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

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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

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