Bubble Growth in Superheated Solutions with a Non-Volatile Solute

BUBBLE GROWTH IN SUPERHEATED SOLUTIONS WITH A NON-VOLATILE SOLUTE

OSAMU MIYATAKE Department of Chemical Engineering, Kyushu University, Fukuoka 812, Japan

ITSUO TANAKA Division of Biological Production System, Gifu University, Gifu 501-1 I, Japan

and

NOAM LIOR! Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315. U.SA

(First received 30 April 1993; accepted in revised form 9 September 1993)

Abstract-Growth of single bubbles in a uniformly superheated binary solution containing a non-volatile solute was studied both experimentally, using an aqueous solution of NaCI at mass fractions of 0.05 and 0.20, an initial temperature range of 40-80°C and an applied initial superheat range of 1.7-16.5°C, and theoretically, with very good agreement between the experimental and theoretical results. The experimental technique developed, using an electrolytically nucleated bubble, was found to be excellent for single-bubble growth studies and photography. Some of the principal conclusions are that increasing the concentration of the non-volatile solute at a given initial solution temperature reduces the bubble growth rate when the far-field pressure is dropped to a fixed value. The effect of concentration on the bubble growth rate becomes smaller when the far-field pressure is reduced to bring the solution to either a fixed superheat, or to a fixed initial pressure difference between the bubble interior and exterior, leading to a conclusion that it is preferable to study and correlate bubble growth rates by using the superheat and this pressure difference as parameters.

I. INTRODUCTION solutions have been learned, such as the basic ap Bubble growth in a superheated liquid is a key con proach to modeling and the fact that the addition of struct of the flash evaporation process. A considerable non-volatile solutes reduces the bubble growth rate, amount of theoretical and experimental work on such the amount of experimental data is limited to a few bubble growth has been conducted [cL review by solutions of volatile binary components, systematic Plesset and Prosperetti (1977), van Stralen and Zijl parametric experimentation to allow the evaluation of (1978) and Prosperetti (1982)J, mostly considering the influence of the major process parameters has not pure liquids, and would not be summarized here. Very been conducted, approximate analytical solutions little is known, however, about bubble growth in have been derived for simplified problems, all but one superheated solutions with a non-volatile solute, focusing only on the heat and mass transfer controlled a topic of both fundamental and practical importance, regime and limited by the need for empirical para with applications including a wide variety of sep meters, most thermophysical properties in the ana aration processes such as water desalination, and lyses were considered independent of temperature and energy conversion processes such as ocean-thermal concentration, and the specific case of a solution with energy conversion, geothermal power generation, and a non-volatile solute, such as water and salt, has not nuclear reactor safety. been addressed. Amongst the small number of papers on bubble The primary objective of this paper is to improve growth in superheated binary solutions, Scriven the fundamental understanding of the effects of a (1959) has described the general approach to non-volatile solute on single-bubble growth from the modeling uniformly heated spherically symmetric time of nuclei;ltion and on, by a thorough experi bubble growth of both pure liquids and binary mix mental study and by analysis, and to formulate a suf tures, and has derived approximate asymptotic solu ficiently accurate method for prediction, taking into tions in the heat- and mass-transfer controlled regime, consideration, for example, the dependence of the followed by studies by Skinner and Bankoff (1964), thermophysical properties on temperature and Yatabe and Westwater (1966), van Stralen (1968) and concentration. Moalem-Maron and Zijl (1978). Summarizing the state of the art, while many basic 2. THE EXPERIMENTAL APPARATUS AND PROCEDURE aspects of bubble growth in superheated binary Bubble growth in a superheated pool of aqueous NaCI solution was observed photographically through a systematic experimental parametric study t Author to whom correspondence should be addressed. aimed at determining the influence of the major para-

l301 tES 49: 9-B 1302 OSAMU MIYATAKE el al. meters on this process. Experiments on this topic temperature. Temperature uniformities in the bath which were reported in the literature [cf. Hooper and and in the chamber were monitored by thermocouple Abdelmessih (1966), Hewitt and Parker (1968), Kosky rakes (items 8 and 4, respectively), each composed of (1968), Florschuetz et al. (1969), and Gopalakrishna three 0.5 mm diameter sheathed Chromel-Alumel et al. (1987)] typically observe bubble growth in thermocouples (in item 4, spaced ~ 20 mm apart). a vessel, subject to sudden decompression. In that case To prevent bubble nucleation on surface irregular the pressure, and thus the superheat, changes with ities of the thermocouple rake in the bubble observa time, and bubbles grow at unpredictably arbitrary tion chamber, it was pulled above the surface of the times and positions in the vessel. It is much easier to solution after the temperature was recorded and observe a bubhle which grows at one location under found to be uniform, and the pressure in the bubble conditions of constant and uniform superheat. This observation chamber was then reduced slowly by was accomplished in this experiment by slowly de a gradual opening of the needle valve (item 12) con compressing a pool of the solution in a container to necting the chamber to a large vacuum tank (item II) the new constant pressure corresponding to the which had been maintained at a prescribed lower desired superheat, and then, under constant sur pressure, until the pressure in the chamber became rounding pressure conditions, generating a bubble equal to that in the vacuum tank and the solution nucleus at the tip of a cathode wire by electrolyzing attained the desired degree of superheat. Pressures in the superheated solution. Unlike the situation in past the vacuum tank and the chamber were measured experiments, the absence of other bubbles in the vessel using mercury manometers (item 13). The levels of also allows the elimination of their influence on the mercury in the manometers and those of the liquid (oil observed bubble, another significant advantage of this and solution) in the bubble observation chamber were technique. measured using a telescopic cathetometer. As described in Fig. 1, a degassed aqueous NaCI A d.c. potential of 100-130 V for pure water and solution (item 5 in Fig. 1) of known concentration was a few volts for the aqueous NaCI solutions was then contained in a cylindrical glass vessel (item I) 50 mm switched on (by item 14) between the electrodes, elec in diameter and 200 mm high, which served as the trolytically forming a hydrogen bubble at the tip of bubble observation chamber. Immersed in the solu the cathode. As the bubble diameter approached its tion were two electrodes (item 2) made of 0.24 mm critical value, it began acting as a nucleus for a water diameter copper wires which, except for their flat vapor bubble. Once the vapor bubble started grow ends, were insulated from the solution electrically. ing, its growth rate was much more rapid than that of A layer of paraffin oil (item 6) was floated on the the hydrogen bubble [cr. Miyatake and Tanaka solution surface to prevent droplets of condensate ( 1982)]. falling from the vessel top from disturbing the solu A high-speed motion-picture camera was used to tion, and to prevent solution top-surface evaporation photograph the bubble growth, and a film analyzer and heat losses and thus improve temperature uni was used to measure the bubble diameter and deter formity in the solution. Analysis has shown that the mine the time intervals between the measured bubble paraffin oil dissolved negligibly in the aqueous NaCI diameters. A rod of 3.00 mm diameter (item 3) was in solution (its solubility is < 10- 10 by mole fraction) the solution near the examined bubble to provide the during the experiments. The bubble observation scale for the diameter measurements. chamber was immersed in a temperature-controlled To ensure that the bubble-nucleation electrode dia water-bath (item 7) having two transparent sides, meter did not have an effect on the results, bubble which was used to bring the solution to the desired growth experiments were at first repeated with e1ec-

/,.__---o0.c. CD Bubble observgti og (1) Electrodes c om er Q) Scole provi di ng rod @ Thermocouples @ Aqueous NoCI soluti on @Paraffinail (j) Thermostatic water bath E @ Thermocouples E o ® Thermometer o @ Vacuum pump N ([j) Vacuum tonk © Needle valve o Manometers o Starti ng button @ D.C. power supply

Fig. I. The experimental apparatus. Bubble growth in superheated solutions with a non-volatile solute 1303 trodes of 0.24, 0.35, and 0.67 mm diameter. The results far-field pressure (Poo), superheat (~Ts), and of the were practically identical, and thus the 0.24 mm dia initial pressure difference between the bubble interior meter electrode was chosen for all of the experiments and exterior (~Po), respectively. The symbols in the reported in this paper. figures indicate the experimental observations and the The experimental errors were ± O.03°C for tem solid lines indicate our numerical solutions (described perature and temperature uniformity, ± 2.7 Pa in Sections 4-6) for the same experimental conditions. ( ± 0.02 mm Hg) for pressure, and ± 0.05 ms for time. The observed behavior agrees with the established While the dimensional measurement error was knowledge of bubble growth in pure liquids [cf. ± 0.02 mm for bubble size, it is noted that the bubble Plesset and Prosperetti (1977)]: after an initial brief was typically not perfectly spherical during its growth. period during which the bubble grew to a size at The asphericity of the bubble, defined by the ratio of which the effect of bubble wall surface tension became the measured largest-to-smallest diameter for a given negligible, bubble growth is dominated by inertial film frame, did not, however, exceed 1.07 up to forces driven primarily by the pressure difference a growth time of 10 ms. The recorded diameter was p,. - Poo between the bubble interior and exterior, that of a perfect sphere having the same volume as the which is also almost equal to ~Po because the measured spheroid. The surface areas of the measured decrease of bubble wall temperature is still comparat spheroid and of the sphere of equal volume, which are ively small, and in that regime the growth history is the dominant geometric factors in bubble growth, linear, R ex t. After some period of further growth, were found to agree within 0.12%. evaporation at the bubble wall causes the temperature Special precautions were taken to avoid the effects there (T;) to drop below the surrounding liquid tem of contaminants on the experiment. The flash cham perature (Too ) to a level high enough for heat transfer ber was cleaned before each experiment with a medi to the evaporating bubble-liquid interface to. start cal-instruments detergent and then with distilled dominating growth due to the consequent addition of water. The water in the solution was prepared by ion vapor to the bubble. This changes the bubble growth 1 2 exchange followed by distiiJation. The NaCI used was relationship to R ex t / , i.e. to a non-linear one. extra.pure chemical reagent grade. The solution was Figures 3-5 show that the transition from the linear replaced with a fresh one after each bubble growth (inertia-controlled) regime to the non-linear (heat observation experiment, typically lasting not more transfer-controlled) one, whose rate is proportional to than 30 min. the rate of the reduction in T;, occurs sooner for larger The experiments were conducted for initial solution Too, and for smaller ~Ts (also see Fig. 10). It is, for temperatures (i.e. the temperature far from the bubble, example, evident from these figures that when Too ) of 40°C and 80°C, superheats (~Ts) of 1.7-16SC Too = 80°C the bubble is brought much more rapidly (depending on T oo ), and for NaCI (solute) mass frac into the heat-transfer-controlled regime characterized 1 2 tions (w oo ) of 0.05 and 0.20. by the R ex t / shape, while at T oo = 40°C the bubble To determine the value of ~Ts, the static pressure at remains in the linear, inertia-controlled regime the cathode-tip level was calculated as the sum of the throughout the investigated period. vapor pressure above the liquid, and the static pres The observed reduction of the bubble growth rates sure rises due to the paraffin oil layer and the solution with increasing concentration for fixed values of layer between the oil-solution interface and the Poo (Fig. 3) is due to the consequent boiling-point cathode tip. elevation, i.e. the reduction of the vapor pressure inside the bubble. The effect of the concentration is 3. EXPERIMENTAL RESULTS seen to be much larger for higher Poo because the Figures 2(a) and (b) show the growth of a bubble for difference between the magnitudes of the bubble initial solution temperatures of Too = 40 and 80°C, growth driving forces (Pv - Pool dominating the respectively. The time elapsed between the conse inertia-controlled regime, and the difference between cutive eight-frame columns in Fig. 2(a) is 2.15 ms and the magnitudes of the bubble growth driving forces in Fig. 2(b) 1.82 ms. The bubble growth is seen to (Too - T;) representing the actual driving force resemble very closely that of a stationary sphere: in (aT/or); dominating the heat-transfer regime, for two deed, past studies (Pinto and Davis, 1971; Gopala given concentrations, are then larger at a given T oo. It krishna and Lior, 1992) show that the rise of such is interesting to note that the effect of concentration is bubbles over a period of 10 ms is smaller than I mm seen to be considerable when Poo is the parameter held and , as mentioned in Section 2, the asphericity was constant (Fig. 3), but that it becomes very small when < 1.07. ~ Ts is the parameter held constant, especially in the [n an attempt to understand the importance of the heat-transfer-controlled bubble growth regime (i .e. at parameters associated with the observed bubble Too = 80°C, Fig. 4), or when ~Po is held constant, growth phenomena, and to discover the parameters especially in the inertia-controlled regime (i .e. at which may lead to better correlation of the results or Too = 40°C, Fig. 5). This observation is shown to be normalization of the variables, the effects of the consistent with the theory, by examining the bubble far-field solute mass fraction (woo) and solution growth driving force (Pv - p oo ) dominating the inertia temperature (Too ) on the bubble growth history are controlled regime, and the driving force (Too - T;) presented in Figs 3-5 for different fixed values of the dominating the heat-transfer-controIled regime, as 1304 OSAMU MIYATAKE et al.

~ (a) 3720 f.p.s. 20mm (b) 4400 f. p.s. 20mm

Fig. 2. Bubble growth as observed by high-speed photography (the time·sequence is from top to bottom in each column, columns a rranged from left to right).

follows. When pet) is fi xed (at constant T ..J both driv- I1T, in the heat-transfer-controlled regime as seen in . ing forces increase when W oo is lower, the first one due Fig. 4 at Too = 80°C. If I1Po is specified, Woo has, by to an increase in p, and the second d ue to a decrease in definition, no effect on bubble growth in the inertia the saturation temperature (which is also the temper controlled regime, as seen in Fig. 5 at 40°C. ature T;). If I1T, = T - T. is fixed, Poo is defined at One of the conclusions from this weak dependence the equilibrium temperature Te and thus both Poo and of bubble growth history and rates on concentration Pv are inversely p roportional to W oo- The dominant for given I1Ts is that experimental results relating the bubble growth d ri ving fo rce ( p" - Poo l in the inertia degree of approach to equilibrium in Hash evap controlled regime c nsequently becomes almost con oration of fresh water [cf. Miyatake et al. (1977), stant, affected only by the relatively weak dependence Lior and Greif (1980), Miyatake et ai, (1985), of Pv(T) on concentra ti on for the cond itions examined Gopalakrishna et al. (1987) and Miyatake et ai, (1992,

in this study. The d ominant d riving force (T"" - T i ) in 1993)] will also be valid for moderate-concentration the heat-transfer-controlled regim also remains al brines and seawater (w oo ~ 0.07 in commercial multi most constant because Ti can va ry only within the stage Hash evaporators for water desalination). It also fixed range from T", to T" and consequently the effect may extend the validity of empirical correlations ob of Woo becomes practically negligible fo r a specified tained for the approach to equilibrium in Hashing of Bubble growth in superheated solutions with a non- vola tile solute 1305

Po kFb W H 8 0

[).• E6 .. 8 E E6 Q:4 E 2 Q:4

2 0

0 8 Po kA.1 WmH o 5.26IC .20 . . ,P..k w- [). 3 . 79 IC ·2C 8 .. ·691( · 5 o ,. E6 • I . ~ E ·26 E6 .. 3· 2 Q:4 E Q:4 2 - Nume-rical soln. 2 - Numerical soln. 2 468 10 t Imsl 2 468 10 Fig. 3. The experimentally observed bubble growth history t Imsl (the solid curves are our numerical solution) for different fixed values of the far-field pressure (poo ), at two values of the Fig. 5. The experimentally observed bubble growlh history solute mass fraction (woo ) and solution temperature (T", ). (the solid curves are our numerical solution) for different fixed values of the initial pressure difference between the bubble interior and exterior (ilPo), at two values of the solute mass fract ion (w« ) and solution tem perature (T.,, ).

• T, - I 8 0 •61 " IS· be seen that bubble growth is inertia-controlled E6 (R c:c I) at the lower solution temperatures and higher E superheats, and that it is heat-transfer-controlled at Q:4 the higher solution tem peratures and lower super heats. The characteristic occurrence of a short delay in 2 bubble growth due to the strong effect of bubble wall surface tension when the solution temperature and 0 superheat are low is also evident (it can also be seen in Fig. 3). 8

E6 4. ANALYTICAL MOOEUNG E Guided by the experimental results and con Q:4 clusions, a theoretical model was develo ped to predict transient bubble growth in superheated solutions with 2 a non-volatile solute. w hich could be validated by the - Numeric.al sol n. experimental results obtained. As commonly done [cf. Scriven ( 1959), Mikic et al. (1 970) and Theofanous and 2 468 10 Patel (1976)] and also confirmed by our and other [cL t Imsl Dergarabedian (1 953) and Hooper a nd Abdelmessih Fig. 4. The experimentally observed bubble growth history (1966)] experiments, it was assumed in the analysis (the solid curves are our numerical solution) for different that the bubble shape is spherical. Derived from the fixed values of the superheat (ilT,), at two values of the equations of continuity and m o tion, the equation for solute mass fraction (w", ) and solution temperature (T", ). the transient bubble radius is [cf. Scriven (1 959)]

d2 R(I) 3 [d R(I)]2 Rt--+-( ) -- saline water at specific concentrations [cL Lior dl 2 2 dt (1986)] to a wider range of concentrations. Figure 6 shows the bubble radius history with more I {[ 2u 4/1-1 dR Ill] } = p,(t) - R (t) - R(l) ~ - Poo (I) values of the superheat as the parameter. It can again p, 1306 OSAMU MIYATAKE el al.

8 8

E6 E6 E E

0::: 4 0::: 4

2 2

0 0

8 8

E6 E6 E E

0::: 4 0::: 4

2 2

4 6 00 2 4 6 8 10 t Imsl t Imsl

W.,=O·05 W.,=O·20

Fig. 6. The experimentally observed bubble growth history (the solid curves are our numerical solution) for various values of the superheat (l1T,), at two values of the solute mass fraction (w",) and solution temperature (T", ). where p" is the vapor pressure in the bubble, p"" the Due to the similarity between the equations of energy pressure in the solution far from the bubble, p, the and mass transfer (for constant density of the solu solution density, 111 the dynamic viscosity of the solu tion), and because the concentration boundary layer tion, and a the surface tension on the bubble wall. is even thinner than the thermal boundary layer, one I Using the expression D,/R ~ Ja- , which relates can derive from eqs (2) and (3) an expression for the the thermal boundary layer thickness D,-bubble mass fraction of the solute at the bubble wall by radius R ratio with the Jakob number Ja (Prosperetti replacing the solution temperature T with the solute and Plesset, 1975), Miyatake and Tanaka (1982) have mass fraction w, and the solution thermal diffusivity shown that for Ja > 10 the radius growth-rate results 0:1 with the solute mass diffusivity D, resulting in obtained by using the thin boundary layer assump tion are indistinguishable from those obtained by Woo --D)1/2 I' [ow(r X) ] Wi(t) = ( F(t, x) --' - dx (4) Daile D onne and Ferranti (1975) for sodium at n 0 or i Ja = 2.98-566, and by Saito and Shima (1977) for where w'" is the mass fraction of the solute in the water at Ja = 1070, from the numerical solution of the solution far from the bubble and F(t, x) was defined full bubble growth problem. Since the numerical by eq. (3). study performed here was in the range of Ja > 36, the The temperature and concentration gradients at the thin boundary layer approximation is valid and thus bubble interface can be obtained from heat and mass the thin boundary layer analytical solution of the balances at that location, respectively: energy and continuity equations obtained by Plesset and Zwick (1952, 1954) for the temperature Ti of the 2 4nR (t)k, [oT(r,-- I)] = hfgw,, (t) (5) solution at the bubble wall (recalling that subscript or i i denotes the bubble wall, i.e. the vapor- liquid inter face), shown in eqs (2) and (3) below, was used: - 4nR2(I)PrD [ow(r, t)] = [ Wi(t) ] w,,(t) (6) or i I - Wi(t) 11. Ti(l) = T"" - )1/ 21' F(t , X) [oT(r -X)] dx (2) ( ~ -~-' n 0 or i where kr is the thermal conductivity of the solution, hf g the latent heat of vaporization, and w" the mass where T co is the temperature of the solution far from rate of evaporation expressed as the bubble, 0:1 the thermal diffusivity of the solution, and F(t , x) the function d [(4/ 3)npv(t)R 3 (t)] w,, (t) = dt . (7) (3) The density Pv of the generated vapor, which is super- Bubble growth in superheated solutions with a non-volatile solute 1307 heated due to the boiling point elevation (Tj - Ts ), iteration number index at a given time step) became can be expressed as < 10 - 5, and t was then increased to the new time step. A variable time-step grid was employed, fine at 273.15+ Ts(t) the beginning and gradually coarser as time increased, p,(t) = 273.15 + Tj(l) (P" sh., (8) by having 91 time steps for each ten-fold increase in I. where (p'" slr, is the saturation density of vapor, Totally a minimum of 370 time steps were used in the 2 evaluated at the saturation temperature Ts of the pure range t = 0-10- s. solvent. As stated above, T ro and Pro, or fI. 1"., or fl.Po were The saturation temperature Ts of the pure solvent, given for each computational run, and a value of 4 which is needed for evaluating p. in eq. (I) and p, in (j = 10- was chosen for use in eg. (10), since it was eq. (8), was determined from the relation found that a deviation of this magnitude in the initial bubble radius had an insignificant effect on the nu p,(I) = Pi-. = PT , . W i (9) merical results. where P* and P are the vapor pressures of pure All properties used in the analytical modeling were solvent and solution, respectively. temperature and concentration-dependent, calculated A bubble which has the critical radius Re is in by empirical formulas either given in the literature or unstable equilibrium. An infinitesimal increase, (j, in developed by the authors from available experimental the bubble radius will thus initiate growth, hence data. The properties and sources for the correlations defining the initial bubble radius, R(O), as are vapor pressure (Fabuss and Korosi, 1966; JSME, 1971), mass diffusivity [using the relationship of Olson R(O) = Re(i + (j) = 2a(1 + (j) (10) and Walton (1951) to correlate the experimental data fl.Po of NRC (1929), Stokes (1950), Fell and Hutchinson, (1971) and Chang and Myerson, (1985)J, viscosity where we recall that fl.Po is the initial pressure differ ence between the bubble interior and exterior defined (JSME, 1962; ORNL, 1966), thermal conductivity by (Yusufova el al., 1978; Ozbek and Phillips, 1980), specific heat (NRC, 1927, 1929; Yusufova el al., 1978), fl.Po = p,(O) - Pro = PT.,. '0. - Peo· (II) vapor density (Sa to, (974), solution density (NRC, 1928a; JSSWS, 1974), latent heat of evaporation The superheat of the solution, fl.Ts, is defined by (JSME, 1980), and surface tension (NRC, 1928b; fl.T$ = T eo - Te ( 12) Yusufova el al., 1978; JSME, 1980). The properties using the equilibrium temperature Te which satisfies a and hJ9' which are used at the bubble interface, were the relation evaluated at temperature T j and concentration Wj . All the other physical properties, which were not indexed ( 13) above by a temperature and concentration, were i.e. the temperature at which the solution of concen evaluated at temperature (Tj + T 00 )/ 2 and concentra tration Weo has the same vapor pressure as the pres tion (Wj + w oo )/ 2. sure Peo in the solution far from the bubble. Given T eo, the solution temperature far from the bubble (i .e. the initial uniform temperature of the 6. THE NUMERICAL SOLUTION RESULTS AND solution), and Pro , the pressure in the solution, or, COMPARISON WITH EXPERIMENTS alternatively, given the superheat fI. T$ or the pressure Figures 3-6 in Section 3, which present the experi difference fl.Po, evaluated from eqs (12) and (13), or mental results, also present the comparison of the eq. (11), respectively, the relationship between the experimental results with the numerically obtained bubble radius R and the time 1 can be determined by results shown by solid lines. The agreement between solving the simultaneous system of equations (1)-(9) the experimental and numerical results is seen to be subject to the initial conditions composed of eq. (10) very good in all the bubble growth regimes, beginning and with the surface-tension-induced initial delay at low radii and solution temperatures and superheats, = [d2R~I)] = 0 (14) dR(t)] through the linear (R ex I) inertia-controlled regime [ dl 1= 0 dl 1= 0 at lower solution temperatures, and into the heat [the second initial condition above needs to be speci transfer-controlled regime occurring at higher fied for the finite difference scheme because in eq. (10) solution temperatures and characterized by R ex 11 /2. (j # 0]. Figure 7 shows the computed bubble radius his tories under two fixed values of the far-field pressure 5. THE NUMERICAL SOLUTION PROCEDURE Poo which correspond to fl.Ts = 5 and 20°C for pure A finite difference forward-matching implicit solvent (water) of a given temperature T "" and for method was used to solve the above system of equa solute mass fractions Woo of 0, 0.05, 0.10 and 0.20; tions iteratively at each time step. The solution was Figs 8 and 9 show the computed bubble radius his iterated at a given time step until the change in the tories for fixed values of the superheat fl.T" and for moduli of Rand p,. ([R(m) - R(m-l)J / R(m - J) and fixed values of the initial pressure difference fl.Po be [p~.. ) - p~m- I)J / p~m-l l, respectively, where m is the tween the bubble interior and exterior, respectively, at 1308 OSAMU MIYATAKE et al. .. 8 8 Ts=80'C i' N

E6 E6 I~ E E ;) Q::4 Q::4

2 2

0 o

E6 E Q::4

8 8

E6 E Q::4

2 2

00 10 2 468 2 468 10 t !msJ t [msJ Fig. 7. The computed bubble growth histories at four solute Fig. 8. The computed bubble growth histories at four solute mass fractions W ro , for two fixed values of the far-field pres mass fractions (w oo ), for three fixed values of the superheat sure Pro which correspond to Il T, = SoC and 20°C for pure (IlT,), at three given solution temperatures (T ro). solvent (water) at three given temperatures T oo .

the same given concentrations Woo and temperatures (p, - Pool, the vapor pressure (p,) is directly related to T oo . The conclusions are the same as those discussed the concentration. Accordingly, when t'1Po is taken as in Section 3 for the experimental results in Figs 3- 5; a parameter, Woo has little effect on the bubble growth Figs 7-9 provide more detail. in this regime. Figure 10 shows the history of the bubble wall As stated in Section 4, this analysis took carefully temperature T; [in terms of the non-equilibrium frac into consideration the effects of temperature and con

tion (T; - T. )/(T 00 - T.)] and the bubble wall solute centration on the thermophysical properties of the mass fraction W; (in terms of W; - woo) for fixed values solution and the vapor. It is interesting to examine the of the superheat t'1Ts' It is seen that the rate of descent effect of this properties' variation on the results. of T; towards T. increases as T oo increases and as t'1Ts Figure II shows the effects of the solute concentration and Woo decrease, while the rate of ascent of W; from its at the bubble wall and of the superheat of the gener lowest value of Woo is proportional to the magnitudes ated vapor due to the boiling point elevation on of T oo , t'1T" and Woo' bubble growth solutions at different values of T oo and Comparison of these results with Fig. 8 shows that t'1Ts ' The solid lines represent the solutions under the slow rate of descen t of T;, ca used by low Too and physically correct assumptions for the bubble wall high t'1T" thus occurs in the inertia-controlled regime, concentration and vapor density, Wi = Wi and in which thus the bubble stays longer because this p, = p" meaning that these variables take the correct slow rate of descent of Ti also produces a relatively values at the interface, as determined by the set of low heat-transfer driving force. This also explains the equations given in Section 4, and by the property fact that the concentration (w oo ) has the largest effect correlations used. at these conditions (Fig. 8): in the inertia-controlled Assuming that the concentration at the bubble wall regime, where bubble growth is driven primarily by does not change during evaporation and remains con- Bubble growth in superheated solutions with a non-volatile solute [309 .08

~ I ,Jl :!1 ~ i I

:~~~ 0 .04 :!1

8 i

E6 E 0 Q::4 .02 :!1 2 20'C I i

:.:.-:..:::.rf _======-=- 0 8 4 6 8 10 t [ms] E6 E Fig. 10. The histories of the non-equilibrium fraction [(T; - Tel/(T", - Tel] and of the change of the bubble wall Q::4 solute mass fraction (w; - Woo l for two values of the solute mass fractions (W ool and the superheat (tl. T,J, at three given w==o 2 0·05 solution temperatures (T,,). 0·10 0·20 2 468 10 (w oo = 0.20), phenomena consistent with the above t Imsl reasoning. It is also seen that the effect of the assump tions W; = W;, pv = (P" ,!r; are small (4.9% at Fig. 9. The computed bubble growth histories at four solute mass fractions (w oo ), for three fixed values of the initial Too = 40°C) in the inertia-controlled bubble growth pressure difference between the bubble interior and exterior region since bubble growth in that region is primarily (tl.Po), at three given solution temperatures (T",). driven by the pressure difference between the bubble interior and exterior and is almost independent of pv'

7. CONCLUSIONS stant at woo, i.e. W; = w oo , pv = P. (the solid symbols in Growth of single bubbles in a uniformly super Fig. 11), results in a solution indicating bubble growth heated binary solution containing a non-volatile rates which are somewhat (8.3% at Too = 80°C) high solute was studied both experimentally (using an er than the correct ones, primarily because in reality aqueous solution ofNael) and theoretically, with very Woo < W; and therefore the vapor pressure in the good agreement between the experimental and theor bubble is lower due to its inverse proportionality to etical results. The experimental technique developed, the concentration. In the third assumption, W; = W;, using an electrolytically nucleated bubble, is new, p. = (P •. ,)Tp the vapor density at the bubble wall is allowing the observation of single-bubble growth at assumed to correspond to that at the saturation con a fixed location. The principal conclusions are as ditions for T; , ignoring the fact that the generated follows: vapor is actually superheated. As shown by the empty symbols in Fig. 11, this assumption results in solu (a) The experimental technique developed was tions indicating that the bubble growth rate is lower found to be excellent for single-bubble growth studies (by 17.8% at Too = 80°C) than the correct one, obvi and photography. ously because the generated vapor density is higher (b) When compared with bubble growth in pure when the superheat is ignored. It can also be seen liquids, the non-volatile solute reduces the vapor pres from Fig. 11 that the effects of the simplifying as sure of the solution (elevates the boiling point), which sumptions W; = Woo and pv = (Pv, ,)T/ are practically is also the vapor pressure in the bubble, and thus negligible (3.5% at Too = 80°C) when the concentra reduces the bubble growth rate. As expected, this tion is low (woo = 0.05), but become significant (17.8% reduction in growth rate increases with the solution at Too = 80°C) when the concentration is high concentration. 1310 OSAMU MtYATAKE et al.

8 . . 8 T. = 80"C T.=80"C ,'.J .",<:;) E6 E6 E E Q::4 Q::4

E6 E Q::4

8 8 T~ =40'C T.=40"C

- Wj=Wj,Pv=Pv 2 - Wj=Wi , Pv=Pv 2 ••• w;=l4..Pv=py ••• WF-I4..Pv=Pv 000 wr=Wi,Pv=(Pv,s)r; 000 Wi=Wj .Pv'=(Pv.• >rj

2 468 10 2 468 10 t [msl t !msl

W ... =O·20

Fig. J J. Comparison of the effects of the solute concentration at the bubble wall and the superheat of the generated vapor due to the boiling point elevation on bubble growth solutions at different values of T oc, and /';. T,. The solid lines represent the solutions under physically correct assumptions for the bubble wall concentration and density. Wi = Wi and p, = p,. Wi = W oo . p,. = p, (e) corresponds to the simplifying assumption that the concentration at the bubble wall does not change during evaporation and remains constant at W oo; Wi = Wi. p, = P,. ,h, (0) corresponds to the simplifying assumption that the vapor density at the bubble wall is that at the saturation conditions for Ti •

(c) This effect of concentration is quite significant bubble growth rates by using the superheat and this when the solution, initially at equilibrium at a pre pressure difference as parameters (rather than using scribed temperature, is brought to the superheated the far-field pressure) in each regime. s ta te by reducing the far-field (or ambient) pressure to (e) Ignoring the effects of the solute concentration a fixed value. The effect of concentration is much at the bubble wall and of the superheat of the gen smaller, however, when the far-field pressure is erated vapor on bubble growth in the theoretical reduced to bring the solution to either a fixed super solution introduces negligible errors at the lower heat, especially in the heat-transfer-controlled bubble concentrations and solution temperatures considered, growth regime, or to a fixed initial pressure difference but these errors increase with concentration and between the bubble interior and exterior, especially in temperature. The latter effect is specially evident in the inertia-controlled regime. the heat-transfer-controlled bubble growth regime, (d) The weak dependence of bubble growth rate on the error becoming as high as 18% at the highest concentration for fixed superheat and fixed initial concentrations and temperatures considered in this pressure difference between the bubble interior and study. exterior, described in conclusion (c), leads to the con (f) The weak dependence of bubble growth history clusion that it is preferable to study and correlate and rates on concentration for a given I'lTs discovered Bubble growth in superheated solutions with a non-volatile solute 1311 in this study indicates that experimental results and Fabuss, B. M. and Korosi, A., 1966, Vapour pressures of correlations relating the degree of approach to equi binary aqueous solutions of NaCI, KCI, Na2S04 and MgS04 a t concentrations and temperatures of interest in librium in flash evaporation of fresh water are also desalination processes. Desalination I, 139-148. valid for moderate-concentration brines and sea Fell, C. 1. D. and Hutchison, H. P, 1971 , Diffusion coef water, and those obtained for saline water at specific ficients for NaCi and KCl in water at elevated temper concentrations would be applicable to a wider range atures. J. chern. Engng Data 16, 427-429. of concentrations. Florschuetz, L. W., Henry, C. L. and Rashid Khan, A., 1969, Growth rates of free vapor bubbles in liquid at uniform superheats under normal and zero gravity conditions, Int . NOTATION J. Heat Mass Transfer 12, 1465-1489. c, specific heat of solution, J/kg K Gopalakrishna, S. and Lior, N., 1990, Bubble growth in flash 2 evaporation, in Proceedings of the 9th International Heat D mass diffusivity, m /s Transfer Conference. Jerusalem, Israel, Vol. 3, pp. 73-78. hJ8 latent heat of vaporization, J/ kg Hemisphere, New York. Ja Jakob number (= p,c, !:1T,/p,hJg ), dimen- Gopalakrishna, S. and Lior, N ., 1992, Analysis of bubble sionless translation during transient flash evaporation. Int. J. Heat k, thermal conductivity of solution, W /m K M ass Transfer 35, 1753-1761. Gopalakrishna, S., Purushothaman, V. and Lior, N., 1987, p, pressure of vapor in bubble, Pa An experimental study of flash evaporation from liquid p,,- pressure in solution far from bubble, Pa, pools. Desalination 65, 139-151. kPa in the figures Hewitt, H. C. and Parker, J. D., 1968, Bubble growth and P vapor pressure of solution, Pa collapse in liquid nitrogen. Trans. ASM E Ser. C 90, 22-26. Hooper, F. C. and Abdelmessih, A. H., 1966, The flashing of p. vapor pressure of pure solvent, Pa liquid at higher superheats, in Proceedings of the 3rd Inter radial coordianate measured from bubble national Heat Transfer COilference, Chicago, pp. 44-50. center, m JSME, 1962, Dennetsu Kogaku Shiryo (Materials of Heat R bubble radius, m Transfer Engineering), p. 215. Japan Society of Mechanical Engineers, Tokyo. R, critical bubble radius ( = 2a/!:1P ), m o JSME, 1971, Jokihyo Sellzu (Steam Tables and Charts), p. 1. t time, s, ms in the figures Japan Society of Mechanical Engineers, Tokyo. T temperature of solution, °C JSME, 1980, Steam Tables, pp. 10 and 123. Japan Society of T, equilibrium temperature of solution corres Mechanical Engineers, Tokyo. ponding to p",,, °C JSSWS, 1974, Kaisui Riyo Handbook (Handbook of Seawacer), p. III and 123. Japan Society of Sea Water T, saturation temperature of pure solvent cor Science, Tokyo. responding to POO ' °C Kosky, P. G., 1968, Bubble growth measurement in uni Wv mass rate of evaporation, kgjs formly superheated liquids. Chern. Engng Sci. 23, 695-706. Lior, N., 1986, Formulas for calculating the approach to Greek letters equilibrium in open channel fla sh evaporators for saline "-, thermal diffusivity of solution, m 2/s water. Desalination 60, 223-249. b infinitesimal deviation, dimensionless Lior, N. and Greif, R., 1980, Some basic observations on heat transfer and evaporation in the horizontal Hash evapor b, thermal boundary layer thickness in solu ator. Desalination 33, 269-286. tion, m Mikii:, B. B., Rohsenow, W. H . and Griffith, P, 1970, On !:1Po initial pressure difference between the bubble bubble growth rates. 1m. J . Heat Mass Transfer 13, interior and exterior [ = p,.(O) - Poo ], Pa 657-666 Miyatake, O. and Tanaka, I., 1982, Bubble growth in uni I1T, superheat ( = T oo - Te ), °C formly superheated water at reduced pressures, 1st Report, p, viscosity of solution, Pa s 2nd Report. Trans. JSM E, Ser. B 48, 355-363, 364-371. p, density of solution, kg/m J Miyatake, 0., Fujii, T. and Hashimoto, T., 1977, An experi p" density of generated vapor, kgjm J mental study of multi-stage Hash evaporation phenomena. Heat Transfer Japanese Res. 6(6), 25 - 35. P".s saturation density of vapor, kg/m J Miyatake, 0., Hashimoto, T. and Lior, N., 1992, The liquid (J surface tension at the bubble wall, N/m How in multi-stage Hash evaporators Int. J. Heat Mass (J) mass fraction of solute in solution, dimen Transfer 35, 3245- 3257. sionless Miyatake, 0., Hashimoto, T. and Lior, N ., 1993, The relationship between How pattern and thermal non-equi librium in the multi-stage Hash evaporation process. Subscripts Desalination 91,51-64. value at the bubble wall Miyatake, 0., Tomimura, T. and Ide, Y., 1985, Enhancement (J) value far from the bubble of spray flash evaporation by means of the injection of bubble nuclei. ASM E J. Solar Energy Engn !! 107, 176-182. 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