179

2008,20(2):179-185

ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF FLOW

DISTRIBUTION IN MANIFOLDS FOR HEAT EXCHANGERS*

LU Fang, LUO Yong-hao, YANG Shi-ming

School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China,

E-mail: lufang@sjtu.edu.cn

(Received April 2, 2007, Revised August 17, 2007)

Abstract: The prediction of flow distribution in flow manifolds is important for the design of heat exchangers. The pressure drop

along the flow in the header is the most influential factor in flow distribution. Various continuous models available in literature have

failed to satisfactorily predict the pressure distribution in the headers of the flow manifolds. In this article, a discrete model matching

the real physical phenomena has been proposed, to predict the pressure distribution in headers. An experimental evaluation of

relevant flow characteristic parameters has been carried out to support the discrete model calculations. The validity of the theoretical

discrete model has been performed with experimental results, under specific conditions. Refined experimental probes, for pressure

heads with ultrasonic measuring devices, have been used to obtain accurate results. The experimental results fully substantiate the

soundness of the theoretical prediction. In addition, the advantage of the ability to accommodate local disturbances in the discrete

model has been pointed out. The effect of some local disturbances may be substantial. As a result of the analysis presented in this

article, improved designs of flow manifolds in heat exchangers can be realized, to assure operation safety under severe operating

conditions.

Key words: heat exchanger, flow distribution , flow manifold, discrete model

1. Introduction

Heat exchangers with flow manifolds have wide

industrial applications. Superheaters and reheaters in

utility boilers, and solar collectors and heat

exchangers in petroleum engineering are some

important examples. The prediction of flow

distribution in the flow manifolds is essential in the

design and operation of these heat exchangers[1]

.

Operation safety of flow manifolds will be seriously

impaired with improper design. The flow schematics

of the manifolds for both the dividing and combining

flows are physically discrete processes. However, a

search in literature has revealed that no general

purpose discrete model is available to analyze the

flow in manifolds. Currently available models

proposed by Bajura[2]

, Bajura and Jones[3]

, Lokshin et

al.[4]

, Shen[5]

, Zhu[6]

, Zhang[7]

, Zhong[8]

, and

* Biography: LU Fang (1970- ), Male, Ph. D. Student, Senior

Engineer

others[9,10]

are all continuous models. Zheng et al.[11]

and Pustylnik et al.[12]

have analyzed two-phase flow

distribution in parallel pipes, in the continuous models.

Zhang et al.[13]

have simulated and analyzed flow

patterns in cross-corrugated plate heat exchangers.

Jones and Lior[14]

have tried to analyze the flow

manifolds in solar collectors with a discrete model,

but the negligence of the velocity heads normal to the

flow direction in headers and other simplifications,

limits the usefulness of their analysis. Fu et al.[15]

have

numerically analyzed the flow in a square cross

section header with two branch tubes, and their

analysis falls in the category of the discrete model

analysis. However, only a special case is considered in

their article and no general formulation is attempted. It

is the purpose of this article to formulate a discrete

model, to generally analyze the flow distribution in

the flow manifolds. Experimental determination of

various characteristic parameters under a wide range

of operating conditions will be carried out to support

the model. A specially designed flow distribution

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experiment will also be carried out to test the validity

of the theoretical discrete model. In addition, through

comparison with the experimental results, the

quantitative inadequacy of the representative

continuous model will be revealed.

2. Formulation of the discrete model

The basic idea for the formulation of the discrete

model is to separately formulate the basic equations

for header sections with outlets or inlets, and header

sections without outlets or inlets. The number of

branch tubes along the circumference of the header

may vary. As the flow distribution is characterized by

average quantities, velocity and static pressure are all

taken as the average value. Single phase and

incompressible fluid flow under isothermal steady

conditions are considered in this article. The flow is

assumed to be under a fully developed condition

before reaching the first row of branch tubes. The

configuration of a dividing flow manifold is shown in

Fig.1. With reference to the control volume of the

header section at the ith outlet, as shown by the

portion enclosed by dotted lines in Fig.1, the

momentum equation in the x direction is

Fig.1 Configuration of a dividing flow manifold

2 2= +dl dr x d d dr dlP i P i P K V i V i

= 1, 2,i n (1)

where denotes the friction loss in the x direction

and K

Px

d denotes the static pressure change coefficient,

which takes into account the losses caused by the

lateral flow through the branch tubes.

With reference to the control volume for the part

of the header without branch tubes, between the

(i-1)th outlet and the ith outlet, the momentum

equation in the x direction is

21( ) = ( 1) ( ) ( )

2

s bdl dr dl d dl

d

L DP i P i i V i

D

= 1, 2,3,i n (2)

The Bernoulli equation for the lateral flow is

1( ) + ( ) = +[1+ + +

2dl dr out p tdP i P i P C C

21( ) ] ( )

2

bb b

b

Li Vb

D= 1, 2,i ni (3)

where Cp denotes the local friction coefficient of the

branch tubes, for example, that caused by the

throttling ring, and Ctd denotes the inlet friction

coefficient of branch tubes.

The continuity equation is

( ) ( ) = ( )d d dl dr b b bA V i V i A V i M

= 1, 2,i n (4)

where Ad and Ab denote the cross section area of the

header and branch tubes respectively, and M denotes

the number of branch tubes at the ith outlet.

( ) = ( 1)dl drV i V i (5) = 1, 2,3,i n

n

The boundary conditions are

(1) =dl dinV V (6)

( ) = 0drV n (7)

Similarly, for combining flow manifolds, the

momentum equation for the control volume in the

header at the ith inlet is written as

2 2( ) ( ) = + ( ) ( )cl cr x c c cr clP i P i P K V i V i

= 1, 2,i (8)

where Kc denotes the static pressure change

coefficient.

With regard to the control volume for the part of

the header without outlets, between the (i-1)th inlet

and the ith inlet, the momentum equation in the x

direction is:

21( ) = ( 1) ( ) ( )

2

s bcl cr cl c cl

c

L DP i P i i V i

D

= 1, 2,3,i n (9)

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The Bernoulli equation for branch tubes reads as

follows:

1( ) ( ) = [0.5 + + +

2cl cr in p tcP i P i P C C

21( ) ] ( )

2

bb b

b

Li Vb

D= 1, 2,i ni (10)

where Pin denotes the environment pressure at the ith

inlet of the branch tubes, and Ctc the friction

coefficient at the exits of the branch tubes.

The continuity equation is

( ) ( ) = ( )c c cr cl b b bA V i V i A V i M

The experimental setup for the evaluation of flow

characteristic parameters is shown in Fig.2. For the

measurement of the water heads with a multitube

manometer, a special computer controlled ultrasonic

static head measuring device was used. This device

was also used in the evaluation of pressure

distribution in headers and will be described in detail

in the next section. All results are shown in Figs.3 - 5.

In these figures, the relative variation of the respective

parameter, with respect to the reference case of V= 1, 2,i n

n

(11)

( ) = ( 1)cl crV i V i i (12) = 1, 2,3,

The boundary conditions are

( ) =cr coutV n V (13)

(1) = 0clV (14)

Equations (1)-(14) represent the complete

formulation of the discrete model. They can be solved

through the iterative method on a computer.

The static pressure change coefficients Kd and Kc,

and branch tube inlet and exit resistance coefficients

Ctd and Ctc are flow characteristic parameters. These

parameters are essential for the solution of the discrete

model equations. They are determined experimentally

and will be discussed in the next section.

3. Evaluation of flow characteristic parameters

For the application of the discrete model,

accurate values of flow characteristic parameters Kd,

Kc, Ctd and Ctc are needed. These values may vary

with different conditions of operation and construction.

Because of the complexity of the flow, these values

have to be evaluated through a large number of

organized experiments. There are three operation and

construction variables relevant to the evaluation of

flow characteristic parameters. The first is the relative

velocity of flow in the branch tube as compared with

the velocity of flow in the header, that is, Vb/VdL for

the dividing flow or Vb/VCL for the combining flow.

The second is the diameter ratio of the branch tube to

the header, that is, Db/Dd for the dividing flow or

Db/Dc for the combining flow. The third is the number

of outlets or inlets in one cross section of the header.

This number will be denoted by M. The ranges of the

construction variables covered in the evaluation are as

follows:

Diameter of header, Dd or Dc: 80 mm, diameter

of branch tube, Db: 14 mm, 24 mm, 35 mm, M: 1, 2, 3,

4, 5, 6, 8, 12.

b/VdL

= 1, Db/Dd = 0.3, M = 1 has been adopted. It may be

pointed out that the effect of M is significant only for

M < 4.

Fig.2 Schematic of experimental setup for flow characteristic

parameters

Fig.3 Effect of the relative velocity of flow in the dividing

flow manifolds

Fig.4 Effect of the diameter ratio in the dividing flow

manifolds

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Fig.5 Effect of the number of branch tubes in one cross

section of the header, in the combining flow manifolds

Fig.6 Comparison of calculation and test of pressure

distribution in dividing header

Fig.7 Comparison of calculation and test of pressure

distribution in combining header

4. Experimental study of flow distribution in

header

For the prediction of flow distribution in flow

manifolds, the pressure distribution in the header and

the inlet and exit resistance coefficients of the branch

tubes need to be known. In this section, the

experimental study of the static pressure distribution

in the header will be considered. The experimental

results provide a crucial test for the validity of the

theoretical model. The same experimental setup in

Fig.2 is used, except that the test section is replaced.

The test section for the dividing flow is shown in Fig.

6. The test section for the combining flow is shown in

Fig. 7. The test section is made of Plexiglas. Air is

used as the flow medium. The diameter of the header

is 80 mm. The diameter of the branch tube is 24 mm.

The length of the branch tube is 330 mm. For the

dividing flow system, there are 15 branch tubes with a

pitch of 92 mm, whereas, for the combining flow

system there are 8 branch tubes with a pitch of 100

mm. The inlet Reynolds number for the dividing flow

header is Rein = 2.6 105. The exit Reynolds

number for the combining flow header is Reout = 1.44

105. The measuring points for static pressure are

located on the part of the wall without outlets or inlets

and at the middle point between neighboring branch

tubes. The flow rates of air are measured with flute

type tubes located in the steady flow area, at the inlet

of the dividing flow system or the exit of the

combining flow system. The flute type tubes are

calibrated with the standard Pitot tube before the test.

All static and dynamic pressure values have been

recorded by a computer controlled ultrasonic water

head measuring device. A schematic of the measuring

device is shown in Fig.8. The working principle of

such a device is to record the traveling time of an

ultrasonic pulse issued from the piezoelectric ceramic

ultrasonic energy converter installed at the base of the

measuring glass tube. The ultrasonic pulse signal will

reflect and travel downwards at the top free surface of

the water column. The reflected signal is picked up by

the energy converter. The elapsed time between the

release and pickup of the pulse signal indicates the

water head Hi . Thirty measurements can be

accomplished within 1 s. The gross average of several

scans will eliminate any unavoidable pulsations of the

water column. The measurement accuracy of the

water head of this device is 0.5 mm. The

application of this device assures the accuracy of the

experimental results. The experimental results

obtained are compared with the theoretical discrete

model calculations in Figs.6-7. In these figures, L

denotes the total length of the header and X the local

distance along the flow direction. The agreement

indicates the soundness of the discrete model. For the

dividing flow header, the maximum deviation of

theoretical prediction is 8.16%, and the rms deviation

is 4.20 %. For the combining flow header, the

maximum deviation of theoretical prediction is 5.79 %,

and the rms deviation is 1.59 %.

5. Comparison with representative continuous

model

A comparison of the continuous model with the

new discrete model is discussed in this section. As the

experimental results agree satisfactorily with the

discrete model, the inadequacy of the continuous

model can be revealed from the differences in

comparison. For the purpose of reasonable

comparison, constant pressure distribution in the

combining header will be assumed in the calculation

of pressure distribution of the dividing header. On the

183

other hand, constant pressure distribution in the

dividing header will be assumed in the calculation of

the pressure distribution in the combining header. As

the pressure distribution in the header is the most

influential parameter for flow distribution, the

comparison will be concentrated on this parameter.

With reference to the flow rate distribution in the

branch tubes, only the minimum flow rate deviation

coefficient will be considered. A representative

continuous model of Lokshin et al.[4]

is selected for

comparison.

Fig.8 Schematic for ultrasonic water head measuring device

Fig.9 Effect of branch tube resistance coefficient on the

pressure distribution in header (Dd=390, Db=44, Lb

=15000(40000), Ls =213, N=29, M=8)

Figure 9 shows the comparison of different

models for the effect of variation of branch tube

resistance coefficients on the pressure distribution for

the dividing header. In this figure, Eu denotes the

Euler number

20.5

i re

in

Euv

f

Fig.10 Effect of pitch variation of branch tube on the pressure

distribution in header (Dd=390, Db=44, Lb =15000,

N=29, M=8)

where i is the static pressure point, ref is the

base static pressure at the inlet tee, is the density,

is the inlet velocity at the inlet tee. H denotes the

branch tube resistance coefficient, which equals the

bracketed term on the right hand side of Eq.(3).

inv

min denotes the minimum flow rate deviation

coefficient. Variation in the branch tube resistance

coefficient can be realized by changing the length of

the branch tubes. Longer branch tubes cause an

increase in the branch tube friction resistance.

Consequently the pressure distribution in the header

tends to be more uniform as a result of an increase in

the system pressure drop. The continuous model,

however, fails to reflect this variation. Figure 10

shows the effect of pitch variation in branch tubes on

the pressure distribution in the dividing header. An

increase in the pitch distance will increase the share of

friction resistance in the decrease of static pressure in

the header along the flow direction. Consequently the

value of the minimum flow rate deviation coefficient

will increase. The discrete model reflects this trend.

However, for the continuous model, both pressure

distribution and the minimum flow rate deviation

coefficient remain unchanged, which is incompatible

with the real situation. Figure 11 shows the

comparison of the effect of variation on the number of

branch tubes, M, and on the pressure distribution in

the dividing header. The increase of M reduces the

flow rate in each branch tube. It causes a reduction of

pressure drop in the system, and consequently

increases the nonuniformity of pressure distribution in

the header. The results of the discrete model reflect

this trend clearly. The results of the continuous model,

however, disregard the variation of M values and are

incompatible to the real situation. Similar comparisons

184

for the combining header are shown in Figs.12 and 13.

In the case of M = 1(see Fig.12), the difference of

these two models is larger in the central part of the

header than in the near end parts. In the case of M =

8,(see Fig.13), the disregard of the variation of the M

value in the continuous model, is indicated by the

differences of the two curves in the figure. It is clear

through these comparisons that the inherent

inadequacies of the continuous model have been

effectively corrected by the discrete model.

Fig.11 Effect of M on the pressure distribution in header

(Dd=390, Db=44, Lb =40000, Ls =213, N=29 )

Fig.12 Effect of large pitch on the pressure distribution in

combining header (Dc=1393.3, Db=44, Lb =40000,

Ls =426, N=29, M=1)

Fig.13 Effect of M on the pressure distribution in combining

header (Dc=390, Db=44, Lb =40000, Ls =213, N=29,

M=8)

As can be seen from the discrete model, it is

crucial to know the static pressure distribution and

entrance resistance coefficient of the branch tube in a

dividing flow header to calculate the flow distribution

of manifolds. For a dividing flow header using tee

structures in inlets and outlets, the static pressure

distribution and entrance resistance coefficient of the

branch tube are strongly influenced, because of the

effect of the vortex flow in the inlet tee structure.

Therefore, an experimental model of isodiametric tee

structure is set up and the static pressure distribution

and entrance resistance coefficient of the branch tube,

Ctd, in the tee vortex area are measured.

The effect of the vortex flow in the inlet tee

region, on the pressure distribution, is shown in Fig.14.

Because of the effect of vortex flow in the inlet tee,

the static pressure distribution is greatly changed.

Change of static pressure in the vortex area is closely

related to the flow ratio between the inlet tee.

However, the position of the lowest static pressure is

basically unchanged around x/D 1.25. As is

presented in the experimental results, the entrance

resistance coefficient of branch tube Ctd is also greatly

changed in the inlet tee vortex influential areas.

Fig.14 Effect of vortex flow in inlet tee on the pressure

distribution

6. Discussion

In addition to the inadequacies of the continuous

model, the drawback in the continuous model also

presents its inability to accommodate local

disturbances. The use of the T-junction as an inlet or

outlet, for instance, presents significant local

disturbances. Drastic changes in the local pressure at

the near junction region, because of recirculating flow,

has been pointed out by Pollard[16]

. The assumption of

parabolic pressure distribution in the header rules out

the possibility of taking care of these local

disturbances. Overlooking the local disturbance in the

T-junction will cause a significant error in the

prediction of flow distribution. On the other hand, it is

the distinct advantage of the discrete model to

accommodate any local disturbances. Different values

can be assigned to a particular location independently.

To perform the calculation, all one needs is the

information of experimental pressure distribution in

the header and experimental inlet and exit resistance

185

coefficients for the branch tubes under local disturbing

conditions. A study with the discrete model for a

particular case of a boiler reheater with T-junction

inlet revealed a relative error of 22.36 % on the flow

rate deviation coefficient by the neglect of local

disturbances[17]

.This again demonstrates the power of

the discrete model to correct the deficiencies of the

continuous model.

7. Conclusion

A discrete model formulation for the calculation

of flow distribution in flow manifolds has been

proposed from first principles. Experimental

evaluation of flow characteristic parameters has been

carried out to support the discrete model. Experiments

have been conducted to obtain the pressure

distribution in the header under specific conditions.

The satisfactory agreements between experimental

and theoretical pressure distribution in headers

confirm the validity of the theoretical discrete model.

The advantage of the ability to accommodate local

disturbances in the discrete model has been also

pointed out. As a result of the analysis proposed in

this article, improved design of flow manifolds in heat

exchangers can be realized, to ensure operating safety

under severe operating conditions.

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