THERMAL EFFECTIVENESS OF A SPLIT-FLOW HEAT EXCHANGER

by

M. IQBAL

Thesis submitted in partial fulfilment of the

requirements for the degree of Master of Engineering.

Department of Mechanical Engineering,

McGill University,

Montreal.

April 1961.

- i -

SUMKARY

A mathematical expression has been derived for the

effectiveness of a split-flow exchanger in terms of C /C min. max.

and NTU Curves have been drawn for f. versus NTU for max. max.

various values of C i /c m n. max.

The effectiveness of a split-flow exchanger has been

compared with that of a reverse flow exchanger and it is observed

that for low values of NTU below 2, there is practically no. max.

difference between the two exchangers.

However for values of NTU greater than about 2 max.

the curves for effectiveness of a split-flow exchanger start

drooping significantly from those of the reverse flow exchanger.

Another interesting result of this study is the fact that the

effectiveness of a split-flow exchanger is improved by having

the heat capacity rate (W Cp) of the shell side fluid smaller

than that of the tube side fluid.

A split-flow exchanger was also built and some points

on the effectiveness curves were checked.

ii

Acknowledgements

The author wishes to express his sincere gratitude to

Professor J.W. Stachiewicz for his guidance and advice.

Thanks are also due to Mr. F. Corrick for his assistance

in the construction of the apparatus.

The financial assistance given by the National Research

Council is gratefully acknowledged.

iii

TABU: OF CONTENTS

Summary

Acknowledgements

Table of Contents

Nomenclature

Introduction

Assumpt ions

Development of an expression for effectiveness of a split-flow

exchanger.

Effectiveness of L.H.S. of exchanger

Effectiveness of R.H.S. of exchanger

Overall effectiveness of the split-flow exchanger

Determination of correction factor Ft

Experimental equipment

Precautions

Test Resulta

Sample Calculation

Discussions and Conclusions

References

Curves of effectiveness of split-flow exchanger

Curves of effectiveness of split-flow exchanger vs. reverse-

flow exchanger

Curves of correction factor Ft for split-flow exchanger

APpendix: Equation (2-31)

Equation (2-3la)

ii

iii

iv

1

13

12.

14

26

44

58

61

69

71

72

74

77

78

79

80

81

82

iv

NOMENCLATURE

Roman Letter Syuabols

A

c

e

q

t

u

w

~otal heat transfer area of the ,tplit .. fl'* ex changer

half of the total beat transfer area of the split-flow exchanger.

flow stream capacity rate (W.~)

specifie beat at constant pressure

baae of natural system of logarithms

temperature correction factor

heat transfer rate

temp-erature

unit overall thermal conductance

- mass flow

Roman Letter Constants

B = ..J!.... 2Ct

D u

= ~ sh

F = tsl - ttl

G = ts3 - t to

H = ts3 - tt3

J = tsl - t t2

·~ 2

ft

ft2

BTU/hr oF

BTU/lb oF

RTU/hr.

lbs/br.

Kl' ~, K3

, K4

, KS' K6' "! and K8 = constants in differentiai equations.

L = -V e

=

v

M 0 "/ -Y = (e - e )

N = (1 - p) e"~ + (1 + p) e -Y

n = NTUmax

tt3 - tto p =

~s:l. - tto

tsi - t ct R = sg. =

tt3 - tto cs

s = 2 eDAh

t ts2 + ts3

= 2 sq

-t.:rr.Umax}l

2

x = 1 + +(cmi~)

e cmax

1 emin

x = +--cmax

-ImJmaxjl 2

+(Cmin) y = 1 - e Cmax

~in y = 1 - Cmax

Greek Letter Syœbols

E 2sh

é.s

d•notes difference

exchanger effectiveness dimensionless

effectiveness of L.H.S. of split-flow exchanger

.effectiveness of R.H.S. of split-flow exchanger

effectiveness of R.H.S. of split-flow exchanger when ct = emin.

effectiveness of R.H.S. of split-flow exchanger when csh = emin•

effectiveness of complete exchanger based on él and é2 when ct = emin•

effectiveness of complete exchanger based on é. 1 and ê.. 2 when C8 = emin.

vi

Greek Letter Constants

o( B + D = Js2 + o2

B -t D = J s2 + o2

= AhjB2 + D2

? B = Ja2 + o2

t D =

J + D2 s2

" = D ~ _ft2 + o2

Dimensionless Groups

= c.,. Flow stream capacity rate ratio

~ heat transfer effectiveness of an exchanger; a function

of NTU, CminlCmax' and flow arrangement.

NTUmax - n&~ber of heat transfer units of an exchanger, a heat

transfer parameter. (UAh \ or (JlL_) Cmi~} ~in

Subscripts

Ah hal f are a,

a left end of exchanger

b middle of exchanger

c counterflow

f r ight end of exchanger

p parallel flow

vii

r ratio

s shell

sb shell half

t tube

t .c tube .. counterflow

tp tube - parallel flow

tr tube .. end where flow reverses .

w weighted

0 1 2 3

for temperature subscript notation refer to each

figure individually

- 1 -

INTRODUCTION

In a heat exchanger the rate of heat flow from the hot

to the cold fluid is proportional to the temperature difference

between the two. For design purposes it is essential to know the

mean difference in temperature between the inlet and exit tempera-

tures.

The Log-Mean Temperature Difference Aperoach.

The classical approach to the performance of a beat

exchanger is in terms of log-mean temperature difference (LMTD)

and a non-dimensional correction factor Ft• Expressed in the

usual nomenclature, the rate of heat transfer for unit time is

given by

q = U.A.Ft. LMTD (counter-flow).

If the exchanger is actually a counter-flow unit the factor Ft

is unity. For all other flow arrangements Ft is less than unity.

In tubular exchangers the simplest case of counter-flow

or parallel-flow is that of two concentric pipes(Figure 1 and

Figure ~· The LMTD for these two cases is easily derived and is

given by:

- 2 -

GOgN!ER-FLOW EXCHANGER

c

ct~ ---Ct

lj--:! 1 ===~-======r-I Cs

SYSTEM DIAGRAM AND

EXPECTED TEMPERATURE DISTRIBUTION

FIGURE - 1

- 3 -

PARALLEL-FLOW EXCHANGER

c

~-----~-----'1 (1 --

SYSTEM DIAGRAM AND

EXPECTED TEMPERATURE DISTRIBUTION

FIGURE - 2

LMTD (counter-flow)

LMTD (parallel-flow)

- 4 -

= (tsl- h·3) -( fj4- tf:o) iM. Es,- tt"I.

ts4 - é to

= (ti~-tto) -(Cs4-tt:?>) 1.,..,_ t ~ 1 - t to

ts.4- tB For any set of terminal temperatures, the LMTD for parallel-flow is always

less than that for counter-flow unless the temperature of one flu!Ld stream

is constant throughout the exchanger.

In practice, in the majority of industrial installations, the

counter-flow beat exchanger is not as economical as multi-pass or cross-flow

units. In multi-pass exchangers the flow is partly counter-flow and partly

parallel-flow and as a result the LMTD lies somewhere between LMTD (counter~

flow) and LMTD (parallel-flow). The same is true for cross-flow excb.a.n.gers.

For multi-pass exchangers. Nagle (1) derived in detail the equations

for mean temperature difference (MID) for the 1-2 exchanger, i.e. having one

pass shell-side and two passes tube-side. Figure - 3. He also extended the

derivations for the 2-4 exchanger, i.e. having two passes shell-side and four

passes tube-aide. Graphical and trial and error solutions were obtained for

these equations and curves of Ft versus dimensionless temperature ratios

R t~, - t!:.4

- tt!> - E ~o tt"!> - t to

t~, "'"""~l:o

were plotted.

* Underwood (2) solved Nagle's equations for MTD. Later Bowman (3)

extended Nagle and Underwood's work by developing equations for any number

of shell passes and plotted curves for Ft for 3-6, 4-8 and 6-12 exchangers.

* This paper was not available, however, the information is based on

reference (7) .

- 5 -

1 - 2 EXCHANGER

(Reverse-Flow Exchanger)

~----------------------------~-:_.__ ct ~-------------------------------~ Ct

SYSTEM DIAGRAM AND

EXPECTED TEMPERATURE DISTRIBUTION

FIGURE - 3

- 6 -

In cross-flow exchangers, the fluids flow at right angles to each

other, therefore, neither the counter-flow nor the parallel~flow equations

for LMrD can be applied. Mathematical analysis of the MTD in single pass

* cross-flow and double pass cross-flow were given by Nusselt (4) and Smith (6).

Bowman-Mueller-Nagle (7) co-ordinated the results of all the above

mentioned types of shell-and-tube exchangers and cross-flow exchangers, and

presented in a single paper the correction factor curves for various surface

and flow arrangements.

The NTU Aperoach

In the thermal analysis of the various types of beat exchangers

presented above, the equation~= UA Ft LMTD (counter-flow) is used and is

found convenient when all the terminal temperatures necessary for the

evaluation of the appropriate mean temperature are known. There are however

numerous occasions when the beat transfer area A and the overall beat trans-

fer coefficient U are known or the latter can at least be estimated, but the

temperatures of the fluids leaving the exchanger are not known. This type

of problem is encountered in the selection of a beat exchanger or when the

unit bas been tested at one flow rate but the serviae conditions require

different flow rates for one or both fluids. The outlet temperatures and

the rate of beat flow can only be found by a rather tedious trial and error

procedure if the charts of Ft are used. In such cases it is desirable to

circumvent entirely any reference to the logarithmic or any other mean

temperature difference.

* This paper was not available, however, the information is based on

reference (7).

- 7 -

This method bas been proposed by Nuaselt (5)* and later by Ten

UA Broeck ( 9-) who showed that there is a relation involving Ct' R, P and Ft

in the equations

also

q = UA Ft LMTD (counter-flow)

= UA. Ft . (ts1- b:-s )-lts4- t~) l tsl - t~

t$4 - tt:o

therefore from the above three equations one can Write

UA _ k( ';:.R:) Ct ft l \- R. )

which connecta the four dimensionless parameters, and is independent of any

considerations of flow arrangement.

From Underwood's equation of Ft for one shell pass and two tube

passes,

Ten Broeck obtained

~t = jRl-+1 ~Ll1-P)/(1- RP)J

(R _1 )l ?.-P(R+1-)R'+1 ) 2- Pl Rt1 +)R';-1 )

l_ 2 -P ( R-t 1-JR~+1 ) 2 - P l R + 1 +) Rl.-+ 1 )

connecting the three dimensionless parameters, and plotted the curves for

UA P versus Ct with R as the third parameter. P the dimensionless temperature

UA ratio was called a thermal efficiency and Ct bas been subsequently termed

as NTU, the number of beat transfer units. Ten Broeck also plotted similar

curves for 2-4 exchangers.

* This paper was not available, however, the information is based on

reference (8}

- 8 -

Defining again the thermal effectiveness as

~ _ actual energy transferred ~ - maximum possible energy transferred

_ Ct ( b:"3 ~ b:o) C& ( ts.,- t ~4) - C~~~~ l ts, - t to) CW\i'r\ l t~, - t to)

and NTU max

UA c min

c and third parameter min, c max

for a large number of flow arr~ngements 1 Kays and London (10) have given the

curves and in some cases also the mathematical expressions for effectiveness

versus N'l'Umax in terms of the third dimensionless parameter ~in. ~ax

For some of the more common flow arrangements the expressions for

effectiveness are reproduced below:

Counter-flow - HTUh\LtX ( 1 -

E-=- 1-e ( . -NTU~ ( 1-

1-~e. c~

Parallel-flow

-NTUmo.x. ( 1 +

E 1- e. ---------~-------1 -t c "WW~

Parallel-Counter Flow c.~

One shell pass, two tube passes

-NTV~:~}+ (~~~ \2. 1- e. ~J

- 9 -

Cross-Flow

for Cmin = Cmixed

and ~ = Cunmixed

for Cmax = Cmixed

and Cmin = Cunmixed

E. = Cw.ox c~w.

Split-Flow Exchanger

Split-flow exchangers are often used when the permisaible pressure

drop of the shell-side fluid is so small that the fluid cannot be permitted

to travel the full length of the shell. The fluid is admitted at the centre

of the shell and the flow is split as shown in Figure 4.

The pressure drop through the shell of an exchanger may be approxi-

mately considered as directly proportional to the length of the path and to

the square of the maas velocity of the shell aide fluid.

Sinca in a split-flow exchanger, both the maas valocity and the

length of the path are reduced by a factor of 2, the resultant presaure drop

is approximately one eighth that of a conventional exchanger.

- 10 -

SPLIT-FLOW EXCHANGER

ts2

ts3

tt21 tt3

t tr LI

SYSTEM DIAGRAM AND .

EXPECTED TEMPERAXURE DISTRIBUTION

FIGURE - 4

- 11 -

Kern and Carpenter (ll) darived an expression for the mean

temperature difference in a split-flow exchanger. This is an implicit.·

and tedious expression and lends itself to solution by a trial and error

method only. In practice thus, an engineer seldom can make use of it

and generally employa the curves of l-2 exchangers.

Schindler and Bates (12) have drawn Ten Broeck charts for a

1-2 divided-flow beat exchanger, however, a study of the effectiveness

versus NTUmax relation and correction factor of this relatively impor­

tant split-flow exchanger bas not appeared in the literature heretofore.

- ll -

Development of an Expression for the Effectiveness of a

Split-Flow Exchanger as a Function of Capacity

Rate Ratio and Number of Heat Transfer Units

The system diagram and expected temperature distribution for a

split-flow exchanger is shown in Figure - 4.

The split-flow exchanger is es•entially a combination of two

independent counter-flow parallel-flow exchapgers: · For thé mathematical

derivation, the two halves, (L.H.S. and R.H.S.), are considered separately

and then a combined effectiveness is derived.

\

- 13 -

ASSUMPTIONS

The following assumptions are made in development of an expression

for the effectiveness of a split-flow exchanger.

1. Equal weight flow of shell fluid in right band side and left band aide

of the exchanger.

2. Constant specifie beats for both fluids on both the left and the right

band side of the exchanger.

3. Equal heat transfer area in each pass and on both aides of the exchanger.

4. Constant and equal overall coefficient of heat transfer U on both aides

of the exchanger.

5. Shell fluid completely mixed at any cross section.

6. No phase changes within the exchanger.

7. Negligible heat losses to the surroundings.

8. The excha~ger operating at steady conditions.

- 14 -

1 - Effectiveness of L.H.S. of Exchanger

This derivation is based mainly on an unpublished report of

c.e. Wright (13) excepting that the shell-side fluid flows from right

to left.

Consider the energy balance and rate equations over the area

dA, Figure - 5.

Energy balance:

Rate equations:

dq,. = dqs = dqc. + dq,.

d\ = C.s.J-. Jts

dq~ = Ct dJ. tc. = U ~A (A tc.)

dq~ = -Ct d.t tp = Uf' (6. t ~)

(1-1)

(1-la)

(l-1b)

(1-lc)

Substituting the rate equations (1-1b) and (1-1c) in (1-l)

d.~A' =Ct lJttc. - ~t tp)

since

and from Figure - ~

he nee

or

also

= UiA ( b tp + ôte.) ..... (l-l)

clt tc. - ~t tp = J.l b:c_ - t '=~J t tc. - tt~ : At p - A tc

c~ J (~tr -A~c.) = u~ lAtp + Ab:) dJAt~ - 6tc.) _ JL ( Atp + b.h:_)

cl.A - 2Ct \.

C~ J.ts = Ct ( J..ttc. -dJt~)

d.ts • ~ ( dtt( -Jtt ~) ..... (1-4)

- 15 -

L.H.S. OF SPLIT-FLOW EXCHANGER

t----- A ---- --t--

TEMPERATURE DISTRIBUIION

FIGURE - 5

- 16 -

Subtracting dttc from both sides of (1-4)

dis -Jttt :: ~ clltt- Ss_ cll.k:.p - Jk tc. (1-5) '-~ c.~

aince dJ:~ - d.t tC. :. J l t~- ltc. J ::. J.~l:c..) and from (l-1b) d.t~ = Ud.A A ~c.

2Ct

from (1-1c) d.( ~~ ~ - UJ.A Â 1:. b 2,Ct r

Subatituting the above three equations in (1-5)

Now substracting dtt from both sides of (1-4) p

J.b,,- J.l:tt e ~~ Jtt-c - t_ dl:t~ _dhp • • • •• (1~7)

Since JJ:s - dtt~ = cllb~ -t~p) = cl(_A~~) and from (1-1 q,) Jt tC. = UJ..A. ~tc.

2G-

from (.1-1 c) c{h~ =- ~~ Al:~

- 17 -

bence substituting these three equations in (1~7)

u u Subatituting B = 2Ct and D = 2Csh in (1-3), (1:'"6) and (ltr-8)

J. ~~~) - J.~;-) = l'> (_Al:~) + ~ (t>A:c) ••••• (1-3)

••••• (1-6)

• • • • • (1-8)

àifferentiating the above equation

d_ ( .ô.t:-~) __ ( D-t J cÀ. ( t:kc.) \ dt(~ dA - ]) 7 dA + J)ll. d..A"L

Subatituting the a'bova two equations in (1•3) and •imp1~fying

or 00000 (1-9)

.... 18-

Similarly from •quation (1-8)

Equations (1~) ~nd (1•10) are differentia1 equations whose solution is

( 1\ ~~

.À~l.DÀ-B J K l. =o

where

( lH·,.{B.':l.+~ )A (f) -.Jal--+'D ... )A A~<. :. k., e.. -T K:t e..

••••• (1-11)

(~ +)&l.-t'Ji1- )~ (D -J!i>-+~1)A A~ p = K ~ e. + Ktt t. ••••• (1-12)

The constants ~ 1 K2, IS and K4 may be determined from the boundary conditions.

From Figure -+ .S: when A = 0 1 Ab:. = A~p = f~l. _ t t'f ••••• (1-13,14)

-

- 19 -

Also from ~quation (1-6)

••••• (1-15)

Also when A = o, equation (1.-U) gives

~t =K +K-=t ... t c 1 -~ s2 tr

••••• (1-16)

Note:· In the foregoing and subsequent equations in th.is article, A does

.»Dt .represent the total area of the split-flow exchanger, but representa

bhe variable area A, which can assume any value 0 - ~·

Equatiug equations (1•15) and (1-16) 1

(2!>-S)(h.l.-rt--t) = ~,(D+)B'"Jrll"l. )~~l.(~.JB,.~l>-a.)

-:::. D ( \.<' -t K~) -t l:f::.' - 'f:-"1.)} 'è;}· -t r:?

= J) (t~2.-~tt) +(t,-'t(~)Js~ 1?

or ~-B) (hl.-~~1) =J P.}·+ ri" ( \<\ - K0 \<, -\<1. = .D-B (h~1. -tt-'~-)

J B"' ;- J.)"l-••••• (1-17)

- 20 -

• • • • • (1-18)

••••• (1-19)

Wow K3 and K4 must be eva1uated.

a1so equation (1-8) at this boundary condition becomes

: (?.J> 1-B)( ts~- ~ tYJ oH o o (1-20)

A:o

Differentiating (1-12), putting A= 0 and equating it to (1-20) in a simi1ar

way as before, it can be shown that,

(zl -+S)( t~1 - trt) == K~ (:D +)B~Jf-) + \(lf \_}-)B,_-+~1.)

or

as

. ..

(D +B )( ts2.- t~t) = jB~+y. (~1.- KttJ

K ~ - KLJ c l> + ~ 1 c ~2- tt -t \ j~} -t-~2. \ J

K'l+ ~ : tsz. - ttt K = t~- b~Y \ 1 + 1) + E> 1

~ ~ l j·p/·.JrJ>4 j

\( = ts>.- !:tv ( 1 _ b-I-t'> (

~ 2 l J ~l.~~ j

• • • • • (1-21)

• • • • • (1-22)

Substituting the values of the constants K11 E11 K3

and K4 in (1•11) and (1•12)

(, ~ ( V-Y, ~ ]),4. \ J !>~-+D1 A -)B .. ~l)" " }

e. \\+ ~ e_ ~ ' - J~}··H):a.) f(_

- 21 -

the ab,.-ve. are general equations where A is at ill a variable.

The a'bo.ve temperature difd!al!'enee ·equations for the centre of the exchanger

i.e. at b when A = ~~ill be

B+D Subatituting o( = -;=::==

)B4 -+b'il..

..... (1-23)

• • • • • (1-24).

The next step is to write expressions for effectiveuess in term.

of capacity rate ratios and temperature difference and then to link the last

two equations.in order to find effectivenass in terms of dimensionless

parameters.

- 22 -

f 1 =The effectiveness of L.H.S. of the split-flow exchanger

= Actual beat transfer rate. Maximum possible beat transfer rate

Maximum possible beat transfer rate = ~min(t 81 - ttl)

:. E 1 ~ Ct (tu. - t ~~ )

C .... w.(ts,- ~t-r)

"" L E., c"""~ L L L.S~ - J...\'\ Cû.. l!>,- Ltl = --------- - b:1

: · ts, - t t 1 :.

1- é1 c""~ Cu.

a1so from (1•25) tt2. =tt\ t (ts, _tt,) E, C~ Ct

Subtracting (1•28) from (1-26) and using (1-27)

- ••••• (1-25)

••••• (1-26)

• • • • • (1 ... 27)

• • • • • (1-28)

- 23 -

l:st-l:tl.- ( h,_ h,) - (t .. - h~ f, C~­

" ( h,. - ~h) ( \ - é:, ('t) . ~- b-2.

1 C"Mk...

• • Cs\ - b-t = - é.' Ct

From equations (l-23) and (1-24) and Figure - 5

Equating (1-29) and (1-30)

1~ ) y -.Y j _ E C \-\.\iM _ ~ - p e. + ( l + ~) e.. 1 1 4- - t -'1

~ +<t:)e_ + (t-oC )e. . y ~

:. E. c\M~ - 1- (1-~)e. + (\+ ~)<L ·l ct - "" y ù +~)e. + (\ - oC)~

:. [, =

y -Y. (-y -~ o<::(e. -e )+~ e.- e.)

( y .. '{) ("" -'1 e. +~ +oé' e.-e.)

:: ----:----~~-

~ + [ l + 'ë_2.i J -2."(

1- e.

q( + ~

[

-"LY I 1-t- e. -1y

1- e.

•.••• (1-29)

••••• (1-30)

- 24 -

B +D Cw.w.. -;::::::== '1(.. --J &' + t?· Ct-

:--------~~----------~~~ -UAtjq -+~

- +- ""'"" + ~ 1(-1{ --\-~ (u U)C· (~ Up;;:fl 24 2.~ 4 Ct- 2 Ct2. CsL.

2

When C = C ~ C = C , then t min sh max

é,-= ----.---::a.~---N~--J-;::\=+=(=~::=.=)~:::a.'

1+(~~) IH

2

1+ (~MW, + c~

-Hru,_J '+ ( ~~ ' -e.

\+ e.

••••• (1-31)

- 25 -

and Ct = Cmax' the resu1t (1-31) remains unchanged.

emin The curves of effectiveness with the parameters ~

max

and NTUmax have been drawn and are avai1ab1e in referenc~ (1~).

- 26 -

II Effectiveness of R.H.S. of Exchanger

This section is a further development of Section I and is based

on similar treatment as before.

Considering the temperature distribution diagram Figure - 6 and

establishing the energy balance and rate equations in this case in a similar

manner as in Chapter I.

dqA. = dqs = dqc + dq~

dq~ = -c hdt s s

d<lc = -Ctdttc = U~A(.otc)

dq p = Ctdttp = U~A(Atp)

Substituting (2-lb) and (2-lc) into (2-1)

dqA = Ct (dttp - dt tc) = U~A(Atp

since dt - dtt = d(tt - ~ ) tp c p tc

+At ) c

and from Figure - 6 b:p - t I:C. :. A tc - At~ bence Ct J. Cô.tc-At~)=~ (ôf:c+At p)

(2-1)

(2-la)

(2-lb)

(2-lc)

..... (2-3)

Now again substituting (2-la), (2-lb) and (2-lc) into (2-1),

-Cu JJs :-Ct Jlt<. + 4 vltt~

or JJ~ ~ ct CJJ:tc. -J.b:~)

Subtracting dttc from both sides of (2-4)

clt~ -dltc. = g_ J.t~:c.- .ft cfrtt> -cll:tc. Cs~.._ Cu.., 1

since dis -dl tc c:. ol (Cs: - ttc.)

••••• (2-4)

••••• (2 .. 5)

- 27 -

R.H.S. OF SPLIT•FLOW EXCHANGER

TEMPERATURE DISTRIBUTION

FIGURE - 6

- 28 -

and from Figure ... 61 t - tt =At s c c

also from equations (2-lb) and (2-lc)

hence substituting the above in (2-5), in a similar manner as (1-6)

Now substracting dt from both sides of equation (2-4) tp

0 • • • 0 { 2co6)

dls- cLtt:r = _s_ oLt tc. - .fL dtt~ _ dl tp Cst. Cu.

since Jls -clttr ~ J. (ts- C~~) and from Figure - 6 ts.- t tf :. At f

also from equation (2-lb) and (2-lc) JJ:: tc. c:- UJ..A Al:c. 2.Ct

and cll tf = ~~ t::.tr

hence substituting these values in (2-7) and simplifying .in a stmilar way

as (1-8)

• • • • • ( 2-8)

u u Substituting B • 2C and D = ~ in (2-3), (.Z-6) and (2-8).,

t .sh

J. (A tc.) J( .6t~) aA - dA ~

(2-3)

• • • • • ( 2-6)

• • • • • ( 2-8)

- 29 -

from (2·6) h.l p = (B;!> ) A~c - t. cA. ~c)

' d_~A(:A p) = ( B_:l)) rL~~) _ ~ ~ ~~~~c) differentiating this ;_: \. JJ ~ ~... ~-

Substituting these two expressions in (2~3) and simplifying

• • . . . (2•9)

In a similar manner (2•8) and (2-3) yield

• • • • • (2~10)

The solution of the two differentiai equations (2·9) and 2·10) is

(-1> +Jf/·~l)l. )A ( -))-jS~+~ )A ~tc.= ks e_ + ~6 (L • • • • • (2-11)

(-l> +J B·~+ff )A (:-J> -J'6 l. +~~ ) lA

6tr = 1<1 e.. + \:::e ~ • • • • • (2-12)

wbare A as in section I is a.ny area 0 to ~·

The constants Ks' K6, ~ and K8 may be determined from tbe

boundary conditions.

When A= 0; from F~gure- 6 and equations (2·11), (2-12) and

(2-6) the fo11owing re1ationships are obtained:

Atc = ts., -tt-\ - K~ + Kf.

At p = tt1 - tt2. • K7 -t K~

(2-13)

(2.;.14)

(2 .. 15)

.. 30 -

Now differentiating (2-11) and eva1uating for A = o,

= ks t:D+jB'~l>' )+ K, ~J):iB~J>'-) ...... (2-16)

A=o

~quating (2-15) and (2-16)

lB-:b )l t~,- rt-t) -1> ( t-s;,_ h-2..) =~s; +~) (:-1>)+6< ~- ~)J"{f.;i

OJ." 1) ( t~\ - h0 ... 1> ( ~~ - b-2.) -= ( ~s - ~)j B 1. -T :rl

• . . Ks -'\(~ = I!>(h:.,- tt:1) -]) ( h,-h-2..) ) B ,_+.I>l

..... (2-17)

... Ks = (t~, - h-v(Jf01..rJI t~) -(t-~·-h-2.)D ~ B'2..·Jd>"l.

(2-18)

k6 - (t~, - h-JU B'+!l' - 'B)- (t~\ -h·2.Jl> ...•. (2~19) - '21-J f3 1:-T l>'

By a simil•at: proeess the other two constants are obtained:

K;z=(~, - h2.)0>S?- - tt,)-:)(t-s.,-th) ..... (2·21)

2 Y-. J 'B"L.-t !)2..

Ka _lh,.-fn)(j B"l.~D:l. +~)+t>(ht-tb) ..... (2• 22)

2. x.} ~2..-+J>:t. Subatituting these v~lues of constants i n (2-11) and (2• 12),

l-J) -+)'B~r>~- )A

- 31 -

su:bst it ut ing

(2-23)

Similarly it can be shown that

• • • • • (2-24)

The above are general expressions of temperature difference where A is

any area () to ~·

These temperature differences at the extrema right hand ~d at

point f where A = ~ can be obtained by substituting

in (2-33)and (2-24) , so that

-+ e. J ] -i1 t

. ,. .. ·:·

- 32 ...

• • • • • (2-Zla)

Similarly

-l>A.t...)l ~ --1 \ f: ~ 1 (Atr \ = ~ e. l 1-t-t +Je. -Ill+ l'+ :r o~i"Y j

from Figure - 6 ~t~).y -= ts~- t~; -: ~ J

In a similar way as (2-23a), it can be shown that

Substituting

)

the equations (2-23a) and (2-24a) can now be written in a more compact form,

L M G=-F .. -J s s

N M H=-J,.-F s s

. ••••• . (2.-2Ja•i)

• • • • • (2-24a-i)

.. 33 ..

As in the last chapter, the next step is to write expressions

for ef'fectiveness in terms of capacity rate ratios and temperature differences

and then to link the last two equations in order to find the effectiveness

in terma of dimensionless parameters.

62 =The effectivenesa of R.H.S. of the split .. flow exchanger.

Actual heat transfer rate • Maximum possible beat transfer rate

Actual beat transfer rate

= csh(tsl- ts3) = ct(<ttl _ .. tto) + (tt3- tt2))

Maximum possible heat transfer rate = C i (t 1 - tt ) mn s o

- Ct l (tt\- tt-~t(t~-~t-2 ~ - (~ ( t~,- ~~) .....

From (2•25) the following relationship is obtained:

(2-25)

••••• (2-26)

• • • • • (2•26a)

Subtracting (2-26a) from (2-26)~

1:,3-t~?> : t"- hl.+ btl- tro- (t-~,-h.) (s.?~ -tf .. ~)

= (t.,-tto)ll-é,_ ~~ -E,_ ~~""1-(~-lt.) ..... <2-27l

.. 34-

Subtracting t from both sidès of (2-26), to

l,~ - bo = ( ~" - t •o) \1 - f.. t:) Dividing (2-27) by (2-27a),

••••• (2-27a)

( \ -t:2

Cw..~ _ éz. C~;..)- tu- tt-\ \ (~ c~ t~, - ho - • • • • • (2-28)

\-El.. c~ Cs.t.,

lt can b.e written, tt2 - ttl = (t81 - ttl) .. (t 81 - tt 2) = F - J ••••• (2-28a)

and (2g27a) can also be written as

ts~- k .. o G

1-E.. ~ = • • • • • (2-27b)

Subst1tut1ng the values of (2-28a) and (2-27b) in right hand side of (2·28),

(2-29)

from (2·24a-1) and (2·23a-1)

N M S J .. S F H ts3 .. tt3 L M =c;=t -t i F - S J s3 ·· to

• • • • • (2-29a)

Equating (2·29) and (2-29a)

• • • • • (2-29b)

- 35 -

Cross multiplying and rearranging (2-29b),

(1-é. ~-:- E, ~-:-il~ F- ~ J)- (F-J) ( 1-E. t:-) = ( 1-t, ~K ~ L ~ ~

or -E.l ~:+Ct)( 1;~=- ~ ~+E,~-:r)t' +~ ~~(~J-~i)

t ( N-S tM)- (L-~+M') ------------------------------------------

.!~I'J ~ M (c~ Cw..:..) lf. ~ ~j c~ (~ ~~ F 11-+ -G +~ + --\ S- -M--l-~-(~ ~ ~...+- J . C~ (~ 4t..4

• • • • • (2-30)

J Now values of F' L, M, N, S are to be converted in terms of

capacities, capacity rate ratios and NTU and then are to be substituted max .

in (2-30), y -'{

. J ~ - tl-'2.. U -~) e + ( \ -+ ~) ~ from ( 1-30) 1 F = = ---=------{-:---------_-::-:yr

Cs, - t ~ (!-+ o() l. + ( \ _ ~) e.

- 36 -

-2Y \+ .e.. ~ -

-2Y L_ ,_ e. F -If

\ + ~ +cf:.. -tv \ - e..

where

1 + 9:._ - c.SL...

~ J \+ (~J

or

Similarly, 13 =

- 1- ~ J l+(~y

eSt.. - i or ~

c:. ~::=;::::====-

}(~} +1

• • • •• (2-30a)

or

u~ = c

t

- 37 -

.----2.

1 + (~)

_u~JI+ (~Î 1- ~ l+e. -

-~}1+ (~J j 1 +(tl ..... (2-30b) • J = ___!_:' -~e.:_--;J===::;:/".=::r;:---~-

• • F _ UAe.. 1 + { ~ \ + G-lt \ C.st.. C.st..

or =

1 + e.. +

-'*'Jh (:iJ J 1 + ( ~J 1 - e..

-~J l+l~t ~- \ 1

+~~.JI+(~)' - j \+ ~l .... , (2-30c) 1 e..

-~~r ~ -t\ c;j'~\~1 c~ 1 + ~ + )----=~, 1 -~~}+ (~]'" 1+ (~)

- 38 -

Now expanding the other values:

'V -Y y -Y "V -'v) L = (1 t f') ~ -t ( \- f') e. : ( e -+ <2. ) -rf ( t - e

- 39 -

N = 't -Y

(1 -l') <L + ( 1 -+ ,P) ~

.:DAt. s = 2 e.

[])A~}~: ~1 ] j ~~ -t\ = 2 e_

•

or =

}f;-tJ 2e.

~c; +G~ =2e..

J Substituting the above values of F' L, M, N, S in terms of

ct ch c: ~, u~ and cts' u~ in (2-30), equations are obtained for62t and c2sh"

sh

These equations (2-31) and (2-Jla) are placed in the appendix (see pages

81 and 82). as .t~ey :are too lengthy to include in the text.

- 40 -

These equations can be simplified somewhat by dividing them by

(eV - ëY) throughout in numerator and in denominator and then putting

ct emin csh emin --- =---- and --- =---- respectively it is finally obtained, c c c c ,

sh max t max

where n = NTU max

t. J 1+ (t;J c

C _ min

r - C max

J f~c .. ~ ... Y -NTUw.._ f-t- \ ~J

x - 1 + e_

Ct C i mn Equation (2-32) applies only when--- = ~. csli cmax

J

• • • • • ( 2-32)

x..-'

c~ + -­c~

- 41 -

csh emin Similarly when C = -C-, the second equation gives:

t max

y--t- 'i +-r- y-+-r- ~ + T 1 + t ~ c ... 21?..\ll-t) '1 t'{.. c, 2 ~'~-~) 1 l ~ + t

E2~~------~----------------------~~--------------" ~'\--t\ ~+a y T

~+x '1 t

1.~ _ ~~ + ; + _Y_-t_T_)L __ 1 ?.. ~ ., \ J t x + ~ "1

'1 1 • • • • • (2-33)

For the special case of isothermal shell-side fluid i.e. for

c ct ~in = 0 = ~~ equation (2-32) reduces to max .sb

:NTU~) c.. ••••• (2-34)

while (1-31) for same condition gives

(2-35)

The temperature diagram for the complete exchanger for such a

case will be as in Figure- 7.

Taking the second special case of isothermal tube-side fluid

i.e. Cm.n Cab

for r. _ = 0 = c:-' equation (1-31) gives the same result as (2~33) -max t

-HTU~

i.e. E. 1 == t.~SJ,.._ = 1 - e..

The temperature distribution diagram for this case for the

complete exchanger will be as in Figure - 8.

- 42 -

SPLIT-FLOW EXCHANGER

t tsl t 82~--------~--------~------------~------, s3

----r--jt3

TEMPERATURE DISTRlBUTION

FOR

ISOTHERMAL SHELL-SIDE FLUID, Ct = 0 cs

FIGURE - 7

to

- 43 -

SPLIT-FLOW EXCHANGER

s3

ttrr-------------------~------------------~tt3 tto

TEMPERATURE DISTRIBUTION

FOR c

ISOTHERMAL TIJBE·SIDE PLUID, ·~. = 0 ct

FIGURE - 8

- 44-

III - Overall Effectiveness of the Split-Flow Exchanger

Refer to Figure - 4 for the analysis in this section.

There are fiv.e distinct possibilities of capacity rate ratios in

complete exchanger.

Possibility 1:

Cs > ct

csh / ct

a) ~total based on shell-side flow,

C, ( ~.~- t,.,. i (o,) t.t =

G: (Es, - tto)

;:::.

=

:: é2.t + [, ts..- tt., E~, .... t-,.o

<4.. ( ts., - tr.l.) 4 (~,-ho)

b) t total based on tube-side flow,

tt = 4(t~- tt-o) Ct (~~-ho}

• • • • • (3-1)

• • • • • (3-2)

• • • • • ( 3-3)

- 45 -

&_ _ Ct_ (t., -lt>)+(h,-l:.to)+ ( t~1- tt,)} Ct tb,- ttc> J

c c t\1 - t~, '=-l.'lt. +c.,

t!.~ - b·o (as before) .•••• (3-4)

Possibility 2:

Csh- Ct

a) <: total based on shell-side flow,

c. ( t,., - t.,. ~ 1-.3 ) Et =

Ct ltSl- t~o) ..... (3-5)

=

4Ll l:s,- ts1) Cg__ l ts, - ts1) = Ct l ~~ - b:o) + C.t-lrs.,- h-o)

..... (3-6)

b) E total based on tube-side flow,

..... (3··1)

(as before) (3-8)

- 46 -

the above equations (3-2), (3-4) and (3-8) apply for any value of

or ct c = 0- .5

8

Possibility 3

a) E total based on shell-side flow,

. ~ l ts, - h.,; Cs.1)

C~ lts,- t~) z:s =

b) t total based on tub~-side flow,

Es = Ct:(tt~- Cro) (~ (h, - b.fl)

• • • • • (3-9)

• • • • • (3-10)

• • • • • (3-11)

• • • • • (3-12)

- 47 -

The expressions (3-10) and (3-12) will apply in the limits

or

Possibility 4:

c = c s t

a) [total based on shell-side flow with Cs

Cs (t .. - b-;_ t.~} ~ ( t~, - tt:c)

~ 4-( t .. - t.,_~(.,,) - 2 est. Cr~, _th)

. 1 l4J.. ( t:., - h.1.) ~ ( k~, - l:Sl.) 1 = 2 est.( tr.,- h·o) + Cse.lt~, -h(lj

in denominator,

01001 (3-13)

• 0 0 0 0 (3-14)

- 48 -

b) é total based on tube-s ide flow with C8 in denominater,

==

ct( h-~ - h-o) Cs ( h.t- ho J ••••• (3-15)

••••• (3-16)

Now redoing the above by taking Ct in the denominator.

a) 6 total based on shell-side flow, with ct in denominator,

c.sl~<l- ts,~ t .. ) Ct (t~\- b-o)

( ( L _ t!>,±t,ll

2 ~\\.SI 2 Ï 4l~~\ -h-o)

Cg__ l ts., - Cs.?> ) ~ l t~, - h . .,_) 4 lh.,- ho) + 4(1:-..,,- b-o)

••••• (3-17)

• • • • • (3-18)

The expressions on the R.H.S. of (3-18) cannot be formed into

effectiveness since Ct is not Cmfn for the individual exchangers. However,

Since 2C8 h is equal to Ct, substituting it in the denominatàr of the above

expression gives:

- 49 -

• • • • • (3-19)

which is the same as equation (3-16).

b) é total based on tube-side flow, with Ct in denominator,

The expressions on the R.H.S. of (3-20) cannot be formed into

effectiveness , however, by substituting 2Csh for Ct as before gives:

.E = c{ ll-t• -l:n) + ln., -n.)} + 4U ... - b-,) s Z Cst..l b.., - tt-t>) '2-(st.l k:s.\ -ho)

~-rf ~( (ln - tb) -t( f:t,_ tto) -+

l eSt.. ( t~\ ~h-o)

which is the same as equation (3-16) •

• • • • • (3-21)

The expressions (3-14) to (3-21) will apply in the limits

or

Possibility 5:

cs/ ct

but csh ..(ct

- 50 -

a) é total based on shell-side flow,

c~ l ts, - ts.). :t~~) Et= -----:-----::-

Ct- l t~,- tt-c) • • • • • (3-22)

=

• • • • • (3-23)

Since Ct is not emin for individual sections of the heat exchanger,

therefore, the expression (3-23) does .not represent the sum of the individual

effectivenesses of the exchangers.

The required result can he obtained by multiplying (3-23) thrpugh-

out in the denominator and numerator by Csh'

c~ c~t. . .C t~,- t ~~ ) = Cst._ ')(. (~ ( k-s,- ho)

Cst. -c~

~ c~Ct~- bol.) t Cg_ >< Ct ltS\- tt-6)

••••• P-24)

-51. -

b) é total based on tube-aide flow,

[_ . = Ct (tt~- b-o) t Lt- ( ts, - tto )

• • • • • (3-25)

Ct t(h-~ -tt'l-)+(b-,- ho)} Ct( tt-l.- f:~) = + -----

4( t~,- tM~) Ct ltcra,- t~:o) • • • • • (3-26)

This again is not applicable, but by a similar process as for

(3-23), multiplying the denominator and numerator throughout by C8h, gives

us,

• • • • • (3-27)

which is the same as equation (3-24).

The expressions (3-24) and (3-27) being applicable in the limit

csh - = 0.5 - 1. Ct

- 52 -

It will be interesting to note that in case of

or

= 1, the expressions (3-24) and (3-27)

reduce to

l.. [ ~ l:s.\ - b, .. , \ Es = 2 C..2.st.._

4 é., fs, - b.-o j which is the same as for

Possibility 4, and confirms equations (3-14), (3-16), (3-19) and (3-21).

Confirming (3-24) and (3-27) for the other limit,

Csh ct = 1,

for which the same e~ressions reduce to

which is the same as for

Po_ssibility 2, and confirma equations (3-6) and (3-8).

In all the foregoing expressions .in this article, for the total

tsl - ttl effectiveness of the split-flow exchanger, a common factor . t is

tsl - to encountered which must be expressed in terms of dimensionless parameters

emin r.--- and NTUmax, in order that a complete expression in terms of these îllax

dimensionless quantities may pe obtained.

:DA 2G e... from (2-23a)

from (2-27b)

- 53 -

"' Ct )c;+ct ( c~)

z e. \ ' - E.'Z. <4. ==-----__;_--------~~ . . . . . (3-28)

Dividing (3-28) by ey in the numerator and denominator,

=

- 54 -

J In equation (3-29) the value ofF will be (2-30b), while in equation

J (3-30) the value ofF will ' be (2-30c).

tsl - ttl Before substituting these values of in the equations of

tsl - tto overall effectivenesses, the various possibilities represented by (3-2) to

(3-27) may be summarized~

Ct for --- = 0 - 1,

csh

csh for --- = 0 - .5

ct

csh for --- = 0. 5 - 1

ct

• • • • • (3-31)

••••• (3-32)

• • • • • (3-33)

The equation (3-33) shall not be used since in .order to calculate

r:8

, equation (3-32) is sufficient.

A very vital point to be noted in (3-31) and (3-32) is that while

calculating the total weighted effectivenesses, in case of (3-31) the values

of [ 2t and f:1 are to be calculated with half of the total area of the split­

flow exchanger, i.e. with Ah in NTUmax. On the other band, in case of (3-32),

the values of[ 2sh andE 1 are to be calculated with total area of the split­

flow exchanger, i.e. using A in NTUmax•

Bearing in mind the above, the equations (3-34) and (3-35) for the

total effectiveness can be written as given in the next pages.

.v

ct _ Catin.. for Csh - Cmax and where NTUmax =

UAh ~in (also in E2t and El)

tKTU-( ~~ -}+ (~n ( 1 _ E1t ~~) ~ ~ é, 2 e_ -KTIJ~I+ ('d tl-= élt +· ---

-tm~_Jt+(~J 1 + e..

-IITU-Ji~ \-e. +-

}tr---+ (-t:-T 1-

~ \- c~

t+e. . ,_ -} (~\~ -ffrU~t+ (~::) h -~- x~

1 ~ ~

-N~I+ (ê.,_j 1 ~ +-(~ 1+ e....

-~;::::=::=:==---- + --

-NTYI+ (t:l } +(~ '- Q_

(3-34)

VI VI

csh emin for-= r. -ct -max

c é:2Sl.. + cs:c:-~

·~·

and where NTUmax "" ~n (also in[ZSh andE 1)

+ rrry,_( 1-)1+ (~J)

E., e_

-liTt)._}+ (~S -HT~JI+(~ t-e..

~+Q.. +-----

\+ (é)~

( \- é2.~

c~ -c~

J rc~r -tmJ~ 1+\~ ~ -1 ~

1+ e. c.,.;..\ ... -J tC....\ -IITU-d 1+ (c....J 1+ IJ;.:}

1-!L

~l+(~f ~+1 c~

'+L +--

. -N~t+(§f )+(?J 1-~

•••••••••• (3-3S)

VI o-

- 57 -

The values of effectiveness have been found out for various

values of NTUmax and Cmin for (l-31), (2-32), (3-34) and (3-35). These Cm a x

values were obtained with the help of Fortran programmes, run in McGill

University's Computing Centre, and the effectiveness curves are plotted.

(Pages 78 and 79). Cmin From the curves of effectiveness we note that for ---- = 0 1 ~x

the effectiveness of the reverse flow exchanger is the same as that of

the split-flow exchanger at any given value of NTUmax· For all other

Cm in values of r.---1 the effectiveness is greater for reverse flow exchanger

"1JlaX

than for split-flow exchanger and this becomes quite marked beyond NTUmax

of about 2, .

In the split-flow exchanger itself, the values of effectiveness

Cs for-= 0

Ct Ct

and Cs = 0 at all values of NTUmax· The same is are the same

Cs true for -ct

Ct = 1 and C = 1 at all values of NTUmax•

s However, for any

other value of capacity rate ratios the effectiveness of a split-flow

exchanger is higher for

Ct value of C and NTUmax •

s values of ~ greater

Cs any value of - and NTUmax than for the same ct This difference becomes quite pronounced for

than about 3.

From the above it can be concluded that from the point of view

of thermal effectiveness, the lawer capacity fiuid should be placed on

the shell side of the split-flow exchanger.

- · 58 -

IV - Petermin~tion of Correction Faètor Ft•

Having found the final thermal effectiveness êt and Es, it is now

quite simple to calculate the Ft from basic rate equations.

q = UA Ft L.M.T.D. (counter-flow)

= UA F t (!:s,- tr~)-( h4- b:o)

l ts,- tt~ tsLt- tto

where t ~ ;- ts~ 9.4 = 7.

...... (4-1)

(4-2)

• • • • • ( 4-3)

• • • • • (4-4)

:. from the above four equations,. it can be derived that

1-RP UA ln ï:'P

= Ct Ft (1-R) • • • • • ( 4-5)

• • • • • ( 4-6)

and

..... (4-7)

- 59 ..

bence substituting (4-2), (4-6) and (4-7) in (4-5),

• • • • • ( 4~8)

For the case when Ct = Cmin and Cs = Cma:x

• • • • • ( 4-9)

For the case when Cs = ~in and Ct = ~ax

((~ ) NrU --\ 'h\.M(. c~

emin when -.-- = 1, the ab.ove two equations become

. cmax

f '-t

••••• (4-11)

- 60 -

Values of Ft in both the cases have been computed and curves

f F h b d ith NTU b i .and Cmin as the third o t ave een rawn w max as a sc ssa Cmax

parameter. (Page 80)

- 61 -

V - Experimental Equipment

G~ne;al

A split-flow exchanger of ·single shell and 4 tubeet .2-pass .design

was built and set up in the research laboratory. of the Mechanical De~art~ent • . ·

It was decided to have air in the shell-side a.a well aa the -tube-

side of the exchanger. The maximum ~ir supply c.apacity of the ·:laboratory'a

compressors is about 1,000 lbs/hr. With this amount of air supply1 sufficiently

high values of the overall beat transfer coefficient U ·could be obtained with

a .. shell ·aiZe of 1 3/4 inch diameter .( 16 BWG),and four 1/2 inch , \20 :st<m)tubes

inside the shell as shown in the cross-section Figure - 9. -To provide

suffi.cient beat transfer area (to obtain a high value .of NTUmax ~ ~n) .the,

exchanger was made 12 feet long.

Exchanger Construction

The exchanger was made from braaa tubes and brasa parts to . faci~it•te

machining and soldering.

Baffles:

40tfo eut segmental baffles from .1/1.6'' thic:k s\leet were rQ&de. The

baffles were spaced 6 inches apart, held securely .by means of baffle spacers

which consist of 3/16" thick thr.ough-b.olts serewed into the tube sbeet and

6" long 3/16" diameter 21 :SWG spacer tubes •

. Air System

The main air was .supplied .at about 100 lbs/S.q." gauge pressure.

Ai r Filter:

In order t~ pnrify the air from atmospheric dirt, rust carried from

old pipe-lines and moisture, an air filter 1 foot diameter and 2 1/2 feet long

packed with ~otton-waste was f itted in the main line.

- 62 -

SPLIT-FLOW BEAT EXCHANGER

1 3/4" diameter, 16 BWG Shell

- 1/2" diameter, 20 BWG Tubes

Equally spaced on 7/8" Pitch Circ1e

Cross-section of She11 and Tube Arrangement

FIGURE - 9

- 63 -

Air Heater:

The air heater was made out of 4 inch diameter, 5 feet long

standard pipe. Ten tubular heating elements of 750 watts each, cQnnected

in parallel with a terminal voltage of 120 volts D.C. were fitted in the

pipe. This total capacity of 7.5 kw was sufficient to raise the temperature

of 600 lbs/br of air through about 200 °F.

Hot air was run in the tube-side of the exchanger.

Air metering:

The:outlets of both the shell aides were connected to a single pipe

and led to exhaust through 3" diameter 6 feet long standard pipe •. At the

centre of this exhaust pipe sharp-edged orifice with D and D/2 tappings was

made. Orifice plates of sizes 3/4", 1", 1 1/4",_ 1 1/2" and 1 3/4" diameter.

were made to insert the suitable aize to give a reasonable difference of

water column in the manometers depending upon the flow rate.

For tube-side, a similar arrangement as above was made for the

flow measurement.

The orifiees were made according to British Standard Code

B.S.l042: 1943, and they were not calib~ated.

A line diagram of the flow system is shawn in Figure - 10.

Temperature Measurement

The temperatures were measured by 19 gauge iron-constantan thermo­

couples connected to a precision potentiometer. The thermocouples were

calibrated in ice-cold water as well as boiling water and a curve was pre­

pared for each thermocouple to effect adjustment:.

- 64 -

A line diagram of the apparatus indicating the thermocouple

stations is shown in Figure - 11.

Insulation

The exchanger was insulated with lagging material of about

0 0.028 BTU/hr. sq.ft F/ft. thermal conductivity.

Ge~eral views of the beat exchanger set up are shown in

Figures 12 and 13.

SPLIT-FLOW HEAT EXCHANGER

AIR FLOW ARRANGEMENT

7.5 kW Tube-Side Air Heater

/Tube Inlet +

Air Main

Tube-Side Gate Valve

+ 120 V D.C. Supply

Shell Out let ____,

-=-===--=----===-== ===-::: =-=-==--=--= =-~ -=-=-..: = =-:.-::::---~= .=.=.:.:;o Orifice Shell Inlets ~~ ~4 _ l/20 20 BWG Tubes Sliding Tube E

3/40, 16 BWG Shell 1

ff) 1

1

/

\Tube Out let

Tube Flow Control Valves

FIGURE - 10

Air Filtér

Main Air Gate Valve

t

"' Ut

SPLIT-FLOW HE.AT EXCHANGER

16

8

.5

4 5

~~~------------------~~~----------------~~J~l2

LOCATION AND NUMBERING .OF THERMOCOUPLES

FIGURE - 11

0\ 0\

- 67 -

SPLIT-FLOW HEAT EXCHANGER

Precision potentiometer for thermocouples

Carbon resistance control for the air heater

Voltmeter and

Manometer Board for orifices and pitot tubes.

Shell-side flow control valves

Tube-side flow control valves

CONTROL PANEL

FIGURE - 12

- 68 -

SPLIT-FLOW HEAT EXCHANGER

air heater

in let

TUBE-SIDE INLET AND OUTLET

FIGURE - 13

an er

carbon resistance control for heater

out let

" .

3.

- 69 -

VI - Precautions

The following main precautions were observed in running the tests:

1. The main air supply was from three compressors, two of 250 lbs/hr.

eacp and one of 500 lbs/hr. capacity. All the three compressors

when switched on are controlled bY aut.omatic switches adjusted for

100 lbs/sq.in. gauge pressure of the receiver tank.. Unless full

air is utilised, due to off-and-on of the compressors, the air

supply is fluctuating. Therefore, a bleed valve was fitted on the

line and adjusted in such a way that full supply i .s drawn and all

the three compressor.s remain operating.

2. In order to have equal flow in bath aides of the shell, _one valv.e

in each outlet was fitted. Pitot tubes were fitted 6" before the

valves. Pitot probes were made to determine the flow in each half

and thereby establish equal flow in both halves of the shell by

regulating it with the valves.

However, since the traverse was over a short length of 1.049" (the

inside diameter of l" standard pipe) and the traverse being manually

operated, an accuracy of not more than about 5% could be achieved in

equally dividing the shell-side flow.

The temperature values required in the exchanger were for the bulk,

therefore, the sensing elements of the thermocouples were made large

enough to c_over t'he full stream. At sorne stations (3-4, 5-6,

17-18, 19·20) two thermocouples were fitted to give a more average

value of the temperature.

- 70 -

At the centre of the tube lengths 1where there could b~ maximum

variation in temperature across the cross-section, a special

c.onstruction of the form shown below was made to create turbu-

lence in the fluid in order to measure the average temperature

of the fluid.

small dive plate

diameter tub.e

FIGURE - 14

4. Each set of observations wàs made after steady-state conditions

had reached about two hours.

- 71 -

VII - Test Resulta

Following are the sample results of a number of tests performed.

c min/C ; max Effectiveness Effectiveness

No. Ct/Cs cs/ct NTUmax Experimental Theoretical deviation

1 0.42 2.51 o. 742 0.745 0.40i

2 0.36 2.·97 o. 753 0.795 5.30

3 0.354 2.76 0.756 0.79 4.20

4 0.428 2.65 o. 730 0.745 2.00 '

5 0.560 i 2.24 0.670 0.690 2.90

6 0.91 1.63 0.580 0.555 4.5

7 o. 708 1.76 0.648 0.615 5.37

' 8 0.432 2.25 ' 0.768 0.730 5.20

9 0.288 3.37 o. 790 0.825 4.25

10 0.345 3.04 0.772 0.80 3.50

11 0.386 2.84 o. 750 0.775 3.225

12 0.460 2.55 o. 717 0.735 2.45

13 0.462 2.70 0.694 0.740 6.22

14 ' 0.326 2.34 0.817 o. 78 4.75

15 0.825 1.98 0 .• 600 0.602 0.33

- 72 -

Samp1e Ca1culation

Following is a typical data of one of the runs with air flow:

Shell side flow W8 = 314 lbs/hr.

C8

= W8 x Cp = 314 x .24 = 75.5 BTU/hr °F

Tube aide flow Wt = 381 lbs/hr.

- 0 ct = wt x cP = 381 x .24 = 91.5 BTU/hr F.

Temperatures at various cardinal points are as indicated below:

241.5

184.5

175.8·----- 163.4 164

He at Balance: FIGURE - l5

Shell side: q = Cs (ts4 - tsl)

= 75.5 (184.5 ; 164. - 74.3)

= 7550 BTU/hr.

Tube si de: q = ct (tt3 - tto)

= 91.5 (241.5 - 163 .4)

= 7140 BTU/hr

... 73 -

Percentage difference in heat balance = 5.3.

Heat transfer area of the exchanger = 6.04 sq.ft.

Overall heat transfer coefficient U:

The overa11 heat transfer coefficient in both ba1ves of the

exchanger is assumed to be the same. U for the 1eft band side can be

calculated aa fo11ows:

NTU : max

Log-mean temperature difference = 57 .6

p tube side temperature difference 36.8 = total temperature difference = 124.7 = •296

R = shell side temperature difference 89.7 2 46 tube side temperature difference = 36.8 = •

Ft (1-2 exchanger) = .79

u = csh (ts2 - tsl> .LMTD x Ft x Ah

314/2 x.2.4 (164 ~ 74.3) 57.6 x .79 x 3.02

= 24.5 BTU/hr ft 2 °F.

= 314 x .24 = 75.5 BTU/hr °F

Ct = 381 x .24 = 91.5 BTU/hr °F

Cmin = 75.5 BTU/hr °F

emin c · max

75.5 = 9ï':"5 = .825

UA =--emin

24.5 x 6.04 = 75.5

= 1.985

Effectiveness:

actual heat transferred = maximum possible heat transferred

= 75.5 184.7 ; 73.3 - 23.5}

75.5 (116.3 - 23.5)

= 0.60

- 74 -

VIII - Discussions and Concl~sions

From the curves of effectivenes.s Figure - 19 and Figure - 20,

the following observations are made:

1. The effectiveness of a split-flow exchanger is practically the

same a.s that for a reverse-flow exchanger for values of NTUmax

below about 2. However, for values of NTU a greater than about mx

2, the effectiveness of a split-flow exchanger is less than that

of. a reverse flow exchanger.

2. In case when one fluid is isothermal, whether it is in the tube-

side or in the shell-side the effectiveness of a split-flow

exchanger is the same a.s that of a reverse-flow exchanger or a

pure counter-flow exchanger.

3. In a split-flow exchanger, for NTU more than about 2.5, the max

effectiveness is greater when the smaller capacity fluid is i .n

the .shell and greater eapacity in the tubè-side. This is a very

useful rèsult since most_of the split-flow exchangers are liquid-

gas excpangers, and the shell-side always carries the gas . for

pressure drop considerations and fortunately this arrangement

givea .a better ~hermal effeetiveneas as well.

4. Thermodynamically, a split-flow exchanger gives highest effective-

ness in the region of NTU between about 3 and 5 and decreases max

beyond that. The reason for this is .as follows:

- 75 -

Consider the temperature distribution as shown in Figure - 16.

FIGURE - 16

The ultimate temperature difference is tsl - tto• The maximum heat

exchange takes place in sections of greatest temperature difference. Consider­

ing the tube side, the highest heat exchange is in Section I and lowest in

Section IV.

If the beat transfer area of the beat exchanger be increased beyond

a certain value, thereby increasing NTUmax' most of the heat energy having been

exchanged in Sections I, II and III, the Section IV might actually be cooled

instead of being heated particularly if there is a temperature cross as shown

in Figure- 17. This means that the additional beat transfer area

FIGURE - 17 tto

in Section IV bas actually reduced the total amount of beat transferred thereby

reducing the overall effectiveness.

5. The performance of a split-flow exchanger is not altered by reversing

the flow directions or by causing the shell to carry the cold stream

and the tubes to carry the hot stream. This is due to the fact that

the effectiveness is a function of the exchanger configuration, ~ax'

- 76 -

Cmin Cmax' and reversing the fluid direction& .. and causing either of the

shell or tube-side to carry hot or cold stream does not change any

of the variables and therefore cannat alter the effectiveness.

From the experimental resulta, the the.oretic.al curves of effectiveness

+ ti. have been checked and found correct with an average of about - 3.510 error. This

confirmation, however, is true only for values of NTUmax below about 3. With

air-flow, NTUmax values of greater than about 3.5 could not be obtained due t.o

small overall heat tranafer coefficiento It is therefore recommended that .with

the same exchanser water-water combinat ion should b.e tried. For thi.s certain

alterations will be necessary e.g. iron-constantan thermocouples will have to

be changed to avoid corrosion or the thermocoup-le tips will have to be protected

from contact with water, a water heater will be required etc.

The Correction Factor Ft

The graph of correction facto.r Ft 1 Figure - 21 1 shows. that for the

same values of Rand P, there is very little difference, "ab.out 1/2~ or less,

between a split-flow exchanger and a reverse-flow exchanger for values of Ft

greater than about 0.8. This, of course, is due to the fact that there is

practically no difference in effectiveness between the two exchangers for NTUmax

values of le.ss than about 2.

Further Developments

1. In this analysis, a two tube pass exchanger hàs been deal~.witb. · -It is

possible to extend the w.ork to cover 4 or more tube passes.

2. A divided-flow exchang~r with two or more tube passes can also be

ana~ysed in a similar way as the split-flow exchanger.

FIGURE - 18

1.

2.

3.

4.

5.

6.

Nagle, W.M.

Underwood, A.J.V.

Bowman, R.A.

ftusselt, W.

Nusselt, W.

Smith, D.M.

- 77 -

References

Ind. Eng. Chem. Vol. 25 1 p. 604 (1933)

J. Inst. Petroleum Tech. Vol. 20, p. 145 (1934)

Ind. Eng. Chem. Vol. 28, p. 541 (1936)

Zeitschrift des Vereines deutscher lnginieur

Vol. 55, p. 2021 (1911)

Technische Mechanik und Thermodynamik Vol •. l,

p. 417 (1930)

Engineering Vol. 1381 p. 479 and p. 606 (1934)

7. Bowman, R.A.; A.C .• Mueller and W .M. Nagle

8.

9.

10.

Kreith, F.

T·en Broeck, H.

Trans. Am. Soc. Mech. Engrs. Vol. 62, p. 283 (1940)

Principles of Heat Transfer, p. 453

International Text Book Company, Scranton

Ind. Eng. Chem. Vol. 30, p. 1041 (1938)

(1960)

Kays, W.M. and A.L. London

Compact Heat Exchangers, p. 7

McGraw-Hill Book Company, N.Y. (1958)

11. Kern, D.Q. and C:~L. Carpenter

Chem. Eng. Progress Vol 47, p. 211 (1951)

12. Schindler; D.L. and H.T. Bates

13. Wright, c.e.

Chem. Eng. Progress Symposium Series Vol. 561

p. 203 (1960)

Para11e1-counter flow shell-tube beat exchangers.

Pe~formance in terms of effectiveness, capacity

rate ratio and number of beat transf er units 1

unpublished Stanford University Mechanical

Engineering report (1954)

0.7

0.6 'fil. l.tJ • • u

~ -::!0.5 u u ... ... ~

0

0.3

o·.2

0

0.1

- 78 -

Thermal Effectiveness

SPLIT-FLOW EXCHANGER

NTUmax

Cs -Ct

Ct --- c.

No. of Transfer Unite, NTUmax = AU/Cmin

UA a Catin

Cm in = c_-;

Cmin c Snax

t/<.0.6

UJ • • ~ 1>

:0.5 u

" ... ... ...

0.4

1

1'

.,

! 1.1

1;· ,,

1 1 ;

1 l' . 1'

- 79 -

,,

•l·

' 1 •1.11 ' ~ : :11:1;: ~::. ;~

,:

1 1 1 ,

Il 1 1

1 ,. .

Ill 11- IHI Il Ill

Thermal Effectiveneel

SPLIT-FLOW IXCHANGIR v1. REVERSI-FLOW EXCHANGER

l C1 C...in

.,,,,_,,,. ......... --------- :: -:::: c. • Ciaax

c. ct Cain ! Reveree-Flow Excbanaer • • • • • • • • • • Ct • Ce • Cux

1.

l-1 o. 0 .u (J

~

""' Q) (J

s:: Q)

~o. ~ ~ ..-1 ~

Q)

l-1 :l ., Qj

~o. ~ Q)

E-4

"' .u

""' o.

o.

"

~ =""-- ,.. "" ~ l'.:'

9 rnrn~~~*ffimR~~~~ffmmm ...

~,rnrn~~~~ttm~8~~#m~ttm ~œrnmm~#m~mH~lliW~#m~mm

1 ~ ~

7 ' ' ' l

_l j_,

: 1 1 1 ' j_ · - -~ i ' 1 - 0

1 ' ' 1 • ' ' .... . _.~ 1 ..... .o

t-+-+-1-~1-U-'-"' H o a fd.-_ ~ , • ~ 1.0 _ ~

6 1

5·.·--~··· 0 0.1 0.2 0.4 0.6 0.3 0.5

p

0.7 0.8

~T~! IXl'D CORRECTION . FACTOR, Ft

J SPLIT-FLOW EXCHANGER

~ -t2

~1 Tl - T2 t2 - tl

' T2 = ts2 + ts3 R = p = ta3 t2 - tl Tl - tl

t•2 2

FIGURE - 21

0.9 1.0

j

(X) 0

_u~(c .. )2 q'+tc;_ 1 + e..

~=-­_UA~Ç2l' c:-)2

(~ ] l+l~

j_~

. . ,· :~··?~-~-· . . . . -.:· .. ~

., .'

. ;

•

l- c~ ( Cst. !

) \...

------ ---------·- -·--· --------- - --

y -Y C (e_ - a_) ~

f+ .. .

' ( .. + ~tJ 1

l

J ·-------

_UAL Ç(S!:)i. t4 _ t cs,t.. P;- '4 c~

1-t-e... ___ - --

-UA~ 1-+ (Cs..t )-:;_ J l(<t)z. ~ c.. 1 + \C.~

1- Q.

-fuJI+ fC~t...)L Gt.. c~.._ \ct ( +1

e_ t 1 +- ------- + -==

.J!&_ 1 CSl-.)7. r. j '_Q_ c"' '+ (c,:- J 1 +~?:

f. = --- ------------------- ------;!_ - uA._I + (~j cs.(

c.~l c~ - -1 Ct-

\+e.. --- - --

-~Fl~) }~cii:l ·-~

•

!

y

~ - -· ... ----~---- --- .. ------------- .. ----

)

:. --. ~ ~:· - ~ ' .· ~,. ~ ~~ ,,. .,.