Second-Law Analysis in Heat Transfer and Thermal Design

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Second-Law Analysis in Heat

Transfer and Thermal Design

ARTICLE · DECEMBER 1982DOI: 10.1016/S0065-2717(08)70172-2

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2/59

ADVANCES IN

HEA T

TRANSFER, VOLUME I5

Second-Law Analysis

in

Heat Transfer

and Thermal Design

ADRIAN

BEJAN

D ep a r tmen t

of

Mechunicul

Engineering . Univers i ty

of

Colorudo.

Boulder.

Colorado

I.

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

A. Secon d-Law Analysis in Engineering . . . . . . . . . . . . . . . . . . . .

2

B. Secon d-Law Analysis in Heat Transfer . . . . . . . . . . . . . . . . . . . 3

(Lost Exergy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

A . O pe ns ys t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

B. Th e Absolute Temperature Factor To . . . . . . . . . . . . . . . . . . . . 8

111. Local

Entrop y Generation in Convective H eat Transfer

. . . . . . . . . . . .

I1

A.

Conductive versus Viscous Effects

. . . . . . . . . . . . . . . . . . . . .

12

B. Entropy Generation Profiles and Maps . . . . . . . . . . . . . . . . . . . 13

11. Irreversibil ity, Entropy Gene ration, and Lost Available Work

C. The En tropy Generat ion Num ber Ns

. . . . . . . . . . . . . . . . . . . . 16

D. Th e Impact of Heat T ransfer Augmentation

on

Entrop y Generation

21

IV.

Entropy Generation Minimization in Heat Exchanger Design

. . . . . . . . .

25

27

B. Heat Exchanger Geo metry fo r Minimum Irreversibil ity . . . . . . . . . . 29

C.

Sensible Heat Units for Energy Sto rage

. . . . . . . . . . . . . . . . . . 34

38

A. Minimization of Entro py Generation in an Insulation Syste m

B.

Engineering Applications

. . . . . . . . . . . . . . . . . . . . . . . . . .

43

5

1

Nomenclature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Note Added in Proof

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

. . .

A .

Heat Exchangers wi th Zero Pressure Drop . . . . . . . . . . . . . . . . .

V.

Thermal Insulation System s

. . . . . . . . . . . . . . . . . . . . . . . . . . .

of Fixed Identity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

VI.

Concluding Remark s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. Introduction

Th e objective of this contribution

is

to summ arize an important contem-

porary tren d in th e field

of

heat transfer and thermal design. This trend

is

repres ented by the infusion

of

the secon d law of therm odyn am ics and its

I

Copyright @ 1982 by Academic &s s , Inc.

AU t i f i ts of reproduction in

any

form reserved.

ISBN -0- 12-020015-5


3/59

2

A D R I A N

B E J A N

design-related concept of entropy generation minimization. This new

trend is important and, at the same time, necessary,

if

the heat transfer

community is to contribute to a viable engineering solution to the energy

problem.

A. SECOND-LAW

N A L Y S l S

I N ENGINEERING

At the root of the growing interest in the thermodynamic irreversibility

of heat transfer lies the emphasis placed today on energy conservation

and the efficient use

of

energy. In any power plant, for example, the ther-

modynamic nonideality (irreversibility) of any of its engineering compo-

nents causes a decrease

in

the net power output of the cycle. Likewise, in

a refrigeration plant the irreversibility accumulated over various compo-

nents leads to an increase

in

the mechanical power input required by the

refrigeration cycle. Either way, the thermodynamic irreversibility of com-

ponents such as heat exchangers, mixers, turbines, and compressors

amounts to a penalty in otherwise available work or, on a unit time basis,

available mechanical power. From an engineering standpoint, it makes

good sense to first identify the irreversibility associated with various com-

ponents and, second, to design for less irreversibility in order to avoid the

imminent loss of available mechanical power.

The above conclusion follows directly from the sirrzultaneous consider-

ation of the first and second laws of thermodynamics,* as we demonstriate

in greater detail in the next section. This is certainly not a new conclusion,

its first statement and engineering use dating back to

t h e

work of Gouy

[ I ]

and Stodola

[2].

Since then, the same principle was restated by others,

who in the process clarified its implications and made it more accessible

to engineering practice

[3-61.

Today, there is a growing consensus that irreversibility analysis is a

powerful approach, in fact, the only one, to deciding which installation or

process is efficient or inefficient [7, 81. In addition, through irreversibility

minimization the engineer can make specific design decisions aimed at

conserving available work. Citing only a few examples, this approach and

its relation to cost minimization was exploited by Tribus and Evans

[9]

in

a cornerstone study of the thermoeconomics of seawater desalination.

More recently Kestin c’t

( I / .

[10, 113 established the thermodynamic faun-

dations for evaluating the available work potential of geothermal installa-

tions. Gaggioli, Wepfer, and Elkouh

[

121 relied on second-law concepts to

show the great margin for improvement present in the contemporary de-

* In the thermal engineering literature, this procedure is recognized simply as “second-

law analysis.”


4/59

St-cO N I ) - L A WNALYSIS

3

sign of heating, ventilating. and air-conditioning (HVAC) systems. Lei-

denfrost [ 13. 141 used the second law

to

analyze the potential for con-

serving available energy in a variety o f power consuming processes.

The growing interest in irreversibility analysis in engineering is paral-

leled by a new emphasis placed on

t h e

teaching of the second law and its

applications in engineering thermodynamics. A significant effort

in

this di-

rection was made by Kestin

[IS.

161. Lu [17], and CravalhoandSmith[18].

B . S E W N D - L A W N . A L ~ Y S I SN HF.A T R A N S F E R

The place occupied by heat transfer and thermal design in the greater

picture described

so

far is central. Engineering components and devices

for heat transfer are inherently irreversible. For example. a two-fluid

counterflow heat exchanger draws its irreversibility from two distinct

mechanisms, namely. heat transfer across the stream-to-stream tempera-

ture difference and fluid friction (pressure drop) in the two flow passages.

We shall consider this example

in

detail

in

Section I l l . For now, it is nec-

essary

to

keep in mind how common and indispensable heat exchangers

are in power systems and in many other applications. This underlines the

important connection which muht be made between heat transfer and fluid

mechanics, on the one hand, and thermodynamic irreversibility. on the

other.

The ultimate motive behind the infusion of entropy generation analysis

in heat transfer and thermal design is economic. Consider for a moment

the many factors which affect the decision of whether one thermal system

design is better than another

[

191. A number of these factors are political

in nature , but, to a large extent, the decision is the result of a cost-benefit

analysis, which takes into account the expense associated with manufac-

turing the device (capital cost ) and the expense associated with running

the device (operating cost) . The combination

of

design parameters which

yields the minimum cost subject to various constraints represents the eco-

nomic optimum design.

A n

important component in the cost analysis is

the degree of thermodynamic ineffectiveness exhibited by the device.

This component is the cost of electrical power required by the device

when i t functions in t h e sense o f a wor-k-absorbing thermal system. o r the

revenue derived from the sale

o f

electrical power when the device func-

tions in the work-producing mode. I t is clear that minimizing irrevers-

ibility in the thermal system yields a decrease in the operating cost. This

effect is usually accompanied by a parallel increase in the capital cost,

which demonstrates that the least irreversible design is not necessarily the

economic optimum. However.

in

order to make

a

sound economic deci-


5/59

4 ADRIAN EJAN

sion, the thermal designer must understand the true thermodynamic per-

formance of the heat transfer device; in other words, the designer must

know

t h e

irre versibiliry pic tu re

.

The work which lies ahead of heat transfer engineers, researchers, and

educators is to finally make the connection, to

fill

the gap, between the

traditional practice of heat transfer and the contemporary implications

of

the second law. This is an activity which must be pursued in engineering

schools as well as

in

industrial circles, at the fundamental and applied

level as well. In order to be able to minimize the thermodynamic irrevers-

ibility of heat transfer equipment, engineers must first understand the fun-

damentals of the entropy generation mechanism. Also, they must under-

stand precisely where in the heat transfer device irreversibility is being

produced, and how much is being produced locally. This requirement

is

very similar to the relevance of local Nusselt number and skin fric-

tion information to the conceptual design of efficient and compact heat ex-

changers.

In

this contribution the reader is exposed to a summary of first steps in

the direction of second-law analysis in basic heat transfer and thermal tie-

sign. The presentation

i s

intended to introduce the inexperienced reader

to the new methodology of irreversibility analysis and irreversibility mini-

mization in heat transfer. At the same time, the article is intended to show

the experienced heat transfer researcher the wealth of research opportu-

nity which exists

in

this growing sector of heat transfer. The monograph

stops short of investigating the relationship between irreversibility mini-

mization in thermal design and economic optimization of heat transfer

equipment. However, a number of thoughts in this direction are offered

in

Section

VI.

In writing the article. this author made a special effort to bring together

as much of the existing heat transfer irreversibility work as possible.

Despite the relative absence of publications on this subject, it is possible

that many workers

in

heat transfer have entertained second-law ideas

over the recent past.

I f

so, it is hoped that through this article a more

ef-

fective dialogue is initiated.

11. Irreversibility, Entropy Generation, and

Lost

Available Work

(Lost Exergy)

It

has been pointed out already that the concept of thermodynamic irre-

versibility and its relation to the one-way destruction of available work

are not new. Brief expositions of this subject are found in some of the


6/59

SEC

ON I I - LAW

ANALYSIS

5

most popular engineering thermodynamics textbooks

[20-221.

Howe ve r ,

its use in engineering is still sp or ad ic, particularly in heat transfer and in

therm al design . For this reason this au tho r finds it nece ssary to review the

irreversibil ity conc ept a n d i ts implications in the ar ea of energy cons er-

va t ion .

A .

O P E N

SYSTEMS

Consider the open thermody namic sys tem shown schematica lly

in

Fig.

1, which is the m ost general model of

a

heat transfer device such

as

a heat

exch ange r. T he sy stem is said to op era te in s tead y state and steady flow.

This means tha t the thermodynamic s ta te of matter surrounded by the

control surface does not vary with t ime, al though it can vary from one

point

t o

an oth er inside the control surf ace . In addit ion , the mass s tre am s

piercing the control surface.

mi

and m k , are cons tant in t ime . The the rmo-

dynam ic s ta te at each of the inlet or outlet ports is represe nted by prop er-

t ies averaged over the port cross section; in other words, the bulk flow

model ap plies . Th e energy tran sfer interactions exhibited by the system

are shaft (sh ea r) work transfer at a rate W , and heat transfer at a rate

Q o .

The posit ive sense of these interactions

is

assumed as shown in Fig. I .

The sys tem is in thermal co mm unica tion with the env iron m en t, which is

modeled as

a

heat reservoir o f tempera ture T o . Most important ly , how -

ev er, the contro l surface is drawn to include the actua l heat transfer de-

vice

p/rr.s

those immediately adjacent par ts of the e nviron me nt affected by

the functioning of the dev ice. This choice m akes the heat transfe r interac-

tion Qo revers ible . as it takes place across an infinitely small temperature

difference.

As

will soon become app are nt, this choic e is motivated by the

c w

S Y S T E M

I

outlet

m,

--

inlet

H E A T

,

(I) rnz -- T R A N S F E R (k)

D E V I C E

> Qo

To

.~

~

F I G .

I .

Open thermodynamic system f o r the second-law analysis of a heal transfer

device.


7/59

6 A D R I A N E J A N

need to identify all the thermodynamic irreversibility associated with the

devic e. In the wo rds

of

various w riters , th e irreversibility

internul

and ex-

term1

to the actual heat transfer device is incorporated in the aggregate

system delineated by the contro l surface. (F or an instructive discussion o f

this proce dure, see Van Wylen and S onn tag

[21].)

With all these assum ptions m ade , we a re s ti ll not in a position to

add ress the question of wh at impact the sy stem irreversibili ty ha s on the

work t ransfer exper ienced by the sys tem . We m ust make addit ional as-

sump tions , for example , tha t the inlet and out le t thermod ynam ic s ta tes d o

not change with the varying degree of irreversibility of the sys tem. In

other words, the property pairs ( h ,

s ) ~

nd ( h ,

s ) k

are considered fixed.

Th e freedom exercised in making this selection is what makes the notion

of lost available work a rrlat ive con cept . We re turn to th is commen t la ter

in this section.

The s ta tements of the first and secon d laws of thermodyam ics for the

system of Fig. 1 are , respect ive ly ,

In w rit ing the first law we assum ed that th e cha nge s in kinetic energy and

gravitational potential energy ex perienc ed by str eam s

m i

and

mk

are neg-

ligible com par ed with the e nth alpy chan ges retained in

Eq.

(2 .1) .Th e neg-

ative of the right-hand 4ide of E q . (2.2) is the net rate

of

entrop y genera-

tion in the system,

S,,,, ,

a quantity which is always positive and in fhe

reversible limit equal to zero:

(2.3)

,en =

C

( m s ) k - C (ms)i - Q o / T o

k 1

Eliminating Qobetween s ta tements

(2.1)

and

(2 .2)

one finds

(2.4)

I= k=1

describing an u pper bound fo r the wo rk tran sfer * of which the system of

Fig.

1 is

capable . As indicated in E q . (2.4), the algebraically maximum

work W,,, is achieved through r eve rsib le op erati on , i .e. ,

mk(h - T o S ) k

,,, = 9

i(h

-

T,S)i

-

( 2 . . 5 )

= I

k= l

*

In what follo ws we use the terms work transfer and heat transfer for

W

and

Q o ,

respectively, instead

of

the unit time terminology of mechanical pow er transfer and heat

transfer r ate .


8/59

SECONI)-LAWANALYSIS

7

This brings us back to the original question addressed by the analysis,

namely. the magnitude of the available work lost as a result of system irre-

versibility. Writing for

lost

ciwiltihlc~work

WLa

=

w,,,

-

w

(2.6)

and combining Eqs. (2.

I )

and (2 .2) with notation (2.3) yields. finally.

Wl,a, =

TJ, (2.7)

Expression (2.7) is a well-known result discussed

in

the Introduction, re-

sult referred to by some authors

as

the Goiry-Stodolci rheorm [23]. We

use the same name in

this

monograph. There exists a general lack of

agreement concerning the terminology associated with result

(2.7) .

For

example, for the difference (W,,,,,, - W ) Keenan [24] introduced the term

irrtwrsihili ty.

This term

will

not be used here

in

order

to

avoid confusing

the irreversible features of a thermodynamic system with the quantity

irreversibility whose units are [W ] .

Some authors prefer to discuss the above concepts by making extensive

use of the property availability. defined

as

b = h - T,s. From Eq.

(2.51,

the maximum available work becomes equal to the drop in availabil-

ity across the system. Once more, the lost available work is that share of

the maximum available work no longer present due to system irrevers-

ibility. The situation is shown schematically in Fig.

2.

Such diagrams are

usually

found

in

availability analyses of complex engineering systems,

where the lost available work can be broken down according to the irre-

versibility of individual components [23].

The property availability. made popular in the United States through

the work of Keenan. has a number o f equivalents proposed overseas.

In

Europe. the term exergy introduced by Rant [25] finds wider accep-

AVAI LAB1

LIT

Y

IN

A C T U A L

WORK

'LOST

A VA ILA B LE

WORK

A V A I L A B I L I T Y

OUT

Y A X I Y U Y

A VA ILA B LE

WORK

FIG.

.

Schematic representation

of

availability analysis

of a

thermodynamic system.


9/59

8 A D R I A N E JA N

tance. Earl ier , the thermody nam ics l iterature wa s acquainted with such

te rms as “die

griisste Nuf,- .arbeit”

(maximum useful wor k, Clausius

[26]),

“dip technisc he Arhei ts f2 ihigkeit”

(capabili ty of performing work,

BOS-

njacovi t

[27]),

and

“c>tiorgir utilisahle”

(useful energy, Danieus

[28]).

Th e last term is perhap5 responsible in part for today’s fashionable refer-

ence to an “energy cr i s is” or to “energy conservat ion” when, in effect ,

we all mean “available work”

or

“exergy” instead of “energy” [9].

In this review article we zero in on result (2.7) and conclude that the

available w ork lost ( de stro ye d) throu gh the irreversible o pera tion of the

system

of

Fig.

1

is proportional to the rate of entropy generation in the

syste m . Th e proport ionali ty facto r in this cas e is the absolute tempe rature

To of the heat reservoir (environm ent) with w hich the system exc hanges

heat .

From a thermal design optimization point of view, the direct route

toward minimizing the

loss

of available work

is

through the systematic

elimination of irreversibility sources in the system, i.e., through mini-

mizing

S,,,

. This is the important conclusion which motivates the work

summarized in this ar t icle. How eve r, a thermal designe r

is

usually not sat-

isfied with simply minimizing the ra te of en trop y gene ration . Equally im-

portant is the price paid a s lost available wo rk, at the e nd of the entrop y

generation minimization effort . In the next paragraph we demonstrate

that the simple statement

(2.7) ,

used for calculating

W,ost,

obscures

a

num ber of sub t let ies relat ive to th e meaning and the value of the tem pera-

ture factor

T o .

These observations are usually not present in published

discussions of the Gouy -Stodola theo rem .

B.

T H E A B SO L UT E E M P E R A T U R EACTOR o

Instead of Fig. 1 , let

u s

focus o n a sy stem in communicat ion with m ore

than one heat reservoir . T he new syste m is shown in Fig. 3, wh ere i t is as-

sumed tha t the system e xec utes an integral num ber of cycles experiencing

the net energy interact ions

Q i

i

=

0 ,

1,

...,

n )

and

W .

For

simplicity, we

consider a closed system. Again, the system bou ndary is chosen such that

all irreversibilities,

if

present , are located

inside

the sys tem.

The analysis presented below is based on a c om mu nicat ion by Jackson

[29].

In a m ann er identical to the pre ced ing ana lysis, we app ly the first and

second laws of thermodynamics to obtain

W = i Q i

i = l

i = O

(2.9)


10/59

S

t

c or.

1)- L. 4

w

A N A L Y S I S

Fit,.

3 .

Closed system operating

111 c y c l e 5

while in thermal commun ication wilh

n

heat

reservoir\.

At this point we arbitrarily el iminate t h e jt h heat transfer interaction , Q,,

between Eqs.

(2.8)

and

2.9).

This op eration yields

and

Wma, 1 t ( 1 1 , / 7 1 )

+

C Qi(1 - Tj/’Zi) ( 2 . 1 )

1 = i

T he

lost

available w ork. W , , , ~ W , follows directly from Eq5. ( 2 . 8 ) nd

(2.11):

( 2 . 1 2 )

WI,,s,j

= 7iLYgt.,,

( . j =

0,

1 , .

n )

(2 .13)

We co nclud e that “lost available w ork ”

is

a quantity which dr,pends on

our choice of reference heat reservoir , hence , su bs cr ip t j in E q . (2 .13) . In

fac t , Eq .

(2 .13)

allows us to calculate not one but

( n

+ I ) magnitudes, all

represen ting available work lost with respect to a succes sion of

( n

+

I )

absolute temperature levels . Since the rate of entropy generation is the

same in all cases. result ( 2 . 1 3 ) implies

w , # l S , . , = (7J /To)w, 0Sf , o (2 .14)


11/59

I0

A D R I A N

E J A N

Equation

(2.14)

constitutes a device for converting one lost available

work value into another, depending on the choice of reference tempera-

ture. Furthermore, since one or more

of

the heat transfer interactions Ql

of Fig.

3

may be zero, according to Eq. (2.14) it is possible to define lost

available work relative to a heat reservoir with which the system does not

interact at all. Therefore, based on convention alone,

it

is possible to refer

lost useful work to a universal absolute temperature for all systems which

may enter

our

consideration. This is actually what is done, a popular

choice of universal temperature being the absolute temperature of the

standard atmosphere

(298.16

K , i.e., 25°C). In cryogenic heat transfer the

reference temperature i s usually taken as To =

300 K .

In

conclusion, the important thought in maximizing available work is

t o

reduce

the

ent rop y gciiirrution in

the>

t h e rm o d y n a m i c s y s t e m .

The notion

of lost available work carries with

it

the specification of an absolute tem-

perature level relative to which the lost work

is

defined.

Since the lost availahle work is proportional to entropy generation, this

article focuses on entropy generation and its minimization through

thermal design.

As

summarized by London [30], the list of components

and phenomena responsible for entropy generation in engineering

systems is practically open-ended.

A

partial listing of entropy generation

sources includes [30] the following:

1 .

flow impact, shock, and

fluid

friction;

2 . solid friction;

3 .

free expansion of a gas (blowdown, explosion);

4 . flow throttling;

5 . mixing of dissimilar fluids, where the fluids can be dissimilar with

respect to temperature, pressure, or composition;

6 .

heat transfer across a finite temperature difference;

7. phase change where the initial conditions are not conditions of

equilibrium, e.g.. supercooled liquid, superheated liquid, super-

cooled vapor;

8 .

solution

of

a solid in a liquid;

9 .

plastic (inelastic) deformation of a solid;

10. electromagnetic histeresis;

11. joule ( P R ) heating in an electrical conductor;

12.

virtually all chemical reactions that occur with any rapidity.

In this article we focus on the mechanism commonly responsible for en-

tropy generation

in

heat exchange processes, namely, heat transfer across

a finite temperature difference and fluid friction. We also discuss specific

analytical methods aimed at minimizing entropy generation in engineering

components for heat exchange processes.


12/59

SkC.ONI)-LAW

A N A L Y S I S

I I

111.

Local

Entropy Generation in Convective Heat Transfer

It is instructive to examine first the entropy production mechanism at

the continuum level. before tackling the more complicated design task of

entropy production minimization at the heat transfer component level. To

do this we focus on convective heat transfer, the heat transfer mode most

commonly encountered

in

heat transfer equipment such as heat ex-

changers. The thermodynamics of irreversible processes in continuous

systems is well established and, tor a consistent exposition of the subject,

the reader is urged to get acquainted with any of the contemporary books

on irreversible thermodynamics [ 3 1 , 321.

A

very good introduction can be

found in Chapter 24, Irreversible Processes in Continuous Systems. in

Kestin's thermodynamics treatise (331.

Consider the local rate of entropy production inside a fluid engaged

in

convective heat transfer without internal heat generation. If the fluid is

Newtonian and incompressible. and if

i t

obeys the Fourier law of heat

conduction, the volumetric rate of entropy generation i n Cartesian coordi-

nates is [34]

( 3 .1 )

Function @ appearing

in

Eq.

( 3 . 1 )

is the viscous dissipation function,

better known from the energy equation for convective heat transfer [35]:

S"' = ( k / 7 9

[(dT/d.r)' t (i,T/il?.Y

+

(dT/dz)2]

+

(p /T )@

( 3 . 2 )

In Eq . ( 3 . 2 ) . s, r u , and

I > ,

are the local velocity components in the Carte-

sian system.

From examining

Eqs . ( 3 .

) and

( 3 . 2 )

i t is evident that high temperature

and velocity gradients are the features responsible for entropy generation

in the convective heat transfer situation considered here. Another impor-

tant effect likely to influence S"' .. is the variation in absolute temperature

through the continuum. For the same temperature and velocity gradients,

S"' increases as the temperature of the medium decreases. This effect is

exploited in Section

V .

in the minimization of entropy generation in

systems exposed to large variations in absolute temperature. In most heat

transfer applications. however. i t is safe to replace T appearing as denom-

* No t e t hat f o r t he r ema i nde r of the

al-trcle

w e

a r e d r opp i ng t h e subsc r r p t "gen" i n the

symbo l f o r e n t r opy ge ne r a t i on r a t e . The p r rn l es i nd i ca t e t he

local

c ha r a c t e r of

S.

1.e. . per

unit

v o l u m e S ' ( W / m . ' ) .

per uni t area

.Y'

( W l m ' ) .

or

per uni t length S ' t W / m ) .


13/59

12 ADRIAN EJAN

inator in Eq. (3.1) by an average, characteristic, absolute temperature of

the medium. This approximation is consistent with the constant property

model relied upon

in

writing Eq. (3.1). It should also be pointed out that

the temperature gradients appearing in Eq.

(3.1)

may be written in terms

of a relative temperature 0

= T - T*,

where T * is the reference absolute

temperature characteristic to the system.

A. CONDUCTIVEE R S U S

VISCOUS

EFFECTS

An important property made visible by the volumetric entropy genera-

tion formula (3. l) is the separation of the two entropy generation mecha-

nisms present, heat transfer in the direction of

a

nonzero temperature gra-

dient and fluid friction. The relative importance of the two contributions

to

S"'

requires special attention. Interpreting Eq.

(3 .1)

as

S '

= SEnduct ive

+

s:l;,,,,,

(3.3)

one can show that the relative order of magnitude of the two terms is

[34]

o(

Irls,,,,/s~,~,idurtive) =

E c h / ~ (3.4)

Ec =

( u * ) * / c p 0 * ,

the Eckert number

3.5)

Pr

=

v / a ,

the Prandtl number

(3 .6)

with the following notation:

T = 0 * / T * ,

the temperature difference number

(3.7)

Here, u* and

0*

are the characteristic velocity and temperature difference

for the convective heat transfer system at hand. The temperature dif-

ference number, Eq.

( 3 . 7 ) ,

is a dimensionless parameter always present

in

the second-law treatment of heat transfer processes.

In

most applica-

tions of engineering interest we find

T + 1.

In heat transfer analyses it

is

often convenient and possible

to

neglect

the viscous dissipation term in favor of the conduction and convection

terms

in

the energy equation

[35]:

This is particularly true

in

the case of subsonic gas flow. It is easy to show

that the order of magnitude of

p@

relative to the conduction part in Eq.

(3.8)

is dictated by the dimensionless group EcPr

[36].

In other words,

in

convective heat transfer problems in which the energy conservation state-

ment may be considered without the viscous dissipation term, the group

EcPr is much smaller than unity.


14/59

St-c

ON

[ > - L A W Y A L Y S I S

13

Comparing this conclusion w i t h Eq. (3 .4) we see that the energy argu-

ment by which Eq. ( 3 . 8 ) is regularly simplified has absolutely no bearing

on

the question on whether or not the viscous dissipation effect is neg-

ligible in the makeup of entropy generation. It is quite possible to have a

heat transfer problem where, although the energy equation can be simpli-

fied according to EcPr I , the entropy generation figure is in fact domi-

nated by viscous effects. This is the case

in

which T is so small that the

ag-

gregate group EcPr/T is actually of order one. This observation is moti-

vated by the fact that sometimes, perhaps suggested by first-law rules of

thumb and the desire to simplify t h e analysis, the viscous effect is

neglected

( I

priori

in

the second-law treatment of convective heat transfer

processes.

B .

E N I

O P Y

G E N F R A I

O N

h o t

I t . s

A N D

M A P S

The local entropy generation formula. Eq.

(3.

I ) , or equivalent forms for

other systems of coordinates [

371.

can be applied to known solutions for

the velocity and temperature

fields

in convective heat transfer. Thus. en-

tropy generation profiles or maps may be constructed, maps illustrating

the areas and features of the flow which act a s sources of thermodynamic

irreversibility. Below, we reproduce two examples which are based on

known textbook solutions

in

laniinar heat transfer.

A s

a first example, consider the Poiseuille flow through a round tube

with uniform heat

flux

4 around its circumference (see insert of Fig. 4).

The velocity and temperature profiles for this flow are particularly

simple [ 3 7 ] :

3.9)

(3.10)

u, = L., , , , , , [ I

-

(r/r,,Y]

8 =

(4 r,/X) [ - 4 ( . u / x o )

-

( r / r o ) *+ +(r/ro)4]

where

L ,,,,,

(ro2/4p)

-

t l P / d . r )

(3 .11)

and

x,, /ro = r~ , i~L~, , , J (~Pe, the Pkclet number ( 3 .12 )

Combining these expressions with the equivalent of Eq. ( 3 . )

in

cylin-

drical coordinates, one obtains the entropy generation profile in the tube

.~ 'kT*2/y 2( 2 R R )' + 16/Pe2 + ( 4 E c P r / ~ ) '

(3.13)

where R

=

r / r o .

In

writing Eq. (3.13) we made the assumption that the

temperature variation across the pipe cross section is small compared

with the local absolute temperature 7 . The local entropy generation rate


15/59

14

A D R I A N E J AN

0 0.5

1

R

F I G .4. Entropy generation profiles for heat transfer to laminar pipe

flow

[34].

depends on the radial position

R ,

on the Peclet number Pe, and on the

group EcPr/7, which determines the relative importance of viscous ef-

fects. The Peclet number governs the relative importance of irreversibility

due to conduction

in

the longitudinal direction. We see

that

when Pe <

4,

the axial conduction contribution dominates the radial conduction effect.

Figure

4

displays a family of entropy generation profiles in the cross

section, for cases

in

which the

axial

conduction effect is negligible,

Pe

4.

The value of ECPr/T increases gradually to the point where viscous

ef-

fects take over.

In

all cases, the pipe

wall

region acts as a strong source of

irreversibility. When E3cPr/T = 0, the maximum S”’ occurs inside the

fluid, at

R = f , it

coincides with the wall.

A s

a second example, consider the development of laminar momentiim

and thermal boundary layers along a flat plate. The situation is shown

schematically in the horizontal plane of the isometric drawing

of

Fig.

5 .

At

a

sufficiently large distance away from the solid wall the velocity and tem-

perature are uniform,

v X , =

and

T , .

The wall temperature is constant, I”*.

The entropy generation surface presented in Fig.

5

is based on the c1,as-

sical Blasius-Pohlhausen solution

[38]

which, for the special case

Pr

=

1,


16/59

S t C ' O N l ) - l , , 4 W A N A L Y S I S

F I G . 5 . Entropy generation \urface

foi-

laminar boundary

flow

and heat transfer over a

flat plate [34].

reduces

to

H I H

=

Ltfldr)

( 3 . 1 4 )

V J 1 J . X

=

(ltldr) ( 3 . 1 5 )

The similarity variable

r

is equal

to v ~ c ~ , ~ / ( u x ) ] ~ ~ * ,

hile f ( q )

\

the func -

tion tabulated by

H o w a r t h [39]

Combining solutions (3 .14 ) and ( 3. 15 )


17/59

16 A D R I A N E JA N

with the S"' formula (3.11, an d neglecting the terms asso ciated with gra-

dients in the longitudinal direction x, yields

( S ' / k )

vT*/f3muz,,)2 = ( 1 + Ec/7)fl12/Rez (3.16)

H ere , Re, is the Reynolds num ber u, , ,x /v . T he three-dimensional display

of Fig. 5 was c ons t ru c ted us ing (xz),,,/v) and

( y u , , , / v )

in the horizontal

plane andfl12/Re, in the vert ical direc tion. I t is evident that the entro py

generation is limited to the boundary layer. In the

y

direction, S s

highest, reaching a m aximum a t the solid wall. T he longitudinal variation

of S"'

is

a s l / x , indicating tha t like all gradients in the boundary layer so-

lution S"' blows up at the origin.

Entropy generation profiles of the type p resented in Figs. 4 and

5

may

be constru cted for ot her basic laminar flow solution s. More exam ples of

suc h plots may be found in Bejan

[34].

Ho we ver instructive, this visuali-

zation technique is limited to laminar solutions for which analytical ex-

pressions (exact

or

approximate) for velocity and temperature may be

available. In turbulent flow one has to rely on average (integral) heat

transfer and frict ion results , con den sed in ex perim ental correlat ions. 'The

method by which the ex per im enta l hea t trans fer and fluid friction informa-

tion is combined

to

reveal the irreversibility picture

is

presented in the

next paragraph.

c .

T H E E N T R O P Y tV tR ATlO N N U M B E R s

Consider a duct of arbitrary cross section, shown schematically as a

round tube in Fig. 6. Th e sk etch rep rese nts an infinitesimally short pas-

sage in a heat ex cha ng er, for exam ple. The h eat f lux per unit length q ' , he

mass flow rate m, he wetted perimeter p , and the cross-sectional area A

are specified. In steady state a finite temperature difference

AT

exists

between the wall and the bulk temperature of the fluid. The schematic

Fig. 6 also app lies to du cts with the c ro ss secti on varying periodically in

W A L L T E M P E R A T U R E ,

T + A T

-

A

,,-q

1

m,P,T ''

m , P -d P ,T + d T

\

,

P

F I G . 6. Infiiiitesimally short duct in a heat transfer dev ice.

1 - d x - I


18/59

St

C O N 0-1

A W A N A L Y S I S 17

the flow direction. I n such ciiseh, q ' represents the heat

flux

averaged

over a length L containing

an

integral number of periods, whereas A be-

comes the minimum flow area.

It can be shown t h a t for ( i t / \ '

prrrc'

. s r r h s r m n c . c ~ flowing through the

system, the rate of entropy generation per unit length is [40]

(3.17)

We recognize here the contribution due to fluid friction in the duct. the

first term

in E q .

(3 .17) , and the irreversibility due to heat transfer across a

nonzero temperature difference. I n many cases the temperature dif-

ference number 7 =

1777'

s much smaller than unity; hence. one can

write approx imately

(3 .18)

Although the A P and A 7 contributions appear separate in E q s . ( 3 . 1 7 )

and ( 3 . 1 8 ) . hey are in fact tightly interrelated through the geometry and

flow parameters of the system. The relations establishing this interdepen-

dence are the standard definitions [ 4 1 ]

h;,, = q ' / ( pA r )

( 3 . 19 )

- dP/i l . \ - =

2fG2//pD (3 .20)

Re =

1)G/p ( 3 . 2 1 )

St = / I : , ,

c,G (3.32)

and

I ) = 4 t , r / G p ( 3 . 2 3 )

where h,,,

,

I ) , G . . and St are. respectively, the average heat transfer

coefficient. hydraulic diameter

( 4 A I p ) .

mass velocity, friction factor. and

Stanton number. Therefore, if the friction and heat transfer information is

available (f'and St). i t

is

possible based on Eq. (3 .17) to evaluate the rate

of entropy generation integrated over the duct cross section. This state-

ment holds for both laminar

( i t i d

turbulent flow.

Since the momentum transfer information is condensed in dimen-

sionless groups such as friction factor. skin friction coefficient. or drag

coefficient, and since heat transfer results are expressed similarly in the

form of Nusselt or Stanton numbers, i t is appropriate to define a dimen-

sionless group for second-law analysis

in

heat transfer, the

ctirrop\' gc'tic'r-

n t i o t i n i r t t i h r r

N , . This new group is defined in

a

manner similar to friction


19/59

18

A D R I A N E J A N

factor and Nusselt num ber:

actual en tropy generation rate

charac ter is tic entropy t ransfer ra te

\. =

(3.24)

Th us , if in the system of Fig.

6

the heat transfer rate

is

fixed, the chara c-

teris tic en tropy transfer interaction acr oss the system bou ndary is

q ' / T .

Co nsequ ently, the entropy generation num ber derived from Eq. (3.1'7) is

N.5,

==

( r n / p q ' ) ( - d P / d ~ ) 7 /

1 + T) (3.25)

Similarly, local entropy generation numbers Ns- can be introduced based

on

Eqs.

(3.13) and ( 3 . 1 6 ) .

There are sys tems

in

which it is possible to identify a cha ract eristic en -

tropy generation rate , for exam ple, the minimum

S

in a thermal insulation

system of fixed geometry (Section V ) . In such cases, a more appropriate

way to define the entropy generation num ber is

actual entro py generation rate

charac teris t ic entrop y generation rate

,

= (3.26)

An important cha rac teristic of th e local entr op y generation rate for a

general duct is that its dependence on various design parameters is

n o n -

monolonic ' .

Th is feature is exemplified by the fact that a p ara m eter varia-

t ion which yields a decrease in one of the tw o contribution s to

S '

is lik.ely

to induce a n increase in the remaining con tribution . For this reason , i t is

difficult to predic t

in

adva nce the ch ange s induced in S ' o r

Nst

through the

implementation of a certain design modification. Care must be exercised

before deciding whether a parameter change is indeed beneficial with

regard t o m inimizing entropy g enera tion.

As a first example. consider the net effect of reducing the wall-fluid

temp erature difference. For this , i t is conven ient to com bine Eqs.

(3. .25)

and (3 .19) - (3 .23) into an expre ssion showing

N s ,

as an explicit function of

7 =

A T / T .

The end result is

N,s .

= (5'/32) ( fRez/St3) / . r3+

~ / 1

+ T)

(3 .27)

where parameter J plays the role of "duty" param eter for the duct:

J = q'p/mp(c,n7': (3 .28)

I t is app arent tha t

t he

A P and AT te rms in Eq. (3.27) are coupled via the

temp erature difference number T. Co nsequ ently, a minimization of the en -

tropy generation nu mber requires th e optimum selection of 7 with resp ect

to the sum of friction and heat tran sfer c on tribu tion s. In the limit

A

,


20/59

SI- O N I -

I

, 4W

A N A L

Y S1

S

\

Fie;. 7 .

The

entropy generation nunihcr

f o r :I

general d uct. a s a function

temperature difference a nd the cornhind parameter A . Eq. ( 3 . 3 0 , [40].

ot

t h e

will-fluid

the optimum is described by

Top1

==

A I12

(3.29)

1V,y,.niiii

?iA

( 3 . 3 0 )

where the dimensionless parameter A . not to be confused with the flow

area

of

Fig. 6, is

A

= [ ( 3 / 3 7 )

.f/St)]l'2 Re/St)J

( 3 . 3 1 )

For a given duty parameter

. I .

parameter A varies approximately as

Re/% . This is due to the fact that for common duct geometries the

Reynolds analogy between momentum and heat transfer holds to

t h e

ex-

tent that the group (JYSt)'" may be regarded a s constant over the range

10' < Re < lo5 [42]. Figure 7 is a three-dimensional logarithmic plot of

Eq.

(3.27). When T

< T , , ~ , ~ .

he heat transfer A T losses are small compared

with the fluid friction losses which account for most of N s . . In this region,

if the combined parameter A remains constant, the entropy generation

number N , s , increases sharply

as T

decreases:

Nhv,

T . Conversely,

when T

>

T,,,,~,

,

N , s .

s dominated by losses due to inadequate thermal con-


21/59

20

A D R I A N

E J A N

tact. In this region, as A is kept constant , N s t varies roughly as Nst -- T .

The minimum is thus shallower (less critical) on the T

>

T~~~ side of the

Nst

surface .

T oo ofte n, designers aim at maximizing the ratio h, ,/(pumping po w er)

in ord er to improve the performance

of

a heat transfer passage.

As

a sec-

ond exam ple, we cri tically ex am ine the meaning of th is procedure f rom

the point of view of’minimizing en trop y generatio n. We show tha t , by i t-

self , this proc edure is irrelevant (o r, a t be st , incom plete) with respect to

minimizing N s r

A dim ension less grou p proportional to the rat io of heat tran sfer coeffi-

cient

to

pumping power is

R

= (h,,pTp/m)

( -

d P / d x ) - ’

( 3 . 3 2 )

Exp ressing the entropy generation n um ber in terms of

R

and

A

as inde-

pendent parameters, we find

N s r

= 3”2/R2’3 A (R /3 ) ” ’

[ l

+ A ( R / 3 ) ” ‘ ] - ’

( 3 . 33)

Th is result is sho w n plotted a s a three-dimensional sur face in Fig.

8 .

T h e

features of this su rface are very s imilar to the feature s presented in Fig.

7.

Thu s , for cons tant A , there is an o ptimu m ratio R for which

Nst

is a min-

- 2

~

- 4i

FIG. 8 . The entropy generation number as a function of A and the ratio of heat

coefficient

to

fluid pumping power

R ,

Eq.

(3 .31 )

[40].

transfer


22/59

imum: for sufficiently small value\ of

A .

the optimum ratio i \

R,,p, 3 I A (3.34)

Based

o n

Fig.

8

and Eq.

( 3 . 3 3 ) .

we conclude that increasing the ratio of

heat transfer coefficient to pumping power

( R ) s not ,s i i , j f ic . ic , i ir to

ensure

improved thermodynamic performance, Since N s , depends

o n

more than

just R . the true effect of a proposed design change can only be evaluated

by estimating the changes induced

in R

and A, and eventually

in N S , .

D. T H I - MPACT O F -

H ~ . A I

’H&\scr H A U G M E N T A IO N

ON EN R O P Y GFN t R 4 I O N

The entropy generation number

i s

an important parameter in deciding

the true merit of a proposed design change aimed at enhancing heat

transfer in a heat exchanger apparatus. The research on heat transfer aug-

mentation techniques

i s

advancing rapidly, the more recent advances

having been reviewed by Bergles [43-451. According to the systematic or-

dering proposed by Bergles [44 .

451,

these techniques belong to two large

classes: I ) passive techniques requiring no external mechanical power,

and

( 2 )

active techniques which

do

require the use of external mechanical

power. ‘The most popular passive techniques are those employing treated

surfaces, roughened surfaces. extended surfaces, swirl flo* devices. dis-

placed promoters of heat transfer. and. finally, additives for liquids and

gases. The active techniques rely o n mechanical aids such as rotating heat

exchangers, surface and fluid vibration. electrostatic fields. and injection

or suction.

The task of evaluating the worth of a proposed augmentation technique

may be as important as conceptualizing and developing the technique. Of

the many evaluation criteria proposed ( see, e. g. , Webb and Eckert [46]

and Bergles r f a / . [47] ), the ratio of heat transfer coefficients (augmented

surface/reference surface) evaluated

at

constant fluid pumping power is

commonly preferred. However. a s demonstrated in the preceding para-

graph, this procedure

is

only partially relevant in that it focuses only

on

the possible payoff derived from an increased heat transfer coefficient,

saying nothing about the importance of

a

possible penalty associated with

the fluid pumping power

loss.

We can assess the merit of a given augmentation technique by analyz-

ing its effect on the degree

of

thermodynamic irreversibility character-

izing the heat exchanger

in

which the technique will be incorporated.

In

this sense,

if a

technique leads

to

reduced entropy generation. the tech-

nique is effective. The evaluation procedure consists of comparing the


23/59

22

A D R I A N E J A N

- x i

- i-

X , i

( 0 )

l b )

F I G .

9. Heat transfer augmentation by

finning

a bank

of

tubes in

cross flow: (a) smooth

tubes; ( b) finned tubes.

rate of entrop y generation p resent in the “a ugm ented” du ct ,

Sa’,

with the

entrop y generation

in

t h e

re fe rence , “un touched ,” duc t ,

S o ’ .

In the ex-

am ple of Fig.

9,

the finned bank of tu bes represen ts the augmented design,

vis

B

vis the bank of smo oth tub es used a s referenc e.

Th e param eter describing the effect of augmentation o n irreversibility is

the entrop y generation num ber Ns,a,which follows from definition (3 .26) ,

Ns,a =

Sk/SA

(3 .35)

Subscript “ a” is used here to den ote the fact t h a t N s is associated

sole/.y

with the effect of augmentation. The reference and the augmented duct

are identical except for those parameters affected by the addition

of

the

heat transfer augmentation fe ature. F or exam ple,

in

Fig. 9 the spacing and

tube diameters are the sam e in both case s , and so is the w orking fluid, th e

flow rate, m , and the heat transfer rate per unit length in the direction

of

flow, q ’ . Affected by the addition of cir cula r fins is the hydraulic diame-

ter , the w et ted per imeter , the Reyno lds number , the f r ic tion fac tor , and

the S tanton number .

A more useful version of Eq. (3 .35)

is

Ns,a = N.t .A*

+

[+o/(l

+

)I

(Ns, i ip

- N S A T )

(3.36)

with

N s , A T = S h . A * / S A , a T =

( S b / s t a ) DalDo

(3.37)

(3.38)

s , A p = S L , A , / S b

=

C f J f o (Do/Da> (&/A,)*

$o = Sh,u,/S, , ,AT = 2 fStRe‘(p2cpT/p2D4) ( w z / ~ ‘ ) ~

(3.39)

F orm s (3 .37) - (3 .39) are all based on the s tanda rd definitions for a d uct ,

Eqs.

(3 .19) - (3 .23) ,

plus the simplified expression

(3 .18) .

A special posi-

tion is occupied by pa ram eter

+o ,

which d esc ribe s the role played by fluid

friction in the irreversibility of the re fere nce desig n. As shown below,

is a crucial parameter, which, in addition to the specific augmentation


24/59

technique, determines whether augmentation will indeed decrease the

rate of entropy generation in the duct (i.e., N.y,a<

I ) .

The entropy generation number. E q .

(3 .36) ,

has been arranged to show

the special forms assumed by

N,,,;, n

the two extremes,

4,

-

and do

+

=.

As

expected,

in

a situation dominated by heat transfer irreversibility

(& -+ 0) he entropy generation number will be proportional among other

things to the ratio of heat transfer coefficients, Eq. ( 3 . 3 7 ) .Conversely,

when the irreversibility is dominated by

AP

effects (4,

x ) , Ns.,

will

vary as the friction factor ratio. I t is necessary to point out that both

ratios, St,(Re,)/St,( Re,) and .til( e21)Lfi,( e,), are function4 of the refer-

ence Reynolds number Re, since. for constant m ,

( 3 .4 0 )

Relations

( 3 . 3 6 ) - ( 3 . 4 0 )

re general and can be used to evaluate the

im-

pact of augmentation on the irreversibility of ducts

of

diverse geometries.

As an

example , we illustrate this procedure by considering the augmenta-

tion technique presented in Fig. Y . There are three finned-tube designs

( a l . a2. a3) being compared with the reference bank of smooth tubes. The

pertinent geometric parameters have been summarized in Table I [48-50].

Despite a conscious effort to compare tube bank geometries which differ

only with regard to t in geometry. it was impossible to find friction and

heat transfer data for smooth and finned tubes having exactly the same

array geometry

( X , ,

A ,).

he cases considered in Table

I

have approxi-

Refe ire nce Augmented \ u r f a c e

( \mooth )

surface (tinned

De\cription

0

11 I

1 1 2

( I 3

X , , transverse-tube pitch ratio 2 00 I .91 1.91

1.91

X , , longitudinal-tube pitch

ratio 1 HI

2.09

2 00 2 . 0 1

Fin density number of fins per

one-diameter unit length 4 . 5 2 5

h l 9.01

X , .

fin height. relative

to

rube

diamete I- - I .74 I 74 1.70

Fin thickne\s. relative to tube

diameter

0.0155

0 0 155 0.01

17

Source of friction and heat S u i - l x e n o . Surface no . Surface no Surface no .

transfer data

10-

I

I

I>

10-79

10-80

,A

10-92

A

148.

491

(48. 501 [4x. 501 [48,

501


25/59

24

ADRIAN

EJAN

o,l ,

1 l L l L

~ i - 1L :

2 10

100

Re,

x

I

0-3

F I G . 10.

Entropy generation number due t o heat transfer au gmentation,

N,,,,

versus ref-

erence surface parameters Re, and 4, .

mately the same array geometry, an inconsistency which does not affect

the main conclusions of this example.

The results of evaluating the available work conservation potential of

finning are summarized

in

Figs. 10 and 11. The entropy generation

number, Eq. (3.36) ,

w a s

plotted

in

Fig. 10 based on friction factor and

Stanton number data compiled by Kays and London [48]. It is evident

that, depending on the value of

& ,

the augmentation techniques under

study can either decrease

or

increase the rate of available work destruc-

tion in the heat transfer device. There exists

a

special class of reference

a1I’

< -

’

2

2 10 100

Re, x I O - ~

F I G . 1 1 . The function ,,(Ke,) for Ns.a= I . corresponding to three different finning

techniques.


26/59

S I - C O N I ) - L A W A N A L Y S I S 2 5

designs (Re,, +o) in which the implementation of a given augmentation

technique has absolutely no effect on irreversibility. This class is defined

by setting Ar,s,a=

I in

Eq.

( 3 . 3 6 )

nd Fig. 10. The result is represented in

Fig.

1 1

as a curve &(Re,) for each of the finned designs under consider-

ation. The curve &(Re,,) divides the field +o

-

Re, into two distinct do-

mains: only below this curve heat transfer augmentation is thermody-

namically advisable.

In conclusion, we find that the evaluation of an augmentation technique

without specifying the design which is to benefit from the technique, is

in c*orn p / r t i , rom the second-law viewpoint of conserving available work.

Whether an augmentation technique is valuable depends on the design

in

which i t is

to

be incorporated. This means that Re, and

6,

r 4 ' / m ,

L)

Eq.

( 3 . 3 9 ) ]

must be known before deciding the merit of augmenting heat

transfer

in

a duct.

The second conclusion is that graphs such as Figs.

10

and 1 I can be

used t o assess the relative effectiveness of different augmentation tech-

niques wi th respect

to

conserving available work. In Fig. I

I .

or example,

three fin designs were considered for the

job; in

addition, we could have

considered the impact of other techniques, such as roughening the tube

surface or using pin fins. for the same design.

\

IV.

Entropy Generation Minimization in Heat Exchanger Design

In this section we increase the degree of complexity of the heat transfer

apparatus, by addressing the question

of

irreversibility minimization

in

heat exchangers. The topic of heat exchanger design, however, suffers

from a traditional bias toward first-law anlaysis and away from second-

law considerations. The very name heat exchanger is suggestive of the

fact that the function of the apparatus might be to transfer a certain

amount of heat between two bodies (fluid streams, most often) at different

temperatures. This

is

not generally true. For example,

in

power and

refrigeration cycles the true function of heat exchange equipment is to

allow various components of the cycle to communicate with one another

in the least irreversible way possible (see Section

V . B . 3 ) .

This observa-

tion is supported directly by the Gouy-Stodola theorem

( 2 . 7 ) .

n connec-

tion with this observation, i t i4 instructive to first consider the following

example.

Figure 12 shows schematically a Brayton cycle heat engine with regen-

erative heat exchanger. The high-temperature end of the cycle (heater.

expander) must communicate with

t h e

low-temperature end (cooler. com-

pressor) in order to exchange low-pressure fluid for high-pressure fluid.

The most efficient communication is established when:

( 1 ) there

is

no


27/59

26

A D R I A N E J AN

p2

HEAT ER EXPANDER

C O O L E R

R E G E N E R A T I V E

H E A T E X C H A N G E R

C O M P R E S S O R

E N T R O P Y

FIG. 2. Countertlow heat exchanger for a Brayton cycle heat engine [42].

pressure drop in the regenerative heat exchanger; (2) the inlet to the

heater

is

already

as

hot

as

possible, i.e., as hot as the expander outlet; and

(3)

the inlet to the cooler is as cold as the compressor outlet. This limiting

case corresponds to a completely reversible regenerator (AT =

0,

A P =

0). From this example

it

is apparent that the effective stream-to-stream

heat exchange

in

the regenerator is only one of the by-products of its true

function, that of allowing the hot and cold ends of the power cycle to corn-

municate (trade fluid) in the least irreversible manner. As discussed in

Section V , B , 3 , the function of the regenerative heat exchanger is to

practically

insulate

the hot end of the cycle from the cold end.”

McClintock [51] appears to have been the first to recognize that the

concept of irreversibility minimization has a definite place in heat ex-

changer design. In a generally unnoticed conference article, McClintock

establishes the connection between Keenan’s “irreversibility” quantity

as a means of measuring thermodynamic nonideality [24] and the engi-

neering task of designing efficient heat exchangers. He discusses the irre-

*

This remark is amplified by the often-quoted example of a naturally occurring counter-

flow heat exchanger, the blood counterflow in the long legs of wading birds [ 5 2 ] . More vis-

ibly than anywhere else, the

job

of

the counterflow

is

not to exchange heat but to permit the

bird‘s body and its foot to exchange oxygenated blood for oxygen-depleted blood, as re-

quired by metabolism. There are reasons to believe that the bird‘s metabolism is ultimately

geared onto the conservation of available work

in

its organs, particularly in the legs.


28/59

S E C ~ N I ) - I _ A WN A L Y S I S

27

versibility minimization procedure in terms of a heat ex chan ger passage

of infinitessimally short flow length. Assuming that t h e wall-fluid temper-

a ture di f ference

A 7

is fixed, M cClintock d em on str ate s analytically the

existence of clear trade-offs regarding the selection of duct geometric

parameters such as hydraulic diameter. His analytical conclusions are

qualitatively similar t o those presen ted graphically

in

Figs. 7 and 8. F rom

a practical engineering point of view , his design con clu sion s are not

immediately applicable d ue to the artificial natu re of the co nsta nt A 7 con-

straint placed at the basis of his study.

Le Foll [S3] used an irreversibility analysis similar to McClintock's to

evaluate the thermody namic effec t iveness of convect ive heat t ransfer

from gas-cooled nuclear re acto rs . Le Foll prop ose s the use

of

St"/J'as a

performance evaluation criterion (figure of merit).

A . Ht- .41 E X C H A N G E R SI I H

% i . ~ o

RESSURE

D R O P

The simplest irreversibility analysis of the process in a heat exchanger

is based on the assumption

of

Lero pressure drop in the flow passages.

Accordingly, the only source o f entropy generation appearing

in

the anal-

ysis is the transfer of heat

across

the nonzero temperature difference

which exists between the two fluid s tream s wh en the heat exchang er area

is finite. Although the practice of neglecting the fluid friction irrevers-

ibility is not generally su pp orte d by o rd er of magnitude arg um en ts such as

the discussion c entered around E q . (3 .41, it is instructive to first cons ide r

this highly idealized

limit

for the insight it provides into the relationship

between the thermodynamic irreversibil i ty and the choice of heat ex-

changer des ign parameters .

Irreversibil ity analy ses of two -stream (parallel and coun terflow ) heat

exchangers with zero pressure drop have been reported by a number of

au thors [23, 30,

54.

551. Here we summarize an example i l lustrated by

Tribus

[54],

who considered the irreversibility in a heat exchanger with

equal and constant capacity rates ,

C

=

Cmin

=

mc,

(F ig .

13).

T he

overall heat tran sfer coefficient I/ is also a cons tant in the analysis. T ribus

sh ow s that in this limit the rate of entrop y ge neration take s a part icularly

simple form:

( 4 .1 )

where T I , Tz are the in le t absolute tem pera tu res of the tw o s t reams and

N,, is the number of heat transfer units UAlmc,. This result is shown

plotted

in

Fig.

13

a s

Sarn/tnc.,,

versus the heat ex chan ger effectiveness

E,

which in the example shown here is given by

€

= Nt,/( 1 + N,, )

( 4 . 2 )


29/59

28

A D R I A NB E J A N

Nt.

0

0 2

0.5

I 2 4

10

0 0.2 0.4

0.6 0.8

1.0

e

FIG.

13 .

Entropy generation rate in a balanced counterflow heat exchanger with

zero

pressure drop . (Afte r Tribus

[54].)

The symmetry

of E q . (4.1) is

reflected graphically in Fig.

13.

The en -

tropy generat ion rate is invariant to the t ransformation

T 1 / T z

+

T z / T , ,

meaning that in Fig.

14 the

absolute temperature may be measured in

ei ther direct ion on the abscissa. In other words, Tl may be assigned to

eithe r the w arm inlet or the cold inlet without changing the app eara nce

of

result

(4.1).

The ent rop y generation ra te reaches

a

clear maximum when

N , ,

=

1 ,

maximum given by

(4.3)

This feature is not intuit ively obvious since we would expect the heat

S e c / l n ( ' P =

In[* +

i(Tl/TZ

+

Tz/T,)I

S T R E A M

2

CONTROL

VOLUME

/

ii

T R E A M

1

TI

T*

T E

M

P E R

A T U R

E

F I G .14. Schematic of counterflow heat exchanger with imbalanced capacity rates [42].


30/59

S

c Y

N I)- 1,A w

A NAL Y

s s

29

tran sfer irreversibility to decrease monotonically with the increasing heat

exchanger a rea ( N , , ) .

n

the range N , ,

>

1, the behav ior is as expecte d,

the entropy genera t ion ra te decreas ing in the direction of smaller

s t ream-to-s t ream temp era ture di f ferences (higher N , , ' s ) . Below N , , = 1 ,

ho w ev er, the s trea m-to -stream tem perature difference is of the order of

IT1

-

T,I

and

is

relatively insensitive t o changes in N , , . Conseq uent ly , as

Nl,

decrea ses the net hea t exchan ge between the two s t reams decrease s ,

and since the tem peratu re difference is roughly c on stan t , so does the en -

tropy generation rate .

Golem and Brzustowski [5S] examined the irreversibility of heat ex-

changers using the Reistad effectiveness

eK

defined as [%]

(4 .4 )

vailabili ty (ex erg y) gained by the cold s tream

availabili ty (ex erg y) lost by the w arm stream

K =

In the limiting case of reversible heat excha nger op era tion . eH s equal to

unity. Neglecting the irrever5ibility associated with frictional pressure

drops, Golem and Brzustowski showed t ha t the Reistad effectiven ess re-

duc es to

where subscr ipts C. H . out , and in refer to the cold s tream, hot s tream.

ou tlet , and inlet, respectively. T he

+

sign applies to counteiflow and

the

L b _

sign to parallel f low. l h i s expre ssion holds fo r ideal gases and

incompressible l iquids. The sam e autho rs extend ed the

eR

concept to the

local level, showing that when the longitudinal temperature variations

Tc( . r ) ,T&)

are know n. one can evaluate locally the relative loss of avail-

abil ity (ex erg y) du e to heat t ransfer acro ss the s t ream-to-s t ream tempera-

ture d i f ference .

B.

H t . 41 E X C H A N ( ; t R

G F . O M I I

K Y

F-OK

M I N I M U M

I R R E V t R S I R I I . I I Y

A mo re realis tic a ppro ach to the second-law analys is and synthe sis of

heat exchan gers must take i n t o ac co un t the irreversibility d ue to fluid fric-

t ion, focusing on the strong ( , o / ~ p / i ~ i ghich exists between heat transfer

and fluid friction irreversibilities. T his w as don e in a rec ent article by this

au thor 1421. This i s the first instance in which the procedure of entropy

generation minimization is pre sen ted

in

the context of a com plete heat ex-

changer sys tem, namely, the c lass o f c oun t e dow he a t e xc ha nge r s fo r

gas-to-gas applications.

A

new design method, thr, tirrnihrr (? I c'ritropy

gericrrrtiou

rrri i ts

N.$.

s proposed

in

lieu

of

the traditional number of heat

transfer units

N , ,

[48].

Co nsider the sche ma tic represe ntation of Fig.

14,

in wh ich indices 1 and


31/59


2

have been associated with the minimum and maximum capacity rates

(rnc,)

in

counterflow. Defining th e en tro py generation n um be r N, in the

m ann er indicated in Eq.

(3.24),

the rate of destruct ion

of

available work in the heat exchanger can be

wri t ten as

N s =

(Cmin /Crnax)

MTz/’T,) +

-

C m i n / C m a x ) ( 1 -

T,/Td]

( C m i n / C m a x Y

( 1

- C m i n / C m a x )

(1

- T , / T J 2

+

x

e x ~ [ - N t u ( l -

C m i n / C m a x ) l

x

exP[-Ntu(l

- Crnin/Cmax) l }

+ ( C r n i n / C m a x ) ( R / ( ’ p ) l (Af‘/P)I

+

(R/cp)Z ( A P / P ) z

[ I

- (Crn in /Crnax) ( 1 -

T1/TdI (1

- (Cmin /Crnax)

(4.’7)

Here ,

R

is the ideal gas con stant , the rest of the sym bols having been d e-

fined in Fig.

14

and

in

the Nom enclature .

Equation

(4 .7)

is based on the assumption that the heat exchanger is

“nearly ideal ,” in other words,

(AP/P)l,z 1

and

1 - E 6 1 ,

where

E

is

the effectiveness (Tl,out- T1)/(Tz

- T I ) .

his assu m ption m ake s visible in

Eq. (4.7)

the two mechanisms responsible for entropy generat ion. The

first two terms account for irreversibil i ty due to heat transfer betwee:n

st reams, across

a

nonzero temperature difference. The last two t e rms ,

individually, represent the fluid friction effect in each of the two ducts of

the heat exchanger.

I t

is easy to s ee that in the

Ntu+

l imit the sec ond

term va nishes, and the

A T

irreversibility is due solely to flow imbalance

(Cmax

> Cmin). igure 15 shows how the imbalance contr ibut ion to N F ,

th e first ter m in Eq.

(4.71,

varies with the capacity ratio and the absolute

temperature ratio. From a practical fiesign viewpoint, it is important

1.0

know the magnitude of the imbalance contr ibut ion when one seeks to

minimize N, by increasing the N,, an d by decreasing t he friction A P ’ s : he

imbalance com ponen t tel ls the designer when he or she has reached the

point of diminishing re turn s in

the m inimization of overall Ns , sinc e

in

the

limit N , ,

-+ 00,

APl,z + 0. the imbalance term is the sole survivor in

Eq.

(4 .7) .

A

furt he r simplification

of

the Ns Eq.

(4.7)

is achieved when on e con-

siders the cas e of nearly balanced cap acity rates,

Cmin= CmaX

Applying

the calculus l imits as

C,,in

-

C, , , ,

the number of entropy generation

uni ts becomes


32/59

St,c

O N I F L A WA N A L Y S I S

31

10

’

I

10’ 1 1 ,

0 5

1

2

1, 1

F I G .

IS.

Th e entropy generation d u e l o capacity rate imbalance in a counter f low heat ex-

changer [4?].

where

NS.irnhalancv

=

[ ( ~ n l a , / ( ’ , n l n )

-

I

I

[ ( ] ; /TI)

- I

-

In 7 2 / 7 1 ]

+

(c’min/crnax)

K l c , , , ) , A P / P ) ,

( 4 .9 )

N , y , = [ ( T 2 / T 1 ) 1 2

-

(7‘,/7;)1’’]2 N,’,

(4 .10)

N,?, = (C’ , , , , , , /C, , , , , , )( 7 2 / 7 ’ 1 ) 1 ’ 2

-

(T1/T2)”*12N +

( R / ( . , ) ,

A P / P ) ,

(4 .11)

In this limit Ns as separate terms describing the rate

of

entropy genera-

tion o n each of the two sides of the heat exchanger. Moreover, the irre-

versibility of each side N s , , ,appears

a s

the sum of one contribution due to

heat exchanger A T and one contribution due to frictional

AP

losses:

N . 5 ,

2 :

Ns.A~r,:?

NS,AP,.P

(4 .12)

This form is similar to the f o r m o f

N, ,

for an elementary heat exchanger

passage.

Eq .

(4. I . Equation ( 4 . 8 ) is pivotal in the design process, as it

permits the minimization of N , , for each side of the heat exchanger. sepa-

rately. Furthermore. the analytical form of

N,,

and

N

s similar (iden-

tical in the balanced f low

case).s o

that the design procedure for each side

is the same.

1 . Nirttr twr c f E I I / ~ o ~ \ ~< , t l ( , / i l / i o t l 1 J t l i t . y

.fbr

OtIc S i t l ~ .

,,,

I t

is

instructive to examine the manner

in

which various heat exchanger

parameters affect the irreversibility

o f

one side. Recalling that f rom the


33/59

3 2 A D R I A N

B E J A N

definitions of number of heat transfer units and friction factor we can

write

[48]

Ntm,,

=

(4L/D)

S t

(4.13)

(AP/f‘)i,z

=

f(4L /D ) G/2pf‘ (4.14)

the irreversibil i ty per side assumes the general form

4L

“1,270 + 61.2 ;)l,, f 1 2

(D)*.,:,2

(4.15)

Ns1 2=

(4L/D)l,,Stl,2

Equat ion

(4.15)

contains the usual heat e xchang er notat ion, whe re, in ad-

dition, we defined

a,

=

1 ,

a2

= Cmin/Crnax

(4.16)

b1

= Crnin/Cm x

b,

=

1

(4.17)

7 0 = [ ( T , / T , Y -

(Tl/T2)”’]2,

g =

G / ( 2 p P ) ” ’

(4.18)

Th u s , g plays the role of dime nsionless mass v eloci ty, while

T~

has the

same significance as the tempe rature difference num ber

7 )

relative to the

difference between inlet temperatures, T I and T,.

The dependence of N s , , ,on design parameters such a s Re,

4L/D,

andl

g

is

shown qualitatively on the three-dimensional logarithmic plot of Fig.

16.

Th e graphic c onstruct ion of Fig.

16

is

actually based on empirical da ta

on turbulent flow ins ide smooth tubes , where both St and f a r e propor-

tional to Rep0.,; one can construct qualitatively similar three-dimensional

plots for other heat ex change r surfaces.

2 .

Opt imum Duct Geonrrtry, 4 L I D

From Fig.

16

an d Eq .

(4 .15)

t i s evident that num ber of entro py genera-

t ion uni ts always increases when

g

increases, with

4L/D

and Re re-

maining fixed. Unlike g , the du ct slenderness rat io

4 L / D

plays a definite

trade-off role: for constant

g

and Re, there exists a clearly defined op-

timum

4L/D

for which th e resulting N8,,* s a m inimum. The opt imum con-

dition for each side is describ ed by

(4.19)

and

=

2g[abr0

(R/c,) (jC/St)]1/2

(4.20)

F or comm on heat exchanger sur faces the group ( f /St )1 /2 s only a w eak

function of the Reynolds number

[42].

Therefo re , Eq .

(4.20)

establ ishes

a


34/59

SECONI)-LAWN A L Y S I S

3 3

F I G .

Ih. Number of entropy generation u n i t s

for

one side of the heat exchanger, as

a

function

of

L / r , , .

g.

and

hRt.

hence

r , ,

=

11/41

[4 ].

one-to-one correspondence between the m ass ve locity g and the lowest

ra te

of

entropy generation achievable in the heat exchan ger duct .

There are var ious ways

in

which the

N s

optimization of a heat ex-

changer may b e con ducte d. Th e three dimensionless design parameters

for on e s ide of the heat exchanger , 4 L / D , g , and Re (F ig . 16). define

a

three-d imen sional space of possible design co ndit ion s. If the degree of

thermodynamic irreversibility o f each side N, , , , is specified in advance ,

then , via E q . (4.15). the number of unkno wns for each s ide i \ reduced to

tw o. If , as in Eqs.

(4.19)

and

( 4 . 2 0 ) .

N S , , *s not specified b ut. for a given

g

and Re , the ratio 4L/D is chose n such that N,, , , is minimized. the number

of design unknowns per s ide is again reduced to two. In practice, the

number of independent design parameters may be less than two per side

du e to addit ional design constrain ts . The design p roced ure subject to

two

constraints . constant heat transfer area and constant heat exchanger vol-

ume, is presented

in

Bejan [42]. I n addit ion, Bejan [42] develops the

com plete design procedu re for minimum heat transfer area subject to fixed

N,, , ,

.

This procedure is applied eventually

to

design of a specific regen-

era tor for a Brayton cycle .


35/59


C . SENSIBLEE A I U U I T SFOR E N E R G Y T OR A GE

As

a s eco nd e xam ple con side r the minimization of irreversibility i.n a

sensible heat unit for energy storage. Traditionally, the thermal design

and optimization of a sensible heat stora ge unit relies on the view that the

system thermal Performance can be assessed based on how much thermal

energy the unit can s to re. In sh or t, a unit is considered m ore efficient than

anoth er if-for the same heat input and the same am oun t of s torage

material-it is cap able of storing more thermal e ner gy . This point of view

is generally accepted and serves as basis for testing and evaluating the

thermal performance of sensible heat (fluid and solid) storage units

[57].

Bejan

[ S 8 ]

analyzed

t h e

performance of such units by treating them as

systems intended to store available work, the function they perform in

most pow er appl ica t ions .

Consider , for example ,

t he

system shown schematically in Fig. 17. It

cons is ts of a large liquid bath

of

mass M and specific heat C placed in an

insulated vessel . Hot ga s ente rs the sy stem through on e port , is cooled by

flowing through a gas-liquid heat exc han ger imm ersed in the bath , and is

eventually discharged into the atm osp her e. Gradually, th e bath te mp era-

ture T a s well as the gas outlet tem peratu re To,, r ise , approaching the hot

gas inlet tempe rature

7

.

I t

is assu m ed initially t hat the bath tem per atur e

equals the environment temperature

T o .

The bath is filled with an incom-

pressible l iquid such a s w ater or oil . Th e s tream m c arries an ideal gas, for

exam ple, high-temperat ure s team or a ir . Th e s tream of hot gas is supplied

continuously at T , and P o ; before entering the u nit , the s tream is com -

pressed to

Po +

A P in order to overcome the pressure drop caused by

friction in the heat exchanger.

The t ime depen dence of the bath temperature and the gas outlet tem-

perature ca n be derived analytically an d th e result is available in the engi-

neering literature

[S9].

Of interest here is the total amou nt of entrop y gen-

erated from the beginning of the ch arging pro ces s until an arbitrary time

r

( S / m c , t )

=

( R / c , , )

In( l

+

A P / P o )

+

T

-

1

+

T)

Documents

Second-Law Analysis in Heat Transfer and Thermal Design