Effect of accumulators on cavitation surge in hydraulic systems

Transcription

1 Effect of accumulators on cavitation surge in hydraulic systems Donghyuk Kang, Satoshi Yamazaki, Shusaku Kagawa, Byungjin An, Motohiko Nohmi, Kazuhiko Yokota ISROMAC 6 International Symosium on Transort Phenomena and Dynamics of Rotating Machinery Hawaii, Honolulu Aril -5, 6 Abstract The analysis of cavitation surge was erformed to investigate the effect of accumulators on cavitation surge. The accumulator was modeled by using the momentum equation with a mass and a daming and a stiffness coefficients. The mass, daming and stiffness coefficients were associated with a ie length between an accumulator and a main ie, a valve resistance and a comliance of fluid in an accumulator, resectively. The ustream accumulator with the valve resistance had the stability effect and caused the increase of the angular velocity of cavitation surge. The downstream accumulator had the stability effect at small mass flow gain factors and caused the increase/ decrease of the angular velocity of cavitation surge at low/ large cavitation comliances. The amlitudes of flow and ressure oscillations can be reduced by the installation of the ustream accumulator. Keywords Cavitation surge Lum arameter models Accumulator Deartment of Mechanical engineering, University of Aoyama Gakuin, Kanagawa, Jaan Fluid Machinery at Systems Comany, EBARA Cororation, Chiba, Jaan Corresonding author: kang@me.aoyama.ac.j INTRODUCTION Since cavitation surge is caused by the interaction of cavitation with hydraulic systems [], an installation of an accumulator in hydraulic systems is lanned for suression of cavitation surge. To redict the effect of accumulators on cavitation surge, the stability analysis of cavitation surge was erformed for the hydraulic systems. The hydraulic systems consisted of an ustream tank, an inlet ie, an ustream accumulator, a cavitating um, a downstream ie, an outlet ie, a downstream accumulator, an outlet ie and a downstream tank. The effect of the accumulators was qualitatively discussed in the ranges of mass flow gain factor and cavitation comliance shown in the exeriments [, 3, 4, 5]. NOMENCLATURE A :cross sectional area [m ] c :dimensionless comliance = D :diameter of imeller [m] K :cavitation comliance [-] l l l :dimensionless ie length :ie length [m] v [-] l / D [-] :dimensionless chord length of imeller l / D [-] l :chord length of imeller [m] M :mass flow gain factor [-] :ressure [Pa] td :total ressure at um downstream [Pa] tu :total ressure at um ustream [Pa] Q :flow rate [m 3 /s] R :flow gain [-] S :ressure gain [-] t :dimensionless time t U / D [-] t :time [s] U :ti velocity of imeller [m/s] v v :dimensionless fluid volume = [-] A D v :fluid volume [m 3 ] :secific heat ratio [-] :valve resistance [-] :ie loss coefficient [-] :density of working fluid [m 3 /s] u :cavitation number ( ) U [-] u v / : flow coefficient Q / U [-] :dimensionless ressure U [-] :dimensionless comlex angular velocity= D /U [-] :comlex angular velocity [m/s] I :daming rate [-] R :angular velocity [-] Suerscrit - :time averaged comonent ^ :fluctuating comonent /

2 Article Title c au c ad au ad l l au l ad l i u Pum d o l u l d Subscrit :inlet ie :outlet ie ad :downstream accumulator au :ustream accumulator c :cavitation d :downstream ie i :ustream tank o :downstream tank u :ustream ie. METHODS The dynamic of hydraulic systems were treated in terms of lumed-arameter models [6, 7] which simlifies the descrition of the hysical effects between two measuring oints. The lum-arameter model is usally considered valid when the dimensions of a hydralic system are shorter than the acoustic wave length at the considered frequency. For simlicity of the analytical model, the following assumutions are adoted in the resent analysis. () The flow is one-dimensional. () The flow and ressure oscillations are exressed as a temorally harmornic oscillation and infinitesimal. Thus, the second or higher order quantities are negligible. (3) The working fluid is incomressible. The ressure loss is considered as the loss coefficient. (4) The elastic deformation of all ies is negligible and the cross-sectional area of a ie is constant. (5) The comliance of tanks is large and thus the ressure oscillation inside tanks is negligible. (6) The cross-section area of all ies connected to a main ie from an accumulator is the same as the horizontal section area of accumulators. Under the above assumtions, the dimensionless ressure and flow oscillations can be written in the comlex form as follows. jt jt ( t) e, ( t) e () / Here, U and Q / A U are the steady ressure and flow coefficients, resectively. / U and v c Figure. Schematic of the analytical model in the resent study Q / A U are the comlex amlitudes of the unsteady ressure and flow coefficients, resectively. t t U / D is the dimensionless time and D /U is the dimensionless angular frequency. A is the cross-sectional area at the suction ie and D is the diameter of the imeller. is the angular velocity and U is the ti velocity of the imeller. is the density of the working fluid and j is the imaginary unit. First, we consider the flow through the ies shown in Fig.. By utting Eq.() into the dimensionless unsteady continuity and momentum equations and subtracting the steady terms of them from themselves, the following equations can be introduced. i u () o (3) Table. Default analytical arameters Inlet ie length, l Ustream ie length, l u 3 Downstream ie length, l d 3 Outlet ie length, l 3 Chord length of a imeller, l.3 Flow rate,.577 Pie loss coefficients, u d. Flow gain, R - Pressure gain, S Comliance of accumulator, c i l jl (4) u uluu jluu (5) d dldd jldd (6) uluu jluu (7) Here, the subscrits i, u,, d,, o indicate the inlet tank, the inlet ie, the ustream ie, the downstream ie, the outlet ie and the outlet tank, resectively. is the loss coefficient. l= l /D is the dimensionless ie length. The first and second terms of the right side in Eq.(4) indicate the hydraulic resistance and inertance, resectively. Next, we consider the flow through the accumulator shown in

3 Fig.. The momentum equations of the accumulator can be exressed as l au j au auau au, (8) l ad j ad adad ad. (9) Here, is the valve resistance. The subscrits au and ad indicate the ustream and downstream accumulators, resectively. The continuous equations for the accumulators can be exressed as u au, () d ad. () With resect to the fluid inside the accumulators, the flow coefficients au and ad can be obtained as following equations. vau au j au cau j au () au vad ad j ad cad j ad (3) ad Here, v v / A D is the dimensionless fluid volume of the accumulator. is the secific heat ratio and c is called the comliance. By utting Eqs.() and (3) in Eqs.(8) and (9), the momentum equations with the mass and daming and stiffness coefficients were introduced as follows. l au au au j au ( au ) (4) cau l ad ad ad j ad ( ad ) (5) cad The mass, daming and stiffness coefficients corresonding to the first, second and third terms of the left sides are associated with the ie length, the valve resistance and the comliance of fluid, resectively. These arameters will be examined in the resent study. A cavity volume v c is assumed to be formed ustream of the cavitating um. Then, the dimensionless continuity equation can be exressed as dv ( t) ( t) ( t) c d u. (6) dt c Here, vc v / A D is the dimensionless cavity volume. The change of the cavity volume dvc can be considered to be functions of the ustream cavitation number u ( u v )/ U (7) and the ustream flow coefficient u Q u / AU. (8) Thus, it can be written as dvc vc d u u vc u d u u u Md (9) u Kd where M and K are mass flow gain factor and cavitation comliance, resectively. The ressure rise sulied from the cavitating um can be exressed as the total ressure rise coefficient. The total ressure rise coefficient is defined as u Article Title 3 ( td tu ) / U. () Here. tu and td are the total ressures at the ustream of the cavitating um and the downstream of the cavitating um, resectively. Using the Bernoulli equation with the ressure rise of the cavitating um, the following equation is obtained. d u d u jl d () Here, l =l /D is the dimensionless inertial length of the cavitating um. l is the mean value of the chord length. The first term of the right side indicates the unsteady total ressure rise of the cavitating um. The second term of the right side shows the unsteady dynamic ressure rise. The last term of the right side indicates the inertia term. The ressure rise of the cavitating um can be considered to be functions of the ustream cavitation number u and the discharge flow coefficient d on the assumtion that cavitation occurs at the ustream of the cavitating um and that the total ressure rise deends on the discharge flow rate. Thus, we can reresent the unsteady ressure rise as d u R d S u d u. () u d Here, R and S are called flow gain and ressure gain in the resent study, resectively. From above formulations, we obtained the homogeneous linear equations. The homogeneous linear equations have the comlex angular frequencies exressed as R I. (3) Here, R and I show the angular velocity and the daming rate, resectively. For I <, the infinitesimal amlitude grow, which means cavitation surge. The zero daming rate indicates the onset boundary of cavitation surge. Table shows the tested arameters. These arameters are used in the resent aer as the default arameters.. RESULTS AND DISCUSSION Figure (a) shows the stability ma of cavitation surge with the ustream accumulator for various comliances c au with au= and l au=. The abscissa and the ordinate show mass flow gain factor and cavitation comliance, resectively. The uer and lower regions of the onset boundary of cavitation surge indicate the stable and cavitation surge regions, resectively. All results show that the lines, the onset boundaries of cavitation surge, indicate the ositive sloe. That is, the increase of cavitation comliance has the stabilizing effect and the increase of the mass flow gain factor causes cavitation surge. Point A is located in the uer region of the onset boundary of cavitation surge for c au=. This means that the state of Point A is stable. As c au is increased u to, Point A is located in the lower region of the onset boundary of cavitation surge and thus its state becomes cavitation surge. Assuming that cavitation comliance and mass flow gain factor are not changed by the modification of the hydraulic systems in the resent study, we can say that the ustream accumulator has the instability effect. For c au=, the cavitation surge region is more widened than that for c au=.

4 Cavitation comliance, K Cavitation comliance, K Article Title 4 c au = c au = c au = c au = c au = c au = A Cavitation surge... Angular velocity, R.. Cavitation comliance, K Figure. Onset condition of cavitation surge and angular velocity for the ustream accumulator with au = and l au = c ad = c ad = c ad = c ad = c ad = c ad = Cavitation surge... Figure (b) shows the angular velocities at the onset boundary of cavitation surge with the ustream accumulator for various comliances c au with au= and l au=. The horizontal and vertical axes reresent cavitation comliance and the angular velocity, resectively. All results show that angular velocity decreases with the increase of cavitation comliance. The ustream accumulator causes the increase of the angular velocity of cavitation surge. This is believed to be due to the decrease of the ie length by the installation of the ustream accumulator. Namely, it says that the ustream accumulator acts as the inlet tank. Figure 3(a) shows the stability ma of cavitation surge with the downstream accumulator for various comliances c ad with ad= and l ad=. For c ad=, the stable region is widened mainly at small mass flow gain factors as comared to the result for c ad=. Thus, the downstream accumulator has the stability effect at small mass flow gain factors. For c ad=, the onset boundary of cavitation surge at small mass flow gain factors is similar to that for c ad= and the onset boundary of cavitation surge at large mass flow gain factors aroaches to that for c ad=. Figure 3(b) shows the angular velocities at the onset boundary of cavitation surge with the downstream accumulator for various comliances c ad with ad= and l ad=. For c ad= and c ad=, the angular velocities increases/.. Cavitation comliance, K Figure 3. Onset condition of cavitation surge and angular velocity for the downstream accumulator with ad = and l ad = Angular velocity, R decrease at small/ large cavitation comliances. This is different from the result of the ustream accumulator that the angular velocity increases at all cavitation comliances. Figure 4(a) shows the stability ma of cavitation surge with the ustream accumulator for various resistances au with c au= and l au=. For au=.5, the onset boundary of cavitation surge is shifted to the cavitation surge. This indicates that the stable region becomes large and that the ustream accumulator with the valve resistance has the stability effect. As the valve resistance is increased u to au=., the onset boundary of cavitation surge at small mass flow gain factors is moved to the right side but the onset boundary of cavitation surge at large mass flow gain factors aroaches to the result without the accumulator. Figure 4(b) shows the angular velocities with the ustream accumulator for various resistances au with c au= and l au=. For au=.5, the angular velocity is slightly decreased at large cavitation comliances as comared to the result for au=. For au=, the angular velocity is almost the same as that without the accumulator. Figure 5(a) shows the stability ma of cavitation surge with the downstream accumulator for various valve resistances ad with c ad= and l ad=. For ad=.5, the onset boundary of cavitation surge is slightly shifted to the left side from that for ad=. For ad=, the onset boundary of cavitation surge at

5 Cavitation comliance, K Cavitation comliance, K au = au =.5 au = au = au =.5 au = Cavitation surge... ad = ad =.5 ad = ad = ad =.5 ad = Cavitation surge... large mass flow gain factors aroaches to the result without the accumulator. All results show that the valve resistance of the downstream accumulator has the instability effect. Figure 5(b) shows the angular velocities with the downstream accumulator for various resistances ad with c ad= and l ad=. The tendency, the increase/ decrease of the angular velocity at low/ large mass flow gain factors, is not changed by the valve resistance. Figure 6(a) shows the stability ma of cavitation surge with the ustream accumulator for various ie lengths l au with c au= and au=. For l au=3, the stable region becomes large as comared to the result for l au=. For l au=, the stable region is more widened. Figure 6(b) shows the angular velocity of cavitation surge with the ustream accumulator for various ie lengths l au with c au= and au=. For l au=3, the angular velocity is smaller than the result for l au=. For l au=, the angular velocity becomes smaller. Figure 7(a) shows the stability ma of cavitation surge with the downstream accumulator for various ie lengths l ad with c ad= and ad=. For l ad=3, the cavitation surge region increases as comared to the result for l ad=. For l ad=, the cavitation surge region at low mass flow gain factors Article Title 5.. Cavitation comliance, K Figure 4. Onset condition of cavitation surge and angular velocity for the ustream accumulator with c au = and l au = Angular velocity, R.. Cavitation comliance, K Figure 5. Onset condition of cavitation surge and angular velocity for the downstream accumulator with c ad = and l ad = Angular velocity, R becomes larger. All results of Figs 3(a), 5(a) and 7(a) show that the onset boundary of cavitation surge at large mass flow gain factors is not affected by the downstream accumulator. Figure 7(b) shows the angular velocities of cavitation surge with the downstream accumulator for various ie lengths l ad with c ad= and ad=. As shown in Figs.3(b) and 5(b), the angular velocity increase/decrease at low/large mass flow gain factors. We observed the intersections of the lines for l ad=, l ad=3 and l ad= and the line without the accumulator. The intersections for l ad=, l ad=3 and l ad= are shown by, and 3, resectively. As l ad is increased, the cavitation comliance of the intersection increases. Figure 8 shows the amlitude ratios and hases of the flow and ressure oscillations with the ustream accumulator. The three cases were examined. The first case is without the accumulator, the second case is with the ustream accumulator for c au= au= and l au= and the final case is with the ustream accumulator for c au= au= and l au=. Without the accumulator, the amlitudes of and u are larger than those of d and as shown in Fig.8(a) and the hases of d and are about -35 degrees as shown in Fig.8(b). However, the amlitudes of all ressure oscillations

6 Cavitation comliance, K Cavitation comliance, K l au = l au =3 l au = l au = l au =3 l au = Cavitation surge... l ad = l ad =3 l ad = l ad = l ad =3 l ad = Cavitation surge... are almost equal and nearly 5 times higher than the amlitude of u as shown in Fig.8(c). It can be observed that the hase differences between the ressure and flow oscillations are about -9 degrees due to the inertia effect as shown in Fig.8(d). For c au= au= and l au=, the amlitudes of, d and and all ressure oscillations are very lower than the amlitude of u. This indicates that the amlitudes of the flow and ressure oscillations can be reduced by the installation of the ustream accumulator. The hases of d and are about 5 degrees. For c ad= ad= and l ad=, all magnitudes and hases are similar to the results without the accumulator excet for the hase of u. Figure 9 shows the amlitude ratios and hases of the flow and ressure oscillations with the downstream accumulator. For c ad= ad= and l ad=, the amlitudes of d and are slightly larger than those without the accumulator as shown in Fig.9(a). This result shows that the flow induced from the change of the cavity volume easily flows to the downstream of the cavitating um. The hases of d and are about 84 degrees as shown in Fig.9(b). The amlitudes of d and are largely decreased as comared to those without the downstream accumulator as shown in Fig.9(c). All result Article Title 6.. Cavitation comliance, K Figure 6. Onset condition of cavitation surge and angular velocity for the ustream accumulator with c au = and au = Angular velocity, R.. Cavitation comliance, K Figure 7. Onset condition of cavitation surge and angular velocity for the downstream accumulator with c ad = and ad = Angular velocity, R 3 show that the ustream flow and ressure oscillations are not sinigificantly affected by the downstream accumulator. For c ad= ad= and l ad=, the amliutudes and hases of the ressure and flow oscillations are similar to the results without the accumulator. 3. CONCLUSION The effects of the accumulators on cavitation surge based on the one dimensional linear stability analysis by the lumarameter models were investigated. The following conculusions are obtained. () The ustream accumulator with the small valve resistance has the stability effect and causes the increase of the angular velocity. () The downstream accumulator at low mass gain factors has the stability effect and causes the increase/ decrease of the angular velocity at low/ large cavitation comliances. (3) The valve resistance of the downstream accumulator has the instability effect at low mass flow gain factors. (4) The onset boundaries of cavitation surge at large mass flow gain factors are not affected by the downstream accumulator. (5) The amlitudes of the flow and ressure oscillations can

7 Amlitude ratio Amlitude ratio Phase [degree] Phase [degree] Amlitude ratio Amlitude ratio Phase [degree] Phase [degree] (M=.87, K=) c au =, au = and l au = (M=.78, K=) c au =, au = and l au = (M=.58, K=) u d (a) Amlitude of flow oscillation (M=.87, K=) c au =, au = and l au = (M=.78, K=) c au =, au = and l au = (M=.58, K=) u d (c) Amlitude of ressure oscillation (M=.87, K=) Article Title 7 c au =, au = and l au = (M=.78, K=) c au =, au = and l au = (M=.58, K=) u d (b) Phase of flow oscillation (M=.87, K=) c au =, au = and l au = (M=.78, K=) c au =, au = and l au = (M=.58, K=) u d (d) Phase of ressure oscillation Figure 8. Amlitude ratios and hases of the flow and ressure oscillations for the ustream accumulator.5.5 (M=.87, K=) c ad =, ad = and l ad = (M=.58, K=) c ad =, ad = and l ad = (M=.4, K=) u d (a) Amlitude of flow oscillation (M=.87, K=) c ad =, ad = and l ad = (M=.58, K=) c ad =, ad = and l ad = (M=.4, K=) u d (b) Phase of flow oscillation 6 4 (M=.87, K=) c ad =, ad = and l ad = (M=.58, K=) c ad =, ad = and l ad = (M=.4, K=) u d (c) Amlitude of ressure oscillation (M=.87, K=) c ad =, ad = and l ad = (M=.58, K=) c ad =, ad = and l ad = (M=.4, K=) u d (d) Phase of ressure oscillation Figure 9. Amlitude ratios and hases of the flow and ressure oscillations for the downstream accumulator be reduced by the installation of the ustream accumulator. (6) The flow induced from the change of the cavity volume easily flows to the downstream of the cavitating um by the installation of the downstream accumulator. (7) The ustream flow and ressure oscillations are not sinigificantly affected by the downstream accumulator. We are rearing the exeriment to investigate the effect of

8 the accumulators on cavitation surge observed in an double suction centrifugal ums [8]. The ustream and the downstream accumulators with the valve resitances are effective for avoidance of cavitation surge. Article Title 8 REFERENCES [] W.E. Young. Study of Cavitating Inducer Instabilities. Final Reort NASA-CR-3939, 97. [] C.E. Brennen. The Bubbly Flow Model for the Dynamic Characteristics of Cavitating Pum. Journal of Fluid Mechanics, 89,.3~4, 978. [3] S. Rubin. An Interretation of Transfer Function Data for a Cavitating Pum. 4th AIAA/ASME/SAE/ASEE Joint Proulsion Conference AIAA-4-45, 4. [4] K. Yonezawa. J. Aono. D. Kang. H. Horiguchi. Y. Kawata. Y.Tsujimoto. Numerical Evaluation of Dynamic Transfer Matrix and Unsteady Cavitation Characteristics of an Inducer. International Journal of Fluid Machinery and System. 5,.6~33,. [5] D. Kang. S. Hatano. K. Yokota. S. Kagawa. M. Nohmi. Estimation of the Dynamic Characteristics of a Double- Suction Centrifugal Pum in Cavitation Surge. Vol ,5 (in Jaanese). [6] A. Cervone. Y. Tsujimoto. Y. Kawata. Evaluation of the Dynamic Transfer Matrix of Cavitating Inducer by Means of a Simlified "Lumed-Parameter" Model. Journal of Fluid Mechanics Vol ~.43-9, 9. [7] D. Kang. K. Yokota. Analytical study of Cavitation Surge in Hydraulic system. Journal of Fluid Mechanics Vol.36 No...3-3, 4. [8] S. Hatano. D. Kang. K. Yokota. S. Kagawa. M. Nohmi. Study of Cavitation Instabilities in Double Suction Centrifugal Pum. International Journal of Fluid Machinery and Systems. Vol.7. No , 4.