AREN 2110 Second Law: Entropy, and Cycle Formula Summary Sheet
ENTROPY BALANCES
I. CLOSED SYSTEM
A. Irreversible processes:
δQ
⎛ kJ ⎞
+ S gen ⎜ ⎟
1 T
⎝K⎠
2
ΔS = S 2 − S1 = ∫
B. Reversible processes: Sgen = 0
δQ ⎛ kJ ⎞
⎜ ⎟
1 T ⎝ K ⎠
2
S 2 − S1 = ∫
C. Adiabatic irreversible processes: Q = 0
⎛ kJ ⎞
S 2 − S1 = S gen ⎜ ⎟
⎝K⎠
D. Adiabatic and reversible processes
S 2 − S1 = 0
E. Cycle
Q
δQ
⎛ kJ ⎞
⎛ kJ ⎞
0=∫
+ S gen ⎜ ⎟ = ∑ k + S gen ⎜ ⎟
T
⎝ K ⎠ k Tsurr
⎝K⎠
II. OPEN SYSTEM (CONTROL VOLUME)
A. Non-steady-state and irreversible control volume
⎛ Q& ⎞
dS cv
⎛ kw ⎞
= ∑ ⎜⎜ ⎟⎟ + ∑ m& i s i − ∑ m& e s e + S& gen ⎜ ⎟
k
e
dt
⎝K ⎠
⎝ T ⎠k i
subscript i is for inlet and e is for outlet
B. Steady-State and irreversible control volume
⎛ Q& ⎞
⎛ kw ⎞
∑ m& e s e − ∑ m& i si = ∑ ⎜⎜ ⎟⎟ +S& gen ⎜ ⎟
e
i
k
⎝K⎠
⎝ T ⎠k
C. Steady state and one inlet (1) and one outlet (2) and irreversible control volume
⎛ Q& ⎞
⎛ kw ⎞
m& Δs = m& (s 2 − s1 ) = ∑ ⎜⎜ ⎟⎟ +S& gen ⎜ ⎟
k
⎝K⎠
⎝ T ⎠k
D. Steady-state, one inlet and outlet and adiabatic control volume
⎛ kw ⎞
m& (s 2 − s1 ) = S& gen ⎜ ⎟
⎝K⎠
E. Steady state, one inlet and outlet and reversible control volume
⎛ Q& ⎞ ⎛ kw ⎞
m& (s 2 − s1 ) = ∑ ⎜⎜ ⎟⎟ ⎜ ⎟
k
⎝ T ⎠k ⎝ K ⎠
F. Steady-state, one inlet and outlet adiabatic and reversible control volume
⎛ kw ⎞
m& (s 2 − s1 ) = 0⎜ ⎟
⎝K⎠
G. Steady-state cycle comprised of open system (control volume) processes
⎛ Q& ⎞
⎛ kw ⎞
0 = ∑ ⎜⎜ ⎟⎟ +S& gen ⎜ ⎟
k
⎝K⎠
⎝ T ⎠k
Can substitute specific properties (kJ/kg-K) for each term above (divide by mass or mass flow rate)
CALCULATING ΔS (KJ/KG-K)
⎛T ⎞
A. Ideal (incompressible) liquids and solids: s 2 − s1 = C v ln⎜⎜ 2 ⎟⎟ where T in Kelvin and CV = CP
⎝ T1 ⎠
NOTE: isothermal processes in ideal liquids or solids are isentropic.
⎛T ⎞
⎛P ⎞
⎛T ⎞
⎛v ⎞
B. Ideal gases:
s 2 − s1 = C v ln⎜⎜ 2 ⎟⎟ + R ln⎜⎜ 2 ⎟⎟ OR s 2 − s1 = C P ln⎜⎜ 2 ⎟⎟ − R ln⎜⎜ 2 ⎟⎟
⎝ T1 ⎠
⎝ P1 ⎠
⎝ T1 ⎠
⎝ v1 ⎠
Equivalent formulas for ideal gases where T in Kelvin and CP = CV + R
Problem solving hint: can use isentropic ideal gas process formula to find T given P or v values
to then use in 1st law calculation of energy terms.
C. Non-ideal pure substances use tables for water/steam (A-4 – A-6) or R-134a (A-11 – A-13)
(i) smixture = x(sfg) + sf (at appropriate Tsat or Psat)
(ii) scompressed liquid ≈ sf at system T = Tsat
Multiply by mass or mass flow rate for extensive property, kJ/K or kw/K
Entropy Balance Problem solving.
1. Find appropriate entropy balance formula to account for process conditions (e.g., reversible,
adiabatic, etc.)
2. Calculate heat (Q) term using the first law, if necessary
3. Calculate change in entropy property term (Δs) using appropriate formula or table
T-S DIAGRAMS
T (o C)
General. For internally reversible processes, area under process line = Q (+ for increasing entropy and
– for decreasing entropy during process). Line A is isothermal process. Line B is isentropic process.
For Cycle, area enclosed by lines = net heat transfer and Qnet = Wnet (1st Law)
120
100
80
60
40
20
0
-20
-40
-60
A
Cycle
B
0.0
0.2
0.4
0.6
s (kJ/kg-K)
0.8
1.0
1.2
o
T ( C)
T-s diagram for Carnot Cycles consisting of two isothermal processes (one expansion and one
compression) and two isentropic processes (one compression and one expansion)
120
100
80
60
40
20
0
-20
-40
-60
Carnot
Heat
Engine
0.0
0.2
Carnot
Refrigerator or
Heat Pump
0.4
0.6
0.8
1.0
1.2
s (kJ/kg-K)
Heat engine: signs for Qnet = Wnet terms will be positive
Refrigerator/Heat Pump: signs for Qnet = Wnet terms will be negative
HEAT ENGINES AND REFRIGERATION CYCLES
I. Heat Engines: Produce work from heat in cycle with some heat rejected.
A. All Heat Engines
η=
Wnet wnet W& net
W& net
Q
q
Q&
=
=
= 1− L = 1− L = 1− L =
QH
qH
Qnet
qH
Q& H
Q& H m& ( q H )
B. Carnot Heat Engines
ηcarnot = 1 −
TL ( Kelvin )
and all the relations in part IA.
TH ( Kelvin )
II. Refrigerators and Heat Pumps: Transfer heat from low temperature reservoir to high temperature
reservoir with work input.
A. All refrigerators
COPR =
Q L Q& L q L
m& q L
1
1
1
=
=
=
=
=
=
Win W& in win ⎛ QH
⎞ ⎛ Q&
⎞ ⎛q
⎞ W& in
⎜⎜
− 1⎟⎟ ⎜⎜ H − 1⎟⎟ ⎜⎜ H − 1⎟⎟
⎝ QL
⎠ ⎝ Q& L
⎠
⎠ ⎝ qL
B. Carnot refrigerator
COPR ,Carnot =
1
⎛ TH
⎞
⎜⎜
− 1⎟⎟
⎝ TL
⎠
and all the relations in part IIA
C. All heat pumps
COPHP =
QH Q& H q H
1
=
=
=
Win W& in win ⎛ QL
⎜⎜1 −
⎝ QH
⎞
⎟⎟
⎠
=
1
⎛ Q& L
⎜⎜1 −
&
⎝ QH
⎞
⎟⎟
⎠
=
1
⎛
q
⎜⎜1 − L
⎝ qH
⎞
⎟⎟
⎠
=
m& q H
=COPR + 1
W&
in
D. Carnot heat pump
1
and all the relations in part IIC
⎛ TL ⎞
⎟⎟
⎜⎜1 −
⎝ TH ⎠
IDEAL RANKINE CYCLE (internally reversible processes okay to evaluate on T-s diagram)
COPHP ,Carnot =
1.
2.
3.
4.
Two isentropic processes (pump and turbine)
One Isobaric process (boiler)
One isothermal and isobaric process (condenser)
Typically, unless otherwise specified, saturated liquid at pump inlet, compressed liquid at
boiler inlet, superheated vapor at turbine inlet, mixture at condenser inlet.
T oC
3
2
1
4
s (kJ/kg-K)
Rankine Cycle Characteristics
Process
1Æ2
2Æ3
3Æ4
4Æ1
State
1
2
3
4
Device
Pump
Boiler
Turbine
Condenser
Phase
Saturated
liquid
Compressed
Liquid
Superheated
Vapor
Saturated
Mixture
Condition
Isentropic
Isobaric
Isentropic
Isothermal and Isobaric
Energy
Work Input, wP
Heat Input, qH
Work Output, wT
Heat Output, qL
T (oC)
T1 = Tsat @ P1
P (kPa)
P1 = Pcondenser
H (kJ/kg)
hf @ P1
s (kJ/kg-K)
s1 = sf @ P1
~ T1
P2 = Pboiler
h2 = h1 + v (P2-P1)
s2 = s1
T3
P3 = P2
h3
s3
T4 = Tsat @ P4
P4 = P1
h4 = x4(hfg)+hf
s4 = s3
Typically, the pressures for the boiler and condenser and the temperature at the turbine inlet are given.
Quality of mixture is found from isentropic turbine, given inlet temperature.
1st Law statements
-wP = h2 – h1
qH = h3 - h2
-wT = h4 – h3
qL = h1 – h4
qNET = wNET (with appropriate sign conventions)
qH + qL = wP + wT (with appropriate sign conventions)
Efficiency (efficiency < 1 and efficiency < ηcarnot)
η=
W& net
m& [( h3 − h4 ) − ( h2 − h1 )]
W& net
q
(h −h )
=
= 1− L = 1− 4 1 =
qH
( h3 − h2 ) m& ( h3 − h2 )
Q& H
Q& H
Entropy (entropy generated in surroundings > 0)
(h −h ) (h −h )
q
q
⎛q⎞
s gen = − ∑ ⎜ ⎟ = − H − L = − 3 2 − 1 4
k ⎝ T ⎠k
TH TL
TH
TL
⎛ kJ ⎞
⎜⎜
⎟⎟
⎝ kg − K ⎠
and
⎛ kw ⎞
⎟
⎜
⎝ K ⎠
given temperature of high- and low-temperature reservoirs (heat source and sink)
& s gen
S& gen = m
IDEAL VAPOR-COMPRESSION REFRIGERATION (VCR) CYCLE (internally reversible processes except
for throttle, still okay to evaluate on T-s diagram)
1.
2.
3.
4.
5.
One isentropic process (compressor)
One isobaric process (condenser)
One isothermal and isobaric process (evaporator)
One isenthalpic process (throttling valve)
Typically, unless otherwise specified, saturated vapor at compressor inlet, superheated vapor at
condenser inlet, saturated liquid at throttling valve inlet, saturated mixture at evaporator inlet.
T oC
2
3
1
4
s (kJ/kg-K)
VCR Cycle Characteristics
Process
1Æ2
2Æ3
3Æ4
4Æ1
State
1
2
3
4
Phase
Saturated
vapor
Superheated
vapor
Saturated
liquid
Saturated
Mixture
Device
Compressor
Condenser
Throttling Valve
Evaporator
Condition
Isentropic
Isobaric
Isenthalpic
Isothermal and Isobaric
Energy
Work Input, wC
Heat Rejected, qH
-Heat Input, qL
T (oC)
T1 = Tsat @ P1
P (kPa)
P1 = Pevaporator
h (kJ/kg)
hg @ P1
s (kJ/kg-K)
s1 = sg @ Tsat
T2
P2 = Pcondenser
h2
s2 = s 1
Tsat @ P3
P3 = P2
hf @ P3
sf @ P3
T4 = Tsat @ P4
P4 = P1
h4 =h3
s4 = x4(sfg) + sf
@ P4
Typically, the pressures for the boiler and condenser are given. T2 and h2 are found from isentropic
compressor relation and interpolation in table to match s2 = s1.
1st Law statements
-wC = h2 – h1
qH = h3 - h2
h4 = h3
qL = h1 – h4
qNET = wNET (with appropriate sign conventions)
qH + qL = wC (with appropriate sign conventions)
COPR
COPR =
Q& L q L ( h1 − h4 )
m& ( h1 − h4 )
1
1
=
=
=
=
=
W& C wC ( h2 − h1 ) ⎛ q H
W& C
⎞ ⎛ ( h − h3 ) ⎞
⎜⎜
− 1⎟⎟ ⎜⎜ 2
− 1⎟⎟
⎝ qL
⎠ ⎝ ( h1 − h4 ) ⎠
COPHP
COPHP =
Q& H q H ( h2 − h3 )
1
=
=
=
W& C wC ( h2 − h1 ) ⎛
q
⎜⎜1 − L
⎝ qH
⎞
⎟⎟
⎠
=
1
⎛ ( h1 − h4 ) ⎞
⎜⎜1 −
⎟⎟
−
(
h
h
)
2
3 ⎠
⎝
=
m& ( h2 − h3 )
W&
C
Entropy (entropy generated in surroundings > 0) for both VCR refrigerator and heat pump
( h − h2 ) ( h1 − h4 )
q
q
⎛q⎞
s gen = − ∑ ⎜ ⎟ = − H − L = − 3
−
k ⎝ T ⎠k
TH TL
TH
TL
⎛ kJ ⎞
⎟⎟
⎜⎜
⎝ kg − K ⎠
and
& s gen
S& gen = m
⎛ kw ⎞
⎟
⎜
⎝ K ⎠
given temperature of high- and low-temperature reservoirs (heat source and sink)