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A general solution for one-dimensional multistream heat

exchangers and their networks

Xing Luo

a,b

,Meiling Li b ,Wilfried Roetzel

a,*

a Institute of Thermodynamics,University of the Federal Armed Forces Hamburg,D-22039Hamburg,Germany

Institute of Thermal Engineering,University of Shanghai for Science and Technology,Shanghai 200093,China

Received 2October 2001

Abstract

A mathematical model for predicting the steady-state thermal performance of one-dimensional (cocurrent and countercurrent)multistream heat exchangers and their networks is developed and is solved analytically for constant physical properties of streams.By introducing three matching matrices,the general solution can be applied to various types of one-dimensional multistream heat exchangers such as shell-and-tube heat exchangers,plate heat exchangers and plate–?n heat exchangers as well as their networks.The general solution is applied to the calculation and design of multistream heat exchangers.Examples are given to illustrate the procedures in detail.Based on this solution the su-perstructure model is developed for synthesis of heat exchanger networks.ó2002Elsevier Science Ltd.All rights reserved.

Keywords:Heat exchangers;Heat recovery;Optimization

1.Introduction

Multistream heat exchangers are widely used in pro-cess industries such as gas processing and petrochemical industries to exchange heat energy among more than two ?uids with di?erent supply temperatures because of their higher e?ciency,more compact structure and lower costs than two-stream heat exchanger networks.A multi-stream plate–?n heat exchanger can even handle up to 10process streams in a single unit [1].The use of multi-stream heat exchangers is more cost-e?ective and can o?er signi?cant advantages over conventional two-stream heat exchangers in certain applications,especially in cryo-genic plants [2–4].However,the investigation on syn-thesis of heat exchanger networks using multistream heat exchangers is still limited because of lack of suit-able calculation methods for the thermal performance of general multistream heat exchangers.

The multistream heat exchangers can be classi?ed into two categories.One is multichannel heat exchanger in which there is no thermal interconnection between the walls separating the ?uids,such as shell-and-tube heat exchangers and plate heat exchangers.The other is multistream plate–?n heat exchanger.The mathematical model and its analytical solution for the thermal per-formance of one-dimensional multistream plate–?n heat exchangers was ?rst proposed by Kao [5].Haseler [6]de?ned a bypass e?ciency which describes heat transfer between non-adjacent layers in a plate–?n heat ex-changer to illustrate the bypass e?ect.For multichannel heat exchangers a general solution of the temperature distributions was proposed by Wolf [7].Many signi?-cant discussions on the general solution have been made [8–12].Based on the pioneering research work of Kao [5]and Wolf [7],the thermal design problems of multi-stream plate–?n heat exchangers were solved by Luo et al.[13].By introducing three matching matrices Roetzel and Luo proposed a general form of the ana-lytical solution for various types of one-dimensional multistream heat exchangers and their networks [14].In the present paper,their method is further developed and applied to the thermal calculation and design

problems

https://www.wendangku.net/doc/4c1845025.html,/locate/ijhmt

Corresponding author.Tel.:+49-40-6541-2624;fax:+49-40-6541-2005.

E-mail address:wilfried.roetzel@unibw-hamburg.de (W.Roetzel).

of multistream heat exchangers and their networks. Examples are given to illustrate the procedures.

2.General mathematical model and its solution

Consider a generalized N-stream heat exchanger which consists of a bundle of M parallel channels (M P N).The?uid?owing through a channel exchanges heat with the?uids in all other channels.It is assumed: (1)The longitudinal heat conduction in the solid wall can be neglected.(2)There is no heat loss to the envi-ronment.(3)The heat transfer coe?cients and the properties of the?uids and wall materials can be con-sidered constant within each channel.The general mathematical model can be written as

_W i d t i

d x

X M

j?1

U ij t j

àt i

ei?1;...;MTe1T

with U ij?U ji and U ii?0.

It is convenient to rewrite Eq.(1)into a matrix form

d T

d x

?AT;e2Twhere A is an M?M matrix A?

à1_

P M

l?1

U1l U12

áááU1M_

à1_

P M

l?1

U2láááU2M_

.....

U M1

W M

áááà1_

W M

P M

l?1

U Ml 2

:e3T

The positive value of_W i indicates that the?uid?ows in the positive direction of the spatial coordinate and vice versa.If_W i and U ij are constant in each channel(they may vary from channel to channel),the above ordinary di?erential equation system is linear and can be solved analytically.According to the theory of linear algebra the general solution of Eq.(2)is obtained in the matrix form as

T?U e K x De4Tin which e K x?diag f e k i x g is a diagonal matrix and k iei?1;...;MTare the eigenvalues of matrix A.U is an M?M square matrix whose columns are the eigenvec-tors of the corresponding eigenvalues.Eq.(4)is valid only if the eigenvalues di?er from each other.It has been proved that all eigenvalues of matrix A are real,how-ever,Eq.(4)might have multiple eigenvalues

[9–11]. 2696X.Luo et al./International Journal of Heat and Mass Transfer45(2002)2695–2705

A practical method to avoid multiple eigenvalues is to add very small random deviations to the input param-eters such as _W

i or U ij .Such small deviations have al-most no e?ect on the results.

The coe?cient vector D in Eq.(4)is determined by the boundary conditions.To get a general expression of the boundary conditions,we introduce the following three matching matrices:

Interchannel matching matrix G .It is an M ?M matrix whose elements g ij are de?ned as the ratio of the thermal ?ow rate ?owing from channel j into channel i to that ?owing through channel i .

Entrance matching matrix G 0.It is an M ?N matrix

whose elements g 0

ik are de?ned as the ratio of the thermal ?ow rate ?owing from the entrance of stream k to channel i to that ?owing through channel i .

Exit matching matrix G 00.It is an N ?M matrix

whose elements g 00

ki are de?ned as the ratio of the thermal ?ow rate ?owing from channel i to the exit of stream k to that ?owing out of the exit of stream k .

From energy balance at the boundaries,i.e.,the en-trances of M channels,we have T x 0àá

?G 0T 0tGT x 00eTe5Tin which

T x 0àá?t 1x 01àá;t 2x 02àá;...;t M x 0M àá??T ;e6TT x 00eT?t 1x 001

àá;t 2x 002àá;...;t M x 00M àá??T

:e7T

x 0and x 00are the coordinate vectors of the entrances and exits of M channels,respectively.

Substitution of the boundary conditions,Eq.(5),into Eq.(4)yields

T ?U e K x V 0 àGV 00 à1

G 0T 0;

e8T

where V 0and V 00are two M ?M matrices,whose ele-ments are given as v 0ij ?u ij e k j x 0

i ;

e9Tv 00ij ?u ij e

k j x 00i

;e10T

respectively.The outlet ?uid temperatures of the ex-changer can then be expressed explicitly as T 00

?G 00

V 00

V 0

àGV 00 à1

G 0T 0:

e11T

Eq.(11)is general for one-dimensional heat exchangers.The input data are the heat transfer parameters and thermal ?ow rates given in A ,the ?ow arrangement set by G ,G 0and G 00and the coordinates given in x 0and x 00.The coe?cient matrix A also depends on the type of the exchanger to be considered.

3.Applications of the general solution

To use the general solution one should at ?rst divide

the exchanger into several sections according to the construction of the exchanger.Each section contains several channels.The ?uids ?ow through the channels and exchange heat with the ?uids in other channels.The sections should be divided such that there are no en-trances or exits of streams inside the sections and the ?uid properties in each channel can be considered con-stant.After the channel con?guration has been made,it is easy to get the matching matrices G ,G 0and G 00.

The major task to use the general solution is the calculation of the coe?cient matrix A .For multichannel heat exchangers Eq.(3)can be used directly to calculate the coe?cient matrix A .A lot of elements of A become zero because there is no heat exchange between corre-sponding channels.However,the mathematical model of temperature distribution in a multistream plate–?n heat exchanger,which also contains energy equations of separating plates and ?ns,di?ers from Eq.(1).By eliminating the temperatures of separating plate and ?ns in the energy equation of ?uids,Luo et al.[13]trans-formed the governing equation system into the form of Eq.(2).The corresponding coe?cient matrix A should be specially calculated.

In the following examples it will be illustrated in detail how to determine the matrices A ,G ,G 0and G 00.The ex-amples also show how to use the general solution to solve the design problems of multistream heat exchangers.3.1.Shell-and-tube heat exchangers

In a multistream shell-and-tube heat exchanger each tube-side ?uid exchanges heat only with the shell-side ?uid.There is no direct thermal contact between any two tube-side ?uids.The input matrices of an N -stream E-type exchanger (one shell pass,arbitrary number of tube passes,no split)have been given in [14].Here a more complicated example will be discussed.

3.1.1.Example 1

In this example,a three-stream E-type shell-and-tube heat exchanger is used to heat two cold streams with one hot stream.The data taken from [2]are presented in Table 1.The exchanger is divided into three sections and seven channels,as shown in Fig.1,which yields

Table 1

Problem data for Example 1Stream

T s (K)T t (K)W (W/K)1H1420

370)80002C130035040003C2

280

320

5000

k ?1:1kW/m 2K for all matches

X.Luo et al./International Journal of Heat and Mass Transfer 45(2002)2695–27052697

àU H1C2

000U H1C2

0àU H1C2

000U H1C2

00àU H1C1

00U H1C1

000àU H1C1

00U H1C1

C1 U H1C2

W H1

000àU H1C2

0U H1C2

W H1

00àU H1C2tU H1C1

000U H1C1

00àU H1C1

;

e12T

0000000 1000000 0000000 0010000 0000010 0000001 0000000 2

;

G0?

001 000 010 000 000 000 100 2

;

G00?

0000100

0001000

0100000

75;

e13T

x0?0;x1;x1;x2;x1;x2;L

? T;x00?x1;x2;x2;L;0;x1;x2

? T: By setting L?1,for given values of variables x1,x2, U H1C1and U H1C2,the outlet stream temperatures can be calculated.The problem given in Table1for minimum heat transfer area becomes

min Leàx1TU H1C1=ktx2U H1C2=k

s:t:t t;C1àt00

2?0;t t;C2àt00

?0;x1àx260;

06U H1C1;06U H1C2;06x16L;06x26L:

e14TThe results are x1?0:2703,x2?0:5483,U H1C1?3:438 kW/m K and U H1C2?4:294kW/m K,which gives the minimum heat transfer area of4.42m2.This exchanger is equivalent to the two-stream heat exchanger network shown in Fig.2.

3.2.Plate heat exchangers

A plate heat exchanger consists of a number of par-allel channels formed by a stack of heat transfer plates. According to the combination of the plates with holes or blanks located at the four corners of the plate and the additional manifold axes if necessary,various?ow pat-terns may be created in a multistream plate heat ex-changer,which can be classi?ed into three categories: series?ow pattern,parallel?ow pattern and complex ?ow pattern.It is assumed that in the plate heat ex-changer the?uid in each channel has thermal contact only with the two adjacent channels.The corresponding coe?cient matrix of the governing equation system reads

àU12_

U12

0ááá0

U21

àU21tU23

U23

0..

00U Mà1;Mà2

Mà1

àU Mà1;Mà2tU Mà1;M

Mà1

U Mà1;M

Mà1 0ááá0U M;Mà1

àU M;Mà1

M 2

;

e15Twhere M is the number of channels.

3.2.1.Example2

As an example,a three-stream plate heat exchanger with countercurrent parallel arrangement shown in Fig. 3is taken for the analysis.The data presented in Table1 are used again.The numbers of channels for C1and C2 are M C1and M C2,respectively.Thus,M H1?M C1tM C2t1,M?M H1tM C1tM C2.Since the values of k H1C1and k H1C2given in Table1are constant,k H1C1?k H1C2?k,we have U?kF p=L for all plates in which F p is the e?ective heat transfer area of one

plate.

Fig.2.Equivalent two-stream heat exchanger

network.

From Fig.3we have,

x0 i ?

L;i is odd;

0;i is even;

x00

0;i is odd;

L;i is even:

e16T

It is further assumed that the thermal?ow rates are uniformly distributed in their channels.Thus,the ther-mal?ow rates in each channel are given as

_W i ?

=M H1;i is odd;

=M C1;i is even and i62M C1;

=M C2;i is even and i>2M C1:

:e17T

According to the channel connection shown in Fig.3,we also have G?0.The non-zero elements of G0and G00 are given by

g0 ik ?1if

k?1;i is odd;

or k?2;i is even and i62M C1;

or k?3;i is even and i>2M C1;

g00 ki ?

1=m H1;k?1and i is odd;

1=m C1;k?2;i is even and i62M C1;

1=m C2;k?3;i is even and i>2M C1:

:e18T

By setting L?1,for given values of integer variables M C1and M C2,the outlet stream temperatures can be calculated.The design problem given in Table1becomes min M C1tM C2

s:t:t t;C1àt00

260;t

t;C1

àt00

60;

e19T

The results are show in Table2for F p?0:2and0:1m2, respectively.3.3.Plate–?n heat exchangers

A plate–?n heat exchanger consists of?ns separated by?at plates,clamped and brazed together,as shown in Fig.4.The plates separating two?uids function as the primary heat transfer surface.The?n sheets between the adjacent plates hold the plates together and form a secondary surface for heat transfer.The space of?n sheets between two plates forms a?ow channel and is known as a layer.A multistream plate–?n heat exchanger contains more than two streams?owing through di?erent layers and sections of the exchanger. The exchanger usually consists of many passage blocks which are repetitively arranged.Each block consists of n layers.Since there is a very large number of layers in an exchanger,we usually assume that the behaviour of a block can adequately describe that of the entire ex-changer,therefore only n layers need to be analysed. There are two kinds of block arrangements.One is se-quential arrangement and the other is symmetrical ar-rangement.For the sequential arrangement of the blocks,e.g.,

áááA B C D

|??????{z??????}

Block jà1

A B C D

|??????{z??????}

Block j

A B C D

|??????{z??????}

Block jt1

ááá;

the layer number i?nt1points to the?rst layer in the upper block(i?1);the layer number i?0points to the n th layer in the lower block(i?n).For the symmetrical arrangement,e.g.,

áááD C B A

|??????{z??????}

Block jà1

A B C D

|??????{z??????}

Block j

D C B A

|??????{z??????}

Block jt1

ááá;

the layer number i?nt1points to the same layer in the upper block(i?n);the layer number i?0points to the same layer in the lower block(i?1).The sym-metrical arrangement also means that the block is thermally insulated at the upper and lower surfaces.If the whole exchanger is analysed,the symmetrical ar-rangement should be adopted.

Table2

Results of Example2

Stream F p?0:2m2F p?0:1m2

T00(K)M T00(K)M 1H1365.414367.925 2C1352

.57350.613 3C2325.36322.911 F(m2) 5.2m2 4.8m

Consider a block in a multistream plate–?n heat ex-changer,which has n layers and is divided along the

exchanger length into m sections according to the inlet and outlet positions of the streams as shown in Fig.5.Therefore the whole exchanger consists of mn channels.The elements of the mn ?mn coe?cient matrix A are given by Luo et al.[13]for both sequential and sym-metrical block arrangements as

a i à1eTm tj ;i à1eTm tj

?àU p ;ij tg ij U f ;ij _W

ij ?1

à12p i à1eTm tj ;i à1eTm tj

àtp I i t1eTà1? m tj ;i à1eTm tj á

!;e20a T

a i à1eTm tj ;l ?

U p ;ij tg ij U f ;ij 2_W

ij p i à1eTm tj ;l à

tp I i t1eTà1? m tj ;l

ál e?1;...;mn ;l ?i eà1Tm tj T

e20b T

for i ?1;...;n and j ?1;...;m ,in which

I ei T?

1;i ?n t1and sequential arrangement ;

i ;others

e21T

and the ?n e?ciency g ij is de?ned as

g ij ?tanh ???????Bi ij p =2àá=???????Bi ij p =2àáe22T

with

Bi ij ?h ij ààd ij á

a f ;ij F f ;ij =A f ;ij k f ;ij :

e23T

To calculate the plate temperatures the ?n bypass e?-ciency introduced by Haseler [6]is used which is de?ned

l ij ?2

???????

Bi ij p sin h ???????Bi ij

p :e24T

P is the coe?cient matrix of plate temperatures,T p ?PT :

e25TFor sequential block arrangement P ?Q à1C ;

e26Twhere P ,Q and C are mn ?mn matrices.The non-zero elements of Q and C are given as follows:i ?1;j ?1;...;m :

q j ;n à1eTm tj ?àl nj U f ;nj ;e27a T

q jj ?U p ;1;j tg 1;j àtl 1;j áU f ;1;j tU p ;nj tg nj àtl nj á

U f ;nj ;

e27b T

q j ;m tj ?àl 1;j U f ;1;j ;e27c Tc j ;n à1eTm tj ?U p ;nj tg nj U f ;nj ;e27d Tc jj ?U p ;1;j tg 1;j U f ;1;j :e27e T

i ?n ;j ?1;...;m :

q n à1eTm tj ;n à2eTm tj ?àl n à1;j U f ;n à1;j ;

e28a Tq n à1eTm tj ;n à1eTm tj ?U p ;nj tg nj àtl nj á

U f ;nj tU p ;n à1;j

tg n à1;j àtl n à1;j á

U f ;n à1;j ;e28b Tq n à1eTm tj ;j ?àl nj U f ;nj ;

e28c Tc n à1eTm tj ;n à2eTm tj ?U p ;n à1;j tg n à1;j U f ;n à1;j ;

e28d Tc n à1eTm tj ;n à1eTm tj ?U p ;nj tg nj U f ;nj :e28e Ti ?2;...;n à1;j ?1;...;m :

q i à1eTm tj ;i à2eTm tj ?àl i à1;j U f ;i à1;j ;

e29a T

q i à1eTm tj ;i à1eTm tj ?U p ;ij tg ij àtl ij á

U f ;ij tU p ;i à1;j

tg i à1;j àtl i à1;j á

U f ;i à1;j ;e29b Tq i à1eTm tj ;im tj ?àl ij U f ;ij ;

e29c Tc i à1eTm tj ;i à2eTm tj ?U p ;i à1;j tg i à1;j U f ;i à1;j ;e29d Tc i à1eTm tj ;i à1eTm tj ?U p ;ij tg ij U f ;ij :

e29e T

For symmetrical block arrangement P is an m en t1T?mn matrix

p l ;i à1eTm tj ?p ?

l ;i à1eTm tj ;i ?1;...;n à1

p ?l ;n à1eTm tj tp ?

l ;nm tj ;i ?n (l e?1;...;m n et1T;j ?1;...;m T;e30T

where P ??Q à1C

e31Tand Q and C are m en t1T?m en t1Tmatrices whose non-zero elements for 1

q jj ?U p ;1;j tg 1;j àtl 1;j á

U f ;1;j ;e32a Tq j ;m tj ?àl 1;j U f ;1;j ;

e32b

Fig.5.Arrangement of the streams,layers and sections in a plate–?n heat exchanger.

2700X.Luo et al./International Journal of Heat and Mass Transfer 45(2002)2695–2705

c jj ?U p ;1;j tg 1;j U f ;1;j :e32c T

i ?n t1;j ?1;...;m :

q nm tj ;n à1eTm tj ?àl nj U f ;nj ;

e33a Tq nm tj ;nm tj ?U p ;nj tg nj àtl nj á

U f ;nj ;e33b Tc nm tj ;n à1eTm tj ?U p ;nj tg nj U f ;nj :

e33c T

The matrices G ,G 0and G 00and the vectors x 0and x 00

should be set according to the particular con?guration of the exchanger.

3.3.1.Example 3

Take a four-stream aluminium plate–?n heat ex-changer as an example,of which the experimental data were given by Li et al.[15].The exchanger is used to cool the product stream A and heat the product stream D to given temperatures.The arrangement of the exchanger is B A C/D A B A C/D A B A C/D A B A C/D A B.However,only one block B A C/D A in sequential ar-rangement is taken for the calculation.The channel ar-rangement is shown in Fig.6.In the exchanger the hot water stream A is cooled by the cold water streams B,C and D.O?set strip ?ns (h ?4:7mm,s ?2:0mm,d ?0:3mm)are used for channels A,B and C,and perforated rectangular ?ns (h ?4:7mm,s ?4:2mm,d ?0:6mm)for channel D.The parameters for the i th layer and j th section can be calculated by U f ;ij ?2a ij eh ij àd ij TW =s ij ;U p ;ij ?2a ij es ij àd ij TW =s ij ;

Bi ij ?2a ij eh ij àd ij T2=ek f d ij T;

where W is the width of the exchanger,W ?130mm.The heat conductivity of ?ns k f ?191:58W/m K.The heat transfer coe?cients and thermal ?ow rates are

given in Table 3.Thus,the coe?cient matrix A can be obtained.According to Fig.6,the coordinate vectors and matching matrices are given as follows:x 0?0:925;1:24;0;0:925;0:925;1:24;0;0:925? T em T;x 00?0;0:925;0:925;1:24;0;0:925;0:925;1:24? T em T:The non-zero elements of G ,G 0and G 00are g 12?g 43?g 87?1;

g 022?g 031?g 053?g 064?g 0

71?1;g 0014

?g 0018?0:5;g 0021?g 0035?g 0046?1:

Table 3also gives the comparison between the measured outlet ?uid temperatures and the calculated ones.A good agreement is achieved between them.3.4.Heat exchanger networks

The general solution can also be applied to the net-works of two-stream heat exchangers and one-dimen-sional multistream heat exchangers by considering the network as a general multistream heat exchanger.However,if the network contains a large number of exchangers,the coe?cient matrix of the governing equation system would be enlarged,which might cause di?culties in calculating its eigenvalues and eigenvec-tors.

Let us consider a network with N streams and R heat exchangers.From Eq.(11)we have already obtained the temperature coe?cient matrices of R individual ex-changers V r ?

G 00r V 0r

V 0r

G r V 00r

à1

G 0r

er ?1;2;...;R T:e34T

We assume that each stream in an exchanger occupies one channel.Therefore,the network consists of M channels (M ?P R

l ?1N l )and the channel number of the n th stream in the r th exchanger can be set as m ?n tP r à1l ?1N l where N l is the number of streams in the l th exchanger.Thus,according to the energy balance at the entrance of each channel,the outlet stream tem-perature of the network can be expressed as T 00?G 00V I eàGV Tà1G 0T 0e35T

which

Table 3

Comparison of predicted outlet ?uid temperatures with the experimental data of a four-stream plate–?n heat exchanger

Stream _W (kW/K)a ekW =m 2K TT in (°C)T out ;exp :(°C)T out ;cal :(°C)A

1.354 1.64441.93

32.4332.53B )0.9604 1.79134.9339.40

39.03C )0.59021.46531.0639.6239.39D

)0.80150.8189

21.98

27.23

26.82

X.Luo et al./International Journal of Heat and Mass Transfer 45(2002)2695–2705

2701

V10

0V R

75:e36T

3.4.1.Example4

The example given in Table4is taken from[16].Its temperature–enthalpy diagram is shown in Fig.7.Ac-cording to Fig.7,we?rst consider a network consisting of?ve multichannel heat exchangers and one two-stream heat exchanger,as shown in Fig.8.The streams in each exchanger are arranged as follows:

EX1:H1C3H2,symmetric,

EX2:H1C1H2,symmetric,

EX3:H1C1H2C2,sequential,

EX4:C1H1C2,symmetric,

EX5:C1H3C2,symmetric,

EX6:H1C1,sequential.

The heat transfer parameters are calculated by

U iàj?

F iàj

L1=a it1=a j

àá;e37T

where F iàj is the heat transfer area between streams i and j.The design problem is to?nd F iàj for all matches so that the sum of them reaches minimum min

F iàj

s:t:t00

H k

àt t;H k?0ek?1;2;3T;

t t;C kàt00

C k

P0ek?1;2T;06F iàj:

e38T

To calculate the outlet stream temperatures,the neces-sary matrices of each exchanger are given as follows, respectively,in which the length of exchangers is set to be L?1.

A1?

àU1–3_

W H1

0U1–3

0àU2–3

U2–3

W H2

U1–3

U2–3

àU1–3tU2–3

75;

A2?

àU4–6_

W H1

0U4–6

0àU5–6

U5–6

W C1

àU4–6tU5–6

75;

A3?

àU7–9tU7–10

0U7–9

U7–10

0àU8–9tU8–10

U8–9

W H2

U8–10

W H2

U7–9

W C1

U8–9

W C1

àU7–9tU8–9

U7–10

W C2

0àU7–10tU8–10

C2 2

; A4?

àU11–12tU11–13

W H1

W C1

àU11–12

U11–13

0àU11–13

75;

A5?

àU14–15tU14–16

W HU

U14–15

W C1

àU14–15

U14–16

0àU14–16

75;

Table4

Problem data for Example4

Stream T s(°C)T t(°C)_W(kW/K)a(kW=m2K) 1H115060)200.05

2H29060)800.4

3HU181********

4C120125250.1

5C22

5100300.6

6CU101580

0.6

Fig.8.Heat exchanger network of Example4. 2702X.Luo et al./International Journal of Heat and Mass Transfer45(2002)2695–2705

A6?

àU17–18

U17–18

W HU

U17–18

W C1

àU17–18

x0 1?x0

?1;1;0

? T;x0

?1;1;0;0

? T;

x0 4?x0

?1;0;0

? T;x0

?1;0

? T;

x00 1?x00

?0;0;1

? T;x00

?0;0;1;1

? T;

x00 4?x00

?0;1;1

? T;x00

?0;1

? T;

G i?0;G0

i ?G00

?Iei?1;2;...;6T:

The non-zero elements of the matching matrices of the whole network are given as,

g1;4?g2;5?g4;7?g5;8?g7;11?g9;6?g12;9?g13;10?g14;17?g15;12?g16;13?g18;15?1;

g0 3;6?g0

6;4

?g0

8;2

?g0

10;5

?g0

11;1

?g0

17;3

?1;

g00 1;1?g00

2;2

?g00

3;14

?g00

4;18

?g00

5;16

?g00

6;3

?1:

Solving Eq.(38)we obtain,

EX1:F H1–C3?128:96m2;F H2–C3?6:02m2,

EX2:F H1–C1?45:58m2;F H2–C1?0m2,

EX3:F H1–C1?346:53m2;F H1–C2?0m2;

F H2–C1?269:27m2;F H2–C2?384:10m2,

EX4:F H1–C1?282:48m2;F H1–C2?218:01m2,

EX5:F H3–C1?37:81m2;F H3–C2?5:08m2,

EX6:F H3–C1?101:48m2.

The total heat transfer area is1825:32m2.Since F H2–C1 in EX2and F H1–C2in EX3are zero,EX2reduces to a two-stream heat exchanger and EX3should be sym-metrically arranged.

This network is equivalent to the network consisting of11two-stream heat exchangers.According to the cost equation[16]:

cost?8:6t0:67Area0:83earea in m2;cost in k$T

e39Tthe cost of exchangers is578.76k$.This value is much smaller than that given by Briones and Kokossis[16]. For the same problem,the application of their model yielded a network consisting of?ve exchangers.They did not give the structure of the network.In their ex-ample,the matches of design A are given as:H1–C1, H1–C3,H2–C1,H2–C2,H3–C2.The total heat transfer area is3314.1m2and the cost of the exchangers is699.2 k$.It should be pointed out that in this problem there are?ve outlet stream temperatures to be targeted(the sixth one is determined by the energy balance of the whole network).Therefore the degree of freedom is equal to the number of variable parameters minus?ve. If only?ve exchangers are used,the heat transfer area of each exchanger is?xed for given structure of the net-work.The possible structure of their design A is shown in Fig.9.The heat transfer area of each exchange can be obtained as:F H1–C3?151:59m2,F H2–C1?162:19m2, F H2–C2?519:30m2,F H1–C1?2462:94m2,F H3–C2?29:44 m2.The total area and total cost of exchangers are 3325.46m2and700.80k$,respectively,which are close to the results given in[16].

Now we consider the minimum total cost of heat ex-changers as the object function.Starting from a general multichannel heat exchanger network illustrated in Fig. 10,synthesis of the two-stream heat exchanger network by using the present general solution for Example4with minimum total cost of heat exchangers yields a net-work with six exchangers shown in Fig.11,whose areas are F H1–C3?151:59m2,F H1–C1?552:34m2,F H2–C1?344:11m2,F H2–C2?425:87m2,F H1–C2?238:80m2and F H3–C1?157:56m2,respectively.The total heat trans-fer area is1870.27m2and total cost of exchangers is 516.46k$.

The synthesis method used here is based on a stage-wise superstructure[2]and the whole temperature

?eld Fig.9.Heat exchanger network of Example4according to

[16].

Fig.10.Start structure of the multichannel heat exchanger network of Example4with minimum cost of heat

exchangers.

Fig.11.Two-stream heat exchanger network of Example4 with minimum cost of heat exchangers.

X.Luo et al./International Journal of Heat and Mass Transfer45(2002)2695–27052703

of the network calculated with the general solution.The advantage using this method is obvious because the synthesis problem reduces into a general non-linear optimization task with outlet temperature constraints and other additional constraints if needed.The binary variables determining whether the corresponding ex-changer exists are not necessary because usually the value of n in the cost equation

cost?atbF ne40Tis less than one.It is convenient to use

cost?

atbF n;F P F min;

atbF n

min

àá

F=F min;F

e41T

In this case the optimization will automatically yield both the area and cost of an unnecessary exchanger to zero.

4.Conclusions

A general form of the analytical solution for various types of one-dimensional multistream heat exchangers and their networks is proposed.The methods for the use of the general solution are illustrated in detail for one-dimensional?ow shell-and-tube heat exchangers,plate heat exchangers,plate–?n heat exchangers and heat exchanger networks.The solution is valid for any types of two-stream heat exchangers by introducing the cor-rection factor of logarithmic mean temperature di?er-ence.

The outlet temperatures of the streams in a multi-stream heat exchanger or heat exchanger network are explicitly given by Eq.(11)or Eq.(35).Therefore it is easy to obtain the outlet stream temperatures for an existing heat exchanger or heat exchanger network. However,for design problems of multistream heat ex-changer and their networks no simple relationship for the calculation of heat transfer area is avail-able.Therefore a constrained optimization algorithm is needed to determine unknown heat transfer areas and other parameters under a set of constraints,as shown in Examples1,2and4.

The general solution is also applied to the synthesis of heat exchanger networks.Based on this solution the stage-wise superstructure method is developed to solve the synthesis problem of two-stream as well as multi-stream heat exchanger networks.For Example4a better network than that of[16]is obtained.

The present solution can be applied to the case of variable physical properties by dividing the?ow pas-sages into several channels and assuming that in each channel the?uid properties are constant.Iterations are needed because the?uid properties should be calculated according to the mean temperatures of the?uids in the corresponding channels.

Acknowledgements

The present research work belongs to the project ‘‘Optimal design,?exibility analysis and dynamic simu-lation of multistream heat exchanger networks’’(No. RO294/9)supported by the Deutsche Forschungsgeme-inschaft(DFG)and the project‘‘Field synergetics and control of heat transfer processes with multiple streams’’(No.G2000026301)supported by the National Devel-opment Program of China for Key Fundamental Re-searches.The authors would like to acknowledge the above?nancial supports.

References

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their Speci?cation and Use,HTFS,Oxon,UK,1987. [2]T.F.Yee,I.E.Grossmann,Z.Kravanja,Simultaneous

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[8]A.Settari,Remarks about‘‘General solution of the

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[9]T.Zaleski,A.B.Jarzebski,Remarks on some properties of

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