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Summary of Useful Formulae ME 24-221 Thermodynamics I Some Useful Formulae 1 Control Mass Continuity Equation m constant First Law £ 2 ¡ 2 £ ¢ ¢ m V V ¤ 2 1 ¡ ¡ U ¡ U mg Z Z Q W 2 1 2 2 1 ¤ 1 2 1 2 Compression-expansion work 2 ¦¥ 1W2 PdV 1 For polytropic procees, PV n c , P V ¡ P V 2 2 1 1 W n § 1 1 2 1 ¡ n V 2 P1V1ln n 1 V1 Second Law 2 δ Q ¢ ¨¥ S2 ¡ S1 1 S2 gen 1 T For isothermal process 2 δQ Q ¥ 1 2 1 T T For reversible process 1S2 gen 0 For adiabatic process 1Q2 0 Therefore, for a reversible adiabatic process S2 ¡ S1 0 ¡ s2 s1 0 Therefore, a reversible adiabatic process is an isentropic process. 1 2 Control Volume 2.1 Steady State Steady Flow (SSSF) Continuity ∑m˙ i ¡ ∑m˙ e 0 i e First Law 2 2 ¢ ¢ ¢ ¢ ¢ V V ¡ ∑m˙ h i gZ ∑m˙ h e gZ Q˙ ¡ W˙ 0 © i © i i e e e cv cv i 2 e 2 Second Law Q˙ ¢ ¢ j ¡ ˙ ∑m˙ isi ∑m˙ ese ∑ Sgen 0 i e j Tj Reversible process S˙gen 0 Adiabatic process Q˙ 0 For one inlet-one outlet device, a reversible adiabatic process is therefore an isentropic process, with si se 2.2 Uniform State Uniform Flow (USUF) Continuity £ ¡ m ¡ m ∑m ∑m 2 1 ¤ i e i e First Law 2 2 2 2 ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ V V V V ¡ ¡ m u 2 gZ ¡ m u 1 gZ ∑m h i gZ ∑m h e gZ Q W © © © 2 © 2 2 1 1 1 i i i e e e cv cv 2 2 i 2 e 2 Second Law t ˙ ¢ ¢ Q ¥ cv ¡ ¡ m2s2 m1s1 ∑misi ∑mese dt 1 S2 gen i e 0 T 3 Gibbs Equation ¢ T ds du Pdv dh ¡ vdP This equation holds true for all simple compressible substances. 2 4 Properties of Pure Substances 4.1 Vapor-Liquid Phase Equilibrium For a specific property φ (such as h,u,v,s etc) under the dome ¢ φ φ φ f x f g 4.2 Ideal Gas Ideal Gas Equations of State Pv RT du CvdT 2 ¥ u2 ¡ u1 CvdT 1 £ C T ¡ T if C is constant v 2 1 ¤ v dh CpdT 2 ¥ ¡ h2 h1 CpdT 1 £ C T ¡ T if C is constant p 2 1 ¤ p Specific Heats and Ideal Gas Constants Cp ¡ Cv R R R M Cp k Cv Entropy Relationships dT dP ds C ¡ R p T P 2 dT P ¥ ¡ ¡ 2 s2 s1 Cp Rln 1 T P1 T P 2 ¡ 2 Cpln Rln if constant Cp T1 P1 P ¡ s0 ¡ s0 Rln 2 otherwise T2 T1 P1 dT ¢ dv ds C R v T v 2 dT ¢ v ¥ ¡ 2 s2 s1 Cv Rln 1 T v1 T ¢ v 2 2 Cvln Rln if constant Cv T1 v1 3 Isentropic Process for Ideal Gas For variable specific heats P P 1 r1 P2 Pr2 v v 1 r1 v2 vr2 For constant specific heats Pvk constant P v k 1 2 © P2 v1 k 1 T P k 2 2 © T1 P1 T v k 1 2 1 © T1 v2 The above four relationships also hold for a reversible polytropic process with the polytropic exponent n replacing k. 4.3 Incompressible Substance Equation of State du dh CdT 2 ¥ ¡ ¡ u2 u1 h2 h1 CdT 1 £ C T ¡ T if C is constant 2 1 ¤ Entropy Relationships du ds T 2 dT ¥ s2 ¡ s1 C 1 T T Cln 2 if C is constant T1 C=constant can usually be assumed for incompressible substances. 4 5 Heat Engines, Heat Pumps and Refrigerators Thermal efficiency of heat engine W η net th QH Q ¡ Q H L QH Coefficient of Performance (C.O.P) of Heat Pump Q β H Wnet Q H QH ¡ QL Coefficient of Performance (C.O.P) of Refrigerator Q β L Wnet Q L QH ¡ QL Carnot Cycle Q T H H QL TL T η ¡ L th 1 TH T β H TH ¡ TL T β L TH ¡ TL 6 Isentropic Efficiency of Engineering Devices w η turbine ws w η s compressor w 2 V 2 η e nozzle 2 Ves 2 5 7 Irreversibility and Availability Availability: ¡ ¡ Ψ ¡ h T s h T s 0 re f 0 re f Irreversibility: ˙ ˙ I T0Sgen ˙ ˙ Work in an SSSF process with heat gain QH from TH and heat loss Q0 to T0: T ¢ ¡ ¡ 0 ¡ W˙ Q˙ 1 m˙ Ψ Ψ I˙ net H © i e TH Second Law Efficiency (Effectiveness) Turbine: W˙ η net 2 Ψ ¡ Ψ m˙ i e Compressor: Ψ ¡ Ψ m˙ η i e 2 W˙ net 8 Power and Refrigeration Cycles 8.1 Ideal Rankine Cycle p=c p=c 3 T 2 1 4 s 1-2:Isentropic compression in pump 2-3: Constant pressure heat addition in boiler 3-4: Isentropic expansion in turbine 4-1: Constant pressure heat rejection in condenser Working fluid is usually water or other two-phase substance 6 Work and heat transfer formulae: ¡ wp h1 h2 £ v P ¡ P 1 1 2 ¤ ¡ qb h3 h2 ¡ wT h3 h4 ¡ qc h1 h4 Also s1 s2 s3 s4 8.2 Ideal Air Standard Brayton Cycle p=c 3 T 2 p=c 4 1 s 1-2:Isentropic compression in compressor 2-3: Constant pressure heat addition in heat exchanger 3-4: Isentropic expansion in turbine 4-1: Constant pressure heat rejection in heat exchanger Working fluid idealized to be air Work and heat transfer formulae: ¡ wC h1 h2 ¡ qh h3 h2 wT h3 ¡ h4 ¡ qc h1 h4 Also s1 s2 s3 s4 7 For constant Cp £ w C T ¡ T C p 1 2 ¤ £ q C T ¡ T h p 3 2 ¤ £ w C T ¡ T T p 3 4 ¤ £ q C T ¡ T c p 1 4 ¤ Also k 1 T P k 2 2 © T1 P1 k 1 T P k 4 4 © T3 P3 For variable Cp P P r2 2 Pr1 P1 P P r4 4 Pr3 P3 Thermal efficiency of Brayton cycle (For constant Cp) T η ¡ 1 th 1 T2 k 1 P k 1 ¡ 1 © P2 8.3 Ideal Air Standard Otto Cycle v =c 3 T 2 v = c 4 1 s Otto cycle is a piston-cylinder cycle (control mass) Models spark ignition (SI) engines 1-2:Isentropic compression 8 2-3: Constant volume heat addition 3-4: Isentropic expansion 4-1: Constant volume heat rejection Working fluid idealized to be air Process 1-2: s1 s2 k 1 T P k 2 2 i f C constant © p T1 P1 T V k 1 2 1 i f C constant © p T1 V2 ¡ ¡ u2 u1 1w2 £ C T ¡ T i f C constant v 2 1 ¤ p Process 2-3: ¡ u3 u2 2q3 £ C T ¡ T i f C constant v 3 2 ¤ p Process 3-4: s3 s4 k 1 T P k 4 4 i f C constant © p T3 P3 T V k 1 4 3 i f C constant © p T3 V4 ¡ ¡ u4 u3 3w4 £ C T ¡ T i f C constant v 4 3 ¤ p Process 4-1: ¡ u1 u4 1q4 £ C T ¡ T i f C constant v 1 4 ¤ p Thermal efficiency of Otto cycle (For constant Cp) T η ¡ 1 th 1 T2 V 1 k 1 ¡ 1 © V2 1 k ¡ 1 rv where rv is the compression ratio. Mean Effective Pressure: w mep net ¡ v1 v2 9 8.4 Ideal Air Standard Diesel Cycle 3 p=c 2 T v = c 4 1 s Diesel cycle is a piston-cylinder cycle (control mass) Models compression ignition (CI) engines 1-2:Isentropic compression 2-3: Constant pressure heat addition 3-4: Isentropic expansion 4-1: Constant volume heat rejection Working fluid idealized to be air Process 1-2: s1 s2 k 1 T P k 2 2 i f C constant © p T1 P1 T V k 1 2 1 i f C constant © p T1 V2 ¡ u2 ¡ u1 1w2 £ C T ¡ T i f C constant v 2 1 ¤ p Process 2-3: h3 ¡ h2 2q3 £ C T ¡ T p 3 2 ¤ Process 3-4: s3 s4 k 1 T P k 4 4 i f C constant © p T3 P3 T V k 1 4 3 i f C constant © p T3 V4 ¡ ¡ u4 u3 3w4 £ C T ¡ T i f C constant v 4 3 ¤ p 10 Process 4-1: u1 ¡ u4 1q4 £ C T ¡ T i f C constant v 1 4 ¤ p Thermal efficiency of Otto cycle (For constant Cp) £ C T ¡ T v ¤ ¡ 4 1 η 1 £ th C T ¡ T p 3 2 ¤ 8.5 Ideal Vapor Refrigeration Cycle p=c p=c 2 T 3 1 4 s Simply Rankine cycle run backwards Note how points 1,2,3,4 have been shifted 1-2:Isentropic compression in compressor 2-3: Constant pressure heat loss at high-temperature source 3-4: Isentropic expansion in expansion valve 4-1: Constant pressure heat gain in evaporator Working fluid is usually refrigerant or other two-phase substance 11.
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