THE PORT DISCHARGE DISTRIBUTION OF A MULTIPLE-PORT DIFFUSER

 

HSIEN-TER CHOU , RAY-YUAN CHENG

 

Department of Civil Engineering, National Central University,

Chung-Li, Taiwan 32054.

886-3-4252960(fax), htchou@cc.ncu.edu.tw(e-mail)

 

 

ABSTRACT

Analytical solutions for the port outflow distribution were derived in this study when either the wall friction parameter, ,or the port momentum parameter, , is dominant. The numerical solutions agree well with both analytical solutions and experimental data. For the uniformity of the outflow distribution of all ports, it is also found that and should be within a suitable range, i.e., near, or the diameter of the diffuser should be close to the value of .

 

Keywords: port discharge, flow distribution, diffuser.

 

INTRODUCTION

Multiple-port diffusers, often referred to as manifolds, are used in sprinkling infiltration systems(McNown,1954),gas pipe burners (Keller, 1949),thermal discharges (Vigander et al., 1970) and ocean outfalls (Lee and Yau,1996). The two main concerns for the diffuser hydraulics are firstly, to obtain a uniform flow distribution for all the ports, and secondly to minimize the total head loss in the pipe system. In this paper, the dimensionless parameters controlling the distribution of the port discharge are analyzed by employing theoretical analysis, numerical simulation and experimental work. Such dimensionless parameters are useful for the hydraulic design of diffusers.

 

THEORETICAL AND NUMERICAL ANALYSIS

 

GOVERNING EQUATIONS

Given a straight porous diffuser with a closed downstream end as shown in Figure 1, one can derive the dimensionless axial(x-direction) momentum equation for the isothermal, incompressible fluid(Bajura, 1971; Shen, 1992).

FIG. 1. A schematic sketch of a multiple-port diffuser

 

, (1)

where and represent the dimensionless cross-sectional average velocity of the diffuser and distance along the flow direction, relative to the initial pipe velocity,, and the diffuser length, L, respectively.

, (2.a)

, (2.b)

where depicts the frictional effect of the diffuser pipe, while represents the port momentum effect, =friction coefficient of the diffuser pipe,=diameter of the diffuser,=hydrostatic recovery coefficient. H depicts loss factor due to both port and lateral pipes as follows,

(2.c)

where, =local head loss at the junction, =length of the lateral pipe, d =diameter of the port. denotes the ratio of port area as follows;

(2.d)

In case of a port discharge without lateral pipes, the contraction effect at the ports should be encountered, i.e.,

(2.e)

(2.f)

= port discharge,=port area(),=the specific energy inside the diffuser pipe. The boundary conditions for equation (1) are facilitated as and. For simplicity, all the superscripts for dimensionless parameters, and are dropped. Since equation (1) is a nonlinear ordinary differential equation, it is quite difficult to obtain the exact analytical solution.

 

PERTURBATION METHOD FOR PORT DISCHARGE DISTRIBUTION

If the frictional effect is relative small, such as for a short, smooth diffuser, then the value of parameter is small. The first approximation assumes in equation (1), to explore the effect of parameter, i.e.;

, (3)

with the boundary conditions, and. One can easily find its solution(Bajura, 1971);

(4)

By assuming that the value of is small, one can evaluate the effect of the friction parameter,,by the perturbation method. Thus the flow velocity in the diffuser can be defined as;

, (5)

Substituting equation (5) into equation (1) and neglecting higher order terms than, one can obtain;

, (6)

with the boundary conditions of;

,. (7)

The dimensionless port discharge,, is defined as;

. (8)

So, we can define a uniformity parameter for the port discharge,, as

. (9)

 

On the other hand, if the port momentum is small relative to the wall friction parameter, such as for a long diffuser, one can approach equation (1) by assuming. i.e.,

. (10)

 

The solution for equation (10) is ;

. (11)

According to the boundary conditions at x=0, U(0)=1and x=1,U(1)=0,one can integrate equation (11) to obtain the the c versus relationship as

. (12)

The uniformity coefficient for the diffuser thus reads;

. (13)

The uniformity coefficient, ,based on equation (13), thus depends on the value of . As the value of increases, such as with an increase of the friction coefficient or the diffuser length, the port discharge at the downstream end will be less than that at the upstream end.

 

NUMERICAL ANALYSIS FOR PORT FLOW DISTRIBUTION

The shooting method is employed to solve equation (1) by transforming the boundary value problem into the initial value problem. By trying an initial slope, i.e., derivative of the velocity in the diffuser, at the upstream end, one can utilize the fourth order Runge-Kutta method to find the flow velocity as well as its derivative ,stepwise along the flow. The initial slope will then be modified according to the shooting method if the predicted flow rate at the downstream end is greater than the specified tolerance (10^-7,say) ( Press et al.,1986).

 

EXPERIMENTAL MEASUREMENT

An acrylic diffuser 2m long, and 2 cm in diameter was installed horizontally in the Environmental Fluid Laboratory of the National Central University. A 5mm port was made at 10-cm intervals along the diffuser in the direction of the flow by careful drilling and surface polishing work. The inlet flow obtained a maximum Reynolds number of 90000. Upstream of the inlet, the flow passes a straight PVC tube 2 m long with the same diameter, in order to reduce the flow disturbance. The pressure inside the diffuser was measured by using the pressure transducer (Druck, model PDCR910pressure range = 100Kpa,error range=1Pa). The measurement of contraction coefficient is thus obtained according to equation (2.f). The flow measurement at all the open ports was made with leveled containers beneath the ports. To ensure the accuracy of the data, every flow rate measurement was repeated three times.

 

RESULTS AND COMPARISONS

 

COMPARISONS BETWEEN THEORETICAL AND NUMERICAL RESULTS

As described by equation (4), the port discharge increases downstream in the case where the port momentum parameter,, is dominant (=0). The results obtained by the small perturbation method (i.e., equation (9) and numerical simulations in the range of are shown in Figure 2.

 

(a) Perturbation method (b) Numerical Simulation

 

FIG. 2. The uniformity contours for the port discharge distribution in terms of perturbation method (a) and numerical simulation (b).

 

In the figure, the uniformity parameters,, obtained by both methods are identical when, and very close to each other in the range of . As, decreases as increases. In other words, In the range where the port momentum parameter is dominant, the port discharge becomes more uniformly distributed when the wall friction parameter increases. The uniformity parameter, , equaling unity can be obtained under the condition . According to equations (2a) and (2b), the corresponding diameter of the diffuser should be . If the diameter D is greater than D_h, then the flow distribution for the ports will increase downstreamly. On the other hand, if D < D_h, the outflow will decrease downstreamly.

 

COMPARISONS BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS

Figure 3 represents the 12-port discharge distribution, providing =0.469, =0.625,=0.524 and=0.663. The numerical results agree well with the experimental data, while Shen's (1992) solution shows an obvious discrepancy with the experimental data.

 

FIG. 3. The port discharge distribution of the experimental data and the numerical simulation (12 ports,Re=53363)

 

CONCLUSIONS

1.      The two controlling parameters for the port discharge distribution are the wall friction parameter, , and the port momentum parameter,. As the value of is negligible for such a short diffuser, the port discharge increases downstream, having a maximum value at the downstream end. On the other hand, if is the dominant parameter such as for a long diffuser, then the port discharge decreases downstream. The maximum port discharge occurs at the upstream end.

2.      The increase of enhances the port discharge distribution causing it to be more convex, while the increase of makes the flow distribution more concave. In order to make the port discharge even, the values for and should be kept within a certain range, such as around.

 

ACKNOWLEDGEMENTS

The writers would like to express their gratitude to Professor H.C. Lei of National Central University for his valuable comments on the theoretical analysis , and to Mr. K.L. Chen and Mr. Y.Y. Chang for their help in the experimental work. The financial support from the National Science Council of Taiwan, R.O.C. (NSC86-2611-E008-002) is also gratefully acknowledged.

 

REFERENCES

McNown, J.S. (1954)." Mechanics of Manifold Flow.", Transactions ASCE, Vol. 119, pp.1103-1142.

Keller, J. D.(1949). "The Manifold Problem," Journal of Applied Mechanics, Vol. 71, March, pp.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, " 4^th Environmental Engineering Specialty Conference, CSCE, Edmonton, Canada, pp. 383-394.

Fischer, H.B, E.J. List, R.C.Y. Koh, J. Imberger, and N.H. Brooks(1979). Mixing in Inland and Coastal Waters, Academic Press, New York.

Bajura, R. A. (1971). "A Model for Distribution in Manifolds", Journal of Engineering for Power, pp.7-12.

Vigander S., R.A. Elder, and N.H. Brooks(1970).. "Internal hydraulics of thermal discharge diffusers,", Proc. ASCE, Vol.96, No. HY 1, pp.509-527.

Shen, P. I. (1992). "The Effect of Friction on Distribution in Dividing and Combining Flow Manifolds," Journal of Fluids Engineering, Vol. 114, March, pp. 121-123.