M inimum wall pressure coefficient of orifice plate energy dissipater

Wan-zheng Ai*，Jia-hong WangSchool of Shipping and Ports Architecture Engineering，Zhejiang Ocean University，Zhoushan 316000，PR China Received 10 September 2013；accepted 15 June 2014 Available online 7 February 2015

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M inimum wall pressure coefficient of orifice plate energy dissipater

Wan-zheng Ai*，Jia-hong Wang
School of Shipping and Ports Architecture Engineering，Zhejiang Ocean University，Zhoushan 316000，PR China Received 10 September 2013；accepted 15 June 2014 Available online 7 February 2015

Abstract

Orifice p late energy dissipaters have been successfully used in large-scale hydropower projects due to their sim ple structure，convenient construction procedure，and high energy dissipation ratio.Them inimum wall pressure coefficient of an orifice p late can indirectly reflect its cavitation characteristics:the lower theminimum wallpressure coefficient is，the better theability of theorifice plate to resistcavitation damage is.Thus，it is important to study them inimum wallpressure coefficientof the orifice plate.In this study，this coefficientand related parameters，such as the contraction ratio，defined as the ratio of the orifice plate diameter to the flood-discharging tunnel diameter；the relative thickness，defined as the ratio of the orifice plate thickness to the tunnel diameter；and the Reynolds number of the flow through the orifice plate，were theoretically analyzed，and their relationshipswere obtained through physicalmodel experiments.It can be concluded that them inimum w all pressure coefficientismainly dom inated by the contraction ratio and relative thickness.The low er the contraction ratio and relative thicknessare，the larger theminimum wall pressure coefficient is.The effects of the Reynolds number on theminimum wall pressure coefficient can be neglected when it is larger than 105.An empirical expression was presented to calculate them inimum wall pressure coefficient in this study. ?2015 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license（http:// creativecommons.org/licenses/by-nc-nd/4.0/）.

Orifice plate；M inimum wall pressure coefficient；Cavitation；Contraction ratio；Relative thickness；Energy dissipater

1.Introduction

Orifice plate energy dissipaters w ith sudden-contraction and sudden-enlargem ent forms have been successfully used in large-scale hydropower projects due to their sim ple structure，convenient construction procedure，and high energy dissipation ratio.For the M ica Dam in Canada，the flow velocity of the flood-discharging tunnel was decreased from 52 m/s to 35m/sat the head of 175m，due to the use of two plugsw ith lengthsof 49m and 37m，which aresimilar to theorifice plate（Russell and Ball，1967）.In the Xiaolangdi Hydropower Project in China，three orifice plates installed in the flooddischarging tunnel obtained an energy dissipation ratio of 44%，and effectively controlled the flow velocity through the gate less than 35m/s under the condition of a head of 145 m（Aiand Zhou，2014）.

For a flood-discharging tunnel w ith orifice plate energy dissipaters，the cavitation characteristics of the orifice plate energy dissipater directly affect the safety of the flood-discharging tunnel.Thus，it is necessary to obtain the relationshipsbetween the cavitation characteristicsof the orifice plate energy dissipater and correlative factors，such as thegeometric parameters of the orifice plate and flow conditions.The contraction ratio（β），defined as the ratio of thediameter（d）of the orifice plate to the diameter（D）of the flood-discharging tunnel，is an important index affecting the critical cavitation num ber of the orifice plate，which can show the cavitation characteristics of the orifice plate（Ai and Wu，2014）.Kim et al.（1998），Takahashi and Matsuda（2001），and Zhang（2003）concluded that the critical cavitation number decreases w ith the increase of the contraction ratio.Qu et al.（2001），Zhang and Cai（1999），and Ball etal.（1975）indicated that theReynoldsnumber has little impacton cavitation characteristics of orifice p late energy dissipaters.

As stated above，research conducted in the past focused mainly on the effects of the contraction ratio and flow conditions on the cavitation characteristics of the orifice plate energy dissipater based on research of the critical cavitation number of the orifice plate.As amatter of fact，the effects of the orifice plate thickness，which can affect the flow regime around the dissipater and energy loss，on the cavitation characteristics of the orifice plate，are also remarkable.Thus，it is necessary to investigate the effectsof orifice plate thicknesson the cavitation characteristics of the orifice plate.

Because cavitation around the orifice p late often occurs first at the position of them inim um wall pressure，the m inimum wall pressure coefficient of the orifice plate can indirectly reflect the cavitation characteristics of the orifice plate（Zhang and Cai，1999），and is also an important index for design of the orifice plate（Aiand Ding，2010）.The objective of this study，therefore，was to investigate the effects of all related factors，especially the orifice plate thickness，on the minimum wall pressure coefficient of the orifice plate，to establish an empirical expression for the minimum wall pressure coefficient of the orifice p late，and to analyze the effects of related factors on the cavitation characteristics of the orifice plate.

Fig.1.Flow through orifice plate.

2.Definition ofm inim um pressure coefficien t

The sketch of the flow through an orifice plate in the flooddischarging tunnel is shown in Fig.1，where T is the thickness of the orifice plate，and Lbis the length of the vortex-ring region.Vortex-ring regions exist in front of and behind the orifice plate due to the sudden-contraction and suddenenlargement geometry of the orifice plate，and those vortexring regions are the important regions of the energy dissipation.Them inimum wall pressure coefficient cpcan be defined as

where p0is the pressure on a non-disturbed section in front of the orifice plate，which can be regarded as the section located at least 0.5D in front of the orifice plate；pminis the minimum wall pressure；ρis the density of water；and u is theaverage flow velocity in the tunnel.Eq.（1）shows that the sm aller pminis，the larger cpis，and them ore easily cavitation occurs.The m inimum wall pressure coefficient cpshows the status of the m inimum wall pressure of the orifice p late. Thus，it can indicate the cavitation characteristics of the flood-discharging tunnelw ith orifice plates.The larger cpis，the lower the capacity of orifice plate to resist cavitation damage is.

3.Theoretical considerations

Them inimum wall pressure coefficientof the orifice plate is related to geometric parameters and hydraulic param eters，including the density of waterρ（kg/m3），the dynam ic viscosity of waterμ（N·s/m2），the tunnel diameter D（m），the orifice plate diameter d，the orifice plate thickness T（m），the average flow velocity in the tunnel u（m/s），and the deviation between the pressure on the non-disturbed section and minimum wall pressure p0-pmin（Pa）.All the above parameters arew ritten into a formula as follows:

According to the dimensionalanalysis，D，μ，andρare three basic parameters of the seven.A non-dimensional equation can be obtained using theπtheorem as follows:

Eq.（3）can be rew ritten as follows:

Combining Eq.（1）w ith Eq.（4），we can obtain

where Re is the Reynolds number；andαis the relative thickness，andα=T/D.Eq.（5）indicates that theminimum wall pressure coefficientof the orifice plate cpisa function of β，α，and Re.The follow ing study procedure wasmeant to determ ine the effects of param etersβ，α，and Re on cp，according to Eq.（5）.

4.Model experiment

The experimentalset-up of the physicalmodel consisted of an intake system，a tank，a flood-discharging tunnel w ith an orifice plate energy dissipater，and a return system w ith a rectangular weir（Fig.2）.The diameter（D）of the tunnel model was 0.21 m，and the length of the tunnelmodel was 4.75m，i.e.，the distance from the intake to the pressure tunnel outlet controlled by a gatewas about 22.6D.The orifice plate energy dissipater was p laced at the position of 10.0D away from the tunnel intake and 12.6D away from the outlet.A water head of about 10.0D could be provided by the intake system and the tank.The opening of the gate could be changed conveniently.There were 35 pieces of small plastic tubeinstalled along the tunnelwall，whichwereutilized tomeasure the wall pressure.Because flows change violently in the vicinity of the orifice plate，in the region from 0.5D in front of the orifice plate to 4.0D behind the orifice p late，the plastic tubes were densely installed，w ith a interval of 0.25D.The physicalmodelexperimentswere conducted at theHigh-speed Flow Laboratory of Hohai University.The geometric parameters of the orifice plate and flood-discharging tunnel in each case are shown in Table 1.

According to Eq.（5），the effects of the contraction ratio β，relative thicknessα，and Reynolds number Re on the m inimum wall pressure coefficient cpwere exam ined through physical model experiments.The experiment arrangement was as follow s:First，the m inimum wall pressure coefficient cpwasmeasured in cases 1 through 5 when βand Re varied andαdid not vary，and the effects of the contraction ratioβand Reynolds number Re on the m inimum wall pressure coefficient cpwere examined；second，the m inimum wall pressure coefficient cpwasmeasured in cases 6 through 10 whenαand Re varied andβwas constant，and the effects of the relative thicknessαand Reynolds number Re on the minimum wall pressure coefficient cpwere exam ined.

Fig.2.Experimentalmodel.

Fig.3.Wall pressure distributions along tunnel forβ=0.70 and α=0.20

5.Results and discussion

Themeasured resultsof thewallpressure distribution along the tunnelwhenβis 0.70 andαis 0.20 are shown in Fig.3，where P is the wall pressure expressed by the height of the water column measured using a piezometer（m），X is the distance from the tank along the flow direction，and R is the ratio of themaximum water level to the diameterof the flooddischarging tunnel.The orifice plate is located between X=10D and X=10.2D.Fig.3 shows that the lowestwall pressure occurs in the vicinity of the orifice plate，which approaches the contraction section.The experimental results of theminimum wall pressure coefficient are shown in Table 2 and Table 3.

It can be seen from Tables 2 and 3 thatwhen the Reynolds num ber Re is less than 105，the m inimum wall pressure coefficient cpincreases slightly with the Reynolds number Re，butwhen the Reynoldsnumber Re ismore than 105，ithasno impact on theminimum wall pressure coefficient cp.

Fig.4 and Fig.5 are drawn using the data in Tables2 and 3，respectively，when the Reynoldsnumber is1.20×105.Fig.4 shows that theminimum wallpressure coefficient cpdecreases drastically w ith the increase of the contraction ratioβwhen the relative thicknessαis constant.Fig.5 demonstrates that them inimum wall pressure coefficient cpalso decreases w ith the increase of the relative thicknessαwhen the contraction ratioβis constant，indicating that the effect of the relative thicknessαon the minimum wall pressure coefficient cpis remarkable，which isoften ignored in previous research.From this analysis，it also can be concluded that，the relative thicknessαhas important effects on the cavitation characteristics of the orifice plate，and the risk of cavitation damage occurring at the orificeplate decreasesw ith the increaseof the contraction ratioβand relative thicknessα.By fitting the curves in Figs.4 and 5，the follow ing empirical expression for them inimum wall pressure coefficient of the orifice plate can be obtained:

This expression is valid for 0.40≤β≤0.80，0.05≤α≤0.50，and Re>105.

Table 1 Geometric parametersof orifice plate and flood-discharging tunnel in each case.

6.Conclusions

The minimum wall pressure coefficient cpof an orifice plate energy dissipater isa function of the contraction ratioβ，the relative thicknessα，and the Reynolds number Re of the flow on the basis on Eq.（5）.The effects of Re on cpcan be neglected when Re is larger than 105.

The contraction ratioβand relative thicknessαare the key factors that dom inate the m inimum wall pressure coefficient cp.The lower the contraction ratioβand the relative thicknessαare，the larger theminimum wall pressure coefficient cpand the risk of cavitation damage occurring at the orifice platew ill be.The relationship between cp，β，andαcan be expressed through Eq.（6）when 0.40≤β≤0.80，0.05≤α≤0.50，and Re>105.

Table 2 Variation of cpw ith Re andβforα=0.10.

Table 3 Variation of cpw ith Re andαforβ=0.70.

Fig.4.Relationship betw een cpandβfor Re=1.20×105and α=0.10

Fig.5.Relationship between cpandαfor Re=1.20×105and β=0.70

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This work was supported by the Zhejiang Provincial Natural Science Foundation（Grant No.Y 15E090022）.

*Corresponding author.

E-mail address:aiwanzheng@126.com（Wan-zheng Ai）.

Peer review under responsibility of HohaiUniversity.

http://dx.doi.org/10.1016/j.w se.2014.06.001

1674-2370/?2015 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license（http:// creativecommons.org/licenses/by-nc-nd/4.0/）.