Post on 05-Jul-2016
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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