Verification of additional losses in complex flow fields

 

F. VALENTIN

 

Professor, Civ. Eng. Dep,. Technische Universität München

Lehrstuhl für Hydraulik und Gewässerkunde

Technische Universität München, D-80290 München

0049 89 2892 2583, 0049 89 2892 8332, valentin@hydra.bauwesen.tu-muenchen.de

 

 

ABSTRACT

Additional loss coefficients are commonly used to simplify the calculation of losses to an appropriate level. The most important disadvantage of this method is the lack of tranparency with regard to the quality of this kind of parameters. Based on a series of experiments the flow in combining manifolds is considered to estimate the total losses for this type of flow. The experimental set-up was changed to different configurations of the pipe itself like the number and the spacing of openings.

The evaluation of the numerous data is based on a simulation model with describes in detail the lateral inflow to the pipe and the axial flow in the pipe itself. The model used for this investigation separates the total losses into the friction losses for a fully developed turbulent flow and the above mentioned additional losses which are created by the disturbance of the velocity profile by the lateral inflow. From the experiments only an information on the total losses was available. One of the main problems during the evaluation was to question the physically correct interpretation of the results of various assumptions which describe the pressure distribution along the pipe and, therefore, the inflow to the pipe at different ports.

The variation of the additional loss coefficient for the discontinuous flow with lateral inflow depends on a number of parameters. There is a main dependence on the ratio of the inflow velocity to the mean pipe velocity. This leads to a first assumption with respact to the variation of the local loss coefficients along the pipe which is characterised by a nonlinear decrease with increasing number of ports with a maximum value z for the first port. To show the influence of the port spacing best fit simulations of the experimental data were done by varying the discharge coefficient m for the inflow and the additional loss coefficient z. These tests resulted in increasing values of m with an increasing number of ports which is in contradiction to reality.

Smaller values of the discharge coefficients can be obtained by increasing differences in pressure head at the ports. Therefore, the additional losses have to be increased at the beginning of the pipe and decreased towards the end of the pipe. Simulations with this typ of variation showed only small variations of the inflow distribution which allowed the assumption of constant losses due to pipe friction under these conditions. A constant value of the total sum of additional losses enables the reduction the variation of z in dependence of the spacing ratio to mean values of

 

INTRODUCTION

Combining manifolds find an increasing application in discharging purified water from final settling tanks in waste water treatment plants[1]. The design of such an outflow system requires detailed knowledge of the resistance behavior to ensure an equally distributed outflow for instance in circular tanks. One of the crucial points in the calculation of discontinuous flow in pipes is the choice of correct values for the additional losses created by the lateral inflow. Over the years a series of experiments have been carried out in in this context at the Laboratory of the Institute of Hydraulics and Hydrology, Technische Universität München.

 

EXPERIMENTS

In the experiments the diameter of the pipe was D = 0,20 m and the spacing of the openings could be varied from 0,5 to 3,0 meters which means values D between 2,5 and 15,0. The shape of the ports was a slot of 10 mm width and semicircles at both ends forming an inflow area a = 6,35 cm². The combining manifold line was composed of nine pipe units, each of the pipes having an overall length of 3.0 m and a maximum number of 6 ports with a minimum spacing of 0, 5 m. The spacing and the number of ports were changed by blocking intermediate openings. Details of the experimental set-up and the performance of the experiments are described in [1]. In Table 1 several runs are presented with the number n of ports, the spacing ratio , the maximum discharge Q at each run and the maximum total loss D measured by pressure transducers.

 

 

Table 1: Main experimental data

 

Decreasing flow rates during the experiments mean decreasing values of the total loss measured and, therefore, decreasing confidence in the parameters calculated from these data with respect to the limited accuracy of data. It is to be expected that for small differences in pressure head coefficients determined under these conditions will show considerable scattering.

 

NUMERICAL MODEL

In the description of the flow in the combining manifold the governing equations are the continuity equation and the Bernoulli-equation. For steady state flow the continuity equation states that in the vicinity of an arbitrary port i

 

Q_i = Q_i-1 + q_i

 

where Q_i is the flow rate in the pipe behind the port i and q_i is is the lateral inflow at the port i. At the first port Q₀ = 0 and at the end of the pipe with n ports

 

 

The lateral inflow is induced by the difference in pressure head between the pressure head h_i at the orifice and the water level h₀ above it. The sketch in Figure 1 demonstrates that h_i is the mean pressure head at the orifice taking into account that the mean flow velocities upstream and downstream of the orifice are V_i-1 and V_i, respectivly.

 

Figure 1: Energy and pressure line in the vicinity of a port in a combining manifold

 

The lateral inflow is expressed by

 

 

where m is the discharge coefficient which depends on the shape of the orifice and the flow field outside the pipe. Contrary to earlier suggestions for internal flow losses the present investigations confirm a nearly constant value of m » 0,650. The energy line along the pipe is influenced by the friction losses and the additional losses due to the disturbance of the flow profile by the lateral inflow. Despite the fact that normally in a combining manifold the formation of a fully developed turbulent flow profile is impossible, the friction losses are calculated using the Darcy-Weisbach-equation

 

 

where l_i is the friction factor estimated by use of the Prandtl-Colebrook-equation in the pipe section considered, is the spacing of the ports which remains constant for each of the runs. The additional shear stresses created by the laterally entering jet are taken into consideration by an additional flow loss

 

 

where z_i is the additional loss coefficient at the port i which is to be determined out of the experimental data. With respect to the upstream and downstream boundary conditions for each run H₁ is the energy head upstream of the first opening and h_r is the pressure head at a distance DL behind the last opening. Thus

 

 

where V is the mean flow velocity behind the last port. The velocity head of the inflowing jet v_i²/2g is totally destroyed as the flow direction is perpendicular to the axial flow in the pipe. For each run the pressure heads H₁ and h_r are recorded for different values of V. Pipe velocities V_i are then determined by summing up the inflow rates which can be calculated for given distributions of the additional flow losses.

 

EVALUATION OF ADDITIONAL LOSSES

The sum in the term for the total losses is known from the experiments, its maximum value for each run is given in the last column of Table 1. To arrive at the additional losses the friction losses have to be subtracted from the total losses. It is shown in [2] that a linear increase of the lateral inflow towards the end of the pipe would imply an unrealistic distribution of the discharge coefficients with a maximum value in the first half of the manifold. Therefore, there is a nonlinear increase of the lateral inflow along the pipe with an increasing gradient (Figure 2). Combined with the nonlinear resistance behavior this means that the friction losses will be reduced by this kind of distribution of q_i. From this point of view the system combining manifold tends towards a maximum outflow by this nonlinear inflow distribution.

 

Figure 2: Comparison of linear and lateral inflow distribution for m = const.

 

The interdependence of the inflow distribution and the system parameters discharge coefficient m and additional loss coefficients z_i requires numerous simulations under differing assumptions. These simulations lead to a best fit process between the experimental data and the results of the simulation. Input data for the calculations were the water level h₀ in the tank and the initial energy head H₁ in the pipe together with the configuration of the pipe system like number and spacing of orifices. Previous investigations [3] have shown a strong dependence between the different additional loss coefficients z_i and the ratio v_i/V_i of the resistance influenced inflow velocity v_i to the pipe velocity V_i. For only one orifice the inflow rate is the same as the discharge in the pipe. In this case the above mentioned ratio becomes

 

 

where A is the area of the pipe. Ratios v_i/V_i have a maximum value behind the first port and decrease with increasing number of ports. It was obvious to combine the additional loss coefficient with this velocity ratio into the formula

 

 

This formula allows the reduction of the varying values z_i to a single value z₁ which represents the loss coefficient of a single port. As an example for a manifold with 54 ports the development of the individual loss coefficients z_i for a given value z₁ = 1,6 is demonstrated in Figure 3.

 

Figure 3: Distribution of z_i-values along the pipe starting at z₁ = 1,6

 

The prediction of the coefficients z_i allows the calculation of the lateral inflow rates q_i and, furthermore, the friction losses over the whole length of the pipe. Based on these assumptions the total losses can be subdivided into the sum of friction losses and the sum of additional losses. The friction factor according to the Prandtl-Colebrook-equation was calculated for each section between the openings assuming an equivalent sand roughness of 0,15 mm for the stainless steel pipe and a kinematic viscosity of 10^-6 m²/s at a water temperature of 20°C. The result of a simulation with the measured values described above is a best fit of the parameters z₁ and m which guarantee a coincidence of the total discharge Q and the pressure head h_r downstream of the last opening. With

 

 

z is a representative value for the additional loss coefficient, which is related to the mean velocity head at the end of the pipe. It should be stated again that both the loss heights are influenced by the inflow distribution to the pipe.

 

SENSITIVITY ANALYSIS

Simulation of several runs of the experiments over the whole range of the investigated modifications of the manifold showed a very good agreement between calculated and measured discharges. This is surprising taking into account that the additional loss coefficients had been correlated only with the velocity ratio. The extremely nonlinear decrease of the individual coefficients along the pipe may be the reason for this result. Looking for further possible influences of geometric and dynamic characteristics of the flow in the manifold system on the additional losses the following relation could be constructed

 

 

where b is the width of the slots. The last dimensionless ratio was varied during each run in the range from 2,0 > v²/2gD > 0,05 for each port of the pipe without any indication of a pronounced variation in z. Small variations in the mean value of z could be observed for variations of DL/D. Using the relation between the velocity ratio and the additional loss coefficient z₁ best fit simulations with variations of m and z₁ were run over the whole range of 15,0 > DL/D > 2,5. The results of these simulations are shown in Figure 4. It is obvious that z₁ increases with increasing values of DL/D. On the other hand small values of z₁ are combined with increasing values of the discharge coefficient m. This result contradicts the physics of flow. There is no reason for such an increase under the same flow field.

 

Figure 4: Optimum z₁- and m-values deducted from simulations for different spacing ratios

 

For constant values of the discharge coefficient the lateral inflow at the beginning and the end of the pipe will be reduced according to the reduction in m. To guarantee the same discharge over the length of the pipe the lateral inflow rates at the other ports must increase. The only way to reach this condition is to increase the additional losses at the beginning of the pipe connected with a decrease towards the end. One plausible relation to obtain this result is

 

 

where z₁ is chosen above the value of the best fitting result and z_n beyond the corresponding one. The discharge coefficient was held constant at a value m = 0,65 for all the runs. The coefficients z₁ and z_n were varied in the simulation process to obtain the correct values of the discharge and the downstream pressure head of the experiments. In the following Table 2 the lateral inflows at different ports are compared with the simulation with constant and variable m, respectively, for the experiments with 54 ports and a spacing ratio of DL/D = 2,5.

 

Port nr.	1	15	27	38	54	Parameter
qi in l/s	0,628	0,658	0,738	0,864	1,165	z1=2,292	z54=0,0621	m=0,6724
qi in l/s	0,608	0,651	0,749	0,883	1,133	z1=5,380	z54=0,0200	m=0,6500

 

Table 2: Lateral inflow at selected ports for different distributions of z_i

 

Although the sum of the friction losses depends on the distribution of the lateral inflow the friction losses are nearly the same for both the distributions. This can be explained by the very small differences in q_i over the whole length of the pipe. Identical friction losses consequently mean unchanged additional losses under the same boundary conditions. To consider the influence of different spacings it is sufficient to change only the overall coefficient z related to the end velocity of the pipe. Figure 5 shows the mean values z in relation to the spacing ratio DL/D. Despite the scattering of the z-values for DL/D > 7,5 it is obvious that in the range of DL/D < 7,5 the additional loss coefficients can be reduced considerably. This result now is in agreement with the physics of flow. Shorter spacings of the ports prevent the extent of the disturbance of the flow field by the lateral inflow over the whole lenght.

 

Figure 5: Representative z-values for different spacing ratios

 

CONCLUSION

The method of using additional loss coefficients in the calculation of the resistance behavior of complex three-dimensional flow fields was demonstrated for a combining manifold. One main result is the statement that the discharge coefficient for the orifices in the manifold pipe remains practically unchanged which contradicts the common practice which relates this type of coefficients to flow in branched pressurized pipe systems. The lateral inflow along the pipe is dominated by a considerably nonlinear distribution of the additional loss coefficient with a maximum value at the upstream end. For variable spacing ratios of the ports it was shown that the mean value for these coefficients can be reduced for ratios DL/D < 7,5. Based on the evaluations of the described experiments combining manifolds can be designed with a high degree of accuracy.

 

REFERENCES

[1] Valentin, F. and Basha, H.: Calculation of flow in combining manifolds for final settling tanks. Proc. XXVIIth IAHR-Congress San Francisco, Proc. Vol. 1, pp 263-268.

[2] Valentin, F. and Merlein, J.: Abfluß im Tauchrohr. Mitt. Hydraulik u. Gew.kde. TU München Nr. 62, München 1996.

[3] Valentin, F.: Widerstandsverhalten der diskontinuierlichen Strömung - Beispiel Tauchrohr. Wasserwirtschaft 87(1997) Nr. 10/97, S. 448-453.