# 03 cycles aux

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• 8/3/2019 03 Cycles Aux

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Independent Properties

Simple Compressible Pure Substance

(Absence of motion, gravity, surface, magnetic, electrical effects)

States are defined by 2 independent properties

Saturated liquid, saturated vapor are in diff state but at same P, T

In a saturated state, Press & Temp are dependent

Superheated vapor: any two of P, v, T define the state

(P, v), (P, T), (v, T)

Saturated state: the following would define the state but not (P,T)

(P, v), (v, T)

(T, x), (P, x)

Example

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P-V-T Behavior of Low- and Moderate-Density Gases

Ideal

gas eqn

(See next slide)

cr

cr P

P

PT

T

T==

, Generalized compressibility chart

ZRTPvRTPvZ == OR Compressibility factor: Unity for ideal gas

22dbcbvv

abv

RTP++

=

Ideal gas can beassumed at low andmoderate densities(See next slide)

)(

,

)KJ/mol3145.8(

,

MRR

RTPvmRTPV

Mmn

R

TRvPTRnPV

=

==

=

=

==

Reduced properties

Cubic eqn of state

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T-v Diagram of water (Opt)

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N2

Compressibility Factor for N2 (Opt)

Ex. 3.8, 9

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==

=

=

==

2

1

2

121 PdVWW

++

::

::

WdV

WdV

Work done during the process from state 1 to state 2

Integral can be:1. Graphically or experimentally evaluated OR2. Analytical evaluated when the P-V relation is

known

Work done at the Piston

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Work = Areain a P-V diagram

=

2

121 PdVW

Graphical Approach to Find Work

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During any cycle a system (control mass)undergoes, the cyclic integral of heat is equal to

the cyclic integral of work.

= WQ

The First Law of Thermodynamics for a

Control Mass Undergoing a Cycle

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Consider different processes between two states

= WQ 121

,

BCA

The First Law of Thermodynamics for a

Change in State of a Control Mass

depends only on the initial and finalstates not on the path.

212112 WQEE

WQdE

=

= Energy = +in out

( )WQ

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Energy is present in various forms

Net change in energy of the system is exactly

equal to energy transfer as work or heat

PEKEUE

E

++=

++= EnergyPotentialEnergyKineticEnergyInternal

mgZPEVVmKE == ,2

1 rr

212112

2

1

2

21212 )(

2)(

2

WQZZmgVVmUUEE

mgdZVVm

ddUdE

=++=

+

+=

rr

WQPEdKEddUdE =++= )()(

Conservation of Energy

m

Uu

U

=

InternalEnergy

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Consider a constant-pressure process withnegligible kinetic energy and potential energy

Enthalpy

12

111222

11221221

1221

211221

)()(

)(

HHVPUVPU

VPVPUUQ

VVPW

WUUQ

=

++=

+=

=

+=

Pvuh

PVUH

+=

+=

Thermodynamic Property Enthalpy

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Recap 1st law of thermodynamics

Definition of specific heats at const volume/press

For solids and liquids, specific heat reduced to:

Little volume change

VdPdHQ

PdVdUWdUQ

=

+=+=

PPP

P

vvv

v

T

h

T

H

mT

Q

mC

T

u

T

U

mT

Q

mC

=

=

=

=

=

=

11

11

CdTdudh

vdPduPvddudh

++= )(

Specific heat atconst volume

Specific heat atconst pressure

pv CCC ==

Const-Vol, Const-Press Specific Heats

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In general, ),( vTuu =

However, for a low-density gas(Confirmed by steam table)

)(~ Tuu

It is known for an ideal gas)(Tuu =

dTmCdU

dTCdudT

duC

v

v

v

0

0

0

=

=

=

Similarly, for an ideal gas)(Thh =

dTmCdH

dTCdhdT

dhC

p

p

p

0

0

0

=

=

=

Specific heats for an ideal gas:

RTuPvuh +=+=

U, H, Cv, Cp of Ideal Gases

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Specific heats for an ideal gas are also functionsof temperature only like internal energy andenthalpy

)(),( 0000 TCCTCC ppvv ==

RCC

RdTdTCdTC

RdTdudh

RTuh

vp

vp

+=

+=

+=

+=

00

00

Specific heats for an ideal gas are related as

follows:

Specific Heats of Ideal Gases

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The inequality of Clausius:

It is a consequence of the 2nd law ofthermodynamics

The equality holds for reversible cyclesThe inequality holds for irreversible cycles

0 TQ

The Inequality of Clausius: Second Law

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Consider reversible cycles between 1 and 2

+

==

1

2

2

10

BA T

Q

T

Q

T

Q

+

==

1

2

2

10 BC T

Q

T

Q

T

Q

For a cycle along A and B:

For a cycle along C and B:

=

2

1

2

1CA T

Q

T

Q

We can thus define a property entropy like

revT

QdS

revT

Q

: path independent

=

2

112

revT

QSS

Entropy A Property of a System

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Entropy: Thermodynamic propertyTherefore, once entropy is obtained for areversible process, the same entropy can be

used for an irreversible as long asthermodynamic states are the same

Zero entropy

1. All pure substances in the (hypothetical)ideal-gas state at absolute zero temperaturehave zero entropy

2. However, it is not practically easy to definezero entropy (or reference state) so anarbitrary state is chosen for zero entropy. Water: Saturated liquid at 0.01C Refrigerants: Saturated liquid at -40C

Entropy A Property of a System

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Extensive property S Intensive property s(specific entropy)

At saturation ( )

fgf

gf

xss

xssxs

+=

+= 1

Temperature-entropy diagramEnthalpy-entropy diagram (Mollier diagram)

Entropy of a Pure Substance

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0

0

41

434

334

23

212

112

=

=

=

=

=

=

SS

T

Q

T

QSS

SS

T

Q

T

QSS

Lrev

Hrev

Consider a heat engine on the Carnot cycle

Rev. isothermal

Isentropic

Rev. isothermal

Isentropic

1-a-b-2-1area

1-4-3-2-1area==

H

netth

Q

W

TH th and TL thSimilar argument for a refrigerator

(see text)

Entropy Change in Reversible Processes

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Consider a reversible heat-transfer process

12: The change of state from saturated liquid tosaturated vapor at constant pressure

Heat transfer = area 1-2-b-a-1

T

h

T

q

T

qsss

fg

rev

fg ==

== 21

2

112

23: The change of state fromsaturated vapor to superheatedvapor at constant pressure

Heat transfer = area 2-3-c-b-2

==3

2

3

232 Tdsqq

Entropy Change in Reversible Processes

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PdvduTds +=

+=

+=

==

2

11

2012

0

0

ln

gas,idealFor

vvRdT

TCss

dvv

RdT

T

Cds

v

R

T

PdTCdu

v

v

v

=

=

2

1 1

20

12

0

ln

P

PRdT

T

Css

dPPRdT

TCds

p

p

Similarly,

Method of integration1. Constant specific heat2. Functional form known

3. Tabulated Standardentropy

=T

T

p

T dTT

Cs

0

00

Entropy Change of An Ideal Gas

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Entropy Change of An Ideal Gas

const

or2

1

1

2

1

2

1

1

2

1

1

2

1

2

1

2

1

20

=

=

=

=

=

k

kk

k

k

C

R

Pv

v

v

P

P

v

v

T

T

P

P

T

T

P

P

T

T p

In an isentropic process, 0ln2

11

20

12 == PP

RdTT

Css

p

When the specific heat is constant, this equation becomes:

0lnln1

2

1

20 =

PPR

TTCp

k

k

kC

CC

C

R

p

vp

p

111

0

00

0

==

=

Ratio of specific heats is defined as

0

0

v

p

C

Ck=

Ideal gas equation

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Reversible Polytropic Process for an Ideal Gasnnn

VPVPPV 2211const ===Polytropic process:

nn

n

n

VV

PP

TT

V

V

P

P

=

=

=

1

1

2

1

1

2

1

2

1

2

1

2

In a polytropic process for an ideal gas,

n

TTmR

n

VPVP

dV

V

PdVWn

=

=

==

1

)(

1

1const

121122

2

1

2

1

21

for n1