entropy for a control mass -...

Prof. Siyoung Jeong

Thermodynamics I

MEE2022-02

Fundamentals of Thermodynamics

Chapter 6

Entropy for a Control Mass

Thermal Engineering Lab. 2

• 열역학 2법칙의 양적인 Formulation을 위해 Entropy라는 개념을 도입

1. 모든 System은 extensive property, S를 갖는다.

2. Entropy의 변화

a. Heat transfer

b. Mass flow

c. Irreversible process in the system

3. Heat transfer에 의한 양은

4. Irreversible process에 의한 Entropy 생성

• S = SA + SB + SC

• s = S / m

Chapter 6. Entropy for a Control Mass

Q

QS

T

0irrS


• For closed system


,

( )

Inequality of Clausius: 0

0

Q irr irr

Q rev

Q irr irr

iirr

i

irr

QdS S S S

T

dS dS

dS QS S S

d T

QdSS

d T

QS

T

Q

T

rev

rev

T

QSS

T

QdS

12

0irrST

Qδ

0T

Qδ


6.1 The Inequality of Clausius



rev

rev

T

QSS

T

QdS

12

6.2 Entropy – a property of a system



6.3 The entropy for a pure substance

T-s Diagram



h-s Diagram



6.4 Entropy change in reversible process

2

2 11

3 2

4

4 33

1 4

( 0)

0

( 0)

0

HH

rev H

LL

rev L

QQS S Q

T T

S S

QQS S Q

T T

S S

Carnot cycle

1 → 2 : rev. isothermal QH

2 → 3 : rev. adiabatic Working Fluid : TH→TL

3 → 4 : rev. isothermal QL

4 → 1 : rev. adiabatic Working Fluid : TL→TH



3

223

12

1212

2

112

2

1

122

112

)(

Tdsq

hh

hh

puu

pduuq

T

h

T

q

T

qss

fg

fg

Constant pressure process



Ex. 6.1 Consider a Carnot-cycle heat pump with R-134a as the working fluid.

Heat is absorbed into the R-134a at 0℃, during which process it changes

from a two-phase state to saturated vapor. The heat is rejected from the

R-134a at 60℃ and ends up as saturated liquid. Find the pressure after

compression, before the heat rejection process, and determine the COP

for the cycle.



Ex. 6.2 A cylinder/piston setup contains 1 L of saturated liquid refrigerant R-

410A at 20℃. The piston now slowly expands, maintaining constant

temperature to a final pressure of 400 kPa in a reversible process.

Calculate the work and heat transfer required to accomplish this process.



6.5 The thermodynamic property relation

( )

irr

rev

rev rev

QdS S

T

Q TdS

Q dU W

TdS dU PdV

TdS d H PV PdV

dH VdP

Tds du Pdv

Tds dh vdP

Entropy : Property ; Independent of Integration Path



6.6 Entropy change of a solid or liquid

1

212 ln

T

Tcss

dTT

C

T

du

dvT

P

T

duds



Ex. 6.3 One kilogram of liquid water is heated from 20℃ to 90℃. Calculate the

entropy change, assuming constant specific heat, and compare the result

with that found when using the steam tables.



6.7 Entropy change of an ideal gas


2

11

2012

0

0

ln

gas idealfor

v

vRdT

T

css

dvv

RdT

T

cds

v

R

T

PdTcdu

dvT

P

T

duds

PdvduTds

v

v

v


1

200

12

00

1

2

1

2012

1

2

1

2012

1

22

1

012

ln)(

lnln

lnln

heat specific const.

ln

12

0

P

PRssss

dTT

cs

P

PR

T

Tcss

v

vR

T

Tcss

P

PRdT

T

css

vdPdhTds

TT

T

T

PT

P

v

P




0

2 22 1 0

1 1

2 2

1 0 1

2 2

1 1

0 0

0 0

1

2 2

1 1

Isentropic Process: const

ln ln

ln ln

1 1 1

P

k

P

P

R

c

P v

P P

k

k

Pv

T Ps s c R

T P

T PR

T c P

T P

T P

c cR k

c c k k

T P

T P

For an Isentropic process



For an ideal gas: RTPv

1

2 2 2 2

1 1 1 1

1

2 2

1 1

2 1

1 2

1 1 2 2

const

k

k

k

k

k k

k

T P v P

T Pv P

v P

v P

v P

v P

Pv P v

Pv


Ex. 6.4 Consider Example 3.13, in which oxygen is heated from 300 to 1500 K.

Assume that during thiss process, the pressure dropped from 200 to 150

kPa. Calculate the change in entropy per kilogram.



Ex. 6.5 Calculate the change in entropy per kilogram as air is heated from 300 to

600 K while pressure drops from 400 to 300 kPa. Assume:

1. Constant specific heat.

2. Variable specific heat.



Ex. 6.6 One kilogram of air is contained in a cylinder fitted with a piston at a

pressure of 400 kPa and a temperature of 600 K. The air is expanded to

150 kPa in a reversible adiabatic process. Calculate the work done by the

air.



6.8 The reversible polytropic process for an ideal gas


1 1 2 2

ln

ln

ln ln 0

constn

n n

d Pn

d V

d P nd V

PV

PV PV C



2

121

2

121

2 2 1 112

2 1

212 1 1

1

2 1

1 2

and

( 1)

1

( )

1

( 1)

ln

ln ln

n

n

w Pdv Pv C

dvw C

v

n

P v Pvw

n

R T T

n

n

vw Pv

v

v PRT RT

v P


Ex. 6.7 In a reversible process, nitrogen is compressed in a cylinder from 100

kPa and 20℃ to 500 kPa. During the compression process, the relation

between pressure and volume is PV1.3 = constant. Calculate the work and

heat transfer per kilogram, and show this process on P-v and T-s

diagrams.



6.9 Entropy change of a control mass during an irreversible process


for reversible process

for irreversible process

irr

QdS S

T

QdS

T

QdS

T


6.10 Entropy generation and the entropy equation


L W 0 ( 0)

( )

irr

irr

L

irr

irr

QdS S

T

TdS Q T S

T

TdS dU PdV

Q dU PdV W

W

dU Q W

Q PdV T S


6.11 Principle of the increase of entropy


irr

surr

irrsystem

irrnet

surr

irrsystem

SQTT

TTT

QdS

ST

QdS

SQTT

dS

TTT

QdS

ST

QdS

11

011

0

0

0

0

0

0


Ex. 6.8 Suppose that 1 kg of saturated water vapor at 100℃ is condensed to a

saturated liquid at 100℃ in a constant-pressure process by heat transfer

to the surrounding air, which is at 25℃. What is the net increase in

entropy of the water plus surroundings?



6.12 Entropy as a rate equation


. .

1

1

gen

c mgen

SdS Q

t T t t

dSQ S

dt T


Ex. 6.9 Consider an electric space heater that convers 1 kW of electric power

into a heat flux of 1 kW delivered at 600 K from the hot wire surface. Let

us look at the process of the energy conversion from electricity to heat

transfer and find the rate of total entropy generation.



Ex. 6.10 Consider a modern air conditioner using R-410A working in heat pump

mode, as shown in Fig. 6.21. It has a COP of 4 with 10 kW of power

input. The cold side is buried underground, where it is 8℃, and the hot

side is a house kept at 21℃. For simplicity, assume that the cycle has a

high temperature of 50℃ and a low temperature of -10℃ (recall Section

5.10). We would like to know where entropy is generated associated with

the heat pump, assuming steady-state operation.



6.13 Some general comments about entropy and chaos


lnS k w

entropy for a control mass -...

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