
=
1 
L10
ln
R90
For L50 = 5 × L10 ,
a=6.84
b=1.17

F L1/n = C
n=3 for ball bearing
n=3.33 for roller bearing
C: Dynamic load rating
L: Life in million revolutions
60N L
L10 =
106
N: Mean rpm
Equivalent bearing load
Fe = S[XV Fr + Y Fa ]
S: service factor
X: Radial load factor
V: Race load factor=1 inner race rotates, 1.2=outer race rotates
Y: Thrust factor 1/n
ΣPin Ni
Mean load Pm =
ΣNi
Ni : Number of revolutions
Bearing designation SKF XDD
X: Bearing series
DD: ×5 = diameter
1DD: Extra light series
2DD: Light load
3DD: Medium load
4DD: heavy load
[44]
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4.9. CLUTCH
Stribeck’s Equation
-gives static load capacity of the bearing
Kd2 Z
C0 =
5
Z: number of balls
d: ball diameter
K: Constant
C0 : Static load
Light loads
Ball bearings
More accurate centering
Anti-friction bearings
Very high radial load
Deep groove bearing
Both radial and axial load
Only Axial load
Thrust bearing
Oscillatory motion.
Can be mounted directly on shaft
Needle roller bearing
(piston pin bearing, rocker arms,
universal joint)
Double row roller bearing
Can carry radial and axial loads
Should be preloaded
Tapered roller bearing
Spherical roller bearings
4.9
Self-aligning property
Clutch
Uniform pressure
3
(new clutch)
r1 − r23
2
T = µW
3
r12 − r22
2
W = P π (r1 − r22 )
Power = T ω
2 r13 − r23
Friction radius =
3 r12 − r22
Uniform wear (old clutches, usually used for calculations)
Pr = C
r1 + r2
T = µW
2
r1 + r2
Friction radius =
2
W = 2πC(r1 − r2 )
Centrifugal clutch
T = µmrg rb (ω22 − ω12 )× Number of shoes
m: Mass of one shoe
Single-plate clutch : Occupies large space (Trucks)
Multiple clutch
: Occupies less space (Scooter)
Centrifugal clutch : Starts after reaching a critical velocity (mopeds)
Jaw clutch
: Low speeds only (Rolling mills)
It is more logical and safer to use uniform wear theory in the design of clutches.
[45]
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4.10
4.10. BRAKES
Brakes
Drum brake Clockwise rotation
b + µC
F =N
a
Drum brake Counter clockwise rotation
b − µC
F =N
a
Band brake
T1
= eµθ
T2
T = (T1 − T2 )r
T1
= eµθ/sin α
For V belt
T2
4.11
Gear
Working depth = sum of addendum of gear and pinion
d
module m=
T
πd
Circular pitch=
T
T
Diametral pitch =
d
Addendum = m
Dedendum =1.157m
Ft = bmσb Y
(bombaY)
t2
Lewis form factor Y=
6hm
Ft ht/2
My
= 3
σb =
I
bt /12
Max torque
Service factor Cs =
Rated torque
3
Velocity factor kv =
3+v
[46]
www.eggsam.com
Pitting
Scoring/Scuffing
Plastic flow
Abrasion
Spur gear
Bevel gear
Spiral bevel
Mitre gears
Worm gear
Helical gear
Herringbone gear
Hypoid gear
Zerol gear
Crown gear
4.12
4.12. POWER SCREW
-surface
-fatigue failure
-small cavities
-near pitch surface
Insufficient lubrication + metal-metal contact
Yielding of surface under heavy loads
Erosion due to foreign abrasive particles
Connects two parallel coplanar shafts
Connects two non-parallel, intersecting coplanar shafts
Bevel gear with helical teeth
Non-intersecting shafts which are perpendicular
Identical bevel gears intersecting at right angles
Skew shafts
Helix angle=90-lead angle
Connects two parallel coplanar gears with teeth inclined to the
axis
Mirror like combination of Helical gears with no thrust load on
shaft
A type of spiral bevel gear, Non-parallel, non-intersecting
an intermediate type between straight and spiral bevel gears.
(contrate gear) teeth project at right angles to the face of the
wheel.
-bevel gear with the pitch cone angle = 90 degrees.
Power screw
D: Nominal diameter:Largest
Dc : Core diameter: smallest diameter
L: lead
Helix angle tan α = L/πDmean
For Lifting
µ cos α + sin α
cos α − µ sin α
Torque T = W tan(φ + α) × D/2
Force F = W
For Lowering
µ cos α − sin α
Force F = W
cos α + µ sin α
Torque T = W tan(φ − α) × D/2
Self locking
tan φ ≥ tan α
Efficiency η =
[47]
WL
tan α
=
2πT
tan(φ + α)
www.eggsam.com
4.13. MISC
π φ
For maximum efficiency α = −
4
2
1 − sin φ
ηmax =
1 + sin φ
ACME threads
µ
µ0 =
cos β
4.13
Misc
Cotter Joint
Knuckle joint
Turn buckle
To connect two joints in
tension/compression
Connects two joints and allows
angular movement at the joint
Join two rods having threads with the
help of coupler nut
Make leak proof joints. (Boilers)
Riveted joints
Knuckle joint
-used to transmit axial tensile force.
-unsuitable to connect two rotating shafts, which transmit torque.
Flexible shafts
-low rigidity in bending.
-high rigidity in torsion.
[48]
Chapter 5
Production Engineering
[The short notes are provided for easy revision. Do not use this for first time study.]
5.1
Metrology
Active inspection or online inspection: Checking when the product is being produced
Passive inspection: Inspecting already produced products
Accuracy: Correctness
Precision: Repeatability
H: hole basis system
h: shaft basis system
Three kinds of fit
Clearance fit
Transition fit
Interference fit
Dimensions of machined parts follows normal distribution
Tolerance: Total deviation permitted to the dimension.(Difference between extreme material limits.)
Allowance: Either minimum clearance or maximum interference
Fundamental deviation: How far tolerance zone from basic size
i = 0.45D1/3 + 0.001D µm
D in mm
1
Work tolerance
10
1
Wear allowances =
Gauge tolerance
10
Gauge tolerance =
Workshop gauges: never sell anything bad
Inspection gauge: Never reject anything good
ISO gauges: Go gauge is from work shop gauge and No-go gauge is from Inspection gauge
49
www.eggsam.com
5.2. CASTING
[50]
General purpose gauge
Materials for gauges
Hard
Low thermal expansion
Low density
High corrosion resistance
High machinability
Ex: En 24 (High carbon steel), Inver, Elinver (42% Ni), Glass
Interferometry
To measure flatness
5.2
Casting
Mould box
Cope
Cheek
Drag
Pouring basin → Sprue → Splash core → Runner (Trapeziodal) → Skimbob → Ingate
→ mould cavity, Riser
Advantages of casting
-Complex shapes can be made
-Cheaper
-Both ductile and brittle can be produced
-Large sized parts can be made
Disadvantages
-Surface is not smooth
-Time consuming process
-Non-uniform properties due to non-uniform cooling
5.2.1
Allowances
Machining/finish allowance
Allowance for finishing operations
Daft/Taper allowance
For easy removal of pattern without affecting the mould
Shrinkage/contraction allowance
To compensate for solid shrinkage
Shake/rapping allowance
For clearance between mould and pattern
Distortion/Camber allowance
V or U shape castings
Highest liquid+solidification shrinkage: Aluminum
Highest total shrinkage: Steel
Shrinkage allowance of steel 1mm per 20mm (approx)
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5.2.2
5.2. CASTING
[51]
Parts
Patterns
Solid or single piece pattern
Split piece pattern
Gated pattern: Gate and runner are included in pattern
Loose piece pattern: For parts with internal webs
Match plate pattern
Sweep pattern
Flowbard pattern
Sprue
Best shape is parabolic tapered
Straight tapered is used
Riser
Compensates shrinkage in liquid phase and shrinkage during solidification
Vents
For easy escape of air
Cores
To produce internal cavities
Chills
To avoid hot tear
Directional solidification
Uniform solidification
Paddings
Avoid Erosion
Maximize heat transfer
Directional solidification
Chaplets
To support core
Directional solidification
Molding sand
Refractoriness: Ability to withstand high temperature
Green strength: strength of moulding sand with moisture
Collapsibility: Ability of the mould to not to resist the shrinkage of the metal. Adding
saw dust improves collapsibility
Dry strength: Strength of the mould after the mould cavity dries when the molten metal
is poured.
Permeability: Ability to allow gases to escape (up to 8% water, permeability increases
and then decreases.)
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5.2. CASTING
VH
Permeability number P N =
P AT
V=2000cc
H=5.08cm
P=10g/cm2
A=20.26cm2
3007.2
PN =
T
Time in minutes
Flowability
Strength
Hardness
Adhesive property
Cohesive property
Conductivity
Thermal expansion
Collapsibility
Additives
Saw dust, wood flour
Improves green strength and collapsibility
Starch and dextrin
Organic binders
Improves skin hardness
Iron oxide and
aluminum oxide
Improves hot strength
Coal dust, sea coal,
silica flour
improves surface finish and resistance to metal
penetration
Solidification
By skin forming or dendritic growth
Casting yield =
Casting volume
Casting volume+ gating volume
[52]
www.eggsam.com
5.2. CASTING
Gating system
Ferrous castings
V
√m
Ag 2gh
Turbulence and splashing
Top gate
Time for filling t =
Bottom gate
Non-ferrous castings
Time for filling
√
Am 1 √
√ ( h − h − hm )
t=2
Ag 2g
No splashing or turbulence
Parting gate
No splashing or turbulence
Step gate
Multiple ingates
Fast
No splashing or turbulence
Gating ratio = AS :AR :AG = Sprue:Runner:Ingate
Non-pressurized: 1:4:4 or 1:2:2 (For highly reactive metals like Al, Mg,...)
Pressurized: 1:2:1
5.2.3
Riser design
Necessary condition: Vriser ≥ 3Vshrinkage
Sufficient condition: Cooling time riser ≥ cooling time casting
M=
V
A
Chorinov’s equation
2
V
Cooling time t = k
A
V
Shape
A
Cube
a/6
Sphere
d/6
Caine’s method
Freezing ratio X =
Y =
X=
Vr
Vc
a
−c
Y −b
Modulus method
Mr = 1.2Mc
Mr
Mc
[53]
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5.2. CASTING
Novel research method
5.2.4
Casting defects
Blow holes
open holes/ gas bubbles
Scar
pinhole porosity
Blister
Drop/dirt
Irregular projection on product due to sand dropping
from cope
Scab
Thin layer of Protrusion on roof of product
Rat tail
Due to compression failure of moulding sand
Shrinkage cavity
Due to the shrinkage of metal
Misrun
Non-filling of furthest point in the cavity due to lack of
fluidity and early solidification
Cold shut
Two streams of molten metal does not fuse properly
due to cooling
Hot tear
Hot cracking
residual stresses in the material causes the casting to
fail as it cools
Mold shift
Due to misalignment between two halves
Core shift
Core moves from its position
Buckle
Sand expands on heating and buckles creating V
shaped notches on product surface.
Swell
Metal displaces sand and creates a bulge due to high
pressure
5.2.5
Moulds
Expendable moulds
Sand moulding
Shell moulding
Investment moulding
Full moulding
CO2 moulding
Permanent moulds
Centrifugal
Die casting
Slush casting
Squeeze casting
[54]
www.eggsam.com
5.2. CASTING
[55]
Type
Key words
Application
Shell moulding
Metallic pattern
Fine grain silica
Phenol formaldehyde acts as thermosetting
resin.
Alcohol resin
good surface finish
Expensive process
Difficult for large parts
Cylinder block of IC engine
Rocker arm
Piston rings
Valve plates of refrigerators
Gear blanks
Cylinder head
Investment Casting
Lost wax process
Pattern: Wax, Rubber, Plastic, mercury
The pattern is dipped into slury of
refractory material (fine silica, water, ethyl
silicate and acids)
Slurry coating - Stucco coating - Pattern
melt out - Shake out
Very high accuracy and surface finish
For Complicated and thin parts
Gas turbine blade
Jet engine parts
Medical implants
Dentures
Gold ornaments
Gears, cam, valves,
ratchets...
Full moulding
Lost foam process
Cavityless moulding
Evaporative pattern
casting
Expandable pattern
Plastic patterns
Polystyrene patterns
Foam pattern
Thermocol pattern
Motor casing
Cylinder head
Crank shafts
Aluminum engine blocks
CO2 moulding
Sodium silicate
Very strong
Turbine housing
gearbox housing
Machine tool beds
Gear blank
Centrifugal casting
No core is used
Lighter impurities are collected at the center
Coarse grains ouside and finer grains
towards the center: Jagged surface
Fast cooling
Fine grain structure
Strong and hard
Accurate and good surface finish
Hollow cylindrical pipes
Semi-centrifugal
casting
Axis of rotation is vertical
Slush casting
Thin castings
Thin hollow castings
ornaments and toys
low MP
Blow moulding
Air is blown into the plastic/glass to make
the shape of it match the mould shape
Hollow Plastic or glass
Bottles
Bulbs, ..
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5.2. CASTING
[56]
Gravity die casting
Mass production
Fast cooling
Fine grain
High strength and hardness
Aluminunm Piston in automobile
For making simple shapes
Pressure die casting
Injecting molten metal
under high pressure
High production rate
High accuracy
Can make complicated
shapes
wheels, blocks, cylinder heads,
manifolds
Hot Chamber die
casting
High production rate
High accuracy
For low melting point metals and
alloys
Lead, Zinc, Tin, Magnesium
Melting furnace is not a
part
High melting point,
Non-ferrous
High melting point alloys of Al,
Cu, Brass, Magnesium
Carburators
Crank case and crank shaft
valve bodies
Fuel injection pump parts
Cold chamber die
casting
Aluminum Brake shoe
brass/bronze bushes
Squeeze casting
Continuous casting
Strand casting
Cleaning of castings
Fettling
Shot/ Sand blasting
5.2.6
Cupola
Advantage: Continuous melting
Low cost of melting
Chemical composition can be controlled
Good temperature control
Very long rods
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5.3
5.3. METAL CUTTING
[57]
Metal Cutting
Chuck
Holding workpiece
Tool post
Holding tool
Head stock
Houses the main spindle, speed change
mechanism, and change gears.
Saddle
Tool motion device
Bed
Base that connects to the headstock and
permits the carriage and tailstock to be
moved parallel with the axis of the spindle.
Cast Iron
Casting
Lead screw
For feeding the cutting tool parallel to the
axis of rotation
Brass, stainless steel
Thread rolling, Thread
machining
Guide ways
For guiding the motion of different parts
High carbon steel
Flame hardening
Turning: Process for making external surfaces (Cylindrical)
Boring: Process for making internal surfaces
Facing: Feed motion is radial
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5.3.1
5.3. METAL CUTTING
[58]
Cutting tool materials
Name
Properties
Components
High Carbon Steel
5-6m/min
Vickers Hardness 750HV
Up to 250o C
Preferred for wood working
High Carbon Steel
High Speed Steel
HSS
40-50m/min
Vickers Hardness 850HV
Up to 600o C
Preferred for high carbon steels
Power saw blades
18% Tungsten/Molybdenum
4%Chromium
1%Vanadium
Cemented carbides
150m/min
Up to 1000o C
Powder Technology is used for
production
Cannot be regrinded
Tungsten carbide
2000 HV
WC
Titanium carbide
3100 HV
TiC
Tantalum Carbide
1800 HV
TaC
UCON
200 BHN
Nitrogen is diffused to produce a
hard surface
50% Columbium
30% Titanium
20% Tungsten
Sialon
300m/min Up to 1800o C
Si-Al-O-N
Ceramics
400-500m/min
2200HV
Up to 1200o C
Lower chance for BUE
For brittle
Aluminum oxide
CBN
600-700m/min
4700HV
Substitute for diamond for
cutting steel
Boron
Nitrogen
Diamond
1000-2000m/min
7500HV
High thermal conductivity
Low thermal expansion
Carbon
Hot hardness: Minimum temperature above which the increase in temperature causes
sudden decreases in hardness
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5.3. METAL CUTTING
HCS < HHS < Carbide < Cermet < Ceramic < Borazon
Effect on tool life
Depth of cut < Nose radius < feed < Cutting speed
HSS
W, Mo: To increase hot hardness of material
Cr: Increase strength of resistance to deformation
V: Increase wear resistance
W based HSS has more wear resistance than Mo based
Carbide tools
P type: For ferrous
K type: For non-ferrous, non-metal, cast iron
Low number: Finish machining
High number: Rough machining
Coating: Aluminum and Zirconium
Cermets
Ceramics + Metals
5.3.2
Tool Signature
Lip angle= angle between face and flank
f = Feed
d = Depth of cut
w = width of cut
ASA Tool Signature
BRA-SRA-ERA-SRA-ECEA-SCEA-R
αb − αs − θe − θs − Ψe − Ψ − r
Normal or Orthogonal Rake System (ORS)
I − α − θs0 − θe0 − Ψe − λ − r
I: Angle of inclination
αn : Normal Rake Angle
λ: Approach angle

 


tan I
cos Ψ − sin Ψ
tan αb

=


tan αn
sin Ψ cos Ψ
tan αs

 


0
cot θs
cos Ψ sin Ψ
cot θs

=


0
tan θe
− sin Ψ cos Ψ
cot θe
[59]
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5.3. METAL CUTTING
[60]
Back Rake Angle
αb
Angle between line parallel to
the tool axis and the rake face
and measured in a plane
perpendicular to the base
For ductile, increasing BRA makes
chip flow easier, reduces diffusion
wear, increases tool life.
Smaller rake angles when machining
stronger or brittle materials.
Negative rake angles when machining
ceramics and carbides
Zero rake angle for Brass and CI
Side Rake Angle αs
Angle between the rake face and
the line passing through the tip
perpendicular to the too axis
and measured in a plane
perpendicular to tool axis
5-15o
End Relief Angle
θe
Angle between the minor flank
and the line passing through the
tip perpendicular to the base
and measured on a plane parallel
to the tool axis
Clearance angle
Prevents rubbing of the machined part
on the flank
Larger clearance is required for ductile
materials
Side Relief Angle
θs
Angle between the side flank and
the line passing through the tip
perpendicular to the base and
measured on a plane
perpendicular to the base.
5-15o
To prevent rubbing of the workpiece’s
machined surface in the flank
End Cutting Edge
Angle Ψe , Ce
Angle between end cutting edge
and line passing through the tip
perpendicular to the tool axis
and measured in a plane parallel
to the base
8-15o
Large values of ECEA - large forcechatter
Side Cutting Edge
Angle Ψ, Cs
Angle between the side cutting
edge and the line extending the
shank measured in a plane
parallel to the base
Increase in ψ increases cutting force,
heat is distributed over large area,
increases tool life (up to a limit)
Increase in ψ increases of chatter
vibrations
Nose Radius R
Larger nose radius gives better surface
finish and longer tool life. But
increases chatter and cutting force.
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5.3. METAL CUTTING
[61]
Continuous chips
Discontinuous chips
Build up edge (is prominent when)
Ductile workpiece
Brittle material
Ductile workpiece
High Back Rake Angle
Low Back Rake Angle
Low Back Rake Angle
High speed
Low speed
Low speed
Low feed
High feed
High feed
Low depth of cut
High depth of cut
High depth of cut
No cutting fluid
Serrated or segmented chips
Non-homogeneous chips
Semi continuous chips
From metals of low thermal conductivity
Titanium
Surface roughness
f2
8R
Hmax
Center line average value Ra =
4
True feed = f cos Ψ
f
Hmax =
tan Ψ + cot Ψe
Peak to valley height Hmax =
Ψ : Major cutting edge angle
Forces
Thrust/radial force Fy = Ft cos λ
Axial/ feed force Fx = Ft sin λ
Merchant’s Analysis
1
t2
cos(φ − α)
Chip thickness ratio= =
=
≥1
r
t1
sin φ
r cos α
tan φ =
1 − r sin α
r : Chip reduction ratio
Shear strain =cot φ + tan(φ − α)
cos α
V
Shear strain rate =
cos(φ − α) ∆y
∆y: Shear zone thickness
wt1
Area of shear plane =
sin φ
wt1
Shear force Fs = τ
sin φ
t1 : Uncut chip thickness
=feed × sin λ (Turning)
=depth of cut (orthogonal machining)
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5.3. METAL CUTTING
Fs = Fc cos φ − FT sin φ
Ns = FT cos φ + FC sin φ
N = FC cos α − FT sin α
F = FC sin α + FT cos α

 


cos φ − sin φ
F
F
 C 
 S =
sin φ cos φ
FT
NS


 

N
cos α − sin α
F

=
 C 
F
sin α cos α
FT
FT
+ tan α
FC
tan β =
FT
1−
tan α
FC
F
µ = tan β =
N
1
ln
If β > 45o : Classical friction theorem µ = π r
−α
2
For orthogonal cutting FR = 0
Assumptions for Merchant’s circle
Cutting edge is sharp and straight
Rigid, perfectly plastic, homogeneous material
Orthogonal cutting
Shear zone is approximated by a straight line
No BUE
Cutting force 66%
Axial force 27%
Radial force 7%
Radial force 10%
Cutting Power=Fc Vc
FC
wt1
Ernest and Merchant Theory
Minimum power consumption during machining
2φ + β − α = 90O
2φ + β − α = cot−1 K (Modified Merchant’s theory, Mohr’s theory)
K: Machining constant
Specific cutting power =
Lee and Shaffer (Slip line field theory)
φ + β − α = 45O
Stabler Theory
2φ + 2β − α = 90o
[62]
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5.3. METAL CUTTING
[63]
Velocity
V
Vc
Vs
=
=
sin(90 − (φ − α))
sin φ
sin(90 − α)
VS : Shear velocity
VC : Chip velocity
sin(φ + β − α)
Contact length lf = t1
sin φ sin β
Cutting fluids
Increase heat dissipation and decrease power consumption
Act as lubricant
Properties
-High conductivity
-Should not fume, foam
-Should not react with workpiece or tool
-Low viscosity
Cast iron
Steel
No cutting fluid
Low speed
: Neat oils + EP Additives
Medium speed
: 1:10 water emulsion + EP additives
High speed
: 1:100 water emulsion
Aluminum
Neat oil + EP additives at low speed. No cutting fluid at high speed
Magnesium
Only neat oils
brass/Bronze
Only neat oils
Tool failure
Diffusion wear
Adhesion wear (Spot welds)
Abrasion wear (Faylite pockets)
Fatigue wear
Plastic deformation: Due to the high temperature (Temperature > Hot hardness temperature)
Mechanical breakage: Due to impact loads
Oxidation wear
Flank wear
Crater wear
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5.4. MACHINING
Orthogonal cutting
Oblique cutting
Cutting edge ⊥ Velocity vector
Cutting edge not perpendicular to Velocity
Two components of force
Three components of force
2D cutting
3D cutting
Chip curls into flat spiral
Chip curls in helical path
Less tool life
More tool life
[64]
Jackplane in carpentry
Parting off in turning
Broaching
Sawing
Misc
√
Maximum temperature on Rake face ∝ f eed
If cutting speed is increased, cutting force remains same. (But due to heating, it can
reduce slightly)
Broaching Super finishing operation, multipoint cutting operation, For making key ways
and internal gears
5.4
5.4.1
Machining
Tool life
Taylor’s Tool life equation V T n = C
V = Cutting velocity in m/min
T = Tool life in minutes
n = Taylor’s tool life exponent
C = Taylor’s constant
n = 0.05-0.1 HC steels
0.1-0.2
0.2-0.4
0.4-0.6
0.7-0.9
HSS
Carbides
Ceramics
CBN
πDl
fV
Idle cost: C2 = Cm × Tm
1/n
πDl
V
Tool cost: C3 = Ce
fV
C
1/n
πDl
V
Tool changing cost: C4 = Cm
Tc
fV
C
Tc : Tool changing time
Machining cost: C1 = Cm
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5.4. MACHINING
Total cost =C1 + C2 + C3 + C4
1. Cutting speed for minimum total cost
Ce
1
Topt =
+ TC
−1
Cm
n
C
n
Vopt = 1
Ce
+ TC
−1
Cm
n
2. Cutting speed for Maximum productivity
1
Topt = TC
−1
n
C
n
Vopt = 1
− 1 Tc
n
Modified Taylor’s tool life equation
V T n f p dq = C
f = Feed in mm/min
t = Depth of cut in mm
Machinability
Tool life
Surface finish
Cutting forces
MRR
Specific cutting energy
Shear angle (Higher shear angle − > Better machinability
Vt
Machinability index = 100
VS
VS : Cutting speed of standard free cutting steel for 60 minute tool life
Vt : Cutting speed of metal for 60 minute tool life
5.4.2
•
•
•
•
•
•
•
Grinding
Surface grinding
Cylindrical grinding
Centre-less grinding
Form grinding
Abrasive belt grinding
Manual grinding
Creep feed grinding
Structure: Distance between two cutting edges
Open structure: Used for ductile
Closed structure: Used for brittle and hard materials
[65]
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Grinding ratio=
5.4. MACHINING
[66]
Volume of material removed
Volume of wheel wear
ISO Designation
45-A-G-H-S-B-20
First and last numbers
–
A: Abrasive type
A: Aluminum oxide (Al2 O3 , Soft and tough work pieces)
B: Boron Carbide (B4 C)
C: Silicon Carbide (SiC, Hard and brittle work pieces)
D: Diamond
G: Grain size
10-24 Roughening
30-60 Medium
70-180 Finishing
220-600 Super finishing operation
H: Hardness
A-H Soft wheels
I-P Medium wheels
Q-Z hard wheels
S: Structure
0 Dense
16 open
B: Bond type
V Vitrified
S Silicate
B Resinoid
R Rubber
M Metal bond
Open structure: Ductile workpiece
Closed structure: Brittle workpiece
Wheel Truing: A redressing process by which the wheel is restored to its true shape
Wheel dressing: The process of making new sharp edges on grains. Required due to
grazing.
Friability: Ability of abrasive grains to fracture into small pieces. (Enables self sharpening)
Grinding wheel wear
Grain wear, grain fracture and bond fracture
Creep feed grinding
-Low feed and high depth of cut
Finishing operations
Honing: To make fine surface finish for holes
Lapping: Finishing operation for flat surfaces
Polishing
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5.4. MACHINING
[67]
Buffing
Deburring
5.4.3
Drilling
Drills are made by forging
HSS is the tool material
Lip angle + Lip relief angle + Helix angle =90o
d
Width of chip W =
2 sin β
2β: Point angle
Rake angle: Angle formed between a plane containing the drill axis and the leading edge
of the land. Positive for right hand flute, negative for left hand flute, zero for parallel
flute.
Point angle or cutting angle. Small point angles are used for cutting ductile materials
and large point angles are used for cutting brittle materials
Drilling
To make a hole
Drill is the cutting tool used
Oblique cutting process
HSS
Boring
Enlarging an existing hole+Better finish
Accuracy =0.125mm
Cannot increase the length of the hole
Reaming
Finishing process
Surface finish ±0.005mm
Negligible change in diameter
Reamer has multiple cutting edges
Pack drilling
Multiple pieces in one go
Core drilling
Hollow cutting tool, usually cylindrical
Trepanning
Tube shaped drill
Periphery is the cutting edge
Used in gun barrel manufacturing
Counter boring
Make hole larger
Done by end milling
To make place for bolt heads
Counter sinking
Make holes tapered in the beginning
5.4.4
Milling
Peripheral milling
Slab milling
End milling
Gang milling
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5.4. MACHINING
Straddle milling
Upmilling
No backlash
s
sin θC = 2
d
D
d
1−
D
s
Mean chip thickness =ft
d
D
d
1−
D
d: depth of cut
D: Diameter of cutting tool
Plain milling cutter
fm p
d/D
Average uncut chip thickness tm =
NZ
2fm p
Maximum uncut chip thickness=
d/D
NZ
MRR=tm bfm
Down milling
Also called climb milling
Backlash can affect the process
Better tool life
Good surface finish
Face milling
p
1
Compulsory approach = (D − D2 − wi2 )
2
wi = width of work + 2Offset
Slab milling cutter p
Compulsory approach= t(D − t)
5.4.5
Gear manufacture
Gear shaper
Cutter reciprocates rapidly
Rack type or pinion type cutter
Pinion type can cut internal gear
Only spur gears can be cut
Can cut internal gears
Hear hobbing
Fast process
Cylindrical tool with slots and gashes
Looks similar to a worm gear
Rotates continuously. A continuous process
Cannot cut internal gears
Helical, worm and spur gears can be made
[68]
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5.5. METAL FORMING
Gear milling
5.4.6
Planning
More then one single point cutting tool
Work piece is reciprocating
5.4.7
Shaping
One single point cutting tool
Tool reciprocates
5.4.8
Powder metallurgy
Bulb filament
Cutting tool, grinding wheel
Metal powder → Blending →Compaction →Sintering →Sizing
Powdering
Metal crushing and pulverizing
Atomization
Corrosion
5.4.9
Jigs and fixtures
-Used in mass production
Fixtures: Locate work piece, milling, shaping,...
Jigs: Locate and guides tool, drilling, boring, reaming,
3-2-1
3 pins at bottom arrest 5 dof
2 pins on side arrest 3 dof
1 pin on the third prevents 1 dof
5.4.10
Screw thread manufacturing
Thread chasing: Using single point cutting tool
Die threading
Tapping
Thread milling: For internal and external threads
Thread rolling
Thread grinding
5.5
Metal forming
Ex: Connecting rod
Hot working: metal is worked above the recrystallization temperature
Cold working: metal is worked below the recrystallization temperature
Brittle materials can be hot worked
[69]
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5.5.1
5.5. METAL FORMING
Rolling
∆h
cos α = 1 −
D
∆h: Reduction in thickness
For unaided entry µ ≥ tan α
∆hmax = µ2 R
σ0 : Flow strength
σyt ≤ σ0 ≤ σut
Maximum bite angle = tan−1 µ
A0
= 2.71
Amin
Rolling defects
Wavy edges
Spread
Crocodile crack
Alligatoring
5.5.2
Forging
Fullering or swaggering, Flattering, Finish, Cut off
Flash: extra material deposited in gutter
Drop forging
Open die forging
Cogging
Press forging (Used for making coins)
Roll forging
Precision forging
Impression forging
Forging defects
Cracks, Fold, Barrelling
5.5.3
Extrusion
Extrusion ratio =
True strain =ln
do
df
2
Ao
Af
Extrusion strain =
ho
hf
Ao
Af
Forward or direct extrusion
Backward extrusion or indirect extrusion
Hydrostatic extrusion
Impact extrusion: Tooth paste tube
Force required =KAo ln
[70]
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5.5. METAL FORMING
Johnson’s equation
Ainitial
σd = σ0 a + b ln
Af inal
Defects
Pipe defect, tail pipe, Fishtailing
Surface cracking
Internal cracking
5.5.4
Wire drawing
Ainitial
Wire drawing force F = σavg Af inal ln
Af inal
"
B #
B
1+B
A1
A0
σ = σy
1−
+ σb
B
A0
A1
B = µ cot α
α : Half angle
σb : Back pull
For maximum possible reduction, σ = σy
When µ = 0,
A0
σ = σy ln
A1
Tube drawing
2B !
h1
1+B
1−
σ = σy
B
h2
Defects
Center line cracks
seams
residual stresses
5.5.5
Sheet metal operations
Fmax kt
kt + I
Fmax = Length of cut × thickness × shear stress
k: penetration
I: shear on the punch or die
Force F =
Punching
To punch holes
A shearing operation
[71]
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5.6. WELDING
[72]
Shear is given on punch
Punched out material is waste
Punch=Size of hole
Die= punch size + 2 radial clearance
Blanking
Blanked out material is the product
Shear is given on die
Punch= die size - 2 radial clearance
Die=Size of product
Deep drawing
Press working
For making steel tumblers
Defects of drawing
Flange Wrinkle/earing: due to insufficient blank holder pressure.
Earing: due to anisotropy induced by rolling operation or due to non-uniform clearance
between tools.
Wall wrinkles
Fracture
Miss strike
Orange peal
5.6
Welding
Solid state welding
Liquid state/fusion
Solid/liquid
Autogeneous
Homogeneous
Heterogeneous
Explosive
Resistance
Brazing
Ultrasonic
Chemical reaction
Soldering
Friction
-gas
Forge
-thermite
Diffusion
Arc
SMAW, TIG, MIG, PAW
Hand peening is a stress relieving process and it consists of hammering the weld along
the length with the peen of the hammer while joint is hot.
Weld pool
Weld bead
Reinforcement
Penetration
Root gap
Toe
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5.6. WELDING
[73]
Root
Throat
Deposition rate
5.6.1
-
Arc welding
e− moves from cathode to anode
-ve cathode
+ve anode
2/3 of heat is generated at anode
In case of AC arc welding, equal heat on both the sides
DC Straight polarity
- Work piece is positive
- more depth of penetration
- Weld deposition rate is less
DC Reverse polarity
- Workpiece is negative
- Less depth of penetration
- Weld deposition rate is high
- Used for thin sheets
- Constant current type
- Constant voltage type
Vt
It
+
=1
IShort circuit VOpen
OCV
I
- V=OCVSCC
Arc on time
- Duty cycle =
Arc on time + idle time
Arc blow
- Deflection of electric arc due to the magnetic field formed in the material during welding
- Weld Splatter
- Provide flux coating to reduce arc blow
Flux coating
Electrode designation
TFEPSX
T: Type of electrode manufacturing
F: Type of flux coating
E: Position of electrode
P: Polarity
S: Strength of electrode
X: Specific information about electrode
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5.6.2
5.6. WELDING
Shielded gas welding
Straight polarity: Workpiece positive (Deep penetration)
Reverse polarity: Electrode positive (Less penetration)
SMAW
Shielded Metal Arc Welding
Most commonly used
Electrode coating provides the shielding gas.
TIG, GTAW
Tungsten Inert Gas
Gas Tungsten Arc Welding
Non-consumable tungsten electrode
Carried out in inert atmosphere. (He, Ar, Ne, CO2 , N2 )
Thorium and Beryllium are added to increase the thermal resistance of Tungsten
For Al and Mg alloys, AC welding is used
For all other materials, Direct current straight polarity is used
Used for thin welding
Applications: Aerospace and automobile industries
MIG, GMAW
Metal inert gas
Gas Metal Arc Welding
Consumable electrode (wire) is converted to molten drops
DC reverse polarity or AC are used for welding Al, Mg, Cu,...
Used for thick welding
Can be easily automated
For welding Stainless steels, Al, Mg, Cu, Ni alloys
Application aircraft and automobile industries
5.6.3
Submerged arc welding
Consumable electrode (wire)
Weld arc is shielded by granular flux (Silica, Manganese oxide, calcium fluride,...)
Automated, downhand position welding
Used on flat surfaces
High HAZ
Thick plate welding in ship or pressure vessels, nuclear reactors, pipes
Used to make LPG cylinders
5.6.4
Plasma Arc Welding
Non-consumable Tungsten electrode
High depth of penetration
For thick and high MP metals
DCSP or AC are used
[74]
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5.6. WELDING
[75]
Titanium, Ni, Stainless steel
Applications: Aeronautical industry, Jet engine manufacturing, precision instruments
manufacturing.
5.6.5
Gas Welding
Oxy-Acetylene
Type
Volume ratio
Temp
Flame
Workpiece
Oxydizing flame
1:1.15-1.5
3300o C
Long flame
Cu, Zn, Brass, Bronze
Neutral flame
1:1
3200o C
Short flame
Mild steel, Low C steel, Al alloys
Carburizing flame
1:0.85-0.95
2900o C
Medium length
High carbon steel, CI, Ni-alloys
Creates hard and brittle weld
bead
Oxygen valves are made of brass
Acetylene valves are made of steel
Acetylene is stored in acetone
C2 H2 + O2 → 2CO + H2 ↑ +heat
2CO + 2H2 + 2O2 → 2CO2 + H2 O ↑ +heat
For complete combustion of 1 unit volume of acetylene, 2.5 unit volume of acetylene
is required
1 part of it is provided from the cylinder and 1.5 is obtained from atmosphere.
Gas cutting
Cutting using oxidation of Iron
Al cannot be cut using gas cutting
5.6.6
Thermite welding
8Al + 3Fe3 O4 → 9Fe + 4Al2 O3 + Heat
Fe: Filler
Al2 O3 : Slag
>3000o C
Application: rail, pipes, thick steel sections
5.6.7
Atomic Hydrogen Welding
4000o C
5.6.8
Resistance welding
Heat generated = I 2 Rt
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5.6. WELDING
[76]
Spot welding
Spot welding is adopted to weld two overlapped metal pieces between two electrode
points.
Indentation is created by the force from the electrodes
The lapped pieces of metal
√ are heated in a restricted area.
Diameter of nugget d = 6 t
Mainly used for lap welding thin sheets
Automobile and refrigerator bodies
Seam welding
Electrodes are in the form of wheels
Process is similar to spot welding
Wheels roll creating series of spot welds
Creates leak proof joint
Projection welding
There are projections on one plate made by embossing
Copper plates are used instead of electrodes
Can be used to weld nuts and bolts to plates
Used to join a network of wires
Flash welding
Flash butt welding
Arcs form and soften the metal as the members come closer.
Force is applied to weld the parts by plastic deformation
Very high current 10,000A
Mild steel, Medium carbon steel, Alloy steels, Al, Ti
5.6.9
Electroslag welding
Also called Electro gas welding
Starts with electric arc
Welds by Resistance heating effect of slag materials
Welding progresses on vertical direction
Water cooled Cu shoes are provided on the sides to prevent spillage
AC and DC are used
1000A current
Used to weld very thick plates (up to 900mm)
Applications: Nuclear rector vessels, Ship welding
5.6.10
Electron beam welding
Beam of electrons is used to melt the material
Magnetic focusing lenses are used for focusing the beam
Very small heat affected zone and deep penetration
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5.6. WELDING
[77]
Vacuum is required for the process
Tungsten electrode (cathode) creates electrons
5.6.11
Laser beam welding
Laser beam is used for welding
Used for welding Cu and Al alloys in electronics industry
No need for vacuum
Difficult to weld highly reflective surfaces
Aluminum, Titanium, Ferrous metals, Copper, super alloys, refractory materials.
5.6.12
Explosive welding
Thick plate: Target plate
Thin plate: flyer plate
For welding dissimilar metals
Used in Heat exchanger plug tubes
5.6.13
Friction welding
It is a solid state welding process
Mechanical energy converted to heat energy
5.6.14
Utra-Sonic welding
Heat affected zone is minimum
High frequency is used
Application: Thin sheets/wires same or different material, plastics
5.6.15
Soldering
Solder is an alloy of Lead and tin
Used in electronics industry
Solder melts at relatively low temperature
5.6.16
Brazing
Material is a alloy of Copper, Zinc and silver. It is called Spelter.
Stronger than soldering
Used to connect pipes and make leak proof joints
5.6.17
Defects
Porosity
Due to entrapment of gas bubbles
Can be reduced by proper selection of filler material, preheating the weld area, cleaning
the weld area and reducing the welding speed
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5.7. NON-TRADITIONAL MACHINING
[78]
Slag inclusion
Caused by materials getting trapped in the weld. (Electrode coating materials, oxides,...)
To prevent slag inclusion, clean the weld surface before next layer is deposited, provide
shield gas
Incomplete fusion or penetration
The melting does not reach till the full thickness of the plates
Due to insufficient heat
Undercut
Incomplete fusion
Overlapping
Weld spatter
Weld cracks
Weld decay
5.7
5.7.1
Non-traditional Machining
Electrochemical Machining
AI
ρZF
A: Gram atomic weight of ions
I: Current
ρ: Density
Z: Valency
F: Faraday’s constant= 96500
M RR =
Material removal by electro chemical process
For extremely hard materials
No wear for the tool
No direct contact between tool and work material so there are no forces and residual
stresses.
The surface finish produced is excellent.
Less heat is generated.
Work piece” +ve terminal, Anode (Erosion happens here) Tool: -ve terminal, cathode.
Electrolyte
-Large electric conductivity
-Good chemical stability
-Inexpensive
-Should not cause corrosion
-Low viscosity
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5.7. NON-TRADITIONAL MACHINING
[79]
-Non toxic
-Chloride solution in water (salt solution)
Applications
Turbine blades
Large through cavities
Blind complex cavities
5.7.2
Electrochemical Grinding
Same principle as electro chemical machining
Small chance of material loss from the tool
5.7.3
Electro Discharge Machining
Spark
Dielectric fluid
Also known as spark machining, spark eroding, burning, die sinking, wire burning or wire
erosion
Limited to materials that are conducting in nature
Produces very smooth surface
Discharge voltage
Vd = V (1 − e−t/RC )
−t/RC 2
2
1
−
e
V
Power P = 12 CV 2 =
2RN
V : Supply voltage
For maximum power, Vd = 0.72V
5.7.4
Electric Discharge Grinding
Same principle as EDM
Spark occurs between a wheel and workpiece
5.7.5
Ultrasonic Machining (USM)
-Almost no noise
-Good surface finish
-High accuracy
-Low MRR
-Used for making dies
-Used for machining hard glasses and precious stones
5.7.6
Abrasive Jet Machining
Uses high velocity stream of abrasive particles
Material removal through erosion
Abrasive water jet machining: Uses water as medium for abrasive particles
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5.7.7
5.7. NON-TRADITIONAL MACHINING
Laser Beam Machining
Used for welding, cutting, localized heating and etching.
No need for vacuum
5.7.8
Water Jet machining
A jet of water is used for cutting
For cutting soft materials like plastic, rubber, wood,...
If abrasives are added, then it is called Abrasive Water Jet Machining (AWJM).
5.7.9
Plasma Arc Machining
Plasma arc is used to cut metals
5.7.10
Electron Beam Machining
Magnetic lenses are used to focus the electron beam
Vacuum is required for electron beam to pass through
[80]
Chapter 6
IM & OR
Productivity =
6.1
Output
Input
CPM / PERT
CPM
PERT
Deterministic
Probabilistic
Activity oriented
Event oriented
Single time estimate
Three times estimate
Usually considers cost
Usually ignores cost
Float is used
Slack is used
For repetitive jobs
For non-repetitive jobs
Ex: construction project
Ex: Research projects
β distribution
Critical Path Method
Crash cost − Normal cost
Cost Slope=
Normal time − Crash time
Total Float
= (Lj − Ei ) − Tij
Free Float
= (Ej − Ei ) − Tij
Independent Float
= (Ej − Li ) − Tij
Slack
= Li − Ei
Total Float
Extra time without delaying project
Negative: Insufficient resources, activity may not complete on time
Zero: Activity can just finish on time
Positive: Surplus resources, can be distributed to other activities
81
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6.1. CPM / PERT
[82]
Critical path: Float =0
Super critical path: Highest positive float
Sub-critical path: Highest negative float path
Free float
Amount of time the activity can be delayed without affecting the succeeding activity
Independent float
Time by which an activity can be adjusted without affecting the preceding or succeeding
activity
Program Evaluation and Review Technique (PERT)
Z value
Percentage area
on the left side
0
50
1
84.13
1.28
90
2
97.72
3
99.87
To + 4Tm + Tp
Te =
6
To − Tp
Standard Deviation σ =
6
Variance= σ 2
√
Net Standard variation= Σσ 2
Activity on Node: Dummies are not used
Activity on arc: Dummies are used
Resource allocation or loading
-sharing or allocating labour between activities and/or projects
Resource optimization
-Manipulate network to balance resources required and available are in balance.
Resource leveling
-adjust the resource against possible floats wherever possible and modify irregularity in
the histogram
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6.2. INVENTORY CONTROL
Resource smoothening
Resource leveling
Infinite resources
Limited resources
Project completion time does
not vary
Project completion time may
increase
Activities are shifted to Total
float, extra resources are
provided from outside
Activities are shifted to total
float, if more resources are
needed, the duration of some
activities may increase.
6.2
[83]
Inventory control
Lead time: Gap between placing an order and time for inventory on hand to be consumed
Re-order point: The point at which an order must be made
Safety stock: The √
extra inventory to protect against unexpected stock outs
Safety stock = K × Average consumption during lead time.
Seasonal inventory
Anticipatory inventory
Decoupling inventory
Transit or pipeline inventory
Direct inventory: Inventories that are directly a part f production and become a part
of the final product.
Ex: Raw materials, In process inventories, Purchased parts, Finished goods
Inventory review system
Fixed order quantity system
Periodic review system
Order quantity is fixed
Re-order data is fixed
Q System
P System
Order is made when the inventory
reaches re-order point
Re-order quantity depends on the size
of inventory at the time of order
Suitable when carrying cost is
measurable and significant
Suitable when carrying cost is
insignificant
Preferred when supplier has a
minimum order quantity restriction
Supplier will supply only on fixed
dates
Suitable for A class items
B and C class items
also called Fixed internal syatem
For perishable products
SS-System: Optimum order
Fixed order system
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6.2. INVENTORY CONTROL
[84]
Fixed Order is placed when inventory falls to a fixed value
-Order size is fixed
-Order time is variable
Fixed period system
Periodically inventory is replenished
-Order size is not fixed
-Order time is fixed
Price break model
DDLT: Demand during lead time
Inventory cost
Order cost = Number of orders × Cost per order (Co )
Setup cost = Number of setups × Cost per setup
Purchase cost = Number of units × Unit cost
Holding/carrying cost = Average inventory level × Carrying cost per piece per unit time
(Cc )
Shortage/stock-out cost = Average shortage × Shortage cost per piece per unit time
6.2.1
Service level model
Number of units supplied without delay
Service level =
Number of units Demanded
Service level = 1-Probability of stock out
CC Q
Service level factor = 1 −
CS D
CU S
Service level =
CU S + COS
COS
Stockout risk=
CU S + COS
CUS=SP-CP
COS=CP-Rebate
6.2.2
Economic Order Quantity / Harris-Wilson model
D
Co
Q
Q
Carrying cost = Cc
2
D
Q
Total Cost= Co + Cc + DxCu
Q
2
D
Number of orders per year =
Q
Avg inventory cost
Inventory Turnover =
cost of goods sold
Ordering cost =
D:
Co :
Q:
Cc :
Cu :
TVC:
OS:
p:
d:
Annual demand
Ordering cost per order
Number of pieces ordered
Carrying cost per piece
Unit price
Total Variable cost
Optimum shortage
production rate
depletion rate
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Situation
Without Shortage
instant production
Production model
With shortage
6.2. INVENTORY CONTROL
[85]
Equations
r
2DCo
Q=
Cc
√
TVC= 2DCo Cc
r
r
2DCo
p
Q=
Cc
(p r
− d)
√
(p − d)
TVC= 2DCo Cc
p
r
r
2DCo (Cc + Cs )
Q=
Cc
rCs
√
Cs
TVC= 2DCo Cc
Cs + Cc
r
r
2DCo
Cc
OS=
Cc
(Cc + Cs )
Model sensitivity
T V C(Q)
1 Q
Q∗
=
+
T V C(Q∗ )
2 Q∗
Q
Demand-profit / Static inventory
perishable items
6.2.3
Inventory classification and control
ABC
Always Better Control
A is more important than B than C
A: Small quantity large price, small inventory, frequent review, frequently ordered in
small quantity
C: Large quantity, small price, large inventory, reviewed rarely
-70-90-100
-Vital few to Trivial many
-Consumption analysis
Pareto’s chart
HML
High Medium Low
-based on the unit price of the product
VED
Vital Essential Desirable
-Based on importance of the product
SDE
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6.3. FORECASTING
[86]
Scarce Difficult Easy
-Based on availability
XYZ
Based on inventory value
X: Items with high inventory value
Z: Items with low inventory value
FNSD
Fast Normal Slow Dead moving items
Based on the speed of usage of items
EOQ : Economic order quantity
EBQ : Economic Batch Quality
Cost of goods sold
Inventory turnover ratio =
Cost of average inventory
6.3
6.3.1
Forecasting
Qualitative methods
Educated Guess
Based on a person’s judgment based on experience and intuition
Delphi method
Panel of experts
Questionnaires are used.
Long range
For new product, technology, changes in society,...
Survey of sale force
Survey of customers
Historical analogy
Market research
Market trial
6.3.2
Quantitative methods
SimpleP
average Method
Ft+1 = t1=t+1−n Di
Weighted
P moving average
Ft+1 = t1=t+1−n Wi Di
Exponential smoothing
D: Actual demand
F: Forecasted demand
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6.4. BREAK EVEN ANALYSIS
Ft = Ft−1 + α(Dt−1 − Ft−1 )
Ft = Ft−1 + α(error)
Linear Regression
Σy = na + bΣx
Σxy = aΣx + bΣx2
y = a + bx
P
(Yc − Ȳ )2
Coefficient of determination r = P
(Y − Ȳ )2
Coefficient of correlation =r
2
Least squares technique
Exponential smoothing with trend
Double moving average method
6.3.3
Error
n
P
(Dt − Ft )2
Mean Square Error MSE=
1
n
Mean Absolute Deviation MAD =
n
P
(Dt − Ft )
1
Bias =
n
n
P
|Dt − Ft |
1
n
Cumulated deviation
=
M AD
Bias × n
Tracking signal =
M AD
√
Upper limit for tracking signal =3 M SE
Tracking signal =
6.4
n
P
(Dt − Ft )
1
M AD
Break even analysis
At break even point,
Fixed cost
F
N=
=
Selling price - Variable cost per piece
S−V
F: Fixed cost
S: Selling price
V: Variable cost per piece
F +P
To get profit P, Np =
S−V
Angle of incidence: Angle at which total sales line cuts total cost line
Contribution margin= Total sales - Total variable cost
Margin of safety= Output at full capacity - BEP output
[87]
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6.5
6.5. QUEUEING THEORY
[88]
Queueing theory
Kendall’s Notation
A/D/N : P/S/C
A : Arrival Pattern (Arrival distribution)
D : Departure Pattern (Service distribution)
N : Number of servers
P : Priority rule
S : System Capacity
C : Calling population
Arrivals are Poisson distributed
Service time is negative exponential distributed
Balking: Some customers leave without joining the queue
Reneging: Leaves the queue after being in the queue for some time due to impatience
or any other reason.
System: Queue + The person being served
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6.6. LINEAR PROGRAMMING
Mean arrival rate
[89]
: λ
Mean service rate : µ
Inter arrival time : a(T ) = λe−λt
1
Mean Inter arrival time :
λ
1
Variance Inter arrival time : 2
λ
Traffic intensity factor / utilization factor / : ρ = λ
µ
channel efficiency
Probability of system being empty : P0 = 1 − ρ
Probability of N customers in the system : PN = (1 − ρ)ρN
−λt
ne
:
P
(t)
=
(λt)
Probability of n customers in the queue
n
n
N
Probability of system size being ≥ N : ρ
λ2
Length of queue : Lq = Ls ρ =
µ(µ − λ)
Ls
λ
= Ws ρ =
Time in queue : Wq =
λ
µ(µ − λ)
ρ
λ
=
= Lq + ρ
Length of system : Ls =
1−ρ
µ−λ
Ls
1
=
Waiting time in system : Ws =
λ
µ−λ
Expected waiting time of one who has to wait : Ws
1
Expected length of non empty queue :
1−ρ
ρ
Variance of queue length :
1−ρ
R∞
Probability of waiting time in queue ≥ W : w λ(1 − ρ)e−(µ−λ)w dw
R∞
Probability of waiting time in queue ≤ W : w λ(1 − ρ)e−(µ−λ)w dw
6.6
Linear Programming
Feasible region is convex
Decision variables: The stuff we want to find
Objective function: Equation that shows relationship between the decision objective and
decision variables
Simplex method
n!
Maximum number of iterations ≤
or n Cm
(n − m)!m!
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Basic
Z
6.7. TRANSPORTATION
Variables
RHS
[90]
Ratio
Entering variable
Basic
↓
variables
Pivot element →Leavingvariable
Feasible solution: Any values of the basic variable that obey the constraints
Unique solution: The number of zeros = number of basic variables
Multiple solution: Number of zeros > number of basic variables
Unbounded solution: All numbers in replacement ratio column is negative or infinite
No solution: Artificial variable remains in the final solution
Degenerate solution: One or more basic variable becomes zero.
6.7
Transportation
Degeneracy: Occupied cells < m+n-1
6.7.1
Initial solution
Northwest corner rule
Least cost cell method
Vogel’s approximation method VAM (Penalty method)
6.7.2
Optimality test
Stepping stone method
Modified Distribution Method MODI
ui + vj = cij , For occupied cells
∆ij = cij − (ui + vj ), For unoccupied cells
If all ∆ij > 0, Optimum unique solution
If ∆ij ≥ 0, Optimum non-unique solution
If any ∆ij < 0, Not Optimum solution
Find the cell with most negative ∆ij value.
Draw te closed loop.
Put + and -, and reallocate
Repeat
6.8
Assignment
Hungarian method
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6.9
6.9. WORK STUDY
[91]
Work study
Time study and method study
Method study
SREDIM
Select: Select the problem, man to solve the problem, machine to solve the problem,
material, working conditions
Record: Record facts
Examine: Examine recorded facts
Develop: Develop most efficient alternative
Install: Implement the plan and install the alternative
Maintain: Maintain the new system
Operation
Inspection
Transport
D
5
Delay
Storage
Micro-motion study
Therbling: Micro-motions
Observed time: Actual time taken
Actual available time
Observed Time OT =
No. of units to be produced
Normal time: Time taken by normal worker in normal time
Normal time = OT × Performance rating factor
For machines, NT = OT
Standard time = NT + Allowances (ST =1.2 NT if data is not available)
6.10
Scheduling and loading
Scheduled date - Today’s date
Days needed to finish the job
Earliest Due date (EDD) Reduces mean tardiness
Shortest processing time (SPT)Reduces inventory cost, mean flow time, mean lateness
Tardiness is the positive lateness
Job flow time: Time from starting to end of a job
Make Span Time(MST): Time from first job to the end of the second job
Tardiness or lateness: Delay in the job
Critical ratio=
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6.11. LINE BALANCING
SPT
Shortest processing time
Reduces Inventory cost, mean flow
time, mean lateness,
EDD
Earliest Due Date
Reduces mean tardiness
CR
STR
6.11
[92]
Critical ratio rule
Due date
CR=
Processing time
Slack time remaining
Due date - Processing time
Line Balancing
To reduce idle time
Task time Ti : Time to complete a work element
Station time TS : Time in a work station
Total work content (TWC): Time to complete one set of job
n: number of workstations
Cycle time TC : Time between two products
nTC − T W C
Total idle time
=
× 100
Total time in assembly
nTC
WTC
Line efficiency/ balance efficiency=
× 100=100-BD
nTC
pP
(max station time − ith station time)2
Smoothness index =
TWC
Minimum number of work stations required=
= Theoretical number of work
TC
stations
time to assemble one unit
ηTheoretical =
Theoretical no of work stations x cycle time
Σti
ηactual =
100
yc
y=actual number of work stations
Balance delay=100-η
Balance delay(BD):
6.12
Material Requirement planning
MRP : Materials requirement planning
CRP : Capacity requirement planning
MPS: Complete timetable of future production. Decides which, how much and when to
produce.
Advantages of MRP
Reduce inventory
Decides when and how much to order
help to avoid delay in production
expected delivery time
Chapter 7
Material Science
Strength
Strength to resist external load without failure.
Stiffness
Stiffness to resist elastic deformation
Toughness
Ability to absorb energy before fracture
Resilience
Ability to absorb energy without permanent deformation
Proof resilience
Energy stored up to elastic limit
Hardness
Hardness to resist indentation
Elasticity
Ability to regain shape after deformation
Plasticity
Ability to stay deformed after deformation
Ductility
Ability to elongate under tension (% reduction in area)
Malleability
Ability to deform under compressive force. (use a mallet)
Brittleness
Ability to break with relatively less plastic deformation
Creep
Time dependent increasing deformation under constant load
Fatigue
Material behavior under repeated load
Machinability
Easiness of machining
93
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7.1
7.1. TESTS
Tests
Test name
Details
Tensile testing
On universal testing machine
Compression
test
On universal testing machine
Izod Test
Measures fracture toughness
Sample is fixed at one end and the other end is free
Non-uniform stress
Charpy Test
Measures fracture toughness with better accuracy
Sample is fixed at both the ends
Herbert
cloudburst
Hardness test
A shower of metal balls
To find defects
Spiral test
Fluidity
Cupping test
Formability
Dye penetrant
method
To find surface defects
To find hardenability
Austenite to martensite
50% - 50% Pearlite and martensite, Jominy distance
√
Gauge length =5.65 A0
Jominy end
quench test
[94]
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Hardness Tests
Test name
7.1. TESTS
Details
Mho Test/ Scratch
Test
A qualitative test
Used as a preliminary test
Used for Low-medium-high hard materials
Brinell Hardness Test
Used for Medium hard materials
Indenter: Spherical, d=10mm
P=50-120kg
2P
√
BHN =
πD(D − D2 − d2 )
UTS=3.6 × BHN, for normalized plain carbon steels
UTS=3.2 × BHN, for tempered plain carbon steels
Rock Well Test
Vickers Test
Applicable to all types
Indenter: 120o diamond cone (brail)
P=1-150kg
1
Hardness ∝
t
For Medium hard materials
Indenter: 136o diamond square pyramid
P=50-120kg
1.854P
V HN =
davg
davg : Average diagonal indentation
Knoop Test / Micro
Hardness test
For Low hard materials (Si, Ge, Ga, As,...)
177o Bi-pyramidal base indenter
P=1-1000g
KHN = 14.22P/L2
Shore Method
Shore’s Scleroscope
For soft plastic, thermo plastic, rubber, thin sheets ...
Diamond tipped indenter (hammer) in a glass tube
Height of reboundness ∝ hardness
For hard plastics, thermosets, composites,...
Spring reading
P : Load at which indentation is produced (kg)
D: Diameter of indenter (mm)
d: Diameter of indentation (mm)
Barcol Method
[95]
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7.2. PLASTICS
[96]
Non-destructive testing
Visual inspection
Hammer test
Hang the casting in the air
Gently strike with a
hammer and listen to the
sound
Radiography
x-rays and γ rays
Liquid penetrant test
Defects that are open to
the surface
Die penetrant test
Ultra-Sonic inspection
Hardness
Diamond > Silicon > Quartz > Topaz > Feldspar > Apatite > Fluorite > Calcite >
Gypsum > talcum
Ductility
Al > Cu > Zn > Mild steel
Elastic failure
Necking → Formation of small cavities → Cavities combine together → crack propagation
→ Fracture (Cup-cone)
7.2
Plastics
Thermoplastics
Thermosetting plastics
Become soft on heating
Becomes hard on heating
Recyclable
Non-recyclable
Linear structure
Cyclic structure
addition polymerization
Polyvinyl Chloride (PVC)
Epoxy
Polypropylene(PP)
Polyester
Polyethylene(PE)
Phenol formaldehyde(Bakelite)
Polystyrene(PS)
Poly Tetra Fluoro Ethylene
(PTFE, TEFLON)
Acrylic
Molecular mass
mer mass
Volume of crystal region
Degree of crystallization fc =
Total volume of specimen
Degree of polymerization =
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7.2. PLASTICS
[97]
Conformation:A single polymer chain can take different 3D shapes. Makes polymers soft.
Glass transition temperature: Reversible transition in amorphous materials from
hard and brittle to soft rubbery state.
Thermoset resins
Compression moulding
Transfer moulding
Injection moulding
Linear polymers v/s the other thing
Fiber reinforced plastic
Thermosetting plastics + glass fiber
Anisotropic
Condensation polymerization
Produces water or ammonia as by-product
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7.3. CERAMICS
ABS
Terpolymer
Polyethyene
Addition polymerization
Polycarbonate
Addition polymerization
Polystyrene
Addition polymerization
Polyamide
Natural (proteins) and artificial (Nylon)
Poly propylene
Excellent fatigue strength
PTFE
Low coefficient of friction
PVC
Synthetic polymer, pipes, bottles,...
PMMA
Poly Methyl Methacrylate
PEEK
Polyether ether ketone,
bearings, piston parts, pumps, High-performance liquid
chromatography columns, compressor plate valves, and electrical
cable insulation.
Nylon
Fabrics
Polyurethene
Low-density flexible foam
Cyano-acrylate
Adhesives
Neoprene
Oil seal
Bakelite
Electric switches
Araldite
Adhesive
SBR
Styrene buta diene rubber , Tyres
Kevlar
Bullet proof vests
7.3
Ceramics
Extremely brittle
High thermal stability
High chemical stability
Corrosion resistance
High hardness
Silica (SiO2 )
Alumina
Tungsten carbide Drawing dies
Silicon nitride
Pipes for conveying liquid metal
Aluminum oxide
Abrasive wheels
Silicon carbide
Heating elements
[98]
www.eggsam.com 7.4. CRYSTAL STRUCTURE AND DEFECTS
[99]
Fast cooling of silica gives Glass
Very slow cooling for silica gives quartz
Devitrification of glass: The process of changing from amorphous to crystal over time
Glass transition temperature: The temperature where silica during quenching is neither
liquid nor solid. Depends on cooling rate.
Static fatigue: Fails on the same static load after some time
Crystal structures of ceramics
AX-Type: Number of cations=number of anions (NaCl)
CeCl- Structure: one iron at center like BCC.
Zinc Blende structure: tetrahedral
Metallic glass
Very fast cooling of metal
Used in transformer cores to reduce eddy current loss
Metals
-Has free electrons
7.4
Crystal structure and defects
Atomics Packing factor AP F =
Crystal Systems
Natoms Vatoms
Vunit call
Cubic
a=b=c
α = β = γ = 90o
Rhombohedral
a=b=c
α = β = γ 6= 90o
Tetragonal
a=b6=c
α = β = γ = 90o
Orthorhombic
a6=b6=c
α = β = γ = 90o
Hexagonal
a=b6=c
α = β = 90o , γ = 120o
Monoclinic
a6=b6=c
α = γ = 90o , β 6= 90o
Triclinic
a6=b6=c
α 6= β 6= γ 6= 90o
Crystal structure
Natoms
APF
CN
Example
Simple cube
1
0.52
6
Polonium
BCC
2
0.68
8
Cr, Mo, V, W, Mn, Ta, Nb, Na
FCC
4
0.74
12
Cu, Al, Pb, Ag, Au, Ca, Ni, Pt
HCP
6
0.74
12
Graphite, Be, Mg, Zn, Cd, Ti, Zr
Diamond
8
0.34
4
CN: Coordination number
For an ideal HCP crystal structure, height/side = 1.633
www.eggsam.com 7.4. CRYSTAL STRUCTURE AND DEFECTS
[100]
Directions
Plane-()
Line -[]
"
Angle between lines θ = cos−1
h1 h2 + k1 k2 + l1 l2
p
p
h21 + k12 + l12 h22 + k22 + l22
#
Linear density
Number of effective atoms in the unit length in the given direction
Planar density
Number of effective atoms in the unit area of the given plane
Burger’s Vector
|~b| = 0: No defect
|~b| = 1: Point defect
|~b| =>: Line defect
Interplanar distance
a
d= √
2
h + l2 + k 2
Bragg’s law = 2d sin θ = nλ
7.4.1
Point Defects
Name
Reason
Effect on
strength
Vacancy
Atom goes missing
Reduces
Displacement
Atom moves from lattice site to
another
No change
Ex: doping in
semiconductors
Substitution
inclusion
Foreign atom occupies a lattice
point.
Diameter of foreign atom ≈
Diameter of lattice atom
No change
Chromium in steel
Interstitial
inclusion
Foreign atom occupies
interstitial positions.
Diameter of foreign atom<<
Diameter of lattice atom.
Valency of impurity > Valency
of lattice atom
Increases
Addition of carbon to
iron
Frenkel
defect
Lattice atom (cation) goes to
interstitial position
Slight
change
Seen in ionic crystals.
Ex: Silver halides,
CaF2
Schottkey
defect
pair of atoms missing. No
change in charge of crystal.
Reduces
Seen in ionic crystals
Ex: Alkali halides
Notes
www.eggsam.com 7.4. CRYSTAL STRUCTURE AND DEFECTS
Line defects
Edge dislocation
[101]
Screw dislocation
Glide
Climb
Dislocation lines are perpendicular to
Burger’s vector
Lies parallel to burger’s vector
Direction of movement of edge dislocation is
in the direction of Burger’s vector
Direction of movement is perpendicular to
Burger’s vector
Movement of edge dislocation is fast
Movement of dislocation is slow
Explains plastic deformation
Explains plastic deformation and crystal
growth
Less shear force is required to make this
defect
High shear force is required to make this
defect
Tensile, compressive and shear stress fields
can be present
Only shear stress field is present
Termination of atomic plane in the middle
of a crystal
Will change surface properties significantly
Whisker
Movement of atomic planes is translation +
rotation
7.4.2
Surface defects
Grain Boundary defect
Tilt boundary defect
Twin boundary defect
7.4.3
Volume defects
Stacking faults
7.4.4
Plastic deformation
Plastic deformation by slip
-Occurs by pure shear stress
-One atomic plane moves with respect to the bottom plane under the effect of a tangential
force.
-Line defect or planar defect
-Usually in BCC or FCC
-Less stress is required to propagate slip
-Appear as thin lines in microscope
Ex: Forging
www.eggsam.com
O
O
O
O
O
O
O
O
O
O
O
O
7.5. ALLOYS
[102]
O
OOOO
O −→ O O O O
O
OOOO
O
OOOO
Plastic deformation by twinning
-Surface defect
-Lattice splits
-Force applied at an angle
-Usually in HCP
-Twinned crystal lattice is mirror image of the original
-More stress than slip is required
-Less stress is required to propagate twinning
-Appear as thick lines in microscope
Grain boundaries restricts the motion to dislocation. It makes the material stronger.
Permanent deformation in metal or alloy is caused by movement of dislocations.
Amorphous solids
No regular arrangement of atoms
No sharp melting point
7.5
Alloys
Hume Rothary rules/Conditions
Difference in atomic radius of both atoms must be less than 15%
Valency of both the atoms should be same
Electro-negativity and electron affinity of both the atoms should be compatible
Atoms at grain boundary has more energy. So oxygen attacks there and causes corrosion.
Chromium reacts with oxygen to produce Cr2 O3 , this gets in grain boundaries and prevent corrosion by blocking oxygen
Weld decay: Corrosion at welded parts due to lack of Cr2 O3 , due to formation of
Chromium carbide during welding
Corsing or Miscibility gap: Due to sudden cooling, no time for diffusion, so concentration gradient, causes cracks on hot working
Ni is added to stabilize austenite phase
Cr is added to stabilize ferrite phase
Carbon equivalent = %C + 31 %(Si+P)
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7.5. ALLOYS
Effect of alloying elements in CI
Element
Effect
Carbon
Steels → Cast iron
Ductility ↓
Brittleness ↑
Chromium
Carbide stabilizer, increases strength and wear-resistance
Copper
Promotes formation of graphite
Magnesium
Increases ductility
Increases strength in tension
Nodular CI can be obtained from Grey CI with addition of Mg
Manganese
Hardens to CI by promoting carbide formation
To remove the effects of Sulphur
Produces MnS with high melting point
Increases machinability
Further addition of Mn increases strength of material
12% Mn material called Hadfied steel
Molybdenum
Improves tensile strength, toughness, machinability hardenability
Nickel
Graphitiser, resists corrosion
Phosphorus
Increases fusibility and fluidity
Increases brittleness
Silicon
Soft and Machinable iron
Kish (Carbon comes out of red hot CI)
Increases carbon equivalent
Fe-C Phase diagram shifts left
Promotes graphite flake formation, improves machinability
Increases fluidity of molten metal
Sulphur
FeS, Hard and Brittle (0.1%)
Causes Brittle failure on hot working
Hot-shortness or Sulphor embrittlement
Mn is added to get ride of sulphur
Vanadium
Increases machinability
Iron ores
Magnetite F e3 O4
Hematite F e2 O3
[103]
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7.5. ALLOYS
Alloying of steel
Aluminum
To make fine grain structure and control growth
De-oxidizer
Boron
Increases hardenability
Carbon
Increases strength, elasticity and hardness. Reduces ductility and
impact strength
Chromium
Increases hardness, corrosion resistance and toughness
Cobalt
increases ferrite and increases red hardness
Copper
Increases tensile strength
Increases yield strength
Anti-corrosive agent
Lead
Machinability
Manganese
Removes S
Acts as de-oxidizing agent
Increases strength and hardness
Molybdenum
Forms abrasion resistant particles
Improves creep properties, tensile strength and hardenability
Nickel
Increases toughness, corrosion resistance, shock resistance and
deep hardening
Phosphorus
Reduce toughness
Increases brittleness
coldshortness
Increases tensile strength
Reduces impact strength and ductility
Increase machinability
Silicon
Removes oxygen to produce killed steel
reduces chance of becoming porous
Graphitization
Sulphor
Improves machinability
Tungsten
Increase hot hardness
Deoxidizes
Fine grain structure
Vanadium
Increases fatigue strength
Increases tensile strength in MCS
Increases hot hardness
Presence of hydrogen in steel: Embrittlement
High strength low alloy steel
Cu, V, Ni, Mo
[104]
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7.5. ALLOYS
[105]
Copper improves corrosion resistance by 3 times that of chromium
Vanadium increases hardness and promotes fine grain structure
Ni increases tendency to retain austenite
Mo resists corse grain formation and increases hot hardness
Free cutting steel
High machinability
Sulphur (upto 0.05%) improves surface finish
Phosphorous (upto 0.05%) reduces brittleness
Lead (2-4%) (Adition of Lead beyond 4% reduces melting point)
Tool steel
Steel Alloys
18% W or Mo
4% Cr
1% V
0.67% C
Cr: Reduces scaling
V: Abrasion resistance
Tungston: Hot hardness
Mo: Hardenability
Co: Hot hardness and wear resistance
Hadfield Manganese steel
≈13% Ma
High wear resistance
High toughness
Bulldozer blades
Magnet steel
15-40% Co
0.4-10% W
Magnets
18/8 Stainless steel
18% Cr
8% Ni
Knives, Forks, spoons
HSS
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7.5. ALLOYS
[106]
Admiralty metal
Cu,Zn, Sn
Corrosion-resistant
Babbit
88% Sn
Cu, Pb, Sb
Used in brass/bronze bearing to increase wear
resistance
Good embedability
Bronze
88% Cu
12% Tin
Utensils, bearings, bushes, wires,...
Brass
Cu
Zn
Musical instruments
Catridge brass
70% Cu
30% Zn
Ductile
Chromel
90% Ni
10% Cr
Thermocouple
Constantan
55% Cu
45% Ni
Thermocouple
Duralumin
94% Al
4% Cu
Cooking utensils, tubes, rivets, sheets,...
German silver
60 % Cu
20%Ni 20%
Zn
Gun metal
88% Cu
10% Sn
2% Zn
Bearings, Bushes, beam glands,...
Inconel
75% Ni
15% Cr
9% Fe
Oxidation and corrosion resistant
Suited for in extreme environments subjected to
pressure and heat.
Invar
Ni
Fe
Negligible thermal expansion
Clocks and scientific instruments.
Imitation gold
Aluminum Bronze
Lead/Solder
Lead, Tin
Soldering
Monel metal
63-70% Ni
2.5% Fe
2% Mn
Si, C, S, Cu
Corrosion resistance in salt water, valve parts for super
heated stream, turbine blades, pumps,...
Nimonic alloy
>50% nickel
>20% Cr
Ti, Al
Gas turbine blades
Phosphor Bronze
P < 0.1%
Bearings, Springs, Fasteners, Acoustic guitar
Sialon
Si,Al,O,N
Cutting tool
Stellite
Mo, Co, Cr
Saw teeth, Valves, acid resistant
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7.6
7.6. PHASE DIAGRAMS
[107]
Phase diagrams
System: the portion of the universe that is being studied
Phase: Chemically uniform, physically distinct and mechanically separable portion of a
system.
Allotropy: Property of some chemical elements to exist in two or more different forms
in the same physical state
Isomorphism: The existence of same phase in Liquid and solid
Invariant points
At these points, the physical variables like temperature, pressure, concentration, etc are
fixed
Ex: Triple point, Eutectic point,...
Triple point of water: 0.006atm and 0.01o C
Unary phase diagram: Has only one component. Example: water
Binary phase diagram: Has two components
Binary phase of first kind
Completely soluble in liquid and solid phase
Ex: Ni-Cu
Binary phase diagram of type 2
Completely soluble in liquid phase, partially soluble in solid phase
Ex: Pb-Sn
Liquidus: The line between Mushy zone and liquid
Solidus: the line between Mushy zone and solid
Lever rule
cs − c0
ml =
cs − cl
c0 − cl
ms =
cs − cl
Phase rule F=C-P+2
P: Number of phases
C: Number of components
F: DOF
Eutectic
Liquid *
Liquid 4.3%C
) Solid1 +Solid2
1150o C
*
) γ + Fe3 C
1493o C
Peritectic
Liquid + Solid1 *
) Solid2
L + δ0.18%C
Eutectoid
Solid *
) Solid1 +Solid2
γ0.8%C *
) α + Fe3 C
Peritectoid
Solid1 +Solid2 *
) Solid3
Monotectic
Liquid *
) Liquid1 +Solid2
723o C
*
) γ
Due to large
difference in MP
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7.6. PHASE DIAGRAMS
Curie Point
-no change in crystal structure.
-Magnetic properties are changing
Ferro magnetism
Sometimes paramagnetic and sometimes diamagnetic
[108]
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7.6. PHASE DIAGRAMS
[109]
Name
Properties
Austenite (γ)
FCC
non-magnetic
soft
Not stable below 725o C
Solid solution of Ferrite+Iron carbide
in gamma iron
Mn, Ni, Si are austenitic stabilizers
Ferrite δ, α
BCC
Highly magnetic
soft
Ductile
0.02% Carbon
δ
BCC
1410o - 1540o
0.1% Carbon
Cementite
Orthorhombic
Extremely hard and brittle
Magnetic below 200o C
Fe3 C , 6.67% Carbon,
Pearlite
α +Fe3 C
Ferrite(87%)+Cementite
(13%)
Phase mixture
Ledeburite
Austenite+Cementite
4.3% carbon Mix
Bainite
Hard
Brittle
cooled slower than the rate required to
form martensite but faster than the
rate that would be required to form
pearlite.
Martensite
Hardest and brittle
Ferrite + Cementite
Rapid cooling of HCS
Troosite
Lower hardness and
brittleness than martensite
Formed by heat treatment of
martensite
Sorbite
Lower hardness and
brittleness than troosite
Formed by heating martensite
www.eggsam.com
7.6. PHASE DIAGRAMS
[110]
Brittle
> 6.67%C or slightly less
But actually 2.4-4%
Obtained by Slow cooling
Carbon in graphite flake form acts
as lubricants and damper
Machine beds, Piston
rings, Ingots, moulds,
pistons, machine
castings, automobile
cylinders
White cast iron
Very hard
Brittle
6.67%C
Carbon in form of cementite
Obtained by rapid cooling
Does not rust easily
Rolls, dies wearing
plates, stamping shoes
Malleable CI
Hard, Brittle
Carbon in combined form
valve bodies, hinges,
machine castings
Chilled CI
Hard
Rapid cooled to be white instead of
gray
Camshafts, crankshafts,
railway wheel
Ductile
Produced through heat treatment.
Heat treat chilled CI in presence of
Mg or Ce just below 1150o C and
slow cooling
parts subjected to
vibration and bending,
Pipe fittings(elbow, tee,
union,...)
Gray Cast Iron
Spheroidal CI
Ductile iron
Nodular CI
Carbon in rosette form
Mechanite CI
Meehanite CI
High strength
Ductile
Easily machinable
camshaft, crankshaft
Wrought iron
Tough,
malleable
99% iron
Does not melt on heating
Becomes soft on heating
Gates, Eiffel tower
Low carbon steel
<0.3% C
Medium carbon steel
0.3%<0.7% C
High carbon steel
>0.7% C
Mild steel
Screw driver
Tool steel
Blanking dies, Ball bearings
Medium carbon steel
Crane hooks
High carbon steel
Commercial beams
www.eggsam.com
7.7. HEAT TREATMENT
[111]
Colourful
Unpaired electrons
Weakly attracted in external magnetic
field
Small positive magnetic susceptibility
Ex: Alkalies, Alkaline earths,...
Colourless
Paired electrons
Weakly repelled in external magnetic
field
Small negative magnetic susceptibility
Ex: Cu, Ag, Au, Bi
Ferromagnetic
Dipoles in same
direction
All dipoles in same direction
Strong attraction to magnetic fields
Strongly magnetized in external fields
Large positive magnetic susceptibility
Ex: Fe, Co, Ni
Anti-ferromagnetism
Dipoles in alternate
directions
Colombium
Ferrimagnetic
Ferromagnetic+
Anti-ferromagnetic
Paramagnetic
Diamagnetic
7.7
Heat treatment
Time Temperature Transformation (TTT)
C-curve
S-curve
Bain’s curve
For stability ∆G > 0
All lines on TTT diagram shows decomposition of austenite into some other structure.
It cannot be reconverted.
Adding impurity shifts TTT diagram towards right
CCR: critical cooling rate, it just touches the nose of TTT diagram
Any cooling rate ≥ CCR will not produce pearlite
But produces martensite
colloidal solution of carbon or ferrite in iron
hardest phase of iron
Austempering
Quench below nose of TTT
but above martensite start
line (220o C ), hold it
Austenite → Bainite (100%)
Ductility, impact strength and
toughness increases
Martempering
Quench below nose of TTT
but above martensite start
line, hold it, move to room
temperature
Quenching in two medium
Austenite → Martensite
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7.7. HEAT TREATMENT
[112]
Annealing
Heat austenite temperature and cool slowly
in furnace
Reduce Hardness
Increase ductility
Improve machinability
Relieve internal stresses
Refine grain size
Full Annealing
Heat steel to 50o C to 70o C above the upper
critical temperature, Hold there, Slowly
cool in furnace.
In hypoeutectoid steels, Austenite becomes
coarse pearlite and ferrite structures.
In hyper eutectoid steels, Austenite →
Pearlite+Cementite
Reduce
Hardness/brittleness
Increase
ductility/toughness
Process Annealing
Heating below lower critical temperature,
usually used in low carbon steels
To remove effects of cold
working (relieve stress)
Make soft
Spheroidise annealing
Heat near lower critical temperature, slow
cooling in furnace
To increase machinability
in MC or HC steels
Increases ductility
Diffusion annealing
1150o C and slow cooling
homogenizing
To make uniform
composition
Usually done after welding
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7.7. HEAT TREATMENT
[113]
Normalizing
Final heat treatment process
40-50o C above Austenite is
stable, Hold it, cool in air
To make hard surface and tough core
Hardening
Heat to austenite temperature,
hold, quench in water, oil, or
molten salt baths. (Equal to or
greater than critical cooling rate)
30-50o C above critical
temperature
Martensite formed
Very hard, brittle
Tempering
Hardening is followed by
tempering.
Heat to below lower critical
temperature, hold, cool slowly
Relieve residual stresses
improve ductility
increase toughness
High temperature
tempering
(500-o C 650o C )
makes sorbite
Medium tempertaure
tempering
(350o C - 500o C )
makes troosite, used in making springs
(250o C )
No structural change, only stress
relief, used for making agricultural
tools and metrology stuff
Widmanstatten structure
Low temperature
tempering
Case hardening
Carburizing
Nitriding
Cynaniding
Make surface hard
Using free carbon
Carbon monoxide
Methane
NaCN
On the surface
Mild steel
Pack carburizing:
Liquid carburizing
gas carburizing
NH3
NaCN
Flame hardening
Guideways of lathe
Induction hardening
To harden surface
Hardness order
Nitriding > Cyaniding > Carburizing
Brine < Water +NaOH < Water < Oil < Air
Cooling rate
Air < Oil < Water < Brine
<3000o C Carburizing
3150o C Neutral
3480o C Oxydizing
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7.8. NANOMATERIALS
[114]
Age hardening or Precipitation hardening
-for Al alloys
Overaging (coarsening of precipitate particles)
Artificial aging
Misc
The iron-carbon diagram is determined under equilibrium and TTT curve is determined
under nonequilibrium condition.
The martensitic transformation is a process of shear, that occurs without any need for
diffusion so there will be no change in composition in this process.
7.8
Nanomaterials
At least one dimension less than 100nm
Zero size: Particles
One dimensional: only one large dimension
Two dimension: sheets
Three dimension: Cubes
Manufacturing
Top-down approach
Bulk material is converted to nano
Mechanical grinding,atomization
Bottom-top approch
Atoms combine to produce nano
Sol-gel technique
Physical/Chemical vapor deposition
7.9
Misc
Cottrell atmosphere: Due to diffusion, the interstitial Carbon gets accumulated in
dislocation sites in iron. (More energy is needed to break it and it causes upper yield
point)
Bauschinger effect:Unload the materials from the region of work hardening- reverseload again, Tensile yield strength increases, compressive decreases
Nitrizing produces harder materials than carburizing.
Strain hardening
σf = Kn
n: work hardening exponent (n=0.3 for steel, n= 0.05 for Al)
K: strength coefficient
at UTS = n
www.eggsam.com
7.9. MISC
[115]
Usually 0 < n < 1
For perfectly plastic, n=0
Dislocation forest
Bauschinger Effect
Higher value of n means more dislocations
Cold working
below re-crystallisation temperature Strength and hardness of a cold worked component
increases
Ductility and toughness decreases
strain hardening is due to dislocation
Surface hardening
Shot blasting: For heavy material, steel balls
Shot peening: For small size material, manual hammering
Sand blasting: For thin material,
Hall-Patch Equation
K
σy = σ0 + √
d
σy : Yield strength
σ0 : Base strength of material
k: Constant
d: Grain diameter
Corrosion
Tin plated iron sheet: tin is anodic to iron
Galvanized: zinc is cathodic to iron
Chapter 8
Mechatronics and robotics
8.1
Microprocessors
Microprocessor is a controlling unit of a micro-computer, fabricated on a small chip capable of performing ALU (Arithmetic Logical Unit) operations and communicating with
the other devices connected to it.
Microprocessor consists of an ALU, register array, and a control unit.
Arithmetic and Logic Unit (ALU)
All the computing functions are maintained in this unit. (+,-,*,/,%, AND, OR, NOT,
XOR, etc)
Control Unit (CU)
Coordinates and times the CPUs functions, and it uses the program counter to locate
and retrieve the next instruction from memory.
controls the data flow between microprocessor and peripheral devices/peripheral chips.
Registers
Store the data temporarily during the execution/runtime of the program
Memory
stores the information (data& instructions) in binary form.
Read Only Memory (ROM)
Stores items that the computer needs to execute when it is first turned on
Random Access Memory (RAM)
Stores user programs and datas temporarily. RAM is a volatile memory.
System Bus
1) Control bus
2) Data bus
3) Address bus
116
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8.1.1
8.2. MICROCONTROLLER
[117]
Advantages of microprocessor
Small in size
Low Power Consumption
Versatility
high speed
high accuracy and reliability
used to perform multitask operations
8.1.2
Disadvantages of a Microprocessor
Highly sensitive to thermal and electric variations
Do not have internal memory(RAM&ROM)
Do not have input/output ports inside the microprocessor
No timers, interrupts inside the microprocessor
Make a system expensive even though microprocessor itself is cheep
Need interfacing components for functioning
8.1.3
Types
RISC: Reduced Instruction Set Computer
CISC: Complex Instruction Set Computer
8.2
Microcontroller
Microcontroller is integration of all microprocessor and input and memory other peripherals in a single chip.
Arduino is a microcontroller
Atmega: Microcontroller
8.2.1
Components
Central processing unit(CPU)
Random Access Memory)(RAM)
Read Only Memory(ROM)
Input/output ports
Timers and Counters
Interrupt Controls
Analog to digital converters
Digital analog converters
Serial interfacing ports
Oscillatory circuits
www.eggsam.com
8.2.2
8.2. MICROCONTROLLER
Advantages of Microcontrollers
Acts like a microcomputer
Reduces cost and size of the system.
Simple to use
Easy to troubleshoot
Most of the pins are programmable
Easily interface additional RAM, ROM,I/O ports.
Low time required for performing operations.
8.2.3
Disadvantages of Microcontrollers
Complex architecture than that of microprocessors.
Only perform a limited number of executions simultaneously.
Mostly used in micro-equipments.
Cannot interface high power devices directly.
Microprocessors
Microcontrollers
It is only a general purpose computer
CPU
It is a microcomputer itself
Memory, I/O ports, timers, interrupts
are not available inside the chip
All are integrated inside the
microcontroller chip
This must have many additional
digital components to perform its
operation
Can function as a microcomputer
without any additional components.
Systems become bulkier and
expensive.
Make the system simple, economic
and compact
Not capable for handling Boolean
functions
Handling Boolean functions
Higher accessing time required
Low accessing time
Very few pins are programmable
Most of the pins are programmable
Very few number of bit handling
instructions
Many bit handling instructions
Widely Used in modern PC and
laptops
Widely in small control systems
E.g. INTEL 8086,INTEL Pentium
series
INTEL8051,89960,PIC16F877
[118]
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8.3
8.3. STEPPER MOTOR
Stepper motor
No of poles on stator
No of teeth on rotor ≥
2
-Three types
Variable reluctance type
Permanent magnet type
Hybrid type
8.3.1
Variable Reluctance type Stepper Motor
Stator acts as electromagnet
Stator pitch
-Angular separation between two successive poles
360o
θs =
Number of stator ploes
Rotor pitch
360o
θr =
Number of teeth on rotor
Full step angle
Angle of rotation of rotor when only one switch is activated
θf s = θr − θs
Half step angle
Rotation of rotor when two switches are activated
θf s
θhs =
2
8.3.2
Permanent magnet type Stepper motor
Stator is same as VRSM
Rotor is permanent magnet
Holding torque is high
Low speed, high torque
Consumes less power than VRSM
360o
Step angle =
no. rotor poles x no. of phases
Has the highest step angle
8.3.3
Servo motor
Closed loop control algorithm which makes comparator output zero.
Any motor in servo mechanism in servo motor
Also called control motors
8.3.4
Hybrid type Servo motor
Used where minimum step angle is needed
[119]
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8.4
8.4. OPTICAL ENCODER
[120]
Optical encoder
Digital transducer to measure angle or position
Incremental encoder
Single track incremental encoder
360o
Resolution =
Number of holes
Multi track incremental encoder
Speed and direction can be measured
More resolution
Absolute encoder
Position known more accurately
360o
Resolution = N
2
8.5
Hall sensor
Based on hall effect
Hall effect: Voltage developed on the surface of a conductor/semi-conductor carrying a
current placed in a perpendicular magnetic field
Current density J = neAVd
~ = eVd B
Force acting on charge particle = q(V~d × B)
1 IB
VH =
ne t
IB
VH = K
t
K: Hall coefficient
8.6
V =
Electromagnetic induction
dφ
dt
Resolver
position sensor
Angular position, angular velocity
Produces analog output
Analog to digital converter required
Inductosyn
Position sensor
Linear and angular displacement
Based on electromagnetic induction
Piezoelectric accelerometer
Accelerometer
State space Representation
Controllability
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8.7. ACTUATORS
[121]
determinant of controllability matrix non-zero
Observability
If we can calculate the state variables of a system at any particular time from the output
of the system, then it is observable
Determinant of observability matrix is non-zero
8.7
Actuators
Hydraulic actuators
Liquid pressure energy to mechanical power
Power = Pin Q
8.8
Robotics
The notations used follows Prof. Ashitava Ghosal ’s NPTEL course.
(https://nptel.ac.in/courses/112/108/112108093/)
Students are strongly advised to visit the course at least once OR read the book
https://www.amazon.in/Robotics-Fundamental-Concepts-Ashitava-Ghosal/dp/0195673913
Asimov’s three laws of robotics
• First law (Human safety):
A robot may not injure a human being, or, through inaction, allow a human being to
come to harm.
• Second law (Robots are slaves):
A robot must obey orders given it by human beings, except where such orders would
conflict with the First Law.
• Third law (Robot survival):
A robot must protect its own existence as long as such protection does not conflict with
the First or Second Law.
Cartesian or Gantry robot(3P)
- Arm has three prismatic joints, whose axes are coincident with a Cartesian coordinator.
- Uses: pick and place work, application of sealant, assembly operations, handling machine tools and arc welding.
Cylindrical robot(R2P)
- Axes form a cylindrical coordinate system.
- Uses: assembly operations, handling at machine tools, spot welding, and handling at
die casting machines.
Spherical or Polar robot(2RP)
- It’s a robot whose axes form a polar coordinate system.
- Uses: handling machine tools, spot welding, die-casting, fettling machines, gas welding
and arc welding.
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8.8. ROBOTICS
[122]
Articulated or Revolute or Anthropomorphic Robot(3R)
- It’s a robot whose arm has at least three rotary joints.
- Uses: assembly operations, die casting, fettling machines, gas welding, arc welding and
spray painting.
- Ex. PUMA560
SCARA robot(3R1P)
-Selective Compliant Assembly Robot Arm or Selective Compliant Articulated Robot
Arm.
- It’s a robot which has two parallel rotary joints to provide compliance in a plane
- Uses: pick and place work, application of sealant, assembly operations and handling
machine tools
- Can move very fast.
- Best suited to planner task
Parallel robot
Used as a mobile platform handling cockpit flight simulators
PUMA
- Industrial robot.
- Programmable Universal Machine for Assembly, or Programmable Universal Manipulation Arm
- Functions like a human arm.
- A total of 6 variables are required, for specifying the position and orientation of a rigid
body in space.
- PUMA has 6 axis of rotation
Transformation
matrix

1
0
0




Rx =  0 cos θ − sin θ 


0 sin θ cos θ


cos θ sin θ 0




Ry = 
0
1
0 


− sin θ cos θ 0


cos θ − sin θ 0




Rz =  sin θ cos θ 0 


0
0
1
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8.9. TRANSDUCER
[123]


Cθi
−Sθi
0
ai−1






i−1[T ] =  Sθi Cαi−1 Cθi Cαi−1 −Sαi−1 −Sαi−1 di 


i
 Sθi Sαi−1 Cθi Sαi−1 Cαi−1 Cαi−1 di 


0
0
0
1
Transformation based on current axis
0
0
1
2
3 [T ] =1 [T ]2 [T ]3 [T ]
Transformation based on a fixed axis
0
1
2
3
0 [T ] =3 [T ]2 [T ]1 [T ]
8.9
Transducer
The device which converts the one form of energy into another is known as the transducer.
Active transducer does not use any external power source for producing the output.
Passive transducer requires the additional energy source for working.
Piezoelectric transducer
d
V =
tP
r 0
d
g=
r 0
t: Thickness of crystal
d: charge density of crystal
P: pressure on crystal
g: Voltage sensitivity of crystal
Photoelectric Transducer
Converts the light energy into electrical energy.
Made of semiconductor material. Photoemissive Cell
Photoconductive Cell
Photo-voltaic cell
Photodiode
photo-diode
Phototransistor
Chapter 9
Fluid Mechanics
9.1
1
1
1
1
1
Hydrostatics
Torr= 1mm Hg
bar=100kPa
poise=0.1Ns/ m 2
Stoke=10−4 m 2 / s
kgf=9.81N
Mass
Volume
Weight
Specific weight =
Volume
1
Specific volume =
Density
Density =
Density
Density of standard fluid
Mean free path
Knudsen number Kn =
Characteristic length of flow
Specific gravity or relative density =
Kn < 0.01 : Continuous fluid
Kn > 0.01 : Continuum does not hold
Isothermal Compressibility (β)
1 dV
1
β=−
=
V dP
K
K: Bulk modulus of elasticity or Coefficient of compressibility
V dP
ρdP
K=−
=
dV
dρ
K is a function of temperature and pressure
Surface tension (σ)
Unit: N/m
Due to cohesion
Surface tension of water-air interface σ = 0.073N/m
Tensiometer or Stalagmometer: surface tension measurement.
2σ
Excess pressure inside a jet of a liquid =
d
124
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9.1. HYDROSTATICS
[125]
4σ
d
8σ
Excess pressure inside a bubble =
d
Capillary raise/fall
Due to Cohesion and Adhesion
4σ cos θ
h=
ρgd
4A
Hydraulic diameter =
P
Hydraulic diameter
Shape
Excess pressure inside a drop =
Circle with diameter D
D
Annulus
Doutside − Dinside
Square of side a
a
2ab
a+b
4ab
2a + b
Rectangle of sides a×b
Rectangular channel with
one ’b’ side open
9.1.1
Buoyancy and flotation
Buoyant force = Volume immersed × density of fluid × g
I
Meta-centric height GM = − BG
V
I
Meta-centric radius BM =
V
V: Immersed volume
s
K2
Period of rolling T = 2π
gGM
Increase in GM =⇒ Decrease in time period oscillation =⇒ Increase in stability =⇒
Decrease in comfort
Decrease in GM =⇒ Increase in time period oscillation =⇒ Decrease in stability =⇒
Increase in comfort
K: Radius of gyration about axis of rolling
I: Least area moment of inertia of the body at water surface
h: Position of Center of gravity from the surface
Equilibrium
Floating object
Submerged object
Stable equilibrium
M above G
B above G
Neutral equilibrium
M and G coincide
B and G coincide
Unstable equilibrium
M below G
G above B
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9.1.2
9.2. VISCOSITY
[126]
Pressure (P)
Pascal’s law: For a stationary fluid, at a point, Px = Py = Pz
Force
Pressure P =
Area
Pressure at depth h, P = hρg
IGG sin2 θ
Center of pressure yp = h +
Ah
h : Center of gravity from surface
A : Area of the surface
IGG : Area moment of inertia about axis through CG, parallel to surface
θ : Angle of the surface with horizontal
Absolute pressure = Gauge pressure + Local atmospheric pressure
P − Pv
Net positive suction head
Thoma’s cavitation number σc =
=
2
ρV /2
ρV 2 /2
Patm,abs
Pvappressure
NPSH=
− hsuction − hLoss −
ρg
ρg
Shape
Rectangle
5 Triangle
4 Triangle
Circle
¯ Semi-Circle
∪
Trapezium
COP from
water surface
2
h
3
1
h
2
3
h
4
5
d
8
3π
d
32
CG from surface
a + 3b h
a + 2b 2
2a + b h
a+b 3
h/2
h/3
2h/3
d/2
2d
3π
Force on submerged bodies
Fx = Ax h̄ρg
Ax : Area of the body projected on a vertical plane
Fy =p
Weight of fluid directly above the body
F = Fx2 + Fy2
F = h̄ρgA
9.2
Viscosity
du
Newton’s law of viscosity τ = µ
dy
µ
Kinematic Viscosity=
n ρ
∂u
τxy = A + µ
∂y
Moment of inertia
(IGG )
bh3
12
bh3
36
bh3
36
πd4
64
−
a2 + 4ab + b2
36(a + b)
h3
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9.3. KINEMATICS OF FLUID FLOW
[127]
Name
A
µ
n
Examples
S
Ideal Solid
-
-
-
-
T
Thixotropic
τyield
µ
<1
Printer ink, lipstick
B
Bhingam Plastic
τyield
µ
1
Sewage sludge, tooth paste, drilling mud
R
Rheopectic fluid
τyield
µ
>1
Gypsum
P
Pseudo Plastic
0
µ
<1
Blood, milk, suspension paints, paper pulp
N
Newtonian
0
µ
1
Air, water
D
Dilatant fluid
0
µ
>1
Butter, Rice starch, Sugar sol
Ideal fluid
0
0
-
-
I
Thixotropic and Rheopectic fluids have time dependent viscosity.
Bhingam
plastic isalso called Ideal plastic
∂u ∂v
τxy = µ
+
∂y ∂x
B/T
For Liquids µ = Ae
(Andrade’s equation)
√
a T
For Gases µ =
(Sutherland equation)
1 + b/T
A, B, a and b are constants.
T is absolute temperature
Rheology: Study of non-Newtonian fluids
Sound
r
r
K
∂P
Velocity of sound in fluid =
=
ρ
∂ρ
9.3
Kinematics of fluid flow
Steady flow: At a given point, all flow characteristics remains constant over time.
Unsteady flow: At a given point in fluid flow, flow characteristics might vary with time.
Uniform flow: Velocity at every point in the flow at a given time is same.
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9.3. KINEMATICS OF FLUID FLOW
[128]
Non-uniform flow: Velocity may vary from one point to another in the flow.
Local Acceleration: Acceleration due to change in velocity with respect to time. (Temporal acceleration)
Convective acceleration: Acceleration due to change in velocity with respect to position
Streak line: Locus of points of all the fluid particles that have passed continuously
through a particular spatial point in the past
Stream line: Curve that is instantaneously tangent to the velocity vector of the flow
Path line: Trajectory of a fluid particle
Cartesian coordinates
Continuity
equations
ρ1 A1 V1 = ρ2 A2 V2
∂ρ
+ ∇ · (ρV ) = 0
∂t
∂u
∂u
∂u
∂u
+u
+v
+w
∂t
∂x
∂y
∂z
∂v
∂v
∂v
∂v
ay =
+u
+v
+w
∂t
∂x
∂y
∂z
∂w
∂w
∂w
∂w
+u
+v
+w
az =
∂x
∂y
∂z
p∂t
a = a2x + a2y + a2z
ax =
Acceleration
Stream
function
∂Ψ
=v
∂x
∂Ψ
= −u
∂y
Line of constant stream
dx
dy
function
=
u
v
Potential
function
∂φ
= −u
∂x
∂φ
= −v
∂y
∂φ
= −w
∂z
dy
u
Equipotential line:
=−
dx
v
Cauchy
Reimann
equations
∂u
∂v
=
∂x
∂y
∂v
∂u
=−
∂x
∂y
Cylindrical coordinates
1 ∂(rur ) 1 ∂uθ ∂uz
+
+
=0
r ∂r
r ∂θ
∂z
∂vr
∂vr vθ ∂vr
∂vr vθ2
+ vr
+
+ vz
−
∂t
∂r
r ∂θ
∂z
r
∂vθ
∂vθ vθ ∂vθ
∂vθ vr vθ
aθ =
+ vr
+
+ vz
+
∂t
∂r
r ∂θ
∂z
r
∂vz
∂vz vθ ∂vz
∂vz
+ vr
+
+ vz
az =
∂t
∂r
r ∂θ
∂z
ar =
1 ∂Ψ
= ur
r ∂θ
∂Ψ
= −uθ
∂r
∂φ
= ur
∂r
1 ∂φ
= uθ
r ∂θ
∂vr
∂vθ
r
=
∂r
∂θ
∂vr
∂vθ
= −r
∂θ
∂r
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9.3. KINEMATICS OF FLUID FLOW
Rotation of fluid particles
1
ω = ∇×V
2 1 ∂w ∂v
ωx =
−
2 ∂y
∂z 1 ∂u ∂w
−
ωy =
2 ∂z
∂x
1 ∂v ∂u
−
ωz =
2 ∂x ∂y
Vorticity = 2ω
Circulation = Vorticity
×Harea
H
~ = (udx + vdy + wdz)
Circulation Γ = V~ · ds
Irrotational flow
ω=0
∇2 φ = 0
∇2 Ψ = 0
φ exists
Possible,steady,incompressible, irrotational flow if ∇2 φ = 0
Possible case of flow if Ψ exists
Potential flow
∇2 φ = 0
∇2 Ψ = 0
Ψ⊥Φ
Irrotational flow
Laplace equation in cylindrical coordinates
1 ∂ 2φ
∂ 2 φ 1 ∂φ
+
+
=0
∂ 2 r r ∂r r2 ∂θ2
Divergence of V~ = ∆ · V~
Linear strain rate
∂u
˙x =
∂x
∂v
˙y =
∂y
∂w
˙z =
∂z
Shear strain
rate
1 ∂v ∂u
˙xy =
+
2 ∂x ∂y 1 ∂z ∂w
˙zx =
+
2 ∂x
∂z 1 ∂v ∂w
˙yz =
+
2 ∂z
∂y
νxy = µxy
[129]
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9.4. FLOW DYNAMICS
Vortex flow
v2
dr − ρgdz
r
Forced vortex
Free vortex
v =ω×r
ω 2 r12 P1
ω 2 r22 P2
z1 −
+
= z2 −
+
2g
ρg
2g
ρg
Doesn’t follow Bernoulli equation
vr = const
v 2 P1
v 2 P2
z1 + 1 +
= z2 + 2 +
2g ρg
2g ρg
Obeys Bernoulli equation
External force is required
No external force
Bucket on rotating table
Bottle, hole at bottom
Washing machine
Kitchen sink
dp = ρ
Whirlpool in a river
Tornado
ω2 2
Equation of free surface of fluid in a rotating cylinder Z = h0 − (R − 2r2 )
4g
h0 : Height of fluid at rest
ω 2 R2
Maximum height difference ∆Z =
2g
Where h0 is the initial height of fluid
Z is the height of fluid from the bottom of the cylinder
9.4
Flow Dynamics
g gravity
p pressure
v viscosity
t turbulence
c compressibility
Fx = (Fg )x + (Fp )x + (Fv )x + (Ft )x + (Fc )x
Reynold’s equation of motion: Fx = (Fg )x + (Fp )x + (Fv )x + (Ft )x
Navier Stokes equations
Fx = (Fg )x + (Fp )x + (Fv )x
∂ V~
~ + ρ~g + µ∇2 V~
ρ
= −∇P
∂t
2
∂u
∂u
∂u
∂u
∂p
∂ u ∂ 2u ∂ 2u
=−
ρ
+u
+v
+w
+ ρgx + µ
+ 2+ 2
2
∂x
∂y
∂z ∂x
∂y
∂z ∂t
∂x
2
2
∂v
∂v
∂v
∂v
∂p
∂ v ∂ v ∂ 2v
ρ
+u
+v
+w
=−
+ ρgy + µ
+
+
∂x
∂y
∂z ∂y
∂x2 ∂y 2 ∂z 2
∂t
∂p
∂w
∂w
∂w
∂w
∂ 2w ∂ 2w ∂ 2w
ρ
+u
+v
+w
= − + ρgz + µ
+
+
∂t
∂x
∂y
∂z
∂z
∂x2
∂y 2
∂z 2
[130]
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9.5. FLOW MEASUREMENT
[131]
Euler’s equation of motion
Fx = (Fg )x + (Fp )x
dp
+ gdz + vdv = 0
ρ
∂u
∂u
∂u
∂p
∂u
+u
+v
+w
=−
+ ρgx
ρ
∂x
∂y
∂z ∂x
∂t
∂v
∂v
∂v
∂v
∂p
ρ
+u
+v
+w
=−
+ ρgy
∂t
∂x
∂y
∂z
∂y
∂w
∂w
∂w
∂p
∂w
+u
+v
+w
= − + ρgz
ρ
∂t
∂x
∂y
∂z
∂z
Assumptions:
• Incompressible
• Inviscid
• Homogeneous
Bernoulli’s equation (Energy equation)
p
v2
+
+ z = constant
ρg 2g
Assumptions: -ideal, irrotational, inviscid, steady, incompressible, homogeneous, continuous flow along a streamline
Water hammer
2L
Critical closure time Tc =
C
ρV L
Gradual closing of valve, Pressure rise =
T
Sudden closure of a rigid pipe, Pressure rise=ρvC
C: Speed of sound
9.5
Flow measurement
Pitot tube
Stagnation pressure = static pressure + dynamic pressure
ρV 2
Stagnation pressure: P +
2
Static pressure: P
ρV 2
Dynamic pressure:
2
Pitot tube: Stagnation pressure - static pressure
P
Piezometric head= + Z
ρg
Differential head: Change in Piezometric head
Piezometer tube
Direct pressure, no role for velocity
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9.6. LAMINAR FLOW
[132]
ρm
−1
U-tube manometer P = ρf gh
ρf
ρm
Inverted U-tube
P = ρf gh 1 −
ρf
manometer
√
Cd A1 A2 2gH
Qth = p 2
Venturi meter
2
A1 −
√A2
Cd A1 A2 2gH
Qth = p 2
Orifice meter
A2
s A1 −
2
ρm
V = Cv 2gh
−1
Pitot tube
ρf
s
ρm
Inverted pitot tube V = Cv 2gh 1 −
ρ
f
√
8
θ
Q = Cd 2g tan
H 5/12
Triangular weir
15
2
√
2
Rectangular notch Q = Cd L 2gH 3/2
3
ρm : Density of fluid in Pitot tube
ρf : Density
of flowing
fluid
ρm
H=h
−1
ρf
9.6
Laminar flow
V: Mean velocity
u: Velocity at a point
V ∗ : Shear velocity or friction velocity
ρV D
Reynolds number Re =
µ
Hydrodynamic entrance length
Laminar flow
0.05ReD
Re=2000
100D
Turbulent flow
10D to 40D
For laminar flow
Case
Re <
Pipe internal
2000
Parallel Plate
1000
Open channel
500
Sphere
1
Over flat plate
5 × 105
Cd :0.62-0.65
measures flow velocity
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9.6.1
9.6. LAMINAR FLOW
[133]
Laminar flow Inside a circular pipe
1 dp 2
[R − r2 ]
4µ dx
1 dp 2
umax = −
R
4µ dx
umax
V =
2
32µV L
f lV 2
Head loss hf =
[Hagen Poiseuille Formula]
=
ρgD2
2gD
∂τ
∂p
=
∂y
∂x
∂p r
du
τ =−
= −µ
∂x 2
dr
dp R
τmax = −
dx 2 P2
P1
dp =
+ z2 −
+ z1 Change in pressure due to change in piezometric head
ρg
ρgp
p
Friction velocity V ∗ = τ0 /ρ = V f /8
64
f=
Re
ur = −
9.6.2
Laminar flow between parallel plates
B=gap between the plates
1 dp
[By − y 2 ]
uy = −
2µ dx
y2
y
uy = 4umax
−
B B2
2
V = umax
3
1 ∂p
τ =−
[B − 2y]
2 ∂x
∂τ
∂p
=
∂y
∂x
∂p B
τwall =
∂x 2
12µV L
hl =
ρgB 2
9.6.3
Couette flow
Uy
1 dP
−
[By − y 2 ]
B
2µ dx
U
B 2 dP
V = −
2
12µ dx
12µ (V − U/2) L
hf =
ρgB 2
U= Velocity of the top plate
u=
www.eggsam.com
9.7. TURBULENT FLOW
KE correction factor
KE based on actual velocity
1 R 3
α=
=
u dA
KE based on average velocity
AU 3 A
For uniform flow, α = 1
For laminar flow inside a circular pipe α = 2
For turbulent flow inside a circular pipe α = 4/3
Momentum correction factor
1 R 2
Momentum based on actual velocity
=
u dA
β=
Momentum based on average velocity
AU 2 A
For uniform flow, β = 1
For laminar flow inside a circular pipe β = 4/3
For turbulent flow inside a circular pipe β = 1.2
Flow with free surface
3µV L
hl =
ρgδ 2
Stokes law
Terminal velocity V = (ρsolid − ρf luid )
gD2
18µ
Valid for Re < 1
9.7
Turbulent flow
Colebrook-white formula: turbulent flow regime in commercial pipes.
y 1/7
u
=
U
δ
k : Average height of irregularities
δ 0 : Laminar sublayer height
V : Average velocity
V ∗ : Shear velocity
r
f
∗
V =V
p 8
V ∗ = τ0 /ρ
11.6µ
δ0 =
ρV ∗
∗ 0
ρV δ
= 11.6
µ
1
Shear stress at pipe surface τ0 = f ρV 2
8
du
dv
τ =µ +η
dy
dx
2
du
du
τ̄ = µ + ρl
dy
dx
Prandtl’s universal velocity distribution equation: u = umax + 2.5V ∗ lne (y/R)
Velocity defect =u − V ∗
[134]
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9.8. LOSSES
[135]
V ∗ kρ
µ
k
V ∗ kρ
δ0
µ
<0.25
<4
Roughness Reynolds number =
Smooth Boundary
Transition
0.25 - 6.0
4 - 100
>6
>100
Rough Boundary
y =R−r
Turbulent flow in smooth pipes
u
ρV ∗ y
+ 5.55
=
5.75
log
10
V∗
µ
V
ρV ∗ R
+ 1.75
=
5.75
log
10
V∗
µ
0.316
f=
[4000 < Re < 105 ]
Re1/4
0.221
[105 < Re] (Blasius)
f = 0.0032 +
Re0.237
Turbulent flow in rough pipes
u
= 5.75 log10 (y/k) + 8.5
V∗
V
= 5.75 log10 (R/k) + 4.75
V∗
1
√ = 2 log10 (R/k) + 1.74
f
(Karman-Prandtl equation for the velocity distribution near hydro-dynamically rough
boundaries.)
9.8
Losses
Name
Friction(Major loss)1
Sudden expansion(Minor loss)
Sudden contraction(Minor loss)
Entrance loss(Minor loss)
Exit loss(Minor loss)
Bends and other fittings
1: Darcy-Weisbach formula
Equation
f lV 2
hf =
2gd
(v1 − v2 )2
hl =
2g
(vc − v2 )2
hl =
2g
v2
hl = 0.5
2g
v2
hf =
2g
V2
hf = K
2g
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9.9. BOUNDARY LAYER FLOW
[136]
Coefficient of friction f 0 = 4f
Equivalent pipe
The equivalent pipe should have same head loss and flow rate of the earlier combination.
Li
Le
When pipes are in series 5 = Σ 5
De
Di
D
When similar pipes are parallel, d = 2/5
n
Power transmission through pipes
Power transmitted = ρg(H − hf )Q
H − hf
Efficiency =
H
H
Maximum efficiency at hf =
3
Maximum efficiency= 66.67%
9.9
Boundary Layer Flow
δ : Boundary layer thickness
ρU∞ x
Rex =
µ
Displacement thickness: Distance by which a surface would have to be moved in
the direction perpendicular to its normal vector away from the reference plane in an inviscid fluid stream of velocity u0 to give the same flow rate as occurs between the surface
and the reference plane in a real fluid.
Z δ
u
∗
dy
δ =
1−
U∞
0
Momentum thickness: Distance by which a surface would have to be moved parallel
to itself towards the reference plane in an inviscid fluid stream of velocity u0 to give the
same total momentum as exists between the surface and the reference plane in a real
fluid.
Z δ
u
u
θ=
1−
dy
U∞
0 U∞
Energy thickness: Distance by which a surface would have to be moved parallel to
itself towards the reference plane in an inviscid fluid stream of velocity u0 to give the
same total kinetic energy as exists between the surface and the reference plane in a real
fluid.
"
2 #
Z δ
u
u
δ ∗∗ = δE =
1−
dy
U∞
0 U∞
δ∗
Shape factor =
θ
Von-Karman momentum integral equation
τ0
∂θ
=
2
ρU∞
∂x
Von Karman momentum equation is used to find the frictional drag on smooth flat plate
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9.9. BOUNDARY LAYER FLOW
for both laminar and turbulent flows.
Coefficient of drag
τ0
local: Cx = 1 2
ρu
2
FD
Average: CD ∗ = 1
ρAU 2
2
RL
ρU 2
,
Drag force FD = 0 τx bdx = Cf x Bx
2
Bx = Ax
Drag force ∝ ρL2 V 2
1
Friction drag force FD = CD ρU 2 A
2
Boundary conditions
At y = 0, u = 0
y = δ, u = U∞
du
y = δ,
=0
dy
9.9.1
Laminar Boundary layer
Rex < 5 × 105
Kx
δ=√
Rex
Local friction coefficient, Cf x =
τ0
ρv 2 /2
Blasius solution
5x
δ=√
Rex
0.664
(Local skin friction coefficient)
Cf x = √
Rex
1.328
CD = √
(Average drag coefficient)
ReL
τ0 : Shear stress on surface
Cubic
Linear
9.9.2
u
3 y 1 y 3
=
−
U
2δ 2 δ
u
y
=
U
δ
K=4.64
K=3.46
Turbulent Boundary layer
y 1/n
u
=
U∞
δ
Logarithmic velocity distribution
5 × 105 ≤ Re < 107
n=1/7
0.37x
δ=
1/5
Rex
[137]
www.eggsam.com
Cf =
CD =
9.10. MISC
0.059
1/5
Rex
0.072
1/5
Rex
9.9.3
Boundary Layer Separation
dp
>0
Adverse pressure gradient:
dx
∂u
< 0: Flow separated
∂y y=0
∂u
= 0: On the verge of separation
∂y
y=0
∂u
> 0: Flow not separated
∂y y=0
9.10
Misc
Energy Gradient Line
v2
P
EGL = z +
+
2g ρg
Hydraulic Gradient Line
P
HGL = z +
ρg
Shows piezometric head
[138]
139
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10.1. IMPACT OF JETS
Chapter 10
Fluid Machinery
10.1
Impact of jets
[140]
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10.2. TURBINES
[141]
V Jet velocity
u Plate velocity
Vr1 Relative velocity of entering jet
Vr2 = KVr1 Relative velocity of leaving jet
Vw1 = Vr1 cos β1 + u1 Whirl velocity of entering jet
Vw2 = Vr2 cos β2 − u2 Whirl velocity of leaving jet
Vf 1 Flow velocity/ velocity of flow
V = V1 Absolute velocity of inlet jet
V2 Absolute velocity of leaving jet
β1 Angle of blade at inlet
β2 Angle of blade at outlet
α Nozzle angle
K Blade friction coefficient
10.2
Turbines
Hydraulic energy → mechanical energy → electrical energy
Hydraulic Power station
Reservoir → Penstock → Surge tank → Scroll casing → Guide wheel → Turbine runner
→ Daft tube
Surge tank: To reduce water hammer effect
Cavitation: More probability of occurrence at outlet of the runner or Entrance of the
daft tube or suction part of pump
Wicket gate: Used in Francis turbine to control flow of water
Volute tube: Converts velocity head to pressure head
Daft tube: Converts Kinetic energy head to pressure head
Pelton
Pelton2
Francis
Kapplan
Trend
8.5-30
30-60
60-300
300-800
Ns
Increasing
>250
250-60
<60
H
Decreasing
.
..
....
......
Q
Increasing
Impulse
Impulse
Reaction
Reaction
Tangential Tangential Mixed flow Axial flow
Radial
Ns ∝ no.
Adjustable
Inward
of jets
Blades
flow
Pelton2 : Pelton with more than one jet
Propeller turbine: Axial
√ flow Reaction turbine
N P
Specific speed Ns =
H 5/4
N is in rpm
P is in kW
H is in m
p
N P/ρ
Dimensionless specific speed=
(gH)5/4
The specific speed of a turbine is defined as the speed of operation of a geometrically
similar model of the turbine which produces unit power (1 kW) when operating under
unit (1 m) head.
Specific speed of a turbo machine remains constant over different working conditions
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10.2. TURBINES
Unit quantities
For a single turbine working under different conditions
N
Nu = √
H
Q
Qu = √
H
P
Pu = 3/2
H
For Similar turbines
To compare model and prototype
Hm
Hp
= 2 2
2
2
Nm Dm
Np Dp
Qm
Qp
=
3
Nm Dm
Np Dp3
Pm
Pp
= 3 5
3
5
Nm Dm
Np Dp
Specific speeds will be also same.
Prototype is the large one and model is the small scaled one
Degree of Reaction =
Static pressure drop inside runner
Total Energy change inside runner
Vf 1
Flow ratio Ψ = √
2gH
U1
Speed ratio Φ = √
2gH
10.2.1
Impulse turbines
Euler Turbine equation Power P = ṁ(U1 Vw1 + U2 Vw2 )
1
Dynamic component: (V12 − V22 )
2
1
Centrifugal component: (u21 − u22 )
2
1
Accelerating component: (Vf21 − Vf22 )
2
[142]
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10.2. TURBINES
Pelton Turbine
Vw1 = V1
F = ρQ[V1 − u](1 + k cos φ)
Power P = ρaV (V − u)(1 + k cos φ)u
u
u
Efficiency η = 2
1−
(1 + k cos φ)
V
V
1 + cos φ
Maximum efficiency =
2
1
u
=
Maximum efficiency happens when
V
2
10.2.2
Reaction turbines
Francis turbine
Power P = ρQ(Vw1 U1 + Vw2 U2 )
For maximum output Vw2 = 0
Vw1 U1 + Vw2 U2
ηh =
gH
P0
ηm =
ṁVw1 U1
Pout
ηoverall =
ρgHQ
ηoverall = ηh ηmech
Vw1 U1 + Vw2 U2 V22
H=
+
g
2g
H=power produced+outlet water energy
Q = (1 − blade thickness ratio)πD1 B1 Vf 1
Vf 1
Flow ratio Ψ = √
2gH
U1
Speed ratio Φ = √
2gH
∆Prunner
Degree of reaction R =
∆Prunner + ∆PGuidewheel
Maximum number of jets = 6 (Normally)
Kaplan turbine
π
Q = (Do2 − Db2 )Vf 1
4
Speed ratio for highest efficiency = 1.4 to 2
Kaplan turbine has high design efficiency and
it has constant efficiency over a wide range of
design regulation
10.2.3
Draft tube
V12 − V22
− hf
2g
V2−V2
Efficiency of draft tube η = 1 2 2
V1
Head recovered in draft tube =
[143]
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10.3
10.3. PUMPS
Pumps
Foot valve + Strainer →
√ Suction pipe → impeller → Delivery pipe
N Q
Specific speed Ns =
H 3/4
10.3.1
Centrifugal pump
Maximum efficiency when blades are bent backwards
Power P = ṁ(Vw2 U2 − Vw1 U1 )
H
gH
=
ηm =
Vw2 U2
He
H = Hm , manometric head
Q = πD1 B1 Vf 1 = πD2 B2 Vf 2
Vw2 U2 − Vw1 U1
Hm
Euler head He =
=
g
ηm
Manometric head Hm = He − loss
Vf 1
Flow ratio Ψ = √
2gH
U1
Speed ratio Φ = √
2gH
Pvappressure
Patm,abs
− hsuction − hLoss −
NPSH=
ρg
ρg
Net positive suction head
P − Pv
Thoma’s cavitation number σc =
=
2
ρV /2
ρV 2 /2
Volute casing
To collect fluid and deliver at constant velocity
To increase the efficiency of the pump
To reduce the loss of head in discharge
Uniform flow of fluid coming out of impeller
Forward curved blades
Has obtuse angle in velocity triangle
β2 > 90o
Backward curved blades
Highest efficiency
10.3.2
Reciprocating pumps
High head and small flow rate
Air vessels on suction and delivery side
Q = ALN/60
ρgALN
(hsuction + hdelivery )
Power P =
60
ld A 2
Acceleration head during delivery stroke had =
ω r cos θ
g ad
ld : Length of delivery pipe
ad : Delivery pipe area
[144]
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Slip S =
10.3. PUMPS
Swept volume-Actual discharge
Swept volume
Air vessel
-Reduces possibility of cavitation (In suction pipe)
-Makes it possible for the pump to run at a higher speed (In suction pipe)
-Makes it possible to increase the suction head (In suction pipe)
-Reduces frictional loss (In delivery pipe)
-Smoothen the flow (In delivery pipe)
[145]
146
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11.1. NAMED NUMBERS
[147]
Chapter 11
Heat Transfer
11.1
Named numbers
Name
Mach number
Thermal diffusivity
Biot number
Fourier number
Nusselt number
Prandtl number
Reynolds number
Stanton number
Grashof number
Rayleigh number
Peclet number
Strouhal Number*
Lewis number*
Schmidt number*
Lorenz number*
Equation
V
M=p
k/ρ
k
α=
ρC
hLc
Bi =
ksolid
αt
Fo = 2
Lc
hL
Nu =
kf luid
µCp
Pr =
kf luid
ρvD
Re =
µ
Nu
St =
ReP r
ρ2 βg∆T D3
Gr =
µ2
Ra = GrP r
P e = ReP r
nD
S=
V0
hm L
Le =
µD
Sc =
ρD
-
k
ρCD
mcp
Graetz number*
Gz =
Lk
V
Froude number*
Fe = √
Lg
P 1 − P2
Euler’s number*
Eu =
ρV 2
*Not important for GATE
Sherwood number*
Extra notes
speed of a body
=
speed of sound
thermal conductivity
=
density × specific heat capacity
Internal conductive resistance
=
Convective resistance
=
=
=
=
=
convective heat transfer
conductive heat transfer
ν
Kinematic viscosity
=
α
Thermal diffusivity
Inertial force
Viscous force
h̄
Actual heat flux of the fluid
=
ρCp U
Heat flux capacity of the fluid flow
Buoyancy force x Inertia force
(Viscous Force)2
Sc
Pr
=
Thermal conductivity
Electrical conductivity
Sh =
Pressure forces
Inertial forces
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11.2
11.2. CONDUCTION
Conduction
Conduction happens by collision of particles and movement of electron
Diathermic: Allows heat flow.
kdiamond > kAg > kCu > kAl > kF e > kSteel
Kpure metal > kits alloy
1
For metals, k ∝
T
For alloys, k ∝ T
1
For liquids, k ∝
T
For non-metals, k ∝ T
r
T
For gases, k ∝
Molecular Mass
1
k ∝
M
dT
Fourier’s law of conduction: Q ∝ A
dx
dT
Q = −kA
dx
δT
Thermal resistance in conduction R =
Q
l
l:Thickness of the slab
Plane slab R =
kA
ln(r2 /r1 )
Cylinder
R=
r2 : Outer radius
2πkl
r2 − r1
Splere
R=
r1 :inner radius
4πkr1 r2
Variable conductivity
If k = k0 (1 + βt)
(t1 − t2 )
(t2 + t1 )
Then Q = k0 1 + β
A
2
L
1 TR2
kavg =
kT dT
T2 − T1 T1
General heat conduction equations
Cartesian
coordinates
∂
∂T
∂
∂T
∂
∂T
∂T
kx
+
ky
+
kz
+ qg = ρC
∂x
∂x
∂y
∂y
∂z
∂z
∂t
Cylindrical coordinates
1 ∂T
∂ 2T
qg
∂ 2T
1 ∂ 2T
ρC ∂T
+
+
+
+
=
∂r2
r ∂r
r2 ∂φ2
∂z 2
k
k ∂t
Spherical coordinates
1
∂ 2T
∂
∂T
1 ∂
qg
1
ρC ∂T
2 ∂T
+
sin
θ
+
r
+
=
2
∂θ
r2 ∂r
∂r
k
k ∂t
r2 sin θ ∂φ2 r2 sin θ ∂θ
1 ∂
∂T
q
ρC
∂T
g
r2
+
=
r2 ∂r
∂r
k
k ∂t
[148]
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11.3. CONVECTION
qg : Rate of heat generation.
11.3
Convection
Newton’s law of cooling: Q ∝ A(T − T∞ )
Q = hA(T − T∞ )
δt
= P r−1/3
δ
k
Critical radius of cylinder rc =
h
2k
Critical radius of sphere rc =
h
Up to critical radius, the heat transfer rate increases
Ao − Ai
Logarithmic mean area for hollow cylinder Am =
ln(A√
o /Ai )
Logarithmic mean area, hollow sphere Am = 4πro ri = Ao Ai
Case
Plane wall
Cylinder
Sphere
Equations
x: distance
from T1 , Thickness
of wall = 2L
qg
T2 − T1
Tx =
(2L − x) +
x + T1
2k
2L
When T1 = T2
qg L
+ T∞
Twall =
h 2
qg L
qL
Tmax =
+
+ T∞
2K
h
qR
Twall =
+ T∞
2h 2
qg R
qR
Tmax =
+
+ T∞
2h
qg 4k2
Tr =
[R − r2 ] + Twall
4k
qR
Twall =
+ T∞
3h 2
qg R
qR
Tmax =
+
+ T∞
3h
qg 6k2
Tr =
[R − r2 ] + Twall
6k
[149]
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11.3. CONVECTION
[150]
Forced convection
Case
Laminar flow over flat
plates and walls
Laminar flow inside tubes
Turbulent flow over flat
plates
Turbulent flow in tubes
Equations
N ux = 0.332(Rex )1/2 (P r)1/3 (Const. temperature)
N u = 0.664(ReL )1/2 (P r)1/3 (Const. temperature)
N ux = 0.453(Rex )1/2 (P r)1/3 (Const. heat flux)
N u = 0.68(ReL )1/2 (P r)1/3 (Const. heat flux)
N u = 4.36 ≈ 48/11 Uniform heat flux
N u = 3.66 ≈ 48/13 Constant wall temp
N ux = 0.0296(Rex )0.8 P r1/3
N uav = 0.036(ReL )0.8 P r1/3
N uav = 0.023Re0.8 P rn
n=0.3 Cooling of fluid
n=0.4 Heating of fluid
Free convection
Case
General form
Vertical wall
Vertical plates &
cylinders
Horizontal plates
Horizontal cylinders
Equations
N uav = C(GrP r)n
N ux = 0.378Gr1/4
Laminar flow
N uav
Turbulent flow N uav
Laminar flow
N uav
Turbulent flow N uav
Laminar flow
N uav
Turbulent flow N uav
Combined free and forced convection
Gr
≥1
Free convection
Re2
Gr
Mixed convection
≈1
Re2
Gr
Forced convection
≤1
Re2
N u = f (ReGrP r)
Reynold’s Analogy
Cf
St =
(Assuming Pr=1)
2
Chilton and Colburn Analogy
Cf
f
St.P r2/3 =
=
2
8
f: Friction factor
= 0.59(GrP r)1/4
= 0.10(GrP r)1/3
= 0.54(GrP r)1/4
= 0.14(GrP r)1/3
= 0.53(GrP r)1/4
= 0.13(GrP r)1/3
104 < GrP r < 109
109 < GrP r < 1012
105 < GrP r < 2 × 107
2 × 107 < GrP r < 3 × 1010
104 < GrP r < 109
109 < GrP r < 1012
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11.4. BOILING
11.4
Boiling
11.5
Fins and Transient heat transfer
Transient heat transfer
T − Ta
hAt
= exp(−BiF o) = exp −
T0 − Ta
ρV C
Can be considered as a lumped system if Bi<0.1
Fins
Tw = Wall temperature or temperature at the base of the fin.
Ta = Ambient temperature
θ = T − Ta
θw = Tw − Ta
d2 θ
− m2 θ
dx2
θ = c1 emx + c2 e−mx
θ = Arcosh mx + B sinh mx
hP
m=
kA
Actual heat transfer
Fin Effectiveness =
Heat transer without fins
Actual heat transfer
Qf in
Fin efficiency η =
=
Maximum heat transfer
(Tw − Tα )P lh
[151]
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11.6. HEAT EXCHANGER
Case
Equations
Finite fin
Fin with insulated tip
Infinitely long fin
11.6
Heat exchanger
Heat capacity ratio R =
NT U =
h
cosh[m(L − x)] +
[sinh m(L − x)]
T − Ta
km
=
h
Tw − Ta
sinh(ml)
cosh(ml) +
km


h
tanh(ml) +
√

km 
Q = hP kA(Tw − Ta ) 

h
tanh(ml)
1+
km
T − Ta
cosh[m(L − x)]
=
Tw −√Ta
cosh(ml)
Q = hP kA(Tw − Ta ) tanh(ml)
tanh(ml)
Fin efficiency ηf in =
ml
Pl
Fin effectiveness f in = ηf in
A
ml > 2.646, tanh(ml) ≈ 1
T − Ta
= e−mx
Tw −√Ta
Q = hP kA(Tw − Ta )
1
Fin efficiency ηf in =
mlr
Pk
Fin effectiveness f in =
hA
UA
Cmin
Cmin
Cmax
Actual heat transfer
Maximum possible heat transfer
Ch (Th1 − Th2 )
Cc (Tc1 − Tc2 )
ε=
=
Cmin (Th1 − Tc1 )
Cmin (Th1 − Tc1 )
Heat transfered Q = U Aθ
Effectiveness ε =
Parallel flow heat exchanger
∆T1 − ∆T2
LMTD θ =
∆T1
ln
∆T2
∆T1 = Thot,inlet − Tcold,inlet
∆T2 = Thot,outlet − Tcold,outlet
1 − e−N T U (1+R)
=
1+R
[152]
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11.7. RADIATION
Counter flow heat exchanger
∆T1 − ∆T2
∆T1
ln
∆T2
∆T1 = Thot,inlet − Tcold,outlet
∆T2 = Thot,outlet − Tcold,inlet
1 − e−N T U (1−R)
=
1 − Re−N T U (1−R)
If ∆T1 = ∆T2 , then LMTD θ = ∆T1
Condenser or Evaporator
R=0
= 1 − e−N T U
Regenerater
R=1
NT U
=
1 + NT U
LMTD θ =
Types of heat exchangers
Direct contact heat exchangers
Indirect contact heat exchangers
Regenerators
Recuperators
Fouling factor F =
11.7
1
1
−
0
U
U
Radiation
Irradiation (G): Rate of energy received per unit area of the surface.
Emissive power (E): Rate of energy emitted per unit area of the surface
Radiosity (J): Rate of Total energy leaving unit area of the surface
h = 6.625 × 10−34 Js
σ = 5.67 × 10−8
Grey body
α, ρ&τ same for all wavelength
α+ρ+τ =1
1 R R cosθ1 cosθ2 dA1 dA2
View factor F1−2 =
A1 A1 A2
πr2
F1−2 ⇒ From 1 to 2
Q1−2 = Q1 F1−2
F1−2 + F1−3 + F1−4 + ... = 1
A1 F1−2 = A2 F2−1
Total Emissive power E = πI = σT 4
Stefan-Boltzman law of radiation
Eb = σAT 4
E = σAT 4
E = Eb
Kirchoff ’s law
[153]
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11.8. MISC
[154]
α=
q = E = σA(T14 − T24 )
J − Eb
Irradiation G =
1−
Radiosity J = E + ρG
Eb1 − Eb2
σ(T14 − T24 )
=
Q1−2,net =
1 − 1
1
1 − 2
Rnet
+
+
A1 1
A1 F1−2
A2 2
1−
Surface resistance =
A
1
Space resistance =
A1 F12
Infinitively large parallel plates
σ(T14 − T24 )
Q1−2,net =
1
1
+ −1
1 2
Infinitely long concentric cylinders
σA1 (T14 − T24 )
Q1−2,net =
A1 1
1
+
−1
1 A2 2
Small body (1) in a large enclosure(2)
Q1−2,net = 1 σAT 4
Wien’s displacement law: λmax T = 2898µmK
λmax : Wavelength corresponding to maximum spectral emissivity
Lambert’s cosine law Eθ = E cos θ
Total emissive power Eθ from a radiating plane surface in any direction is directly proportional to the cosine of the angle of emission
11.8
Misc
Fourier number ratio of rate at which heat is conducted through a body to at which
heat is stored in body.
Chapter 12
Thermodynamics
12.1
Basic Principles
ln 2 = 0.693
log 2 = 0.301
ln 10 = 2.301
P V γ γ=const
P T 1−γ =const
T V γ−1 =const
Nozzle: increases velocity at the expense of pressure
F − 32
C −0
=
212 − 32
100 − 0
Triple point of water T=273.16K
Ice point, T=273.15K
Specific
heats
du
R̄
=
Cv =
dT v=c M (γ − 1)
dh
Cp =
dT p=c
Cp − nCv
Cn =
1−n
Cp
Adiabatic Index γ =
Cv
Cp − Cv = R (Meyer’s relation, ideal gases)
2
γ =1+
x
x
Internal energy u = RT [x: dof of molecule]
2
Monoatomic gas x=3
Diatomic gas
x=5
Triatomic gas
x=6,7
Change in internal energy ∆u = Cv ∆T
Change in enthalpy ∆h = Cp ∆T
155
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12.1. BASIC PRINCIPLES
Polytropic efficiency =
Gas Laws
Name
Boyle’s Law
Charles’ Law
Gay-Lussac’s Law
Avogadro’s Law
[156]
γ−1 n
γ n−1
Eqn
PV=Const
V∝ T
P∝ T
V∝ n
Condition
T=Const
P=Const
V=Const
P,T=Conts
Ideal gas equation
P V = nR̄T
Universal gas constant R̄=8.314J/molK
R̄
R=
Molecular mass
Van-der-Waal’s
equation
a P + 2 (V − b) = RT
V
27R2 Tc2
RTc
a=
,b=
64Pc
8Pc
Vc
b=
3
8a
R=
9Tc Vc
Compressibility factor
Actual volume
Z=
Volume predicted by ideal gas equation
Pv
Z=
RT
Z = 1 Ideal gas at all temperatures
Actual pressure less than ideal gas pressure,
Z<1
intermolecular forces play important role
Actual volume greater than ideal gas
Z > 1 volume, Volume of molecules play
important role
Law of corresponding states states that all gases when considered at the same values
of reduced pressure and reduced temperature will have same compressibility factor.
Dalton’s law (also called Dalton’s law of partial pressures) states that in a mixture
of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases.
Amagat’s law states that the volume of a gas mixture is equal to the sum of volumes of the component gases, if the temperature T and the pressure p remain the same.
Gouy Stodola theorem states that rate of reversibility is proportional to the rate
of entropy generation
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12.2. WORK AND HEAT
Case
Open system
Closed system
Isolated system
Energy transfer
Yes
Yes
No
Intensive property
Independent of mass
Density
Melting point
Boiling point
Resistivity
Specific properties
12.2
[157]
Mass transfer
Yes
No
No
Extensive property
Depends on mass
Mass
Volume, length, area
Work and Heat
• Work is a path function
• Work is energy in transit
Work done by the system
: Positive
Work done on the system
: Negative
Heat added to the system
: Positive
Heat removed from the system : Negative
∆U = Q − W
U: internal energy
Non-flow process
R2
W = P dV
1
R2
Q = CdT
1
dU = mCv (T2 − T1 )
Process
n
W
Q
dU
Isochoric
∞
0
mCv (T2 − T1 )
Q
Isobaric
0
mCp (T2 − T1 )
Q-W
Isothermal
1
W
0
Adiabatic
γ
P (V2 − V1 )
V2
mRT ln
V1
P1 V 1 − P2 V 2
γ−1
P1 V 1 − P2 V 2
n−1
Polytropic
Cn =
n
0
γ−n
Wpoly
γ−1
W
dS
T2
mCv ln
T1 T2
mCp ln
T1
V2
mR ln
V1
0
Q-W
Cp − nCv
1−n
Internal energy change from Van-der-Waal’s equation
1
1
dU = mCv (T2 − T1 ) + a( − )
V1 V2
mCn ln
T2
T1
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12.3. LAWS OF THERMODYNAMICS
[158]
Flow process
R2
W = V dP
1
H = U + PV
dH = mCp (T2 − T1 )
Process
n
W
Q
Isochoric
∞
V (P1 − P2 )
dH − W
Isobaric
0
0
mCp (T2 − T1 )
Isothermal
1
Adiabatic
γ
Polytropic
12.3
n
mRT ln
V2
V1
γ
(P1 V1 − P2 V2 )
γ−1
n
(P1 V1 − P2 V2 )
n−1
W
0
dS
T2
mCv ln
T1 T2
mCp ln
T1 V2
mRT ln
V1
0
dH − W
mCn ln
T2
T1
Laws of thermodynamics
Zeroth Law
• When two bodies are independently in equilibrium with a third body, then the two
bodies are also in equilibrium.
• Basis of temperature measurement
• Deals with thermal equilibrium
First law
• Whenever a system undergoes a cyclic change, the net work done on the system is
always equal to the amount of energy removed from the system as heat
• There exists a property of system called E such that a change in its value is equal to
the
between the heat supplied and the work done during any change in state.
H difference
H
dQ = dW
• In an isolated system, the energy of the system remains constant
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12.3. LAWS OF THERMODYNAMICS
[159]
• Energy can neither be created nor be destroyed
• Introduces concept of internal energy
• u=q−w
• Perpetual motion machine of first kind works without power input
Second law
• Introduces entropy
• Kelvin-Plank Statement: It is impossible to operate a cyclically operating device which
produces no effect other than the extraction of energy as heat from a single reservoir and
performs an equivalent amount of work
• Clausiu’s Statement: It is impossible to operate a cyclically operating device which
produces no effect other than the transfer of energy as heat from a low temperature body
to a high temperature body.
• PMMSK: Absorbs energy and converts the full energy to work.
Third law
• The entropy of a pure crystalline substance at absolute zero temperature is zero.
• It is impossible to achieve zero Kelvin temperature in a finite number of processes.
• The entropy measured relative to pure crystalline substance at absolute zero temperature is called absolute entropy
• Defines datum for measuring entropy
• perpectual motion machine of third kind has no friction
Carnot principle
• No heat engine operating between two given thermal reservoirs can be more efficient
than a reversible engine operating between the same two reservoirs.
• The efficiency of all reversible heat engines acting between same temperatures are same.
Steady flow process
mV22
mV12
+ mz1 g + Q = H2 +
+ mz2 g + W
H1 +
2
2
dq = dh + vdv + gdz + dw
Turbines
Adiabatic
W = H1 − H2 +
m(V12 − V22 )
2
Diffuser
V22 − V12
H2 = H1 −
2
Throttling
• Isenthalpic process
∂T
Joule-Thomson coefficient µJT =
∂P h
Slope of constant enthalpy line in T-P diagram
µJT = 0: Inversion point
µJT > 0: Cooling
µJT < 0: Heating
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12.4. ENTROPY
Throttling of steam
Pressure decreases
Temperature increases
Dryness fraction increases
Entropy increases
Enthalpy is constant
Specific volume increases
QH − QL
TH − TL
W
=
=
QH
QH
TH
TL
QL
=
COP of refrigerator COPr =
QH − QL
TH − TL
QH
TH
=
COP of heat pump COPh =
QH − QL
TH − TL
Efficiency of Carnot engine η =
12.4
Entropy
H δQ
≤ 0 (Clausius inequality)
For a system
T
H δQ
< 0 : Irreversible process
T
H δQ
= 0 : Reversible process
T
H δQ
> 0 : Impossible process
T
• Entropy is a point function and a property
• ∆Sgen (≥ 0) is not a point function
• All adiabatic processes are NOT isentropic
• All isentropic process are NOT adiabatic
• Reversible adiabatic process are isentropic
General equations
T2
V2
+ R ln
s2 − s1 = Cv ln
T1
V1
T2
P2
s2 − s1 = Cp ln
− R ln
T1
P1
γ − n R P1
s2 − s1 =
ln
γ − 1 n P2
For melting and vaporization
h1 − h2
sf g =
T
Mixing of different gases
dS = −Ru (Σni ln Xi )
Mixing same type of gas
Tf
dS = Σmi ci ln
Ti
[160]
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12.5. AVAILABILITY
Entropy of disorder
P2
s2 − s1 = K ln
P1
T-ds equations
T ds = du + P dv
T ds = dh − vdP
SdT = −dA − P dv (Not important for GATE/ESE)
SdT = −dG + vdP (Not important for GATE/ESE)
Maxwell’s
equations
∂P
∂T
=−
∂V
S
∂S
V
∂P
∂S
=
∂V
∂T V
T
∂S
∂V
=−
∂P T
∂T
P
∂V
∂T
=
∂S P
∂P S
12.5
•
•
•
•
•
Availability
Availability is a property of system+surroundings
High grade energy : Can be completely converted into useful work
Low grade energy : Cannot be completely converted to work
Exergy never increases in a process.
Maximum possible work = Change in availability
Second law efficiency =
ηII =
Actual output
Reversible output
or
Reversible input
Actual input
η
ηrev
T0
Available energy or exergy= 1 −
δQ
T
T0
Unavailable energy or anergy= δQ = T0 ds
T
T
Exergy=mCp (T − T0 ) − mCp T0 ln
T0
Availability function for a non-flow process φ = U + P0 V − T0 S
φ2 − φ1 = (u2 − u1 ) + P0 (v2 − v1 ) − T0 (s2 − s1 )
Wmax = (U1 − U0 ) − T0 (S1 − S0 )
Availability = Wusef ul = Wmax − P0 (V0 − V1 )
Availability function for a flow process Ψ = H − T0 S
V22 − V12
ψ2 − ψ1 = (h2 − h1 ) − T0 (s2 − s1 ) +
+ g(z2 − z1 )
2
Irreversibility I = T0 ∆Su
Irreversibility I = T0 (∆Ssys + ∆Ssurroundings )
[161]
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12.6. PURE SUBSTANCES
[162]
Gibbs function
G=H-TS
dG=VdP-SdT
Gibbs Helmholtz function
F=U-TS
dF=-Pdv-SdT
12.6
Pure Substances
A pure substance is homogeneous and invariable in chemical composition along with no
change in properties of the chemical elements constituting the substance.
Gibbs Phase rule F + P = C + 2
mf
Wetness fraction =
mf + mg
mg
Dryness fraction or quality of steam x =
mf + mg
Priming = (1 − x)100
Sub-cooled liquid at temperature T
u = h − Pv
h = hf − Cp
(Tsat − T)
Tsat
s = sf − Cp ln
T
Saturated liquid at temperature Tsat
h = hf
s = sf
Liquid-vapor mix at temperature Tsat
h = hf + x(hg − hf )
s = sf + x(sg − sf )
v = vf + x(vg − vf )
Saturated vapor at temperature Tsat
h = hg
s = sg
v = vg
Super heated vapor at temperature T
h = hf + Cp (T− Tsat)
T
s = sf + Cp ln
Tsat
T
v = vf
Tsat
Clausius-Clapeyron Equation
d(ln P )
hf g
=
dT
RT 2
∂P
hf g
=
∂T
T vf g
Used for Liquid-Vapor transition
Kirchoff relation
Solid-Liquid phase transition
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12.6. PURE SUBSTANCES
[163]
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12.7
Air cycles
12.7. AIR CYCLES
[164]
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12.8. PSYCHROMETRY
re = expansion ratio, rk =Compression ratio, rc =Cut off ratio
Carnot engines in series
η = η1 + η2 − η1 η2
For same work output from both the engines T2 =
Refrigerators in series
1
1
1
1
1
=
+
+
COP
COP1 COP2 COP1 COP2
For same work output from both the engines T2 =
12.8
T1 + T3
2
T1 + T3
2
Psychrometry
Specific humidity
mv
kg of vapour
ω=
=
ma
kg of dry air
Pv
Pv
ω = 0.622
= 0.622
Pa
Patm − Pv
Patm = Pa + Pv
Degree of saturation
ω
µ=
ωsat
Relative humidity
Pv
vsat
mv
φ=
=
=
Psat
vv
msat
µ
φ=
Ps
1 − (1 − µ)
Patm
Enthalpy of moist air
h = ha + ωhv
h = 1.005tdb + ω(2500 + 1.88tdb )
Cp = Cpa + ωCpv = 1.005 + 1.88ω ≈ 1.021kJ/kg-dry air-K
Tcoil − T2
Bypass factor =
Tcoil − T1
Contact factor = 1 - BPF
fg
Lewis number L =
kw cphs
For air-water mix, L=0.945
Wet bulb depression= DBT-WBT
Psychrometric processes
1 : Humidification
1’: Dehumidification
2 : Sensible heating
2’: Sensible cooling
[165]
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12.9. RANKINE CYCLE
[166]
Cooling tower
Approach: Temperature difference between cooled outlet water and WBT of entering air
Minimum temperature to which water can be cooled = WBT of air
Range: Temperature difference between exit cold water and incoming condenser hot water.
R = Tw1 − Tw2
A = Tw2 − Twbt1
12.9
Rankine cycle
WT − WC
WT
WC
=1-Work ratio
Back work ratio =
WT
Net work Wnet = WT − WC
Work ratio =
Mean temperature of heat addition Tm =
h3 − h2
s3 − s2
T1
Tm
Net work ratio is almost 1 for Rankine cycle
Thermal efficiency η = 1 −
Reheat cycle
Regenerative cycle
Pump work
Constant
Decreases
Turbine work
Increases
Decreases
Net work
Increases
Decreases
Dryness fraction
Increases
Decreases
Condenser load
Increases
Decreases
Work ratio
Increases
Decreases
Steam rate
Decreases
Increases
Thermal efficiency I or D or C
Increases
Rankine cycle is preferred for waste heat recovery
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12.10. STEAM TURBINE
[167]
Internal efficiency: Product of stage of steam turbine efficiency and reheat factor
Stage efficiency: Ratio of adiabatic heat drop to isentropic heat drop per stage of a turbine
Rankine efficiency: Ratio of isentropic heat drop in prime mover to the amount of the
heat supplied per unit mass of steam
Rankine cycle with infinite series of regenerative feed heating has efficiency almost equal
to Carnot efficiency
12.10
Steam Turbine
Degree of reaction=
∆hmb
∆hmb + ∆hsb
Compounding of steam turbines reduces turbine speed
Impulse turbines
De Laval : Single stage blade wheel
Curtis: Velocity compounded
Rateau: Pressure compounding
Zoelly: Pressure compounding
Velocity compounding
Moving and fixed blades
Curtis
Pressure compounding
Moving blades and fixed nozzles
Rateau and Zoelly
Force F = ṁs (Vw1 + Vw2 )
Power P = F u
2u(Vw1 + Vw2 )
Work done on blade
=
Blade efficiency ηb =
Energy supplied to the blade
V12
cos β2
ηb = 2s(cos α − s) 1 + K
cos β1
u
Blade speed ratio s =
V1
Vr2
Blade velocity constant K =
Vr1
Frictionless blades Vr1 = Vr2
cos α
For maximum efficiency, s =
2
Maximum efficiency η = cos2 α
Maximum work Wmax = 2ṁu2
α1 : Nozzle
α2
β1 : Blade angle
β2 : Blade angle, Discharge angle, exit
angle, blade outlet angle
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12.11. BRAYTON CYCLE, GAS TURBINES
V12
Nozzle efficiency ηN =
2(h0 − h1 )
Stage efficiency = ηb ηN
Axial flow turbine
β = 90, Vw2 = 0
Reaction turbine
Q̇ = πDhVf 1
Power P = ṁ(Vw1 + Vw2 )U
2
1 + 2s cos α1 − s2
For maximum efficiency s = cos α
2 cos2 α
Maximum efficiency =
1 + cos2 α
(Vw1 + Vw2 )U
Diagram efficiency η = V 2 V 2 −V 2
1
+ r2 2 r1
2
V 2 − Vr22
Energy lost in moving blades = r1
2
-Adiabatic expansion
Efficiency η = 2 −
50% reaction, or identical bladings, or Parson’s turbine
V1 = Vr2 V2 = Vr1 α1 = β2 α2 = β1
12.11
Brayton cycle, Gas Turbines
[168]
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12.11. BRAYTON CYCLE, GAS TURBINES
P2
P3
=
P1
P4
V1
Compression ratio rk =
V2
V4
Expansion ratio re =
V3
T1 γ−1
WT − WC
= 1 − rp γ
Work ratio =
WT
T3
WC
Back work ratio =
=1-Work ratio
WT
Net work Wnet = WT − WC
1
1
Wnet
= 1 − γ−1 = 1 − γ−1
Efficiency η =
Qs
rk
rp γ
γ
Tmax γ−1
For maximum efficiency, rpmax = ηT ηC
Tmin
Brayton cycle is not as efficient as Rankine cycle
Optimum work
√
rpopt = rpmax
√
T2 = T4 = √ T1 T3 √
2
Wopt = C(
r T3 − T1 )
T1
η =1−
T3
Pressure ratio rp =
Regeneration
Thermal efficiency increases
No change in Turbine work
No change in compressor work
heat supplied reduces
Mean temperature of heat addition increases
Mean temperature of heat rejection reduces
Tmin γ−1
Ideal regeneration η = 1 −
rp γ
Tmax
Reheating
Turbine work increases
Thermal efficiency may or may not increase
Intercooling
Increases the net work output
Compressor work reduces
Turbine work constant
Heat supply increases
Thermal efficiency decreases
√
Perfect cooling Pi = P1 P2
[169]
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12.12. NOZZLE
Reversed Brayton cycle
When working as refrigerator, COP = 12.12
P2
P1
1
γ−1
γ
−1
Nozzle
Adiabatic, W=0
V1 <<<
p V2
V2 = v2(H1 − H2 )
"
#
u
n−1
n
u 2n
P
2
V2 = t
P1 v 1 1 −
n−1
P1
r
2n
P1 v1
V2,max =
n−1
12.13
Jet Propulsion
Turbo jet engine
Thrust =(ṁa + ṁf )Ve − ṁa Vi
Thrust power =[(ṁa + ṁf )Ve − ṁa Vi ]Vi
[(ṁa + ṁf )Ve − ṁa Vi ]Vi
Propulsive efficiency =
V2
V2
(ṁa + ṁf ) e − ṁa i
2
2
2Vi
Propulsion efficiency η =
(Assuming small mass for fuel)
Vi + Ve
Vi : Flight velocity
Ve : Relative jet velocity
Turbojet has no power for take-off
Efficiency continuously increases with speed
Turboprop engine
Has high power for take-off
Low efficiency at high altitudes and high velocity
Efficiency first increases with speed and then decreases
Flight velocity cannot exceed jet velocity
Rocket
2V0 Ve
η= 2
V0 + Ve2
Ve : Jet velocity
V0 : Vehicle velocity
Jet velocity of a rocket is independent of forward motion
Thrust=mp Ve + Ae (Pe − Pambient )
mp : mass rate of flow of propellant
Ve : Jet exit velocity
[170]
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12.14. RECIPROCATING COMPRESSOR
Nitric acid: Oxidizer
Hydrogen: Fuel
Fuming nitric acid hydrazine: Hypergolic propellant
Methyl nitrate methyl alcohol: Compounded liquid propellant
Ethyl alcohol: Liquid fuel
Nitrocellulose: Solid fuel
Ammonium perchlorate: Solid oxidant
Hydrogen peroxide: Liquid oxidant
12.14
Reciprocating compressor
Low volume
High pressure ratio
Brake power=Indicated"power+ Friction
# power
n−1
n
P2
n
−1
P1 (V1 − V4 )
W =
n−1
P1
V2 − V1
V4
ηvol =
=1+C −C
Vs
V
1/n 3
P2
ηvol = 1 + C − C
P1
Multi-staging increases efficiency
Power consumed is least for isothermal compression
Puppet valve is used in reciprocating engines
Compression work per kg of air is independent of clearance
volume
Inter-cooling/Multi-staging
#
" n−1
P2 nN
nN
P1 (V1 − V4 )
−1
W =
n−1
P1
Work is reduced
Weight of compressor is reduced
Flywheel weight is reduced
Volumetric efficiency is increased
In perfect inter-cooling, work is equal in both the stages
To increase efficiency
Decrease clearance ratio
Decrease delivery pressure
Multistage
[171]
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12.15
12.15. CENTRIFUGAL COMPRESSOR
[172]
Centrifugal compressor
β > 90o : Forward curved
β = 90o : Radial blades
β < 90o : Backward curved
Vw2
2u2
Used in large refrigeration plants
Low head, high flow rate
Degree of reaction =1 −
Stalling: Separation of flow from the blade surface
Surging: Complete breakdown of flow. Physical damage due to impact loads and high
frequency vibration
Choking: mass flow rate is highest in choking condition
Power required per kg =U 2
12.16
Axial flow compressors
Degree of reaction =1 −
∆TA
Enthalpy drop in rotor
Vw2
=
=
2u2
∆TA + ∆TB
Enthalpy drop in stage
Naxial > Ncentrif ugal > Nreciprocating
12.17
Fuels and combustion
CO2
: 44g
CO
: 28g
O2
: 32g
H2 O
: 18g
H2
: 2g
By volume, dry air contains 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon
dioxide
Semi-bituminous coal: Power plants
Biogas: Carbon dioxide and methane
LPG: Propane and butane
Lignite:
Anthracite: hard and high heating value
Bituminous: High ash content
Coke: Derived from coal
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12.17. FUELS AND COMBUSTION
Bomb Calorimeter
Const Volume, HCV, liquid and solid fuels
Exhaust gas calorimeter
Specific heat
Junkers gas calorimeter
Const Pressure
Throttling calorimeter
Const Enthalpy
Separating calorimeter
Bomb calorimeter
Constant volume
High calorific value
Solid and liquid fuels
[173]
Isobaric
Primary fuels
Wood, coal, natural gas,...
Artificial or secondary fuels
Charcoal, coal gas, coke, kerosene, diesel, petrol,...
O
O
100 8
C + 8H − + S = 11.6C + 34.8(H − ) + 4.35S
Theoretical air required=
23 3
8
8
kg-air/kg-fuel
11
Mass of CO2 formed = C×
3
Mass of H2 O formed = H2 × 9
Higher/gross calorific value
Lower/net calorific value
LCV=HCV-Latent heat of water formed
Coal that does not cake: Free burning cake
Expansion in volume during combustion: swelling index
Grindability index
Weatherability
a[O2 +3.76N2 ]
Volatile matter is responsible for flame length
FC: Fixed carbon
VM: Volatile matter
M: Moisture
Proximate analysis
FC+VM+M+Ash=100%
Step 1: 1g at 105o C for 1 hour. Loss in weight=M
Step 2: 950o C, 7 min, covered platinum crucible. loss in weight = M+VM
Step 3: 720o C, complete burning, uncovered crucible. Remaining mass= Ash
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12.18. IC ENGINES
Ultimate analysis
C+H2 +O2 +N2 +S+M+Ash=100%
Mass percentage
Dulong and Petit’s formula
HCV = 33, 800C + 144, 450(H − O/8) + 9, 380S kJ/kg
Orsat Apparatus
Volumetric analysis
Dry analysis
KOH (Caustic soda): CO2
Pyrogalic acid: O2
Cuprous chloride: CO
Remaining volume: N2
12.18
IC Engines
Petrol Engines(SI)
Diesel Engines(CI)
Compression ratio
5-10
14-22
Speed
High
Relatively low
Weight
Light
Heavy
Peak pressure
Low (To avoid self ignition)
High
Thermal efficiency
Low (Due to low CR)
High
Cycle
Otto cycle
Diesel cycle
Fuel
Petrol, Gasoline
Diesel
Highly volatile
less volatile
Ignition
Spark plug is used
Compression ignition
Fuel injection
Air+ Fuel mixture during
suction stroke
Fuel alone at the end of
compression
Carburetor is used
Fuel pump and injector is
used
Load control
Quantity of air-fuel mixture
is controlled using throttle
Quantity of fuel alone is
controlled
Vehicles
Usually light vehicles
Usually heavy vehicles
Specific output
Higher
lower
For the same pressure ratio, Otto cycle is more efficient than Diesel cycle
[174]
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12.18. IC ENGINES
[175]
Four stroke engine
Two stroke engine
Number of crank shaft
rotations per power stroke
2
1
Number of strokes per cycle
4
2
Turning moment
Non-uniform
Uniform
Flywheel
Heavy
Lighter
Engine size and weight for
a given power
Heavier, bulkier
Lighter, compact
Need of cooling and
lubrication
Relatively Less
More
Rate of wear and tear
Low
Higher
Valves
Present
No valves. Has ports.
Volumetric efficiency
Higher due to higher time
for induction
Lower due to lower time for
induction
Thermal efficiency
Higher
Lower
Part load efficiency
Higher
Lower
Use
When efficiency is preferred
over weight
When weight is to be
reduced
Application
Car, trucks,...
Mopeds, scooters, hand
sprayers.
Air-Fuel Ratio
Case
AFR
Cold start
9:1
Very rich
Idling
10:1
Vehicle not moving
Full load
12.5:1
Cruising
14:1
Best economy
Part load 16:1
For gasoline engines, the stoichiometric A/F ratio is 14.7:1
For diesel Engines, the stoichiometric A/F ratio is 14.5:1
Combustion chamber
Pre-combustion chamber
Combustion induced swirl
Turbulent chamber
Compression swirl
open combustion chamber
Masked inlet valve
F-head combustion chamber
Spark ignition
www.eggsam.com
12.18. IC ENGINES
[176]
How to reduce knocking
Parameter
SI engine
CI engine
Delay period
Increase
Reduce
Ignition lag
Increase
Reduce
Self Ignition temp
Increase
Reduce
Engine speed
Increase
Reduce
Air-Fuel ratio
Richer
Lean
Load
Reduce
Increase
Compression ratio
Reduce
Increase
Spark advance
Reduce
Increase
Spark
Retard
Inlet temp
Reduce
Increase
Inlet pressure
Reduce
Increase
Supercharging
Reduce
Increase
Wall temp
Reduce
Increase
Cylinder size
Reduce
Increase
Power output
Reduce
Increase
Knock happens at the late part of combustion in SI and early part of combustion in
CI
10% Richer flame has highest flame propagation speed and less knocking
Octane number
Iso-octane and normal heptane percentage by volume
In SI engine
BS IV: 81
BS VI: 81/85
Research Octane Number (RON) 91 91/95
Cetane number
CI engine
n-hexadecane (cetane) α methyl naphthalene
BS 4 and 6: 51
Cetane index: 46
An ignition delay in a CI engine is the time taken by the fuel to auto-ignite after
being injected into the engine cylinder.
Suction-Compression-Expansion-Exhaust
Super charging
To increase inlet air density
www.eggsam.com12.19. REFRIGERATION & AIR CONDITIONING
[177]
Increases power output
Uses supercharger
Best used in CI
In SI Engine, increases the chance of knocking
Ignition systems
Battery Ignition
-Battery, ignition switch, ignition coil, breaker points, condenser, distributor and spark
plugs
Efficiency
BP
IP
Brake thermal efficiency
ηrelative =
Air standard efficiency
BP
(ηth )brake =
ṁf × CV
PM EP LA N2 n
IP =
60
Actual work
Diagram factor=
Theoretical work
Vswept + Vclearance
Compression ratio =
Vclearance
ηmech =
Brake thermal efficiency CI> SI>2 stroke SI
Photo Chemical smog
HC and NOx
3 way catalytic converters
HC: follows U shaped curve (Platinum used)
NOs: ∩ shaped (Rhodium used)
CO: negative exponent (Palladium used)
12.19
Refrigeration & Air Conditioning
1TR= 12000BTU/hr
1TR= 3000kcal/hr
1TR= 3.517kW (Use this value in calculations)
1TR= 3.88kW
Lice−water = 336kJ/kg
R50X: Azeotropic Mixture
R7XX: Inorganic Refrigerant. XX-molar mass
R(m−1)(n+1)p : Cm Hn Fp Clq : n + p + q = 2m + 2
R1(m−1)(n+1)p : Cm Hn Fp Clq : n + p + q = 2m
www.eggsam.com12.19. REFRIGERATION & AIR CONDITIONING
Ammonia
R717
Ice plants
Reacts with copper and alloys
Requires larger displacement per TP
Has higher compressor discharge temperature
Smell or Sulphor candle test to find leakage
Refrigerant absorber, ammonia - water
Reciprocating compressors
Freon 12
CCl2 F2 , R12
Window type units
Halide torch to find leakage, green flame
Inflammable, Non-toxic, Chemically stable
Low refrigerating effect
Freon 22
CHClF2 , R22
Low temperature cold storage
Halide torch to find leakage, green flame
Freon 11
CCl3 F , R11
Centrifugal systems
Halide torch to find leakage, green flame
Air
Aircraft air conditioning
CO2
Direct contact freezing food
Reciprocating compressors
Lithium Bromide
Water
Absorption refrigerations, solar refrigeration
Azeotropes
Refrigerant mix, acts like pure substance
QL
TL
COP of refrigerator COPr =
=
QH − QL
TH − TL
Refrigerators in series acting between T1 & T and T & T2
1
1
1
1
1
=
+
+
COP
COP1 COP2 COP1 COP2
T1 + T2
For equal work input T =
2
√
For equal COP T = T1 T2
Thermostatic expansion valve
Maintains constant degree of super heat at the end of the expansion valve
Ensures the evaporator completely filled with refrigerant of the load
RSH
RSH + RLF
RSH = 0.0204(ti − tADP )V̇
RLH = 50(ωi − ωADP )V̇
RT H = 0.02(hi − hADP )V̇
V̇ : air flow in m3 /min
RSHF =
[178]
www.eggsam.com12.19. REFRIGERATION & AIR CONDITIONING
Vapor compression refrigeration
Throttle valve instead of expansion cylinder
Evaporator → Compressor → Condenser → Throttle valve
Reversed Rankine
Bell-Coleman or Joule cycle (Reverse Brayton)
Vapor absorption refrigeration system
Can use solar energy directly
Refrigerant
Low boiling point
Low freezing point
Low specific volume
Low viscosity
Low specific heat
Low positive operating pressure
High latent heat
High thermal conductivity
High density
High critical temperature
Inflammable
Non-toxic
Winter air conditioning
Heating-humidifying-heating
Air conditioning
Air velocity: 6-7m/s
Air per person: 0.25 m3 /min
Sensible heat factor 0.7
Comfort conditions: 22o C and 60% RH
Air change: Air changed per hour
RSHL= 0.0204 V̇ ∆T
RLHL= 50 V̇ ∆ω
V̇ is in m3 /min
Answer is in kW
Ozone layer is in stratosphere
Vortex tube refrigeration
No moving parts
Air is used as refrigerant
[179]
www.eggsam.com
12.20
12.20. BOILER
[180]
Boiler
Output of a boiler is normally stated as evaporative capacity in tonnes of steam at 100o C
that can be produced from 100o C water.
Drum
Setting
Confine heat to boiler and form a
passage for gases
Grate
CI, above which fluid is burned
Furnace
Above grate and below boiler shell
Burning happens here
Flue gas
Hot mix of products of combustion
Flue
Passage for flue gas
Stocker
regulates fuel usage
Water wall
Water space
Volume occupied by water
Steam space
Shell except water tubes
Feed water
Water supplied to the boiler
Economizer
Water supplied to the boiler is
preheated using the waste hot gases
before reaching chimney.
Placed before Air preheater
Air Preheater
fresh air going to furnace is heated
from hot waste water
Super heater
Heats the saturated steam
Above furnace
No change in pressure
Safety device that is used to protect
the boiler when the water level falls
below a minimum level
Flow rate of saturated water in down comers
Circulation ratio =
Flow rate of steam released from drum
Fusible plug
Flue gas
Boiler furnace → Super heater → Reheater → Economizer → Air preheater → Electrostatic precipitators → induced draft fans → Chimney
Water
Economizer → Boiler drum → Water walls → Boiler drum → Super heater → Turbine
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12.20. BOILER
Lancashire
Horizontal double fire tube
Cornish
Horizontal single fire tube
La-Mont
High pressure water tube
Cochran
Vertical multiple fire tube
Babcock and Wilcox
Horizontal Water tube
Benson
High pressure boiler
Once through flow
Stirling
Bent tube, water tube
Locomotive
Fire tube boiler
[181]
Chapter 13
Renewable sources of Energy
Solar Energy
Wind Energy
Biomass Energy
Geothermal (0.05 W/m2 )
Tidal energy
Ocean thermal energy conversion
Advantages
Good for environment
Unlimited supply
Cheaper
Disadvantages
Dilute form of energy
Depends on whether and location
13.1
Solar energy
1 Langley=1cal/cm2 = 1.163 ×10−2 kWh/m2
182
www.eggsam.com
13.1. SOLAR ENERGY
[183]
θ
Angle of incidence
Angle between incident beam and normal to the plane
θz
Zenith angle
Angle between the beam and normal to the horizontal plane
αa
Solar altitude angle
90o − θz
φ
Latitude
-90o to 90o
+ve for Northern hemisphere.
δ
Declination
Angle made by the line connecting the centers of earth and sun
with its projection on the equatorial plane of earth.
-23.45o to 23.45o (South-North)
γ
Azimuth angle
Angle between the projection of the normal to the surface on a
horizontal plane and meridian (longitude)
-180o to 180o (east-west)
γs
Solar azimuth angle
Angle between projection of solar beam on the horizontal and
meridian
Hour angle
Angular measurement of time
Angle of rotation of earth since solar noon
-180o to 180o (Morning-evening)
β
Slope of the plane
Angle between collector plate and horizontal
0o − 180o
(0-towards equator-90-Away from equator-180)
Vertical surface, β = 90o
Ψ
Longitude
Measured from Greenwich
Positive eastward
ω
cos θ = sin φ(sin δ cos β +cos δ cos γ cos ω sin β)+cos φ(cos δ cos ω cos β −sin δ cos γ sin β)+
cos δ sin γ sin ω sin β
Solar constant ISC = 1367W/m2
n: day ofthe year
360
In = ISC 1 + 0.033 cos
n
365 360
Declination angle δ = 23.45o sin
× (284 + n) (Cooper’s relation)
365
On March 21 ans september 21, δ ≈ 0
Hour angle ω = 15(tsolar − 12)
ω = 15(tzone − 12) + (Ψ − Ψzone ) + ωeq
Local Apparent Time: = Standard Time + 4(Ψlocal − ΨStandardtime ) + ωeq
1
Air mass (m)=
cos θz
θz : Zenith angle: Angle between the beam and normal to the horizontal plane
www.eggsam.com
13.1. SOLAR ENERGY
Spectral intensity Iλ (λ, T ) =
λ5
2hC 2
hC
−1
exp
λkT
[184]
Plank’s law
k: Universal Boltzman constant
C: Speed of light
h: Plank’s const
Spectral emissive power of black body Eb = πIλ (λ, T )
For sun rise and sun set, ωs = cos−1 (− tan φ tan δ)
Perihelion: Closest to the sun
Aphelion: Maximum distance from the sun (June 21)
Irradiance: Total rate of radiant energy incident on a unit area of surface
Beam radiation (Ib ): Solar radiation received directly from sun. Also called direct
radiation
Diffuse radiation (Id ): Radiation reaching a surface after scattering in the atmosphere.
Also called indirect radiation
Global radiation: Ib + Id
Pyranometer: Measures global radiation
Pyrheliometer: Measures direct/beam radiation
Albedo meter: Reflected radiation
Spring equinox: March 21
Longest day: June 21
Autumn equinox: September 21
Shortest day: December 21
13.1.1
Solar Thermal Energy
cos θ
ratio of beam falling on the tilted surface to horizontal surface
cos θz
1 + cos β
Tilt factor for diffused radiation rd =
2
1 − cos β
Tilt factor for reflected radiation rr =
2
Tilt factor:=
Components of solar thermal
Selective coating: Applied on the absorber plate for high absorbility in short wave region (solar radiation) and low emissivity in long wave region (Re-radiation from absorber
plate) (Black Nickel, Black copper, Black chrome, Commercial coating)
Absorber plate:Absorbs solar radiation and transfers heat to the fluid (Cu, Al, Brass,
steel, silver)
Riser tube: To absorb maximum heat from from absorber plate and transfer it to the fluid.
www.eggsam.com
13.1. SOLAR ENERGY
[185]
Transparent cover: To reduce re-radiation. Uses glazed glass. High transmissivity for
for short waves and high reflectivity for long waves.
(τ α)net =
τα
1 − (1 − α)ρ
Useful energy gien to the fluid
Total energy reaching the collector
Effective area of apperture
Concentration ratio =
Area of collector
Collector efficiency=
Adding more layers of covers reduces τ α product. Reduces heat loss through convection. Reduces efficiency.
Flat plate collector
No optical concentration
No need for solar tracing
Commonly used for water heating
Concentration ratio = 1
Max Temperature ≈100o C
Parabolic collectors
Line concentrating
Tracing mechanism is necessary (In one axis)
Concentration ratio ≈ 100
Up to 300o C
Central tower collector
Dual axis solar tracking
Point concentrating
Maximum temperature ≈ 600o C
Concentration ratio ≈ 1000
Paraboloid dish collector
Dual axis solar tracking
Point concentrating
Maximum temperature ≈ 900o C
Concentration ratio ≈ 10000
13.1.2
Solar thermal energy storage
Sensible heat storage device
No phase change
Latent heat storage
Heating with phase change
Hydrated salt: N a2 SO4 .10H2 O *
) N a2 SO4 + 10H2 O
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13.2. WIND ENERGY
Thermo-Chemical storage
Endothermic and exothermic reactions to store and extract energy
13.2
Wind energy
Indirect solar power
Wind velocity ∝ H 1/7
1
Power in wind E = ρAV 3
2
E
Power density =
A
1
Bernaulli’s equation P1 − P2 = ρ [V12 − V22 ]
2
1
V1 + V2 2
Power produced = ρAturbine
[V1 − V22 ]
2
2
V1
For maximum power, V2 =
3
Maximum efficiency = 59.3% (Lanchester-Blitz limit)
1
16
Maximum power Emax = ρAturbine V13
2
27
2ρ 2
Maximum horizontal force Fx = πD V1
9
Maximum torque =Fx R
Power extracted
Power available in the wind
Lift
Lift coefficient CL =
Wind force
Drag
Drag coefficient CD =
Wind force
ωR
Tip speed ratio λ =
V1
Ablade
Solidity γ =
Aswept
Power coefficient Cp =
Cut-in speed: Minimum speed when power production starts
cut-off/cut-out speed: Speed at which power production is stopped
Solidity ratio: Ratio of blade area to rotor circumference
Horizontal axis machine
Vertical axis machine
Cup anemometer
Savonius rotor
Darrieus rotor
13.3
Biomass energy
Biogas
Cx Hy Oz →Cx Hy + O2
[186]
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13.4. TIDAL POWER
Anaerobic fermentation
1 kg dry cattle dung ≈ 1m3 of biogas
1 kg fresh cattle poop ≈ 0.9Litre
1 kg fresh cattle dung 8% biodegradable dry matter
1 kg fresh cattle dung requires same volume of water
Retention time ≈ 40 days
Thermo-Chemical Method
Biomass Gasification
Drying → Pyrolysis → Oxidation → Reduction
Produces Producer gas: CO2 , CO, N2 , H2
Partial combustion of biomass
Bio-ethanol and Bio-diesel
Produced through fermentation of certain biomass.
13.4
Tidal power
Tidal range (R): Difference between consecutive high and low tides
Ebb: Low tide
M
R∝ 3
D
M: Mass of the body causing tide
D: Distance to the body causing tide
Spring tides: Earth, sun and moon are in a line.
Neap tide: Earth, sun and moon are in perpendicular lines.
Rspinrg > Rneap
Tidal power generated =
1
ρgAR2
2
Single basin Single action tidal power plant
Single basin double action tidal power plant
Multi basin tidal power plant
13.5
Fuel cells
-Direct conversion to electrical energy
-No moving parts
-No vibration or sound
Fuel-Electrode-Electrolyte-Oxidant
[187]
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13.5. FUEL CELLS
Hydrogen-Oxygen FC
Alkaline FC (AFC)
Hydrogen-air FC
Phosphoric acid FC (PAFC)
Hydrazine-oxygen FC
Polymer electrolytic membrane FC (PEMFC)
Hydrocarbon-air FC
Molten carbonate FC (MCFC)
Synthesis gas-air FC
Solid oxide FC (SOFC)
[188]
Ammonia-air FC
Fuel cells are classified based on fuel/oxidant, electrolyte, working temperature HydrogenOxygen fuel cell
Fuel: Hydrogen (At negative electrode, Anode)
Oxygen (At positive electrode, Cathode)
At anode: H2 → 2H + + 2e−
1
At cathode: 2H + + 2e− + O2 → H2 O
2
Electrolyte
Conductive to ions and non-conductive to electricity
Should not get charged
Electrode
Good conductor of electricity
Should not corrode on contact with electrolyte
Stable at high temperatures
Chapter 14
Engineering Mathematics
Important
I personally guarantee 100% failure for anyone who blindly memorizes equations from
this chapter. Never use this chapter for first time study. This is solely for the purpose of
quick revision.
14.1
Linear Algebra
A,B,C,..: Matrices
X,Y,Z: Vectors
k: Scalar Constant
m: number of rows
n: number of columns
ρ: Rank of the matrix
O: Null matrix
I: Unit matrix
n: Order of a square matrix (m=n)
m×n matrix ⇒ m rows and n columns
k(A + B) = kA + kB
(k + l)A = kA + lA
189
www.eggsam.com
Row matrix
Column matrix
Square matrix
Singular matrix
Symmetric matrix
Skew symmetric matrix
14.1. LINEAR ALGEBRA
Matrix with just one row (1×n)
Matrix with just one column (m×1)
m=n
Determinant =0
AT =A
AT =-A
Orthogonal matrix
AAT =AT A=I
Hermitian matrix
A = A¯T = Aθ
Skew-hermitian matrix A = −A¯T = −Aθ
Unitary matrix
AAθ =A
Periodic matrix
Ap+1 =A
p: Period of A (When n is the smallest such number)
Diagonal matrix
All elements except diagonal elements are zero
Diagonal matrix is both upper triangular and lower
triangular matrix
Scalar matrix
A diagonal matrix with all diagonal elements same is
called scalar matrix
Identity matrix -Also called unit matrix
-Diagonal matrix with all elements one.
Upper triangular matrix Square matrix with all the elements below principle
diagonal equal to zero.
Lower triangular matrix Square matrix with all the elements above principle
diagonal equal to zero.
Strictly triangular matrix
Nilpotent matrix
Idempotent matrix
Upper or lower triangular matrix with the diagonal
elements zero
Aq =0
q is a positive integer
When ’q’ is the such smallest number, it is called the
index of A
A2 =A
If A is idempotent, then I-A is also Idempotent
[190]
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14.1.1
14.1. LINEAR ALGEBRA
Matrix addition
i.
A+B=B+A
Commutative
ii.
A+(B+C)=(A+B)+C
Associative
iii.
A+O=O+A=A
Additive identity (O=null matrix)
iv.
A+(-A)=(-A)+A=O
Additive inverse
14.1.2
Matrix multiplication
i.
AB6=BA
Not Commutative
ii.
A(BC)=(AB)C
Associative
iii.
A(B+C)=AB+AC
Distributive over addition
Trace of a matrix
Trace=Sum of diagonal elements
tr(A+B)=tr(A)+tr(B)
tr(A-B)=tr(A)-tr(B)
tr(AB)=tr(BA)
tr(kA)=ktr(A)
tr(AB)6=tr(A)tr(B)
Transpose of a matrix
(AT )T =A
(A+B)T =AT +BT
(A-B)T =AT -BT
(AB)T =BT AT
(kA)T =kAT
Symmetric matrix
AT =A
If A and B are symmetric matrices
A+B
Symmetric
A-B
Symmetric
AB or BA
Need not be symmetric
AB+BA
Symmetric
AB-BA
Skew-symmetric
An
Symmetric (n is a natural number)
kA
Symmetric
[191]
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14.1. LINEAR ALGEBRA
Skew symmetric matrix
AT =-A
If A and B are skew-symmetric matrices
A+B
Skew-Symmetric
A-B
Skew-Symmetric
AB or BA
Need not be Skew-symmetric
A2 , A4 ,...,A2N
symmetric
A,A3 ,A5 ,...,A2N −1
Skew-Symmetric (N is a natural number)
kA
Skew-Symmetric
Let A be any square matrix
A+AT is symmetric
A-AT is Skew-symmetric
AAT and AT A are symmetric
Orthogonal matrix
AAT =AT A=I
A=A−1
If A and B are orthogonal matrices, then AB and BA are orthogonal
|A| = ±1
Conjugate matrices
All the elements are replaced by their conjugate
Conjugate(A) = Ā
Ā¯ = A
(A ± B) = Ā ± B̄
Transposed conjugate of a matrix
Aθ = (Ā)T
(Aθ )θ = A
(A + B)θ = Aθ + B θ
(zA)θ = z̄Aθ (z: complex number)
(AB)θ = B θ Aθ
14.1.3
Determinants
1. |A|=determinant of A
2. |AB|=|A||B|
3. |Ak | = |A|k
4. If |A| =
6 0, then |A−1 |=
1
|A|
[192]
www.eggsam.com
14.1. LINEAR ALGEBRA
[193]
5. |AT |=|A|
6. |k A | = k n |A|
7. If any two columns or rows are proportional, then |A|=0
8. If all the elements of any row or column are zero, then |A|=0
9. If any two rows or columns are interchanged, then the sign of |A| changes
10. If all the elements of a row or a column is multiplied by k, then the determinant
will become k times.
11. Adding a row/column to another after multiplying with a constant will not change
the value of the determinant
12. Determinant of triangular matrix is product of its diagonal elements
13. Determinant of an orthogonal matrix is ±1
14. Determinant of a hermitian matrix is real number
15. Determinant of an idempotent matrix is 0 or 1
16. Determinant of s skew-symmetric matrix of odd order=0
14.1.4
Inverse
Minor: Aij
Cofactor:Cij = (−1)i+j Aij
Cofactor matrix: Matrix of cofactor elements
Adjoint matrix: Transpose of cofactor matrix
1. A Adj(A) = Adj(A) A = |A|In
2. Adj(O) = O
3. Adj(I) = I
4. Adj(AT ) = (Adj (A))T
5. Adj(AB) = Adj(B)Adj(A)
6. If |A| = 0 then |Adj(A)| = 0
7. |Adj(A)| = |A|n−1
8. Adj(Adj(A)) = |A|n−2 A
2
9. |Adj(Adj(A))| = |A|(n−1)
10. Adj(Aθ ) = (Adj(A))θ
11. If A is diagonal then Adj(A) is also diagonal
12. If A is symmetric then Adj(A) is also symmetric
13. If A is hermitian then Adj(A) is also hermitian
www.eggsam.com
14.1. LINEAR ALGEBRA
[194]
Inverse of a matrix
1
Adj(A)
A−1 =
|A|
1. Inverse is unique
2. (AB)−1 =B−1 A−1
3. (AT )−1 =(A−1 )T
4. If A is symmetric, A−1 is also symmetric
5. If A is orthogonal, A−1 is also orthogonal
6. (kA)−1 =
1 −1
A
k
7. If |A|6= 0, then |A−1 |=
1
|A|
Rank of a matrix(ρ)
Order of the highest singular minor of a matrix
1. ρ(O)=0
2. ρ(In )=n
3. If |An×n |=0, then ρ(A)<n
4. If |An×n | =
6 0, then ρ(A)=n
5. ρ(AT ) = ρ(A)
6. ρ(AB) ≤ min{ρ(A),ρ(B)}
7. ρ(A+B) ≤ ρ(A)+ρ(B)
8. If ρ(A) = n, then ρ(Adj A) = n
9. If ρ(A) = n − 1, then ρ(Adj A) = 1
10. If ρ(A) = n − 2, then ρ(Adj A) = 0
Elementary matrix
Matrix obtained by doing one row or column operation on a unit matrix
Row Echelon form
Number of zeros before the first non-zero element in a row is less than that of the next
row
Zero rows (if any) will follow non-zero rows
Number of non-zero rows in row echelon form is equal to rank
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14.1.5
14.1. LINEAR ALGEBRA
[195]
Vector
Linearly dependent vectors
If one vector can be expressed as a scalar multiplication of another vector, then those
two vectors are linearly dependent. Else they are linearly independent
Inner Product
X · Y = XY T = Y T X
If X · Y = ±1, X and Y are parallel
If X · Y = 0, X and Y are orthogonal
Length√or norm of a vector
||X|| = X · X
Orthogonal vectors
If X · Y = 0, X and Y are perpendicular
Orthogonal set
n vectors of order n that are perpendicular to one another
Orthonormal vectors/Orthonormal sets
Xi XjT = 0 for all the vectors
||Xi || = 1 for all the vectors
Orthogonal matrix
All columns of the matrix are perpendicular to each other
14.1.6
Linear Equations
Linear non-homogeneous equations
AX=B
Augmented Matrix
[A|B]
Consistent system
Consistent of the system has at least one solution
A and [A|B] has same rank
ρ(A) = ρ([A|B]) = no of variable =⇒ Unique solution
ρ(A) = ρ([A|B]) < no of variable =⇒ infinite solution
Indeterminate system
System has more than one solution
Inconsistent system
A and [A|B] does not has same rank
No solution
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14.1. LINEAR ALGEBRA
[196]
Free variables
Also called nullity, dimension of null space, number of independent variables or dimension
of space of solution
Nullity = Total number of variables - rank
Linear homogeneous equations
Always consistent
AX=O
X=O, is always a solution called trivial solution
Number of independent solutions = m-ρ
Here m is the number of rows of A
If A is singular, then there will be only trivial solution
14.1.7
Eigen values and Eigen vectors
Characteristic polynomial: A-λI
Characteristic equation: |A-λI|=0
Eigen values: Roots of |A-λI|=0
Eigen vector: (A-λI)X=0
Algebraic multiplicity: Number of times a particular eigen value is repeated
1. Sum of eigen values = Trace of the matrix.
2. Product of eigen values = Determinant of the matrix.
3. If A is singular, then at least one eigen value must be zero.
4. If A is non-singular, then all the eigen values must be non-zero.
5. If λ is the eigen value of A, then
(a) λm is eigen value of Am
(b) kλ is eigen value of matrix kA
(c) λ+k is eigen value of A+kI
6. If A is a real matrix, then complex eigen values appear in conjugate pairs
7. If A is real symmetric or Hermitian, then λ is always real
8. If A is real skew symmetric or skew hermitian, then λ is zero or purely imaginary
9. Eigen vectors of A and Am are same
10. Eigen vectors of A and kA are same
11. Eigen vectors of A and kAm +kA are same
12. Eigen values of A = Eigen values of AT
13. Eigen vectors of A and AT are NOT same
Clayley-Hamilton theorem
Every square matrix satisfies its own characteristic equation
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14.2
14.2. CALCULUS
Calculus
Even function
f(-x)=f(x)
Odd function
f(-x)=-f(x)
Closed interval
[a, b] =⇒ a ≤ x ≤ b
Open interval
(a, b) =⇒ a < x < b
14.2.1
Continuity
If f (x) and g(x) are two continuous functions, then
f + g is continuous
f − g is continuous
f × g is continuous
f
, g 6= 0 is continuous
g
14.2.2
Differentiation
f (x) − f (a)
x→a
x−a
f (x + h, y) − f (x, y)
∂u
= Lt
∂x h→0
h
∂X
Convention used: Xu =
∂u
f 0 (a) = Lt
Homogeneous function
-Degree of each term in the function is same.
Extreme point
Point at which maximum or minimum appears in a function
At extreme point, f 0 (x) = 0
At maximum, f 0 (x) = 0, f 00 (x) < 0
At minimum, f 0 (x) = 0, f 00 (x) > 0
Point of inflection
A stationary point at which neither maximum nor minimum happens.
Curve changes from concave to convex or reverse
f 0 (x) = 0
f 00 (x) = 0
f 000 (x) 6= 0
[197]
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14.2. CALCULUS
Extreme points for functions with two variables
∂z
∂z
, q=
∂x
∂y
∂ 2z
∂ 2z
∂ 2z
, t= 2
r= 2 , s=
∂ x
∂x∂y
∂ y
At stationary points, p=q=0
If rt − s2 > 0 and r > 0, then relative minimum
If rt − s2 > 0 and r < 0, then relative maximum
If rt − s2 < 0, then saddle point
If rt − s2 = 0 No conclusion
d
c=0
dx
d
x=1
dx
d n
x = nxn−1
dx
d √
1 1
x= √
dx
2 x
d x
x
e =e
dx
d x
a = ax ln a
dx
d
1
ln x =
dx
x
1
d
loga x =
dx
x ln a
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec2 x
dx
d
cot x = −cosec2 x
dx
d
1
sin−1 x = √
dx
1 − x2
−1
d
cos−1 x = √
dx
1 − x2
d
1
tan−1 x =
dx
1 + x2
d
f g = f g0 + f 0g
dx
d
f 0g − g0f
f /g =
dx
g2
p=
Roll’s theorem
f : [a, b] → R is a continuous derivable function
a<b
f (a) = f (b)
Then, there exists at least one ’c’ such that,
f 0 (c) = 0
[198]
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14.2. CALCULUS
[199]
Lagrange’s Mean Value theorem
f : [a, b] → R is a continuous derivable function
a<b
Then, there exists at least one ’c’ such that,
f (b) − f (a)
f 0 (c) =
b−a
Cauchy’s mean value theorem
f : [a, b] → R is a continuous derivable function
g : [a, b] → R is a continuous derivable function which is not zero inside the interval
a<b
Then, there exists at least one ’c’ such that,
f (b) − f (a)
f 0 (c)
=
0
g (c)
g(b) − g(a)
14.2.3
Taylor’s series
(x − x0 ) 1 0
(x − x0 ) 2 00
f (x)|x=a = f (x0 ) +
f (x0 ) +
f (x0 ) + ...
1!
2!
n
∞
P (x − x0 ) N
f (x) =
f (x0 )
n!
n=0
Maclaurin’s series
Taylor series expansion about origin
z0 = 0
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14.2. CALCULUS
Important expansions
x1 x2 x3
+
+
+ ...
1!
2!
3!
x 3 x5 x7
sin x = x −
+
−
+ ...
3!
5!
7!
x2 x4 x6
cos x = 1 −
+
−
+ ...
2!
4!
6!
x5
x3
+ 2 + ...
|x| < π/2
tan x = x +
3
15
1 x 7x3
cosecx = + +
+ ...
x 6 360
x2 5x4
+
+ ...
sec x = 1 +
2
24
1 x x3
cot x = − −
+ ...
x 3 45
x3 x5
sinh x = x +
+
+ ...
3!
5!
x2 x4
+
+ ...
cosh x = 1 +
2!
4!
x2 x3 x4
log(1 + x) = x −
+
−
+ ...
|x| < 1
2
3
4
x 2 x3 x4
−
−
− ...
|x| < 1
log(1 − x) = x −
2
3
4
1
= 1 + x + x2 + x3 + x4 + ...
|x| < 1
1−x
1
= 1 + 2x + 3x2 + 4x3 + ...
|x| < 1
(1 − x)2
c,a: Constants
f & g are functions of x
ex = 1 +
[200]
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14.2.4
14.2. CALCULUS
Integration
Integration by parts
R
R
R
R
uv = u v − (u0 v)
R
sin xdx = − cos x + C
R
cos xdx = sin x + C
List
R
Cdx = Cx + C1
R
x2
xdx =
+C
23
R 2
x
+C
x dx =
3
R 1
1
+C
dx
=
−
x2
x
R√
2
xdx = x3/2 + C
3
R 1
√
√ dx = 2 x + C
x
R n
xn+1
x dx =
+C
n+1
R 1
dx = log |x| + C
x
R
(ax + b)n+1
(ax + b)ndx =
a(n + 1)
R
1
1
dx = ln |ax + b| + C
ax + b
a
R
tan xdx = ln | sec x| + C
R
cot xdx = ln | sin x| + C
R
sec xdx = ln | tan x + sec x| + C
R
ln xdx = x ln x − x + C
xn+1
xn+1
xn ln xdx =
ln x −
+C
n+1
(n + 1)2
R
R
ex dx = ex + C
R x
ax
a dx =
+C
ln a
R
sinh xdx = cosh x + C
R
cosh xdx = sinh x + C
R
1
dx = tan−1 x + C
1 + x2
R
1
√
dx = sin−1 x + C
1 − x2
R
a−x
1
1
dx = tan−1
+C
2
2
x +a
a
a+x
Order of selecting the first
ILATE: (Integer, logarithm, Algebra, Trigonometry, Exponential)
s
2
x2
R
dy
Length of a curve =
1+
dx
dx
x1
s
2
θR2
dr
2
Length of a curve =
r +
dθ
dθ
θ1
Jacobian
Xu Xv
J=
Yu Yv
∂X
Xu =
∂u
∂Y
Yv =
∂v
X = f (u, v)
Y = g(u, v)
Euler’s theorem
u = f (x, y)
Homogeneous function of degree n
xux + yuy = nu
x2 uxx + 2xyuxy + y 2 uyy = n(n − 1)u
[201]
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14.3
∇ = î
14.3. VECTOR CALCULUS
Vector Calculus
∂
∂
∂
+ ĵ
+ k̂
∂x
∂y
∂z
∂φ
∂φ
∂φ
+ ĵ
+ k̂
∂x
∂y
∂z
Gradient gives the normal vector to the surface φ(x, y, z) = c, at the given point
Gradient of φ = ∇φ = î
Directional derivative = (∇φ) ·
F̄
|F̄ |
−1
Angle between two surfaces θ = cos
∇φ1 · ∇φ2
|∇φ1 ||∇φ2 |
Divergence = ∇ · F~
î
~ × F~ = ∂
Curl ∇
∂x
Fx
ĵ
∂
∂y
Fy
k̂
∂
∂z
Fz
∇ × (∇φ) = 0
∇ operator on a vector function
~a = xî + y ĵ + z k̂
~a
∇(f (~a)) = f 0 (~a)
|~a|
~a
∇2 (f (~a)) = f 00 (~a) + 2f 0 (~a)
|~a|
Solenoidal vector
∇ · ~a = 0
~a is a solenoidal vector
14.3.1
Vector Integration
Line
integration
R
R
F~ · d~r = Fx dx + Fy dy + Fz dz
Green’s theorem H
R R ∂N
∂M
M dx + N dy =
−
dxdy
∂x
∂y
C
R
Gauss
divergence
HH
R R Rtheorem
~
(F · ~n)dS =
(∇ · F~ )dV
S
V
Closed surface
Stokes
theorem
H
RR
~
F · d~r =
(∇ × F~ ) · ~n dS
C
S
Open surface
[202]
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14.3.2
14.4. PROBABILITY AND STATISTICS
Fourier Series
∞
∞
nπx X
nπx a0 X
f (x) =
+
an cos
+
bn sin
2
L
L
n=1
n=1
1
a0 =
L
an =
bn =
1
L
1
L
c+2L
Z
f (x)dx
c
c+2L
Z
f (x) cos
c
c+2L
Z
f (x) sin
nπx L
nπx L
dx
dx
c
If f (x) is an even function, then b terms become zero
If f (x) is an odd function, then a terms become zero
14.4
Probability and Statistics
φ : Null set
Sample space: Set of all possible outcomes
Mutually exclusive events: A ∩ B = φ
0 ≤ P (A) ≤ 1
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
P (A ∩ B) = P (A)P (B), If A and B are independent events
Conditional Probability
P (A/B) =
P (A ∩ B)
P (B)
Bayes’ theorem
P (Ei )P (A/Ei )
P (Ei /A) = P
P (Ei )P (A/Ei )
i
Expectation
Expectation E(x) =
i
P
xi P (xi ) =
∞
R
xf (x)dx
−∞
E(cx) = cE(x)
E(x + y) = E(x) + E(y)
E(xy) = E(x)E(y) If x and y are independent
[203]
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14.4. PROBABILITY AND STATISTICS
[204]
Variance
Var(x) = E[(X − µ)2 ]
σ 2 = E(x2 ) − [E(x)]2
Var(cx) = c2 Var(x)
Var(x + y) =Var(x)+Var(y)
Var(x − y) =Var(x)+Var(y)
If x and y are independent
If x and y are independent
Binomial distribution
Mean µ = np
√
Standard deviation σ = npq
Variance = σ 2
p: Probability of event occurring (success)
q: Probability of failure
Probability of event occurring n times in N trials =N Cn pn q N −n
Poisson distribution
X: Discrete random variable
x: 0,1,2,...
λx e−λ
P (X = x) =
x!
Mean µ = λ
√
Standard deviation σ = λ
Limiting case of Binomial distribution with very large number of tries and small value of
p
λ = np
Normal distribution
1 x − µ !2
−
1
σ
f (x) = √ e 2
σ 2π
µ: Mean
σ 2 : Variance
Number of trials is infinitely high
Neither p nor q is small
Bell shaped and symmetric about mean
Mean, mode and median are same
Exponential distribution
f (x) = θe−θx for x ≥ 0
1
Mean E(x) =
θ
1
Variance = E(x2 ) − E(x)2 = 2
θ
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14.5. DIFFERENTIAL EQUATIONS
Uniform distribution
Uniform distribution in the interval [α, β]
1
P (x) =
β−α
α+β
Mean =
2
(β − α)2
Variance =
12
x1 + x2 + ... + xn
Mean=
n
Median= Value at the middle of a sample when sorted
Mode= Most frequent sample
Mean - Mode = 3(mean - median)
rP
(x̄ − xi )2
Standard deviation =
n
14.5
Differential Equations
∂M
Mx =
∂x
f, g: Functions
v, x, y, z: Variables
a, b, c, a0 , b0 , c0 , m: Constants
Order
Order of the highest order derivative in the equation is called order
Degree
Degree of the highest order term in the equation is called degree of the equation
The equation should be free from partial powers
14.5.1
First order differential equations
dy
= f (x, y)
dx
M dx + N dy = 0
M and N are functions of x and y
[205]
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14.5. DIFFERENTIAL EQUATIONS
Type
Form
Solution
Variables
separable
f (y)dy = g(x)dx
Integrate both the sides
Homogeneous
Equations
Eqns Reducible
to homogeneous
dy
f (x, y)
=
dx
g(x, y)
f (x, y) & g(x, y) are
homogeneous functions of x and
y with same degree
dy
ax + by + c
= 0
dx
a x + b0 y + c 0
Eqns reducible
to exact form
Regroup the terms and
d(xy) = xdy + ydx
b
a
If 0 6= 0
a
b
Put x= X+h, y=Y+k
aX + bY
dY
= 0
Such that
dX
a X + b0 Y
Now substitute Y = V X and solve
b
1
a
= 0 =
0
a
b
m
Put ax + by = z
a0 x + b0 y = mz
Use variable separable method
M dx + N dy = 0
LHS is exact differential of some
function u(x, y)
My = Nx
R
R M dx (Treating y as constant) +
(Terms of N independent of x)dy = c
M dx + N dy = 0
IF found by inspection
Homogeneous eqn
M dx + N dy = 0
IF=
1
Mx + Ny
Not homogeneous
f1 (xy)ydx + f2 (xy)xdy = 0
IF=
1
Mx − Ny
My − Nx
= f (x)
N
Linear DE
Put y = vx
dv
dy
=v+x
dx
dx
Separate variables v and x and solve
If b = −a0
If
Exact DE
[206]
Nx − My
= f (y)
M
dy
+ Py = Q
dx
dy
+ f (y)P (x) = Q(x)
dx
Reducible to
Linear form
f 0 (y)
Bernoulli’s Eqn
dy
+ P y = Qy n
dx
IF= ef (x)dx
IF= ef (y)dx
R
IF= e PRdx
y(IF)= Q(IF)dx + c
Put f (x) = z
Put y 1−n = z
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14.5.2
14.5. DIFFERENTIAL EQUATIONS
[207]
Higher order Linear DE
Constant coefficients
dn y
dn−1 y
dn−2 y
+
f
+
f
+ ... + fn y = X(x)
1
2
dxn
dxn−1
dxn−2
dn y
Put n = Dn y
dx
f (D)y = X(x)
Auxiliary equation f (m) = 0
m1 , m2 , ... are the solutions
Find Complimentary function (CF)
m1 , m2 , ... are real and distinct
CF= c1 em1 x + c2 em2 x + c3 em3 x + ...
Two roots are same
CF= (c1 + c2 x)em1 x + c3 em3 x + c4 em4 x + ...
n roots are same
CF= (c1 + c2 x + c3 x2 ... + cn xn−1 )em1 x + cn+1 emn+1 x + ...
Complex roots (a + ib, a − ib, m3 , ...) CF= eax (c1 cos bx + c2 sin bx) + c3 em3 x + ...
Two complex roots are same
Find Particular integral
X
PI=yp =
f (D)
Equation type
X = eax
X = sin(ax + b)
X = xn
n: natural number
X = eax V
X = xV
X = f (x)
CF= eax ((c1 + c2 x) cos bx + (c3 + c4 x) sin bx) + c5 em5 x + ...
Solution method
eax
if f (a) 6= 0
yp =
f (a)
ax
xe
yp = 0
if f (a) = 0, f 0 (a) 6= 0
f (a)
x2 eax
yp = 00
if f (a), f 0 (a) = 0, f 00 (a) 6= 0
f (a)
so on...
2
Replace D2 with
−a , (Denominator not zero) Else,
1
yp = x 0
sin(ax + b) and replace D2 with −a2
f (D)
If denominator again zero, then take f 00 (D) and repeat
yp = [f (D)]−1 X n
Expand [f (D)]−1 using binomial theorem and integrate
1
ax
yp = e
V
f
(D
+
a)
0
1
f (D)
yp = x
V −
V
f (D) (f (D))2
1
Resolve
into partial fractions
f (D)
R
1
X = eax Xe−ax dx
D−a
Final solution y=CF+PI
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14.5.3
14.5. DIFFERENTIAL EQUATIONS
[208]
Method of variation of parameters
y 00 + py 0 + qy = X
p, q, X, y1 , y2 : Functions of x
CF=C1 y1 + C2 y2
R y1 X
R y2 X
dx + y2
dx
PI=−y1
W
W
y1 y2
Wronskian W =
y10 y20
14.5.4
Euler-Cauchy’s homogeneous linear equation
n−1
n−2
dn y
y
y
n−2 d
n−1 d
+
k
x
+
xk
+ ... + kn y = X
1
2
n
n−1
n−2
dx
dx
dx
Put x = ez
xn
z = ln x
d
D=
dz
dy
= Dy
x
dx2
dy
x2 2 = D(D − 1)y
dx
3
3d y
x
= D(D − 1)(D − 2)y
dx3
14.5.5
Partial differential equation
∂ 2u
∂ 2u
∂ 2u
∂u
∂u
A 2 +B
+C 2 +D
+E
+ Fu = G
∂x
∂x∂y
∂y
∂x
∂y
Elliptic
B 2 − 4AC < 0
Parabolic
B 2 − 4AC = 0
Hyperbolic
B 2 − 4AC > 0
Important: Many books use ’2B’ instead of ’B’ in the PDE. In such cases, the conditions will change to B 2 − AC. Do not get confused.
One dimensional wave equation
2
∂ 2y
2∂ y
=
c
∂t2
∂x2
One dimensional heat equation
∂T
∂ 2T
= c2 2
∂t
∂x
Law of natural growth
dx
= kx
dt
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14.6
14.6. LAPLACE TRANSFORM
Laplace transform
F (s) : L(f (t))
G(s) : L(g(t))
(
0 t<a
Unit step function: u(t − a) =
1 t>a
∞
R
F (s) = f (t)e−st dt
0
L[δ] = 1
Unit impulse function
1
L[1] =
s
1
Unit Step function
L[u(t)] =
s
L[δ(t − a)] = e−as
Dirac Delta function
1
L[u(t − a)] = e−as
Delayed unit step function
s
1
L[t] = 2
s
n!
L[tn ] = n+1
s
1
L[eat ] =
s−a
n!
L[tn eat ] =
(s − a)n+1
a
L[sin at] = 2
s + a2
s
L[cos at] = 2
s + a2
a
L[sinh at] = 2
s − a2
s
L[cosh at] = 2
s√− a2
√
π
L[ t] = 3/2
2s
√
1 · 3 · 5 · · · ·(2n − 1) π
n−1/2
L[t
]=
2n sn+1/2
Linearity
L[c1 f (f ) + c2 g(t)] = c1 F (s) + G(s)
First Shifting theorem
L[eat f (t)] = F (s − a)
Second Shifting
L[u(t − a) f (t − a)] = e−at F (s)
Change of scale
1 s
L[f (at)] = F
a
a
[209]
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14.6. LAPLACE TRANSFORM
Differentiation
n
d
L
f (t) = sn F (s) − [sn−1 f (0) + sn−2 f 0 (0) + sn−3 f 00 (0) + ... + f N −1 (0)]
dtn
Integration
R R R
F (s) f N −1 (0) f N −2 (0)
f 0 (0)
+
+
...
+
L
... f (t)dtn = n +
s
sn−1
sn−2
s
Initial value theorem
lim f (t) = lim sF (s)
t→0
t→∞
Final value theorem
lim f (t) = lim sF (s)
t→∞
t→0
14.6.1
Inverse Laplace transformation
L−1 [L(f (t))] = f (t)
L−1 [1] = δ
−1 1
=1
L
s
L−1 e−as = δ(t − a)
1
−1
L
=t
s2
tn−1
1
−1
L
=
sn
(n − 1)!
√ √
π
t
L−1
=
2s3/2
1
L−1
= eat
s−a
a
−1
L
= sin at
s 2 + a2
s
−1
= cos at
L
s 2 + a2
a
−1
L
= sinh at
s 2 − a2
s
−1
L
= cosh at
s 2 − a2
Dirac Delta function
Linearity
L−1 [aF (s) + bG(s)] = aL−1 [F (s)] + bL−1 [G(s)]
Change of scale
1
L−1 [F (as)] = L−1 [F (s)]t→t/a
a
[210]
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14.7. COMPLEX VARIABLES
First Shifting
L−1 [F (s − a)] = eat L−1 [F (s)]
Second shifting
L−1 [e−at F (s)] = f (t − a)(u − a)
Multiplication by sn
dn
L−1 [sn F (s)] = n L−1 [F (s)]
dt
Assumption f (0) = f 0 (0) = f 00 (0) = ... = 0
n
Division
by
st t
RR
F (s)
...L−1 [F (s)]dtn
L−1
=
sn
0 0
Differentiation
n
of L−1
d
L−1
(F (s)) = (−1)n tn f (t)
dsn
−1
Integration
on
L
∞
R
f (t)
L−1 F (s)ds =
t
s
14.7
Complex variables
√
i = −1
z = x + iy
Conjugate of z = z̄ = x − p
iy
Modulus of z = r = |z| = x2 + y 2
y
Amplitude θ = tan−1
x
Polar form: z = reiθ
Equality of complex numbers
z1 = z2
Iff, x1 = x2 , y1 = y2
Or, r1 = r2 , θ1 = θ2
Addition of complex numbers
z1 + z2 = (x1 + x2 ) + i(y1 + y2 )
Multiplication
z1 z2 = (x1 x2 − y1 y2 ) + i(x1 y2 + x2 y1 ) = r1 r2 ei(θ1 +θ2 )
Division
z1
r1
= ei(θ1 −θ2 )
z2
r2
Properties
[211]
www.eggsam.com
14.7. COMPLEX VARIABLES
[212]
|z| ≥ 0
z1
|z1 |
=
z2
|z2 |
|z1 z2 | = |z1 ||z2 |
|z1 + z2 | ≤ |z1 | + |z2 |
|z1 − z2 | ≤ |z1 | + |z2 |
|z1 + z2 | ≥ |z1 | − |z2 |
|z1 − z2 | ≥ |z1 | − |z2 |
z z̄ = |z|2
sin iz = i sinh z
cos iz = cosh z
tan iz = i tan hz
cosh iz = cos z
sinh iz = i sin z
tanh iz = i tan z
14.7.1
Analytic function
If a function is differentiable at a point and its neighborhood, then it is analytic at at
that point
Cauchy-Riemann Equations
Necessary condition
f (z) = u(x, y) + iv(x, y)
f (z) is analytical if
∂u
∂v
=
∂x
∂y
∂u
∂v
=−
∂y
∂x
In polar coordinates,
1 ∂v
∂u
=
∂r
r ∂θ
∂u
∂v
= −r
∂θ
∂r
Singularity
The point where the function is not analytic is called singularity
Pole of a function
The point at which the function becomes infinite
Zero of a function
The point at which the value of the function becomes zero.
www.eggsam.com
14.7. COMPLEX VARIABLES
[213]
Entire function
A function that is analytic in the entire complex plane.
Argand plane: Complex plane
Harmonic function
Functions that satisfy Laplace equation
∇2 u = 0
∇2 v = 0
Residue
Res(f (z)z=z0 ) = lim
z→z0
14.7.2
1
dm−1
((z − z0 )m · f (z))
(m − 1)! dz m−1
Complex integration
Z
Z
f (z)dz =
(u + iv)(dx + idy)
Z
Z
= (udx − vdy) + i (vdx + udy)
14.7.3
Cauchy’s integral theorem
If a function f (z) is analytic and f 0 (z) is continuous everywhere inside and on a closed
curve
C, then,
H
f (z) dz = 0
C
14.7.4
Cauchy’s Integral formula
φ(z)
(z − z0 )n
H
2πi
Then f (z) dz =
φN −1 (z0 )
(n
−
1)!
C
φ(z)
f (z) =
(z − z0 )
H
f (z) dz = 2πiφ(z0 )
Let, f (z) =
For counterclockwise
C
φN −1 : N-1
th
14.7.5
Power series
∞
P
derivative of φ
an (z − z0 )n = a0 + a1 (z − z0 )1 + a2 (z − z0 )2 + ...
n=0
Convergence of a power series
1
Radius of convergence r =
Lt |an |1/n
n→∞
www.eggsam.com
14.7. COMPLEX VARIABLES
[214]
Circle of convergence |z − z0 | = r
Region of convergence |z − z0 | < r
14.7.6
Taylor’s series
(z − z0 ) 2 00
(z − z0 ) 1 0
f (z0 ) +
f (z0 ) + ...
f (z) = f (z0 ) +
1!
2!
n
∞ (z − z )
P
0
f (z) =
f N (z0 )
n!
n=0
Maclaurin’s series
Taylor series expansion about origin
z0 = 0
14.7.7
f (z) =
∞
X
Laurent’s series
n
an (z − z0 ) +
n=0
1
an =
2πi
Z
1
2πi
Z
∞
X
bn
(z − z0 )n
n=0
f (z)dz
, n = 0, 1, 2, ...
(z − z0 )n+1
C1
bn =
f (z)dz
, n = 1, 2, 3, ...
(z − z0 )−n+1
C2
Part with a terms are called analytical parts
Part with b terms are called principal parts
Removable singular point: z or power of z does not occur in the denominator of the
expansion
Essential singular point: z or power of z occur infinite times in the denominator of the
expansion
Pole: z or power of z occur finite times in the denominator of the expansion
www.eggsam.com
14.7.8
14.8. NUMERICAL METHODS
[215]
Important expansions
z1 z2 z3
+
+
+ ...
1!
2!
3!
z3 z5 z7
sin z = z −
+
−
+ ...
3!
5!
7!
z2 z4 z6
+
−
+ ...
cos z = 1 −
2!
4!
6!
z3
z5
tan z = z +
+ 2 + ...
|z| < π/2
3
15
1 z 7z 3
+ ...
cosecz = + +
z 6 360
z 2 5z 4
sec z = 1 +
+
+ ...
2
24
z3
1 z
+ ...
cot z = − −
z 3 45
z3 z5
sinh z = z +
+
+ ...
3!
5!
z2 z4
+
+ ...
cosh z = 1 +
2!
4!
z2 z3 z4
log(1 + z) = z −
+
−
+ ...
|z| < 1
2
3
4
z2 z3 z4
log(1 − z) = z −
−
−
− ...
|z| < 1
2
3
4
1
= 1 + z + z 2 + z 3 + z 4 + ...
|z| < 1
1−z
1
= 1 + 2z + 3z 2 + 4z 3 + ...
|z| < 1
(1 − z)2
ez = 1 +
14.8
Numerical methods
Intermediate value theorem
If a continuous function has values of opposite sign inside an interval, then it has a root
in that interval (Bolzano’s theorem).
Bisection method
Only real roots can be found
If the root of an equation lies in the interval (a,b),
a+b
x1 =
2
x1 + b
If f (x1 ) < 0, then x2 =
2
a + x1
If f (x1 ) > 0, then x2 =
2
If f (x1 ) = 0, solved.
b−a
Error e < n
2
www.eggsam.com
14.8. NUMERICAL METHODS
Newtons- Raphson method
f (x) = 0
xn+1 = xn −
f (xn )
f 0 (xn )
Secant method or Modified regula falsi or intermediate value theorem
xn+1 =
xn−1 f (xn ) − xn f (xn−1 )
f (xn ) − f (xn−1 )
14.8.1
Numerical Integration
Trapezoidal rule
h
I = [(y0 + yn ) + 2(y1 + y2 + y3 + ...)]
2
Gives exact value for polynomial of degree 1 or 0
M2 (b − a)3
|E| ≤
12n200
M2 = max(|f (x))|
Error is in the order of h2
Simpson’s 1/3 rule
h
[(y0 + yn ) + 4(y1 + y3 + y5 + ...) + 2(y2 + y4 + y6 + ...)]
3
M4 (b − a)5
|E| ≤
180n4
M2 = max(|f IV (x)|)
Error is in the order of h4
I=
Simpson’s 3/8 rule
3h
[(y0 + yn ) + 3(y1 + y2 + y4 + ...) + 2(y3 + y6 + y9 + ...)]
8
Error is in the order of h5
I=
14.8.2
Numerical solution of differential equations
Euler’s method
dy
= f (x, y)
dx
Initial condition y(x0 ) = y0
Find y(xn )
Step 1. Divide the interval (x0 , xn ) into n equal parts of width h
Step 2. y1 = y0 + hf (x0 , y0 )
x1 = x0 + h
y2 = y1 + hf (x1 , y1 )
y3 = y2 + hf (x2 , y2 )
.
[216]
www.eggsam.com
14.8. NUMERICAL METHODS
.
.
yn = yn−1 + hf (xn−1 , yn−1 )
xn−1 = x0 + (n − 1)h
Stable if 1 + h
∂f
<1
∂y
Heun’s Method, Modified Euler method, R-K second order
dy
= f (x, y)
dx
Initial condition y(x0 ) = y0
y1p = y0 + hf (x0 , y0 )
h
y1c = y0 + [f (x0 , y0 ) + f (x1 , y1p )]
2
h
c
yn = yn−1 + [f (xn−1 , yn−1 ) + f (xn , ynp )]
2
y p : Obtained by Euler’s method
Runge’s method (R-K third order method)
k1 = hf (x0 , y0 )
k2 = hf (x0 + h/2, y0 + k1 /2)
k 0 = hf (x0 + h, y0 + k1 )
k3 = hf (x0 + h, y0 + k 0 )
k1 + 4k2 + k3
k=
6
y1 = y0 + k
Runge-Kutta method (R-K fourth order method)
k1
k2
k3
k4
= hf (x0 , y0 )
= hf (x0 + h/2, y0 + k1 /2)
= hf (x0 + h/2, y0 + k2 /2)
= hf (x0 + h, y0 + k3 )
k1 + 2k2 + 2k3 + k4
k=
6
y1 = y0 + k
Milne-Simpson method
Multistep method
[217]
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14.9
14.9. MISC
Misc
a(rn − 1)
GP
Sum =
r−1
a
Infinite GP Sum=
1−r
n
AP
Sum = [2a + (n − 1)d]
2
[218]
Thank you
.
I would like to thank all of you for your support. I extend my sincere thanks to all my
social media supporters. The constant support from the followers is what motivated me
in making this book a reality.
I especially thank the following people who have helped me in making this equation
book by pointing out errors and mentioning the missing equations.
Anshuman Sarma
Er Yogendra Pratap Singh
Satya Panda
Hemant Sukhija
Raja Radha
Malaya ranjan
Arjun Bajpayee
Deepak Kumar
Somesh Akolkar
3 Anonymous contributors
219