sizing of a crossflow compact heat ex changer

P1: IML/OVY P2: IML/OVY QC: IML/OVY T1: IML

GRBT056-29 Kutz-2103G GRBT056-Kutz-v4.cls July 27, 2005 17:33

Chapter

29Sizing of a Crossflow

Compact Heat Exchanger

Dusan P. SekulicDepartment of Mechanical EngineeringUniversity of KentuckyLexington, Kentucky

Summary

This text offers a rigorous, step-by-step methodology for calculatingcore dimensions of a compact heat exchanger. Considering the analyt-ical complexity of implemented calculations, the most intricate basicflow arrangement situation in a single-pass configuration would be acrossflow in which fluids do not mix orthogonal to the respective flowdirections. Such a flow arrangement is selected to be considered in thiscalculation. The set of known input data is provided in the problemformulation. Subsequently, calculations are executed using an explicitstep-by-step routine. The procedure follows a somewhat modified ther-mal design (sizing) procedure derived from the routine advocated inShah and Sekulic (2003). The modification is related to the fact thatiterative steps are listed and executed for both intermediate and over-all refinements of all assumed and/or estimated entities. The overallpressure drop constraints are ultimately satisfied without further opti-mization of the design. The main purpose of the calculation sequence isto illustrate the procedure, usually hidden behind a user-friendly, butcontent-non-revealing, platform of any existing commercial softwarepackage. Such a black-box approach executed by a computer, althoughvery convenient for handling by a not necessarily sophisticated user, isutterly nontransparent for a deeper insight. Consequently, this example

29.1

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Source: Heat-Transfer Calculations



29.2 Heat Exchangers

COLD-AIR FLOW

HOT-AIR FLOW

Lgas

Lstack

Lair Primary heat-transfer surface

Secondary heat-transfer surface

Figure 29.1 Schematic of a cross-flow heat exchanger with both fluids unmixed.

calculation is not intended to discuss a particular design; rather, it il-lustrates the procedure. Therefore, the numerical values calculated donot have importance for any particular design.

Problem Formulation

A task at hand is to design (“to size”) a heat exchanger, specifically, to de-termine principal heat-exchanger core dimensions (width, length, andheight of the core having specified heat-transfer surfaces) (see Fig. 29.1).The heat exchanger has to cool a hot-air gas stream, available at anelevated temperature, with a cold-air stream, available at a signifi-cantly lower temperature. Terminal states of both fluid streams areknown, except for the outlet of the hot stream as follows:

Cold fluid Hot fluidFluidProperty Symbol Unit Value Symbol Unit Value

Inlettemperature

Tc,i K 500 Th,i K 700

Outlettemperature

Tc,o K 620 Th,o K —

Inletpressure

pc,i kPa 500 ph,i kPa 100

Mass flowrate

mc kg/s 20 mh kg/s 20

Pressuredrop

�pc kPa 5 �pc kPa 4.2

Fluidtype

Air — — Air —



Sizing of a Crossflow Compact Heat Exchanger



Sizing of a Crossflow Compact Heat Exchanger 29.3

Given data

The following information is provided: cold fluid—air; hot fluid—air, Tc,i,Th,i, Tc,o, pc,i, ph,i, mc, mh, �pc, �ph; heat exchanger flow arrangementtype—single-pass, crossflow unmixed-unmixed flow arrangement.

Find

The principal dimensions of the core must be determined: (1) fluid flowlengths (core dimensions) in directions of hot and cold fluid flows and(2) the dimension of the stack of alternate layers of flow passages in thedirection orthogonal to crossflow planes.

Assumptions

Determination of the core dimensions assumes an a priori decision re-garding selection of heat-transfer surface types on both sides of a heatexchanger. This selection is, as a rule, within the realm of an engineer’sdecisions for any sizing problem; thus, it is not necessarily given in theproblem formulation. In the present calculation, a decision regardingthe surface selection will be made at a point when geometric and heat-transfer and/or hydraulic characteristics of the core need to be assessedfor the first calculation iteration. That decision may always be modi-fied and calculation repeated. The types of heat-transfer surfaces willbe selected, and data involving geometric, heat-transfer, and hydraulicproperties will be obtained from a database given in Kays and London(1998). The assumptions on which the calculation procedure is basedare listed and discussed in detail in Shah and Sekulic (2003, chap. 3,p. 100) and will not be repeated here (standard assumptions for design-ing a compact heat exchanger).

Calculation Procedure

Design procedure for a sizing problem features two distinct segmentsof calculation. The first one delivers the magnitude of the thermal sizeof the core, expressed as an overall heat-transfer area A, and/or for-mulated as a product of the overall heat-transfer coefficient and theheat-transfer area UA. Determination of this quantity should be basedon an application of thermal energy balance [i.e., the heat-transferrate delivered by one fluid is received by the other; no losses (gains)to (from) the surroundings are present]. Formulation of this balance in-volves a fundamental analysis of heat-transfer phenomena within theheat exchanger core, which can be summarized through a concept ofheat exchanger effectiveness, Shah and Sekulic (2003). The resultingdesign procedure is the “effectiveness number of heat-transfer units”method. The effectiveness is expressed in terms of known (or to be







determined) inlet/outlet temperatures, and mass flow rates (for knownfluids). The unknown temperatures (for some problem formulations)must be determined (i.e., they may not be known a priori—such as theoutlet temperature of the hot fluid in this problem), and any assumedthermo-physical properties should be re-calculated multiple times (i.e.,an iterative procedure is inherent). This feature of the calculation isonly one aspect of the design methodology that ultimately leads to aniterative calculation sequence. The second reason for an iterative na-ture of this procedure is, as a rule, inherently transcendent structure ofthe effectiveness-number-of-heat-transfer correlation (for the crossflowunmixed-unmixed arrangement, as in the case that will be revealedin step TDG-8 in the tabular list below). Finally, the third and mainreason for an iterative procedure is a constraint imposed on pressuredrops. The magnitudes of pressure drops must be obtained from thehydraulic part of the design procedure. The hydraulic design part ofthe procedure cannot be decoupled from the thermal part, which leadsto the calculation of pressure drops after thermal calculations are com-pleted, and hence is followed by a comparison of calculated pressuredrops with the imposed limits. As a rule, these limits are not neces-sarily satisfied after the first iteration, and subsequent iterations areneeded.

In this routine calculation presentation, determination of the thermalsize of the heat exchanger will be termed the “targeting the design goal”(TDG) procedure. Each step will be separately marked for the purposeof cross-referencing. The second segment of the calculation is devotedto the determination of actual overall dimensions of the core, in a man-ner to satisfy the required overall heat-transfer area and to achieve theoverall heat-transfer coefficient, that is, to satisfy the required thermalsize. This segment is inherently iterative because it requires a satisfac-tion of pressure drop constraints as emphasized above. This segmentof calculation will be termed “matching (the design goal and) geometriccharacteristics” (MGC) procedure.

Both procedures will be organized as a continuous sequence of cal-culations and presented in a tabular format for the sake of compact-ness and easy access to various steps. The most important commentswill be given as the notes to the respective calculation steps immedi-ately after the equation(s) defining the step. A detailed discussion ofnumerous aspects of these calculations, and the issues involving re-laxation of the assumptions, are provided in Shah and Sekulic (2003).The reader is advised to consult that source while following the step-by-step calculation procedure presented here. Some numerical values ofthe derivative variables presented may differ from the calculated valuesbecause of rounding for use elsewhere within the routinely determineddata.






Targ

etin

gth

ed

esig

ng

oal

(TD

G)

pro

ced

ure

Ste

pC

alcu

lati

onV

alu

eU

nit

s

TD

G-1

aT c

,ref

=T c

,i+

T c,o

2=

500

+62

02

560

K

TD

G-1

bT h

,ref

=T h

,i=

700

700

K

To

init

iate

the

iter

ativ

epr

oced

ure

for

asi

zin

gpr

oble

mli

keth

eon

egi

ven

inth

ispr

oble

mfo

rmu

lati

on,a

dete

rmin

atio

nof

refe

ren

tte

mpe

ratu

res

ofbo

thfl

uid

sis

nee

ded.

As

afi

rst

gues

s,ei

ther

anar

ith

met

icm

ean

ofte

mpe

ratu

rete

rmin

alva

lues

ora

give

nte

mpe

ratu

reva

lue

(if

sin

gle)

for

each

flu

idm

aybe

sele

cted

.

TD

G-2

ac p

,c=

c p,a

ir(T

c,re

f)=

c p,a

ir(5

60)

1.04

1kJ

/kg

K

TD

G-2

bc p

,h=

c p,a

ir(T

h,re

f)=

c p,a

ir(7

00)

1.07

5kJ

/kg

K

Th

esp

ecifi

ch

eat

ofei

ther

ofth

etw

ofl

uid

s(f

orsi

ngl

e-ph

ase

flow

s)is

dete

rmin

edat

the

calc

ula

ted

refe

ren

tte

mpe

ratu

res.

Sin

cebo

thfl

uid

sar

ega

ses

inth

isca

se,a

nd

sin

cebo

thar

eco

nsi

dere

das

air,

anid

ealg

as(s

eean

yid

eal-

gas

ther

mod

ynam

icpr

oper

ties

data

sou

rce)

prop

erti

esw

illb

eas

sum

ed.

TD

G-3

aC

c=

(mc p

) c=

20×

1.04

120

.82

kW/K

TD

G-3

bC

h=

(mc p

) h=

20×

1.07

521

.50

kW/K

Hea

tca

paci

tyra

tes

ofth

efl

uid

stre

ams

repr

esen

tth

epr

odu

cts

ofre

spec

tive

mas

sfl

owra

tes

and

corr

espo

ndi

ng

spec

ific

hea

ts,c

alcu

late

dat

the

esti

mat

edre

fere

nt

tem

pera

ture

s.

TD

G-4

Des

ign

atio

nof

flu

idst

ream

sC

h=

C2>

Cc=

C1

——

At

this

poin

t,it

isco

nve

nie

nt

tode

term

ine

wh

ich

ofth

etw

ofl

uid

stre

ams

has

ala

rger

hea

tca

paci

ty(f

ora

non

bala

nce

dca

se).

Not

eth

atth

ede

sign

ator

1de

not

esth

e“w

eake

r”fl

uid

and

the

desi

gnat

or2,

the

“str

onge

r”fl

uid

.

TD

G-5

C∗

=C

1

C2

=20

.82

21.5

00.

9685

—

29.5






Ste

pC

alcu

lati

onV

alu

eU

nit

s

Th

eh

eat

capa

city

rate

rati

ois

not

equ

alto

1;th

eref

ore,

the

hea

tex

chan

ger

oper

ates

wit

hn

onba

lan

ced

flu

idst

ream

s.

TD

G-6

ε=

T 1,o

−T 1

,i

T 2,i

−T 1

,i=

620

−50

070

0−

500

0.60

0—

Hea

t-ex

chan

ger

effe

ctiv

enes

sre

pres

ents

the

dim

ensi

onle

sste

mpe

ratu

reof

the

wea

ker

flu

id(t

he

one

wit

hC

1=

Cc)

(Sek

uli

c,19

90,2

000)

.Not

eth

atth

ecu

rren

tde

cisi

onon

wh

ich

flu

idis

wea

ker

was

base

don

the

rou

ghes

tim

ate,

nam

ely,

afi

rst

iter

atio

nof

refe

ren

tte

mpe

ratu

res.

Th

ese

are

not

nec

essa

rily

the

best

assu

mpt

ion

s,in

part

icu

lar

for

the

hot

flu

idin

this

case

.S

o,th

eou

tlet

tem

pera

ture

ofth

eh

otfl

uid

mu

stbe

dete

rmin

edw

ith

mor

epr

ecis

ion

(see

TD

G-7

),an

dst

eps

TD

G-1

thro

ugh

TD

G-7

shou

ldbe

repe

ated

.

TD

G-7

T 2,o

=T h

,o58

3.8

K

T h,o

=T c

,i+

(1−

C∗ ε

)(T h

,i−

T c,i

)

=50

0+

(1−

0.96

85×

0.6)

(700

−50

0)

Not

eth

atth

ere

lati

onsh

ipu

sed

for

dete

rmin

ing

the

outl

ette

mpe

ratu

reof

the

hot

flu

idis

ast

raig

htf

orw

ard

con

sequ

ence

ofad

opte

dde

fin

itio

ns

ofth

eh

eat-

exch

ange

ref

fect

iven

ess

and

hea

tca

paci

tyra

tera

tio,

both

expr

esse

das

fun

ctio

ns

ofte

rmin

alte

mpe

ratu

res.

Now

,hav

ing

this

orig

inal

lyu

nkn

own

tem

pera

ture

esti

mat

ed,a

new

valu

eof

the

refe

ren

tte

mpe

ratu

refo

rth

eh

otfl

uid

can

bede

term

ined

;th

atis

,ste

psT

DG

-1th

rou

ghT

DG

-7sh

ould

bere

peat

ed.

TD

G-1

a/2

T 1,r

ef=

T c,r

ef56

0K

T c,r

ef=

T c,i

n+

T c,o

ut

2=

500

+62

02

Th

ese

con

dit

erat

ion

for

T c,r

efis

iden

tica

lto

the

firs

tbe

cau

sebo

thte

rmin

alte

mpe

ratu

res

wer

eal

read

ypr

ovid

edin

the

prob

lem

form

ula

tion

and

anar

ith

met

icav

erag

eof

the

two

was

use

d(i

.e.,

not

,say

,an

inte

gral

mea

nva

lue

for

wh

ich

the

dist

ribu

tion

thro

ugh

out

the

core

wou

ldbe

nee

ded)

.

TD

G-1

b/2

T h,r

ef=

T h,i

n+

T h,o

ut

2=

700

+58

3.8

264

1.9

K

29.6






Not

eth

atth

ese

con

dit

erat

ion

ofT h

,ref

diff

ers

sign

ifica

ntl

yfr

omth

efi

rst

iter

atio

n:6

41.9

Kin

stea

dof

700

K.

TD

G-2

a/2

c p,c

=c p

,air

(Tc,

ref)

=c p

,air

(560

)1.

041

kJ/k

gK

Th

ere

isn

och

ange

inth

isit

erat

ion

for

the

cold

-flu

idsp

ecifi

ch

eat.

TD

G-2

b/2

c p,h

=c p

,air

(Th,

ref)

=c p

,air

(641

.9)

1.06

1kJ

/kg

K

Th

ere

sult

obta

ined

repr

esen

tsa

new

valu

efo

rth

esp

ecifi

ch

eat

ofth

eh

otfl

uid

.

TD

G-3

a/2

Cc

=(m

c p) c

=20

×1.

041

20.8

2kW

/K

Th

ere

isn

och

ange

inth

isit

erat

ion

for

the

cold

flu

id.

TD

G-3

b/2

Ch

=(m

c p) h

=20

×1.

061

21.2

2kW

/K

Th

isis

an

ewva

lue

for

the

hea

tca

paci

tyra

teof

the

hot

flu

id.

TD

G-4

/2D

esig

nat

ion

offl

uid

stre

ams

Ch

=C

2>

Cc

=C

1—

—

Not

eth

atth

eim

plem

ente

dre

fin

emen

tof

the

valu

eof

the

hea

tca

paci

tyra

teon

the

hot

-flu

idsi

dedi

dn

otch

ange

the

desi

gnat

ion

sof

the

flu

ids.

TD

G-5

/2C

∗=

C1

C2

=20

.82

21.2

20.

9814

—

Th

isis

an

ew,r

efin

edva

lue

ofth

eh

eat

capa

city

rate

.Th

en

ewva

lue

issl

igh

tly

larg

erth

anth

eva

lue

inth

efi

rst

iter

atio

n.

TD

G-6

/2ε

=T 1

,o−

T 1,i

T 2,i

−T 1

,i=

620

−50

070

0−

500

0.60

0—

Th

isre

pres

ents

are

fin

edva

lue

ofth

eh

eat-

exch

ange

ref

fect

iven

ess;

ther

eis

no

chan

gein

this

valu

ebe

cau

seal

lth

eva

lues

invo

lved

wer

eal

read

ykn

own

(i.e

.,al

lwer

egi

ven

inth

epr

oble

mfo

rmu

lati

on),

and

are

not

affe

cted

byin

term

edia

teca

lcu

lati

ons.

TD

G-7

/2T h

,o=

T c,i

+(1

−C

∗ ε)(

T h,i

−T c

,i)

582.

2K

=50

0+

(1−

0.98

14×

0.6)

(700

−50

0)

29.7






Ste

pC

alcu

lati

onV

alu

eU

nit

s

We

obta

ined

an

ewva

lue

ofor

igin

ally

un

know

nou

tlet

tem

pera

ture

ofth

eh

otfl

uid

.Th

ecr

iter

ion

for

ate

rmin

atio

nof

the

iter

ativ

epr

oced

ure

may

invo

lve

eith

era

suffi

cien

tly

smal

lch

ange

oftw

osu

cces

sive

valu

esof

this

tem

pera

ture

,or

ach

ange

ofth

esu

cces

sive

valu

esfo

rh

eat-

exch

ange

ref

fect

iven

ess.

Inth

isca

se,t

hes

eco

mpa

riso

ns

indi

cate

that

eith

ern

och

ange

ora

very

smal

lch

ange

take

spl

ace,

and

the

iter

ativ

epr

oced

ure

iste

rmin

ated

atth

ispo

int.

TD

G-8

NT

U=

NT

U(ε

=0.

600;

C∗

=0.

9814

;u

nm

ixed

-un

mix

ed)

1.81

1—

For

the

cros

sflow

un

mix

ed-u

nm

ixed

arra

nge

men

tth

ere

lati

onsh

ipbe

twee

nef

fect

iven

ess

and

the

nu

mbe

rof

un

its

(NT

U)

(exp

lici

tin

term

sof

effe

ctiv

enes

sbu

tn

otex

plic

itin

term

sof

NT

U)

isas

foll

ows

(Bac

lic

and

Heg

gs,1

985)

:

ε=

1−

e−N

TU

(1+C

∗ )∞ ∑ n=

1

( n 1

) (C∗ )

n/2I n

(2N

TU

√ C∗ )

C∗ N

TU

Th

eref

ore,

the

exac

tex

pres

sion

for

the

hea

t-ex

chan

ger

effe

ctiv

enes

sof

anu

nm

ixed

-un

mix

edcr

ossfl

owar

ran

gem

ent

isal

gebr

aica

lly

very

com

plex

,reg

ardl

ess

ofth

efa

ctth

atit

may

bere

pres

ente

din

diff

eren

tfo

rms

(Bac

lic

and

Heg

gs,1

985)

.See

Sh

ahan

dS

eku

lic

(200

3)fo

ran

appr

oxim

ate

equ

atio

nan

dS

eku

lic

etal

.(19

99)

for

met

hod

sof

dete

rmin

ing

the

effe

ctiv

enes

sre

lati

onsh

ips

for

oth

er(n

otn

eces

sari

lycr

ossfl

ow)

flow

arra

nge

men

ts.G

raph

ical

repr

esen

tati

on,a

sw

ella

sta

bula

rda

tafo

rth

ecr

ossfl

owu

nm

ixed

-un

mix

edfl

owar

ran

gem

ent

can

befo

un

din

Bac

lic

and

Heg

gs(1

985)

.It

isim

port

ant

tore

aliz

eth

atre

gard

less

ofw

hic

hco

rrel

atio

nis

use

d,th

ede

term

inat

ion

ofN

TU

vers

us

effe

ctiv

enes

sis

alge

brai

call

yan

intr

acta

ble

task

wit

hou

tso

me

kin

dof

iter

ativ

epr

oced

ure

.In

the

corr

elat

ion

pres

ente

d,I n

,n

=1,

2,..

.,∞

,rep

rese

nt

mod

ified

Bes

self

un

ctio

ns

ofth

ein

tege

rn

orde

r.

TD

G-9

UA

=N

TU

·C1

=1.

811

×20

.82

37.7

1kW

/K

Th

epr

odu

ctU

A,a

lso

term

edth

e“t

her

mal

size

,”is

aco

mpo

un

ded

ther

mal

and

phys

ical

size

ofth

eu

nit

.Th

issi

zein

volv

esth

eph

ysic

alsi

ze(a

rea

ofth

eh

eat-

tran

sfer

surf

ace

A)

and

hea

t-tr

ansf

ersi

ze(U

isth

eov

eral

lhea

t-tr

ansf

erco

effi

cien

t).T

he

subs

equ

ent

proc

edu

re(i

.e.,

MG

C)

un

tan

gles

intr

icat

ere

lati

ons

betw

een

thes

een

titi

esan

dph

ysic

alsi

zeof

the

hea

tex

chan

ger,

lead

ing

toex

plic

itva

lues

ofal

lcor

edi

men

sion

sfo

rse

lect

edty

pes

ofbo

th(h

ot-

and

cold

-sid

e)h

eat-

tran

sfer

surf

aces

.

29.8






Mat

chin

gg

eom

etri

cch

arac

teri

stic

s(M

GC

)p

roce

du

re

Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-1

Det

erm

inat

ion

offl

uid

s’th

erm

oph

ysic

alpr

oper

ties

Flu

ids

1an

d2

�R

efer

ent

tem

pera

ture

T c,r

ef=

T 1,r

ef,T h

,ref

=T 2

,ref

560

642

K�

Inle

tpr

essu

res

p 1,i,

p 2,i

500

100

kPa

�O

utl

etpr

essu

res

p 1,o

,p 2

,o>

495

>96

kPa

�S

peci

fic

hea

tsc p

,c=

c p,1

,c p

,h=

c p,2

1.04

11.

061

kJ/k

gK

�V

isco

siti

es�

1,�

228

.95

31.6

710

6P

as

�T

her

mal

con

duct

ivit

ies

k 1,k 2

4.32

4.80

102

W/m

K�

Pra

ndt

lnu

mbe

rsP

r 1,P

r 20.

698

0.69

9—

�D

ensi

ties

(in

let)

�1,

i,�

2,i

3.48

40.

498

kg/m

3

�D

ensi

ties

(ou

tlet

)�

1,o,

�2,

o2.

782

0.57

4kg

/m3

�D

ensi

ties

(bu

lkm

ean

)�

1,m,�

2,m

3.09

30.

533

kg/m

3

Th

efl

uid

prop

erti

esar

eu

sual

lyde

term

ined

atar

ith

met

ic(o

rin

tegr

al)

mea

nva

lues

offl

uid

tem

pera

ture

s.In

this

calc

ula

tion

,we

wil

lado

ptth

ear

ith

met

icm

ean

valu

esfr

omth

ese

con

dit

erat

ion

.Cer

tain

data

(tem

pera

ture

s,pr

essu

res)

are

prov

ided

inth

epr

oble

mfo

rmu

lati

on(s

eeab

ove)

,an

d/or

devi

sed

from

the

inle

tda

taan

dkn

own

pres

sure

drop

s.S

peci

fic

hea

tsan

dvi

scos

itie

sar

eba

sed

onth

em

ean

tem

pera

ture

s.D

ensi

ties

are

calc

ula

ted

assu

min

gth

eid

eal-

gas

assu

mpt

ion

.Th

em

ean

den

sity

isba

sed

onth

efo

llow

ing

rela

tion

ship

:

�i,

m=

[ 1 2

( 1 �i+

1 �o)] −

1

MG

C-2

NT

U1

=N

TU

c=

2NT

U=

2×

1.81

13.

622

—

NT

U2

=N

TU

h=

C∗ N

TU

1=

0.98

14×

3.62

23.

555

—

Dis

trib

uti

onof

the

tota

ldim

ensi

onle

ssth

erm

alsi

zebe

twee

ntw

ofl

uid

side

sis

dete

rmin

edin

this

step

.Sin

cebo

thfl

uid

sar

ega

ses,

both

ther

mal

resi

stan

ces

are

tobe

assu

med

aseq

ual

inth

efi

rst

iter

atio

n.T

hat

lead

sto

the

give

ndi

stri

buti

onof

NT

U1

and

NT

U2

vers

us

NT

U(S

hah

and

Sek

uli

c,20

03).

29.9






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-3

Sel

ecti

onof

hea

tsu

rfac

ety

pes:

plan

epl

ate

fin

surf

ace

19.8

6(K

ays

and

Lon

don

,199

8)�

Pla

tesp

acin

gb

6.35

0×

10−3

m�

Nu

mbe

rof

fin

sn

f78

2m

−1�

Hyd

rau

lic

diam

eter

Dh

1.87

5×

10−3

m�

Fin

thic

knes

s�

0.15

2×

10−3

m�

Un

inte

rru

pted

flow

len

gth

l f63

.75

×10

−3m

�H

eat-

tran

sfer

area

per

volu

me

betw

een

pass

es�

1841

m−1

�F

inar

eape

rto

tala

rea

Af/

A0.

849

—�

Pla

teth

ickn

ess

a(u

nde

rde

sign

er’s

con

trol

)2

×10

−3m

Th

esi

zes

and

shap

esof

hea

t-tr

ansf

ersu

rfac

esar

eco

rrel

ated

wit

hth

eh

eat-

tran

sfer

and

hyd

rau

lic

char

acte

rist

ics.

How

ever

,th

ese

char

acte

rist

ics

intu

rnar

en

eede

dto

dete

rmin

eth

esi

zes

and

shap

esof

the

hea

t-tr

ansf

ersu

rfac

es!T

his

inte

rrel

atio

nre

nde

rsth

eca

lcu

lati

onpr

oced

ure

iter

ativ

e.A

sele

ctio

nof

the

surf

ace

geom

etry

(i.e

.,se

lect

ion

ofbo

thfl

uid

flow

area

geom

etri

es)s

hou

ldbe

don

efi

rst

apr

iori

.Su

bseq

uen

tly,

calc

ula

tion

ofh

eat-

tran

sfer

and

flu

idfl

owch

arac

teri

stic

sm

aybe

con

duct

edto

esta

blis

hw

het

her

the

surf

aces

sele

ctio

nfi

tsth

eth

erm

alsi

zedi

stri

buti

onan

dth

eov

eral

lth

erm

alsi

ze(b

ut

ina

man

ner

tosa

tisf

yth

epr

essu

redr

opco

nst

rain

ts).

We

wil

lsel

ect

the

surf

ace

desi

gnat

edas

19.8

6(K

ays

and

Lon

don

,199

8)fo

rbo

thfl

uid

side

s.S

eeal

soW

ebb

(199

4)fo

rfu

rth

erdi

scu

ssio

nof

issu

esin

volv

ing

enh

ance

dh

eat-

tran

sfer

surf

aces

.

MG

C-4

Su

rfac

ech

arac

teri

stic

sin

volv

eth

era

tio

ofC

olbu

rnan

dFa

nn

ing

fact

ors

j/f

for

plan

epl

ate

fin

surf

ace

19.8

6(K

ays

and

Lon

don

,199

8).F

orth

issu

rfac

e,th

era

tio

ofj

and

fov

erth

ew

ide

ran

geof

Rey

nol

dsn

um

bers

isap

prox

imat

ely

con

stan

t:

0.29

—

( j f

) c,h

≈co

nst

=0.

29

29.10






Alt

hou

ghse

lect

ion

ofsu

rfac

ety

pes

lead

sto

the

know

nh

eat-

tran

sfer

geom

etri

eson

both

flu

idsi

des,

the

calc

ula

tion

ofj(

Re)

and

f(R

e)pa

ram

eter

s(i

.e.,

hea

t-tr

ansf

eran

dfr

icti

onfa

ctor

sin

dim

ensi

onle

ssfo

rm)

can

not

bepe

rfor

med

stra

igh

tfor

war

dly

atth

ispo

int.

Th

isis

beca

use

the

Rey

nol

dsn

um

bers

for

flu

idfl

ows

are

stil

lun

know

n.T

his

hu

rdle

can

bere

solv

edby

firs

tgu

essi

ng

atth

em

agn

itu

deof

the

rati

oof

jto

f(i

.e.,

j/f)

.Sin

ceth

isra

tio

isn

earl

yco

nst

ant

ina

wid

era

nge

ofR

eyn

olds

nu

mbe

rs,a

un

iqu

eva

lue

can

besu

gges

ted

asin

dica

ted

abov

e.In

the

firs

tit

erat

ion

(wh

ich

wil

lfol

low

)on

lyth

eva

lue

ofj/

f(r

ath

erth

anse

para

tej

and

fva

lues

)w

ould

ben

eede

dfo

rca

lcu

lati

onof

both

flu

idco

rem

ass

velo

citi

es.S

ubs

equ

entl

y,th

ese

core

mas

sve

loci

ties

(see

step

MG

C-6

)w

illb

eu

sed

tode

term

ine

the

firs

tit

erat

ion

ofR

eyn

olds

nu

mbe

rs,l

eadi

ng

toth

eco

rres

pon

din

gva

lues

ofj

and

f.S

ubs

equ

entl

y,th

ese

con

dit

erat

ion

for

j/f

can

beca

lcu

late

dfr

omkn

own

jan

df

valu

es.I

nth

isca

se,

j/f

∈(0

.25,

0.37

)fo

rR

e∈(

500,

4000

),so

aned

uca

ted

gues

sw

ould

lead

toj/

f≈

0.29

.

MG

C-5

An

init

ialg

ues

sof

the

firs

tes

tim

ate

for

the

tota

lsu

rfac

ete

mpe

ratu

reef

fect

iven

ess

for

both

side

ssh

ould

bem

ade

atth

ispo

int

(i.e

.,�

0,c

=�

0,h).

Itis

gen

eral

lyas

sum

edth

atth

eto

tal

surf

ace

tem

pera

ture

effe

ctiv

enes

sfo

ra

com

pact

hea

t-tr

ansf

ersu

rfac

e(f

ora

good

desi

gn)

mu

stbe

wit

hin

the

ran

geof

0.7

to0.

9.L

etu

sas

sum

eth

eva

lue

tobe

atth

eh

igh

end

ofth

isra

nge

for

both

flu

idsi

des

(not

eth

atth

esa

me

geom

etry

was

sugg

este

dfo

rbo

thsu

rfac

es).

0.9

—

MG

C-6

Gc

≈[(

j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 c=

[ 0.29

5×

103

500

×10

3

2×

500

×10

3×

0.9

×3.

093

3.62

2×

0.69

82/3

] 1/253

.22

kg/m

2s

Gh

≈[(

j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 h=

[ 0.29

4×

103

100

×10

3

2×

100

×10

3×

0.9

×0.

533

3.55

5×

0.69

92/3

] 1/219

.94

kg/m

2s

Inth

isst

ep,t

he

firs

tes

tim

ates

ofco

rem

ass

velo

citi

esar

eba

sed

onth

ees

tim

ates

ofj/

fan

d�

para

met

ers

asdi

scu

ssed

inM

GC

-4an

dM

GC

-5.T

his

esti

mat

e,as

give

nab

ove,

isba

sed

ona

sim

plifi

edex

pres

sion

for

Gth

atta

kes

into

acco

un

tth

eas

sum

ptio

ns

asfo

llow

s:(1

)on

lyfr

icti

onco

ntr

ibu

tes

toth

epr

essu

redr

op,(

2)fo

uli

ng

resi

stan

ces

are

neg

lect

ed,(

3)th

erm

alre

sist

ance

ofth

eh

eat-

tran

sfer

wal

lis

neg

lect

ed,a

nd

(4)

ther

mal

resi

stan

ces

cau

sed

byfo

rmat

ion

sof

con

vect

ive

bou

nda

ryla

yers

onbo

thfl

uid

side

sar

eeq

ual

.

29.11






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-7

Re c

=G

cD

h,c

�c

=53

.22

×1.

875

×10

−3

28.9

5×

10−6

3446

—

Re h

=G

hD

h,h

�h

=19

.94

×1.

875

×10

−3

31.6

7×

10−6

1181

—

Not

eth

atu

nce

rtai

nti

esin

volv

edw

ith

anex

peri

men

tald

eter

min

atio

nof

the

Rey

nol

dsva

lues

,an

dsu

bseq

uen

tly

jan

df,

are

±2,

±14

,an

d±

3pe

rcen

t,re

spec

tive

ly.S

o,th

efi

rst

esti

mat

esfo

rR

eyn

olds

nu

mbe

rsm

ust

bere

fin

edla

ter

(in

subs

equ

ent

iter

atio

ns)

up

toth

em

argi

nof

±2pe

rcen

t.O

ne

iter

atio

nw

ould

like

lysu

ffice

.

MG

C-8

j c=

exp(

a 0+

r c(a

1+

r c(a

2+

r c(a

3+

r c(a

4+

a 5r c

))))

)0.

0037

18—

j h=

exp(

a 0+

r h(a

1+

r h(a

2+

r h(a

3+

r h(a

4+

a 5r h

))))

)0.

0050

6—

f c=

exp(

b 0+

r c(b

1+

r c(b

2+

r c(b

3+

r c(b

4+

b 5r c

))))

)0.

0100

6—

f h=

exp(

b 0+

r h(b

1+

r h(b

2+

r h(b

3+

r h(b

4+

b 5r h

))))

)0.

0170

7—

r c=

ln(R

e c)=

ln(3

446)

8.14

5—

r h=

ln(R

e h)=

ln(1

181)

7.07

4—

Th

isex

plic

itca

lcu

lati

onof

the

refi

ned

valu

esfo

rj

and

fis

con

duct

edby

usi

ng

j(R

e)an

df(

Re)

corr

elat

ion

sfo

rth

ese

lect

edge

omet

ryas

list

edin

MG

C-4

(not

eth

atbo

thfl

uid

side

sh

ave

the

sam

ege

omet

ry,b

ased

ona

deci

sion

elab

orat

edin

MG

C-3

).T

he

valu

esar

eca

lcu

late

dfo

rR

eyn

olds

nu

mbe

rsas

dete

rmin

edin

step

MG

C-7

.Th

efo

rmof

the

corr

elat

ion

isdi

ctat

edby

the

curv

efi

ttin

gof

the

expe

rim

enta

lly

obta

ined

data

list

edin

Kay

san

dL

ondo

n(1

998)

.

29.12






MG

C-9

( j f

) c=

0.00

3718

0.01

006

0.36

96—

( j f

) h=

0.00

506

0.01

707

0.29

65—

Refi

ned

valu

esof

j/f

rati

osar

en

owba

sed

onn

ewly

calc

ula

ted

valu

esof

jan

df

fact

ors

for

the

esti

mat

edR

eyn

olds

nu

mbe

rs.

MG

C-6

/2G

c≈

[(j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 c=

[ 0.36

965

×10

3

500

×10

3

2×

500

×10

3×

0.9

×3.

093

3.62

2×

0.69

82/3

] 1/260

.08

kg/m

2s

Gh

≈[(

j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 h=

[ 0.29

654

×10

3

100

×10

3

2×

100

×10

3×

0.9

×0.

533

3.55

5×

0.69

92/3

] 1/220

.16

kg/m

2s

Th

ese

are

the

refi

ned

valu

esfo

rco

rem

ass

velo

citi

es,b

ut

stil

lbas

edon

anap

prox

imat

eG

–(j/

f)re

lati

onsh

ip.

MG

C-7

/2R

e c=

GcD

h,c

�c

=60

.08

×1.

875

×10

−3

28.9

5×

10−6

3890

—

Re h

=G

hD

h,h

�h

=20

.16

×1.

875

×10

−3

31.6

7×

10−6

1194

—

Refi

ned

valu

esof

Rey

nol

dsn

um

bers

can

now

beca

lcu

late

dag

ain

.Not

eth

atR

e ch

asin

crea

sed

inth

isit

erat

ion

;th

atis

,it

diff

ers

from

the

prev

iou

sit

erat

ion

by13

perc

ent—

this

isob

viou

sly

far

mor

eth

ana

stan

dard

un

cert

ain

tym

argi

nof

2pe

rcen

tal

low

edfo

rde

term

inin

gth

eR

eyn

olds

nu

mbe

r.A

lth

ough

Re h

diff

ers

from

the

prev

iou

sit

erat

ion

for

less

than

2pe

rcen

t,w

ew

illc

onti

nu

eit

erat

ion

su

nti

lbot

hR

eyn

olds

nu

mbe

rch

ange

sre

duce

tobe

low

that

mar

gin

.

29.13






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-8

/2j c

=ex

p(a 0

+r c

(a1

+r c

(a2

+r c

(a3

+r c

(a4

+a 5

r c))

)))

0.00

3649

—

j h=

exp(

a 0+

r h(a

1+

r h(a

2+

r h(a

3+

r h(a

4+

a 5r h

))))

)0.

0050

29—

f c=

exp(

b 0+

r c(b

1+

r c(b

2+

r c(b

3+

r c(b

4+

b 5r c

))))

)0.

0097

79—

f h=

exp(

b 0+

r h(b

1+

r h(b

2+

r h(b

3+

r h(b

4+

b 5r h

))))

)0.

0169

2—

r c=

ln(R

e c)=

ln(3

890)

8.26

6—

r h=

ln(R

e h)=

ln(1

194)

7.08

5—

Inth

isst

ep,t

he

nex

tes

tim

atio

nof

the

jan

df

fact

ors

isex

ecu

ted.

Not

eth

atj

fact

ors

diff

erfr

omth

eon

esde

term

ined

inth

epr

evio

us

iter

atio

nfo

rle

ssth

an2

perc

ent,

resp

ecti

vely

,for

both

flu

ids

the

ffa

ctor

sdi

ffer

abi

tm

ore.

As

indi

cate

din

the

MG

C-7

step

,th

eac

tual

valu

esof

jfa

ctor

s(d

eter

min

edex

peri

men

tall

y)u

sual

lyh

ave

am

argi

nof

erro

rfo

ran

orde

rof

mag

nit

ude

larg

erth

anca

lcu

late

dh

ere

intw

osu

cces

sive

calc

ula

tion

s.T

he

ffa

ctor

su

sual

lyh

ave

the

expe

rim

enta

lmar

gin

ofer

ror

atth

eca

lcu

late

dle

vel.

So,

anad

diti

onal

iter

atio

nw

ould

prob

ably

besu

ffici

ent.

MG

C-9

/2( j f

) c=

0.00

3649

0.00

9779

0.37

31—

( j f

) h=

0.00

5029

0.01

692

0.29

72—

Th

isis

the

nex

tit

erat

ion

for

j/f

rati

os.

MG

C-6

/3G

c≈

[(j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 c=

[ 0.37

315

×10

3

500

×10

3

2×

500

×10

3×

0.9

×3.

093

3.62

2×

0.69

82/3

] 1/260

.36

kg/m

2s

Gh

≈[(

j f

)(�

pp i

n

) 2p i

n�

0�

m

NT

UP

r2/3

] 1/2 h=

[ 0.29

724

×10

3

100

×10

3

2×

100

×10

3×

0.9

×0.

533

3.55

5×

0.69

92/3

] 1/220

.19

kg/m

2s

Th

ese

are

refi

ned

valu

esfo

rth

eco

rem

ass

velo

citi

es.

29.14






MG

C-7

/3R

e c=

GcD

h,c

�c

=60

.36

×1.

875

×10

−3

28.9

5×

10−6

3909

—

Re h

=G

hD

h,h

�h

=20

.19

×1.

875

×10

−3

31.6

7×

10−6

1195

—

Th

ese

are

the

new

valu

esof

Rey

nol

dsn

um

bers

.Th

eydi

ffer

wel

lbel

owth

em

argi

nof

2pe

rcen

tfr

omth

epr

evio

usl

yca

lcu

late

dva

lues

(ado

pted

her

eas

acr

iter

ion

for

term

inat

ion

ofth

eit

erat

ive

proc

edu

re).

Th

eref

ore,

this

shou

ldbe

con

side

red

asth

ela

stit

erat

ion

.

MG

C-8

/3j c

=ex

p(a 0

+r c

(a1

+r c

(a2

+r c

(a3

+r c

(a4

+a 5

r c))

)))

0.00

3646

—

j h=

exp(

a 0+

r h(a

1+

r h(a

2+

r h(a

3+

r h(a

4+

a 5r h

))))

)0.

0050

26—

f c=

exp(

b 0+

r c(b

1+

r c(b

2+

r c(b

3+

r c(b

4+

b 5r c

))))

)0.

0097

69—

f h=

exp(

b 0+

r h(b

1+

r h(b

2+

r h(b

3+

r h(b

4+

b 5r h

))))

)0.

0169

1—

r c=

ln(R

e c)=

ln(3

909)

8.27

1—

r h=

ln(R

e h)=

ln(1

195)

7.08

6—

Th

isis

the

last

esti

mat

ion

ofj

and

ffa

ctor

s.C

alcu

late

dva

lues

are

prac

tica

lly

iden

tica

lto

the

ones

calc

ula

ted

inth

epr

evio

us

iter

atio

n.

MG

C-9

/3( j f

) c=

0.00

3646

0.00

9769

0.37

32—

( j f

) h=

0.00

5026

0.01

691

0.29

72—

Th

ese

are

the

last

esti

mat

ion

sfo

rj/

fra

tios

.

MG

C-1

0

T w=

T h,re

f+

NT

UcC

∗

NT

Uh

T c,r

ef

1+

NT

UcC

∗

NT

Uh

=T c

,ref

+T h

,ref

2=

560

+64

22

601

K

29.15






Ste

pC

alcu

lati

onV

alu

eU

nit

s

Th

ete

mpe

ratu

reof

the

hea

t-tr

ansf

ersu

rfac

ew

allb

etw

een

the

flu

ids

isca

lcu

late

dfr

oma

bala

nce

equ

atio

nth

atre

late

sh

eat-

tran

sfer

rate

sde

live

red

from

one

flu

idto

thos

ere

ceiv

edby

the

oth

er.T

hes

eh

eat-

tran

sfer

rate

sar

eex

pres

sed

inte

rms

offl

uid

-to-

wal

lan

dw

all-

to-fl

uid

tem

pera

ture

diff

eren

ces

and

the

resp

ecti

veth

erm

alre

sist

ance

son

both

flu

idsi

des.

Th

ew

allt

empe

ratu

reis

nee

ded

tope

rfor

ma

corr

ecti

onof

ther

mop

hys

ical

prop

erti

es.

Th

isco

rrec

tion

isdu

eto

tem

pera

ture

grad

ien

tsbe

twee

nth

efl

uid

san

dth

eh

eat-

tran

sfer

surf

ace

wal

lacr

oss

the

resp

ecti

vebo

un

dary

laye

rson

both

flu

idsi

des.

MG

C-1

1j c

,co

rr=

j c

( T w T c,r

ef

) n =0.

0036

46( 60

156

0

) −0.1

185

0.00

3616

—

n=

0.3

−[ lo

g 10

( T w T c,r

ef

)] 1/4

=0.

3−

[ log 1

060

156

0

] 1/4−0

.118

5—

Not

eth

atco

ldai

r(i

.e.,

gas)

isex

pose

dto

hea

tin

g,an

dth

atit

sfl

owre

gim

eis

turb

ule

nt.

For

deta

ils

ofth

eal

tern

ate

expo

nen

tde

term

inat

ion

,see

Sh

ahan

dS

eku

lic

(200

3,ta

ble

7.13

,p.

531)

.Th

eco

ndi

tion

sto

besa

tisfi

edar

e1

<T w

,ref

/T c

,ref

<5;

0.6

<P

r<

0.9.

MG

C-1

2j h

,cor

r=

j h

( T w T h,r

ef

) n =j h

=0.

0050

260.

0050

26—

Th

eh

otfl

uid

expe

rien

ces

cool

ing

con

diti

ons.

Th

efl

owre

gim

eis

inth

ela

min

arre

gion

.T

her

efor

e,th

eex

pon

ent

n=

0in

Sh

ahan

dS

eku

lic

(200

3,ta

ble

7.12

,p.5

31).

MG

C-1

3f c

,cor

r=

f c

( T w T c,r

ef

) m=

0.00

9769

( 601

560

) −0.1

0.00

970

—

m=

−0.1

29.16






Col

dfl

uid

ish

eate

d,an

dth

efl

owre

gim

eis

turb

ule

nt.

Th

esu

gges

ted

calc

ula

tion

ofth

eex

pon

ent

inth

eco

rrec

tion

term

isva

lid

for

the

ran

geof

tem

pera

ture

rati

osas

foll

ows:

1<

T w,r

ef

T c,r

ef<

5

Inou

rca

se,t

his

rati

ois

1.07

;th

eref

ore

the

calc

ula

tion

ofth

eex

pon

ent

mis

perf

orm

edas

indi

cate

d.F

luid

atth

eco

ldsi

deis

air;

ther

efor

e,it

istr

eate

das

anid

ealg

as.

MG

C-1

4f h

,cor

r=

f h

( T w T h,r

ef

) m=

0.01

691

( 601

642

) 0.81

0.01

603

—

m=

0.81

For

afl

uid

cool

ing

case

and

lam

inar

flow

,th

eex

pon

ent

iseq

ual

to0.

81.T

he

con

diti

ons

tobe

sati

sfied

are:

0.5

<T w

,ref

/T h

,ref

=0.

94<

1;0.

6<

Pr

=0.

699

<0.

9.

MG

C-1

5h c

=j c

,cor

rG

ccp,

c

Pr2/

3c

=0.

0036

1660

.36

×10

410.

6982

/3

288.

7W

/m2

K

Th

eh

eat-

tran

sfer

coef

fici

ent

for

the

cold

flu

idis

dete

rmin

edfr

omth

ede

fin

itio

nof

the

Col

burn

fact

or.

MG

C-1

6h h

=j h

,cor

rG

hc p

,h

Pr2/

3h

=0.

0050

2620

.19

×10

610.

6992

/3

136.

7W

/m2

K

Th

eh

eat-

tran

sfer

coef

fici

ent

for

the

hot

flu

idis

dete

rmin

edfr

omde

fin

itio

nof

the

Col

burn

fact

or.

MG

C-1

7�

f,c

=ta

nh

(ml)

c

(ml)

c=

tan

h(1

37.8

×0.

0030

2)13

7.8

×0.

0030

20.

9460

—

mc

=√ 2h

c/k�

=√ 2

×28

8.7/

(200

×0.

152

×10

−3)

137.

8m

−1

l c=

b 2−

�=

6.35

×10

−3

2−

0.15

2×

10−3

0.00

302

m

29.17






Ste

pC

alcu

lati

onV

alu

eU

nit

s

Not

eth

at�

and

l cdi

ffer

from

each

oth

erm

ore

than

for

anor

der

ofm

agn

itu

de,s

oth

eex

pose

dst

rip

edge

ofth

efi

nis

not

take

nin

toac

cou

nt

and

the

mpa

ram

eter

isca

lcu

late

din

the

firs

tap

prox

imat

ion

asin

dica

ted

[see

Sh

ahan

dS

eku

lic

(200

3,pp

.280

and

627)

for

anal

tern

ativ

e].T

her

mal

con

duct

ivit

yof

the

fin

isas

sum

edto

be20

0W

/mK

for

anal

loy

atth

egi

ven

tem

pera

ture

.Not

eth

atth

ere

sult

ing

fin

effi

cien

cybe

com

esve

ryh

igh

.Act

ual

fin

effi

cien

cyin

abr

azed

hea

tex

chan

ger

thro

ugh

out

the

core

may

besi

gnifi

can

tly

smal

ler

(Zh

aoet

al.,

2003

).

MG

C-1

8�

f,h

=ta

nh

(ml)

h

(ml)

h=

tan

h(9

4.8

×0.

0030

2)94

.8×

0.00

302

0.97

35—

mh

=√ 2h

h/k

�=

√ 2×

136.

7/(2

00×

0.15

2×

10−3

)94

.8m

−1

l h=

b 2−

�=

6.35

×10

−3

2−

0.15

2×

10−3

0.00

302

m

Not

eth

atth

efi

nge

omet

ryis

the

sam

eon

both

flu

idsi

des;

ther

efor

eth

ele

ngt

hs

are

the

sam

e.

MG

C-1

9�

0,c

=1

−(1

−�

f,c)

Af A

=1

−(1

−0.

9460

)×0.

849

0.95

42—

Ade

tail

eddi

scu

ssio

nof

the

mea

nin

gof

the

tota

lext

ende

dsu

rfac

eef

fici

ency

can

befo

un

din

Sh

ahan

dS

eku

lic

(200

3,p.

289)

.

MG

C-2

0�

0,h

=1

−(1

−�

f,h)A

f A=

1−

(1−

0.97

35)×

0.84

90.

9775

—

Not

eth

atth

eex

ten

ded

surf

ace

effi

cien

cies

mu

stdi

ffer

for

both

flu

idsi

des

rega

rdle

ssof

the

sam

ege

omet

ryof

the

fin

su

sed.

Th

isis

due

toth

edi

ffer

ence

inth

eh

eat-

tran

sfer

coef

fici

ents

.

MG

C-2

1U

=[

1(�

0h)

c+

Ac/

Ah

(�0h)

h

] −1=

[1

(�0h)

c+

1(�

0h)

h

] −1=

[1

0.95

42×

288.

7+

10.

9775

×13

6.7

] −189

.93

W/m

2K

29.18






Bec

ause

ofa

hig

hth

erm

alco

ndu

ctiv

ity

ofw

allm

ater

ial,

ther

mal

resi

stan

ceof

the

wal

lis

neg

lect

edin

this

calc

ula

tion

.Als

o,it

isas

sum

edth

atn

osi

gnifi

can

tfo

uli

ng

ispr

esen

ton

eith

ersi

deof

the

hea

t-tr

ansf

ersu

rfac

e.T

her

efor

e,th

eov

eral

lhea

t-tr

ansf

erco

effi

cien

tis

defi

ned

byh

eat-

tran

sfer

con

duct

ance

due

toco

nve

ctio

non

both

flu

idsi

des

only

.Not

e,ag

ain

,th

atth

eh

eat-

tran

sfer

surf

ace

area

sar

eth

esa

me

onbo

thfl

uid

side

sbe

cau

seth

esa

me

fin

geom

etry

isu

sed.

MG

C-2

2A

c=

Ah

=( m

c p) c

NT

U

U=

20.8

2×

103

×1.

811

89.9

341

9.3

m2

Th

eh

eat-

tran

sfer

surf

ace

area

sar

eth

esa

me

onbo

thfl

uid

side

s.

MG

C-2

3A

0,c

=m

c

Gc

=20

60.3

60.

3313

m2

Th

efr

ee-fl

owar

eaon

the

cold

flu

idsi

deis

dete

rmin

edfr

omth

ede

fin

itio

nof

the

mas

sve

loci

ty,

G=

m/A

0.

MG

C-2

4A

0,h

=m

h

Gh

=20

20.1

90.

9908

m2

Th

efr

ee-fl

owar

eaon

the

hot

flu

idsi

deis

dete

rmin

edan

alog

ousl

yto

the

sam

een

tity

onth

eco

ldfl

uid

side

;see

step

MG

C-2

3.

MG

C-2

5�

c=

�h

=b�

Dh

8(b

+a)

=6.

35×

10−3

×18

41×

1.87

5×

10−3

8(6.

35+

2)×

10−3

0.32

81—

Th

em

inim

um

free

-flow

area

:fro

nta

lare

ara

tio

isth

esa

me

onbo

thfl

uid

side

s(b

ecau

seof

the

sam

eh

eat-

tran

sfer

surf

ace

geom

etry

).N

ote

that

the

equ

atio

nu

sed

repr

esen

tsa

redu

ced

form

ofa

gen

eral

expr

essi

onfo

rth

atra

tio,

assu

min

gth

atN

ppa

ssag

esex

ist

onth

eh

otsi

dean

dN

p+1

pass

ages

exis

ton

the

cold

side

.

MG

C-2

6A

fr,c

=A

0,c

�c

=0.

3313

0.32

811.

0097

m2

Th

efr

ee-fl

owar

eaon

the

cold

flu

idsi

deis

dete

rmin

edfr

omth

ede

fin

itio

nof

the

free

-flow

area

:fr

onta

lare

ara

tio

for

the

cold

flu

idsi

de.

29.19






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-2

7A

fr,h

=A

0,h

�h

=0.

9908

0.32

813.

0198

m2

Th

efr

ee-fl

owar

eaon

the

hot

flu

idsi

deis

dete

rmin

edfr

omth

ede

fin

itio

nof

the

free

-flow

area

:fr

onta

lare

ara

tio

for

the

hot

flu

idsi

de.

MG

C-2

8L

c=

Dh

Ac

4A

0,c

=1.

875

×10

−3×

419.

34

×0.

3313

0.59

3m

Th

efl

uid

flow

len

gth

onth

eco

ldfl

uid

side

repr

esen

tsth

epr

inci

palc

ore

dim

ensi

onin

this

dire

ctio

n.

MG

C-2

9L

h=

Dh

Ah

4A

0,h

=1.

875

×10

−3×

419.

34

×0.

9908

0.19

8m

Th

efl

uid

flow

len

gth

onth

eh

otfl

uid

side

repr

esen

tsth

epr

inci

palc

ore

dim

ensi

onin

this

dire

ctio

n.

MG

C-3

0L

stac

k=

Afr

,c

Lh

=1.

0097

0.19

85.

099

m

Lst

ack

=A

fr,h

Lc

=3.

0198

0.59

35.

092

m

Th

eco

redi

men

sion

inth

eth

ird

dire

ctio

nca

nbe

calc

ula

ted

byu

sin

gth

efr

onta

lare

aof

eith

erth

eco

ldfl

uid

orth

eh

otfl

uid

.If

the

calc

ula

tion

wer

eco

ndu

cted

corr

ectl

y,bo

thva

lues

wou

ldh

ave

tobe

wit

hin

the

mar

gin

ofer

ror

only

asa

resu

ltof

rou

ndi

ng

ofth

en

um

bers

.Not

eth

atn

oco

nst

rain

tre

gard

ing

the

aspe

ctra

tio

ofan

ypa

irof

the

core

side

dim

ensi

ons

isim

pose

d.S

uch

an

onco

nst

rain

edca

lcu

lati

onm

ayle

ad,a

sis

the

case

her

e,to

ah

eat-

exch

ange

rco

rew

ith

rela

tive

lyla

rge

aspe

ctra

tios

.An

opti

miz

atio

npr

oced

ure

wou

ldbe

nee

ded

toex

ecu

teth

isca

lcu

lati

onw

ith

this

con

stra

int

impo

sed.

Inth

atca

se,a

nad

diti

onal

iter

ativ

epr

oced

ure

wou

ldbe

nee

ded.

Su

cha

proc

edu

rew

ould

requ

ire

are

con

side

rati

onof

the

hea

t-tr

ansf

ersu

rfac

ege

omet

ryon

both

flu

idsi

des

(in

this

case

,for

sim

plic

ity,

itis

assu

med

that

thes

ege

omet

ries

are

the

sam

e).

29.20






MG

C-3

1( �

p p i

) c=

G2 c

2(p i

n�

in) c

[ (1−

�2

+K

c)+

fL�

i

r h�

m+

2( �

i

�o

−1) −

(1−

�2

−K

e)�

i

�o

] c

Kc,

c=

f 1(�

c,R

e c,su

rfac

ege

omet

ry)

Ke,

c=

f 2(�

c,R

e c,su

rfac

ege

omet

ry)

Kc,

c=

f 1(�

c=

0.32

81,R

e c=

3909

,su

rfac

ege

omet

ry=

19.8

6)0.

51—

Ke,

c=

f 2(�

c=

0.32

81,R

e c=

3909

,su

rfac

ege

omet

ry=

19.8

6)0.

45—

Th

ere

lati

vepr

essu

redr

opca

lcu

lati

ons

requ

ire

dete

rmin

atio

nof

both

entr

ance

and

exit

pres

sure

loss

coef

fici

ents

Kc

and

Ke.

Th

ese

coef

fici

ents

can

bede

term

ined

from

Kay

san

dL

ondo

n(1

998)

data

for

agi

ven

set

ofin

form

atio

nco

nce

rnin

g(1

)th

era

tio

ofth

em

inim

um

free

-flow

area

toth

efr

onta

lare

a�

c(h)

for

both

flu

idsi

des

(in

this

case

,th

ese

valu

esar

eid

enti

cal;

see

calc

ula

tion

step

MG

C-2

5),(

2)R

eyn

olds

nu

mbe

rs,a

nd

(3)

surf

ace

geom

etry

[i.e

.,(L

/D

h) c

(h)]

.Fan

nin

gfr

icti

onfa

ctor

ssh

ould

,as

aru

le,b

ede

term

ined

byac

cou

nti

ng

for

corr

ecti

ons

for

the

refe

ren

tw

alla

nd

flu

idte

mpe

ratu

res.

Not

eth

atth

ere

fere

nt

wal

lte

mpe

ratu

rem

aybe

calc

ula

ted

byta

kin

gin

toac

cou

nt

ther

mal

resi

stan

ces

onbo

thfl

uid

side

s.In

step

MG

C-1

0,th

ew

allt

empe

ratu

reis

dete

rmin

edin

the

firs

tap

prox

imat

ion

wit

hou

tac

cou

nti

ng

for

this

fact

or(t

her

mal

resi

stan

ces

onbo

thsi

des

wer

eas

sum

edas

equ

al).

MG

C-3

2( �

p p i

) h=

G2 h

2(p i

n�

in) h

[ (1−

�2

+K

c)+

fL�

i

r h�

m+

2( �

i

�o

−1) −

(1−

�2

−K

e)�

i

�o

] h

Kc,

h=

f 1(�

h,R

e h,su

rfac

ege

omet

ry)

Ke,

h=

f 2(�

h,R

e h,su

rfac

ege

omet

ry)

Kc,

h=

f(�

h=

0.32

81,R

e h=

1195

,su

rfac

ege

omet

ry=

19.8

6)1.

22—

Ke,

h=

f(�

h=

0.32

81,R

e h=

1195

,su

rfac

ege

omet

ry=

19.8

6)0.

20—

29.21






Ste

pC

alcu

lati

onV

alu

eU

nit

s

Cor

rect

ion

ofth

em

agn

itu

deof

aFa

nn

ing

fric

tion

fact

oris

con

duct

edas

sugg

este

din

step

sM

GC

-13

and

MG

C-1

4.A

refi

nem

ent

ofth

eFa

nn

ing

fric

tion

fact

orm

ayn

otbe

just

ified

inca

ses

wh

enth

eac

tual

chan

geof

its

valu

eis

belo

wth

em

argi

nof

un

cert

ain

tyty

pica

lfor

anex

peri

men

tale

stim

atio

nof

its

valu

e.W

ew

illa

ssu

me

that

the

mar

gin

isat

the

3pe

rcen

tle

vel.

Not

eth

atth

ege

omet

ryof

the

surf

ace

isa

plan

epl

ate

fin

wit

ha

desi

gnat

ion

of19

.86

(Kay

san

dL

ondo

n,1

998)

.An

addi

tion

alco

mm

ent

rega

rdin

gth

eco

rrec

tion

ofth

eFa

nn

ing

fric

tion

fact

ors

isap

prop

riat

eat

this

poin

t.T

his

corr

ecti

onw

aspe

rfor

med

inst

eps

MG

C-1

3an

dM

GC

-14,

assu

min

gth

atth

erm

alre

sist

ance

son

both

side

sof

the

wal

lare

con

side

red

asbe

ing

the

sam

e.T

his

may

not

beth

eca

se,a

nd

insu

cha

situ

atio

nth

eco

rrec

tion

wou

ldre

quir

ede

term

inat

ion

ofth

ew

allt

empe

ratu

re(s

eest

epM

GC

-10)

,ass

um

ing

that

resi

stan

ces

are

not

the

sam

e,sp

ecifi

call

y

T w=

T h,r

ef+

Rh Rc

T c,r

ef

1+

Rh Rc

wh

ere

Rh

=1/

(�0h

A) h

and

Rc

=1/

(�0h

A) c

.If

this

corr

ecti

onis

not

perf

orm

ed,t

he

Fan

nin

gfr

icti

onfa

ctor

sw

illb

eth

esa

me

asde

term

ined

inst

eps

MG

C-1

3an

dM

GC

-14.

MG

C-3

3( �

p p i

) c=

60.3

62

2(50

0×

103

×3.

484)

×

(1−

0.32

812

+0.

51)+

0.00

970

0.59

3×

3.48

41.

875

×10

−3

43.

093

+2

( 3.48

42.

782

−1) −

(1−

0.32

812

−0.

45)3.

484

2.78

2

0.01

59—

Th

ere

lati

vepr

essu

redr

ops

can

now

beca

lcu

late

dby

usi

ng

allt

he

prev

iou

sly

dete

rmin

edva

riab

les

and

para

met

ers.

Not

eth

atth

eh

ydra

uli

cra

diu

sis

repr

esen

ted

ason

e-qu

arte

rof

the

hyd

rau

lic

diam

eter

.Th

ean

alyt

ical

expr

essi

onfo

rth

isca

lcu

lati

onis

give

nin

the

step

MG

C-3

1(a

bove

).

29.22






MG

C-3

4�

p c=

p i,c

( �p p i

) c=

500

×10

3×

0.01

597.

9kP

a

Fro

mth

ein

put

data

,an

allo

wed

pres

sure

drop

is5

kPa

<7.

9kP

a.T

his

indi

cate

sth

atth

eim

pose

dco

ndi

tion

isn

otsa

tisfi

ed!T

his

prom

pts

an

eed

tore

iter

ate

the

calc

ula

tion

wit

ha

new

valu

eof

the

mas

sve

loci

ty(i

nth

efi

rst

iter

atio

n,t

he

mas

sve

loci

tyw

asca

lcu

late

din

step

MG

C-6

byu

sin

gth

efi

rst

appr

oxim

atio

nba

sed

ona

wea

kde

pen

den

ceof

j/f

onth

eR

eyn

olds

nu

mbe

r).

MG

C-3

5( �

p p i

) h=

20.1

92

2(10

0×

103

×0.

498)

(1−

0.32

812

+1.

22)+

0.01

603

0.19

8×

0.49

8

1.87

5×

10−3

40.

5

0.03

1—

+2

( 0.49

80.

574

−1) −

(1−

0.32

812

−0.

2)0.

498

0.57

4

T

he

hot

flu

idsi

depr

essu

redr

opdi

vide

dby

the

inle

tfl

uid

pres

sure

atth

eh

otfl

uid

side

can

now

beca

lcu

late

din

the

sam

em

ann

eras

for

the

cold

flu

id.O

fco

urs

e,on

em

ayde

cide

toca

lcu

late

the

pres

sure

drop

righ

taw

ay(s

tep

MG

C-3

6).T

he

anal

ytic

alex

pres

sion

for

this

calc

ula

tion

isgi

ven

inth

est

epM

GC

-32.

MG

C-3

6�

p h=

p i,h

( �p p i

) h=

100

×10

3×

0.03

13.

1kP

a

Th

eim

pose

dm

axim

um

allo

wab

lepr

essu

redr

opis

4.2

kPa

>3.

1kP

a.T

her

efor

e,th

isre

quir

emen

tis

sati

sfied

.How

ever

,sin

ceth

epr

essu

redr

opfo

rth

eco

ldfl

uid

was

not

sati

sfied

,th

en

ewit

erat

ion

isn

eede

d,as

emph

asiz

edin

step

MG

C-3

4.

MG

C-3

7

Gc

=

[ 2(p

in�

in)( �

p p i

)] 1/2

c[ (1

−�

2+

Kc)

+f

L�i

r h�

m+

2( �

i

�o

−1) −

(1−

�2

−K

e)�

i

�o

] 1/2 c

47.9

1kg

/m2

s

=

[ 2×

500

×10

3×

3.48

4×

5×

103

500

×10

3

] 1/2 (1

−0.

3281

2+

0.51

)+0.

0097

00.

593

×3.

484

1.87

5×

10−3

43.

093

+2

( 3.48

42.

782

−1) −

(1−

0.32

812

−0.

45)3.

484

2.78

2

1/2

29.23






Ste

pC

alcu

lati

onV

alu

eU

nit

s

Th

en

ewit

erat

ion

loop

star

tsw

ith

the

dete

rmin

atio

nof

the

set

ofn

ewm

ass

velo

citi

es.T

hes

eva

lues

wil

lbe

use

dto

calc

ula

teth

ere

fin

edva

lues

ofR

eyn

olds

nu

mbe

rsin

step

MG

C-7

.S

ubs

equ

entl

y,st

eps

MG

C-8

thro

ugh

MG

C-3

6sh

ould

bere

visi

ted.

Th

en

ewm

ass

velo

citi

esG

shou

ldbe

calc

ula

ted

from

the

exac

tex

pres

sion

for

the

pres

sure

drop

,as

give

nin

step

sM

GC

-33

and

MG

C-3

5,as

sum

ing

Gva

lues

asu

nkn

own

and

the

oth

ern

um

eric

alva

lues

inth

ese

equ

atio

ns

asgi

ven

.Th

eco

nve

rgen

cew

ould

beve

ryfa

st.T

he

new

valu

eof

the

mas

sve

loci

tyof

53.6

1kg

/m2

sis

sign

ifica

ntl

ysm

alle

rth

anth

eas

sess

edva

lue

inth

efi

rst

iter

atio

n(6

0.37

kg/m

2s)

;th

us,

are

duct

ion

ofal

mos

t10

perc

ent

isac

hie

ved.

MG

C-3

8

Gh

=

[ 2(p i

n�

in)( �

p p i

)] 1/2

h[ (1

−�

2+

Kc)

+f

L�i

r h�

m+

2( �

i

�o

−1) −

(1−

�2

−K

e)�

i

�o

] 1/2 h

=

[ 2×

100

×10

3×

0.49

8×

4×

103

100

×10

3

] 1/2 (1

−0.

3281

2+

1.22

)+0.

0160

30.

198

×0.

498

1.87

5×

10−3

40.

5+

2( 0.

498

0.57

4−

1) −(1

−0.

3281

2−

0.2)

0.49

80.

574

1/2

22.9

3kg

/m2

s

Th

en

ewva

lue

ofth

em

ass

velo

city

isin

crea

sed

from

20.1

9to

27.2

4kg

/m2

s,th

atis

,for

rou

ghly

7pe

rcen

t.

MG

C-7

/4R

e c=

GcD

h,c

�c

=47

.91

×1.

875

×10

−3

28.9

5×

10−6

3103

—

Re h

=G

hD

h,h

�h

=22

.93

×1.

875

×10

−3

31.6

7×

10−6

1357

—

Th

en

ewit

erat

ion

cycl

ere

quir

esde

term

inat

ion

ofn

ewR

eyn

olds

nu

mbe

rs(w

ith

chan

ged

Gva

lues

,in

clu

din

gal

lth

eea

rlie

rlo

cali

tera

tion

s;th

isis

the

fou

rth

iter

atio

nof

Rey

nol

dsn

um

bers

).T

his

iter

atio

nre

peat

sst

epM

GC

-7.

29.24






MG

C-8

/4j c

=ex

p(a 0

+r c

(a1

+r c

(a2

+r c

(a3

+r c

(a4

+a 5

r c))

)))

0.00

3777

—j h

=ex

p(a 0

+r h

(a1

+r h

(a2

+r h

(a3

+r h

(a4

+a 5

r h))

)))

0.00

4709

—f c

=ex

p(b 0

+r c

(b1

+r c

(b2

+r c

(b3

+r c

(b4

+b 5

r c))

)))

0.01

034

—f h

=ex

p(b 0

+r h

(b1

+r h

(b2

+r h

(b3

+r h

(b4

+b 5

r h))

)))

0.01

539

—

r c=

ln(R

e c)

=ln

(310

3)8.

040

—

r h=

ln(R

e h)

=ln

(135

7)7.

213

—

Th

en

ewit

erat

ion

for

the

MG

C-8

esti

mat

ion

ofj

and

ffa

ctor

sta

kes

the

valu

esfo

rR

eyn

olds

nu

mbe

rsfr

omM

GC

-7/4

.

MG

C-9

/4( j f

) c=

0.00

3777

0.01

034

0.36

51—

( j f

) h=

0.00

4709

0.01

539

0.30

6—

Th

era

tios

ofC

olbu

rnan

dFa

nn

ing

fact

ors

are

calc

ula

ted

exac

tly

for

new

Rey

nol

dsn

um

bers

.N

ote

that

thes

eva

lues

did

chan

ge,b

ut

asin

any

oth

erit

erat

ion

for

dete

rmin

ing

this

rati

o,th

ese

chan

ges

are

not

sign

ifica

nt

beyo

nd

the

firs

ttw

ode

cim

alpl

aces

.

MG

C-1

0/2

T w=

T h,r

ef+

Rh Rc

T c,r

ef

1+

Rh Rc

=T h

,ref

+1/

(�0h

A) h

1/(�

0h

A) c

T c,r

ef

1+

1/(�

0h

A) h

1/(�

0h

A) c

=64

2+

1/(0

.977

5×

136.

7×

419.

3)1/

(0.9

542

×28

8.7

×41

9.3)

560

1+

1/(0

.977

5×

136.

7×

419.

3)1/

(0.9

542

×28

8.7

×41

9.3)

=64

2+

1.78

5×

10−5

8.65

7×

10−6

560

1+

1.78

5×

10−5

8.65

7×

10−6

586.

7K

Th

ew

allt

empe

ratu

reis

calc

ula

ted

this

tim

eby

taki

ng

into

acco

un

tth

edi

ffer

ence

inth

erm

alre

sist

ance

son

both

flu

idsi

des

(com

pare

this

wit

hst

epM

GC

-10)

.Wit

ha

diff

eren

ceof

slig

htl

ym

ore

than

10K

(vs.

the

last

iter

atio

n)

for

air,

the

corr

ecti

onof

Col

burn

and

Fan

nin

gfr

icti

onfa

ctor

sw

ills

till

rem

ain

rela

tive

lysm

all.

29.25






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-1

1/2

j c,c

orr

=j c

( T w T c,r

ef

) n =0.

0037

77( 58

6.7

560

) −0.0

770.

0037

64—

n=

0.3

−( lo

g 10

T w T c,r

ef

) 1/4=

0.3

−[ lo

g 10

586.

756

0

] 1/4−0

.077

—

Not

eth

atco

ldga

sis

bein

gex

pose

dto

hea

tin

gan

dth

atth

efl

owre

gim

eis

turb

ule

nt.

For

deta

ils

ofth

eal

tern

ate

expo

nen

tde

term

inat

ion

,see

Sh

ahan

dS

eku

lic

(200

3,ta

ble

7.13

,p.

531)

.Th

eco

ndi

tion

sto

besa

tisfi

edar

e1

<T w

,ref

/T c

,ref

<5;

0.6

<P

r<

0.9.

MG

C-1

2/2

j h,c

orr

=j h

( T w T h,r

ef

) n =j h

0.00

4709

—

For

the

hot

flu

id,w

eh

ave

the

flu

idco

olin

gco

ndi

tion

s.T

he

flow

regi

me

isin

the

lam

inar

regi

on.T

her

efor

e,th

eex

pon

ent

n=

0(S

hah

and

Sek

uli

c,20

03,t

able

7.12

,p.5

31).

MG

C-1

3/2

f c,c

orr

=f c

( T w T c,r

ef

) m=

0.01

034

( 586.

756

0

) −0.1

0.01

029

—

m=

−0.1

Col

dfl

uid

ish

eate

d,an

dth

efl

owre

gim

eis

turb

ule

nt.

Th

esu

gges

ted

calc

ula

tion

ofth

eex

pon

ent

inth

eco

rrec

tion

term

isva

lid

for

the

ran

geof

tem

pera

ture

rati

osas

foll

ows:

1<

T w,r

ef

T c,r

ef<

5

Inth

isca

se,t

his

rati

ois

1.04

;th

eref

ore,

calc

ula

tion

ofth

eex

pon

ent

mis

perf

orm

edas

indi

cate

d.F

luid

atth

eco

ldsi

deis

air;

ther

efor

e,it

istr

eate

das

anid

ealg

as.

MG

C-1

4/2

f h,c

orr

=f h

( T w T h,r

ef

) m=

0.01

539

( 586.

764

2

) 0.81

0.01

431

—

m=

0.81

For

the

flu

idco

olin

gca

sean

dla

min

arfl

ow,t

he

expo

nen

tis

equ

alto

0.81

.Th

eco

ndi

tion

sto

besa

tisfi

edar

e0.

5<

T w,r

ef/

T h,r

ef=

0.91

<1;

0.6

<P

r=

0.69

9<

0.9.

29.26






MG

C-1

5/2

h c=

j c,c

orr

Gcc

p,c

Pr2/

3c

=0.

0037

6447

.91

×10

410.

6982

/3

238.

5W

/m2

K

Th

eh

eat-

tran

sfer

coef

fici

ent

onth

eco

ldfl

uid

side

isde

term

ined

from

defi

nit

ion

ofth

eC

olbu

rnfa

ctor

.

MG

C-1

6/2

h h=

j h,c

orr

Ghc p

,h

Pr2/

3h

=0.

0047

0922

.93

×10

610.

6992

/3

145.

4W

/m2

K

Th

eh

eat-

tran

sfer

coef

fici

ent

onth

eh

otfl

uid

side

isde

term

ined

from

defi

nit

ion

ofth

eC

olbu

rnfa

ctor

.

MG

C-1

7/2

�f,

c=

tan

h(m

l)c

(ml)

c=

tan

h(1

25.3

×0.

0030

2)12

5.3

×0.

0030

20.

9549

—

mc

=√ 2h

c/k�

=√ 2

×23

8.5/

(200

×0.

152

×10

−3)

125.

3m

−1

l c=

b 2−

�=

6.35

×10

−3

2−

0.15

2×

10−3

0.00

302

m

Not

eth

at�

and

l cdi

ffer

mor

eth

anfo

ran

orde

rof

mag

nit

ude

,so

the

expo

sed

stri

ped

geof

the

fin

isn

otta

ken

into

acco

un

tan

dth

em

para

met

eris

calc

ula

ted

usi

ng

this

appr

oxim

atio

n[s

eeS

hah

and

Sek

uli

c(2

003,

pp.2

80an

d62

7)fo

ran

alte

rnat

ive]

.Th

erm

alco

ndu

ctiv

ity

ofth

efi

nis

assu

med

tobe

200

W/m

Kfo

ran

allo

yat

the

give

nte

mpe

ratu

re.

Not

eth

atth

ere

sult

ing

fin

effi

cien

cyis

abi

tla

rger

than

inth

epr

evio

us

iter

atio

n(M

GC

-17)

.A

ctu

alfi

nef

fici

ency

ina

braz

edal

um

inu

mh

eat

exch

ange

rm

aybe

sign

ifica

ntl

ysm

alle

rw

ith

inth

eco

re(Z

hao

etal

.,20

03).

MG

C-1

8/2

�f,

h=

tan

h(m

l)h

(ml)

h=

tan

h(9

7.8

×0.

0030

2)97

.8×

0.00

302

0.97

19—

mh

=√ 2h

h/k�

=√ 2

×14

5.4/

(200

×0.

152

×10

−3)

97.8

m−1

l h=

b 2−

�=

6.35

×10

−3

2−

0.15

2×

10−3

0.00

302

m

Not

eth

atth

efi

nge

omet

ryis

the

sam

eon

both

flu

idsi

des;

ther

efor

eth

ele

ngt

hs

are

the

sam

e.T

he

fin

effi

cien

cyis

quit

ela

rge.

29.27






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-1

9/2

�0,

c=

1−

(1−

�f,

c)A

f A=

1−

(1−

0.95

49)×

0.84

90.

9617

—

Adi

scu

ssio

nof

the

mea

nin

gof

the

tota

lext

ende

dsu

rfac

eef

fici

ency

can

befo

un

din

Sh

ahan

dS

eku

lic

(200

3,p.

289)

.

MG

C-2

0/2

�0,

h=

1−

(1−

�f,

h)A

f A=

1−

(1−

0.97

19)×

0.84

90.

9761

—

Not

eth

atth

eex

ten

ded

surf

ace

effi

cien

cies

diff

erfo

rbo

thsi

des

rega

rdle

ssof

the

sam

ege

omet

ryof

the

fin

s.T

his

isdu

eto

the

diff

eren

cein

hea

t-tr

ansf

erco

effi

cien

ts.

MG

C-2

1/2

U=

[1

(�0h )

c+

Ac/

Ah

(�0h )

h

] −1=

[1

(�0h )

c+

1(�

0h )

h

] −1=

[1

0.96

17×

238.

5+

10.

9761

×14

5.4

] −187

.67

W/m

2K

Bec

ause

ofth

eh

igh

ther

mal

con

duct

ivit

yof

wal

lmat

eria

l,th

eth

erm

alre

sist

ance

ofth

ew

alli

sn

egle

cted

inth

isca

lcu

lati

on.A

lso,

itis

agai

nas

sum

edth

atn

osi

gnifi

can

tfo

uli

ng

ispr

esen

ton

eith

ersi

deof

the

hea

t-tr

ansf

ersu

rfac

e.T

her

efor

e,th

eov

eral

lhea

t-tr

ansf

erco

effi

cien

tis

defi

ned

byh

eat-

tran

sfer

con

duct

ance

due

only

toco

nve

ctio

non

both

flu

idsi

des.

Not

e,ag

ain

,th

atth

eh

eat-

tran

sfer

surf

ace

area

sar

eth

esa

me

onbo

thfl

uid

side

sbe

cau

seof

the

use

ofth

esa

me

fin

geom

etry

.

MG

C-2

2/2

Ac

=A

h=

( mc p

) cN

TU

U=

20.8

2×

103

×1.

811

87.6

7

430.

1m

2

Th

eh

eat-

tran

sfer

surf

ace

area

sar

eth

esa

me

onbo

thfl

uid

side

s.

MG

C-2

3/2

A0,

c=

mc

Gc

=20

47.9

10.

4175

m2

Th

efr

ee-fl

owar

eaon

the

cold

side

isde

term

ined

from

the

defi

nit

ion

ofth

em

ass

velo

city

,G

=m

/A

0.

MG

C-2

4/2

A0,

h=

mh

Gh

=20

22.9

30.

8724

m2

Th

efr

ee-fl

owar

eaon

the

hot

flu

idsi

deis

dete

rmin

edan

alog

ousl

yto

dete

rmin

atio

nof

the

sam

een

tity

onth

eco

ldfl

uid

side

;see

step

MG

C-2

3.

29.28






MG

C-2

5/2

�c

=�

h=

b�D

h

8(b

+a)

=6.

35×

10−3

×18

41×

1.87

5×

10−3

8( 6.

35+

2) ×10

−30.

3281

—

Th

isst

epis

iden

tica

lto

the

one

perf

orm

edin

the

firs

tit

erat

ion

(th

eh

eat-

tran

sfer

surf

ace

type

was

not

chan

ged)

.

MG

C-2

6/2

Afr

,c=

A0,

c

�c

=0.

4175

0.32

811.

272

m2

Th

efr

ee-fl

owar

eaon

the

cold

flu

idsi

deis

dete

rmin

edfr

omde

fin

itio

nof

the

free

-flow

area

:fr

onta

lare

ara

tio

for

the

cold

flu

idsi

de.

MG

C-2

7/2

Afr

,h=

A0,

h

�h

=0.

8724

0.32

812.

659

m2

Th

efr

ee-fl

owar

eaon

the

hot

flu

idsi

deis

dete

rmin

edfr

omde

fin

itio

nof

the

free

-flow

area

:fr

onta

lare

ara

tio

for

the

hot

flu

idsi

de.

MG

C-2

8/2

Lc

=D

hA

c

4A

0,c

=1.

875

×10

−3×

430.

14

×0.

4175

0.48

29m

Th

efl

uid

flow

len

gth

onth

eco

ldfl

uid

side

repr

esen

tsth

epr

inci

palc

ore

dim

ensi

onin

this

dire

ctio

n.

MG

C-2

9/2

Lh

=D

hA

h

4A

0,h

=1.

875

×10

−3×

430.

14

×0.

8724

0.23

11m

Th

efl

uid

flow

len

gth

onth

eh

otfl

uid

side

repr

esen

tsth

epr

inci

palc

ore

dim

ensi

onin

this

dire

ctio

n.

MG

C-3

0/2

Lst

ack

=A

fr,c

Lh

=1.

272

0.23

11

Lst

ack

=A

fr,h

Lc

=2.

659

0.48

29

5.50

4m

5.50

6m

Th

eco

resi

zein

the

dire

ctio

nor

thog

onal

toth

efl

owdi

rect

ion

sis

appa

ren

tly

redu

ced

wh

enco

mpa

red

toth

epr

evio

us

iter

atio

n.

29.29






Ste

pC

alcu

lati

onV

alu

eU

nit

s

MG

C-3

1/2

( �p p i

) c=

G2 c

2(p

in�

in) c

[ (1−

�2

+K

c)+

fL�

i

r h�

m+

2( �

i

�o

−1) −

(1−

�2

−K

e)�

i

�o

] c

Kc,

c=

f 1(�

c,R

e c,

surf

ace

geom

etry

)

Ke,

c=

f 2(�

c,R

e c,

surf

ace

geom

etry

)

Kc,

c=

f 1(�

c=

0.32

81,R

e c=

3103

,su

rfac

ege

omet

ry=

19.8

6)0.

52

Ke,

c=

f 2(�

c=

0.32

81,R

e c=

3103

,su

rfac

ege

omet

ry=

19.8

6)0.

42

Th

ech

ange

ofbo

thK

para

met

ers

take

spl

ace

beca

use

ofth

ech

ange

inR

eyn

olds

nu

mbe

rva

lues

inth

isit

erat

ion

.

MG

C-3

2/2

( �p p i

) h=

G2 h

2(p

in�

in) h

[ ( 1−

�2

+K

c) +f

L�i

r h�

m+

2( �

i

�o

−1) −

( 1−

�2

−K

e) �i

�o

] h

Kc,

h=

f 1(�

h,R

e h,

surf

ace

geom

etry

)

Ke,

h=

f 2(�

h,R

e h,

surf

ace

geom

etry

)

Kc,

h=

f(�

h=

0.32

81,R

e h=

1357

,su

rfac

ege

omet

ry=

19.8

6)1.

22—

Ke,

h=

f(�

h=

0.32

81,R

e h=

1357

,su

rfac

ege

omet

ry=

19.8

6)0.

20—

Not

eth

atth

eh

otfl

uid

stay

sin

the

lam

inar

flow

regi

on;t

her

efor

e,th

eK

’sco

effi

cien

tsdo

not

chan

ge.

MG

C-3

3/2

( �p p i

) c=

47.9

12

2( 50

0×

103

×3.

484) ×

(1−

0.32

812

+0.

52)+

0.01

029

0.48

29×

3.48

4

1.87

5×

10−3

43.

093

+2

( 3.48

42.

782

−1) −

(1−

0.32

812

−0.

42)3.

484

2.78

2

0.

0087

—

Th

ere

lati

vepr

essu

redr

ops

can

now

beca

lcu

late

dby

usi

ng

allt

he

prev

iou

sly

dete

rmin

edva

riab

les

and

para

met

ers.

Not

eth

ath

ydra

uli

cra

diu

sis

agai

nre

pres

ente

das

one-

quar

ter

ofth

eh

ydra

uli

cdi

amet

er.

29.30






MG

C-3

4/2

�p c

=p i

,c

( �p p i

) c=

500

×10

3×

0.00

874.

3kP

a

Fro

mth

ein

put

data

,th

eal

low

edpr

essu

redr

opis

5kP

a>

4.3

kPa.

Th

isre

sult

indi

cate

sth

atth

isco

ndi

tion

isn

owsa

fely

sati

sfied

.Con

sequ

entl

y,be

cau

seof

this

pres

sure

drop

,th

ere

isn

on

eed

for

furt

her

iter

atio

ns.

MG

C-3

5/2

( �p p i

) h=

22.9

32

2(10

0×

103

×0.

498)

×

(1−

0.32

812+

1.22

)+0.

0143

10.

2311

×0.

498

1.87

5×

10−3

40.

533

+2

( 0.49

80.

574

−1) −

(1−

0.32

812−

0.2)

0.49

80.

574

0.04

—

Th

eh

ot-s

ide

flu

idpr

essu

redr

opdi

vide

dby

the

inle

tfl

uid

pres

sure

atth

eh

otfl

uid

side

can

now

beca

lcu

late

din

the

sam

em

ann

eras

for

the

cold

flu

id.O

fco

urs

e,on

em

ayde

cide

toca

lcu

late

the

pres

sure

drop

righ

taw

ay(s

eest

epM

GC

-36)

.

MG

C-3

6/2

�p h

=p i

,h

( �p p i

) h=

100

×10

3×

0.04

4.0

kPa

Th

eim

pose

dm

axim

um

allo

wab

lepr

essu

redr

opis

4.2

kPa,

wh

ich

isvi

rtu

ally

equ

alto

the

valu

ede

term

ined

inth

isst

ep.T

her

efor

e,th

isre

quir

emen

tis

also

sati

sfied

.Th

eca

lcu

late

dpr

essu

redr

opis

exac

tly

atth

ele

velo

fth

eal

low

edpr

essu

redr

opli

mit

.Th

eref

ore,

the

calc

ula

tion

proc

edu

reis

com

plet

ed.

29.31







Conclusion

The step-by-step heat-exchanger design procedure clearly demon-strates how intricate the sizing of a compact heat exchanger maybe. However, an inevitably iterative routine converges very rapidly.The main dimensions of this heat-exchanger core are determined tobe Lc = 48 cm, Lh = 23 cm, and Lstack = 550 cm. The core is made ofplane triangular plate fin surfaces [surface designation 19.86 (Kays andLondon, 1998)]. Imposed limitations on the pressure drops are bothsatisfied. If, because of say, space considerations, the core dimensionsmust satisfy certain a priori imposed aspect ratios (fluid flow vs. stacklength), further iterations would be needed. Calculation is presentedin a most explicit manner, by listing each step (regardless of whetherit may consist merely of a repeated calculation, already exercised in aprevious step). All algebraic operations are, as a rule, included. Thisis done keeping in mind a need for full transparency of the calculationalgorithm. Such design is conducted, as a rule, in practice by using acomputer routine (what would never be fully transparent but elimi-nates any calculation errors that may often be present in a calculationas given here). Still, following the procedure as presented here, one caneasily devise such a routine and execute the calculation.

References

Baclic, B. S., and P. J. Heggs, 1985, “On the Search for New Solutions of the Single-PassCrossflow Heat Exchanger Problem, Int. J. Heat Mass Transfer, vol. 28, no. 10, pp.1965–1976.

Kays, W. M., and A. L. London, 1998, Compact Heat Exchangers, reprint 3d ed., Krieger,Malabar, Fla.

Sekulic, D. P., 1990, “A Reconsideration of the Definition of a Heat Exchanger,” Int. J.Heat Mass Transfer, vol. 33, pp. 2748–2750.

Sekulic, D. P., 2000, “A Unified Approach to the Analysis of Unidirectional and Bidirec-tional Parallel Flow Heat Exchangers,” Int. J. Mech. Eng. Educa., vol. 28, pp. 307–320.

Sekulic, D. P., R. K. Shah, and A. Pignotti, 1999, “A Review of Solution Methods for De-termining Effectiveness-NTU Relationships for Heat Exchangers with Complex FlowArrangements,” Appl. Mech. Reviews, vol. 52, no. 3, pp. 97–117.

Shah, R. K., and D. P. Sekulic, 2003, Fundamentals of Heat Exchanger Design, Wiley,Hoboken, N.J.

Webb, R. L., 1994, Principles of Enhanced Heat Transfer, Wiley, New York.Zhao, H., A. J. Salazar, and D. P. Sekulic, 2003, “Influence of Topological Characteristics

of a Brazed Joint Formation on Joint Thermal Integrity,” Int. Mech. Eng. Congress, vol.1, ASME paper IMECE2003-43885, Washington, D.C.




sizing of a crossflow compact heat ex changer

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