UPME – ME 63 – ENQ – 1/06 10–1

10 REVERSIBLE WORK, IRREVERSIBILITY AND AVAILABILITY

This chapter focuses attention on the issue of determining the ultimate potential for doing work for

a) a system or control volume in a given state b) a system or control volume undergoing a process

Relate this potential for doing work to irreversibility and thus present irreversibility in energy terms instead of entropy as previously done

Relate irreversibility to entropy generation

Evaluate performance of devices or processes in terms of 2nd Law efficiency

10.1 REVERSIBLE WORK AND IRREVERSIBILITY

Reversible Work represents the maximum amount of work that could be

produced (or the minimum work that needs to be supplied) whenever a system undergoes a process (i.e., from an initial to a final state and/or given inlet and exit states).

The reversible work could only be obtained if the process the system undergoes is totally (both internally and externally) reversible.

The reversible work is obtained by applying the 1st and 2nd Laws of thermodynamics for a totally reversible process between the same initial, final, inlet, and exit states as in the actual process.

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10.1.1 CLOSED SYSTEM REVERSIBLE WORK AND IRREVERSIBILITY

Consider the control mass undergoing an ACTUAL process described by the

following:

1. Heat transfer Q12 = QH from a thermal reservoir at TH ; Q12 is transferred across portion of control mass boundary at temperature T j not necessarily equal to TH.

2. Change of state from 1 to 2 experiences (U2 – U1), (S2 – S1), etc.

3. No change in KE and PE of the control mass

4. Does a total amount of actual work W12

1st Law: 12 2 1 12HQ Q U U W

Or 12 12 2 1W Q U U (Actual Work) (10-1)

2nd Law: 122 1 gen

j

QS S S

T (10-2)

Approximations to Local Boundary Temperature Tj It is often difficult to determine the local temperature T j at which QH crosses the system boundary. To overcome this difficulty, two approaches may be used.

a) One approach is to use an average boundary temperature Tb when the variation of the local temperature Tj over the system boundary is fairly small. For a control volume, a usual value of Tb is the average of the inlet and exit mass stream temperatures.

/ 2j b i eT T T T

W12 (U2 – U1)

(S2 – S1)

Tj

QH = Q12

TH

Actual Process

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b) Another approach is to locate the control surface where the boundary

temperatures are known. The increase in volume and mass of the control volume should be included in the analysis.

As an approximation, it is often reasonable to imagine the control mass boundary extended with negligible volume and mass change, to touch the reservoir where QH occurs so that

j HT T

With this approximation, the 2nd Law equation (10-2) thus becomes

2 1H

gen

H

QS S S

T (10-3)

W12 (U2 – U1)

(S2 – S1)

QH = Q12

TH

Tj = TH

Extending boundary so that Tj = TH

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Consider now the same control mass undergoing the same change of state through a REVERSIBLE process described as follows:

1. Same heat transfer Q12 = QH from a thermal reservoir at TH

For Q12 to be transferred reversibly, a reversible heat engine is used.

This heat engine will be inside the control mass boundary and rejects heat Qo to a reservoir at temperature To.

The heat engine reversible work output WHE is part of the control mass total reversible work output Wrev.

For Wrev to be maximum for the given QH, WHE should be maximum Qo

should be minimum To should be minimum. Although To is arbitrary, To is usually chosen to be the lowest naturally occurring reservoir temperature. The ambient or atmospheric temperature is typically used as To .

2. Same change of state from 1 to 2 experiences same (U2 – U1), (S2 – S1)

3. No change in KE and PE of the control mass

4. Does a total amount of reversible work Wrev

1st Law: Wrev = (QH – Qo) – (U2 – U1) (Reversible Work) (10-4) 2nd Law: S2 – S1 = (QH/TH – Qo/To) + Sgen (10-5) Noting that Sgen = 0 for a reversible process and multiplying Eq. (10-5) by To gives

To(S2 – S1) = QH(To/TH) – Qo

Or Qo = QH(To/TH) – To(S2 – S1) (10-6)

Wrev (U2 – U1)

(S2 – S1)

QH = Q12

TH

To

Qo

WHE

Reversible Process

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Substituting (10-6) into (10-4) gives the reversible work for the same change of state of the control mass

(10-7)

The irreversibility is

(10-8)

For the more general case of the control mass experiencing different heat transfers Qk with various thermal reservoirs at different temperatures Tk, the reversible work and irreversibility become

2 1 2 1 1 orev o k

k

TW T S S U U Q

T

(10-9)

12 12 2 1k

rev o o net o gen

k

QI W W T S S T S T S

T

(10-10)

2 1 2 1 1 orev o H

H

TW T S S U U Q

T

12 12 2 1H

rev o o net o gen

H

QI W W T S S T S T S

T

Reversible Process with

Multiple Thermal

Reservoirs

Wrev (U2 – U1)

(S2 – S1)

Qk

Tk

To

Qok

WHPk

Q1

T1

To

Qo1

WHE1

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10.1.2 OPEN SYSTEM REVERSIBLE WORK AND IRREVERSIBILITY

Consider the control volume undergoing an ACTUAL process during the time interval t1 to t2 described by the following:

1. Heat transfer Q1, …, Qk from or to thermal reservoirs at T1, …, Tk

respectively ; each Qk is transferred across a portion of control mass boundary at local temperature Tj not necessarily equal to Tk.

2. Change of state from 1 to 2 experiences (U2 – U1)cv , (S2 – S1)cv, etc.

3. With inlet mass flow streams mi1 ,…, mix at inlet states 1, …,x respectively

4. With exit mass flow streams me1 ,…, mey at exit states 1, …,y respectively

5. No change in KE and PE of the control volume

6. Does a total amount of actual work WCV

1st Law:

2 2

2 2 1 1

1 1

1 1

2 2

yx

k i i i i e e e e cvQ m h V gz m h V gz m u m u W

Solving for the actual work gives

2 2

2 2 1 1

1 1

1 1

2 2

yx

cv k i i i i e e e eW Q m h V gz m h V gz m u m u

(10-11)

2nd Law:

2 2 1 1

1 1

yxk

i i e e gen

k

Qm s m s m s m s S

T

(10-12)

Actual Process

mi1

mix

me1

mey

Q1

T1

Qk

Tk

Wcv

(U2 – U1)

(S2 – S1)

Tj1 Tjk

cv

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Consider now the control volume undergoing a REVERSIBLE process during the time

interval t1 to t2 described by the following: 1. Same heat transfer Q1, …,Qk from or to thermal reservoirs at T1, …,Tk

respectively ; each Qk is transferred across a portion of control mass boundary at local temperature Tj not necessarily equal to Tk.

For Q1,…,Qk to be transferred reversibly, reversible heat engines and heat pumps are used. These heat engines and heat pumps will be inside the control mass boundary and reject/absorb heat Qo1,…,Qok respectively to/from the surrounding thermal reservoir at temperature To. The heat engines’ reversible work output WHE1 and heat pumps’ reversible work input WHPk are part of the control volume total reversible work Wrev. For Wrev to be maximum,

WHE1 should be maximum IQo1I should be minimum for a given Q1

To1 should be minimum. Although To1 is arbitrary, To1 is usually chosen to be the lowest naturally occurring reservoir temperature.

WHPk should be minimum IQokI should be maximum for a given Qk

, (Qk = Qok+WHPk) Tok should be maximum. Although Tok is arbitrary, Tok is usually chosen to be the highest naturally occurring reservoir temperature.

A convenient choice of the reservoir temperatures To1,…,Tok is the

ambient or atmospheric temperature To . To1,…,Tok = To

2. Same change of state from 1 to 2 experiences same (U2 – U1), (S2 – S1)

3. With same inlet mass flow streams mi1 ,…, mix at inlet states 1,…,x respectively ; with same exit mass flow streams me1 ,…, mey at exit states 1,…,y respectively

Reversible Process

mi1

mix

me1

mey

Wrev

(U2 – U1)

(S2 – S1)

cv

T1

To1

Qo1

WHE1

Q1

Tk

Tok

Qok

WHPk

Qk

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4. No change in KE and PE of the control volume

5. Does a total amount of reversible work Wrev

1st Law:

2 2

2 2 1 1

1 1

1 1

2 2

yx

k ok i i i i e e e e revQ Q m h V gz m h V gz m u m u W

Solving for the reversible work gives

2 2

2 2 1 1

1 1

1 1

2 2

yx

rev k ok i i i i e e e eW Q Q m h V gz m h V gz m u m u

(10-13)

2nd Law:

2 2 1 1

1 1

yxk ok

i i e e

k ok

Q Qm s m s m s m s

T T

(10-14)

Since To1,…,Tok = To , Eq.(10-14) is multiplied by To and solved for Qok giving

2 2 1 1

1 1

yxo

ok o o i i e e k

k

TQ T m s m s T m s m s Q

T

(10-15) Substituting Eq. (10-15) into Eq. (10-13) yields (10-16) The reversible work can also be grouped and written as

(10-17)

The irreversibility is then (F) (10-18)

2 2 1 1 2 2 1 1rev o o e e i iW T m s m s m u mu T m s m s

2 2

1 1

1 11

2 2

y xo

e e e e i i i i k

k

Tm h V gz m h V gz Q

T

2 2

1 1

1 1

2 2

yx

rev i i i i o i e e e e o eW m h V gz T s m h V gz T s

1 1 1 2 2 2 1 oo o k

k

Tm u T s m u T s Q

T

12 2 2 1 1k

rev cv o o e e i i o

k

o cv surr o net o gen

QI W W T m s m s T m s m s T

T

T S S T S T S

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10.1.3 USEFUL WORK

In many instances, part of the total actual work done by (or on) the control volume Wcv, which includes boundary expansion (or contraction) is work done on (or by) the surroundings Wsurr. The surroundings is assumed to be at constant pressure Po. The rest of the actual total work done by (or on) the control volume can be regarded as its useful work, Wu .

Useful Work – Wu, is defined as the difference between the total work actually done by (or on) the control volume Wcv and the work done on (or by) the surroundings at pressure Po as the control volume boundary expands (or contracts).

(10-19)

Since surr cvV V , then

(10-20)

Recognizing useful work Wu is part of developing the concept of and determining expressions for availability or exergy.

u cv surr cv o cvW W W W P V

u cv o surrW W P V

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A rate expression for the useful work for a general control volume undergoing an actual process is now developed. Consider the control volume below:

1st Law:

2 21 1

2 2

cvcv cv i i i i e e e e

dEQ W m h V gz m h V gz

dt

(10-21) where

cv kQ Q total control volume heat transfer (10-22)

. .cv boundary mechanical electricalW W W W actual total work of c v (10-23)

The control volume work in terms of the useful work from Eq. (10-19) is

cvcv u o

dVW W P

dt (10-24)

Substituting (10-22) and (10-24) into (10-21) gives,

2 21 1

2 2

cv cvk u o i i i i e e e e

dE dVQ W P m h V gz m h V gz

dt dt

(10-25)

2nd Law:

cv ki i e e goen

k

dS Qm s m s S

dt T

(10-26)

Multiplying (10-26) by the surrounding temperature To results into

cv ko o i i o e e o o goen

k

dS QT T m s T m s T T S

dt T

(10-27)

Subtracting (10-27) from (10-25) and re-grouping give the expression for:

General control volume

undergoing an actual process

mi me

Wcv = Wboundary + Wmechanical + Welectrical

Wboundary Wmechanical Welectrical

dEcv / dt

cv dScv / dt

Qk

Tk Tk

Q1

T1 T1

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Control Volume (Open System) USEFUL WORK for an ACTUAL Process (10-28)

The useful work for an internally reversible process is now obtained by noting that the entropy generation Sgen = 0 for such a process. Thus

Control Volume (Open System) USEFUL WORK for a REVERSIBLE Process (10-29)

The actual useful work and reversible useful work are thus related as follows

,u rev uW W (10-30)

The irreversibility can be expressed in terms of either total work

12 rev cvI W W (10-31)

or in terms of useful work

12 ,rev u uI W W

2 21 1

2 2

1

u i i i i o i e e e e o e

o oo cvk o gen

k

W m h V gz T s m h V gz T s

d E PV T STQ T S

T dt

2 2

,

1 1

2 2

1

rev u i i i i o i e e e e o e

o oo cvk

k

W m h V gz T s m h V gz T s

d E PV T STQ

T dt

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10.2 AVAILABILITY

Availability is the maximum work potential of a system at a given state determined by letting the system undergo a reversible process towards a state of equilibrium with the surroundings (called the dead state) while any heat transfer is solely with the surroundings.

Availability of the system in the dead state is zero.

The availability of a system depends on both the state of the system and conditions of the surroundings. It is a property of the system-surroundings combination.

The availabilities of both non-flowing mass (i.e., inside the control volume) and flowing mass (i.e, mass streams entering or leaving the control volume) are identified and corresponding expressions developed.

Availability can be transferred. In general, availability transfers associated with mass flow, heat transfer, and work interaction occur during a process.

10.2.1 THE DEAD STATE

The dead state refers to the state of a system when it is in thermal, mechanical, and chemical equilibrium with the surroundings.

At the dead state,

no further change of state of the system can occur spontaneously no further work can possibly be done

the velocity of a closed system or a fluid stream is zero, potential energy is also zero

properties of the system are given the subscript o and are evaluated at the surrounding pressure Po and temperature To so that

P = Po u = uo = u)To,Po V = Vo = 0 T = To h = ho = h)To,Po z = zo = 0 v = vo = v)To,Po s = so = s)To,Po

availability of the system is zero

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10.2.2 AVAILABILITY OF NON-FLOWING MASS

The availability (or exergy) of NON-FLOWING mass, such as in a closed system or that inside the control volume, at a given state is the maximum useful work WMAX,u that may be obtained from a system-surroundings combination as the system proceeds from the given equilibrium state to the dead state by a process where any heat transfer occurs only with the surroundings.

Consider the closed system at a given state (P, T, V) undergoing a process towards the dead state while interacting only with the surroundings:

The useful work for a control volume is

2 21 1

2 2

1

u i i i i o i e e e e o e

o oo cvk o gen

k

W m h V gz T s m h V gz T s

d E PV T STQ T S

T dt

me and mi are zero for a closed system

The system does maximum useful work if it goes to the dead state through a

reversible process Sgen = 0

Heat exchange only with the surroundings Tk = To so that third term on RHS is zero

Evaluating the above equation over Δt from the given state to the dead state gives the maximum useful work

, ,MAX u rev u o o o o oW W E U P V V T S S

This maximum useful work is therefore the availability of the closed system and is

denoted by at any given state (P, T, V).

Wu = Wcv - PoΔV

Closed System

at T, P, V

Q

Closed system at a given state

and exchanging heat only with

the surroundings

Surroundings

at To , Po

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Thus, Closed System, Control Mass, or Non-Flow Availability

On a per unit mass basis, The change in availability of a closed system as it undergoes a process from state 1 to state 2 is then The reversible work in terms of availability is The irreversibility in terms of availability is

o o o o o o o o o o o oE U P V V T S S E PV T S U PV T S

o o o o o o o o o o o oe u P v v T s s e Pv T s u Pv T s

2 1

2 1 2 1 2 1

2 2 2 1 1 1

o o

o o o o

E E P V V T S S

E PV T S E PV T S

1 2 2 1 1 orev o k

k

TW P V V Q

T

12 1 2 2 1 121 oo k

k

TI P V V Q W

T

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10.2.3 AVAILABILITY OF FLOWING MASS

The availability (or exergy) of fluid in steady flow, (i.e., mass stream entering or leaving the control volume), also known as stream availability, is the maximum useful work WMAX,u that may be obtained as the flowing fluid proceeds from a given equilibrium state to the dead state by a process where any heat transfer occurs only with the surroundings.

An expression for the stream availability can be developed by considering a control volume undergoing a steady-state steady flow process. The useful work for a control volume is

2 21 1

2 2

1

u e e e e o e i i i i o i

o oo cvk o gen

k

W m h V gz T s m h V gz T s

d E PV T STQ T S

T dt

For the SSSF process,

0o o cv

d E PV T S

dt

The maximum useful work is obtained if the

a) process is reversible ToSgen = 0

b) exit stream is at the dead state he = ho, Ve = 0, ze = 0, se = so

The maximum useful work is then

2

,

10 0 1

2

oMAX u e o o o i i i i o i k

k

TW m h T s m h V gz T s Q

T

Note that only the first two terms on the RHS are associated with the mass streams while the last term is associated with the control volume.

Dropping the subscripts for the inlet streams to denote any inlet state, the stream availability per unit mass is then defined as

Flow or Stream Availability

The control volume reversible work for a given process can be expressed in terms of availability as follows

2 21 1

2 2o o o o o o o oh V gz T s h T s gz h h T s s V gz

1 1 2 2 1 1 2 2 1 orev i i e e o k

k

TW m m m m P m m Q

T

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10.2.4 FLOW OF AVAILABILITY OR EXERGY

Availability or exergy transfer is associated with mass, heat, and work interactions during a process. The availability may be thought of as “flowing” during these interactions.

Availability Flow with Heat Transfer

For heat transfer QR from a constant temperature source at TR, maximum work is obtained by transferring QR using a reversible heat engine rejecting heat to the surroundings at To.

The availability transfer is equal to the work output of the reversible heat engine.

For heat transfer Q12 that takes place over varying temperature, e.g., in a constant-pressure process shown below,

the availability transfer is

, 1 oQ R R Carnot R

R

TQ Q

T

20

,12 121

1Q o

TQ Q T S

T

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Availability Flow with Work Interaction

Work interactions by concept are reversible at the point where they occur at the system boundary. The availability transfer associated with work transfer equals the value of the useful work itself.

Availability Flow with Mass Flow

The availability flow associated with mass flow is equal to the stream availability .

10.3 EXERGY BALANCE

The concepts of reversible work, availability, and irreversibility can be expressed in a unified concept through the formulation of an exergy balance for a control volume undergoing a process.

For the mass inside the control volume, its exergy at a given state is

o o o o om m e e P m v v T m s s

The rate of change of exergy becomes

o o o o o o o

d me d mv d msd dm dm dme P P v T T s

dt dt dt dt dt dt dt

(A)

Since

, ,cv cv cvo o o o

d me d mv d msdE dV dSand h e P v

dt dt dt dt dt dt ,

Eq. (A) can be written as

cv cv cvo o o o o

dE dV dSd dmP T h T s

dt dt dt dt dt

(B)

The continuity equation for the control volume is

i e

dmm m

dt (C)

1st Law for the control volume:

2 21 1

2 2

cvk cv i i i i e e e e

dEQ W m h V gz m h V gz

dt

(D)

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2nd Law for the control volume:

cv ki i e e cv gen

k

dS Qm s m s W S

dt T

(E)

Substituting (C), (D), (E) into (B) yields

2 21 1

2 2

1

cvi i i i o i e e e e o e o o o i e

o cvk cv o o gen

k

dm h V gz T s m h V gz T s h T s m m

dt

T dVQ W P T S

T dt

(F)

Or

(G) The above exergy balance equation indicates that for a control volume undergoing a process, the net change in exergy of the control volume

cvd

dt

= net exergy change in the control volume

is equal to the sum of the following

i i e em m net exergy flow due to mass flow

1 ocov

TQ

T

net exergy flow due to heat transfer

cvcv o

dVW P

dt

(-) net useful work

o genT S exergy destruction

1cv o cvi i e e k cv o o gen

k

d T dVm m Q W P T S

dt T dt

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10.4 2nd LAW EFFICIENCY

The 2nd Law efficiency of devices or processes compares the desired output to the supplied availability.

For specific devices, the 2nd Law efficiency will have the following forms:

Turbine

,actual

II turbine

i e

w

Compressor or Pump

, /i e

II comp pump

actualw

Heat Exchanger

1 2 1

,

3 3 4

II HX

m

m