Department of Mechanical Engineering ME 322 – Mechanical Engineering Thermodynamics Lecture 20...

Department of Mechanical Engineering

ME 322 – Mechanical Engineering Thermodynamics

Lecture 20

Entropy Balance Equation

2

The Second Law of Thermodynamics

Entropy is a balanced quantity. Therefore,

Entropy transported

into a system

Entropy transported out

of a system

Entropy produced within

a system

Entropy gained within

a system- + =

TS PS GS

T P GS S S

T P GS S S Total Entropy Form

Entropy Rate Form

Something to think about ... We have already seen that entropy production occurs due to irreversibility. But, how is entropy transported across a system boundary?


3

T P G T P GS S S dS dS dS

First, consider a closed system. For this system, we know that the Second Law is,

Recall that Clausius discovered entropy analyzing a Carnot cycle as a closed system. His discovery was,

E

LT

HT

rev

dQdS

T

Q W

T in this expression is the boundary temperature where Q is being transferred.


4

Q WFor a reversible process, we know that the entropy production is zero. Therefore, for a reversible process, the Second Law says,

T GdS dS

rev

dQdS

T

Now, consider Clausius’ discovery again,

For the closed system that we are analyzing, it is clear that heat and work cross the system boundary. Therefore, we can say that,

and T G sys

rev

dQdS dS dS

T


5

We have just discovered that that the entropy is transported into the system by heat, but NOT by work!

Q W

P sys

dQdS dS

T

For any closed system process, we can write,

This expression can now be integrated between any two states,

2 2 2

1 1 1P sys

dQdS dS

T


6

2 2 2

1 1 1P sys

dQdS dS

T

Analyzing each integral ...

2

1

k

kk

QdQ

T T

Assumption: The system boundary is isothermal.

The summation sign accounts for all heat transfer.

2

2 1 2 11

sysdS S S m s s The entropy change of the system.

2

1???PdS

This is the integral of the entropy production in an irreversible process. But ... how do we evaluate the integral???


7

Entropy production is caused by irreversibilities. Consider two piston-cylinder assemblies. They are identical in every way except that one of them operates reversibly and the other is irreversible.

Q

revW

Q

irrW

reversible irreversible

Questions ...

1. Which system has zero entropy production?

2. Which system delivers more work?

3. Is work a property?

4. Is entropy production a property?


8

Entropy production is NOT a property. It is a function of path, just like heat and work!

P sys

dQdS dS

T

2

,121

P PdS S

Now, we can integrate the entropy production term!

> 0 means that the process is irreversible= 0 means the process is reversible< 0 means the process is impossible


9

For a closed system, the Second Law of Thermodynamics can be written as,

,12 2 1k

Pkk

QS m s s

T T P GS S S

This can be written on a per unit mass basis by dividing both sides of the equation by the mass of the system,

,12 2 1k

Pkk

qs s s

T ,12

2 1

/ Pk

kk

SQ ms s

T m


10

syski i e e P

k i ek

dSQm s m s S

T dt

Incorporating the net entropy transport into the system due to the mass flows gives the complete form of the Second Law of Thermodynamics!

Notice that if the system is closed,

syskP

k k

dSQS

T dt

,12 2 1k

Pk k

QS m s s

T

11

The Laws of the Universe

2 2

2 2sysi e

i i i e e ei ec c c c

dEV Vg gQ W m h z m h z

g g g g dt

sysi e

i e

dmm m

dt

Conservation of Mass – The Continuity Equation

Conservation of Energy – The First Law of Thermodynamics

The Entropy Balance – The Second Law of Thermodynamics

syski i e e P

k i ek

dSQm s m s S

T dt

Special Application – Closed Systems

12

Consider a closed system that is reversible and adiabatic. The second law for such a system is,

,12 2 1k

Pk k

QS m s s

T

00

2 1 2 10s s s s

We have just learned that a reversible and adiabatic process for a closed system is isentropic!

Notice: Reversible does not mean isentropic Adiabatic does not mean isentropic Reversible and adiabatic means isentropic!

We have just learned that a reversible and adiabatic process for a system of this type is isentropic!

Entropy is the gate keeper of the Second Law!

Special Application – Open Systems

13

Consider an open system that is operating in steady state and is reversible and adiabatic with one flow in and one flow out. The second law for such a system is,

00

0i e i em s s s s

syski i e e P

k i ek

dSQm s m s S

T dt

0

Ideal Gases w/constant cp

14

2

1

22 1

1

lnT

vT

vdTs s c R

T v

2

1

22 1

1

lnT

pT

PdTs s c R

T P

2 22 1

1 1

ln lnp

T Ps s c R

T P

We previously derived the following expressions for an ideal gas from the Gibbs Equations,

If the gas is undergoing a process where the heat capacity can be assumed to be constant,

Consider a case where the process is also isentropic. Then,

2 2

1 1

ln ln 0p

T Pc R

T P


15

2 2

1 1

ln ln 0p

T Pc R

T P

Algebra time ...

2 2

1 1

ln lnp

T Pc R

T P 2 2

1 1

ln lnp

T PR

T c P

2 2

1 1

p

R

cT P

T P

Rearrange the exponent,

p v

p p

c cR

c c

/ /

/

p v v v

p v

c c c c

c c

1 p

v

ckk

k c


16

Substitution gives,

1

2 2

1 1

k

kT P

T P

2 22 1

1 1

0 ln lnv

T vs s c R

T v

If we consider an isentropic process with the other Ds equation,1

2 2

1 1

kT v

T v

Therefore, for an ideal gas with constant heat capacity undergoing an isentropic process,

11

2 2 2

1 1 1

kk

kT P v

T P v


17

But wait ... there’s more!!1

1

2 2 2

1 1 1

kk

kT P v

T P v

11

2 2

1 1

kk

kP v

P v

Rearranging ... 1 11 1

2 1 1 2

k kk kk kP v P v

1 11 1

2 2 1 1

k kk kk kP v P v

1 11 1

2 2 1 1

k kk k

k kP v Pv

2 2 1 1k kP v Pv Does this look familiar?

Polytropic Process RelationsconstantnPv

Any fluid model

Ideal Gas Model

Ideal Gas Model – Isentropicprocess, constant heat capacity

(isochoric)n

18

1 1 /

2 2 2 2

1 1 1 1

k k kT v T P

T v T P

1 1 /

2 2 2 2

1 1 1 1

n n nT v T P

T v T P

1/

2 1 2 1

1 2 1 2

n nP v v P

P v v P

/ (ideal gas with constant heat capacity - isentropic process)p vn k c c

1 (ideal gas, isothermal)n0 (isobaric)n

Department of Mechanical Engineering ME 322 – Mechanical Engineering Thermodynamics Lecture 20...

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