my thermodynamics cheat sheets
Nasser M. Abbasi
Sumemr 2004
Compiled on May 23, 2020 at 4:09am
1. all of theormodynamics in one sheet.
(a) PDF
(b) image
2. polytropic process diagrams
(a) PDF
(b) image
3. first and second laws diagrams
(a) PDF
(b) image
4. Gas laws
(a) PDF
(b) image
All of theormodynamics in one sheet
1
2
Entropy change equation
h  u  P
Process that causes irreversibility
Enthlapy definition
Solids/Liquids
1. Friction
Ideal Gas
2. Unrestrained expansion
Solids/Liquids
3. Heat transfer from hot to cold body
Ideal Gas
4. Mixing of 2 differrent substances
dh  du  dP
5. i 2 R loss in electric circuits
dh  du  P d  v dP
6. Hystereris effects
dP  0 (incompressible), and d  0
dh  du
dh  C dT
7. Ordinary combustion
h 2  h 1  C p ln
T2
T1
h 2  h 1  C ln
T2
T1
but dP  0 since incompressible,
and dv is very small, so
First law
ds  C dT
T
Flow
Non-Flow
s 2  s 1  C ln
General equation. Valid at any instance of time. Steady or not steady flow.
Q  W  dE
where E  U  KE  PE
 C.V. m e h  KE  PE 
Q C.V.  m i h  KE  PE i  W
e
Or, it can be written as follows
(ignoring KE and PE changes to the
control mass)
dE
dt
T2
T1
1
How to get this below from the above??
ds  C p dT
 R dP
T
P
Non-steady state (Transient, state change)
Second law
QC.V.  m i h  KE  PE i  WC.V. m e h  KE  PE e  m 2 u 2  m 1 u 1 
Q1,2  W1,2  mu 2  u 1 
Non-flow
Usually Simplifies to
QC.V.  m i h i  WC.V. m e h e  m 2 u 2  m 1 u 1 
As a Rate equation
 
Q  W
dq
(by definition, entropy law)
T
 dw  du  dw  du
T
T
T
 1 dPv  1 dC v T
T
T
C
 1 P dv  v dP  v dT
T
T
 P dv  v dP  C v dT
T
T
T
but Pv  RT, hence
dP
dT
ds  R dv
v  R P  Cv T
v
P
T
s 2  s 1  R ln v 21  R ln 2  C v ln 2
P1
T1
ds 
dq
(by definition, entropy law)
T
 dw  du  dw  du
T
T
T
 1 dPv   1 d C T
T
T
C
1

P dv  v dP  v dT
T
T
 P dv  v dP  C v dT
T
T
T
ds 
P  RT
so
h  u  RT
dh  du  R dT
dh  C v dT  RdT
dh  C p dT
dE
dt
m1
hi
s2  s1 
m2
State 1
q
T
flow
Q
T
ms 2  s 1  
 S gen
transient
steady
 s gen
0  misi  me se 
q
T
0  si  se 
State 2
Q
T
 S gen
 s gen
steady state. m i  m e  m
QC.V.  mh  KE  PE i  WC.V. mh  KE  PE e
QC.V.  mh i  WC.V. mh e
QC.V.  WC.V. mh e  h i 
q  w  h e  h i 
Steady state devices:
Heat exchanges, Nozzle,
Diffuser, Throttle, Turbine,
Compressors and Pumps
hi
he
GAS
Ideal gas, Any process
s2
P2 V2
T1
T2
2
s1
1
s2
Any gas, any process
P1 V1
2
s1
1
C
0
T
C p0
T
dT
Entropy change determination formulas, for an ideal gas, ANY process type
mass constant
Assume
constant
specific
heat
2
R ln
1
dT
R ln
Solving
s1
C
ln
s2
s1
C p 0 ln
0
P2
T2
T1
T2
T1
2
R ln
P1
u1
Cv
T2
T1
h2
h1
C p T2
T1
1
P2
R ln
P1
2
w
P
w
1
Table A.6
Using
Table
A.7 or
A.8
du  C v0 dT
dh  C p0 dT
C p0  C v0  R
s2
u2
s 2  s 1  s 0T 2  s 0T 1   R ln
P2
P1
Ideal Gas Process classification
1
sgen
T
n
w
e
w
1
st
Law Q
W
1 n
nR
P
dU
Gibbs
equations
dH
U
dU
P2
T1
n
V dp
w
T2
T1
n
Shaft work (for
FLOW process only)
0
s2
T1
w
e
i
1
V1
n 1
n
V2
s1
R
Cv0
1
w
R T1 ln
w
2
1
ln
Pi
i
ln
2
R Ti ln
1
R Ti ln
V2i V2e
2
Pe
P2
P1
P2
w
Pi
Pe
e
w
2
P2
1
R T2
k
T1
1
Pi
e
P
1 k
kR
1
Entropy
change
s2
s1
0
 gZ i  Z e 
Figure 1: thermodynamics
2
k
k
w
i
Entropy
change
wtotal    v dP 
T2
T1
i
1
P1
ln
Shaft work (for FLOW
process only)
Boundary
Work
P
P2
R T1 ln
i
n
i
P1
Total specific work for steady state flow process where only
shaft work is involved (no boundary work). Valid for ANY
reversible process (do not have to be polytropic)
i
Pi
n
Shaft work (for
FLOW process only)
s 2  s 1  R ln
Introduction to Thermodynamics,
equations. By Nasser M Abbasi
image2.vsd
August 2004
Ti
n=k, constant entropy
Boundary
Work
T2
T1
Cv0 ln
s2
1 constant T
P
i
Verify this, what
volume is this?
s1
T2
P1
e
w
Pe e Pi
R Te Ti
Entropy
change
Cp0 ln
0
e
Entropy change determination formulas,
for an ideal gas, polytropic process type
e
2
W
Entropy
change
s1
P dV
constant V
W
P2
4
Boundary
Work
P2 2
R T2
n
Pe
Te
PV
Specialized polytropic
processes work formulas
Shaft work
(for FLOW
process
only)
s2
dH
V dP
Boundary
Work
w
2
n
n
3
so
Substitute into
0 constant P
1
formulas for
general
polytropic
process
General
polytropic
relation
5
T dS
n
P dV
constant
T
enthlapy law H
2
T ds
P Vn
P dV
Substitute into
U
General polytropic process
Q
ds
W
1
n
T1
Shaft work (for FLOW
process only)
Shaft Work
General formulas for reversible
compressible processes
Q
P2
1
i
2
2
1
R T2
Boundary
Work
reversible
s1
1
P2
WORK
irreversible
s2
n
k
Pe
e
Te
Ti
Pi
i
m2 s 2  m1 s 1  mi s i  me s e 
q
T
 s gen
1
3
P Vn
We have a total of 7 unknowns. 3 in state 1, 3 in state 2, and n, the polytrpoic
process exponent.
If given any 5 out of these 7, then the remaining 2 can be found.
For example, if we know T1, P1, V1, n, P2, then we can find V2, and T2
P1
V1
T1
constant
P2
V2
T2
Polytropic process
State 1
n
P
COMPRESSIONQU
ADRANT
Heat OUT
processes
State 2
constant V
Process lines to left of adiabatic
line means negative Q (i.e. heat
OUT), on the right are positive Q
(i.e. heat IN) process
Heat IN
processes
Ignore this quadrant in real
engineering equipments
Clock wise,
n=0,1,k,infinity
Initial
state
point
n=0 const
P
n=1,
const T
Ignore this quadrant in real
engineering equipments
EXPANSION
QUADRANT
Heat IN
processes
Heat OUT
processes
n=k, adiapatic, const S
v
n=k const S
T
n
constant V
n=0,
const P
Ignore this quadrant in real
engineering equipments
Initial
state
point
EXPANSION
QUADRANT
n=1 const
T
COMPRESSIONQU
ADRANT
Ignore this quadrant in real
engineering equipments
Clock wise,
n=0,1,k,infinity
s
Polytrpoic process
by Nasser M Abbasi
polytropic.vsd
August 2004
Figure 2: polytropic process diagrams
4
Laws of thermodynamics
First law
Second law
This is also called the law of conservation of energy
The entropy of an isolated system increases in all real processes and is
conserved in reversible processes.
OR
The AVAILABLE energy of an isolated system decreases in all real
processes (and is conserved in reversible processes)
Chapter 5. 1st Law for control mass
Chapter 6
1st Law for control volume
Q12  W 12  E 2  E 1
The 2nd law says that work and heat energy are not the same. It is easier
to convert all of work to heat energy, but not vis versa. i.e. work energy is
more valuable than heat energy. Heat can not be converted completely and
continuously into work.
E  U  PE  KE
Q in  W in  m i h  PE  KE i 
Q out  W out  m e h  PE  KE e  m 2 u 2  m 1 u 1 
Chapter 5.6
Chapter 5.5
enthalpy
H  U  PV
h  u  Pv
Derived from first Law
by setting P constant
Q  W  mu 2  u 1 
Q   P dV  mu 2  u 1 
Q  PV 2  V 1   mu 2  u 1 
q  P 2   1   u 2  u 1 
So, 2nd law dictates the direction at which a system state will change. It
will move to a state of lesser available energy
Cv Cp
Specific heat is the
amount of heat required
to raise the temp. of a
unit mass by one degree
u
T v
C p  h
T p
Solids and liquids: Use
average specific heat C.
Cv 
Steady
state
Entropy is constant in a reversible adiabatic process.
S2  S1  
W net
W lost   T dS gen
hi
Actual boundary work W 12   P dV  W lost
he
Gibbs relations
T ds  du  P dv
T ds  dh  v dp
Ideal Gas
dh  du  dP v
dh  du  v dP  P dv
Solids, Liquids
s 0T  
T
C p0
T0
T
dT
s 2  s 1  s 0T 2  s 0T 1  R ln
For solids and liquids
For ideal gas
For Solids and Liquids
Also for solids and liquids, v is very small, hence
Hence for solids/liquids, dh  du  C dT
s 2  s 1  C ln
T2
T1
s 2  s 1  C p0 ln
s 2  s 1  C v ln
du  C v0 dT
dh  C p0 dT
C p0  C v0  R
P2
P1
P2
P1
v2
v1
 R ln
 R ln
Using table A.7 or A.8
Constant C p , C v
Constant C p , C v
k  C p0
v0
Polytrpic process
PV n constant
P2
P1
P2
P1

V1
V2

T2
T1
n
n
n1
specific work (work is moving boundary work,  P dv
w12 
By Nasser M. Abbasi
Image1.vsd
T2
T1
T2
T1
C
For solids and liquids, dv  0
since almost incompressible, hence
For Solids and Liquids
S gen  0
Hence, S 2  S 1 for a reversible adiabatic process
Note: Even though enthlapy was derived by the
assumption of constant P, it is a property of a
system state regardless of the process that lead
to the state.
dh  du
 S gen
Q  0  adiabatic process
W net  h e  h i
 h2  h1
dh  du  v dP
2 Q
1 T
S gen  0  reversible process
Adiabatic process
q  P 2  u 2   P 1  u 1  C v  C v For solids and liquids
1q2
Transient
State
1
1n
P 2 v 2  P 1 v 1  
w12  P 1 v 1 ln
v2
v1
 RT 1 ln
R
1n
v2
v1
T 2  T 1  n  1
 RT 1 ln
Figure 3: first and second laws diagrams
P1
P2
n1
5
Solids/liquids
ds  C p dT
 R dP
T
P
GAS
Ideal gas, Any process
s2
P2 V2
T1
T2
2
s1
1
s2
Any gas, any process
P1 V1
2
s1
1
C
0
T
C p0
T
dT
Entropy change determination formulas, for an ideal gas, ANY process type
mass constant
2
R ln
1
dT
R ln
s2
Solving
s1
C
s1
ln
0
C p 0 ln
P2
T1
T2
T1
2
R ln
u1
Cv
T2
T1
h2
h1
C p T2
T1
1
P2
R ln
P1
2
w
P1
P
w
1
Table A.6
Using
Table
A.7 or
A.8
du  C v0 dT
dh  C p0 dT
C p0  C v0  R
s2
Assume
constant
specific
heat
T2
u2
s 2  s 1  s 0T 2  s 0T 1   R ln
P2
P1
s1
1
Ideal Gas Process classification
sgen
1 n
nR
P
Law Q
W
dU
Gibbs
equations
U
so
dU
Substitute into
dH
constant V
P2
T1
W
w
Cp0 ln
T2
T1
s2
i
s1
P1
T1
e
i
1
n
n 1
V1
n
V2
R
Cv0
1
w
i
R T1 ln
w
2
1
ln
Pi
i
ln
2
R Ti ln
1
wtotal    v dP 
R Ti ln
P2
P1
 gZ i  Z e 
Figure 4: gas lawss
Pe
P2
w
Pi
Pe
e
w
2
P2
1
R T2
k
T1
1
Pi
e
P
1 k
kR
1
Entropy
change
s2
s1
0
2
k
k
w
i
Entropy
change
V2i V2e
2
T2
T1
i
1
P1
ln
Shaft work (for FLOW
process only)
Boundary
Work
P2
R T1 ln
n
i
Total specific work for steady state flow process where only
shaft work is involved (no boundary work). Valid for ANY
reversible process (do not have to be polytropic)
i
s1
P
P1
s 2  s 1  R ln
Introduction to Thermodynamics,
equations. By Nasser M Abbasi
image2.vsd
August 2004
Ti
Shaft work (for
FLOW process only)
Boundary
Work
T2
T1
Cv0 ln
Te
Pi
n=k, constant entropy
w
Verify this, what
volume is this?
Entropy
change
s1
0
s2
1 constant T
P
Pe e Pi
R Te Ti
0
T2
e
w
e
Entropy change determination formulas,
for an ideal gas, polytropic process type
e
2
W
Entropy
change
V dp
n
Shaft work (for
FLOW process only)
Boundary
Work
P2 2
R T2
P2
4
n
Shaft work
(for FLOW
process
only)
s2
P dV
V dP
Boundary
Work
w
dH
Specialized polytropic
processes work formulas
0 constant P
n
Pe
PV
5
T dS
n
P dV
General
polytropic
relation
3
enthlapy law H
2
T ds
2
n
n
P dV
Substitute into
U
1
formulas for
general
polytropic
process
constant
T
W
st
n
T1
n
w
e
w
General polytropic process
P Vn
Q
ds
1
1
P2
Shaft work (for FLOW
process only)
Shaft Work
General formulas for reversible
compressible processes
T
2
1
R T2
1
i
s2
P2
Boundary
Work
reversible
Q
1
WORK
irreversible
2
n
k
Pe
e
Te
Ti
Pi
i