w := P⋅ v ⋅ ln 
⌠
w :=  P d v
⌡
Closed System,
Reversible,
w := P( v 2 − v 1)
P = Const.
w := R( T2 − T1)
Cp = Const.
 P1 

 P2 
w := R⋅ T⋅ ln 
w :=
P = Const.
I.G.
Heat = Work
1st Law of Thermodynamics
 v2 

 v1 
Closed System:
n := 1
(P2⋅ v2 − P1⋅ v1)
Closed System
q − w := u 2 − u 1 Delta KE ~ 0
Delta PE ~ 0
T = Const.
Reversible
I.G.
Pv^n = Const.
1
n≠1
1−n
Comp.
P
X is Quality
δQ := δW
x :=
massvap
masstotal
Sat.
1 − x :=
massliq
0
masstotal
0
v
v := v f + x⋅ v fg Saturated
u := u f + x⋅ u fg Saturated
h := u + Pv Always
v ≡ v f @ T Compressed
u ≡ u f @ T Compressed
look up h in charts Superheated
look up v in charts Superheated
look up u in charts Superheated
Steady Flow Systems (St. Fl.):
⌠
w := − v d P
⌡
w := n ⋅
Reversible
St. Fl.
∆KE ~ 0
∆PE ~ 0
1−n
Q := q ⋅ m
2
St. Fl.
  − g ⋅ ( z − z ) Reversible
2
1 St. Fl.

 v2 
 Reversible
 v 1  Isothermal
 P1  n = 1
w := R⋅ T⋅ ln ⋅ 

 P2 
w := −v ⋅ ( P2 − P1)
m := ρυA St. Fl.
ρ :=
m :=
v = Const.
P
I.G.
R⋅ T
υA
not an I.G.
v
w := −R⋅ ( T2 − T1) I.G.
1st Law of Thermodynamics
i
Heat Engines:
n =1
qh = heat gained
qH
qL = heat rejected q L
Thermal Efficiency
∑
Q+
St. Fl.
q − w := h 2 − h 1 ∆KE ~ 0
∆PE ~ 0
:=
TH
i
∑
n =1


(υ i)2
mih i +
+ g ⋅ zi
2


St. Fl.
For Pumps or Engines
1
ζth := 1 −
qH
W net
TL
qH
TH
T
0.5
W net := q H − q L
qH
q := T⋅ ( S2 − S1)


(υ i)2
mih i +
+ g ⋅ zi =
2


TL
qL
Assume Carnot!
ζth :=
2
w := P⋅ v ⋅ ln ⋅ 
m = mass flow rate (Const.)
ζth := 1 −
1
q's don't change form St. Fl. to Closed System
  υ − υ
⌠
1
 2
w := − v d P − 
2
⌡

Pv^n = Const.
I.G.
n = 0, P = Const.
n = 1, T = Const.
n = κ, S = Const.
n to ∞, v = Const.
St. Fl.
(P2⋅ v2 − P1⋅ v1)
W := m⋅ w
Super
qL
0
reversible
isothermal
Heat Pumps:
0
0.5
S
1
1
Heat Pumps:
Coefficient of Performance
α :=
β :=
qH
α :=
W net
qL
β :=
W net
qh = heat rejected
qH
qL = heat gained
qL
TH
:=
For Pumps or Engines
TL
1
qH
α − β := 1
qH − qL
qH
W net := q H − q L
T
qL
qH
qH − qL
qH − qL
:=
0.5
TH
qL
TH − TL
0
0
0.5
w := −v ⋅ ( P2 − P1)
( κ − 1)
Other Equations:
 P2 
:= 

T1  P1 
T2
P⋅ V := nR ( bar) T
P⋅ V := mRT
R :=
ρ :=
R( bar)
I.G. (all 5)
M
P
δh
Cp :=
R⋅ T
P < 10 MPa
Pv := RT I.G.
T > 2 Tcrit
Cv :=
κ
Pv^κ = Const. which implies:
δu
δT
⌠
h 2 − h 1 :=  Cp d T

⌡
∆h := Cp ⋅ ∆T
⌠
u 2 − u 1 :=  Cv d T

⌡
Cp − Cv := R
Cp :=
u(T) only
Cp
 T2 
 v2 
+ R⋅ ln 


 T1 
 v 1  C = Const.
p
 T2 
 P2  I.G.
∆S := Cp ⋅ ln 
 − R⋅ ln P 
 T1 
 1
( κ ⋅ R)
κ−1
Pr1
:=
P2
P1
Cp (is not) = Const.
I.G.
adiabatic
h1 = h2
absolutely not reversible
R
κ−1
Isentropic
Cp is not Const.
∆S = 0
0
Throttling:
h ~ hf (@ T) + vf (P - Psat) compressed
or look in compressed tables
2
It's commonly an I.G.
if Cp = Const., then
Cv = Const.
h := u + Pv Always
Pr2
It's commonly an I.G.
:= κ
Cv :=
∆S := Cv⋅ ln 
 P2 
∆S := S2 − S1 − R⋅ ln 

 P1 
h(T) only
Cp = Const.
Cv
TdS := dh − vdP
0
h(T) only
u(T) only
Cv = Const.
∆u := Cv⋅ ∆T
TdS := du + Pdv
S = Const.
Cp = Const.
I.G.
δT
⌠
reversible
q :=  T d S
⌡
q := T⋅ ( S2 − S1) reversible
isothermal
⌠
1
 P2 
∆S :=  Cp ⋅ d T − R⋅ ln 


T
P1 

⌡
1
S
n
n
n
n
= 0, P = const.
= 1, T = const.
= R, S = const.
-> ∞, v = const.
1
T
P
0
S
0
v
1
Devices:
Turbines
Compressor
1
1
T
T
0
0
0.5
0
1
0
0.5
S
S
Adiabatic
Adiabatic
γ tur :=
Steady Flow
in Ideal case, Isentropic
Wa
γ comp :=
Steady Flow
Ws
1
T
No Graph
T
0
0.5
0
1
0
S
Adiabatic
3
kg
0.5
1
S
γ pump :=
Steady Flow
m
Wa
Nozzle
1
v := .001⋅
Ws
W s := h 1 − h 2s
I.G.
Pump
0
1
2
Ws
Adiabatic
Wa
Steady Flow
W s := −v ⋅ ( P2 − P1)
v := v f
@T
3
γ noz :=
νa
νs
2
Diffuser
1
T
0
0
0.5
1
S
Adiabatic
γ diff :=
Steady Flow
∆Pa
ν1 > ν2
∆Ps
P2 > P1
 2
 2
 P1   ν 1 
 P2   ν 2 
 +
 + g ⋅ z1 :=  ρ  +  2  + g ⋅ z2
ρ   2 
 
for ν is less than .5 moch
3 Part Cycle:
Pr2
Pr1
:=
P2
P1
Isentropic
Cp is not Const.
∆S = 0
( k− 1)
 P2 

 P1 
k
T2 := T1⋅ 
Closed System
q − w := ∆u
Closed System
P⋅ v := n ⋅ R⋅ T I.G.
 P2  I.G.

 P1 
w := R⋅ T⋅ ln ⋅ 
USUF:
(uniform state/ uniform flow)
i = in the tank
e = exit the tank
1 = initially in the tank
2 = final in tank

Q + mi⋅ h i +

 ν 2 


i
 2  + g ⋅ zi := W + me⋅ h e +
 ν 2 


e
 2  + g ⋅ ze + m2⋅ u 2 +
4
 ν 2 


2
 2  + g ⋅ z2 + m1⋅ u 1 +
 ν 2 

1
 2  + g ⋅ z1
Definitions:
Isothermal: T = Const.
Isentropic: S = Const. means (Reversible & Adiabatic)
Adiabatic: q = 0
Reversible: Both the system and surroundings can be returned to their initial state
Heat Engines: Thermo cycle with + net heat transfer and + net work
Heat Pumps: Thermo cycle with - net heat transfer and - net work
Steady Flow: No change @ a point with time
Throttling: Adiabatic, h1 = h2 (for single inlet, single outlet only), & Absolutely not Reversible
Pumps: Adiabatic
Turbines: Adiabatic
Compressors: Adiabatic
Nozzles: Adiabatic
Diffusers: Adiabatic
Units Conversions:
Charts:
Description
Air
Critical Properties
Critical Property Graphs
H2O
Propane
R-22
I.G. Properties of Selected Gases
Thermo Properties of selected Subs.
I.G. Specific Heats of common Gases
5
SI
English
p. 840
p. 803
p. 899
p. 804
p. 832
p. 814
p. 842
p. 847
p. 838
p. 890
p. 851
p. 851
p. 852
p. 881
p. 864
p. 892
p. 897
p. 888
Properties of Various Ideal Gases
SI Units
Gas
Chemical Formula
Air
Argon
Butane
Carbon
Dioxide
Carbon
Monoxide
Ethane
Ethylene
Helium
Hydrogen
Methane
Neon
Nitrogen
Octane
Oxygen
Propane
Steam
Molecular Weight
R(
KJ
)
Kg° K
Cp(
KJ
)
Kg ° K
Cv(
KJ
)
Kg ° K
κ
Ar
C4 H10
28.97
39.948
58.124
.28700
.20813
.14304
1.0035
.5203
1.7164
.7165
.3122
1.5734
1.400
1.667
1.091
CO2
44.01
.18892
.8418
.6529
1.289
CO
C2 H6
C2 H4
He
H2
CH4
Ne
N2
C8 H18
O2
C3 H8
H2 O
28.01
30.07
28.054
4.003
2.016
16.04
20.183
28.013
114.23
31.999
44.097
18.015
.29683
.27650
.29637
2.07703
4.12418
.51835
.41195
.29680
.07279
.25983
.18855
.46152
1.0413
1.7662
1.5482
5.1926
14.2091
2.2537
1.0299
1.0416
1.7113
.9216
1.6794
1.8723
.7445
1.4897
1.2518
3.1156
10.0819
1.7354
.6179
.7448
1.6385
.6618
1.4909
1.4108
1.400
1.186
1.237
1.667
1.409
1.299
1.667
1.400
1.044
1.393
1.126
1.327
English Units
Gas
Chemical Formula
Air
Argon
Butane
Carbon
Dioxide
Carbon
Monoxide
Ethane
Ethylene
Helium
Hydrogen
Methane
Neon
Nitrogen
Octane
Oxygen
Propane
Steam
Molecular Weight
R(
ft ⋅ lbf
)
lbm° R
Cp(
Btu
)
lb m°R
Cv(
Btu
)
lbm° R
κ
Ar
C4 H10
28.97
39.94
58.124
53.34
38.68
26.58
.240
.1253
.415
.171
.0756
.381
1.400
1.667
1.09
CO2
44.01
35.10
.203
.158
1.285
CO
C2 H6
C2 H4
He
H2
CH4
Ne
N2
C8 H18
O2
C3 H8
H2 O
28.01
30.07
28.054
4.003
2.016
16.04
20.183
28.016
114.22
32.000
44.097
18.016
55.16
51.38
55.07
386.0
766.4
96.35
76.55
55.15
13.53
48.28
35.04
85.76
.249
.427
.411
1.25
3.43
.532
.246
.248
.409
.219
.407
.445
.178
.361
.340
.753
2.44
.403
.1477
.177
.392
.157
.362
.335
1.399
1.183
1.208
1.667
1.404
1.32
1.667
1.400
1.044
1.395
1.124
1.329
SI: (Cp, C v , & κ are at 300 K), English: (Cp, C v , & κ are at 80 F)
6