TI'1T1 IN IV E S I ' Y OF MICG1 A N College' of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. UMICH 03371-15-T ON THE EFFECTS OF HEAT TRANSFER UPON COLLAPSING BUBBLES (To be Submitted for Publication to International Journal of Heat and Mass Transfer) by Terry Mitchell1 Frederick G. Hammitt Financial Support Provided by National Science Foundation Grant No. GK- 13081 Research Associate (Formerly Doctoral Candidate) Professor-In-Charge January 1972

ABSTRACT The effects upon spherically-symmetric bubble collapse of the various thermodynamic parameters including interfacial non-equilibrium boundary effects are examined numerically, for the case of cavitation or highly subcooled boiling bubbles. Pressures in the surrounding liquid as a function of time and distance are computed. It is shown that the value selected for the evaporationcoefficient, not well known experimentally, has a very strong influence on collapse pressures and velocities. The thermal diffusivity of the bubble contents is also significant but somewhat less important. The effect of non-equilibrium temperature or pressure discontinuities at the interface were found negligible for the cases studied. ACKNOWLEDGEMENTS Financial support for this work was provided under NSF Grant GK-13081. The authors would also like to thank Mr. N. R. Bhatt and Miss Nancy J. Parsons for their assistance in the assembling and typing of the paper.

LIST OF FIGURES Figure Page l(a). TLocation of Variables for Thernmodynamic Effects Study........... 15 l(b). Sample Temperature Profile During Collapse..................... 15 2. Internal Temperature Distributions at Different Radii for Bubbles Containing Nitrogen Gas with and without Water Vapor; o = O.1, Pi =~0.075 atm., p =0.0277 atm., Ap= 2.925 atm. Pi, vo R =0.1cm., T - 528R....................................... 16 o o 3. Effect of Evaporation Coefficientcs on Quantity of Vapor Crndensed During Bubble Colla se; "Standard" Bubble R = 50 x 10 in., T = 5370R,p. =10 atm., p =0.0277atm.,p =latm......... 17 4. Effect of Evaporation Coefficient oC on Bubble Radius Change with Time for "Standard" Bubble (see Fig. 3).........................18 5. Effect of Evaporation Coefficient oC on Internal (Gas + Vapor) Pressure Change with Radius for ''Standard" Bubble (See Fig. 3). 19 6. Effect of Evaporation CoefficientOCon Average Internal Temperature Change with Radius for "Standard" Bubble (see Fig. 3)........20 7. Effect of Assumption of Constant Temperature in Liquid on Internal (Gas + Vapor) Pressure Change with Radius for "Standard" Bubble (see Fig. 3), CC = 0. 1......................... 21 8. Effect of Assumption of Constant Temperature in Liquid on Condensation Rate, Average Internal Temperature, and Liquid Temperature at Interface as Function of Radius for "Standard" Bubble (see Fig 3),OC = 0.1........................................... 22 9. Internal Temperature Distributions at Different Stages of Collapse for "Standard" Bubbles (see Fig. 3) with Different Initial Radii, CC - 0.1...................................................... 23 10. Effect of Initial Bubble Radius on the Internal (Gas + Vapor) Pressure History During Collapse of Otherwise "Standard" Bubbles (see Fig. 3), C = 0.1...................................... 24 11. Effect of Thermal Conductivity of Contents on the Internal (Gas + Vapor) Pressure History During Collapse of "Standard" Bubble (see Fig. 3), o = 0.1...................................... 25 ii~~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Figure Page 12. Effect of Thermal Conductivity of Contents on the Average Internal Temperature and on Vapor Condensation Rate as Function of Radius for "Standard" Bubble (see Fig. 33\, cOC =0. 1....................................................~. 26 13. Effect of Overpressure (4p) on the Dimensionless Internal (Gas + Vapor) Pressure History gf Collapsing Bubbles, R 50 x 10 in., T =537R pi 10 atm., PV =0.0277atm. C= 0 1......... 27 1. 0 14. Effect of Overpressure (Ap) on the Actual Internal (Gas + Vapor) Pressure History of Collapsing Bubbles; Conditions same as for Fig. 13...................................................... 28 15. Pressure Profiles in Liquid at Different Stages of Collapse for "Standard" Bubble (see Fig. 3),(C= 0.1...........................29 LIST OF TABLES Table Page 1 Effects of Parameters on Cavitation Bubble Collapse.............30 iii

NOMENC LATURE Symbol De sc rip ion C( (Consallt v(ollllte Sl)t ecilfic heat, V C: Constant pressure specific heat p hfg Latent heat of evaporation h Saturated steam enthalpy g (J) Work-energy conversion constant kf Liquid thermal conductivity Transport coefficient defined in Ref. 15 k+l Lk Transport coefficient defined in Ref. 15 m Mass p Pressure Pvsat(T) Saturation vapor pressure corresponding to temperature T q Heat condition rate into bubble/unit area R Bubble radius R Gas constant r radial coordinate T( ) Temperature of particular component (absolute) T Dimensionless total collapse time = 0. 91468 T Average gas temperature over bubble volume (absolute) Tgi Gas temperature at interface (absolute) TL. Liquid temperature at interface (absolute) t Time U Bubble wall velocity; total internal energy u Radial velocity ug Saturated steam internal energy vL Liquid specific volume v Gas specific volume w Rate at which work is done on bubble contents per unit surface area Evaporation (or condensation) coefficient iv

ILiquid thermal diffusivity (XC (G1s Ihll,.l'-,1l (elil'l'k!sivily ") ~:F'olytropic exlponent T - TLL o oo ~ '1- ovoL Ft L Liquid viscosity Density -r Surface tension Dimensionle s s time Subscripts g Non-consensable gas + vapor i Non-condensable gas L Liquid v Vapor ox Value at large r o Initial value Superscripts Time derivative Dimensionle s s v

I. INTRODUCTION An accurate modelling of cavitation bubble collapse is contingent on knowledge of the role of a variety of individual effects. A major effect is that of asymmetry, treated elsewhere and including contributions of the present authors(1' ). Also, there are the effects of a number of mechanical and "thermodynamic" parameters to be examined even if the collapse remains spherical. In the first category there are, e. g., the effects of liquid surface tension, viscosity, and compressibility, and these effects have been examined (e. g., 3,4). In addition, other investigators have modelled spherical bubble collapses considering the individual effects of phase change, spatial temperature variation both inside and outside the bubble, the value of the evaporation coefficient, the thermal conductivity of the internal contents, interfacial discontinuities, and the presence of a noncondensable gas (e. g., 5-9, 20). The aim of the present work is to "model", as realistically as possible, spherically-symmetric cavitation bubble collapse including all of the above parameters II. ANALYSIS The problem to be solved is that of a spherical bubble of initial radius R containing both non-condensable (and non-diffusing) gas and saturated 0 vapor. The bubble is located in an infinite mass of incompressible liquid; initially its internal contents are in mechanical equilibrium (pressure and surface tension). The liquid and bubble are initially at temperature T. The initial non-condensable gas partial pressure is Pi, while that of the vapor is p = Pv (T ). At time zero the system pressure is increased o sat instantaneously to p and the collapse ensues. For clarity, the adjective "non-condensable" will always be included when referenc-e its made to only that component. The term "gas" in the subsequent text refers to noncondensable gas plus vapor. Portion of Ph.D. dissertation(2) of first author.

-2 -The additional assumptions made in developing the model for such collapses include: 1) Both non-condensable gas and vapor obey the perfect gas law. The bubble wall will be assumed to remain sufficiently smaller than the sonic velocity in the gas-vapor mixture throughout the calculation so that the vapor and non-condensable gas pressures may be assumed uniform throughout the bubble at any given time. Since in fact asymmetries will prevent the volume reduction ratio proceeding to very small values, this assumption is realistic. 2) The rate of condensation or evaporation at the interface is defined by the kinetic theory equation (Eq. (5)). 3) The heat conducted out into the liquid is equal to the sum of the latent heat component deposited on the wall by condensation and the heat conducted out from the bubble contents. This heat conducted out through the liquid results in the formation of a thin thermal boundary layer in the liquid which was assumed to be of parabolic temperature distribution. Such an assumption is reasonably valid as long as the bubble contents temperature increases monotonically, Once the bubble contents temperature starts to drop, e. g., during rebound, the assumed temperature profile shape is no longer valid. However, calculations were not continued into the rebound phase. 4) There is a thin but finite nonequilibrium region at the interface because of the continuing process of phase change there, which region contains both temperature and pressure discontinuities between the liquid and the vapor. 5) The temperature within the bubble does not remain uniform, In all previous cavitation bubble collpase analyses known to the authors, (6) with the exception of Hickling's treatment of sonoluminescence, the contents of the bubble are assumed to be at uniform temperature at any given time. By assuming that the rate at which work is being

-3 -done on the bubble is constaniit ov-er short tinme segnieints of tht co`lpset and neglecting radiative and conv-ctiv-e heat transfer, the temrevteu'.'' history can be treated as a classical heat conduction problem (over the same short time segments), i.e., that of a sphere with given initial temperature profile and known surface temperature. Fig. l(a) is a sketch of the bubble with surrounding liquid with the temperatures, pressures, and densities at the appropriate points labelled with the same symbols used in the equations which follow. Fig. l(b) shows a typical temperature profile from the bubble center to the liquid far from the bubble at some time after the beginning of the calculation. The details of the derivation of the governing equations are presented elsewhere, and acknowledgement is made to a number of previous (10-14) related works. The complete system of equations follows: 2 2 ru = RU =g(t) (1 u (R) =R (2, 2?RR (A R 1 R )4 RR +z2cR)a 2 _- _ eL 3 2 r (3' R eR 2e~ (R - 2 eL(L 3 L - (r) R2 2RR~v R( 1 G 2(~ RR ) - eL3 eL 2u (R) e(L g K7-LR ('LR

-4 -dm m 2 1 1 dt = OC4TrR L P 2v. R vT sat 1 v L. Pi Pi.RiTg (6' p, = ePRvTg (7) v2 2 2rV V V g O 2 = (T L- R.T) q(6 i gfg 3hfg6L T dtPv v g( RT 3T 2 T2 +q- T (h d —h hfg _h f O-g L 2 T 4irR dt 34 - ZrR3 (R dk The bubble contents temperature gradient calculation (below) requires some explanation. At some time t the bubble has an internal temperature profile given by T (r,t ) with an average temperature T (t ). If the contents g o fg of the bubble are taken as an open thermodynamic system, then the net rate at which work is being done on the system per unit surface area at time t is 0m (9) w J) 4J2 Lg (Tg) (T g giT k +g ( Tv ( hTg - h (RR)

-5 -where the first term on the right is the actual work rate, the second an(l third terms correct for the energy of the vapor mass which leaves the system (condensation), and the last term is the rate of change of the kinetic energy of the bubble contents. If W from Eq. (10) is assumed to remain relatively constant over a small time interval At, then 4Tr R WAt is the total energy input into the bubble over that interval. Part of this energy goes into increasing the internal energy of the contents, the rest being conducted out of the bubble as heat. To make this division, all of the energy input W At first is assumed to go into increasing the internal energy of the contents; the temperature 2 0 profile T (r, t ) is increased uniformly by an incremental AT = 47r R WAt/ g O Ii Cvg where mgCvg C miCvi + mvCvv, so that the local temperature at the beginning of the time step is simply T (r, t ) + AT. The temperature g o at the interface is assumed constant at T (t! over the time interval At, and then the problem can be treated as the classical heat conduction problem mentioned previously, solved inCarslawand Jaeger's classical text(15) The solution for the present case with fixed surface temperature Tgi and initial temperature profile f(r) = T (r,t ) + AT is. g o 2 2At (r, t) -_ Tgi Rr 2 LT n) r T (rt)-T -2 R = exp ( ) sin( 9 gi Rgi R 2 R (11) Bo r f(r') sin ( R ) dr' After Eq. (11) is evaluated for the new temperature profile (at t + At), a new mean temperature can be calculated for the contents of the bubble T = T (t +At). The part of the work done which has gone into increasing g go the internal energy of the gas is then just Au=m C (T (t +At)-T (t) (12) gg v g o g o and the remainder of the energy has been conducted across the interface, so that m C (T (t +At) -AT -T (t) g Vg g o(13) 4Tr R2 At

-6 -In addition to the above equations, we have the following initial and boundary conditions R(O) = R R(O) = 0, T(r,O0) = T for all r, T (oo, t) = T oo L oo (16) (0) V Pv t(T ), and Pi 0) =pi The system of equations and initial and boundary condition are non-dimensionalized using appropriate combinations of R, oL \ (= Po - Pi - ) 0 0 T and CpL. Early calculations in the present study showed that for a cavitation bubble collapsing under one atmosphere overpressure in room temperature water, the interface temperature differential (Eq. (9a)) was ev5-10 F in the late stages of collapse. However, this discontinuity does not affect the results significantly because the total temperature difference between the bubble contents and the liquid in the same stages of collapse is very much larger, i.e, thousands of degrees. The irreversible effects were subsequently excluded from the system of equations especially when it was found that they had a highly unstabilizing effect on some solutions; hence, Eq. (9) was used rather than Eq. (9a). A similar unstabilizing effect was produced on some solutions when Eq. (8) was included in the analysis; as discussed in section (3) of the results T was assumed equal to T Li co throughout many of the cases. The system of equations (Eqs. (1)-(13)) along with the initial and boundary conditions (Eq. (14-) completely describe the collapse. Eqs. (4) and (5) were simultaneously integrated using Hamming's modified predictor-corrector method; Eqs. (6), (7), (9), and (11)-(13) were solved at the same tir:e, with TL and W (Eq. (10)) held constant over each time step. The remainder of the calculation for an individual time step consists of evaluating Eq. (8) for TLi) determining the local pressures and velocities at various points in the liquid from Eqs. (3) and (1), respectively, and

calculating a new value of W front Eq. (10). The whole sequence of calculations is then repeated for subsequent time steps. III RESULTS AND DISCUSSION The list of parameters studied, with effects of phase change, spatial temperature variation inside the bubble, and presence of non-condensable gas all included in the analysis, consists of the following: (1) Proportion of water vapor in the bubble (2) Evaporation coefficient (3) Liquid temperature: variable vs. constant (4) Bubble radius (5) Thermal conductivity of gas-vapor mixture (6') Liquid overpressure (1) Presence of Water Vapor in Bubble The bubble with only non-condensable gas within was studied previously by Hickling(6) His results can be compared with our own which include the presence of some water vapor within the bubble. Both studies have considered a bubble of 1 mm. radius with an internal nitrogen pressure of 0.075 atmospheres, an external overpressure (=Po - Pio- Pvo) of 2. 925 atmospheres, and a uniform initial temperature of 68 0F. Our calculation includes the effect of the vapor, while Hickling's does not. Fig. 2 shows a direct comparison of the temperature profiles within the bubble in the two cases at various present to initial radius ratios. The x's show the actual data points resulting from the water vapor calculation, while the solid line is a smoothed approximation. There is good agreement between the two calculations; one expects, as Hickling has pointed out, somewhat lower temperatures with the water vapor present merely because those molecules are triatomic. The adiabatic state relation, Tv = Constant, which gives an upper limit for the internal temperature rise during collapse, will yield larger values of T for diatomic gases ( '= 1. 4) than for triatomic gases ( S = 1. 33) for a given volume

-8 -reduction. On the other hand, the thermal diffusivity is reduced by the addition of water vapor to the nitrogen gas, tending to produce higher temperatures as the bubble collapses more nearly adiabatically. The polytropic exponent increases only slightly from about 1. 30 near the beginning of collapse to about 1. 34 near the start of rebound. The adiabatic value for a bubble whose initial contents are 85% nitrogen gas and 15% water vapor is 1.38. (2) Evaporation Coefficient The value chosen for the evaporation coefficient oC in Eq. (5) can be expected to have a most important effect. cC is the correction factor known as the absorption or evaporation coefficient, and represents the ratio of experimentally observed evaporation rates to those predicted by the righthand side of Eq. (5) with oC equal to one. cC has experimentally been shown to take on a wide range of values between 0 and 1. Theofanous, Biasi, and (16\ Isbin give a good summary of recent experimental research on this (8) important parameter, and, in a subsequent paper, Theofanous et al ) suggest that 5 x 10 may be an appropriate value for c for water. It depends on various relatively unpredictable physical parameters such as interface contamination, etc. Three values have been assumed for cC in this portion of the analysis: 0. 01, 0.1 and 1. 0. Figs. 3-6 show some of the results from this calculation. The bubble -3 in this case contained air at initial partial pressure of 10 atmospheres and water vapor (saturated at 770F) at pressure of 0. 0276 atmospheres (i. e., initial vapor mass fraction of total = 95%0. The external pressure far from the bubble is assumed constant at one atmosphere. Fig. 3 shows the effect of oC upon the portion of vapor not condensing. For the smallest value of CC (0. 01), about 50% of the vapor would remain at the conclusion of the calculation (at start of rebound), while for oC> 0.1 < 3% of the original vapor would remain at radius ratios (R/R t of 0. 01. For the larger oc(s), the wall velocity continues to increase monotonically throughout the calculation, with virtually all of the vapor condensing before

-9 -high collapse velocities develop. Thus in this case the collapse is essentially that of a bubble containing only the non-condensable gas. For OC 0.01, however, the vapor has a definite retarding effect; the bubble wall reaches a peak dimensionless velocity of 30. 6 at R/R = 0. 057 and rebounds shortly 0 thereafter from a minimum radius of R/R = 0. 041. 0 Fig. 4 confirms the above. There is only a very slight difference (not distinguishable in the figure) between the radius history of the C= 1. 0 case and the radius history of an identical empty Rayleigh bubble down to R/R = 0. 03. Thus, only a single dotted curve is shown for both cases. For o oC = 0. 01 significant retardation of the collapse begins when R/R is still greater then 0. 5, although this does not show in Fig. 4 until later. Fig 5 shows the history of the total (vapor plus non-condensable gas) pressure inside the bubble. The conclusion from these curves is that the lower the contents pressure over the early part of the collapse, the higher will be the peak pressure inside the bubble at the conclusion of collapse, and hence presumably the greater the damage capability of the cavitation bubble upon rebound. Therefore, the more rapidly the water vapor condenses (i.e., the larger the value of cc), the more likely it is that damage will occur. If CC is actually very low, then little of the vapor will condense, and rebound will commence with substantially lower pressures inside the bubble. M6reover, the further that collapse proceeds before the internal gas pressure begins to retard the inward motion of the wall, the greater will be the buildup of inertia in the liquid, and hence the greater the overshoot beyond the bubble equilibrium radius. This effect of the evaporation coefficient seems to establish a paradox if shock waves from the rebounding bubble are in fact a primary damage cause. If a bubble initially contains mostly saturated water vapor with only a little non-condensable gas, then the evaporation coefficient o must be very large (-w*l. 0) to insure high enough peak pressures to subsequently cause damage. Conversely, it appears likely that it is actually much less than 1.0 (as indicated in ref. 8! in low gas content bubbles, since otherwise sonoluminescence would be much more commonly observed in cavitation bubbles collapsing in water.

-10 -Fig. 6 (average gas temperature as function of bubble radius) shows that the more rapidly the vapor condenses, the slower will be the temperature buildup inside the bubble. The primary cause of this variation is most likely the larger thermal diffusivity which arises from the lower contents density in the larger oC cases. The slower temperature buildup produces both lower pressures and eventually lower peak temperatures within the bubble. (3) Variable vs. Constant Liquid Temperature For bubbles only slightly subcooled, it is obvious that the variation in liquid temperature near the interface can be very critical. For highly subcooled bubbles (characteristic of many cavitation cases', the effect of the liquid temperature increase during collapse is not so important, as perhaps best shown analytically by Florschuetz and Chao. Various experiments have also demonstrated this with respect to actual cavitation damage pro(17' duction, including work by one of the present authors. Figs. 7 and 8 show some of the effects of allowing the liquid temperature to vary in the calculation by comparing the resulting collapse conditions with those under a constant liquid temperature assumption, i.e., liquid with infinite thermal diffusivity. The initial bubble conditions are identical to those in the previous section with oCC qual.to 0.1. Fig. 7 shows that the internal pressure of the gas-vapor mixture is only slightly less throughout the collapse if constant liquid temperature is assumed than if the actual variable temperature is used. The difference can be attributed to the higher rate of evaporation from the liquid surface when the temperature increases (hence a lower net rate of condensation (see Eq. (7)), and to the higher temperature of the contents in the latter case. In Fig. 8, the liquid temperature at the interface can be seen nct to change by more than 10% until R/R is less than 0. 3. The increase in TLi for smaller radii causes a corresponding divergence in condensation rates, ias ex)plained above. Because of the only slight differences in all the curves,f Il'ilgs. 7 antd 8, andt b)t'callse ll) re Ilan 8'5, 01 I11w initial vapor will have ce(ttl(nstd whllen RIt/1 (). 3 (R1ef. Fig. 3 I'tt' OC 0. 1, we conclucde that the variation in liquid temperature produced by release of the latent condensation

- 11 -heat and by heat conduction out of the bubble will have a negligible effect on the collapse of a highly subcooled cavitation bubble of the type here studied. (4) Bubble Radius For smaller maximum (initial) bubble radii, one expects the collapse process to become more and more isothermal, at least during the early stages, because the resistance to heat flow from the interior of the bubble out to the liquid is reduced, i.e., the ratio of surface area to volume for small bubbles is more favorable. The initial conditions other than maximum radius are the same for this calculation as in the previous section. Calculation of the polytropic exponent ~ (assuming p/A' = Constant over short segments of the collapse) at various points during the collapse, shows that the collapse becomes more and more adiabatic as it proceeds, approaching as an upper limit the adiabatic exponent (1. 33 for the initial mixture of air and water). 6 takes on values between 1. 00 and 1. 20 during the early stages of collapse for the smallest bubble calculated (R = 0.5 x 10 in'), between -3 0 1. 10 and 1. 20 for a medium bubble (R = 5 x 10 in), and between 1. 24 and 1. 30 for the largest bubble calculated (R = 50 x 10 inm. By the time R/R 0 o reaches 0.05 in the first case, 6 is about 1. 30. The net effect of the lower contents temperature in smaller bubbles on the damage capability of smaller bubbles appears to be negligible. According to Fig. 10, the gas-vapor total pressure does not significantly differ between the three initial size cases throughout their collapse histories. (5) Thermal Conductivity of Gas-Vapor Mixture An additional parameter of interest is the thermal conductivity of the bubble contents. The same initial bubble conditions were used for this -3 -3 calculation as in the previous three sections (R = 50 x 10 in., Pi = 10 atm, 0 o Pv0= Pvsat (77~ F, T 77 F, =- 0. 1), but the actual thermal diffusivities Pvo - Pvsat 00 of the contents was replaced in two different cases by the extrema values, zero and infinity. The first corresponds to the adiabatic case giving uniform but not constant temperature in the bubble,and the second to a constant temperature of bubble contents.

-12 -Figs. 11 and 12 show the effect of this range of thermal diffusivities of the bubble contents on the pressure history of the bubble, on the average temperature of the contents, and on the rate of vapor condensation. The pressure record shows no significant difference between either of the extrenre values and the true value of the thermal diffusivity; in fact, the adiabatic (zero thiermnal diffusivity) collapse pressure history can be represented by the same curve as that for the true value of thermal diffusivity. The essentially adiabatic nature of the typical cavitation bubble collapse is further confirmed in Fig. 12. The average temperature of the conteiity is greater than 90% of the adiabatic temperature at the corresponding radii ratio throughout the collapse. The rates of condensation in the three cases are quite similar, about 80% of the vapor mass having condensed by the time R/R reaches 0. 3 in the infinite diffusivity case; more than 85% in the other two cases. (6) Liquid Overpressure Experiments in this laboratory and elsewhere e.g.) have shown that the greater the difference between the initial bubble internal pressure andt the system pressure far from the bubble, the greater will be the cavitation damage rate. This pressure effect can be explained with the aid of Figs. 13 and 14, which show the internal pressure histories of the "standard" bubble described in the previous sections, collapsing under 1, 2 and 4 atm. ovenrpressure. According to Fig. 13, the higher the overpressure, the lower wil1, be the ratio of internal gas pressure to initial pressure differential at a given radius ratio R/R. The lower this ratio is in the early stages of collapse, the smaller will be the eventual minimum radius and the higher th-e eventual pressure ratio. However, according to Fig. 14, the actual. internal pressure increases with system overpressure for a given R/R. The result is that the greatest initial overpressure produces the smallest rnminimum radius and the largest peak internal pressure, and thus can produce the mrost intensely damaging shock waves (or microjet velocities\ in (18, etc agreemnent with the experiments

-13 - In addition to the above parameters, the results of the analysis include pressure distributions in the liquid at various points during the collapse. In Fig. 15 one such set of profiles are shown, for a bubble with the same initial conditions employed in sections (2) - (6') above. The pro(4) files are quite similar to those reported by Ivany and Hammitt in their analysis which includes compressibility effects but does not include phase change or thermodynamic effects. Substantial pressures (>10 atm.) arise in the liquid at small radii during collapse. Although high enough pressures during collapse to explain observed damage at distances greater than the bubble initial radius are not obtained, a similar calculation by Hickling and Plesset( shows sufficiently high pressures in a shock wave produced upon the rebounding of such bubbles. CONCLUSIONS The effects of the various thermodynamic parameters and of phase change are summarized in Table I. The results given are, in general, for a typical highly subcooled cavitation bubble (e.g., R = 50 x 10 in., pio= -3 (770F), T T 770F~, pm~lat.o 10 3 atm., pv= 0.406 psi= P (77F), T = T 77F)p =1atm. oV~ ~ sat o oo O The effects on damage capability(3rd column of Table I) are based on the hypothesis that damage is caused by a shock wave produced as the bubble begins to rebound from its minimum size. The peak pressure obtained internally and the minimum radius are thus critical factcrs in establishing the strength of such shock waves. Presumably, at least the directions of variations would be the same if microjet impact is the damage mechanism. Actually, in most cases, it is probably due to a combination of both effects. As can be seen from the table, the presence of water vapor coupled with the value of the evaporation coefficient and the pressure difference initially between the liquid far from the bubble and its contents are the most significant variables in the present study. As was pointed out in the previous discussion, if the evaporation coefficientoC is very large ( —1. 0 the effect of the vapor on the collapse will be relatively small since it will

-14 -condense very rapidly, essentially being "blotted-up" by the in-moving (8),-,' wall. For oC of the order of 0. 01, the most likely magnitude for water the vapor will have a significantly retarding effect on the collapse. The thermal diffusivity of the bubble contents has a slightly less important, but still significant effect on the collapse. It appears, however, that cavitation bubble collapses may be assumed adiabatic throughout without disturbing the results very greatly. The other factors investigated, such as initial bubble radius and the variation of liquid temperature, have been shown to have little effect on the collapse velocity or pressure at a given radius ratio for highly subcooled bubbles. The effect of any non-equilibrium temperature discontinuity at the vapor-liquid interface is also negligible. The calculations indicate further that the vapor will be far from saturation conditions in the later stages of collapse, the bulk of it having condensed. The present authors and others have shown in recent papers that cavitation bubbles will collapse asymmetrically under a variety of real flow conditions. The collapse under such conditions produces liquid jets of sufficiently high velocity to cause impact damage and does so with relatively small volume changes (R/R > 0. 25). The effects discussed in the present paper will probably be of only second order when compared to the asymmetric effects which arise in real flow conditions (e. g., steep pressure gradients, proximity to a solid wall. ) No good experimental values for this coefficient for various fluids and Conditions yet exist.

LIQUID THERMAL VAPOR UNIFORM P OUNDARY\ + LAYER,, AS UNIFORM F,PI * TL(Y) T\ VPI PL(r) R+_ I VAPOR A+ A / T= 6T(r) T1 Iir // TL(R+8)= T o ~gi NONEQUILIBRIUM REGION AT INTERFACE Fig. l(a). Location of Variables for Thermodynamic Effects Study T LIQUID VAPOR THERMAL PLUS BOUNDARY L AYER GAS lo Tg i 1 TLi R r RADIAL COORDINATE 3130 Fig. l(bl. Sample Temperature Profile During Collapse

16 9 x xxx 8 X= t/r =I.01 8 6 L HICKLING (6) CALCULATION (NO WATER VAPOR) /'~ I I x x w | ' x K PRESENT CALCULATION z |r |(WITH WATER VAPOR)?- 5 cr r 50 sec aw F 4 I i t/r =.98 3 x x X,x 2 N t/r =.90 t/r =.76, f__t/r -0.47 0 0.2 0.4 0.6 0.8 1.0 RADIAL DISTANCE r/R0 FiR. 2. Internal Temperature Distributions at Different Rtr.ii for Bubbles Containing Nitrogen CGas with and withut Water Va;)(or; ~ ). 1, 3131 0.075 am.. p (). 0277 attr. Ap 2 925 atr,. R. 1 cm. To-"- 528~ R.

I00 I --- - 100 Z 80 ta = 1.0 > 40 -z o 20 O. I.2.3.4.5.6.7.8 31332 DIMENSIONLESS TIME (t.9 1.0 Fig. 3. Effect of Evaporation CoefficientoCon Quaittity of Vapor Condensed During Bubble Collapse; "Standard" Bubble: R = 50x10 in., T = 537 R, P =10 atm.. Pv= O. 0277 atm., p =latm.

1.0 0.8 m o a.6 --— 1.0 and RAYLEIGH INERTIAL COLLAPSE a: |WITHOUT INTERNAL GAS _J Cn.4.2 0 i I..... I I I I. I I O.1.2.3.4.5.6.7.8.9 DIMENSIONLESS TIME (t JAp/pL /Ro ) Fig. 4. Effect of Evaporation Coefficient GC on Bubble Radius Change with Time for "Standard" Bubble (see Fig. 3)

103 19 Io I i\ I' I 0 V) I o \\\ I.01.0 1.0L I ' --- — L 0 0.2 0.4 0.6 0.8 1.0 BUBBLE RADIUS, R/R0 313f Fig. 5. Effect of Evaporation Coefficient ~C on Internal (Gas + Vapor) Pressure Change with Radius for "Standard" Bubble (See Fig. 3)

20 0 WI. 4 6 Li 0 C 0 0 0.2 4 0.6 08 1.0 BUBBLE RADIUS, R / R Fig. 6. Effect of Evaporation Coefficient C Average Internal Temperature Change fe th Radius for fStandard" Bubble (see Fig. 3).

10 21 102 0 --- IVARIABLE LIQUID TEMP. -a \', CONSTANT LIQUID TEMP. I) cIo a.10 0 0.2 0.4 0.6 0.8 1.0 3136 BUBBLE RADIUS, R/Ro Fig. 7 Effect of Assumption of Constant Temperature in Liquid on Internal (Gas + Vapor) Pressure Change with Radius for "Standard" Bubble (see Fig. 3),oC = 0.1

22 3 ---— VARIABLE LIQUID TEMPERATURE CONSTANT LIQUID TEMPERATURE 5 -J z v0 49 2 w 3 8 2. El; o a I I I 1 0u 0 0.2 0.4 0.6 0.8 1.0 BUBBLE RADIUS, R/R0 3137 QC 0.1.

4 -23 RO = 50 x 103 IN. Ro- 5.0 10- 3 IN. -Ro 0.5x 10- 3 IN. T=.91468 i 3 C. H3I-F St/T1.00 2 | t/T-=.95;i.t/T=.98 ~ Jt/T=.97 t/T-.93 t/T-.91 t/T:.76,/T-6 = t/T =0.47 0 0.2 0.4 0.6 0.8 1.0 RADIAL DISTANCE, r/Ro 3138 Fig. 9. Internal Temperature Distributions at Different Stages of Collapse for "Standard" Bubbles (see Fig. 3) with Different Initial Radii, oC = 0.1.

105 24 102 a. 0. cr C) 10 R=0.5ond 5xl03 IN. 1.0... 0 0.2 0.3 0.6 0.8 1.0 BUBBLE RADIUS, R/Ro Fig. 10. Effect of Initial Bubble Radius on the Internal (Gas + Vapor) Pressure History During Collapse of Otherwise "Standard" Bubbles (see Fig. 3\, = 0.1

[O = -O'(, ~ * IA a1as) alqqnii pvepuvjS, |o asd-elloo BuEa-nd Alo sEH ains said.iodeeA + seo) Ieu.a;uI uIaq4uo sua4uoD jo A1A-qznpuoD leuLiiayjL JO ajj 11 *2-E 0 O tU O O?/8 SniaV8 3188nsn 0'1 8'0 9'0 b 'O '0 0 01 0n C) (OliV8VlOV) O= I x), i 1I )P1

=;o '( (2i, a s) alqqnq,,pl-pueS,, xoJ snipvH JouoE43un,{ se asev uoie suapuo3 IodleAuo ain1-e.iaduai T.leu.aIuI a;e.IATla axi uo sua4uauoD Jo A4TAEflnPUOD J[euIajI;L Jo )ajja 'ZI I N Ol/l 'snaICVI 38aen0 0' 1 8'0 9*0 b'0 Z 0 0 0 o;O z 0 09:S3-191 — 08 08 G) rn

10 27 Ap: Pco - Pg Ap= 1 ATM..p = 2 ATM. ---— Ap= 4ATM a. <1 1 aI - C. I...I.I. I 0 0.2 0.4 0.6 0.8 1.0 BUBBLE RADIUS, R/R Fig. 13. Effect ofOverpressure (Ap) onthe Dmension0ess Internal (Gas + Vapor) Pressure Historyof CollapsingBubbles, R =50x10 in., T = 537~R, Pi -103atm., pO=0.0277 atm. 03, 100,-0.

I0 28 Lp: poo-pg~ 1'.. Ap I ATM. w M --- —Ap- 2 ATM. r — U~~~Ap 4 ATM. U% wa I cc w 1. 0 U) a) -2 L I I I l I 0...O. 2 0.4 0.6..0.8..1.0 3143 BUBBLE RADIUS, R/Ro Fig. 14. Effect of Overpressure (Ap) on the Actual Internal (Gas + Vapor) Pressure History of Collapsing Bubbles; Conditions same as for Fig. 3

io5 i02 BUBBLE WALL C, w LrJ w CL Q, 10 n 10 0~ C,) o I-o 103 1.0 2L~~~~ Io I 10 10 I 0 6 1' 1 010 RADIAL DISTANCE, rfR0 Fig. 15. Pressure Profiles in Liquid at Different Stages of Collapse for 'Standard' Bubble (see Fig. 3), oC =0. 1 3 14 4

30 TABLE I EFFECTS OF PARAMETERS ON CAVITATION BUBBLE COLLAPSE Parameter Effect Due to Increased Significant Effect on Value of Parameter Damage Capability Presence of Lower Temperatures of Depends on Actual Water Vapor Contents Value of Evaporation Coefficient Evaporation Faster Condensation, Yes, Increase Coefficient Higher Peak Pressures Assumed Variable Slower Condensation No Temperature of Liquid Initial Bubble Radius Collapse More Nearly Pressure Similar at Adiabatic Similar Radius Ratio, but Larger Region is Affected for Larger Bubble Thermal Conductivity Some Reduction in Con- Moderate Reduction of Contents densation Rates and Contents Temperature Liquid Overpressure Higher Peak Pressure Yes, Increase

31 REFERENCES 1. Mitchell, T. M., Hammitt, F. G., "Asymmetric Cavitation Bubble Collapse, " to be published, ASME (1972). 2. Mitchell, R. M., "Numerical Studies of Asymmetric and Thermodynamic Effects on Cavitation Bubble Collapse, " Ph. D. Thesis Dept. of Mech Engr., Univ. of Mich., Ann Arbor, Mich. December 1970; also available as ORA UMICH Rept. No. 03371-5-T. 3. Hickling, R., and Plesset, M., "Collapse and Rebound of a Spherical Bubble in Water," Phys. of Fluids, Vol. 7, No. 1 (1964). 4. Ivany, R. D., and Hammitt, F. G., "Cavitation Bubble Collapse in Viscous, Compressible Liquids -- Numerical Analysis," Trans. ASME, J. Basic Eng., 87 (1965), 977. 5. Zwick, S. A., and Plesset, M.S., "On the Dynamics of Small Vapor Bubbles in Liquids," J. Math. Phys., 33 (1955), 308-330. 6. Hickling, Robert, "I. Acoustic Radiation and Reflection from Spheres, II. Some Effects of Thermal Conduction and Compressibility in the Collapse of a Spherical Bubble in a Liquid, " Ph. D. Thesis, Cal. Inst. Tech. (1962). 7. Florschuetz, L. W., and Chao, B. T., "On the Mechanics of Vapor Bubble Collapse, " Trans. ASME, J. Heat Transfer, 87 (1965), 209- 220. 8. Theofanous, T., Biasi, L., Isbin, H. S., and Fauske, H., "A Theoretical Study on Bubble Growth in Constant and TimeDependent Pressure Fields, " Chem. Eng. Sci., 24 (1969). 885 -89'7. 9. Canavelis, R., "Contribution a l!1iEude de lErosion de Cavitation dans les Turbomachines Hydrauliques$' Theses, La Faculte des Sciences de l'Universitf de Paris, France, Juin, 1966. 10. Scriven, L. E., "On the Dynamics of Phase Growth," Chem. Eng. Sci., 10 (1959), 1-13. 11. Poritsky, H., "The Collapse of Growth of a Spherical Bubble ori Cavity in a Viscous Liquid, " Proc. First National Congress of Appl. Mech., ASME, (1952) 813-821. 12. Kennard, E. H., Kinetic Theory of Gases, McGraw-Hill, New York, (1938), 68-70

32 13. Bornhorst, W. J., and Hatsopoulos, G.N., "F.-ibble - Growth Calculation without Neglect of Interfacial i sconjltinuities, " Trans. ASME, J. Appl. Mech., 34, Series E (1967), 847 853. 14. Bornhorst, W. J., and Hatsopoulos, G. N. N" -aallysis of a Liquid Vapor Phase Change by the Methods of Ir;re V-1`' — le Thermodynamics," Trans. ASME, J. Appl. Mech., 34, Series 1, i(1967), 840-846. 15. Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, Second Edition, Oxford University Press (]:3'5, 233e 16. Theofanous, T.G., Biase, L., Isbin, H1. c;, and Fauske, H.K., "Nonequilibrium Bubble Collapse -.- A Th't-eei:ical Study, " AIChE Preprint 1, Presented at Eleventh Natiornafl p'1er:: Tlransfer Conference, Minneapolis, August, 1969. 17. CpGarcia, R., and Hammitt, F. G, 'C1a.vi.ta I rn:-atge and Correlation with Material and Fluid Properties," 1 Trans ASf ME, J. Basic Eng., 89, Series D (1967), 753-763. 18. Harnmitt, F. G., and Rogers, D.O., "Effects of Pressure and Temperature Variation in Vibratory Cavitation D.anaage Test, " J. Mech. Eng. Sci, 12, 6, 1970, 432-439. 19. Kling, C. L., and Hammitt, F. G., "A Photographic Study of SparkInduced Cavitation Bubble Collapse," (subin-ittied for ASME presentation and publication), available as ORA UN4MC-Ir 033() 71-4-T (Mod), Nov. 1971. n20, Florschuetz, L. W., and Al-Jouboulri, 'IGe: ntcr ' ciLzed Quantitative Criteria-for Predicting Rate-.Controlling MV4ec]hani sr for Vapor Bubble Growth in Superheated Liquids'' I:t. H eat kMass Transfer, 14, 587-600.

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