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The Numerical Simulation of Collapse Pressure and Boundary of the Cavity Cloud in Venturi* ZHANG Xiaodong (໋Ջ)**, FU Yong (ؑဇ), LI Zhiyi (हᄝ࿌) and ZHAO Zongchang (ვᆗФ)

R&D Institute of Fluid and Powder Engineering, College of Chemical Engineering, Dalian University of Technology, Dalian 116012, China

Abstract The idea that the collapse proceeds from the outer boundary of the cavity cloud towards its center for the ultrasonic cavitation proposed by Hasson and Morch in 1980s is further developed for calculating the collapse pressure and boundaries of cavity cloud at the collapse stage of bubbles for hydraulic cavitation flow in Venturi in present research. The numerical simulation is carried out based on Gilmore’s equations of bubble dynamics, which take account of the compressibility of fluid besides the viscosity and interfacial tension. The collapse of the cavity cloud is considered to proceed layer by layer from the outer cloud towards its inner part. The simulation results indicate that the predicted boundaries of the cavity cloud at the collapse stage agree well with the experimental ones. It is also found that the maximum collapse pressure of the cavity cloud is several times as high as the collapse pressure of outside boundary, and it is located at a point in the axis, where the cavity cloud disappears completely. This means that a cavity cloud has higher collapse pressure or strength than that of a single bubble due to the interactions of the bubbles. The effects of operation and structural parameters on the collapse pressure are also analyzed in detail. Keywords cavity cloud, collapsing layer by layer, hydrodynamic cavitation, collapse pressure, bubble dynamics

1

INTRODUCTION

High local temperature and pressure induced by the collapse of cavity bubbles causes erosion, vibration and noise in many hydraulic structures or machineries, which could result in severe damage of these installations. In the past several decades, many researchers have investigated the mechanisms of hydraulic cavitations in order to prevent the occurrence of the cavitations. However, the high local temperature and pressure due to the collapse of cavity bubbles can induce the cleavage of water molecules and yield free hydroxyl radicals, ·OH, which is an oxidizing agent in many chemical reactions. Hydrodynamic cavitation can therefore be used to enhance many industrial processes such as chemical reactions, sterilization, treatment of organic waste water, and so on. These useful effects of cavitation flow have attracted great attention of many researchers in recent years [16]. Until now most of the researches have focused their study on a single bubble behavior in cavity flow. In real hydraulic structures or equipments, the cavity bubbles, however, will exist in the form of bubble cloud or bubble cluster. Therefore, it is more important to study the dynamic behavior of a bubble cloud in order to understand the mechanism of cavity erosion or enhancing processes by cavity flow. van Wijingaarden [7] was the first one who investigated the collapse of a group of bubbles near a flat wall and considerable increase of collapse pressure at the wall was found due to the bubble interaction. d’Agostino and Brennen [8] studied the linear dynamics of a spherical cloud of bubbles using a continuum

mixture model coupled with the Rayleigh-Plesset equations. In practice, however, the dynamics of an individual bubble and bubble/bubble interaction through the surrounding liquid are highly nonlinear. Kumar and Brennen [9] found weakly nonlinear solutions to a number of cloud problems by retaining only the terms that are quadratic in the amplitude. Kubota et al. [10] simulated the unsteady bubbles flow passing a two-dimensional hydrofoil by solving the N-S equations of bubbly liquid mixture coupled with the Rayleigh equations. The shedding of cavitation cloud and the generation of vortex cavitation were discovered. Based on the energy transfer theory, Morch [1113], Hanson et al. [14, 15] studied the dynamics of cavity clusters under ultrasonic radiation and pointed out that the collapse of a cavity cluster was driven by the ambient pressure and the collapse proceeded from the outer boundary of the cluster towards its center. During the collapse the pressure at the inward-moving cluster boundary increased continuously, and at the cluster center it rised significantly above the ambient pressure. The fully nonlinear solution to dynamic problem of the spherical cavity cloud was obtained by Wang and Brennen [1618]. Their computational results manifested the shock wave phenomena and the idea proposed by Hanson et al [14, 15]. Reisman et al. [19, 20] measured very large impulsive pressures on the suction surface of an oscillating hydrofoil experiencing cloud cavitations and demonstrated that these pressure pulses were associated with the propagation of the bubbly shock waves. Kanthale et al. [21] investigated the dynamics of cavity bubbles in the cavitation flow through an orifice plate and analyzed the effects of operation and

Received 2009-02-17, accepted 2009-09-25. * Supported by the National Natural Science Foundation of China (10472024). ** To whom correspondence should be addressed. E-mail: [email protected]

Chin. J. Chem. Eng., Vol. 17, No. 6, December 2009

system parameters on the collapse pressure. They developed empirical correlations for predicting collapse pressure and active volume of cavitation. The research works reviewed above are, however, based on assumptions that the bubbles inside a cavity cloud are uniformly distributed and the liquid medium is incompressible. Huang and Ni [22] studied the collapse pressure of cavity bubbles in the region of an hydrofoil based on Gilmore’s equations of bubble dynamics, which consider the compressibility of fluid as well as the viscosity and interfacial tension. In addition they adopted the assumption that the collapse of the cavity cloud is considered to proceed layer by layer from the outside layer of the cavity cloud towards its inner part. They found that the collapse pressure of the cavity bubbles in the inner part of the cavity cloud is higher than that in the outside boundary. In this paper the idea that the bubbles collapse layer by layer from outside layer of a cavity cloud towards its inner part has been adopted for calculating the collapse pressure and boundaries of cavity cloud in hydraulic cavitation flow in Venturi based on Gilmore’s equations of bubble dynamics. The random distribution of initial bubble radius in the cavity cloud is also considered. The boundaries of the cavity cloud at the collapse stage are successfully simulated and compared with experimental data for the first time. The effects of various parameters on the collapse pressure of cavity cloud are also analyzed in detail. 2

EXPERIMENTAL

In order to obtain a visual configuration of the cavity cloud or cavitation region formed in the Venturi, the experimental apparatus shown in Fig. 1 is set up. It consists of a scroll pump, a Venturi made of organic glass, the digital camera, the water tank and pressure gauge and so on. The configuration of the experimental Venturi and the visual picture of the cavity cloud in the expansion section of the Venturi are shown in Figs. 2 and 3 respectively. It is clear that the cavity zone is composed of large number of micro-bubbles instead of an individual bubble so that it is called cavity cloud. Its shape looks like a rhombus or two coaxial cones connected base against base, they represent the growth

Figure 1 Schematic diagram of experimental apparatus 1üwater tank ; 2üpump; 3, 4üpressure gauge; 5üVenturi tube; 6üflow meter; 7ücooler; 8ücamera; V1, V2, V3üvalves

Figure 2

897

Configuration of the experimental Venturi

Figure 3 Visible configuration of the cavity cloud in expansion section of the Venturi (p1 0.5 MPa, p2 0.12 MPa)

and collapse processes of bubbles in Venturi respectively. The schematic diagrams of the cavity cloud and the pressure profile in the Venturi are shown in Fig. 4. When the water is pumped into the Venturi, the liquid pressure decreases in the tapering section and reaches the minimum at the end of throat of the Venturi, where the liquid pressure is lower than the saturated vapor pressure of fluid at the operation temperature, so the micro-nuclei in the liquid will start to expand and form cavity bubbles. As cavity bubbles move downstream together with the bulk fluid flow in the expansion section, the expansion of cavity bubbles will be restricted until its surface velocity is zero, at this moment the bubble expansion stops. After that moment a rapid compression and collapse will take place. By measuring size of the visual picture of the cavity cloud and taking the outlet diameter of the Venturi (d2 25 mm) as the measuring reference the real size of cavity cloud can be obtained. 3 3.1

MATHEMATICAL MODEL Collapse model

As shown in the Fig. 4, a thin slice of the cavity cloud adjacent to the maximum diameter, dmax, of the cavity cloud in the collapsed zone is taken as the object, its thickness and position in the x direction are dm and x0 respectively. dm is taken as the mean distance between the bubbles in cavity cloud, and x0 and dmax can be obtained from pictures of the transparent experimental apparatus. In order to obtain the mathematical model for describing the behavior of the cavity cloud, the following assumptions are made. (1) The cavity bubbles in motion are spherical. (2) The relative motion between the liquid and cavity cloud and the effects of the solid wall on the collapse of bubbles are neglected. (3) Initial vapor hold-up or void fraction, D, in the slice of cavity cloud with thickness of dm is constant.

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(a)

(b) Figure 4

Schematic diagram of cavity cloud and pressure profile in Venturi

(4) The distribution of the initial radius of the bubbles in cavity cloud obeys the Rayleigh random distribution. (5) There is a definite boundary between the cavity cloud and the surrounding liquid, and the space occupied by the bubbles will be filled with liquid after these bubbles collapse. (6) The bubbles in cavity cloud subjected to the pulse pressure will collapse completely with no rebound effects. As shown in Fig. 5, the slice of cavity cloud with the thickness of dm in x direction is divided into a series of layers with width of ǻri in the r direction. Here the ǻri is taken as the mean distance between the bubbles in cavity cloud, which depends on the initial mean bubble diameter and vapor void fraction. The collapse proceeds from the outside layer in r direction of this slip of cavity cloud towards its inner layers, that is, the bubbles in outer layer will collapse first

and the induced pulse collapse pressures will propagate inwards with the sound velocity, cc, in the compressible liquid. At the same time its magnitude will decay with 1/r. The induced pulse collapse pressure together with the liquid pressure of the far flow field turn into the driving pressure acting on bubbles in the next inner layer. After these bubbles have collapsed, the space occupied by them will be filled with the liquid. The next inner layer, therefore, becomes the outer layer of the slice of cavity cloud, so the boundary of the cavity cloud will move inward continually. Meanwhile, the slice of cavity cloud moves downstream a distance with bulk liquid flow, so a tapering configuration of cavity cloud will be formed at the collapse stage. 3.2

Gilmore’s equations of bubble dynamics

In the present research, the following Gilmore’s equations of bubble dynamics [23] are adopted for calculating the collapse pressure, which takes account of the liquid compressibility besides viscosity and interfacial tension: §¨1 R ·¸ 3 R 2 §¨1 R ·¸ RR c ¹ 2 © 3c ¹ © § R· R § R · dH ¨1 ¸ H ¨1 ¸ c¹ c© c ¹ dt ©

with Figure 5 The slice of cavity cloud with the thickness of dm divided into a series of layers in the r direction

R

dR , dt

R

d2 R dt 2

(1)

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Chin. J. Chem. Eng., Vol. 17, No. 6, December 2009

H

pR

dp

d

U

³p

4.2 The driving pressure of the bubbles

n 1 ª º cf2 «§ pR B · n » 1 » n 1 «¬¨© pd B ¸¹ ¼

n 1

c

§ p B · 2n cf ¨ R ¸ © pd B ¹

Here c is the local sound velocity in the liquid, c is the sound velocity in far field of liquid, B and n are constants (n 7, B 3000×105 Pa), pd is the driving pressure for the bubble collapse, which is the surrounding liquid pressure of the bubble, while pR is liquid pressure on the wall of the bubble: 2V R 4P (2) pR pi R R where ı is the interfacial tension and ȝ the viscosity of liquid, pi is the inside pressure in bubbles, which equals to the saturated vapor pressure at operating temperature, i.e. pi pv. The collapse pressure of the bubble, pc, is defined as the liquid pressure on bubble surface at the moment of bubbles collapsing, namely pc pR. The driving pressure of bubbles in any layer of the cavity cloud is the sum of the liquid pressure in far field and the propagating pulse pressure generated by all those collapsed bubbles in each outside layer. For instance, the driving pressure for bubbles in jth layer is given as follows: pdj

pf p j

(3)

where pf is the liquid pressure in far flow field, and pj is the pulse pressure for bubbles in jth layer, which equals to the mathematical expected value of the sum of all the collapse pressure propagated from the collapsed bubbles beyond the jth layer. 4

NUMERICAL METHODS

4.1

Initial radius distribution of the cavity bubbles

The collapse pressure of a bubble mainly depends on its initial radius, R0. The sizes of initial bubbles in cavity cloud are random and they have a mean value of R0mean, At the same time there is a critical bubble radius, Rlim, over which the cavitation will occur. The following Rayleigh random distribution for R0 will be adopted in present research: g R0

R R ª R R 2 º lim 0 lim ° 0 » exp « 2 2 ® 2V R VR ¬« ¼» ° ¯0

4.2.1 Liquid pressure in far flow field pf In this paper, the k-H turbulent flow model and the cavitation flow models are adopted for calculating the liquid pressure in far flow field, pf, in Venturi. Fig. 6 shows the profiles of pressure and velocity along the axis in the expansion section of Venturi, the pressure calculated in single liquid phase flow using FLUENT is used as the liquid pressure in far flow field for the Gilmore’s equations of bubble dynamics.

Figure 6 Configurations of liquid pressure and flow velocity in expansion section of Venturi (p1 0.6 MPa, p2 0.12 MPa, ȕ 0.16, L1 30 mm, L0 20 mm, L 150 mm, d1 25 mm)

4.2.2 Pulse collapse pressure In the polar coordinate frame the cross section of the slice of cavity cloud considered are divided into a series of layers as shown in Fig. 5. The radius and width of the ith layer are ri and ǻri respectively. The bubbles in the same layer collapse the same way irrespective of ș. The collapse pressure of the bubbles in ith layer is the pulse function of the time, which is taken from Ref. [22] as shown in Fig. 7:

pci (t )

pmax i f (t )

(5)

Here pmaxi is the maximum value of the pulse collapse pressure of a bubble in the ith layer, which is related to its initial radius R0 and can be obtained by solving the Gilomre’s equations of bubble dynamics. In addition, pmaxi is the random function of R0, and f(t) is the pulse function which takes the same form as the Ref. [22]: f (t )

exp bi t W i

R0 ı Rlim R0 Rlim (4)

with

VR

2 R0mean Rlim S

Figure 7

Sketch of pulse collapse pressure

(6)

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with bi

p 2 ln v ai pmax i

where ai is the interval of time corresponding to pci ı pv , W i ti 0.5ai , and ti is the starting time of the collapse of bubbles in the ith layer. Since all the bubbles in jth layer have the same collapse characteristics, the bubble at the point (rj, 0) in the jth layer is considered. The pulse pressure for this bubble derived from the propagated collapse pressure of the bubble at the point (ri, ș) in the ith layer (i

Here rij

r

i

rj2 2ri rj cos T

2

0.5

is the distance be-

tween the points (rj, 0) and (ri, ș), and Rmini is the final radius of a collapsed bubble in ith layer, which equals to 0.1R0. If the number of bubbles in unit volume is denoted 3 by O0, considering the relation D O0 4SR0mean / 3 , O0 could be obtained from the following equation: 3D 3 R0mean O0 (8) 4S The total pulse pressure acting on the bubble at the point (rj, 0) is the sum of collapse pressures propagated from all of the collapsed bubbles beyond the jth layer: j 1

2S

¦ ³0

p*j (t )

i 1

ª¬ pij (t )d m ri 'ri O0 º¼ dT

j 1

2S

¦ pmax i Rmin i ri dm 'ri O0 ³0 > f t i 1

rij / c / rij º¼ dT

(9)

Since the initial radius, R0, is a random variable, the pulse pressure acting on the bubbles in the jth layer is a mathematical expected value of p*j (t ) as follows: p j (t )

j 1 2S ® g R0 ¦ ³0 ª¬ pij (t )d m Rlim ¯ i 1

f

¦

R0

º ½ ri 'ri O0 » dT ¾ ¼ ¿ j 1 ® g ( R0 )¦ pmax i Rmin i ri d m 'ri Rlim ¯ i 1

f

¦

R0

2S

½

0

¿

O0 ³ ª¬ f t rij / c / rij º¼ dT ¾ 4.3

(10)

Solution procedure

The bubble dynamic Eq. (1) is a second-order

nonlinear ordinary differential equation. It can be turned into a group of first-order differential equations as Eq. (11) and solved by using the fourth-order Runge-Kutta method [24]: R y ° ª 3 2 § 1 R · ° R ¨1 ¸ « °° y f (t , R, R) § R · ¬ 2 © 3c ¹ R ¨1 ¸ ® c¹ © ° ° § · · º § (11) ° ¨1 R ¸ H R ¨1 R ¸ dH » °¯ © c¹ c© c ¹ dt ¼ The initial conditions of Eq. (11) are t 0 , R R0 and y dR / dt 0 . The number of the bubbles, Ȝ0, and their radius, R0, in each layer can be determined by Eqs. (4) and (8) provided the parameters Į, R0mean and Rlim are specified. The calculation starts from the most outer layer (i 1) of the cavity cloud as shown in Figs. 4 and 5 where the driving pressure is pd pf . The maximum pulse collapse pressures of a bubble with the initial radius of R0 in this layer, pmax,1(R0 ) and collapsing time for this layer 't1 can be obtained by solving Eqs. (2) and (11). Meanwhile the slice will move downstream a distance 'x1 v1't1 , where v1 is the velocity of the slice at its original axial position, x0. The sum of the collapse pressure of bubbles in this layer together with the liquid pressure in the far field becomes the driving pressure for the bubbles in the second layer ( i 2 ), and in the same way pmax,2(R0), 't2 and 'x2 v2 't2 can be obtained, and so on. The procedure cannot be terminated until the collapse of the last layer of the slice has been finished. Finally the maximum collapsed pressure and the boundary of the entire collapsed zone can be obtained. The values of some parameters used in simulation are listed as follows: D 0.005 0.05 , R0mean 60 ȝm, Rlim 10 ȝm, dm 0.5 mm, p1 0.4 0.8 MPa, d1 d2 25 mm, ȕ 0.12 0.28, L 80180 mm, ı ˉ ˉ 0.0727 N·m 1, ȝ 1.005×10 3 Pa·s, and ȡ 998 ˉ3 kg·m . As mentioned previously, the value of ǻri depends on the initial mean bubble diameter and vapor void fraction. The value of initial voidage Į was not measured in this work, but its range of 0.0050.05 [22, 25] is relatively narrow, and a typical value of Į 0.01 was adopted in this simulation. For example, in the case of Į 0.01 and R0mean 60 ȝm, ǻri is equal to 0.5 mm [22]. 5

RESULTS AND DISCUSSION

5.1 The boundary of cavity cloud at the stage of bubbles collapse

The evolvement of the cavity bubbles involves the growth and collapse stages. The visualized boundary of the cavity cloud can exhibit the growth and

Chin. J. Chem. Eng., Vol. 17, No. 6, December 2009

collapse processes and can be used to calculate the area of cavity cloud and the strength of cavity flow. Meanwhile, by comparing the predicted boundary with the experimental one, the mathematical model can be validated. As mentioned above, the evolution of the cavity bubble is very complex, since the mechanism of the interaction between the cavity bubbles is not very clear by now. Only the boundary of the cavity cloud at the collapse stage is simulated in the present study. The procedure of calculating the boundary of cavity cloud at the stage of bubbles collapse has been described in Section 4.3. Fig. 8 shows the predicted and experimental boundaries of the cavity cloud at the collapse stage under various inlet pressure conditions and with ȕ 0.16, L 150 mm, d1 25 mm. and Į 0.01 [25], the liquid pressure and flow velocity in expansion section of Venturi has been shown in the Fig. 6. It is revealed that the area of the cavity cloud increases with the inlet pressure and the boundaries of cavity cloud predicted are in good agreement with the experimental ones. Since it is assumed in our model that the col-

(a) p1

0.5 MPa

(b) p1

0.6 MPa

901

lapse is always complete and there is no elastic rebound, the tail flow of the cavity cloud can not be predicted. 5.2 Effects of operating and structure parameters on the collapse pressure 5.2.1 Initial vapor void fraction The effect of initial vapor void fraction on collapse pressure is shown in Fig. 9 with p1 0.6 MPa, ȕ 0.16, d1 25 mm, and L 150 mm. When the radius of a layer becomes smallerˈthe driving pressure on the bubbles in this layer becomes larger. Consequently, the collapse pressure of this layer increases with the decrease of its radius. The maximum collapse pressure is achieved at the center of the slip of cavity cloud. As the slip of cavity cloud moves together with the bulk fluid, the maximum collapse pressure is really located at a point in the axial line, where the cavity cloud disappears, which is clearly shown in Fig. 10. When the initial vapor void fraction increases, the number of cavity bubbles will increase and this leads to the increase of collapse pressure.

Figure 9 Variation of pc with r and Į (p1 0.12 MPa, ȕ 0.16, L 150 mm, d1 25 mm) Į: 1ü0.005; 2ü0.01; 3ü0.03; 4ü0.05

0.6 MPa, p2

Figure 10 Variations of cavity cloud radius and collapse pressure in the axial direction of x (p1 0.6 MPa, p2 0.12 MPa, ȕ 0.16, L 150 mm, d1 25 mm) üü r;Ƶpc

(c) p1

0.7 MPa

Figure 8 Comparison of predicted boundaries of the cavity cloud at collapse stage with experimental ones (p2 0.12 MPa, ȕ 0.16, L 150 mm, d1 25 mm) üü exp.; cal.

5.2.2 Inlet pressure The effect of the inlet pressure on the collapse pressure of each layer is shown in Fig. 11 with Į 0.01, ȕ 0.16, d1 25 mm, and L 150 mm. As shown in Fig. 11 the collapse pressure of each layer increases with the decrease of its radius or the increase of the inlet pressure. When the inlet pressure increases, the pressure at the throat and the cavitation number both decrease, which results in the formation of a larger number of cavity bubbles and a larger cavity zone.

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panding section. As the length of the expanding section increases, both the pressure gradient of the flow field in expansion section and the driving pressures on the bubbles will decrease, which leads to the decrease of the collapse pressure.

Figure 11 Variation of pc with r and p1 (Į 0.01, p2 0.12 MPa, ȕ 0.16, L 150 mm, d1 25 mm) p1/MPa: 1ü0.4; 2ü0.5; 3ü0.6; 4ü0.7; 5ü0.8

Figure 12 shows the variation of the maximum radius of the cavity cloud measured experimentally with the inlet pressure. It is clearly shown that the maximum radius of the cavity cloud increases with the increase of the inlet pressure, which means more cavity bubbles are formed. When the inlet pressure reaches about 0.80.9 MPa, the radius of the cavity cloud increases rapidly and the cavity flow goes into the super cavitation stage at which the Venturi is filled with a large number of bubbles.

5.2.4 The ratio of throat diameter to pipe diameter The effect of the ratio of throat diameter to pipe diameter, ȕ, on the collapse pressure for each layer is shown in Fig. 14 in the case of p1 0.6 MPa, Į 0.01, d1 25 mm, and L 150 mm. It can be found that the collapse pressure for each layer will increase with the decrease of ȕ. When ȕ decreases, the throat diameter decreases too, but the corresponding liquid velocity increases at the same time. The pressure at throat and the cavitation number both decrease, which results in a number of bubbles formed at the expending section and the increase of the collapse pressure.

Figure 14 Variation of pc with r and ȕ (Į 0.01, p1 0.6 MPa, p2 0.12 MPa, L 150 mm, d1 25 mm) ȕ: 1ü0.28; 2ü0.24; 3ü0.20; 4ü0.16; 5ü0.12

Figure 12 Variation of the maximum cavity cloud radius with p1 (Į 0.01, p2 0.12 MPa, ȕ 0.16, L 150 mm, d1 25 mm)

5.2.3 The length of expanding section The effect of the length of expanding section on the collapse pressure of each layer is shown in Fig. 13 in the case of p1 0.6 MPa, Į 0.01, ȕ 0.16, and d1 25 mm. It is found that collapse pressure for each layer decreases with the increase of the length of ex-

5.2.5 The correlation of the maximum collapsed pressure to operating and structure parameters As shown in Fig. 9, the maximum collapsed pressure, pcmax, are located at the axial point (r 0), where the cavity cloud is disappeared, based on the calculating results for pcmax in the Figs. 914 (18 data for pcmax) the correlation of the maximum collapsed pressure in the entire collapsed zoon to operating and structure parameters is given by the nonlinear least square method:

pcmax (MPa) 82.065 u D 0.177 p10.759 L0.314 E 0.810 (12) The average relative error is 4.8% in the operating and structure parameters range for the present work. 6

Figure 13 Variation of pc with r and L (Į 0.01, p1 0.6 MPa, p2 0.12 MPa, ȕ 0.16, d1 25 mm) L/mm: 1ü180; 2ü150; 3ü120; 4ü80

CONCLUSIONS

The present simulation study leads to the following conclusions. (1) The collapse of the cavity cloud proceeds from its the outer layer towards its inner ones. The driving pressures acting on bubbles in the inner layer equals to the sum of all pulse collapse pressures propagated from collapsed bubbles in each outside layer and the liquid pressure in the far flow field. This results in significant increase of collapse pressure for

Chin. J. Chem. Eng., Vol. 17, No. 6, December 2009

those inner cavity bubbles. (2) The simulation indicated that the maximum collapse pressure is as high as several times that at the outside boundary, and it is located at a point on the axial line, where the cavity cloud disappears finally. This means that a cavity cloud has higher collapse pressure or strength than that of a single bubble due to the interactions of the bubbles. (3) The boundaries of the cavity cloud at the collapse stage in Venturi are successfully simulated for the first time and compared with experimental data. The results obtained using the proposed model can explain not only the geometrical characteristics of the cavity cloud but also why the inner cavity bubbles having higher collapse pressures. (4) It is also found that the collapse pressure increases with the inlet pressure and initial vapor void fraction, and decreases with the length of expending section and the ratio of throat diameter to pipe diameter. The inlet pressure and the ratio of throat diameter to pipe diameter have more influence on the collapse pressure.

3

4

5

6

7

8 9 10

11

NOMENCLATURE 12 B c cf dm d0 d1 H L n pc pd pi pmax pR p p1 r R (t) R R Rlim Rmin R0 R0mean x0 Į

E O0 P V I

liquid constant, Pa ˉ local sound velocity in the liquid, m·s 1 ˉ sound velocity in far field of liquid, m·s 1 mean distance of the bubbles in cavity cloud, mm diameter of the Venturi throat, mm diameter of the pipe, mm ˉ liquid enthalpy difference at pressure pR and pd, kJ·kg 1 length of expansion section, mm liquid constant collapse pressure, MPa driving pressure on the bubble, MPa pressure in the bubble, MPa maximum of the pulse collapse pressure, MPa liquid pressure on the wall of the bubble, MPa liquid pressure in far field, MPa inlet pressure, MPa radius of the cavity cloud, mm bubble radius at time t, ȝm ˉ first order differential coefficient of R (t), ȝm·s 1 ˉ second order differential coefficient of R (t), ȝm·s 2 critical radius of the bubble, ȝm final radius of collapsed bubble, ȝm initial radius of the bubble, ȝm mean initial radius of the bubble, ȝm axial position of maximum diameter of cavity cloud, m initial vapor void fraction ratio of Venturi throat diameter to pipe diameter number of bubbles in unit volume viscosity of liquid, Pa·s ˉ surface tension of liquid, N·m 1 expansion angle, (º)

13 14 15

16

17

18

19

20 21

22

23

REFERENCES 1 2

Suslick, K.S., Mdleleni, M.M., Ries, J.T., “Chemistry induced by hydrodynamic cavitation”, J. Am. Chem. Soc., 119, 93039304 (1997) . Jyoti, K.K., Pandit, A.B., “Effect of cavitation on chemical disinfec-

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tion efficiency”, Water Res., 38, 22492258 (2004). Sivakumar, M., Pandit, A.B., “Wastewater treatment: A novel energy efficient hydrodynamic cavitational technique”, Ultrason. Sonochem., 9, 123131 (2002). Zhang, X.D., Li, Z.Y., Wu, J., Liu, X.W., Liu, D.X., Xia, Y.J., “Enhancement of chemical reactions by hydrodynamic cavitation”, J. Chem. Ind. Eng. (China), 56 (2), 262265 (2005). (in Chinese) Zhang, X.D., Yang, H.Z., Li, Z.Y., “Relationship between strength of hydrodynamic cavitations and amount of induced hydroxyl radical”, J. Chem. Ind. Eng. (China), 58 (1), 2732 (2007). (in Chinese) Zhang, X.D., Fu, Y., Li, Z.Y., Zhao, Z.C., “The collapse intensity of cavities and the concentration of free hydroxyl radical released in cavitation flow”, Chin. J. Chem. Eng., 16 (4), 547551 (2008). van Wijngaarden, L., “On the collective collapse of a large number of gas bubbles in water”, In: Proceedings of 11th International Congress of Applied Mechanics, Springer-Verlag, Berlin, 854861 (1964). d’Agostino, L., Brennen, C.E., “Linearized dynamics of spherical bubble clouds”, J. Fluid Mech., 199, 155176 (1989). Kumar, S., Brennen, C.E., “Non-linear effects in the dynamic of clouds of bubbles”, J. Acou. Soci. Amer., 89, 707714 (1991). Kubota, A., Kato, H., Yamaguchi, H., “A new modelling of cavitating flows: A numerical study of unsteady cavitation on a hydrofoil section”, J. Fluid Mech., 240, 5996 (1992). Morch, K.A., “On the collapse of cavity cluster in flow cavitation”, In: Proceedings of the 1st International Conference on Cavitation and Inhomogeneities in Underwater Acoustics, Springer Series in Electrophysics, Springer, Berlin, 95100 (1980). Morch, K.A., “Cavity cluster dynamics and cavitation erosion”, In: Proceedings of ASME Cavitation and Polyphase Flow Forum, Hoyt, J.W., ed., ASME, New York, 1–10 (1981). Morch, K.A., “Energy considerations on the collapse of cavity cluster”, Appl. Sci. Res., 38, 313321 (1982). Hanson, I., Kedrinskii, V.K., Morch, K.A., “On the dynamics of cavity clusters”, J. Appl. Phys, 15, 17251734 (1981). Hanson, I., Morch, K.A., “The dynamics of cavity clusters in ultrasonic (vibratory) cavitation erosion”, J. Appl. Phys., 9, 46514658 (1980). Wang, Y.C., Brennen, C.E., “Shock wave development in the collapse of a cloud of bubbles”, ASME Cavitation and Multiphase Flow Forum, FED, 194, 1520 (1994). Wang, Y.C., Brennen, C.E., “The noise generated by the collapse of a cloud of cavitation bubbles”, ASME/JSME Symposium on Cavitation and Gas-Liquid Flow in Fluid Machinery and Devices, FED, 226, 1729 (1995). Wang, Y.C., Brennen, C.E., “Numerical computation of shock waves in a spherical cloud of cavitation bubbles”, J. Fluid Eng., 121, 872880 (1999). Reisman, G.E., Brennen, C.E., “Pressure pulse generated by cloud cavitation”, ASME Symposium on Cavitation and Gas-Liquid Flows in Fluid Machinery and Devices, FED, 236, 319328 (1996). Reisman, G.E., Wang, C., Brennen, C.E., “Observation of shock waves in cloud cavitation”, J. Fluid Mech., 355, 255283 (1998). Kanthale, P.M., Gogate, P.R., Pandit, A.B., Wilhelm, A.M., “Dynamics of cavitational bubbles and design of a hydrodynamic cavitational reactor: Cluster approach”, Ultrason. Sonochem., 12, 441452 (2005). Huang, J.B., Ni, H.G., “On the pressure generation by the collapse of a travelling cavitation bubble cluster in a compressible liquid”, Adv. Hydrodyn., 2 (3), 111 (1987). Huang, J.T., Fundamental and Application of Cavitation and Cavitation Erosion, Tsinghua University Press, Beijing (1991). (in Chinese) Li, R.H., Feng, G.C., Numerical Solution of Differential Equations, Chinese Higher Education Press, Beijing (1996). (in Chinese) Kanthale, P.M., Gogate, P.R., Pandit, A.B., Wilhelm, A.M., “Cavity cluster approach for quantification of cavitational intensity in sonochemical reactors”, Ultrason. Sonochem., 10, 181189 (2003).

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