Ind. Eng. Chem. Fundam. 1983, 22, 463-471
Literature Cited Lee, 6. I.; Kesler M. 0. AIChE J . 1975, 27, 510-525. Leland, T. w.; Rowlhmn, J. s.; Sather, G. A. T ~~~~~h~ ~ soc, ~ 1968, ~ 64, , 1447- 1460. Mason, E. A-Saxena, S. C. Phys. Fluids 1958, 1 , 361-369. Michels, A.; Sengers, J. V.; Van Der Gullk P. S. PhySlC8 1962, 2 8 , 12 16- 1220. Michela, A.; Sengers, J. V.; Van de Klunderl J. M. physlca 1983, 29, 149-1 60. Misic, D.; ThodOs, G. PhySiCe 1966, 32, 585-899.
463
Stiel, L. I.; Thodos, G. AIChE J. 1964, IO, 26-29, Wasslljewa, A. Phys. 2.1904. 5 , 737-742. YoriWW, M.; Yoshimura, S.; Masuoka, H.; Yoshkla, H. Ind. Eng. C2mm. Fundarn. 1983, preceding paper In this issue.
Received for review December 2, 1981 Revised manuscript received F e b r u a r y 7 , 1983 Accepted June-23, 1983
Flow Distribution in Piping Manifolds Robert L. Pigford,’ Muhammad Ashraf,t and Yvon D. Mlron’ Depadmnt of Chemical Englneerlng, University of Delaware, Newark, Delaware 1971 1
The uniformity of flow rates among the parallel tubes of a piping manifold is governed by the variations in fluid pressure inside the entrance and discharge headers. These result from fluid friction and from loss or gain of fluid momentum at exit and entrance ports. Using hydraulic flow coefficients derived from experiments, a theory for the distribution of velocities is developed for turbulent flow In manifolds having cylindrical headers of equal diameter. The results are confirmed by experiment and are used in two design examples.
Introduction There are many times in the design of process flow equipment when it is necessary to subdivide a large fluid stream into several parallel streams, to process these streams separately, and then to recombine them into one discharge stream before sending the fluid to another step in a process. The entering feed stream is subdivided by a header to which the parallel small tubes are connected at right angles. After treatment, the parallel streams are combined through ports leading into an output header. Aa an example, reactions which are accompanied by evolution or absorption of heat can be carried out in small tubes in order to provide surface for heat transfer. This occurs in furnaces used for heating fluids in petroleum refineries and in some fixed-bed catalytic reactors. A key question which arises in the design of such units is the uniformity of the flow distribution which will be obtained. The greater the pressure drop across the parallel tubes and the smaller the pressure changes inside the two headers to which the tubes are connected the more uniform the flow distribution will be. Headers can have various shapes. This paper assumes cylindrical piping headers. Given the number of tubes to be placed in parallel and the fluid friction coefficient, the length, and the diameter €or each tube, how large must the inlet and exit headers be to provide a flow distribution among the tubes as uniform as may be required? The detailed flow patterns among all the paths of a complex piping system can be computed from straightforward use of the one-dimensional equations of fluid motion if enough information is available on the pressure changes that occur near the points at which streams divide or recombine. For turbulent flow, at least, the computations are expected to be simple because all pressure changes will be nearly proportional to squares of velocities and the percentage variation of flow will be independent ‘ S t a n f o r d Research I n s t i t u t e ,
Palo Alto, CA.
* Hercules of Canada, M o n t r e a l , Canada.
0196-4313/83/1022-0463$01.50/0
of velocity. Nevertheless, the numerical work required may be long and tedious when there are many tubes in parallel. A general solution of the flow equations leading to tables or graphs which are generally applicable should save much effort when a design is needed. Such a solution is presented here, along with some recommended values of empirical coefficients representing wall friction and momentum recovery in headers of uniform diameter. Turbulent flow and essentially constant fluid density are assumed. Vaporization of the fluid is not included. The entrance and discharge headers are assumed to have constant and equal diameters. Distributions in Piping Manifolds Two arrangemenh of two headers and parallel tubes are considered, as illustrated in Figure 1. The one at the top can be called “reverse-flow” because the fluid in the exit header leaves in the direction opposite to its path in the entrance header. It is referred to here as a “U-manifold.” Alternatively, the two flow directions may be the same, as in the parallel-flow “Z-manifold,” also shown in the figure. Among other things, we want to know which design is better, i.e., which gives the more nearly uniform flow distribution among the connecting tubes or requires the smaller total pressure drop when the input flow rates are the same. The flow streamlines in the headers are not simple near side ports but we assume nevertheless that the one-dimensional flow approximation, often called the “hydraulic approximation,” is valid. Thus the fluid pressure in an entrance header will change for two reason: (a) because of wall friction in the straight sections between adjacent side outlets, the fluid pressure will fall in the flow direction; (b) near each exit port in the entrance header the pressure will rise because an opposing force (a fluid pressure difference) is needed to cause the exit fluid to lose some of its forward momentum as it leaves. Similar changes occur in the exit header as entering fluid acquires momentum. The net result is that the pressure in each header both rises and falls, as illustrated in Figure 2. In the exit header, @ 1983 American Chemical Society
484
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983 Un
IVIl
Pb U n r l
Pn
It1
-L!--+q PI
A -
UI
-
L-
:1
L-
_ _ _ _ _ _ _' I
P2
u2
Figure 3. Sketch of a flow exit port in an entrance header.
total momentum flux correctly if it is squared and multiplied by density and cross section, unless the velocity is uniform and unidirectional. In the hydraulic model such deviations are disregarded, their effects being absorbed into empirical coefficients which may vary with flow rates. One of the most important of these coefficients is a momentum recovery factor, k, defined by eq 1. Referring to Figure 3 and using a balance of momentum entering and leaving the dashed control volume we see that Un
Pn un+l
Pn
uN+I'
0
Figure 1. Two arrangements of piping manifolds.
6- 841 t
fc
1
4
6
K
i
I
I
jz.,.I....., 0.2
Y 100
200
300
DISTANCE F R O M ENTRANCE AND E X I T , in
We have assumed in eq 1that all the momentum loss has resulted from pressures acting in the header cross section. In fact, some of the fluid entering the cross channel will retain part of its initial forward momentum as it crosses the control surface and leaves the entrance header. The result is that we need an empirical coefficient k in eq 1 which becomes
We shall find that experimental lz values are greater in exit than in entrance headers. The former values are approximately unity, the latter about 0.4. Note from Figure 2 that observed preasures near the side porta can vary with angular position. The pressures used in eq 2 are values extrapolated to the center line of each exit port, based upon measured pressures in the header segments at a distance. The gradient of such measured values determines the Fanning friction factor for the header, f H .
Figure 2. Observed pressure distribution in the headers of a Umanifold.
(3)
fluid which enters from a side port must be accelerated, requiring a greater average pressure before the port than after, i.e., a pressure fall in the direction of flow. In a U-manifold these pressure changes owing to momentum effects will tend to compensate; in a %manifold, however, the pressure changes tend to reinforce each other and the pressure differences between headers tend to vary more from one end to the other. We have conducted experiments, to be described below, in which we observed fluid pressures in the headers and flow-rate distributions. Reference to Figure 2 will show that our observed header pressures can be approximated by a series of stright line segments between porta and abrupt changes a t the axes of the ports. Thus, the flow phenomena may be divided into fluid friction at tube walls between adjacent porta and momentum effects occurring over short distances at the porta themselves. Because of the absence of uniform fluid velocity over the cross section of a header, the hydraulic assumption may be in error, leading to an abnormal value of the measured friction factor. Moreover, the mean velocity will not give
Both k and f H can be expected to depend on several things. Not only will the Reynolds number be important, but the ratio of diverted flow to in-line flow, the roughness or radius of curvature of the fittings, and the length of a piping segment between successive ports may be significant too. Few measurements of k have been published. The most complete analysis of existing data appears to be that of Miller (1971). Converting values of his differently defied coefficients to our kE and kD for entrance and discharge headers, respectively, we obtain the values shown in Figure 4. Data from other sources are also shown in the figure. Although data from different investigators do not agree, it is clear that values of kD exceed those of kD Thus, a diverted stream may lose a little less than half of its forward momentum (kE 0.4) before it leaves the entrance header, but the same mass of fluid must gain essentially d l of its new forward momentum (kD 1.0)after it joins the fluid in the discharge header. We shall see that this difference has an important bearing on flow distributions. Such differences mean that pressure distributions in the
-
-
Ind. Eng. Chem. Fundam., Vol. 22,
1.2
I
I .o
\\
J
-
\
k,
-
I
I
Similarly
I
un- un+,=
,Average of alldata from Miron Gardel
Miller
Manson
Miller
0.8 Vogel
d= I
Io.
0.6 .
CNown
k, 0.4.-
'Averaae r of % all data 'Average from Miran
d'zD 3
.( g)'.
These two equations can be expressed jointly as 1-s un= u1 + sun 2 where we have used the boundary conditions, uN+1 = 0 at the closed end of the entrance header and where u1is equal to a value known at the entrance. To relate the tube velocities, Vn,to the header pressures one could include a second set of empirical flow coefficients as used by Miller (1971) to account for entrance and exit effeds at the ends of branching tubes. A simpler approach is usually acceptable, however, when the pressure drop across the connecting tubes is caused mostly by friction inside them rather than by end effects. We assume here that the arithmetic averages of header pressures extrapolated from the two sides of the manifold adjacent to the port can be employed and write
d=g
COMBINING FLOW
0.6
No. 4, 1983 465
dzD
Miller
"
I
0
43-
0.2
0.4
0.5
0.6
I.o
41 1% Figure 4. Values of the momentum-change coefficient, kE and k,, from various sources.
These equations give the solution for the pressures and the velocities. As shown in the Appendix, the pressures and discharge-header velocities can be eliminated, obtaining the second-order difference equation for the entrance header
entrance and exit headers will not be parallel, even for U-manifolds. Equations for Flow Distribution In order to find the flow rate through one of the tubes connecting the headers, we must be able to compute the difference in the pressures at its ends. This requires that we express the pressures in the two headers in terms of the friction factors for the connecting segments of the headers and the momentum coefficients at the branches. Referring to Figure 1,we express the pressure loss owing to friction between tubes n and n + 1 in the entrance header as ~ n ' Pn+l
=
4fnu
PU;+~
X D 2gc
(4)
In the discharge header 4fHU PG+l X(5) D 2gc where s is equal to +1 for a U-manifold and -1 for a Zmanifold, respectively. At the points where dividing and combining flow occurs we have, from eq 3 P,' - Pn+l = -s-
P
pn' - P n = k E z ( ~-i ui+J
(6)
and
The number of velocity heads of pressure drop across the tubes, H = Aptub/(pV/2gJ,is introduced because in many applications thisquantity will be more wful than the tube friction factor, fP The entrance-header velocity ,u,, is now dimensionless, the original U-value having been divided by the entrance velocity, ul. Equation 12 is a second-order nonlinear difference equation. Although one particular solution can be found, we have not been able to find the general solution but have had to resort to numerical computations. After numerical values of the ui have been found the values of (d/D)2Vi can be found from eq 8. Comparing these first differences we can compute a "quality index factor," Q, from its defining equation
(7) respectively. A material balance, assuming density constant, is
Moreover, the pressure drop for the whole manifold can be computed from equations given in the Appendix. The differential equation equivalent to eq 12, valid in the limit as n becomes large, can be found by approxi-
468
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
Table I. Summary of Calculated Flow Distributions and Pressure Drops for U-Manifolds
N, no. tubes in parallel 5
50
a
h,, re1 A p between hdrs.
h,= 0
0.4 1 3 10 30 100 300 0.4 1 3 10 30 100 300
60.612.315 34.610.870 14.210.274 4.6/0.080 1.610.026 0.5l0.008 0.21 65.112.40 39.310.880 16.8/0.277 5.610.081 2.010.027 0.6/0.008 0.2 10.003
Q and (APIAPpi - 1)” h,= 1 h,= 3
h,= 10
76.313.85 54.111.64 28.110.571 10.610.174 3.910.058 1.2/0.017 0.41 81.014.98 60.512.11 33.610.131 13.310.223 4.910.014 1.5/0.024 0.51
68.012.96 43.111.16 19.510.377 6.710.112 2.410.037 0.7/0.011 0.21 12.813.37 48.811.32 23.510.433 8.410.129 3.010.043 0.9/0.013 0.31
87.515.18 11.712.88 46.511.147 21.610.379 9.410.131 2.810.040 0.91 91.119.18 17.614.34 53.511.66 26.610.531 12.510.184 3.710.059 1.31
The number before the slant bar is Q in percent; the number after is (APIA&) - 1
mating the first and second differences by the corresponding derivatives with respect to n. The result is headers on uniformity of flow dislribution In u - manifolds I No header wall friction, h2 = 0 1
C
u’u’’- 2Buu’- ;s(l- s ) u ’ 5 ,
A[ u2 + S( 4
+ SU)”]
= 0 (17) t
40)=1 (184 u(N) = 0 (lab) These expressions can be made independent of N by introducing the relative length of the header, [ = n/N, as a new independent variable. The result is
”[ .( 9 .)“I u2
2hl
+
+
1
c 01
01
= 0 (19)
\\\\
N = 2’
1
I
I l l l l l i
IO
I
I
1 lIl11i
,
,
\\\\
\\\
1 IIlIIi
100
1000
LLLj
1000.0
h l = ( 4 f , l / d l l D/d ) 4 N - 2
Figure 5. Effect of pressure difference between headers on uniformity of flow distribution in U-manifolds.
Greater values of hl cause the flow distribution to be more nearly uniform because of greater pressure difference between headers; greater hz results in less uniform flow because of greater pressure variation in the headers.
where
hz = ( 4 f ~ u / D ) N
(22)
Since constant values can be used for k D and k E only the two combinations of parameters given in eq 21 and 22 affect the u(E)function and the constancy of the function V([) when N is larger than about 20. Therefore, grouping numerical computations into sets at constant hl and h2 will make it possible for us to present the results compactly in a form easy to use. Note that hl and h2have physical meaning. If the flow through the tubes were uniform Nd2V* = D2u1,or V*= (D/d)2(ul/N). Expressing the pressure difference between the two headers in units of one entrance velocity head, PU12/2& p - p - (PV/2k!C)(4ftl/d) = hl (23) (PU12/ 2gc) (PU12/2t%) Similarly, the normalized s u m of all the frictional pressure drops in either header is
Numerical Solutions of the Equations a. U-Manifolds. Inspection of eq 12 will show that for U-manifolds (s = +1)the first term on the right reduces to 2 ~ 2 , and + ~ the third term disappears. As a result, the coefficient, C has no effect on the solution and the equation is homogeneous in the squares of velocities. An efficient method of numerical solution is available in which successive approximation is not necessary. Details are given in the Appendix. Figure 5 shows a graph of Q values vs. hl for h, = 0 and various N values. Table I gives numerical values of Q for various hl, hz, and N , it is useful for design calculations. Figure 5 shows that, for hl greater than about 3, Q is inversely proportional to M if N is large enough. For small values of N the dependence is greater. For N equal to 50 the Q values are only 2% smaller than values found from eq 21 for s = 1 (U-manifolds) and hz = 0. Under these conditions
where
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
467
Table 11. Summary of Calculated Flow Distributions and Excess Pressure Drops for Z-Manifolds
N, no. tubes h , , re1 Ap in parallel between hdrs.
10
30
100
1 3 10 30 100 300 1.11 3.33 11.1 33.3 111 100
Q and ( A ~ / A P ~-) la h,= 0 70.9 (1)/1.55 33.9 (1y0.400 12.0 (1)/0.106 4.2(1y0.034 1.3 (1)/0.010 0.4 (1y0.003 69.4 (1)/1.18 32.9(1)/0.309 11.6 (1)/0.082 4.1 (l)/0.026 1.2 (1)/0.008 1.2 (1y0.006
h,= 0.3 67.9 (1)/1.65 33.3 (1)/0.440 11.9 (1)/0.118 4.2 (1y0.038 1.3 (l)/O.Oll 0.4 (1)/0.004 66.5 (1)/1.29 32.4 (1)/0.350 11.5 (1)/0.095 4.0 (1y0.031 1.2 (1)/0.009
h,= 1 61.9 (1)/1.89 32.0 (1)/0.532 11.7(1)/0.146 4.2 (1y0.047 1.3 (1)/0.014 0.4 (1)/0.005
1.2(l)/O.Oll
h,= 3 55.4 (3)/2.59 29.9 (3)/0.795 11.6 (2)/0.227 4.2 (2)/0.074 1.3 (2)/0.022 0.4 (2)/0.007 55.2 (9)/2.31 29.6 (7)/0.721 11.3 (6)/0.208 4.1 (5)/0.068 1.3 (5)/0.020 1.3 (23)/0.021
a Note that the first number is Q in %: the number in Darentheses is that of the tube which has the minimum flow rate; and the number after the slant b& is (A;/AP~) - 1.
Effect of pressure loss in connecttnp tubes on excess pressure drop across u-rnonifalds N o woll friction in heoders, h,=O
; '010 O l
h,:
(T)
N"
Figure 6. Effect of pressure loss in connecting tubes on excess pressure drop across U-manifolds.
When there is wall friction in the headers (h, > 0) the fall of Q with increasing N is always faster than W2.Selecting typical values from Table I one finds that QM is approximately independent of N when Q is small. As indicated in Figure 5 and Table I, the value of Q can be ieduced as much as may be desired in any of several ways. For example, for fixed N , Q is reduced by increasing hl. This is accomplished by increasing the pressure difference between the headers as indicated by H, by increasing the header diameter, D, or by decreasing the distance between tubes, A J ~ . The pressure drop across the whole manifold can be computed from the theory as indicated above. Figure 6 shows a few calculated values and Table I gives numerical results. The ordinate shows the fractional rise in the pressure drop owing to flow nonuniformity. As the flow distribution deteriorates, the pressure drop rises above the value expected for uniform flow distribution. b. Z-Manifolds. When s = -1 the first term on the right of eq 12 contains 2 ~ , , +- ~1 rather than the square of u , + ~and the third term survives. As a result the equation is no longer homogeneous in squares of velocity or of velocity differences. We have found no direct numerical solution but have had to resort to a method of successive approximations. Some details are given in the Appendix. Values of Q and of the excess total pressure drop for 2-manifolds depend on hl and N in a way similar to that shown by Figures 5 and 6. For given hl, h2,and N , both
Q and Ap are greater for a 2-manifold than for a U-manifold. Table I1 lists selected computed values. The flow distribution among the tubes of the symmetrical 2-manifold is qualitatively different from that in a U-manifold. The tube carrying minimum flow may be at either end or between. If h2 = 0 minimum flow occurs in the middle tube and the distribution of flows is nearly parabolic. As h2 increases the minimum flow point moves toward the inlet; as hl increases the opposite occurs. Experimental Section Measurements of pressures inside the entrance and exit headers of several manifolds and flow rates through the connecting tubes were observed with an apparatus constructed from polyvinyl chloride (PVC) pipe. Each horizontal header was made from sections of 4-in. diameter, schedule-40 pipe; the vertical parallel tubes were similar 1-in. pipe and were 4.5 f t long. The measured inside diameters were 4.004 in. and 1.027 in., respectively, corresponding to D / d = 3.899. The tee joints were made from 4-in. X 4-in. schedule-40 PVC tees. These were carefully machined such that, after the header segments had been pushed in, an almost smooth interior surface was obtained. Since the pieces were not cemented together various manifold arrangements could be constructed easily. Each connecting tube was cut in half and an orifice was installed at the center between flanges. This permitted the measurement of each tube's flow rate and, by changing orifices, each tube's pressure drop coefficient, H,could be altered. The orifices were calibrated individually in order that reliable flow distribution data could be obtained when Q was small. Air was supplied to the flow system from a blower capable of discharging 500 ft3/min at 1.25 lb/in., gauge pressure. A calibrated orifice in a long calming section of 4-in. pipe preceding the entrance header permitted measurement of the air velocity, ul,at the entrance. Air pressure differences were measured by means of a micromanometer supplied by Dwyer Instruments Co., having a range of 0.2 in. water and a smallest scale reading of 0.0005 in. water. Larger pressure differences were measured with a standard U-tube manometer using a cathetometer to read either water or mercury columns. Temperatures were observed with chromel-alumel thermocouples. Experiments were carried out using manifolds of four and ten tubes with both the parallel and the reverse-flow arrangements of headers. The center-to-center distance between adjacent parallel tubes was 73.5 in. when four tubes were used and 25.7 in. for ten tubes. Different or-
488
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
Table 111. Summary of Observed Flow Distributions for U-and Z-Manifolds normalized press. diff between headers, press. vel into spacing H=@drop across P ) / ( pV2/ ?6 whole entrance no. between 2gc) cross-flow manifold, header of cross adjacent u , , ftls tubes, N tubes, in. vel heads inequalitya AP, in. Hg
press. drop across whole manifold,
Ap/(puI2/2gc), vel heads
b
0.719 1.075 1.083 1.130 1.615 1.606 0.164 0.322 0.296
38.1 39.8 39.2 88.6 88.2 88.8 7.69 7.80 14.9
1.020 1.084 1.083 1.021 1.039 1.054 1.373 1.327 1.154
1.025 1.568 0.721 1.023 1.071 0.171 0.323 0.548
8810 82.9 39.2 37.1 39.3 7.83 7.80 14.9
1.029 1.042 1.042 1.050 1.042 1.322 1.311 1.140
I. U-Manifolds
33.8 49.9 41.3 27.9 33.3 33.2 36.0 50.1 34.9
4 4 4 4 4 4 10 10 10
73.5 73.5 73.5 73.5 73.5 73.5 25.7 25.7 25.7
2.59 2.54 2.51 6.00 5.87 5.83 2.42 2.54 5.58
26.8 34.0 33.9 41.3 41.0 36.4 50.1 47.3
4 4 4
73.5 73.5 25.7 25.7 25.7 25.7 25.7 25.7
5.93 5.52 2.61 2.45 2.61 2.56 2.57 5.65
2.9 4.6 2.7 1.0 1.0 2.0 14.4 12.5 6.4
11. Z-Manifolds
a
4 4 10 10 10
Q = 100[(Vm,-
Vmin)lVmax].
[ N 'W/(P~ 1 '/2gc)]/ H ( D / C ~ ) ~ .
i
1
0.04
2
3
Table IV. Effects of Pressure Drop Across Manifold and Number of Tubes on Observed Flow Distribution
,,-Entrance
14
0.02 0
I
2
to Header I
Closed End of Entrance Header I
4 6 TUBE NUMBER, n
I
\ ,\I
8
IO
Figure 7. Plot of observed mass flow rates through parallel tubes of a ten-tube U-manifold.
ifices in the vertical tubes were used to obtain two different values of M. A typical set of pressure observations is shown in Figure 2. The corresponding flow distribution data are shown in Figure 7. To find apparent values of the friction factor in the segments of the header pipes between adjacent ports the observed pressures were fitted by least-squares straight lines to determine the pressure gradients. The data included pressures observed on the top and bottom interior surfaces of each header. A few values observed near the tees were not used because they deviated so much from otherwise good straight lines. The derived values of fH were compared with the equation of Drew, Koo, and McAdams (1932) for long, smooth, straight pipes fH
= 0.00140
+ 0.125/Re0.32
where Re is the Reynolds number. Values from our measurements ranged from about 0.5 to 3.0 times the values from eq 27. A few values of pressure gradients
Qav QavlN I. U-Manifolds
N
HT
4 10 4 10
2.55 2.48 5.90 5.58
3.4t 0.6 13.5 i: 0.9 0.8t 0.6 6.4
4 10 4 10
2.56 2.56 5.7 5.7
6.1t 1.1 17.2t 0.3 4.4 7.1
I
1
I
4.4 4.4 7.4 7.0 4.0 16.0 17.5 7.1
0.21 0.14 0.05 0.07
( H / N' ) Q
0.55 0.37 0.3 0.41 0.41av t 0.05
11. Z-Manifolds
0.38 0.17 0.28 0.07
0.98 0.44 1.57
0.40 0.85 av t 0.27
obtained from pressures near the closed ends of manifolds were unreliable because of the small velocities. Some of these values were negative but very nearly equal to zero. On the average the ratio of the measured f values to the values from eq 27 was 1.17 with a standard deviation of the mean equal to 0.04. Measured values of the pressure recovery factor, k,were significantlydifferent in the entrance and exit headers. In the former, kE ranged between extreme values of 0.26 and 0.49 with a mean of 0.423 and a standard deviation of the mean equal to 0.010. In the discharge header, extremes of observed values were 0.91 and 2.4 with a mean of 1.024 and a standard deviation of the mean of 0.019. There was no significant trend of the k values with the spacing between the side ports but there was some evidence of a slight decrease of kE with decreasing velocity in the entrance header and a slight opposite trend in kD Considering all the data available it seems satisfactory to assume that kE = 0.42 and kD = 1.02, O r kD - kE = 0.60. Table 111summarizes the flow distribution and pressure drop data obtained from the experimental 4 and 10-tube manifolds of the two types. The extent of nonuniformity, reflected by the value of the flow distribution parameter, Q, is seen to depend primarily on the number of crossconnecting tubes, N, and the number of velocity heads of
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983 469 1
I
I
I
20%
N o r m o l i t r d Exce88 Pressure Drop, 100
Figure 8. Comparison of observed flow distribution in similar Uand Z-manifolds at equal excess pressure drops.
pressure difference across a single tube, H = 4fTZ/d. Table IV lists averages of the Q values from Table I11 and shows that for constant H, Q is nearly proportional to IV and inversely proportional to H, as expected when wall friction has a small effect. Table I1 also lists the observed overall pressure drops from the inlet of the entrance header to the atmosphere. When expressed as numbers of entrance velocity heads, the values show that the pressure loss was proportional to the square of the flow rate. The last column in Table I11 shows the ratio of the observed pressure drop to that expected if the parallel cross tubes were the only source of pressure change and if the flow had been uniformly distributed. It is seen that there is an excess pressure drop owing partly to the nonuniform flow distribution. Figure 8 is a plot of the experimental data using the variable
where d2V* = D2u1/N. I t is seen that, for comparable pressure-drop values across U- and Z-manifolds of the same dimensions, the Q values were greater in the Z-manifolds. As Table I11 and Figure 7 show, the accuracy of the experimental Q values is not sufficient when the flow distribution was nearly uniform for conclusions to be derived empirically from the data for individual tubes. To obtain the results presented in Table I11 the orifices in all the tubes were calibrated carefully. Even so, calculations based on the flows in the first and last tubes in the sets of four or ten were not as reliable as desired. As a result, the smallest Q values finally presented were computed from least-squares analysis of the flow rates in all the tubes of a set, the maximum and minimum flows being calculated from the empirical equations. Values of Q could also be computed from the measured pressures in the two headers using eq 11 and assuming that H was constant for all the tubes. Occasionally these results were averaged with those found from the readings across the individual tube orifices but the values based on orifice measurements were given greater weight. Comparison of Flow Distribution Data with Results of Calculation Figure 9 compares the values of Q measured experimentally with those calculated from the theoretical flow distribution equations, using Fanning friction factors from the Koo formula for smooth pipe and kE = 0.42 and kD = 1.02. The values of H, specifying the pressure loss through
I0 % CALCULATED VALUE OF 0
5%
0
I5 %
Figure 9. Comprison of observed Q values with theoretical estimates. Table V. Comparison of Q Values from Theory with Measurements of Jones (1974) no. of entrance conQ,% velocity, necting H= ft/s tubes, N 4fTlld obsd calcd I. U-Manifoldsa 52.3 10 4.5 24.8 21.9 52.3 10 12.2 12.3 9.2 54.5 20 4.5 55.3 62.7 53.6 20 12.2 32.4 37.4 46.5 20 453.6 1.5 1.6 55.0 53.5 52.8 54.9 45.3
10 10 20 20 20
11. Z-Manifoldsa 4.5 52.8 12.2 24.8 4.5 74.2 12.2 61.7 453.6 2.4
41.7 19.1 74.8 51.0 2.4
Both manifolds made from 4-in. i.d. headers. Connecting tubes were 1.5-in.i.d., containing orifices 3/8, 1,. or 1*/+ in. diameter. The tubes were spaced at 10.2-in.center-to-center distances.
the parallel connecting tubes, were determined from the experimentally observed header pressures, as listed in Table III. The comparison indicates that the theory works remarkably well, differences more than 2% occurring only once. Table V presents a similar comparison of observed and computed values based on the experiments of Jones (1974), who earlier had used apparatus very similar to ours. Included among his measurements were data for low pressure-drop manifolds having 20 connecting tubes, twice the number used in our work. Jones reported experimental pressures in his two headers but did not measure flow rates in his connecting tubes. Values of Q can be found from his measurements, however, by using eq 11 and assuming H was constant. As Table V indicates, the theory agrees approximately with Jones’ data but differences in Q as large as 10% occur occasionally. Sample Design of a Manifold System As an illustration of the use of the thoery and numerical values of parameters developed in the present work, consider the following example: water at 70 O F is to be passed at a flow rate of 1000 gal/min through a manifold consisting of 30 parallel tubes. Each tube is 20 ft long and
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Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
Table VI. Calculated Flow Distribution Parameters in an Example U-Manifolda assumed header d i m , D, in. 14 18 22
26
h, =
iL, =
(4f~AL1 (4f~Ild) DIN ( D l d ) 4 / N Z Q, % 11.7 9.37 2.12 4.2 25.6 1.74 1.8 57.1 1.49 0.9 111. 1.31
APp/APg=o 1.19 1.06 1.03 1.01
a H = 3.51, corresponding to 30 %in. i.d. tubes, each 20 f t long.
2 in. in i.d. The center-*center distance between adjacent tubes is to be 4 ft, making each header 120 f t long. Find the headers' diameters and the corresponding pressure drops for U- and Zmanifolds that will permit Q to be equal to 1 and lo%, respectively. U-Manifold Design. The volumetric flow rate into the manifold is 2.228 ft3/s, which determines the Reynolds number and the friction factor in each header once a header diameter is chosen. The total flow rate, if evenly distributed among the 30 tubes, fixes the tube velocity at 3.404 ft/s and the tube Reynolds number at 5.373 X lo4. The corresponding friction factor for smooth pipe is 0.00523, giving H = 4fIl/d = 2.51 velocity heads. Adding one extra head loss because of end effects we use H = 3.51. Table VI shows calculations of Q for selected values of D. Interpolation in the table is assisted by the fact that QD2is nearly independent of D. It shows that D = 14.6 in. for Q = 10% and D = 25.2 in. for Q = 1%.The ratios of total tube cross sectional area to that of one header are 0.562 and 0.189, respectively. Obtaining good distribution requires, in this case, that the header cross section be greater than that of all the parallel tubes. A second illustration will show the importance of the pressure drop through the tubes. Use the same total flow rate of water but replace the 2-in. tubes by 30 4-in. i.d. tubes, each containing 5 ft of 1/2-in.spherical packing. The superficial velocity in the average tube becomes 2.10 ft/s and, based on an empirical correlation for packed beds, H = 4713. End effects are of course negligible. The calculation gives D = 5.4 in. and 9.3 in. at Q = 10% and l ?& , respectively. The corresponding ratios of header cross section to total tube cross section are 0.061 and 0.180, respectively. Now, because of the greater difference in pressure between the headers, smaller headers can be used and larger pressure changes inside the headers can be tolerated. In this second case use of a header cross-sectional area equal to the sum of that in 30 4-in. tubes would have required D = 21.9 in. and the value of Q would have been extremely small. Obviously, Q is very strongly affected by the flow resistance between headers, not simply by areas available for flow.
Appendix. Development of Difference Equations for Manifolds Combination of Fundamental Equations for Fluid Pressure Change. We divide eq 4,5,6, and 7 of the text by put/2g, to obtain dimensionless pressures. (These new pressures are identified by the same symbols introduced before.) Similarly, all velocities are divided by ul, the velocity at the entrance of the entrance header, but the symbols are not changed. Pn
- Pn+l = @ f ~ u / D ) u X + l
P,' - P n + 1 = -s(4f~AL/D)U2+1 P,'-
(A-4)
~n
= 2kduX - uX+J
(A-1) (A-2)
(A-3)
1-s u,= + sun 2
H = 4f~l/d is the number of velocity heads of friction loss in each connecting tube. We need to eliminate pressures, U,,'s, and Vn's from the eight equations to obtain a difference equation in u,. The result is
s = +1 for U-manifolds, = -1 for Zmanifolds.
H(D/d)4[(~n- Un+1)' - (Un+1 - Un+J21 (U;+l
(4fHa/O)
X
+ SE+i+ ) kE(Ui+2 - UX) - ~ D ( Z -+ v",)Z(A-9)
Finally, use the material balance, eq A-7, to eliminate the U,, obtaining the nonlinear, second-order difference equation in u,
where A=2
(A-12)
H
and (A-13)
c=-( 2 k
~d
H E
)
(A-15)
Numerical Solution of the Difference Equations A. U-Manifolds. Because of the boundary condition uN+1 = 0, corresponding to the closed end of the entrance manifold having N connecting tubes, eq A-11 for n = N - 1has only two unkown velocities. Moreover, the equation is homogeneous in these and can be divided through by UN to obtain a quadratic equation for the ratio, UN-l/uN in terms of the constants A and B. Similarly, this result can be substituted into eq A-11, written for n = N - 2, and the equation can be solved for u N - ~J UN-1. The procedure ends when u1 = 1is reached. The flow rates through the four parallel tubes, V,, are proportional to the differences, ul- u2, u2 - u3, etc. Numerical results for U-manifolds show that the first tube, n = 1,always carries the greatest flow of any; the last tube, n = N , carries the least. Calculated profiles show
Ind. Eng. Chem. Fundam., Vol. 22, No. 4, 1983
that V , is nearly a linear function of ( N - n)2as suggested by eq 25, a fact that was used in smoothing some of the experimental V , data in determining Q. The total pressure drop across the whole manifold can be found by applying eq A-10 at n = 1. When the result is divided by ( p ~ , 2 / 2 g A ( D / d ) ~ Hwe / Wfind the expression for the relative or dimensionless pressure drop Ap/Apg=o = W[(1 - ~
2
+) B(1 ~ - uZ2)]
(A-16)
The quantity on the left is the ratio of the actual pressure drop to that which would occur if all the V , were equal. B. Z-Manifolds. Because of the presence of the term proportional to u, - u,+~in eq A-14, the solution procedure used for U-manifolds does not apply. A method of successive approximations can be used, however. The left side of eq A-14 is equivalent to c
and is proportional to the difference between u , + ~and the arithmetic mean of the two adjacent values, u, and u , + ~ . This suggests a procedure reminiscent of the Schmidt graphical procedure much used for transient heat conduction calculations before the advent of digital computers. If we solve eq A-14 for u , + ~we find
(A-17) in which we have assumed that trial values of uj will lead to an improved value of u;:;. Assuming a perfect flow distribution to compute the initial values of u,’ on the right, new values of u”+i were calculated iteratively, using a digital computer. The dimensionless pressure drops in Table I1 were computed from a formula somewhat more involved than that for U-manifolds, because the feed and discharge pressures are separated by several pressure changes along the headers. Nevertheless, these can be summed to obtain the result N
i=2
N
uN(I1 - (C+ Bfy:,
+ C)))(A-18)
Nomenclature A , B , C, = constants in eq 12; see also eq 13, 14, and 15 d = tube inside diameter D = header inside diameter
471
= friction factor in header, constant = friction factor in connecting tubes, constant g, = units conversion constant in English system, 32.1740 (lb fH
fT
mass/lb force) (ft/s2) hl = friction parameter for the connecting tubes; cf. eq 21 h2 = header friction parameter, defined in eq 22 H = 4fTl/d, the number of velocity heads of pressure drop through the tubes k = momentum recovery factor in headers; kE applies to entrance header, k D to the discharge header AL = center-to-center distance between adjacent connecting tubes n = number of a tube; the tube next to the entrance is number one N = total number of connecting tubes Pb = barometric pressure, at exit from experimental manifold p L= average fluid pressure in entrance manifold (see Figure 1)
Q = flow distribution quality, equal to difference between maximum and minimum tube flows, expressed as a percentage of the maximum flow (cf. eq 16) Re = Reynolds number for average conditions in the headers s = either +1 for U-manifolds or -1 for Z-manifolds S = cross-sectional area of headers u, = average fluid velocity in entrance manifold before tube n
U, = average fluid velocity in discharge header
V , = average fluid velocity in a tube V* = average fluid velocity in a tube if flow distribution were uniform x = distance along axis of header a = parameter defined in eq 28 for a U-manifold having an infinite number of tubes and no header wall friction p = fluid density, constant 1 = length of tubes connecting the headers A p = pressure drop across manifold from entrance to exit Api = pressure drop across manifold if flow distribution were perfect
Literature Cited Ashraf. M. M.S. Thesis, University of California, Berkeley, CA, 1975. BlaMsel, F. W.; Manson, P. W. “Loss of Energy at Sharp-Edged Pipe Junctions in Water Conveyance Systems”, U S . Dept. Agriculture Tech. Bull. No. 1283, Aug 1963, quoted by Miller (1971). Drew, T. B., et ai. Trans Am. Inst. Chem. Eng 1932, 28, 58-72. Gardei, A. Bull. Tech. Sulsse Romande 1957, 83, 9, 123030 and 10, 143048, quoted by Miller (1971). Jones, E. H. M.S. Thesis in Mechanical Engineering, West Virginia University, 1974. McNown, J. S. Trans Am. SOC.ClvllEng. 1954, 119. 1103-42. MUier, D. S. “Internal Flow: A Guide to Losses in Pipe and Direct Systems,” B.H.R.A. Report, Crenfield, Bedford, England, 1971. Miron, Y. D. M.S. Thesis in Chem. Eng., University of Delaware, Newark, DE, Dec 1978. Vogei, G. wdr. Inst. Techn. Hochschule-Munchen, 1928, 1 , 75090; 1928, 2, 62-64, quoted by Miller (1971).
Received for review April 2, 1982 Accepted August 3, 1983
