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AoRoEoAoEoBo/Rep c-144

ARAB REPUBLIC OF ESfPT

ATOMIC ENERGY ESTABLISHMENT

REACTORS DEPARTMENT

THERMAL DESIGN OF HEAT EXCHANGERS

HAVING NON-B0UAL TUBE PASSES

MoFo EL-FOULY

AoFo EL-SAIEDI

MoAo BARWISH

1972

SCIENTIFIC INFORMATION DIVISION

ATOMIC ENERGY POST OFFICE

vAIiiOy AoHoEo

C O N T E N T S

P a g e

A B S T R A C T „<> „ > <> o o 0 „ o o „

M w A U U E l v X j i l > I U i V * - 4 o o o o o < » o o o o o o o o « o o o o o o o g o o o o o o o o o o o o o o o o o o o o o o o o o o o o A J L

i

INTRODUCTION

I C U U U J L C I o o o o o o o o d o o o o o o o o o o o o o u u o o o o o o o o u u o o o o o o o o u o o o o o u o J o o o u o o Tt

KjWtbUJLiO A H I / 1/J.O vUd&> X U N o o o o o o o o o o o u o o o o o o o o u o j v o o o o u o o u o o i o o u o o o '- •"•

R E F E R E N C E S 0 0 o o 20

eezj J . usa

A B S T R A C T

This is a study on single shell, two tufoe^paesp parallel «©muter

flow heat exchangers having non-equal tube passes0 It aims at deriving

relations between the heat ex©hanger effectiveness, its number of trans­

fer units9 the hot and eold fluids heat eapatxty ratios and thi. pair?ert

age of the first pass 8sros6«~se*tional area to the total mrea of the heat

exchanger» Figures showing such rtel&iions are presented*, Thfes® figures

would be quite usefwl in design opti»iff&ticm of thus type of heat

exchangers0

- ii -

MOMEfrOIAQlUBE

2

A heat transfer area, ft

c specific heat of cold fluid, btu/.lb-°F

C . Specific heat of hot fluid btu/lb~°F P, correction factor defined by equation (1) in

the textj dimensionless

L length along heat exchanger, ft-

LMTD logarithmic mean temperature difference, °P

m flow rate of cold fluid, Ib/hr

M mass flow rate of hot fluid 9 Ib/hr NTU number of transfer units defined by equation (2)

in the test

R heat capacity ratio, dimeusionless

t» temperature of cold fluid in the first pass,?0?

t" temperabur? of the cold fluid in the second pass,0]?

t. temperature of cold fluid at tne end of the first

pass^°F

T temperature of the hot f lu id at any cross-sec t ion ,

o U overall heat transfer coefficient, btu/ft =*hr~°$ x ratio of the first pass tube area to the total

heat exchange area, dimensionless•

0 heat capacity, btu/hr - 0T?

INTRODUCTION

Heat exchangers in nuclear reactor systems and power

plants are considered of the most important constructional

parts* This is due to the fact that heat removal from the

reactor primary system is one of the deciding factors of

reactor power limit* Thus designing an optimum heat exchan­

ger is vital in such systems and accordingly reliable and di­

rect methods would help reach that design*

The known ways for the process design of a heat exchan­

ger are the transfer unit method described by Kays and London^ ̂ (2) and the conventional method described by Kern^ J. The former

method has several advantages over the latter. For example,

in the conventional method, it is necessary to know the outlet

process temperature tp and Tp of the cold and hot fluids res­

pectively ̂in order to calculate the logarithmic mean tempera­

ture difference (IMTlO* The total heat exchanged in this case

is .expressed by

Q a UAFt, (HIED) (1)

Variable's in this equation and others are defined in the nome-

tilature* For "the transfer unit method, the knowledge of such

temperatures is not required. Besides, it is direct and the

heat exchanger performance is easier to predict*

«•» £ mm

iThe transfer unit method is based on a mathematical

relation-ship between number of defined parameters, namely,

KTU number of transfer units, £ effectivness, and fi heat

capacity ratio* These terms are defined as*

MU = 2L&L (2) I *min

Where #-„ is mc if mc < MC

or MC if MC < mc

fi = -SiS (3)

^max

mc as — •

MO if mc < MC

MC if MC < m* mc

and £ = — 3 — = S (4) Lax ^min^l""*!^

for mc < MC *i -*i T - IP

» -i J=- for MC < mc

The most common type of heat exchangers is the 1-2

parallel counter fl©Wt shell and tube heat exchanger (con­

sisting of a number of tubes in parallel enclosed in a

mm "y mm

©ylinderical shell) # The derivation for the value of 3L In

equation (1) is presented by Kenr 2' and based on thi fork

of underwood^',Nagle^4' and Browman et al'^ for 1-2 shell

and tube heat exchangers of two equal tube passes* For the (2)

same case referencev ' presents a figure representing the

relation between the effectiveness and number of tranafer

units for different heat capacity ratios* A very practical

case y for one shell and two tube passes, is that the passes

do not have equal surfaces« A number of tubes are usually

ommitted in the region close to the shell side inlet and out-

let nozzles to ensure good flow paterns • Tubes are also

omitted in the central row9 or the rows near the center, for

two pass arrangement. Sometimes9 the centre of the rows is

offsetting the centre of the shell and this leads to unequal

number of tubes in each pass* The problem is first mentioned

by Kerav '• 0

t

This report considers the case fer 1-2 shell-tube ex-

©hangers of non-equal'tube passes* Here is presented a ma­

thematical relation-ship for the effectiveness 6 , the num­

ber of transfer units HTU| heat capacity ratio £ as defined

earlier, and the percentage of the first pass area to the to­

tal area of the excanger x* figure illustrations of these

aathematical relation-ships are presented*

Theory

Fig 18 Temperature distribution

in the 1~& shell-tube

heat exchanger*

To simplify the analysis a number of assumptions are

useds-

1- The ratio of first pass area to the total area is equal

to x (constant)«

2- Constant over-all heat transfer coefficient, U«

2~ Kegligable heat loss from the exchangero

4- Constant fluid properties o

5~ late of flow of each fluid is constant and no phase

change occurs0

The over-all energy balance

q = mc (tg-t-̂ m W (Tx - Tg) 0 )

For an elemental length dl along the exchanger

the heat transfer area is dA.

ler*<{>'

I

I

0 *t< L f length along the exchanger

-J.

She elemental area in the first pass » x dA

The elemental area in the second pass as (l-x)dA

Assume T9 t8, and t" are the temperature of the hot

fluid, first pass cold fluid, and second pass, cold fluid at

any cross-section at length t •

Make an energy balance for the el&ental area dA

- MG &T a U x dA (5!~t8)+ V(l-*)dfcCf-*M) (6)

nc dt«« U x d& (9Ms») (7)

- me dtw« U (l~x) dA (*D~tn). (8)

An energy balance between erese sections at

and the end of the echanger, i*e«

AG (T-«2 ) • mc (t^t1) + fficCt98-^)

a; mc (t^t*) (9)

Bquations (5)* (9) give

t^tx " $ r * 2

(10)

Equation (6) gives

Equation (11) has three variables $, t'f and t"

rwe of these variables should be eliminated by using the

above relatione

=*6°**

By using equation (8)9 equation (11) becomes

[x(T-Tg) d3? UR UR r ^p^i ^ + — T + — ix(T-T^) -=-*-.-- - t dA me me ^1^2

Differentiate (12) w^roto A§ and substitute R =

3IT UR dT ^ URfx dT

dA - m@ dA meLR dA dA

dt" ]

K 0 (12)

(13)

From equations (7)&(S)

m jt.o mc dw" ^^^ mc T-t* = n— 2r? p and <- dt M T-f

T-T and tt"-t*HS 2 me r dt'* 1 dT

gg r 1 me dt S!

32 SHE J * ETI-x? 31

dtM

dA HMS R ^ &

dT

substitute in (13)

(14)

d̂ g ^ 13 — 2 + -*=-dA me

-l+2x) ̂ $ dA mt ^ *

and § ^ + ̂ (R-l+2x) ̂ - -S-w x (l-x)̂ f = - E ^ x(l-x)T <La mo dA m^o^ m c -2

The particular integral of equation (15) is T̂ c the

solution of the homogeneous equation is

C, «*/• {'j^ [(*-!+**] + /(/?-/ + «2X)* + 4-X(l-X) ]J

where C, and Gp are constants

The general solution of (15) is

tC. «/°f -J^i F(tf-/-^> -/(/?-/+ax/-r^Jt(.'-x) J / ^16)

To evaluate C^ , Cg0 the following boundary conditions are

used 9

at A * 09 T m Tx (1?)

A = At f T a T2 (18)

Prom 16 and (17)

S ^ ^ * C l * % ^19)

ind from (16) and (IB)

•-• **/>[% fi^i+nf* >*«-*) ] C20)

.-. i£ = _ L . & /- £1 \ (ai)

- 8

Subs t i tu t ion i n equation (12) a t A=0, noting t h a t a t

t h i s pos i t i on (A=0) , t ' s t j ^ , t°°=t2 and 0? = T1$ i„e<,

4J\ + UK T + HE. [x/t-t \ - t 7 - o

Tl% [*<''*'*,)•**! '-* &V

^

(24)

(/A =

f(fi-l+ax.fi-*-x.(i-x) <(*-h **) + i(R-i + 2X?T¥x.(i-*)

^V/-x) ![=•£?- " T ^ r

9

Equation (25) i s used to find tfce BTO , R v and £

relation (unit transfer method)*

MW U dLA

UA

min for U « constant

9 MU s Sf

mc <C M0

38 S~~f ~

R ss

NT\J* jf(^-/-^^x)'?-f^x(<-x}

^ (*?-!-***) r/(R-l+2X)>**-X-0-2-)

^ - ' t 2 X ) + ^ . / , f ; ^ + i f / ( H / ] - 2 ( / - X > ./.. c — X

10

(2?)

+ / (ff . /+ax/t^.(i-x) ( **(* ( NTU* jjlR-i +2xft+x.(i-x.))+iJ

- 11 -

RESULTS MP DISCUSSION • 'U« I li III I i ' • I I I I I I I I I

Equation (27) was programmed9 and the effectiveness

was calculated and presented graphically in figures (g)

to (8) as function of the number of transfer units* NTU f

heat capacity ratio9 R9 and the precentage of first tube

pass area to the total area of th« excehai&ger* Three special

cases of can be derived frpm equation (df) , namely?

i - 2?h© case of x«0 9 this represent® a single pass eoun-

ter°»current heat exchangers« A& expected^ £ 9 has

the highest value in this case than any other case*

ii - The case of x s % 9 this represent© 1̂ 2 heat exchanger

$£ equal passes as presented fey &ond©n and Kay°s^ '•

ill- She case of x»l « this represents a parallel flow one

pass heat exchanger where 6 % as expected , is the

lowest of all cases.

It is clear that for liquid-liquid heat transfer * the

effectiveness is improved as x decreases9 i«e» 9 the true

temperature differences is more close t© the logrithmic

mean temperature difference of counter flow*

In nuclear feild$9 1«£ heat exchangers may be desi­

gned for the use of condensing steam as heating fluid*

« L<t °»

The corrosiveness ©$ steam may dictate that the steam flow

in tubes» The large volume rate of the inlet steam as com­

pared with the out-let irate volume of the condensing steam

necessitates larger number of tube passes in the first pass*

In case of using such exchanger for liquids-liquid heat ex­

change, figures C£~8) are used to find the OT{J0 and conse­

quently ®hedfc the suitability of the exchanger*

Finally it ©ay b« noticed that if the shell nozzles

are interchanged,th© ratio x is changed to (l~x)o

<t I

- 32

leO

8

0«2

0*1

1 2 ? 4 5 6

*ig 2S Values of £ at different*values of NTU and 1 for x s Q0e

- 14

i « u

D.9

0.8

0«7

QoS

0.5

n„4. W » ~

0*3

0.2

0 . 1

0 .0

4

* / / /

'

1

<*~~*^

1

E 0.2

*

0.4

0 .6

.0.8

1.0

1

! j l i

xssQ«3 i

i

s*

r ! f ; '

0 1 2 3 4 5 6

tig 3 s Values of £ at different values of and E for x a 0.3

15

do©

Co9

G*8

0*7

Co6

£ ©•5

©«4

*3

@02

Oil

0«0

fig 4 8 Values of £

and H for x.

different values of KTU

©•4

- 26

0,0 ,

58 Yalues @f £ at different values of and E for x = 0,5

- 37

1.0 9%J

Oo9

0a8

©o?

OoS

U « ( b

0«0 0 I 2 5 * \ 5 6

8 Yalu.es ©f 6 a t d i f fe ren t values of HOT and £ for x = 0*6

18

Fig 78 falues of £ at different Yalues and fi foir x s 0.7

of Hf ¥

-19

Oo9

0.8 .

0.7

0.6

0.5

0o4

0.3

0c2

0.1

0.0 L

Fig 3 5 Values of £ at dif f eent values of MTU

and E for x « 100

*~ d\) °"

References a

1- WoMo Kays and A»L« London9 compact heat exchangers,

McGraw-Hill book Go9 Ino» (1958)*

2- D®Mo £©rn9 Process heat transfer

McGraw-Hill book Go9 In©* (1950)•

3- A© Jo B»derw©©&0 loljasto Peta&lltjm f©chn®logy& 20,

145-15® (1934)*

4 - WoMo Haglee I ndus t r i a l Bxagine@ring Chemistry, 25 % 604~

608 (1933)o

5 - EcTo BoT»man9 AoC. Mueller, and W«MaBagle»

Trans, ASMB9 62,283-294, (1940),

6- F«AG Holland, B~M9 Moores> FoA© iatson and JJloWoKinsoi

Heat t ransfe r 9 Heinemann Chemical EJng* s e r i e s , London

(192©)©