ChemE 260 The Clausius Inequality

& Entropy

May 3, 2005

Dr. William BaratuciSenior Lecturer

Chemical Engineering Department

University of Washington

TCD 7: A & BCB 6: 1

The Clausius Inequality

• Cyclic Integrals– Integrate through all the steps in

a cycle and return to the initial state.

• Inexact Differentials: Q & W– Used for path variables, Q and W

• Evaluating Cyclic Integrals– Example 1: Carnot HE

– Example 2: Carnot HE

BaratuciChemE 260May 3, 2005

Q0

T

Hot Reservoir

Cold Reservoir

HER

QH

QC,rev

Wrev

2 4

12 34 H C1 3

Q Q Q Q Q Q Q 0 42

2 4C,rev3 341 12 H

H C H C H C1 3

QQQQQ QQ Q Q

T T T T T T T T T

Clausius: Int. Rev. and Irrev. Cycles• Reversible Cycle, such as Carnot:

– Kelvin: or:

– Therefore:

• Irreversible Cycles:

C,revH

H C

QQQ

T T T

C,rev C

H H

Q T

Q T C,rev H

C H

Q Q

T T

Q0

T

Hot Reservoir

Cold Reservoir

HEIrr

QH

QC,irr

Wirr

rev irr

rev irrW W

H C,rev H C,irrQ Q Q Q

C,rev C,irrQ Q

C,irr C,rev C,irrirr H

H C C C

Q Q QQ Q0

T T T T T

Q0

T

• All Cycles :

– 1st Law :

– Definition of efficiency :

BaratuciChemE 260May 3, 2005

Entropy• Cycle 1-A-B-2:

2 1

1 2A B

Q Q Q0

T T T

• Cycle 1-A-C-2:2 1

1 2A C

Q Q Q0

T T T

• Subtract Eqns:1 1

2 2C B

Q Q

T T

int rev

Q

T

• Does not depend on path !

• It is a state variable or property !

int rev

QdS

T

• Definitionof Entropy:

BaratuciChemE 260May 3, 2005

S: Int. Rev. & Irrev. Processes• Change in Entropy:

2

2 1rev1

QS S S kJ / K

T

2

2 1 2 1int rev irrevrev1

QS S S S

T

• Problem: Since Int. Rev. processes do not exist, how do we evaluate S ?

• Special Case: Reversible, Isothermal Processes

2 2rev

revrev1 1

QQ 1S Q

T T T

Especially useful for evaluating Sreservoir because Treservoir = constantBaratuci

ChemE 260May 3, 2005

Entropy of a Pure Substance• S = fxn( T, P, phase )• NIST Webbook

– Thermophysical Properties of Fluid Systems

– Specific entropy is listed in the thermodynamic data tables

– Observations: sat vap sat liqˆ ˆS S

S with P

S with T All substances at all T :

– Subcooled Liquids: *subcooled sat liq

ˆ ˆS T,P S T,P

Where P* = vapor pressure = Psat

– Sat’d Mixtures: sat mix sat vap sat liqˆ ˆ ˆS x S 1 x S

IG and many real gases :

BaratuciChemE 260May 3, 2005

T-S Diagram

BaratuciChemE 260May 3, 2005

Carnot Cycle• Steps

– 1-2: Isothermal expansion– 2-3: Adiabatic expansion– 3-4: Isothermal compression– 4-1: Adiabatic compression

• Step 1-2:2

2 1rev1

2H

revH1

QS S

T

Q1Q

T T

3

3 2rev2

QS S 0

T

4 434 C

4 3 revrev C C3 3

Q QQ 1S S Q

T T T T

2

2 1rev1

QS S 0

T

• Step 2-3:

• Step 3-4:

• Step 4-1:

Isentropic !

Isentropic !

BaratuciChemE 260May 3, 2005

Heat, Work & TS Diagrams

• 1st Law Cycle

H H 2 1ˆ ˆ ˆQ T S S

C C 3 4ˆ ˆ ˆQ T S S

H Cˆ ˆˆQ W Q

H Cˆ ˆW Q Q

• QH = area under path for step 1-2

• QC = positive area under path for step 3-4

• W = area enclosed by the cycle !BaratuciChemE 260May 3, 2005

Next Class …

• Principle of Increasing Entropy & Entropy Generation– This will let us express the 2nd Law in terms of Entropy

– This very powerful result will let us perform 2nd Law Analysis on processes as well as cycles.

• Fundamental Property Relationships– These are sometimes called the Gibbs Equations

– They show us how entropy is related to other properties, such as H, U, P, V and T

– This will show us how the specific entropy values in the thermodynamic tables were determined.

BaratuciChemE 260May 3, 2005

Example #1

• A piston-and-cylinder device contains saturated R-134a vapor at -5oC. This vapor is compressed in an internally reversible, adiabatic process until the pressure is 1.0 Mpa. Determine the work per kg of R-134a for this process.

Example #2

• Steam enters an adiabatic turbine at 5 MPa and 450oC and leaves at a pressure of 1.4 MPa. Determine the work output of the turbine per kg of steam flowing through the turbine if the process is reversible and changes in kinetic and potential energies are negligible.

Example #3

• Consider a process in which 1.00 kg of saturated water vapor at 100oC is condensed to a saturated liquid in an isobaric process by heat transfer to the surrounding air, which is at 25oC. What is the change in entropy of the water ? What is the change in entropy of the surroundings ? What is the change in the entropy of the universe ?