Influence of design parameters on throttling efficiency of cylindrical

Journal of Hydraulic Research Vol. 47, No. 5 (2009), pp. 559–565 doi:10.3826/jhr.2009.3449 © 2009 International Association of Hydraulic Engineering and Research

Influence of design parameters on throttling efficiency of cylindrical and conical vortex valves Influence des paramètres de conception sur l’efficacité de l’étranglement des vannes cylindrique et coniques de vortex PATRYK WÓJTOWICZ, PhD, Institute of Environmental Protection Engineering, Wrocław University of Technology, Wrocław 50-370, Poland. E-mail: [email protected] (author for correspondence) ANDRZEJ KOTOWSKI, PhD, Institute of Environmental Protection Engineering, Wrocław University of Technology, Wrocław 50-370, Poland. E-mail: [email protected] ABSTRACT Selected hydraulic test results of cylindrical and conical vortex valves in a pilot-plant scale are presented to discuss the effects of the geometric parameters and the spatial orientation of a vortex chamber. Tests performed include the effects of the inlet and outlet diameters, swirl radius, chamber height, and diameter on the hydraulic characteristics. A novel, universal variable K, describing the geometry of vortex valves and grouping their geometrical parameters, is introduced. This constant K increases the quantitative accuracy of the relationship between the discharge coefficient and the geometrical parameters. A test methodology for hydrodynamic vortex regulators has been developed and was applied to environmental engineering, based on dimensional analysis. This research presents the empirical equations enabling to predict the discharge coefficient according to the geometrical parameters of the devices. The investigated relationships allow for a rational parameter selection for hydrodynamic vortex regulators. RÉSUMÉ Des résultats sélectionnées d’essais hydrauliques de vannes de vortex cylindrique et coniques à l’échelle d’une installation pilote sont présentés pour discuter les effets des paramètres géométriques et de l’orientation spatiale d’une chambre à vortex. Les essais réalisés incluent les effets des diamètres d’entrée et de sortie, du rayon du tourbillon, de la hauteur de la chambre, et du diamètre sur les caractéristiques hydrauliques. Une nouvelle, variable universelle K, décrit la géométrie des vannes de vortex et groupe leurs paramètres géométriques. Cette constante K augmente la précision du rapport entre le coefficient de débit et les paramètres géométriques. Une méthodologie d’essai pour les régulateurs hydrodynamiques de vortex a été développée et a été appliquée à la technologie environnementale, basée sur l’analyse dimensionnelle. Cette recherche présente les équations empiriques permettant de prévoir le coefficient de débit à partir des paramètres géométriques des dispositifs. Les rapports étudiés tiennent compte d’un choix raisonnable de paramètre pour les régulateurs hydrodynamiques de vortex.

Keywords: Hydraulics, Hydrodynamic regulator, Liquid flow, Modeling, Throttling

1 Introduction

geometrical device parameters to obtain optimum hydraulic efficiency, i.e. the highest ratio between throttled discharge and free flow resistance. Such devices are used, among others, in bypass channels of electric power stations (Brombach 1972a, Elalfy 1979). Vortex diodes are also used to dampen water level variations in surge tanks and in front of hydrodynamic accumulators (air vessels) to improve the efficiency in reducing water hammer (Giesecke et al. 1988, Haakh 1993, 2003, Huber and Prenner 1999, Karney and Simpson 2007). Vortex diodes are successfully implemented in micro-fluidics as a component of micro-pumps (Anduze et al. 2001). If a vortex device is fed only through one side and has only a throttling function, it is referred to as a vortex flow regulator, hydrodynamic regulator, a throttle or a vortex valve (German

Flow regulators are used to throttle flow. Commonly used throttling devices, such as orifices, reducers, or valves, allow for a discharge limitation by reducing the active regulator crosssection. For polluted liquids (sewage), this may cause clogging (Molinas and Marcus 1998). Vortex flow regulators do not have moving mechanical parts lowering the operational reliability. The prototype of the vortex device was patented by Thoma (1928, 1930), describing the so-called check valve (German Rückstrombremse). Because of operational similarity to a diode, it was also called a “vortex diode” (German Wirbelkammerdiode). The original device was further investigated by Heim (1929) and Zobel (1934). Their research was intended to optimize the

Revision received January 8, 2009/Open for discussion until April 30, 2010. 559

560

P. Wójtowicz and A. Kotowski

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

efficiency. Empirical formulae were developed in which the discharge coefficient is related to device dimensions, the air core diameter da and the spray cone angle γ. Investigations related to vortex regulators are rare in the literature. The pressure loss p in a vortex device depends on the following dimensional variables: liquid density ρ, dynamic water (subscript w) viscosity µw , gravitational acceleration g, volume discharge q, vortex chamber radius R = D/2, inlet swirl radius Ro = R − rin , inlet radius rin , outlet radius rout , vortex chamber height hc , outlet edge thickness s, and regulator wall roughness k. From Buckingham’s Pi-theorem, the pressure loss was determined as Figure 1 Cylindrical vortex valve, scheme and notation

p = ζ

q2 ρ , 2A2in

where ζ is the loss coefficient as a function of 
R Ro rout hc s k , , , , ζ = ζ R, F, , rin rin rin rin rin rin

(1)

(2)

where Reynolds number is R = 2ρq/(πµw rin ), Froude number 5 is F = q2 /(2gπ2 rin ) such that with ζ −1/2 = µ  q = µAin 2gH. (3)

Figure 2

Conical vortex valve, scheme and notation

Wirbeldrossel) (Mays 2001). Vortex regulators are used in hydroengineering as energy dissipators (Brombach 1972b). The vortex chamber is also used as a pre-swirling liquid jet flowing to sewage pumps to decrease their power demand (Elalfy 1979). In water and sewage networks, they are mainly used for throttling liquid flow (Kotowski and Wójtowicz 2008a, 2008b). Originally, hydrodynamic regulators had cylindrical chambers (Fig. 1). Regulators with a conical vortex chamber (Fig. 2) were introduced toward the end of the 1970s (Brombach 1980). Such regulators are characterized by a lower hydraulic resistance at free flow conditions as compared to cylindrical regulators. The current analytical description of vortex regulator operation breaks down to Torricelli’s formula, in which the discharge coefficient µ is determined empirically for each regulator. There is a lack of hydraulic characteristics specifying the quantitative and qualitative relation of construction and operational parameters in terms of the throttling effect, measured with parameters such as the loss coefficient ζ or the discharge coefficient µ. The operational reliability of such devices, especially of large dimensions, are thus difficult to assess.

This is referred in the literature to as Torricelli’s formula (Brombach 1972a, 1980, Elalfy 1979, Tesaˇr 1980). The discharge coefficient µ depends on the angular momentum. Specifically, a higher angular momentum results in a lower µ value and a higher hydraulic resistance ζ. Liquid jets flowing across a tangent inlet connector collide with the liquid layer already swirling inside the chamber. The angle of attack θ of the inflowing jet is defined as (Abramovich 1958, Borodin et al. 1967)
 
 Ro − rin Ro Ro cos θ = = −1 +1 . (4) Ro + rin rin rin The angular momentum relative to the swirl radius Ro is generated at the entry to the regulator vortex chamber resulting in a swirling motion of which the dominant peripheral speed depends 2 on the inlet area rin . The centrifugal force in the swirling motion is inversely proportional to the third power of the outlet hole 3 radius rout (Shuy 1996). The following combination of parameters stated in Eq. (2) defines the geometrical constant K of cylindrical regulators (Fig. 1)
 Ro rin 2 Ro r 2 K= = 3 in . (5) rout rout rout In analogy to ζ, µ takes the final form: 
R Ro rout hc s k . , , , µ = µ R, F, K, , rin rin rin rin rin rin

(6)

For the conical vortex regulator (Fig. 2) the geometrical constant K was defined as 2 Experimentation K= Model tests of cylindrical and conical vortex regulators conducted in a pilot-plant scale are presented to determine the effects of constructional and operational parameters on the throttling

2 Ro cos θ rin 2Ro cos θ din2 = . 3 3 rout dout

(5a)

A modification of the inlet entry angle of the conical regulator to θ < 90◦ results in a reduction of angular momentum at the

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

regulator inlet according to the value of cos θ. For cylindrical regulators, θ = 90◦ and hence cos θ = 0. In addition to K, cos θ was considered. Thus, µ for conical flow regulators takes the final form:
 R Ro rout hc s k µ = µ R, F, K, , , , , , , cos θ . (6a) din din din din din din The effects of these dimensionless parameters on µ or similarly on ζ) was investigated empirically. Regulator models were assembled from components to obtain a wide range of geometric parameters. Particular elements were made of Perspex or stainless steel. The following parameters were varied for 19 models of cylindrical vortex regulators: Inlet diameter din , outlet diameter dout , and chamber height hc . The inner vortex chamber diameter was D = 290 mm, while the outlet edge thickness was s = 10 mm. For conical vortex valves, the inlet entry angle θ was in addition varied relative to the cone base plane. The larger cone diameter was D = 290 mm, while the outlet edge thickness was s = 2 mm. Figure 3 shows the test stand including (1) inflow chamber, (2) outflow chamber, and (3) measuring weir. The feeding system consisted of (4) lower tank, (5) circulating pump and (6) upper tank with (7) surge weir. The discharge was controlled with (8) ball valve located in front of (9) feeding chamber, connected to (1) inflow chamber (Fig. 3). Additional components include (10) telescopic weir, (11) anti-surge baffle, (12) regulator, (13) piezometers, and (14) gauge pressure transmitters.

Influence of design parameters on throttling efficiency of cylindrical

561

3 Discussion of results 3.1 Effects of Reynolds and Froude numbers on µ The effect of R and F on coefficient µ for the cylindrical flow regulator is shown in Fig. 4. It appears that in a vortex flow, µ slightly decreases as both R and F increase. For the limit (subscript lim) numbers Rlim > 30,000 and Flim > 1, µ remains nearly constant for all tested cylindrical flow regulators. Because the tests were performed using scale models of real prototypes, the results obtained for variable R resulting in a practically constant µ, have practical significance for the assessment of the µ value. The average discharge coefficient computed from the entire vortex (subscript v) flow range µv that is, below the limit value of Rlim , was compared with the average value µc computed (subscript c) from the range above Rlim . The mean values differ in the average by 1.6% and by 3.8% in the maximum (Wójtowicz 2007). The dependence of the coefficient µ on R and F for the conical valve is shown in Fig. 5. It appears that µ is practically independent of both R and F for Rlim > 50,000 and Flim > 2 in vortex flow for all tested conical flow regulators. From an analysis of the discharge coefficient (Fig. 5a), its value depends only to a small extent on R in the lower range of the vortex flow in the hysteresis range (R1 ≤ R ≤ R2 ). In Fig. 5, R1 corresponds to the starting point of vortex flow as the inlet pressure increases, whereas R2 marks the point where the vortex flow decays for decreasing inlet pressure.

Figure 3 Experimental setup scheme, for details see main text

Figure 4 Relationship between (a) µ(R) and (b) µ(F) for cylindrical valves

562

P. Wójtowicz and A. Kotowski

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

Figure 5 Relationship between coefficient µ and (a) R and (b) F for conical valve with din = dout = 50 mm, hc = 280 mm and θ = 30◦

The hydraulic characteristics of conical vortex regulators are characterized by a hysteresis occurring in the initial operational range, i.e. for a low total head loss H. The flow characteristics for increasing outflow pressure do not coincide with these for decreasing pressure. Within the hysteresis zone one specific value of R (or F) corresponds to two discharge coefficients (Fig. 5). This hysteresis phenomenon is absent for cylindrical flow regulators (Wójtowicz 2007, Wójtowicz and Kotowski 2008). 3.2 Effect of geometrical parameters on µ For the investigated cylindrical regulators, the influence of construction parameters on µ was analyzed separately. The dimensionless relations dout /din , hc /din , D/din , Ro /din , and K were considered, while k/din and s/din were a priori eliminated. Zobel (1934) and Elalfy (1979) demonstrated that a roughness increase reduces the flow resistance. This is in contrast to the aims put forward for these devices, i.e. a maximum flow resistance for vortex flow. The influence of s on coefficient µ was tested resulting in minute differences due to measurement error (Wójtowicz 2007). According to Fig. 6, µ increases with dout /din , such that the hydraulic resistance of the device is reduced. The plot includes results of Elalfy (1979) for a vortex diode in the throttling

Figure 6 Relationship between coefficient µ and relative outlet diameter dout /din for cylindrical valves with D/din = 5.8 and hc /din = 1.64

Figure 7 Relationship between µ and relative swirl radius Ro /din for cylindrical valves with dout /din = 1

mode for D/din = 5 with D = 75 mm and din = 15 mm, for dout /din = 0.67, 1.0 and 1.33, and from Syred and Beér (1974) for cyclones with D = dout for dout /din = 5.0, with an extreme value of µ = 0.967. Increasing the relative vortex chamber height hc /din increases coefficient µ. Accordingly, for an increasing vortex chamber height hc the hydraulic resistance decreases. Therefore, a significant influence of the boundary layer on the velocity and pressure distributions for flat vortex chambers may be observed for hc /R  1 (Ebert 1977). Figure 7 shows the dependence of µ on the relative swirl radius Ro /din for the regulators tested in runs 1, 5, and 9. The plot includes test data of Elalfy (1979) for cylindrical vortex diodes operating in the flow throttling direction of Ro /din = 0.75, 2, and 3.25 (hc /din = 1, dout /din = 1, and din = 15 mm). Figure 7 also contains data of small-sized models with Ro /in = 0.5 and 1.0 (hc /din = 1, dout /din = 1, din = 10 mm). For low values of Ro /din , the discharge coefficient reaches high values as high as µ ≈ 0.45. Increasing Ro /din , thereby conveying the angular momentum of inflowing liquid, the µ decreases. This causes the device resistance to increase (Fig. 7). In the range of 1.31 ≤ Ro /din ≤ 4.33, µ is approximately constant, particularly if Ro /din > 1.5.

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

Influence of design parameters on throttling efficiency of cylindrical

563

3.3 Empirical equations The relationship among µ and the dimensionless parameters, for the investigated cylindrical hydrodynamic valves, are considered below. The analysis indicated that the influence of K, hc /din and dout /din on µ is statistically relevant. Based on 19 runs a multiple regression analysis with a standard error of estimate SEE = 0.00167 and root mean square percentage error of RMS = 7.30% gives µ = 0.066

Figure 8 Relationship between µ and constant K for conical (θ = 30◦ ) and cylindrical regulators (θ = 0)

dout hc + 0.0089 + 0.362K−0.135 − 0.211. din din

(7)

Equation (7) is valid for the following dimensionless ratios related to cylindrical valves 0.375 ≤ dout /din ≤ 2.67, 1.4 ≤ hc /din ≤ 8.73, 3.63 ≤ D/din ≤ 9.67, 0.457 ≤ K ≤ 49.78, 1.31 ≤ Ro / din ≤ 4.33, 0.40 ≤ da /dout ≤ 0.825, 0.675 ≤ tan(γ/2) ≤ 1.51 (68◦ ≤ γ ≤ 114◦ ) and 1 ≤ F ≤ 64.9. To generalize the results for conical hydrodynamic valves, a model was assumed to describe the dependence of µ on the dimensionless similarity numbers. The best fit of data and calculations resulted in a combination of dout /din , hc /din , and D/din (or Ro /din ), K and cos θ. Including the constant K increases the accuracy of the description. The fit for µ based on 81 runs was established as a result of a statistical analysis by means of a multiple regression by the least squares method with R2 = 0.994 and RMS = 6.2% as (Fig. 10)

For conical vortex regulators the relationships between µ and dout /din , hc /din and Ro /din are similar, as is noted from Fig. 8 (Wójtowicz 2007). The hydrodynamic regulator constant K (Fig. 8) accounts for the geometrical parameters of cylindrical regulators involving din , dout , and Ro as given by Eqs. (5) and (5a). The dependence of µ on K shown in Fig. 8 indicates an increasing K value as µ decreases. Thus, the hydraulic resistance generated by the regulator increases. For a specific value of K, three values of µ take the conical chamber height variations into account, namely hc = 140, 280, and 420 mm for θ = 30◦ , with an increase of hc as µ increases for a given value of K. Figure 8 includes the data for cylindrical vortex regulators for θ = 0◦ , resulting in cos θ = 1 from Eq. (5a). This emphasizes a universal form of the regulator constant K proposed herein. Figure 9 relates the discharge coefficient and the inlet entry angle θ for conical regulators of hc /din = 5.6 and 8.4. It appears that for 30◦ ≤ θ ≤ 45◦ only slight variations in µ occur, whereas µ increases significantly as θ ≥ 45◦ . Figure 9 includes data of conical regulators for θ = 0 obtained at similar to conical devices relations hc /din = 4.44 and 7.24 and identical values of D/din = 5.8.

All regression coefficients are statistically relevant at a 95% confidence level. Equation (8) is valid for conical flow regulators with 0.375 ≤ dout /din ≤ 2.67, 1.75 ≤ hc /din ≤ 14.0, 3.63 ≤ D/din ≤ 9.67, 0.229 ≤ K ≤ 43.1, 1.31 ≤ Ro /din ≤ 4.33, 0.50 ≤ cos θ ≤ 0.87 (30◦ ≤ θ ≤ 60◦ ), 0.58 ≤ tan(γ/2) ≤ 1.92 (60◦ ≤ γ ≤ 125◦ ), including 2 ≤ F ≤ 97.4.

Figure 9 Relationship between µ and inlet angle θ for conical valves with dout /din = 1, D/din = 5.8

Figure 10 Observed versus predicted µ-values based on Eq. (8) for conical valves

µ = 0.660 − 0.067

hc D dout + 0.0068 + 0.0055 din din din

+ 0.553K−0.239 − 0.841(cos θ)−0.015 .

(8)

564

P. Wójtowicz and A. Kotowski

4 Conclusions A methodology and selected laboratory results on the effect of geometrical parameters on the discharge coefficient of hydrodynamic flow regulators including both cylindrical and conical elements is presented. The main findings may be summarized as follows: • Coefficient µ or vortex regulators ranges from 0.0521 to 0.455, corresponding to loss coefficients ζ values from 368 to 5, respectively. Values of µ from Eq. (7) involve an RMS below 8%. • The smallest possible vortex chamber height is preferable hc /din ≤ 2 for a maximum liquid flow throttling effect. This is determined by the inlet connector diameter for the relation of D/din ≤ 4 and an outlet hole diameter dout ≥ din , • For conical vortex regulators the discharge coefficient µ ranges from 0.068 to 0.497, corresponding to loss coefficients ζ values from 216 to 4, respectively. Equation (8) has an RMS of 6%. • A small conical vortex chamber height is preferable hc /din < 3 for a maximum liquid flow throttling effect, provided that D/din < 6 and θ ≤ 30◦ . Empirical formulae for the discharge coefficient allow for a proper assessment of the constructional parameters for both cylindrical and conical hydrodynamic regulators, and aids optimal parameter selection in practical applications, especially in environmental engineering.

Acknowledgments Funding of this study was provided by a grant from the Polish Ministry of Science and Higher Education during the years 2005– 2007 (Grant no. 4T07E 05629).

Notation Ain = Inlet area Aout = Outlet area D = Vortex chamber diameter da = Air core diameter din = Inlet diameter dout = Outlet diameter F = Froude number g = Gravitational acceleration H = Total head loss hc = Height (axial length) of vortex chamber K = Vortex regulator constant p = Total pressure loss q = Discharge rin = Inlet radius rout = Outlet radius ra = Air core radius R = Reynolds number R = Vortex chamber radius

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

Ro = Swirl radius γ = Spray cone angle ζ = Loss coefficient θ = Inlet angle µ = Discharge coefficient

References Abramovich, G. (1958). Angewandte gasdynamik. Verlag Technik, Berlin [in German]. Anduze, M., Colin, S., Caen, R., Camon, H., Conedera, V., Do Conto, T. (2001). Analysis and testing of a fluidic vortex micro-diode. J. Micromech. Microeng. 11, 108–112. Borodin, W.A., Ditiakin, Y.F., Kliachko, L.A., Iagodkin, V.I. (1967). The atomization of liquids. Mashinostroenie, Moscow [in Russian]. Brombach, H. (1972a). Untersuchung strömungsmechanischer Elemente (Fluidik) und die Möglichkeit der Anwendung von Wirbelkammerelementen im Wasserbau. PhD Thesis. Mitteilung 25 Institut für Wasserbau, Universität Stuttgart, Stuttgart, Germany [in German]. Brombach, H. (1972b). Vortex devices in hydraulic engineering. 5th Cranfield Fluidic Conf. BHRA, Fluid Engineering B1, 1–12. Cranfield, Bedford, UK. Brombach, H. (1980). Bistable vortex throttles for sewer flow control. IFAC Pneumatic and Hydraulic Components, Warszawa, 109–113. Ebert, F. (1977). Zur turbulenten Durchströmung einer flachen kreiszylindrischen Kammer. Acta Mechanica 25, 241–256 [in German]. Elalfy, Y.-E. (1979). Untersuchung der Strömungsvorgänge in Wirbelkammerdioden und Drosseln. PhD Thesis. Mitteilung 47, Institut für Wasserbau, Universität Stuttgart, Stuttgart, Germany [in German]. Giesecke, J., Horlacher, H.-B., Wacker, R. (1988). Wirbelkammerdioden als Drosselorgan für Druckluftwasserkessel. Wasserwirtschaft 78(9), 378–382 [in German]. Haakh, F. (1993). Transientes Strömungsverhalten in Wirbelkammerdioden. Mitteilung 81. Institut für Wasserbau, Universität Stuttgart, Stuttgart, Germany [in German]. Haakh, F. (2003). Vortex chamber diodes as throttle devices in pipe systems: Computation of transient flow. J. Hydr. Res. 41(1), 53–59. Heim, R. (1929). Versuche zur ausbildung der thomaschen rückstrombremse. Oldenburg, München [in German]. Huber, B., Prenner, R. (1999). The influence of a vortex-flow throttle on transmission and reflection of pressure wave. 28th IAHR Congress, Graz, Austria (CD-Rom). Karney, B.W., Simpson, A.R. (2007). In-line check valves for water hammer control. J. Hydr. Res. 45(4), 547–554. Kotowski, A., Wójtowicz, P. (2008a). Analysis of hydraulic parameters of cylindrical vortex regulators. Environ. Protection Engng. 34(2), 43–56. Kotowski, A., Wójtowicz, P. (2008b). The basis for design and application principles of vortex flow regulators in water

Journal of Hydraulic Research Vol. 47, No. 5 (2009)

and wastewater systems. Wroclaw University of Technology, Wrocław [in Polish]. Mays, L.W. (2001). Stormwater collection systems design handbook, McGraw-Hill, New York, NY. Molinas, A., Marcus, K.B. (1998). Choking in water supply structures and natural channels. J. Hydr. Res. 36(4), 675–694. Shuy, E.B. (1996). Wall shear stress in acceleration and deceleration turbulent pipe flows. J. Hydr. Res. 34 (2), 173–183. Syred, N., Beér, J.M. (1974). Combustion in swirling flows: A review. Combustion and Flame 25(2), 143–201. Tesaˇr, V. (1980). Superquadratic behaviour of vortex diodes. Proc. IFAC Symp Pneumatic and Hydraulic Components, Warszawa, 79–95. Thoma, D. (1928). Vorrichtung zur Behinderung des Rückströmens. D. Reich, Patent 507713 [in German].

Influence of design parameters on throttling efficiency of cylindrical

565

Thoma, D. (1930). Die Rückstromdrossel. VDI - Zeitschrift 74, 1098 [in German]. Wójtowicz, P. (2007). Influence of design and operational parameters on the hydraulic performance of selected vortex flow regulators. PhD Thesis. Wrocław University of Technology, Wrocław [in Polish]. Wójtowicz, P., Kotowski, A. (2008). Flow throttling effect in cylindrical vortex valves. 4th IWA Young Water Professionals International Conference, University of California, Berkeley, USA (CD-Rom). Zobel, R. (1934). Versuche an der hydraulischen Rückstromdrossel. PhD Thesis. Technische Hochschule München, München [in German].