Analysis of cavitating flow through a venturi

Transcription

1 Vol. 0(), pp. 67-7, June, 0 DOI: 0.897/SRE0.60 Article Number:BFBED8 ISSN 99-8 Copyright 0 Author() retain the copyright of thi article Scientific Reearch and Eay Full Length Reearch Paper Analyi of cavitating flow through a venturi Mohammed ZAMOUM and Mohand KESSAL Laboratoire Génie Phyique de Hydrocarbure LGPH, Faculté de Hydrocarbure et de la Chimie FHC, Univerité M hamed Bougara, Boumerdè, 000, Algérie. Received March, 0; Accepted 6 May, 0 A dynamical tudy of a bubbly flow in a tranveral varying ection duct (Venturi), i modeled by the ue of the ma and momentum phae equation, which are coupled with the Rayleigh-Pleet equation of the bubble dynamic. The effect of the throat dimenion and the uptream void fraction on flow parameter are invetigated. The numerical reolution of the previou equation et let u found that the characteritic of the flow change dramatically with uptream void fraction. Two different flow regime are obtained: a quai-teady and a quai-unteady regime. The former i characterized by a large patial fluctuation downtream of the throat, which are induced by the pulation of the cavitation bubble. The quai-unteady regime correpond to flahing flow in which occur a bifurcation at the flow tranition between thee regime. Thi tranition occur at R c. which correpond to An analytical epreion for the critical bubble ize at the flahing flow point i alo obtained and compared with theoretical data. Key word: Venturi meter, tow-phae flow, cavitation. INTRODUCTION It i well known that the venturi i a robut technique for meauring the flow characteritic of a ingle-phae fluid for high Reynold number. Multiphae flow meauring i generally more difficult. The denity of a ga-liquid miture depend upon the volume fraction of the ga, and the phae denitie. The velocity of the ga within the venturi i likely to be different from that of the liquid. Over the two lat decade, the invetigation of a homogeneou teady-tate cavitating nozzle flow, uing pherical bubble dynamic with a polytropic thermal proce (Wang and Brennen, 998), have hown ome flow intabilitie illutrated by the flahing flow phenomenon. The flow model, generally ued, i a nonlinear continuum bubbly miture which i coupled with the dynamic equation of the bubble. A three equation model wa firt propoed by van Wijngaarden (968, 97), and ha been ued for tudying teady and tranient hock wave propagation in bubbly liquid, by omitting the acceleration of the mean flow. Thi model ha been alo conidered by Wang and Brennen (998), in the cae of converging-diverging nozzle, with an uptream variable void fraction. It wa oberved that ignificant change of the flow characteritic depend trongly on the latter and a critical bubble radiu have been obtained. Conidering the ga nucleation rate, a a Correponding author. m_zamoum000@yahoo.fr, Tel: (00)0668. Author() agree that thi article remain permanently open acce under the term of the Creative Common Attribution Licene.0 International Licene

2 68 Sci. Re. Eay ource term in the ma conervation equation of the bubble, Delale et al. (00) have ued the previou model for the ame converging-diverging nozzle. They have concluded that the encountered flow intability can be tabilied by thermal damping. Several author have alo conidered the bubble dynamic equation under an appropriate form to the choen eample. Among them, Wang and Brennen (999) have epreed the flow equation in time and radial coordinate, for a bubbly miture, where the hock wave have been tudied for pherical cloud of cavitating bubble. Beide, effect of the hock on the bubble interaction have alo been analyed. The ame Rayleigh-Pleet equation ha been ued by Gaton et al. (00) in modelling the bubble a a potential ource. The tream function ha been written in function of patial coordinate and the ource term. They have analyed the effect of comple interaction through a venturi. By introducing liquid quantity and motion equation in a patial Rayleigh-Pleet dynamic relation, Moholkar and Pandit (00) have obtained a global dynamic equation which ha been reolved in a three tep method. In their work they have tudied the effect of the downtream preure, the venturi pipe ratio, the initial bubble ize and the uptream void fraction, on the dynamic of the flow. The reult of the imulation how that the bubble/bubble and bubble/flow interaction through the hydrodynamic of the flow ha important effect on the behaviour of the bubble flow. Conidering a one bubble motion in a venturi, Soubiran and Sherwood (000) have obtained a dynamic equation of the flow, baed on different acting force. More recently, Ahrafizadeh and Ghaemi (0) have eperimentally and numerically invetigate the effect of the geometrical parameter, uch a throat diameter, throat length, and diffuer angle, on the ma flow rate, critical preure ratio and application rang of mall-ized cavitating venturi (CV). The obtained reult how that the CV in very mall ize are alo capable in controlling and regulating the ma flow rate while their characteritic curve are imilar to thoe of ordinary CV with larger throat ize. Alo, by decreaing the throat diameter of CV, the choked mode region, the critical preure and dicharge coefficient decreae. By decreaing the diffuer angle from to in the numerical imulation, the critical preure ratio increae and the dicharge coefficient remain contant. By increaing the throat length of CV, the critical preure ratio decreae while dicharge coefficient doe not hown any change. Alo, a variable area cavitating venturi wa deigned and invetigated eperimentally by Tian et al (0). Four et of eperiment were conducted to invetigate the effect of the pintle troke, the uptream preure and downtream preure a well a the dynamic motion of the pintle on the performance of the variable area cavitating venturi. The obtained reult verify that the ma flow rate i independent of the downtream preure when the downtream preure ration i le about 0.8. The ma flow rate i linearly dependent on the pintle troke and increae with the uptream preure. The dicharge coefficient i a function of the pintle troke; however it i independent of the uptream preure. They concluded that the variable area cavitating venturi can control and meaure the ma flow rate dynamically. Our invetigation i baed on the firt model (a non linear continuum bubbly miture model coupled with the dynamic equation of the bubble), the preent work conider a cavitating flow through a venturi. The effect of the throat diameter of the venturi and the limit of flahing flow occurring for ome uptream void fraction, are analyed, and a critical value of bubble radiu at the flahing flow point i obtained. BASIC EQUATIONS An aiymmetric venturi with cro-ectional area A() i howed in Figure, where the dimenion are reported to the inlet radiu a. The liquid i aumed to be incompreible and the relative motion between the liquid and the duct wall i neglected and the total uptream bubble population i uniform without coalecence and further break up of the bubble in the flow. Ga and vapor denitie are neglected in comparion to one of the liquid. The bubble are aumed to have the ame initial radiu R. Friction between the liquid and the duct wall i neglected and the relative motion between the tow phae ignored. Then the miture denity can be epreed in function of bubble popula tion : ρ ρl ηv Where V πr, t i the bubble volume. Continuity and momentum equation of the bubbly flow (Wang and Brennen, 998) are: t () A ua 0 u u Cp u t ( α) Where α,t πηr / πηr i the bubble void fraction, and u(, t) the fluid velocity. Cp,t p,t p / ρlu i the fluid preure coefficient, and p(, t) the fluid preure, p the uptream fluid preure, and u the uptream fluid velocity. The dynamic of the bubble can be modeled by the Rayleigh-Pleet equation (Knapp et al., 970; Daily and Hammitt, 970; Pleet and Properetti, 977). D R DR R Dt Dt Where σ k DR k R R R Cp 0 Re R Dt We () () D/ Dt / t u / i the Lagrangian derivative,

3 ZAMOUM and KESSAL 69 Flow a a 0 Figure. Schematic of the venture. Fig. Table. Initial condition and water characteritic. Initial parameter Water characteritic at 0 C R =00 µm u =0 m/ ρ L =000 Kglm μ E =0.0 N/m k=. Re= =0.8 We=7 μ L =0.00 N/m S=0.07 N/m σ p p ρ u v L the cavitation number, p v preure of vapor inide the bubble. L the partial E Re ρ u R μ i the Reynold number, μ E the effective vicoity of liquid. L We ρ u R / S i the Weber number, S repreent the liquid urface tenion, and ρ L the liquid denity. Equation (), () and () contitute a imple model of onedimenional two phae bubbly flow bubbly with a nonlinear bubble dynamic equation. Steady-tate olution Auming teady-tate condition, all the partial time derivative term in Equation () to () diappear. Then, Equation et () to () can be tranformed into an ordinary differential equation et, with only one independent variable (): (-)ua=(-)=contant () du dcp u d ( α) d d R du dr u dr Ru u d d d d Re u dr R d The correponding initial condition are: Cp 0 k k We R R R R(=0)=, U(=0)=, Cp(=0)=0 (7) () (6) The aial variation of the cro ectional take the following from: A β β β 0 Where i the dimenionle radiu of the venturi throat, and the ditance along the ai. In the preent work we aumed: =0., =.0, =.7, =6.7, =0. RESULTS AND DISCUSSION Equation et ( to 6) i reolved by the ue of a fourth order Runge-Kutta cheme, with ome flow condition (Table ). Venturi diameter throat effect Three non-dimenional diameter ventuti throat (β) 0., 0.6 and 0.7 were teted numerically. Effect of the bubble radiu evolution i howed in Figure for ome non- (8)

4 R ection A() 70 Sci. Re. Eay, , ,8 0.8 =0.6 =0. = , , , Figure. Venturi ection for variou value of the non-dimenional throat radiu..,,0.0., =0. =0.6 =0.7.0,0.,.0,0 0. 0, 0.0 0, Figure. Bubble radiu for variou value of the non-dimenional throat radiu. dimenional diameter of the venturi throat (Figure ). By increaing the diameter venturi throat, the bubble radiu decreae and the bubble ocillation frequency increae. Fluid aial velocity and preure ditribution are drawn in Figure and, for different throat diameter. It eem that the evolution of thee parameter correpond to the monophaic cae (Figure ). In the Figure 6, the aial bubble radiu gradient give a large value after the throat ection which i due to the inertial phenomena, a eplained in Blak (99) work. A trongly dumping i alo oberved for the ubequent peak. Figure 7, how a part of the previou (Figure 6), correponding to a mall ditance, where the continuity of radiu gradient can be verified.

5 u ZAMOUM and KESSAL 7,0.0,8.8,6.6,.,., ,0.,.0,0 7. 7, 0.0 0,0,..0,0 7. 7, 0,0 0.0 Figure. Fluid velocity for variou value of the non-dimenional throat radiu ,0-0, ,0 C p -. -, -.0 -, , -.0 -,0 0,0,,0 7, 0,0,,0 7, 0, Figure. Fluid preure for variou value of the non-dimenional throat radiu. Uptream void fraction effect Five different uptream void fraction (α ) of the order of 0 - are ued in the computation to tudy, the effect of the uptream void fraction on the flow tructure through the ventuti. The cae of =0 correpond to the incompreible pure liquid flow, the reult are hown in Figure 8, 9 and 0 which correpond to the nondimenional bubble radiu ditribution, fluid velocity and fluid preure coefficient, repectively, an intability

6 -.0 dr/d dr/d 7 Sci. Re. Eay Figure 6. Bubble radiu gradient for variou value of the non-dimenional throat radiu ,6 6, , , 7. 7, 7. Figure 7. A part of Figure 6. inception can be 0.0 remarked in thee figure, which i located jut after the throat, thee reult confirm thoe of Wang and Brennen (998) with no important difference. Figure 8 hown -0.8 that the bubble ize reach the maimum after paing the nozzle throat of the venturi with increae in -.0 the uptream void fraction, the maimum ize of the bubble increae and bubble frequency ocillation decreae, thi maimum ize i hifted further downtream after it reach the critical radiu (intability occur), the bubble growth without bound in the calculation, thi intability occur when the bubble reache a critical value, alo the void fraction growing -.

7 u() R() ZAMOUM and KESSAL 7 6 =0 =0 - =.70 - =.80 - = Figure 8. Aial bubble radiu ditribution for deferent uptream void fraction.,.,0.0,8.8,6.6 =0 =0 - =.70 - =.80 - =0 -,.,., Figure 9. Aial fluid velocity ditribution for different uptream void fraction. lead to large amplitude of the previouly drown parameter, an important remark concern the venturi geometry effect: in the Wang and Brennen (998) work, where cavitation in converging-diverging nozzle bubbly

8 Cp() 7 Sci. Re. Eay 0, , -0. -, , -. -, , -. -,0 -.0 =0 =0 - =.70 - =.80 - =0 - -, Figure 0. Fluid preure coefficient for different uptream void fraction. flow i tudied. It can be oberved that intability occur for an uptream void fraction,0.0-6, which correpond to a critical bubble radiu r c. wherea, for our geometry (Figure ), the ame phenomenon occur for,7.0 -, with r c,. Thi difference i due to the throat nozzle geometry. An other difference between thee geometry concern the numerical implementation in the firt cae (Wang and Brennen, 998) a variable pace tep i required, contrarily to the econd cae where a contant and relatively large pace tep i ufficient in the practice r c correpond the flahing flow inception, which i illutrated by an intability of the parameter flow analytical epreion for r c i obtained by / Wang and Brennen (998), R c (σ / αc ), where c i the uptream void fraction at which flahing flow occur. The fluid velocity i illutrate in Figure 9. The preence of the bubble in the uptream flow reult in the downtream fluctuation of the flow. With increae the uptream void fraction, the amplitude of thi velocity fluctuation downtream increae and it frequency ocillation decreae. However, a a, increae to a critical value of the uptream void fraction, the flahing flow occur, the velocity increae dramatically and the flow become untable. Due to the Bernoulli effect, the fluid preure coefficient varie inverely with the fluid velocity (Figure 0). Figure illutrate the Bubble radiu gradient in the flow for different uptream void fraction. Due to the inertial phenomena, the bubble radiu gradient become a large value after the throat ection of the venturi and a trongly dumping i alo oberved for the ubequent peak. Thee peak are reduced and amortized far further downtream flow. CONCLUSION A teady tate equation et i conidered for a bubbly two phae flow acro a venturi. We have hown the effect of throat diameter and uptream void fraction on the characteritic parameter flow evolution. In the obtained reult, we found that the uptream void fraction trongly affect the tructure of the flow. Two different flow regime are obtained: quai-teady and quai-unteady regime, where the tranition between them i illutrated by a flahing flow inception. The latter phenomenon occur at R c. which correpond to Thi value i compared with the cae of converging-diverging nozzle which indicate that the converging-diverging nozzle preent more tability than the venturi. Thi analytical reult i numerically teted for a venturi. Alo we have hown the infleion point poition and the correponding bubble radiu and void fraction. Conflict of Interet The author have not declared any conflict of interet.

9 dr/d ZAMOUM and KESSAL =0 =0 - =.70 - =.80 - = Figure. Bubble radiu gradient for different uptream void fraction. Nomenclature A: dimenionle cro-ectional area of the Venturi, A A, A: cro-ectional area of the Venturi, A : uptream cro-ectional area of the Venturi, Cp: fluid preure coefficient, p p /ρ L u, R: dimenionle bubble radiu, R R, R c : dimenionle critical bubble radiu at which flahing flow occur, R : uptream bubble radiu, Re: Reynold number, ρ L u R μ E, S: urface tenion of the liquid, We: Weber number, ρ L u R / S, k: polytropic inde for the ga inide the bubble, p : fluid preure, p :uptream preure, p v : vapor preure, T: dimenionle time, t u R, t: time, u: dimenionle fluid velocity, u u, u: fluid velocity, u : uptream fluid velocity, V: volume of the bubble, V πr, : dimenionle Eulerian coordinate, R, : Eulerian coordinate. Greek Letter : void fraction of the bubbly fluid, c : uptream void fraction at which flahing occur, : uptream void fraction, β : dimenionle radiu of the Venturi throat, η : dimenionle bubble population per unit liquid volume, η R, η : bubble population per unit liquid volume, γ : ratio of pecific heat of the ga inide the bubble, μ E : effective dynamic vicoity of the liquid, :dimenionle fluid denity, ρ L : denity of the liquid, σ : cavitation number, p p ρ u. REFERENCES v L Ahrafizadeh SM, Ghaemi H (0). Eperimental and numerical invetigation on the performance of mall-ized cavitating venturi. Flow meaurement and Intrumentation, :6-. Blak FG (99). The tenile trength of liquid. A review of the literature. Harvard coutic Re. Lab. TM 9, June. Delale CF, Okita K, Matumoto Y (00). Steady-State cavitating nozzle flow with nucleation. Fifth international ympoium on cavitation. Oaka, Japan, November -. Gaton MJ, Reize JA, Evan GM (00). Modelling of bubble dynamic in a Venturi flow with a potential flow method. Chem. Eng. Sci. 6:67-6. Knapp RT, Daily JW, Hammitt FG (970). Cavitation. New York: Me Graw Hill. Moholkar VS, Pandit AB (00). Numerical invetigation in the behaviour of one-dimenional bubbly flow in hydrodynamic cavitation. Chem. Eng Sci. 6:-8. Pleet MS, Prooeretti A (977). Bubble dynamic and cavitation. Annual review of fluid mechanic, 9:-8. Soubiran J, Sherwood JD (000). Bubble motion in a potential flow within a Venturi. J. Multiphae Flow. 6: Tian H, Zeng P, Yu N, Cai G (0). Application of variable area cavitating venturi a a dynamic flow controller. Flow Meaurement and Intrumentation.8:-6. van Wijngaarden L (968). On the equation of motion for miture of liquid and ga bubble. J. Fluid Mech. :6-7. van Wijngaarden L (97). One-dimenional flow of liquid containing mall ga bubble. Annual Rev. Fluid Mech. : Wang YC, Brennen CE (998). One-dimenional bubbly cavitating flow through a converging-diverging nozzle. J. Fluid Eng. 0: Wang YC, Brennen CE (999). Numerical computation of hock wave in a pherical bubble cloud of cavitation bubble. J. Fluid Eng. :