Useful Equations for TME2121 (Part 1)
Part 1. Review of Basic Concepts and Properties of Pure Substances
1.1. Properties of Steam:
Specific volume: v =
V
[m3/kg]
m
(Eq 1)
where V is the absolute volume in m3, and m is the mass in kg
In the two-phase-mixture region,
v= v f + xv fg [m3/kg] where x is the quality of the steam (dryness fraction), and vfg = vg – vf (Eq 2)
u= u f + xu fg [kJ/kg] where x is the quality of the steam (dryness fraction), and ufg = ug – uf (Eq 3)
h= hf + xhfg [kJ/kg] where x is the quality of the steam (dryness fraction), and hfg = hg – hf (Eq 4)
=
s s f + xs fg [kJ/kg·K] where x is the quality of the steam (dryness fraction), and sfg = sg – sf (Eq 5)
1.2. Ideal Gas Equation of State:
PV = mRT = nR0T (Eq 6)
PV
PV
1 1
= 2 2 (Eq 7)
T1
T2
where P is the pressure in N/m2, V is the volume in m3,
T is the absolute temperature in Kelvin.
n is the no. of moles, Ro is the universal gas constant = 8.3143 kJ/kmol·K
m is the mass in kg, R is the gas constant in kJ/kg·K
Note: m = n·M (Eq 8) and R = Ro/M (Eq 9) where M is the molar mass in kg/kmol
1.3. Specific Heats:
γ=
Cp
Cv
(Eq 10)
C p − Cv =
R (Eq 11)
1.4. Mass Flow Rate of Fluid
 = ρ ⋅ A ⋅ Velocity [kg/s]
m
where ρ =
(Eq 12)
1
[kg/m3], and v is the specific volume [m3/kg]
v
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Part 2. 1st Law of Thermodynamics
2.1. The first law of thermodynamics is essentially an expression of conservation of energy principle.
It is commonly expressed as
Ein − Eout =
∆E system [kJ] (Eq 13)
where Ein − E out : net energy transfer by heat, work and mass
∆E system = ∆U + ∆KE + ∆PE ≈ ∆U : change in internal, kinetic (negligible) and potential (negligible)
energies
2.2. For a closed-system (no mass transfer) and neglecting changes in KE and PE,
Q12 − W12 =U2 − U1 (Eq 14)
For a closed-system undergoing a complete thermodynamic cycle:
∫ Q = ∫ W
(Eq 15)
2.2.1 Heat transfer during a certain process:
Q12 mC (T2 − T1 ) (Eq 16)
=
2.2.2 For ideal gases, change in internal energy during a process (Joule’s Law):
U2 −=
U1 mCv (T2 − T1 ) (Eq 17)
u = CvT (Eq 18)
2.2.3 Work during a process:
For an isothermal process, T = constant, dT = 0, hence PV = constant
=
W12
2
PdV
∫=
1
 V2 
PV
 (Eq 19)
1 1 ln 
 V1 
For an isobaric process, P = constant, dP = 0, hence V/T = constant
=
W12
2
=
∫ PdV
1
P1 (V2 − V1 ) (Eq 20)
For an isochoric process, V = constant, dV = 0, hence P/T = constant
=
W12
2
PdV 0 (Eq 21)
∫=
1
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T P 
For a polytropic process, PVn = constant (Eq 22), and 2 =  2 
T1  P1 
=
W12
2
PdV
∫=
1
n −1
n
(Eq 23)
PV
1 1 − P2V2
(Eq 24)
n −1
For isentropic process (reversible + adiabatic), n = γ and
T2  P2 
= 
T1  P1 
γ −1
γ
(Eq 25) and s = constant, ds
=0
=
W12
2
PdV
∫=
1
PV
1 1 − P2V2
(Eq 26)
γ −1
2.3. 1st Law of Thermodynamics for Flow Processes
For an open-system (with mass transfer across system boundaries) and neglecting changes in KE and
PE,
dU




C2
C2




Qnet in − Wnet out + ∑ min  h + + gZ  − ∑ mout  h + + gZ  =sys
dt
2
2
in

in out

out
(Eq 27)
For an open-system with steady-flow process (SFEE: Steady-Flow Energy Equation):




C2
C2
 in  h + + gZ  − ∑ m
 out  h + + gZ  =
Q net in − W net out + ∑ m
0 (Eq 28)
2
2
in

in out

out
For an open-system with transient-flow over a finite time interval (finite mass transfer, e.g.
charging of tank), and neglecting changes in KE and PE,
Qnet in − Wnet out + ∑ min hin − ∑ mout hout =
∆Usys =
( m2u2 − m1u1 )sys
in
(Eq 29)
out
For ideal gas, change in enthalpy during a process:
H2 −=
H1 mC p (T2 − T1 ) (Eq 30)
h = CpT (Eq 31)
Part 3. 2nd Law of Thermodynamics
3.1. The thermal efficiency of a heat engine is defined as
ηth =
Wnet out
QH
= 1−
QL
QH
(Eq 32)
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3.2. The performance of a refrigerator or a heat pump is expressed in terms of coefficient of
performance (COP), which is defined as
=
COPR
QL
1
=
Q
Wnet in
H
−1
QL
=
COP
HP
QH
1
=
Wnet in 1 − QL
QH
(Eq 33)
(Eq 34)
3.3. For Carnot cycles:
QH TH
=
QL TL
(Eq 35)
Part 4. Entropy
4.1. Entropy change is defined as:
∆S = S2 − S1 =
 dQ 
 (Eq 36)
1
T rev
2
∫ 
A special case for a constant temperature:
∆Q
∆S =
To
(Eq 37)
Clausius inequality:
∫
dQ
≤ 0 (Eq 38)
T
4.2. Entropy change for ideal gases:

T 
 v 
S2 − S=
m ( s2 − s1=
) m Cv ln 2  + R ln 2   [kJ/K] (Eq 39)
1
 T1 
 v1  


T 
 P 
S2 − S=
m ( s2 − s1=
) m C p ln 2  − R ln 2  
1
 T1 
 P1  

[kJ/K] (Eq 40)
4.3. Entropy balance equation:
4.3.1 For a closed-system, there is no mass transfer across system boundary. Thus, the entropy rate
of change of a closed system is
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Qk
∑T
+ Sgen =
∆Ssys (Eq 41)
k
4.3.2 For an open-system, they involve one more mechanism of entropy exchange: mass flow across
system boundary. Thus, the entropy rate of change of an open system is
∑m
s − ∑ mout sout + ∑
in in
Qk
+ Sgen =
∆Ssys (Eq 42)
Tk
For an open-system in the rate form:
∑ m
 out sout + ∑
s − ∑m
in in
dSsys
Q k 
+ Sgen =
Tk
dt
(Eq 43)
For a steady-flow process, the above equation simplifies to
∑ m
 out sout + ∑
s − ∑m
in in
Q k 
0 (Eq 44)
+ Sgen =
Tk
4.4. An isolated system:
∆Siso− system = ∆Ssystem + ∆Ssurroundings = Sgen
(Eq 45)
4.5. Gibbs equation:
du = Tds – Pdv (Eq 46)
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