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all of thermodynamics in one sheet
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Ideal gas, Any process Any gas, any process
General polytropic process
Ideal Gas Process classification
irreversible reversible
P 1 V 1
T 1
P 2 V 2
T 2
P V n^ constant
n 0 constant P n 1 constant T
Boundary Work
Boundary Work
Boundary Work
Boundary Work Boundary Work
n
n
w R T 1 ln
R T 1 ln
2
1
P 1 1 ln
2
1
1
mass constant
s 2 s 1 1
dT R ln
2
1
s 2 s 1 1
2 Cp 0
T
dT R ln
Assume constant specific heat
s 2 s 1 C 0 ln
R ln
2
1
s 2 s 1 Cp 0 ln
R ln
Table A.
s 2 s 1 sT 2
(^0) s T 1
(^0) R ln P 2 P 1
Using Table A.7 or A. WORK
Shaft work (for FLOW process only)
w
n
1 n
Pe e Pi i
n R
1 n
Te Ti
w P 2 2 P 2 2
R T 2 T 1
Shaft work (for FLOW process only)
Shaft work (for FLOW process only)
e
i
Shaft work (for FLOW process only)
Shaft work (for FLOW process only)
Verify this, what volume is this?
ds
Q T W P dV
1 st^ Law 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 Abbasi image2.vsd August 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
s 2 s 1 Cv 0
R
1 n
ln
T 2
T 1
s 2 s 1 Cp 0 ln
s 2 s 1 0
Entropy change Entropy change
Entropy change
Entropy change
n=k, constant entropy
w i
e P
w 1
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
e v dP
V i^2 V e^2 2
g Zi Ze
w i
e w P i
e P w i
e P
s 2 s 1 R ln
GAS
Solids/Liquids
dP 0 (incompressible), and d 0 dh du dh C dT
Ideal Gas
dh du d P dh du P d v dP
P RT so h u RT dh du R dT dh Cv dT 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. i^2 R loss in electric circuits 6. Hystereris effects 7. Ordinary combustion
h 2 h 1 C ln T T^2 1
h 2 h 1 Cp ln T T^2 1
Entropy change equation
Solids/Liquids Ideal Gas
ds
dq T
(by definition, entropy law)
dw^ ^ du T
dw T
du T 1 T
d Pv 1 T
d Cv T
P dv v dP Cv T
dT
PT dv vT dP Cv dTT
but Pv RT , hence
ds R dvv R dP P
Cv dT T
s 2 s 1 R ln
v 2 v 1 ^ R^ ln^
Cv ln
How to get this below from the above??
ds
dq T (by definition, entropy law) dw^ ^ du T
dw T
du T (^1) T d Pv (^1) T d C T
(^1) T P dv v dP
Cv T dT P T
dv v T
dP Cv dT T but dP 0 since incompressible, and dv is very small, so
ds C dT T s 2 s 1 C ln T^2 T 1
1 Q W dE
where E U KE PE
Enthlapy definition
First law
Non-Flow Flow
Or, it can be written as follows (ignoring KE and PE changes to the control mass)
As a Rate equation
dt
Non-steady state (Transient, state change)
Q ^ C. V. m (^) i h KE PE i W ^ C. V. m (^) e h KE PE e dE dt
QC. V. m h KE PE i WC. V. m h KE PE e QC. V. mhi WC. V. mhe QC. V. WC. V. m he hi
Steady state devices: Heat exchanges, Nozzle, Diffuser, Throttle, Turbine, Compressors and Pumps
QC. V. mi h KE PE i WC. V. me h KE PE e m 2 u 2 m 1 u 1
General equation. Valid at any instance of time. Steady or not steady flow.
Usually Simplifies to
QC. V. mihi WC. V. mehe m 2 u 2 m 1 u 1
steady state. mi me m
q w he hi
hi he
hi m 1 m 2
State 1 State 2
Second law
Non-flow
m s 2 s 1
Q T ^ Sgen s 2 s 1
q T ^ sgen
(^) flow
steady transient
0 misi mese QT Sgen 0 si se qT sgen
m 2 s 2 m 1 s 1 misi mese
q T ^ sgen
Figure 1: thermodynamics
This is also called the law of conservation of energy
Chapter 5. 1 st^ Law for control mass
Q 1 2 W 1 2 E 2 E 1
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
q P 2 u 2 P 1 u 1 1 q 2 h 2 h 1
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.
Specific heat is the amount of heat required to raise the temp. of a unit mass by one degree
Cv u T v
Cp h T (^) p Solids and liquids: Use average specific heat C.
dh du d P 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
Cv Cv For solids and liquids
For solids and liquids
Hence for solids/liquids, dh du C dT
(^) For ideal gas
Chapter 6 1 st^ Law for control volume
Steady state
Transient State
Adiabatic process
Wnet he hi
Wnet
hi he
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 2 nd^ 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, 2 nd^ 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.
S 2 S 1 1
(^2) Q
Q 0 adiabatic process
Hence, S 2 S 1 for a reversible adiabatic process
Wlost T dSgen Actual boundary work W 1 2 P dV Wlost
Gibbs relations T ds du P dv T ds dh v dp
Solids, Liquids
s 2 s 1 C ln
T 2 T 1
Ideal Gas
T 0
T Cp 0 T dT
s 2 s 1 sT^0 2 sT^0 1 R ln
P 2 P 1 Using table A.7 or A. s 2 s 1 Cp 0 ln
T 2 T 1 ^ R^ ln^
P 2 P 1 Constant^ Cp ,^ Cv s 2 s 1 Cv ln T T^2 1
R ln v v^^2 1 Constant Cp , Cv
k
Cp 0 Cv 0 Polytrpic process PVn^ constant P 2 P 1 ^
V 1 V 2
n
P 2 P 1 ^
T 2 T 1
n n 1
w 1 2
1 1 n P^2 v^2 ^ P^1 v^1 ^ ^
R 1 n T^2 ^ T^1 ^ n^ ^1
w 1 2 P 1 v 1 ln
v (^) 2 v (^) 1 ^ RT 1 ln^
v (^) 2 v (^) 1 ^ RT 1 ln^
P 1 P 2^ n^ ^1
By Nasser M. Abbasi Image1.vsd
Figure 3: first and second laws diagrams
General polytropic process
P V
n constant
n constant V
n 1 constant T
Boundary Work
Boundary Work
Boundary Work
W 0
Boundary Work Boundary Work
P 2
P 1
T 2
T 1
n n 1 V 1
V 2
n
2
1
2
1
u 2 u 1 Cv T 2 T 1
h 2 h 1 Cp T 2 T 1
s 2 s 1 1
2 Q
T
sgen
mass constant
1
2
1
1
Assume constant specific heat
s 2 s 1 C 0
ln
T 2
T 1
R ln
s 2 s 1 Cp 0 ln
T 2
T 1
R ln
P 2
P 1
Table A.
s 2 s 1 sT 2
sT 1
R ln
Using Table A.7 or A.
WORK
w
P 2 2 P 2 2
1 n
R T 2 T 1
1 n
w
P 2 2 P 2 2
1 k
R T 2 T 1
1 k
w P 2 2 P 2 2
R T 2 T 1
Shaft work (for FLOW process only)
W 0
Shaft work (for FLOW process only)
w Pi i ln
Pe
Pi
R Ti ln
Pe
Pi
R Ti ln
e
i
Shaft work (for FLOW process only)
w
k
1 k
Pe e Pi i
k R
1 k
Te Ti
Shaft work (for FLOW process only)
n 1
n 1
w Pe e Pi i
R Te Ti
ds
W P dV
st Law Q W U
T ds dU P dV
Substitute into
Substitute into
General formulas for reversible compressible processes formulas for general polytropic process
Introduction to Thermodynamics, equations. By Nasser M Abbasi image2.vsd August 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
s 2 s 1 Cv
R
1 n
ln
T 2
T 1
s 2 s 1 Cv
ln
T 2
T 1
s 2 s 1 Cp 0 ln
T 2
T 1
s 2 s 1 0
Entropy change Entropy change
Entropy change
n=k, constant entropy
w i
e P
w 1
2 P
Shaft Work
wtotal
v dP
g Zi Ze
w i
e
i
e
i
e P
du Cv 0 dT
dh Cp 0 dT
Cp 0 Cv 0 R
s 2 s 1 R ln
ds Cp
dT T ^ R^
dP P
Figure 4: gas lawss
