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Mechanical Engineering – GATE Exam
Thermodynamics
Symbol/Formula Parameter
M Molar mass (M/)
m Mass (M)
M
mn
Number of moles ()
E Energy or general extensive property
m
Ee
Specific molar energy (energy per unit mass) or general extensive property per unit mass
eMn
Ee
Specific energy (energy per unit mole) or general extensive property per unit mole
P Pressure (ML-1T-2)
V Volume (L3);
Specific volume or volume per unit mass, v (L3M-1) and the volume
per unit mole v (L3
-1)
T Temperature (Θ)
Density (ML-3); = 1/v.
x Quality
U Thermodynamic internal energy (ML2T-2);
Internal energy per unit mass, u (L2T-2), and the internal energy per
unit mole, u (ML2T-2
-1)
H = U + PV Thermodynamic enthalpy (ML2T-2);
Enthalpy per unit mass, h = u + Pv (dimensions: L2T-2) and the
internal energy per unit mole h (ML2T-2
-1)
S Entropy (ML2T-2Θ-1);
Entropy per unit mass, s(L2T-2Θ-1) and the internal energy per unit
mole s (ML2T-2Θ-1
-1)
W Work (ML2T-2)
Q Heat transfer (ML2T-2)
uW : The useful work rate or mechanical power (ML2T-3)
m : The mass flow rate (MT-1)
Mechanical Engineering – GATE Exam
2
2V
: The kinetic energy per unit mass (L2T-2)
gz: The potential energy per unit mass (L2T-2)
Etot: The total energy = m(u +
2
2V
+ gz) (ML2T-2)
Q : The heat transfer rate (ML2T-3)
dEcvdt
: The rate of change of energy for the control volume.(ml2t-3)
M Molar mass (M/)
m Mass (M)
M
mn
Number of moles ()
E Energy or general extensive property
m
Ee
Specific molar energy (energy per unit mass) or general extensive property per unit mass
eMn
Ee
Specific energy (energy per unit mole) or general extensive property per unit mole
P Pressure (ML-1T-2)
V Volume (L3);
Specific volume or volume per unit mass, v (L3M-1) and the volume
per unit mole v (L3
-1)
T Temperature (Θ)
Density (ML-3); = 1/v.
x Quality
U Thermodynamic internal energy (ML2T-2);
Internal energy per unit mass, u (L2T-2), and the internal energy per
unit mole, u (ML2T-2
-1)
H = U + PV Thermodynamic enthalpy (ML2T-2); we also have the enthalpy per unit mass, h = u + Pv (dimensions: L2T-2) and the internal energy
per unit mole h (ML2T-2
-1)
S Entropy (ML2T-2Θ-1);
Entropy per unit mass, s(L2T-2Θ-1) and the internal energy per unit
mole s (ML2T-2Θ-1
-1)
Mechanical Engineering – GATE Exam
W Work (ML2T-2)
Q Heat transfer (ML2T-2)
uW : The useful work rate or mechanical power (ML2T-3)
m : The mass flow rate (MT-1)
2
2V
: The kinetic energy per unit mass (L2T-2)
gz: The potential energy per unit mass (L2T-2)
Etot: The total energy = m(u +
2
2V
+ gz) (ML2T-2)
Q : The heat transfer rate (ML2T-3)
dEcvdt
: The rate of change of energy for the control volume.(ml2t-3)
M Molar mass (M/)
m Mass (M)
M
mn
Number of moles ()
E Energy or general extensive property
m
Ee
Specific molar energy (energy per unit mass) or general extensive property per unit mass
eMn
Ee
Specific energy (energy per unit mole) or general extensive property per unit mole
P Pressure (ML-1T-2)
V Volume (L3); we also have the specific volume or volume per unit
mass, v (L3M-1) and the volume per unit mole v (L3
-1)
T Temperature (Θ)
Density (ML-3); = 1/v.
x Quality
U Thermodynamic internal energy (ML2T-2); we also have the internal energy per unit mass, u (L2T-2), and the internal energy per unit
mole, u (ML2T-2
-1)
H = U + PV Thermodynamic enthalpy (ML2T-2); we also have the enthalpy per unit mass, h = u + Pv (dimensions: L2T-2) and the internal energy
per unit mole h (ML2T-2
-1)
Mechanical Engineering – GATE Exam
S Entropy (ML2T-2Θ-1); we also have the entropy per unit mass, s(L2T-
2Θ-1) and the internal energy per unit mole s (ML2T-2Θ-1
-1)
W Work (ML2T-2)
Q Heat transfer (ML2T-2)
uW : The useful work rate or mechanical power (ML2T-3)
m : The mass flow rate (MT-1)
2
2V
: The kinetic energy per unit mass (L2T-2)
gz: The potential energy per unit mass (L2T-2)
Etot: The total energy = m(u +
2
2V
+ gz) (ML2T-2)
Q : The heat transfer rate (ML2T-3)
dEcvdt
: The rate of change of energy for the control volume. (ml2t-3)
Unit conversion factors
For metric units
Basic:
o 1 N = 1 kg·m/s2;
o 1 J = 1 N·m;
o 1 W = 1 J/s;
o 1 Pa = 1 N/m2.
Others:
o 1 kPa·m3 = 1 kJ;
o T(K) = T(oC) + 273.15;
o 1 L (liter) = 0.001 m3;
o 1 m2/s
2 = 1 J/kg.
Prefixes (and abbreviations):
o nano(n) – 10-9
;
o micro() – 10-6
;
o milli(m) – 10-3
;
o kilo(k) – 103;
o mega(M) – 106;
o giga(G) – 109.
o A metric ton (European word: tonne) is 1000 kg.
For engineering units
Mechanical Engineering – GATE Exam
Energy:
o 1 Btu = 5.40395 psia·ft3 = 778.169 ft·lbf = (1 kWh)/3412.14 = (1 hp·h )/2544.5 =
25,037 lbm·ft2/s
2.
Pressure:
o 1 psia = 1 lbf/in2 = 144 psfa = 144 lbf/ft
2.
Others:
o T(R) = T(oF) + 459.67;
o 1 lbf = 32.174 lbm·ft/s2;
o 1 ton of refrigeration = 200 Btu/min.
Concepts & Definitions
Formula Units
Pressure FP
A
Pa
Units 21 1 /Pa N m 51 10 0.1bar Pa Mpa
1 101325atm Pa
Specific Volume Vv
m
3 /m kg
Density m
V
1
v
3/kg m
Static Pressure Variation P gh , Pa
Absolute Temperature ( ) ( ) 273.15T K T C
Properties of a Pure Substance
Formula Units
Quality vapor
tot
mx
m (vapour mass fraction)
1liquid
tot
mx
m (Liquid mass fraction)
Specific Volume f fgv v xv
3 /m kg
Average Specific Volume (1 ) f gv x v xv (only two phase mixture) 3 /m kg
Ideal –gas law cP P cT T
1Z
Equations Pv RT PV mRT nRT
Mechanical Engineering – GATE Exam
Universal Gas Constant 8.3145R /kJ kmol K
Gas Constant RR
M M = molecular mass
/kJ kg K
Compressibility Factor Z Pv ZRT
Reduced Properties r
c
PP
P , r
c
TT
T
Work & Heat
Formula Units
Displacement Work 2 2
1 1W Fdx PdV
J
Integration 2
2 11
( )W PdV P V V J
Specific Work Ww
m (work per unit mass)
/J kg
Power (rate of work) W FV PV T W
Velocity V r /rad s
Torque T Fr Nm
Polytropic Process ( 1)n 1 1 2 2
n n nPV Const PV PV nPv C
Polytropic Exponent 2
1
1
2
ln
ln
PP
nV
V
n=1 1 1 2 2PV Const PV PV
Polytropic Process Work 1 2 2 2 1 1
1( ) 1
1W PV PV n
n
J
n=1 2
1 2 2 21
lnV
W PVV
J
Adiabatic Process 0Q
Conduction Heat Transfer dTQ kA
dx , k =conductivity
W
Convection Heat Transfer Q hA T , h =convection coefficient W
Radiation Heat Transfer 4 4( )s ambQ A T T W
Terminology: Q1 = heat Q2 = heat transferred during the process between state 1 and state 2
Mechanical Engineering – GATE Exam
Q = rate of heat transfer W = work
1W2 = work done during the change from state 1 to state 2 W = rate of work = Power. 1 W=1 J/s
The First Law of Thermodynamics
Formula Units
Total Energy ( ) ( )E U KE PE dE dU d KE d PE J
Energy 2 1 1 2 1 2dE Q W E E Q W J
Kinetic Energy 20.5KE mV J
Potential Energy 2 1 2 1( )PE mgZ PE PE mg Z Z J
Internal Energy liq vap liq f vap gU U U mu m u m u
Specific Internal Energy of Saturated Steam (two-phase mass average)
(1 ) f g
f fg
u x u xu
u u xu
/kJ kg
Total Energy 2 2
2 12 1 2 1 1 2 1 2
( )( )
2
m V VU U mg Z Z Q W
J
Specific Energy 20.5e u V gZ
Enthalpy H U PV
Specific Enthalpy h u Pv /kJ kg
For Ideal Gasses ( )Pv RT and u f T
Enthalpy h u Pv u RT
R Constant ( ) ( )u f t h f T
Specific Enthalpy for Saturation State (two-phase mass average)
(1 ) f g
f fg
h x h xh
h h xh
/kJ kg
Specific Heat at Constant Volume
1 1v
v v v
Q U uC
m T m T T
( ) ( )e i v e iu u C T T
Specific Heat at Constant Pressure
1 1p
p p p
Q H hC
m T m T T
( ) ( )e i p e ih h C T T
Solids & Liquids Incompressible, so v=constant
c pC C C (Tables A.3 & A.4)
2 1 2 1( )u u C T T
2 1 2 1 2 1( )h h u u v P P
Mechanical Engineering – GATE Exam
Ideal Gas h u Pv u RT
2 1 2 1( )vu u C T T
2 1 2 1( )ph h C T T
Energy Rate ( )E Q W rate in out
2 1 1 2 1 2 ( )E E Q W change in out
First-Law Analysis for Control Volume
Formula Units
Volume Flow Rate V VdA AV (using average velocity)
Mass Flow Rate Vm VdA AV A
v (using average values)
/kg s
Power pW mC T
vW mC T
Vmv
W
Flow Work Rate flowW PV mPv
Flow Direction From higher P to lower P unless significant KE or PE
Total Enthalpy 212toth h V gZ
Instantaneous Process
Continuity Equation
. .C V i em m m
Energy Equation
. . . . . .C V C V C V i tot i e tot eE Q W m h m h First Law
2 21 1( )2 2i i i e e e
dEQ m h V gZ m h V gZ W
dt
Steady State Process A steady-state has no storage effects, with all properties constant with time
No Storage . . . .0, 0C V C Vm E
Continuity Equation
i em m (in = out)
Energy Equation
. . . .C V i tot i C V e tot eQ m h W m h (in = out) First Law
2 21 1( )2 2i i i e e eQ m h V gZ W m h V gZ
Specific Heat Transfer
. .C VQq
m
/kJ kg
Specific Work . .C VW
wm
/kJ kg
SS Single Flow Eq.
tot i tot eq h w h (in = out)
Transient Process Change in mass (storage) such as filling or emptying of a container.
Mechanical Engineering – GATE Exam
Continuity Equation
2 1 i em m m m
Energy Equation
2 1 . . .C V C V i tot i e tot eE E Q W m h m h
2 2
2 1 2 2 2 2 1 1 1 11 1
2 2E E m u V gZ m u V gZ
2 2
. 2 2 2 2 1 1 2 1 . .. .
1 12 2C V i tot i e tot e C V
C V
Q m h m h m u V gZ m u V gZ W
The Second Law of Thermodynamics
Formula Units
All ,W Q can also be rates ,W Q
Heat Engine HE H LW Q Q
Thermal efficiency
1HE LHE
H H
W Q
Q Q
Carnot Cycle 1 1L L
Thermal
H H
Q T
Q T
Real Heat Engine
1HE LHE Carnot HE
H H
W T
Q T
Heat Pump HP H LW Q Q
Coefficient of Performance
H HHP
HP H L
Q Q
W Q Q
Carnot Cycle H H
HP
H L H L
Q T
Q Q T T
Real Heat Pump
H HHP Carnot HP
HP H L
Q T
W T T
Refrigerator REF H LW Q Q
Coefficient of Performance
L LREF
REF H L
Q Q
W Q Q
Carnot Cycle L L
H L H L
Q T
Q Q T T
Real Refrigerator
L LREF Carnot REF
REF H L
Q T
W T T
Absolute Temperature L L
H H
T Q
T Q
Mechanical Engineering – GATE Exam
Entropy Formula Units
Inequality of Clausis 0
Q
T
Entropy
rev
QdS
T
/kJ kgK
Change of Entropy 2
2 1
1 rev
QS S
T
/kJ kgK
Specific Entropy (1 ) f gs x s xs
f fgs s xs
/kJ kgK
Entropy Change
Carnot Cycle Isothermal Heat Transfer:
2
1 22 1
1
1
H H
QS S Q
T T
Reversible Adiabatic (Isentropic Process):
rev
QdS
T
Reversible Isothermal Process:
4
3 44 3
3 rev L
QQS S
T T
Reversible Adiabatic (Isentropic Process): Entropy decrease in process 3-4 = the entropy increase in process 1-2.
Reversible Heat-Transfer Process
2 2
1 22 1
1 1
1 1 fg
fg
rev
hqQs s s Q
m T mT T T
Gibbs Equations Tds du Pdv
Tds dh vdP
Entropy Generation gen
QdS S
T
irr genW PdV T S
2 2
2 1 1 2
1 1
gen
QS S dS S
T
Entropy Balance Equation Entropy in out gen
Principle of the Increase of Entropy
. . 0net c m surr gendS dS dS S
Entropy Change
Solids & Liquids 2
2 1
1
lnT
s s cT
Reversible Process: 0gends
Adiabatic Process: 0dq
Mechanical Engineering – GATE Exam
Ideal Gas Constant Volume:
2
22 1 0
11
lndT v
s s Cv RvT
Constant Pressure: 2
22 1 0
11
lndT P
s s Cp RPT
Constant Specific Heat: 2 22 1 0
1 1
ln lnT v
s s Cv RT v
2 22 1 0
1 1
ln lnT P
s s Cp RT P
Standard Entropy
0
00
Tp
T
T
Cs dT
T
/kJ kgK
Change in Standard Entropy 0 0 2
2 1 2 11
lnT T
Ps s s s R
P
/kJ kgK
Ideal Gas Undergoing an Isentropic Process
2 22 1 0
1 1
0 ln lnT P
s s Cp RT P
0
2 2
1 1
RCpT P
T P
but 0 0
0 0
1p v
p p
C CR k
C C k
,
0
0
p
v
Ck
C = ratio of specific heats
1
2 1 2 1
1 2 1 2
,
k k
T v P v
T v P v
Special case of polytropic process where k = n: kPv const
Reversible Polytropic Process for Ideal Gas
1 1 2 2
n n nPV const PV PV 1 1
2 1 2 2 1
1 2 1 1 2
,
nn nnP V T P V
P V T P V
Work 2 2
2 2 1 1 2 11 2
1 1
( )
1 1n
PV PV mR T TdVW PdV const
V n n
Values for n Isobaric process: 0,n P const
Isothermal Process: 1,n T const
Isentropic Process: ,n k s const
Isochronic Process: ,n v const
Mechanical Engineering – GATE Exam
Second-Law Analysis for Control Volume Formula Units
2nd Law Expressed as a Change of Entropy
. .c mgen
dS QS
dt T
Entropy Balance Equation rateof change in out generation
. . . .C V C Vi i e e gen
dS Qm s m s S
dt T
where . . . . ...C V c v A A B BS sdV m s m s m s
and . . ...gen gen gen A gen BS s dV S S
Steady State Process . . 0C VdS
dt
. .
. .
C Ve e i i gen
C V
Qm s m s S
T
Continuity equation i em m m . .
. .
( ) C Ve i gen
C V
Qm s s S
T
Adiabatic process e i gen is s s s
Transient Process . .
. .
C Vi i e e genC V
Qdms m s m s S
dt T
. .2 2 1 1 1 2. .
0
t
C Vi i e e genC V
Qm s m s m s m s dt S
T
Reversible Steady State Process
If Process Reversible & Adiabatic
e is s e
e i
i
h h vdP
2 2
2 2
( )2
( )2
i ei e i e
e
i ei e
i
V Vw h h g Z Z
V VvdP g Z Z
If Process is Reversible and Isothermal . .
. .
. .
1 C Ve i C V
C V
Qm s s Q
T T
or . .C Ve i
QT s s q
m
e
e i e i
i
T s s h h vdP
Mechanical Engineering – GATE Exam
Incompressible Fluid
2 2
02
e ie i e i
V Vv P P g Z Z
Bernoulli Eq.
Reversible Polytrophic Process for Ideal Gas
e
n n
i
w vdP and Pv const C
1
1 1
e e
ni i
e e i i e i
dPw vdP C
P
n nRPv Pv T T
n n
Isothermal Process (n=1) ln
e e
ei i
ii i
PdPw vdP C Pv
P P
Principle of the Increase of Entropy
. . 0net C V surrgen
dS dS dSS
dt dt dt
Efficiency
Turbine a i e
s i es
w h h
w h h
Turbine work is out
Compressor (Pump)
s i es
a i e
w h h
w h h
Compressor work is in
Cooled Compressor Tw
w
Nozzle 2
2
12
12
e
es
V
V Kinetic energy is out
Note:
°F = (°C x 9/5) + 32
°C = (°F - 32) x (5/9)
°K = °C + 273
Q = mC∆T thermal energy = mass x specific heat x change in T
Q = mHf thermal energy = mass x heat of fusion
Q = mHv thermal energy = mass x heat of vaporization
∆L = αLi∆T change in length = coefficient of expansion x initial length x change in T
∆V = βVi∆T change in volume = coefficient of expansion x initial volume x change in T
∆U = Q – W internal energy = heat energy - work
Mechanical Engineering – GATE Exam
Plausible Physical Situations
Name State Variables P V T ΔEth Q Ws W
Insulated sleeve or rapid process
Add weight to or push down on piston
Adiabatic compression
PVγ = Const; TVγ-1 = Const
Up Down Up
nCvΔT > 0
0 -nCvΔT < 0 nCvΔT > 0
Insulated sleeve or rapid process
Remove weight from or pull up on piston
Adiabatic expansion
PVγ = Const; TVγ-1 = Const
Down Up Down
nCvΔT < 0
0 -nCvΔT > 0 nCvΔT < 0
Heat gas Locked piston or rigid container
Isochoric V fixed; P α T
Up Fixed Up
nCvΔT > 0
nCvΔT > 0
0 0
Cool gas Locked piston or rigid container
Isochoric V fixed; P α T
Down Fixed Down
nCvΔT < 0
nCvΔT < 0
0 0
Heat gas Piston free to move, load unchanged
Isobaric expansion
P fixed; V α T
Fixed Up Up
nCvΔT > 0
nCpΔT > 0
PΔV > 0 -PΔV < 0
Cool gas Piston free to move, load unchanged
Isobaric compression
P fixed; V α T
Fixed Down Down
nCvΔT < 0
nCpΔT < 0
PΔV < 0 -PΔV > 0
Immerse gas in large bath
Add weight to piston
Isothermal compression
T fixed at temperature of bath, PV = Const
Up Down Fixed
nCvΔT = 0
nRT*ln(Vf/Vi) < 0
nRT*ln(Vf/Vi) < 0
-nRT *ln(Vf/Vi) > 0
Immerse gas in large bath
Remove weight from piston
Isothermal expansion
T fixed at temperature of bath, PV = Const
Down Up Fixed
nCvΔT = 0
nRT*ln(Vf/Vi) > 0
nRT*ln(Vf/Vi) > 0
-nRT* ln(Vf/Vi) < 0
Unknown Unknown No Name PV/T = Const
? ? ?
nCvΔT ΔEth + Ws
∫PdV = ± area under curve in PV diagram
-∫PdV = ± area under curve in PV diagram