Finite element analysis of shell and tube heat exchanger

I N T : COMM. HEAT M~SSTRANSFER Voi: 15, pp. 151-163~ 1988 ©Pergamon P r e s s

0735:1933/88 S3~00 + :80 Printed z n t h e U n i t e d S h a f e s

plc

FINITE ELEMENT ANALYSIS OF SHELL AND TUBE HEAT EXCHANGER m mm ##I S.G.Ravikumaur , K.N.Seetharamu , P.A.Aswatha Narayan~ m Research Scholar, HTTP Lab., Mech. Engg., IIT Madras wm Professor, HTTP Lab., Mech. EnEE., IIT Madras, India; • wm Asst. Ffofessor, Flui~ Mechanics Lab., IIT Madras, India.

(C~,tunicated by D.B. Spalding) ABSTRACT A finite element model to predict temperature distribution in heat exchangers is reported. The model Can be effectively used to analyse and design the heat exchangers with complex flow arrangements for Which no regular design procedure is available. Illustrations ape ~fSvided to explain the application of the method for the analysis Gf shell and tube heat exchangers;

~roduction Design

procedures

flow arrangements, such for

as E-NTU simple

are well

established

for

exchangers

With

simple

like one shell ~ass and two tube passes. Design charts

curves

types

of

and

LMTD

correction

exchangers.

Similar

factor design

curves charts

are

do

established

not exist

for

designing some exchangers with multiple entries on the shell side and also for complex flow arrangements. Attempts many

researchers

lumped Changer LiOn

to ~enerate earlier.

resistance,

to

He generated

simplify

the use of auxiliary

Saryal

design

the

heat

charts

[I] suggested

resistance-capacltanoe

calculations.

curve

such

and

and

were

electro-analog hybrid

the dimensionless exchanger

curves

model

mJde

models for

heat

by

like ex-

temperature distribu-

calculations

and

showed

that

Curves eliminated iterations in the process of calcula-

151

152

S.G. Ravikumaur,

K.N. Seetharamu and P.A.A. Narayana

tion.

In

and Schlunder

1979,

dimensionless baffles.

temperature

types.

and

Schlunder

analysis

to

the

eases

Mikhailov and

analysis

of

and

curves

Ozisik

introduced

obtain

the

[3]

the

of

of

heat

proposed

in

shell

a cell and

model

tube

to predict

exchangers

for shell and tube exchangers adopted

the

procedures

temperature

of network

assemblies

[2]

distribution

They generated ~-NTU

ous

for

Gaddis

Vol. 15, No. 2

model

distribution.

heat

exchangers.

exchangers

studied

in

They They by

of vari-

proposed

followed

with

by Gaddis

finite

extended

element

the

had

applied

DominEo

[4]

model

for

and

the

found

that their results agreed well with the solution of DominEo [4]. In the present lation

of

element

paper

a new element model

matrices

is

described.

The

is proposed application

and the formuof

the

element

number

of ele-

model in heat exchanger analysis is illustrated and discussed.

Finite Elements in Heat Exchangers The ments.

exchanger

Each element

passin E through element

are

exchanger

as

given

The

matrix

formulation.

heat

first

losses

have

it at least

nodes.

no

for

should

marked

is

taken

analysis

all the fluids

once.

nodes.

The

to

the

The entry

The

in Fig.l.a.

element

shown

Every in

and exit

and

considered no

model

element

Fig.l.b.

axial

into

taking part

diseretized

exchanger ambient

is subdivided

is is

in

in the exchange

of the fluids of

the

a

shell

and

tube

exchanger

has

four

considered assumed

conduction

in the

for to

be

along

the element ideal the

with

tube

or

shell metal. Phase change is neglected in the present analysis.

Formulation of Element Matrix Consider governing as:

the

the heat

element

transfer

shown

in

Fig.l.b.

The

differential

equation

for the hot fluid in the element can be written

Vol. 15, NO. 2

C

dT -dA

c

dt --dA

~J,A~DTUBEHEATEXCHAN(~R

:

153

- U (T-t)

(I)

U (T-t)

(2)

and :

for the cold fluid. Where the to

C and

c represent

element, the

T

and

t

incremental

elements

are

used

the heat represent

areas to

in

capacity their

their

approximate

respective

flow

the

rates

of hot and cold temperatures,

directions.

field

variables

Linear and

fluid dA

in

refers

isoparametric

the area.

Hence

the approximation can be written down as: T

The

T4 N I

+

T3N ~

(3)

t

=

t I N I + t6N 2

(4)

A

=

A i N I + AoN 2

(5)

approximation

(I) and [5],

=

(2).

i.e.,

(3) and

The residual

multiplying

(4) are substituted error

is minimized

the residual

into the governing by weighted

equations

residual procedure

errors by suitable weights,

integrating

over the domain and forcing the integral to zero in every element. I Ae f In

the

dT B i [C ~ + U (T-t)

]

dA

:

0

(6)

dt Bj [C ~ - U (T-t) ]

dA

=

0

(7)

present

collocation)

problem

which

weights

results

in

are two

taken

independent

-D t I - D t 6 + (-C+D) T 4 + (C+D)T 3 (-c+D) t1+ (c+D) t 6 -DT 4- DT 3

to be equal

=

=

to unity

equations

0

0

from

(subdomain (6)

and

(7) (8) (9)

where D = UA/2 Two more equations are formed from the inlet conditions. C

T4

=

Q4

(10)

154

S.G. R a v ~ u r ,

¢ tl

:

K.N. Seetharan~ and P.A.A. Narayana

Vol. 15, No. 2

ql

(11)

where Q and q represent the heat input at the nodes. The final set of equations in element matrix can be written as:

I:iI

~y

c

0

0

0

tI

-c÷D

c+D

-D

-D

tel

C

0

T4

-C÷D

C+D

T3

0

0

-D

-D

(12)

The general equation for the element matrix can be written as

(13)

[ke]{~ ~ : { f ~

Application of the finite, element model to Exchanger analysis: The element model discussed can be used effectively for the analysis and

the design of heat exchangers.

and

the

flow

inherent

flexibility

arrangements,

first

of

the

the

examples

in

To demonstrate the prediction accuracy

the

following

model

numerical

establishes

the

to describe examples

accuracy

various

are

types of

presented.

prediction

and

the

The rest

of the examples bring out the applicability of the model to various types of flow arrangements. Example - I: Shell and tube exchanger I Shell pass - 2 Tube passes The discretized model of the exchanger analysed is shown in Fig.2. The

flow and

the

fluid

properties used

Heat transfer coefficient

U

=

in

the

600 W/m 2 K

Heat capacity rate of cold fluid (tube side)

(c) =

200,000 W

Heat capacity rate of Hot fluid (shell side)

(C) =

17000 W 2 64 m

Total area of exchanger

(A) =

Not fluid inlet temperature

(T34)=

190°C

Cold fluid inlet temperature

(T 1) =

35°C

Number of elements

=

analysis

32

are

as

follows:

Vol. 15, NO. 2

SH'~.ANDTU~EHEATEXCHANGER

Elemental area This model

(A e)

example

proposed

by

predictions

of

expressions

given

is

to the present

analysed

Schlunder

the

:

models,

using

and

2

the

Gaddis

[6,7].

present

[2].

analytical

in

model,

2 m

To

sets

[6,7]

are given

reproduces

the

temperature

in Table-1.

The

distribution

are

of

by

well

the

as

the

accuracy

evaluated

solutions

from

of the

corresponding

[2] and the analytical

results

given

as

compare

solutions

Three

model

the model given by Schlunder

solutions

155

from the

the present

analytical

model

solution

on the shell side flow with 0.4% accuracy. But the results from the Schlunder and Gaddis maximum.

[2] model analysis deviate

Though

the

from the analytical results by 2.25%

2.25% variation may seem to be small,

the flux calcu-

lation show a large deviatin due to the high heat capacity rates involved. The to

derivations evaluate

present

given

in

tube

side

the

analysis

the

Kern

[6]

fluid

tube

side

do

not

give

temperatures. temperatures

an On

are

explicit the

other

known

in

relationship hand, in the

the process

of

analysis. An is

average

calculated

true

from

temperature

total

heat

heat transfer coefficient. temperatures evaluated

obtained

from

the

agree with the

transferred,

for

the

The LMTD is evaluated

from

the

average

temperature correction

difference

true

analysis.

The

temperature

total

of

complete area

and

exchanger the known

from the inlet and outlet LMTD

correction

difference

and

factor

the LMTD.

is The

factor F T obtained from the results of present model

the theoretical F T given in [6] whereas

results

the

Schlunder's

[2]

model

deviates

from

the F T calculated

from

the theoretical F T by

more than 2%. If results

the

from

analysis the

in the analysis

is

model

using

extended

[2]

for

deviates

the model

2 shell

pass and

significantly.

[2] because

The

4 tube pass deviation

of the approximation

the

occurs

involved

156

S.G. Ravikumaur,

in

the

formulation.

assumed an

to

be

element

average

[2,3],

bulk

The

potential

difference whereas

fluid

in an element. dently

the

K.N. Seetharar~ and P.A.A. Narayana

in

between the

temperatures

The

that

comparison

drives the

present

is taken of

the

the

exit

model

Vol.

heat

in

temperature

in Table

shows that the present model can predict

element

is

fluids

in

of

the difference

as the potential

results

an

15, No. 2

between

driving

I and

the heat

Table

the temperature

the

2 evi-

distribution

well in heat exchangers. The method. type

example

The

of

versatality

shell

analysed

above

of

tube

the

establishes present

exchanger

The

method

with

examples

the

of

the

is demonstrated

complex

presented

accuracy

flow

so that any

arrangements

hereafter

are

present

can

be

modifications

I with inlet of the fluids changed or split.

The

ponding

and

confidently.

of example

change

cited

example

properties, figure

number,

properties

of

the fluids inlet nodes,

number

and

the

table

the

fluids

in

exchanger, the

the fluid exit nodes,

in which

results

tabulated

the corresare given

in the table which follows: .............................................................................

Ex-2

Ex-3

Ex-4

Ex-5

Ex-6

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

Discretized Fig. No.

3

4

5

6

7

No. of elements

32

32

32

32

32

No. of Nodes

66

67

67

67

67

Cold fluid inlet(s)

I

I

I

I

I

Hot fluid inlet(s)

66

34,50

67,50

67,66

34,66

Cold fluid exit(s)

33

33

33

33

33

Hot fluid exit(s)

34

67,66

34,66

34,50

67,50

17,000

8,500

8,500

8,500

8,500

200,000

200,000

200,000

200,000

200,000

Heat Capacity Rate Hot fluid (C) Cold fluid (c) Heat transfer coefficient

ex-

(U)

600

600

600

600

600

Elemental Area (A e)

2

2

2

2

2

Table No. of results

3

3

3

3

3

Vol. 15, NO. 2

SHELL A N D ~ X ] ~ E H E A T ~

157

Application to the Problem of Varying Physical Properties In the examples sumed

to be constant,

throughout ties

cited hence

the exchanger.

(like

gases)

above,

the properties

a constant

heat

of the fluids were as-

transfer coefficient was used

If the fluids with temperature sensitive

undergo

larger

changes

in

temperature,

then

properthe

heat

transfer coefficient will be varying from inlet to outlet of the exchanger. In

the

into

number

heat

transfer

in

the

local

present of

elements.

The

of

Hence

coefficients

element. heat

method

every

depending

element

transfer

analysis,

exchanger

element

can

be

has

been divided

assigned

on the local properties

stiffness

coefficients

the

and

matrices local

are

bulk

of the fluids

calculated

mean

different

using

the

properties

of

the

aD,riori

for

all

fluids. The the

local

elements.

calculated

bulk

Hence

mean

temperatures

to start with,

with known

inlet

are

not

a constant

temperatures,

known

heat

is assumed

transfer

coefficient,

to prevail

throughout

the exchanger and the problem is solved to get the temperature distribution in the exchanger.

The local elemental heat transfer coefficients and proper-

ties are calculated using the temperature distribution from the first iteration the

and

used

solution

capacities

in

subsequent

converges.

of

the

To

fluids

iterations.- The ensure

are

energy

assumed

to

iteration

continuity remain

is

at

continued

nodes,

constant

the

till heat

throughout

the

exchanger.

Stability and Conversence In

order

respect

to number

changer

was

the shell

to

test

the

of elements,

analysed

side fluid

with

4,

stability a single 8,

12 and

in the middle

and

convergence

shell

pass and

16 elements.

of two The

of the shell and exit

solution

with

tube pass extemperature

of

temperatures

of

158

S.G. Ravikumaur, K.N. Seetharamu and P.A.A. Narayana

Vol. 15, No. 2

both the fluids are given in Table-4 to show the convergence of the solution with increasing number of elements. along with

the results.

The percentage errors are also tabulated

The basic data assumed

for the present

stability

and convergence analysis are: Heat transfer coefficient

U

=

600.00 W/m2K

Heat capacity rate of Hot fluid

C

=

17000 Watts

Heat capacity rate of Cold fluid

c

=

200000.0 Watts

Total Area of Heat Transfer

A

=

64.0 m 2

Inlet temperature of Hot fluid

T

=

190.0°C

Inlet temperature of Cold fluid

t

:

35.0°C

As we number

of

observe

from

elements

Table-4,

from

4

to

the accuracy

16.

The

increase

when we increase the number of elements beyond of elements must be taken for the analysis. the order of I (linear)

improves as we in

12.

increase

accuracy

is

the

marginal

Thus a reasonable number

The local truncation error is of

[5].

Conclusions

A

finite

element

its applicability

model

in shell

examples.

The

prediction

and

it

able

to model

can

be used

is

to analyse

to

and

analyse

tube exchanger

accuracy any

heat

type

of of

exchangers

proposed

is estimated

in heat

exchangers.

any new design and it can even be used

a network of heat exchangers.

and

analysis is illustrated with

the method flow

is

to be good This

model

to analyse

Vol. 15, No. 2

swarx. A N D T t E E H E A T E M C H A N G ~

159

TABLE I Temperature Distribution on Shell Side (Fig.3)

NODE

ANALYTICAL RESULT °C

PRESENT MODEL °C

REFERENCE (2) MODEL °C

33

46.355

46.357

46.246

36

170.342

170.308

170.980

38

153.276

153.258

154.424

40

138.459

138.419

139.942

42

125.597

125.570

127.334

44

114.431

114.388

116.306

46

104.736

104.704

106.704

48

96.319

96.278

98.306

50

89.013

88.979

90.994

52

82.669

82.629

84.599

54.

77.162

77.127

79.028

56

72.379

72.343

74.159

58

68.228

68.195

69.916

60

64.623

64.589

66.209

62

61.494

61.463

62.976

64

58.776

58.746

60.152

66

56.417

56.387

* 57.689

* (Error - 2.254%)

TABLE 2 Table of Comparison for Example-1 Quantities Evaluated/compared

Theoretical

Present Model

Schlunder's Model

Effectiveness (E)

0.8618

0.8620

0.8536

LMTD Correction Factor F T

0.9208

0.9214

0.8933

Total heat Transferred Hot fluid exit temp. Cold fluid exit temp.

O °C °C

2270911

2271421

2249287

56.417

56.387

57.689

46.355

46.357

46.246

160

S.G. Ravikumaur, K.N. Seetharamu and P.A.A. Narayana

Vol. 15, No. 2

TABLE 3 Temperature Distribution in Exchangers given in FIG. 3, 4, 5, 6, 7. NODE

TEMPERATURE Ex-4

Ex-2

Ex-3

I

35.OOO

35.000

5

35.635

°C Ex-5

Ex-6

35.000

35.000

35.000

37.233

35.816

35.816

37.233

9

36.632

37.992

38.039

38.038

37.992

13

38.268

40.156

40.202

38.785

38.739

17

41.029

40.849

40.893

40.938

40.894

21

43.733

41.541

41.586

43.075

43.030

25

45.205

43.672

43.716

43.714

43.669

29

45.978

44.296

45.786

45.783

43.293

33

46.354

46.360

46.358

46.355

46.358

37

56.418

190.000

56.351

56.350

190.000

104.547

40

64.662

46

77.273

52

96.431

153.205

68.117

77.045

77.044

104.564

125.608

125.607

68.116

153.216

61.489

61.451

58

125.711

88.950

89.062

89.031

64

170.378

61.432

61.471

88.981

153.279

153.268

66

190.000

56.361

56.402

190.000

190.000

67

--

56.331

190.000

190.000

56.331

TABLE 4 Stability and Convergence Test Results

Number of Elements

Shell fluid Temp. °C Exit Temperature °C ................................................................. Middle of Hot Cold shell Error % fluid Error % fluid Error %

4

86.43

2.9

54.59

3-3

46.50

0.3

8

88.47

0.6

55.95

0.9

46.39

0.09

12

88.72

0.3

56.20

0.5

46.37

0.04

16

88.87

0.15

56.30

0.3

46.36

0.02

Vol. 15, No. 2

S~w~.A~D~'dBEHEAT~

161

Nomenclature A

Area

AI,A O

Area at inlet and outlet of an element

Bi,B j

Weighting function

C

heat capacity rate (W/kg) hot fluid

c

heat capacity rate (W/kg) cold fluid

N.

Shape function

Qi

heat input at node i Hot fluid

qj

heat input at node j cold fluid

T

temperature - hot fluid

t

temperature - cold fluid

U

heat transfer coefficient

{f}

set of nodal heat inputs

[K]

global stiffness matrix

{e}

set of nodal temperatures.

1

References I.

N.Saryal, Int. J. Heat Mass Transfer 17, 971, (1974).

2.

E.S.Gaddis

3.

M.D.Mikhailov and M.N.Ozisik, Finite Element Analysis of Heat Exchangers, in S. Kakac, A.D.Bergles and F. Mayinger (ed.) Heat Exchan6ers ~ thermal and hydraulic fundamentals and design, Hemisphere Publishing Co., Washington (1981).

4.

J.D.Domingo, Int. J. Heat Mass Transfer 12, 537, (1969).

5.

J.N.Reddy, An Introduction McGraw Hill, (1985).

6.

D.Q.Kern (1972).

7.

D.Q.Kern, Process Heat Transfer, 6th ed., McGraw Hill, (1961).

and E. Schlunder,

and

A.D.Kraus,

Heat Transfer Engineering ~, 43,

to

the

Extended

Finite

Surface

Elements

Heat

Methods,

Transfer,

(1979).

P.

McGraw

133,

Hill,

162

S.G~ R a v e ,

K.N. Seetharamu and P.A.A. Narayana

Vol. 15,

I d2,T1~ r---~ l~ l

~

i~

cl ,~t

4 l~r

I

i~

l'lz

t~

I

:i0

I

C~

11'3

I

-3

FIG. I a, b. The di~cretized model and first cell

4 nnuuunMuuuu!m ll~ddd~llUm~dJn~ lbhl,Hbl~W*lJbm muHnmnnaUlUaNM

:1

I l ~I

..~L.

FIO, 2 Discretized exchangOr for oxample-1.

4

"l'rl --" =.~,

il't

mi "[-"~ ~

[

~ ,'-! ' i ' ['[ "

"; "'1 '1 --z'l ~i ' J'":['i I °' ,

Ul-["~ ~'l ,l[

"1

D"1

# t- - " t

?" "i

FIG. 3 Discretized exchanger for example-2.

.

NO;

Vol. 15, ~

2

~;~AND

TUBE HEAT ~

,

@÷

163

d ;

i

i

l:l I . ]

ljl!'[.i.~i;,

I

--

I

•

I ,. I L]--.i

'

1

1

•

'

L

I

.

•

L

&

,

I

• i-

FIG. 4 Discretized exchanger for e x a m p l e ~

~ , ;"

7 I~

"~i

"1 I

~

] i!'

T 1 1 1.1 1 ~.~

ii'

~l~"i

I

#

T

T

l

¢

1

]

I

I

....

FIG. 5 Discretized exchanger for example-4.

"]"I

I

i

, l,

T

i

r-'l y'

•

~"

I

I

1

T

k

L.

]

i

i~

I

,I

) j IJ_L ] ....!_._L

=~ ri'L "! 4f ['i

~[

•

L

1

!

1

_

L,,,ll-

FI~. 6 Di~cretized exchanger for example-5.

il

,

1i

l'

i

!

~

LI

.

II

t

~i

v

i

1

I.

_., ?i '[~'] °L "'~ ~::t_'j "°l "i "r''i_ ~""]f~]~-. Lit,-,

I , - .,d~

FIG, 7 Discretized exchanger for example-6.