Outflow Distribution Along Multiple-Port Diffusers: H - T Chou R - Y Cheng
Outflow Distribution Along Multiple-Port Diffusers: H - T Chou R - Y Cheng
Outflow Distribution Along Multiple-Port Diffusers: H - T Chou R - Y Cheng
ROC(A)
Vol. 25, No. 2, 2001. pp. 94-101
ABSTRACT
The outflow distribution along a multiple-port diffuser was explored in this study by employing analytical,
numerical and experimental methods. The effects of two dimensionless parameters controlling the flow distribution,
i.e., the wall friction parameter (α) and the port momentum parameter (β), were evaluated. When the wall friction
parameter is negligible, the port discharge increases downstream, and the flow distribution becomes more uneven
as the value of the port momentum parameter increases. On the other hand, the port discharge decreases downstream
when the momentum parameter is negligible. Analytical solutions for the port outflow distribution were derived for
the cases in which either the wall friction parameter or the port momentum parameter is dominant. The numerical
solutions agree well with both analytical solutions and experimental data. As for uniformity of the outflow distribution
of all ports, it is also found that these two parameters should be controlled within a suitable range (i.e., β ≅
0.5940α), and that the diameter of the diffuser should be close to 0.353 fL .
2 – γd
Key Words: diffuser, port discharge, flow distribution
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Outflow along Multi-port Diffusers
(Bajura, 1971):
1 dP f 2 dU λπ D
ρ dx + 2D U + 2U dx + γ d UV A = 0 , (1)
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H.T. Chou and R.Y. Cheng
β6
B = – [α + (α + )1/2]1/3 ,
2 4
(7c) where U0(x) is the zero-order solution, as shown in Eq. (9),
4 16 27 and U1 is the correction term due to the existence of α.
Substituting Eq. (11) into Eq. (5a) and neglecting order
(A + B)
C1 = , (7d) terms higher than α2, one can obtain
2
U 20(x) {sin[β(1 – x)]}2
C 2 = 3 (A – B) . (7e) U 1′′(x) + β 2U 1(x) = – =
2 2U 0′ (x) 2βsinβcos[β(1 – x)]
(12)
A numerical test showed that Eq. (6) does not agree with
Eq. (5a) (Cheng, 1997). Consequently Eq. (6) is not a valid with the boundary conditions
solution. Since Eq. (5a) is governed by the two parameters
α and β, the effect of these two parameters will be discussed U1(0) = 0, U1(1) = 0. (13)
first based on theoretical analysis of special cases, and will
later be systematically analyzed based on the results of nu- Let
merical simulations.
U 1(x) = U h (x) + U p (x) , (14a)
1. The Perturbation Method for Port Flow Distribu-
tion where
If the frictional effect is relatively small, such as for a U h (x) = c 1cos(β x) + c 2sin(β x) , (14b)
short, smooth diffuser, the value of α is close to zero. When
α = 0, Eq. (5a) is reduced to a simpler form: U p (x) = k 1cos(β x) + k 2sin(β x) , (14c)
d (U ′2 + β 2U 2) = 0 . (8) and
dx 0 0
x
–1 sin2[β(1 – x)]sin(β x)
Equation (8) allows us to explore the effect of the port mo- k 1(x) = dx
2β 2sinβ cos[β(1 – x)]
mentum parameter, β, on the port discharges. Based on the 0
boundary conditions, U0(0) = 1 and U0(1) = 0, one can easily
find the solution (Bajura, 1971): –1 cosβ
= A(x) – 3 B(x) , (14d)
2β 2 2β sinβ
sinβ(1 – x)
U o (x) = . (9)
sinβ
A(x) = [1 (1 – x) – 1 sin(2β) + 1 sin(2β x)] , (14e)
2 4 4β
According to Eq. (3), the dimensionless port discharge,
Vi(x ), that is, the discharge at any location x with respect to B(x) = [ln cosβ – 1 cos2β – ln cos(β x) + 1 cos2(β x)] .
the upstream port discharge, is defined as 2 2
(14f)
V 0(x) U 0′ (x) cosβ(1 – x) Similarly,
V i (x) = = = . (10)
V 0(0) U ′ (0) cosβ x
0
sin2[β(1 – x)]cos(β x)
k 2(x) = 1 dx
According to Eq. (10), the port discharge increases β 0
2βsinβcos[β(1 – x)]
downstream under the condition that the friction loss is com-
pletely neglected (i.e., α = 0). The maximum port discharge cosβ
= A(x) – 1 3 B(x) . (14g)
thus occurs at the downstream end, with a value of Vi(1) = 2β 2sinβ 2β
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Outflow along Multi-port Diffusers
can find the coefficients k1, k2, c1 and c2. The derivatives of dU =1. (22)
the correction term U1 at the boundaries are 3 α 2
– U 3 + 3c
1 2
4sin2β [2β – sin(2β)]
U 1′ (1) = 1 2 [ – The constant c in Eq. (22) is solely determined by α.
8β cosβ sinβ
Since an explicit presentation between c and α is not available,
8(2ln cosβ + sin2β)cosβ theoretically, one can obtain the c versus α relationship by
– ], (15) means of numerical methods, such as Romberg integration.
sin2β
The regression relationship between c and α, with a correlation
– (2ln cosβ + sin2β)(sin2β + 4cos2β) coefficient exceeding 0.99, reads as
U 1′ (0) = . (16)
4β 2sin2β
c = 0.0195α2 + 0.0237α − 0.3387. (23)
In order to quantify the port discharge distribution, a uniformity
parameter, Vr, is defined as the ratio of the port discharges The uniformity coefficient, Vr, for the diffuser thus reads as
between downstream and upstream ends, i.e.,
U ′(1) 1
Vr = = 3 . (24)
U ′(1) U 0′ (1) + α 2U 1′ (1) U ′(0) 1– α
2
V r = V i (1) = = . (17) 6c
U ′(0) U 0′ (0) + α 2U 1′ (0)
The uniformity coefficient, Vr, based on Eqs. (23) and
From Eqs. (9), (15) and (16), one can obtain
2 3
1 + α2 1 [2β – sin(2β)] + 8(2ln cosβ + sin β)cosβ – 4sin β
8β 3 sin β cosβ
U ′(1)
Vr = = . (18)
U ′(0) 1 (2ln cosβ + sin2β)(sin2β + 4cos2β)
cosβ + α 2[ 3 ]
4β sinβ
On the other hand, if the port momentum parameter, β, (24), thus, depends on the value of α. As the value of α in-
is small relative to the wall friction parameter, α, such as for creases, such as with an increase of the friction coefficient
a long diffuser, one can approach Eq. (5) by assuming that or the diffuser length, the port discharge at the downstream
β = 0, i.e., end will be less than that at the upstream end.
The solution for Eq. (19) is The shooting method is employed to solve Eq. (5a) by
transforming the boundary value problem into the initial value
– α U 3 + 3c .
3 2 problem (Cheng, 1997). By trying an initial slope, i.e., the
U′ = (20)
2 derivative of the velocity in the diffuser, at the upstream end,
one can utilize the fourth order Runge-Kutta method to find
According to the boundary conditions at x = 0, U(0) = 1, and the flow velocity as well as its derivative stepwise along the
x = 1, U(1) = 0, one can integrate Eq. (20) as follows: flow. The initial slope will then be modified according to
the shooting method if the predicted flow rate at the down-
U(x)
stream end is greater than the specified tolerance (10−7) (Press
dU et al., 1986).
=x (21)
3 α 2
– U 3 + 3c
1 2 III. Experimental Measurement
with An acrylic diffuser pipe of 2 m in length and 2 cm in
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H.T. Chou and R.Y. Cheng
Fig. 3. The contours of the uniformity index Vr for the port discharge distribution based on results obtained using the perturbation method (a) and numerical
simulation (b).
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Outflow along Multi-port Diffusers
D h = 0.353 fL . (25)
2 – γd
If the diameter D is greater than Dh, then the flow
distribution for the ports will increase downstream. On the
other hand, if D < Dh, the outflow will decrease downstream.
J. Sherman’s experimental data1 were used to verify the Fig. 6. The port discharge distribution of the experimental data and nu-
theoretical and numerical results. The diffuser used in Sherman’s merical simulation (6 ports, Re = 27000).
1
Sherman, J. (1949) Internal Report, Research and Development Center, The Babcock and Wilcox Company, Alliance, OH, U.S.A. (not available in the
open literature but the experimental data were cited in Bajura’s paper (Bajura, 1971)).
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H.T. Chou and R.Y. Cheng
Acknowledgment
References
the values of Ar, α and β all increase accordingly. As shown
in Fig. 4(b), the uniformity parameter, Vr, deviates more Bajura, R. A. (1971) A model for distribution in manifolds. Journal of
obviously from unity when the values of β/α are farther away Engineering for Power, 47, 7-12.
from 0.594, and the deviation grows with increasing values Cheng, R. Y. (1997) Flow Distribution for Multi-port Diffusers. M.S. Thesis.
National Central University, Chung-Li, Taiwan, R.O.C.
of α and β. Consequently, the measured port discharge Fischer, H. B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks (1979)
distribution shown in Fig. 8 is more uneven than that shown Mixing in Inland and Coastal Waters. Academic Press, New York,
in Fig. 6. The increase of the port numbers thus causes outflow NY, U.S.A.
along the diffuser away from uniform distribution unless the Keller, J. D. (1949) The manifold problem. Journal of Applied Mechanics,
diameter of the diffuser is close to Dh. 71(March), 77-85.
Lee, J. H. L. and W. C. Yau (1996) Experimental investigation of sea water
intrusion and purging on the Hong Kong ocean outfall diffuser model.
V. Conclusions 4th Environmental Engineering Specialty Conference, pp. 383-394,
CSCE, Edmonton, Canada.
(1) The controlling parameters for the port discharge dis- McNown, J. S. (1954) Mechanics of manifold flow. Transactions ASCE,
tribution along a multiple-port diffuser are the wall 119, 1103-1142.
Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling (1986)
friction parameter, α, and the port momentum parameter, Numerical Recipes − The Art of Scientific Computing. Cambridge
β. The analytical solutions for the outflow distribution University Press, Cambridge, U.K.
were derived in this study, i.e., Eqs. (18) and (24), under Rawn, A. M., F. R. Bowerman, and N. H. Brooks (1961) Diffuser for disposal
the condition that either α or β is the dominant parameter. of sewage in sea water. Transactions, ASCE, 126(Part III), 344-388.
As the value of α is negligible, i.e., β is dominant, such Shen, P. I. (1992) The effect of friction on distribution in dividing and
combining flow manifolds. Journal of Fluids Engineering, 114(March),
as for a short diffuser, the port discharge increases 121-123.
downstream with a maximum value at the downstream Vigander, S., R. A. Elder, and N. H. Brooks (1970) Internal hydraulics of
end. On the other hand, if α is the dominant parameter, thermal discharge diffusers. Proc. ASCE, 96(HY 1), 509-527.
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Outflow along Multi-port Diffusers
多孔擴散管孔口流量分布之分析
周憲德 鄭瑞元
國立中央大學土木工程學系
摘 要
本文以理論及數值解析探討多孔擴散管之孔口流量分佈,並進行實驗以驗證理論解及數值解之正確性。影響多
孔擴散管之孔口流量分佈之主要物理因子可歸納為管壁摩擦損失係數 α 及孔口動量係數 β 兩個無因次參數。當管壁
摩擦損失係數可忽略時,流量分佈隨孔口動量係數之增加而趨於不均勻,且沿下游遞增以尾端之孔口流量最大;而當
管壁摩擦損失係數為主控因子時,則孔口流量以前端之孔口流量最大,並沿下游呈遞減分佈。本文並以數值分析α、β
二參數對孔口流量分佈之綜合影響,當此二個參數之比在一定範圍時,即β = 0.594 α ,或多孔擴散管之管徑應接近
0.353fL/(2−γd),方能使各孔口流量分佈趨於均勻。
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