AREN 2110 Second Law: Entropy, and Cycle Formula Summary Sheet
ENTROPY BALANCES
I. CLOSED SYSTEM A. Irreversible processes: 2 δQ ⎛ kJ ⎞ SSS 12 =−=Δ ∫ + S gen ⎜ ⎟ 1 T ⎝ K ⎠ B. Reversible processes: Sgen = 0 2 δQ⎛ kJ ⎞ SS 12 =− ∫ ⎜ ⎟ 1 T ⎝ K ⎠ C. Adiabatic irreversible processes: Q = 0 ⎛ kJ ⎞ 12 =− SSS gen ⎜ ⎟ ⎝ K ⎠ D. Adiabatic and reversible processes
− SS 12 = 0 E. Cycle
δQ ⎛ kJ ⎞ Qk ⎛ kJ ⎞ 0 = ∫ + S gen ⎜ ⎟ = ∑ + S gen ⎜ ⎟ T ⎝ K ⎠ k Tsurr ⎝ K ⎠
II. OPEN SYSTEM (CONTROL VOLUME) A. Non-steady-state and irreversible control volume dS ⎛ Q& ⎞ kw cv ⎜ ⎟ ⎛ ⎞ = ∑∑⎜ ⎟ ii −+ ∑ && ee + Ssmsm & gen ⎜ ⎟ dt kiT e K ⎝ ⎠ k ⎝ ⎠ subscript i is for inlet and e is for outlet B. Steady-State and irreversible control volume ⎛ Q& ⎞ ⎛ kw⎞ ⎜ ⎟ ∑ ee − ∑ && smsm ii = ∑⎜ ⎟ +S& gen ⎜ ⎟ e i k T K ⎝ ⎠ k ⎝ ⎠ C. Steady state and one inlet (1) and one outlet (2) and irreversible control volume ⎛ Q& ⎞ ⎛ kw⎞ ⎜ ⎟ && ()ssmsm 12 =−=Δ ∑⎜ ⎟ +S& gen ⎜ ⎟ k T K ⎝ ⎠k ⎝ ⎠ D. Steady-state, one inlet and outlet and adiabatic control volume ⎛ kw⎞ & ()12 =− Sssm &gen ⎜ ⎟ ⎝ K ⎠ E. Steady state, one inlet and outlet and reversible control volume ⎛ Q& ⎞ kw ⎜ ⎟ ⎛ ⎞ & ()ssm 12 =− ∑⎜ ⎟ ⎜ ⎟ k T K ⎝ ⎠ k ⎝ ⎠ F. Steady-state, one inlet and outlet adiabatic and reversible control volume ⎛ kw⎞ & ()ssm 12 =− 0⎜ ⎟ ⎝ K ⎠ G. Steady-state cycle comprised of open system (control volume) processes ⎛ Q& ⎞ kw ⎜ ⎟ ⎛ ⎞ 0 = ∑⎜ ⎟ +S& gen ⎜ ⎟ k T K ⎝ ⎠ 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: ⎜ 2 ⎟ where T in Kelvin and C = C =− v12 lnCss ⎜ ⎟ V P ⎝ T1 ⎠ NOTE: isothermal processes in ideal liquids or solids are isentropic. ⎛ T ⎞ ⎛ v ⎞ ⎛ T ⎞ ⎛ P ⎞ B. Ideal gases: ⎜ 2 ⎟ ⎜ 2 ⎟ OR =− lnCss ⎜ 2 ⎟ − lnR ⎜ 2 ⎟ =− v12 lnCss ⎜ ⎟ + lnR ⎜ ⎟ P12 ⎜ ⎟ ⎜ ⎟ ⎝ T1 ⎠ ⎝ v1 ⎠ ⎝ T1 ⎠ ⎝ P1 ⎠ 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
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. st For Cycle, area enclosed by lines = net heat transfer and Qnet = Wnet (1 Law)
120 100 80 A 60
C) 40 o 20 T ( 0 Cycle -20 B -40 -60 0.0 0.2 0.4 0.6 0.8 1.0 1.2 s (kJ/kg-K)
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
C) 40 Carnot Carnot o 20 Heat Refrigerator or T ( 0 Engine Heat Pump -20 -40 -60 0.0 0.2 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 W w W& Q q Q& W& net net net L L 111 L =−=−=−====η net Qnet qH Q& H QH qH Q& H & H )q(m B. Carnot Heat Engines
TL )Kelvin( carnot 1−=η 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
QL Q& L qL 1 1 1 & qm L COPR ======W W& w ⎛ Q ⎞ ⎛ Q& ⎞ ⎛ q ⎞ W& in in in ⎜ H −1⎟ ⎜ H −1⎟ ⎜ H −1⎟ in ⎜ Q ⎟ ⎜ ⎟ ⎜ q ⎟ ⎝ L ⎠ ⎝ Q& L ⎠ ⎝ L ⎠
B. Carnot refrigerator 1 COP Carnot,R = and all the relations in part IIA ⎛TH ⎞ ⎜ −1⎟ ⎝ TL ⎠
C. All heat pumps
QH Q& H qH 1 1 1 & qm H COPHP ======COPR + 1 Win W&in win ⎛ QL ⎞ ⎛ Q& L ⎞ ⎛ qL ⎞ W&in ⎜1− ⎟ ⎜1− ⎟ ⎜1− ⎟ ⎝ QH ⎠ ⎝ Q& H ⎠ ⎝ qH ⎠
D. Carnot heat pump 1 COP Carnot,HP = and all the relations in part IIC ⎛ TL ⎞ ⎜1− ⎟ ⎝ TH ⎠ IDEAL RANKINE CYCLE (internally reversible processes okay to evaluate on T-s diagram)
1. Two isentropic processes (pump and turbine) 2. One Isobaric process (boiler) 3. One isothermal and isobaric process (condenser) 4. 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
4 1
s (kJ/kg-K)
Rankine Cycle Characteristics
Process Device Condition Energy 1Æ2 Pump Isentropic Work Input, wP 2Æ3 Boiler Isobaric Heat Input, qH 3Æ4 Turbine Isentropic Work Output, wT 4Æ1 Condenser Isothermal and Isobaric Heat Output, qL
State Phase T (oC) P (kPa) H (kJ/kg) s (kJ/kg-K) 1 Saturated T1 = Tsat @ P1 P1 = Pcondenser hf @ P1 s1 = sf @ P1 liquid 2 Compressed ~ T1 P2 = Pboiler h2 = h1 + v (P2-P1) s2 = s1 Liquid 3 Superheated T3 P3 = P2 h3 s3 Vapor 4 Saturated T4 = Tsat @ P4 P4 = P1 h4 = x4(hfg)+hf s4 = s3 Mixture
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)
W&net qL − 14 )hh( W&net &[ 43 −−− 12 )hh()hh(m ] 11 −=−==η = = Q& H qH − 23 )hh( & − 23 )hh(m Q& H
Entropy (entropy generated in surroundings > 0)
⎛ q ⎞ qH qL − 23 )hh( − 41 )hh( ⎛ kJ ⎞ sgen −= ∑⎜ ⎟ −=−−= − ⎜ ⎟ k ⎝ T ⎠k TH TL TH TL ⎝ − Kkg ⎠ and ⎛ kw ⎞ & gen = & smS gen ⎜ ⎟ ⎝ K ⎠ given temperature of high- and low-temperature reservoirs (heat source and sink) IDEAL VAPOR-COMPRESSION REFRIGERATION (VCR) CYCLE (internally reversible processes except for throttle, still okay to evaluate on T-s diagram)
1. One isentropic process (compressor) 2. One isobaric process (condenser) 3. One isothermal and isobaric process (evaporator) 4. One isenthalpic process (throttling valve) 5. 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 Device Condition Energy 1Æ2 Compressor Isentropic Work Input, wC 2Æ3 Condenser Isobaric Heat Rejected, qH 3Æ4 Throttling Valve Isenthalpic -- 4Æ1 Evaporator Isothermal and Isobaric Heat Input, qL
State Phase T (oC) P (kPa) h (kJ/kg) s (kJ/kg-K) 1 Saturated T1 = Tsat @ P1 P1 = Pevaporator hg @ P1 s1 = sg @ Tsat vapor 2 Superheated T2 P2 = Pcondenser h2 s2 = s1 vapor 3 Saturated Tsat @ P3 P3 = P2 hf @ P3 sf @ P3 liquid 4 Saturated T4 = Tsat @ P4 P4 = P1 h4 =h3 s4 = x4(sfg) + sf Mixture @ 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
Q& L qL − 41 )hh( 1 1 & − 41 )hh(m COPR ======W&C wC − 12 )hh( ⎛ qH ⎞ ⎛ − 32 )hh( ⎞ W&C ⎜ −1⎟ ⎜ −1⎟ ⎝ qL ⎠ ⎝ − 41 )hh( ⎠
COPHP
Q& H qH − 32 )hh( 1 1 & − 32 )hh(m COPHP ======W&C wC − 12 )hh( ⎛ qL ⎞ ⎛ − 41 )hh( ⎞ W&C ⎜1− ⎟ ⎜1− ⎟ ⎝ qH ⎠ ⎝ − 32 )hh( ⎠
Entropy (entropy generated in surroundings > 0) for both VCR refrigerator and heat pump
⎛ q ⎞ qH qL − 23 )hh( − 41 )hh( ⎛ kJ ⎞ sgen −= ∑ ⎜ ⎟ −=−−= − ⎜ ⎟ k ⎝ T ⎠k TH TL TH TL ⎝ − Kkg ⎠ and
⎛ kw ⎞ & gen = & smS gen ⎜ ⎟ ⎝ K ⎠
given temperature of high- and low-temperature reservoirs (heat source and sink)