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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
Ideal gas, Any processAny gas, any process
General polytropic process
Ideal Gas Process classification
reversibleirreversible
P1 V1
T1
P2 V2
T2
P Vn
constant
n constant Vn 0 constant P
n 1 constant T
Boundary
Work
Boundary
Work
Boundary
Work
W 0
Boundary Work
Boundary
Work
P2
P1
T2
T1
n
n 1 V1
V2
n
w R T1 lnP1
P2
R T1 ln2
1
P1 1 ln2
1
u2 u1 Cv T2 T1
h2 h1 Cp T2 T1
s2 s11
2 Q
Tsgen
mass constant
s2 s11
2 C0
TdT R ln
2
1
s2 s11
2 Cp0
TdT R ln
P2
P1
Assume
constant
specific
heat
s2 s1 C0ln
T2
T1
R ln2
1
s2 s1 Cp0ln
T2
T1
R lnP2
P1
Table A.6
s2 s1 sT2
0 sT1
0 R lnP2
P1
Using
Table
A.7 or
A.8
WORK
Shaft work (for FLOW
process only)
wn
1 nPe e Pi i
n R
1 nTe Ti
wP2 2 P2 2
1 n
R T2 T1
1 n
wP2 2 P2 2
1 k
R T2 T1
1 k
w P2 2 P2 2
R T2 T1
Shaft work
(for FLOW
process
only)
W 0
Shaft work (for
FLOW process only)
w Pi i lnPe
Pi
R Ti lnPe
Pi
R Ti lne
i
Shaft work (for
FLOW process only)
wk
1 kPe e Pi i
k R
1 kTe Ti
Shaft work (for FLOW
process only)
n 1
n 1
w Pe e Pi i
R Te Ti
Verify this, what volume is this?
dsQ
T
W P dV
1stLaw Q W U
T ds dU P dV
Substitute into
Substitute into
T dS dH V dP
1
2
3
4
5
Gibbs equations
enthlapy law H U P V
so dH dU P dV V dp
Specialized polytropic
processes work formulas
General formulas for reversible compressible processes formulas for
general
polytropic
process
Introduction to Thermodynamics, equations. By Nasser M Abbasiimage2.vsdAugust 2004
Solving
Entropy change determination formulas, for an ideal gas, ANY process type
Entropy change determination formulas,
for an ideal gas, polytropic process type
General
polytropic
relation
s2 s1 Cv0
R
1 nln
T2
T1
s2 s1 Cv0 lnT2
T1
s2 s1 Cp0 lnT2
T1
s2 s1 0
Entropy
changeEntropy
change
Entropy
change
Entropy
change
n=k, constant entropy
wi
e
P
w1
2
P
Shaft Work
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)
wtotal i
ev dP
Vi2Ve
2
2 gZi Ze
wi
e
Pwi
e
P wi
e
P
du Cv0 dT
dh Cp0 dT
Cp0 Cv0 R
s2 s1 R lnP2
P1
GAS
ds CpdT
T R dP
P
Solids/Liquids
dP 0 (incompressible), and d 0
dh du
dh C dT
Ideal Gas
h u P
dh du dP
dh du P d v dP
P RT
so
h u RT
dh du R dT
dh CvdT RdT
dh CpdT
Process that causes irreversibility
1. Friction
2. Unrestrained expansion
3. Heat transfer from hot to cold body
4. Mixing of 2 differrent substances
5. i2R loss in electric circuits
6. Hystereris effects
7. Ordinary combustion
h2 h1 C lnT2
T1
h2 h1 Cp lnT2
T1
Entropy change equation
Solids/Liquids Ideal Gas
ds dq
T(by definition, entropy law)
dw duT
dwT
duT
1T
dPv 1T
dCvT
1TP dv v dP Cv
TdT
PT
dv vT
dP CvdTT
but Pv RT, hence
ds R dvv R dP
P Cv
dTT
s2 s1 R lnv2v1
R lnP2
P1 Cv ln
T2
T1 1
How to get this below from the above??
ds dq
T(by definition, entropy law)
dw duT
dwT
duT
1T
dPv 1T
d C T
1T
P dv v dP Cv
TdT
PT
dv vT
dP CvdTT
but dP 0 since incompressible,
and dv is very small, so
ds C dTT
s2 s1 C lnT2
T1
1Q W dE
where E U KE PE
Enthlapy definition
First law
FlowNon-Flow
Q1,2 W1,2 mu2 u1
Or, it can be written as follows (ignoring KE and PE changes to the control mass)
As a Rate equation
Q W dE
dt
Non-steady state (Transient, state change)
Q C.V. m ih KE PEi W C.V.m eh KE PE
e dE
dt
QC.V. mh KE PEi WC.V.mh KE PE
e
QC.V. mhi WC.V.mhe
QC.V. WC.V.mhe hi
Steady state devices: Heat exchanges, Nozzle, Diffuser, Throttle, Turbine, Compressors and Pumps
QC.V. mih KE PEi WC.V.meh KE PE
e m2u2 m1u1
General equation. Valid at any instance of time. Steady or not steady flow.
Usually Simplifies to
QC.V. mihi WC.V.mehe m2u2 m1u1
steady state. mi me m
q w he hi
heh i
h i m1 m2
State 1 State 2
Second law
Non-flow
ms2 s1 Q
T Sgen
s2 s1 q
T sgen
flow
steady transient
0 misi mese Q
T Sgen
0 si se q
T sgen
m2s2 m1s1 misi mese q
T sgen
Figure 1: thermodynamics
3
P1 V1 T1
State 1
P2 V2 T2
State 2
P Vn
constant
Polytropic process
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
Polytrpoic processby Nasser M Abbasipolytropic.vsdAugust 2004
n=0 const P
n constant VP
v
n=1, const T
n=k, adiapatic, const S
EXPANSION QUADRANT
COMPRESSIONQUADRANT
Ignore this quadrant in real
engineering equipments
Ignore this quadrant in real
engineering equipments
Initial
state
point
n=1 const T
T
s
n=0, const P
EXPANSION QUADRANT
COMPRESSIONQUADRANT
Initial
state
point
n=k const S
Clock wise, n=0,1,k,infinity
Clock wise, n=0,1,k,infinity
n constant V
Ignore this quadrant in real
engineering equipments
Ignore this quadrant in real
engineering equipments
Heat OUT processes
Heat IN processes
Heat IN processes
Heat OUT processes
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
Figure 2: polytropic process diagrams
4
Laws of thermodynamics
First law Second law
This is also called the law of conservation of energy
Chapter 5. 1st Law for control mass
Q12 W12 E2 E1
E U PE KE
Chapter 5.5
enthalpy
H U PV
h u Pv
Derived from first Law
by setting P constant
Q W mu2 u1
Q P dV mu2 u1
Q PV2 V1 mu2 u1
q P2 1 u2 u1
q P2 u2 P1 u1
1q2 h2 h1
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.
Chapter 5.6
Cv Cp
Specific heat is the
amount of heat required
to raise the temp. of a
unit mass by one degree
Cv uT v
Cp hT p
Solids and liquids: Use
average specific heat C.
dh du dP v
dh du v dP P dv
For solids and liquids, dv 0
dh du v dP For Solids and Liquids
Also for solids and liquids, v is very small, hence
dh du For Solids and Liquids
since almost incompressible, hence
Cv Cv For solids and liquids
For solids and liquids
Hence for solids/liquids, dh du C dT
For ideal gas
du Cv0 dT
dh Cp0 dT
Cp0 Cv0 R
Chapter 6
1st Law for control volume
Qin Win mih PE KEi Qout Wout meh PE KEe m2u2 m1u1
Steady
state
Transient
State
Adiabatic process
Wnet he hi
Wnet
heh i
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)
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.
So, 2nd law dictates the direction at which a system state will change. It
will move to a state of lesser available energy
Entropy is constant in a reversible adiabatic process.
S2 S1 1
2 Q
T Sgen Sgen 0
Sgen 0 reversible process
Q 0 adiabatic process
Hence, S2 S1 for a reversible adiabatic process
Wlost T dSgen
Actual boundary work W12 P dV Wlost
Gibbs relations
T ds du P dv
T ds dh v dp
Solids, Liquids
s2 s1 C lnT2
T1
Ideal Gas
sT0
T0
T Cp0
TdT
s2 s1 sT2
0 sT1
0 R lnP2
P1Using table A.7 or A.8
s2 s1 Cp0 lnT2
T1 R ln
P2
P1Constant Cp,Cv
s2 s1 Cv lnT2
T1 R ln
v 2
v 1Constant Cp,Cv
k Cp0
Cv0
Polytrpic process
PVn constantP2
P1 V1
V2
n
P2
P1 T2
T1
n
n1
specific work (work is moving boundary work, P dv
w12 1
1nP2v2 P1v1 R
1nT2 T1 n 1
w12 P1v1 lnv 2
v 1 RT1 ln
v 2
v 1 RT1 ln
P1
P2n 1
By Nasser M. AbbasiImage1.vsd
Figure 3: first and second laws diagrams
5
Solids/liquids
Ideal gas, Any processAny gas, any process
General polytropic process
Ideal Gas Process classification
reversibleirreversible
P1 V1
T1
P2 V2
T2
P Vn
constant
n constant Vn 0 constant P
n 1 constant T
Boundary
Work
Boundary
Work
Boundary
Work
W 0
Boundary Work
Boundary
Work
P2
P1
T2
T1
n
n 1 V1
V2
n
w R T1 lnP1
P2
R T1 ln2
1
P1 1 ln2
1
u2 u1 Cv T2 T1
h2 h1 Cp T2 T1
s2 s11
2 Q
Tsgen
mass constant
s2 s11
2 C0
TdT R ln
2
1
s2 s11
2 Cp0
TdT R ln
P2
P1
Assume
constant
specific
heat
s2 s1 C0ln
T2
T1
R ln2
1
s2 s1 Cp0ln
T2
T1
R lnP2
P1
Table A.6
s2 s1 sT2
0 sT1
0 R lnP2
P1
Using
Table
A.7 or
A.8
WORK
Shaft work (for FLOW
process only)
wn
1 nPe e Pi i
n R
1 nTe Ti
wP2 2 P2 2
1 n
R T2 T1
1 n
wP2 2 P2 2
1 k
R T2 T1
1 k
w P2 2 P2 2
R T2 T1
Shaft work
(for FLOW
process
only)
W 0
Shaft work (for
FLOW process only)
w Pi i lnPe
Pi
R Ti lnPe
Pi
R Ti lne
i
Shaft work (for
FLOW process only)
wk
1 kPe e Pi i
k R
1 kTe Ti
Shaft work (for FLOW
process only)
n 1
n 1
w Pe e Pi i
R Te Ti
Verify this, what volume is this?
dsQ
T
W P dV
1stLaw Q W U
T ds dU P dV
Substitute into
Substitute into
T dS dH V dP
1
2
3
4
5
Gibbs equations
enthlapy law H U P V
so dH dU P dV V dp
Specialized polytropic
processes work formulas
General formulas for reversible compressible processes formulas for
general
polytropic
process
Introduction to Thermodynamics, equations. By Nasser M Abbasiimage2.vsdAugust 2004
Solving
Entropy change determination formulas, for an ideal gas, ANY process type
Entropy change determination formulas,
for an ideal gas, polytropic process type
General
polytropic
relation
s2 s1 Cv0
R
1 nln
T2
T1
s2 s1 Cv0 lnT2
T1
s2 s1 Cp0 lnT2
T1
s2 s1 0
Entropy
changeEntropy
change
Entropy
change
Entropy
change
n=k, constant entropy
wi
e
P
w1
2
P
Shaft Work
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)
wtotal i
ev dP
Vi2Ve
2
2 gZi Ze
wi
e
Pwi
e
P wi
e
P
du Cv0 dT
dh Cp0 dT
Cp0 Cv0 R
s2 s1 R lnP2
P1
GASds Cp
dT
T R dP
P
Figure 4: gas lawss