Modelling of a batch sonochemical reactor

Pergamon

Chemical Engineering Science, Vol. 49. No. 6, pp. 877-888. 1994 Copyright Q 1994 Elsevier Science Lt., Printed in Great Britain. All rights reserved ooo!-2509p4 xi.cQ + 0.00

MODELLING

OF A BATCH

SONOCHEMICAL

D. V. PRASAD NAIDU,’ R. RAJAN,t R. KUMAR,‘* ‘Department

K. S. GANDHI,’

REACTOR V. H. ARAKERI5 and

S. CHANDRASEKARAN” of Chemical Engineering, $Department of Mechanical Engineering, and IIDeparment Organic Chemistry, Indian Institute of Science, Bangalore012, India (Received

2 May

1993; accepted for publication

27 September

of

1993)

Abstmct-Ultrasonication of aqueous KI solution is known to yield I, due to reaction of iodide ions with hydroxyl radicals, which in turn are generated due to cavitation. Based on this conceptual framework, a model has been developed to predict the rate of iodine formation for KI solutions of various ooncentrations under different gas atmospheres. The model follows the growth and collapse of a gas-vapour cavity using the Rayleigh-Plesset bubble dynamics equation. The bubble is assumed to behave isothermally during its growth phase and a part of the collapse phase. Thereafter it is assumed to collapse adiabatically, yielding high temperatures and pressures. Thermodynamic equilibrium is assumed in the bubble at the end of collapse phase. The contents of the bubble are assumed to mix with the liquid, and the reactor contents are assumed to be well stirred. The model has been verified by conducting experiments with KI solutions of different concentrations and using different gas atmospheres. The model not only explains these results but also the existence of a maximum when Ar-0, mixtures of different compositions are employed.

INTRODUCTION Ultrasound is known to enhance the rates of many chemical reactions. In reactions catalysed by a solid phase, it is generally understood that the rate is en-

hanced by ultrasound due to its cleaning action on the catalyst surface. In liquid-liquid reactions, the dispersed phase disintegrates into tiny fragments under the influence of ultrasound, and rates are enhanced due to the increased interfacial area. A more interesting phenomenon is the occurrence of non-catalytic reactions in a single phase only under the influence of the irradiation of sound, and this is the subject of the present investigation. It is reasonably well established that ultrasound does not interact directly with the molecules but brings about the chemical effects through cavitation (Suslick, 1989; Lorimer and Mason, 1987; Ley and Low, 1989). When the liquid medium is irradiated with ultrasound, it causes a series of compression and rarefaction cycles which give rise to areas of high and low local pressures. When the local pressure drops below the saturation vapour pressure of the liquid medium at any location, existing gas nuclei receive the solvent vapour to give rise to cavities in the medium. Under the influence of ultrasound, these vapour, gas-filled cavities show a variety of behaviour patterns according to their sizes (Flynn, 1964; Young, 1989). Transient cavities, having lifetimes of one or at best a few acoustic cycles, expand to many times their initial size and collapse violently producing high local temperatures and pressures, which can be as large as 4000 K and 400 atm (Suslick, 1990). The vapour inside the bubble can decompose to yield radicals under these conditions (Sudick, 1989; Lorimer and Mason, 1987; Ley and Low, 1989). These *Author to whom correspondence should be addressed. Also at Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore. 877

radicals enter the surrounding hquid medium and react with the molecules present there. Non-aqueous media do not respond well to ultrasound because organic solvents often have high vapour pressures and cavitation bubbles do not yield the high temperatures needed for dissociation of the solvent molecules (Lorimer and Mason, 1987). However, if water is sonicated, water vapour inside the cavity yields dH, H, HO, etc. (Weissler, 1959). These radicals can be reduced in the liquid phase if any oxidizable species are present there. Many examples of this class of sonochemical reactions have been discussed by Ley and Low (1989) and Lorimer and Mason (1987). So far, however, an overall quantitative model that predicts the rate at which these reactions occur is not available. The extent of the formation of radicals depends on the extreme conditions developed inside a collapsing bubble. In this work, we predict the conditions at collapse using a bubble dynamics equation and utilize this information to calculate the rate of production of radicals. It is proposed that when the bubble collapses the radicals are released into liquid phase and undergo a variety of reactions there. The kinetics of various reactions of the radicals with the molecules present in the liquid phaseare used to predict the rate of formation of products. It will be shown later that the mechanistic picture with certain empirical inputs can be successfully employed to predict quantitatively the rate of iodine liberation when iodide ion in aqueous phase is oxidized by the radicals. REACTION SYSTEM As mentioned

earlier, organic liquids do not respond to sound very efficiently. Thus, it was decided to select a reaction that occurs in aqueous medium. A large number of such reactions have been reported (Ley and Low, 1989; Lorimer and Mason, 1987). A re-

878

D. V. PRASAD NAIDU

action which has been investigated earlier (Hart and Henglein, 1985) and which offers considerable challenge for quantitative prediction is the formation of iodine by sonication of KI solution. Weissler (1959) reported that sonication of water produced the following radical species: dH, H and H’O,. In addition, 0 atoms may also be produced, especially if O2 is present in the atmosphere over the solution being sonicated. Thus, since the cavities contain water vapour and gases which constitute the atmosphere, the reactions in the liquid phase can be caused only by these radicals. To study the reactions of these radicals in the aqueous phase, Hart and Henglein (1985) conducted extensive experiments by irradiating aqueous solutions of KI in a batch reactor with 300 kHz ultrasound under argon, oxygen and Ar-0, mixtures of different compositions. The products formed were determined to be I, and H,Oz. They also carried out reactions in the presence of ammonium molybdate, which catalyses the oxidation of iodide ion by HZ02 to iodine. In this case, the only product formed was iodine. They found that the rate of formation of I2 remained constant with time. The rate increased with increasing KI concentration, but reached a plateau value at very high concentrations of KI. Interestingly, they found that the rate of IZ formation reached a maximum at a gas composition of 30% oxygen-70% argon, when experiments were conducted under atmospheres of mixtures of these gases. They proposed that the observed intermediate maximum is related to the following fact. The ratio of specific heats (y) for argon is higher than that of O2 and thus, with the increase in 0, content of the bubble, y decreases, and the collapse temperature of the bubble also decreases. This leads to the generation of fewer hydroxyl radicals. At the same time, as the O2 concentration increases, the formation of hydroxyl radicals should also increase. Thus, a composite effect exists at an intermediate point, resulting in a maximum. These findings are interesting and show complex behaviour. No attempt has been made until now to predict these experimental results quantitatively. Any model of this reaction must explain the following observed features. (1) The rate of iodine liberation increases with KI concentration but remains constant with time. (2) Different rates of iodine formation are observed when different gases like N,, OZ and Ar are used. (3) When mixtures of Ar and O2 are used as dissolved gases in water, the reaction rate passes through a maximum at an intermediate composition. Once these basic features are explained, it should be possible to explain the effect of frequency on the rate of reaction (Cum et al., 1992) and the effect of inclusion of other compounds like Ccl,+ (Alippi et al., 1992).

et al.

One of the significant inputs to the model is the amplitude of sound waves generated, and this depends on the equipment used. Thus, it was not possible to test the model against experimental results already reported by others as the necessary information was not available in their publications. Thus, experiments were carried out by sonicating aqueous solution of KI under various gas atmospheres. In some experiments, ammonium molybdate was also used for reasons explained later.

EXPERIMENTAL

Set-up In the present study, a stainless steel low powered ultrasonic cleaning bath (Vibronics Ltd, Bombay) was used for the experiments. The internal dimensions of the bath were: length = 0.227 m, breadth = 0.178 m and height = 0.15 m. Thus, the bath had a floor area of 0.0404 m’. The reactions were carried out in culture tubes placed in the bath. The bath has to be filled with water upto a height of 0.127m which acts as a medium for the propagation of ultrasound generated from the transducers to the reaction vessel. A provision was made for fixing the reaction vessel at a fixed height and location in the bath. Culture tubes of size 25 ml were chosen as the reaction vessels because they gave continuous cavitation and reproducible results. Further, stroboscopic observation revealed that cavitation started at the glass surface and became bulk cavitation for most of the run. The bath operated at 25 kHz. The amplitude of the sound waves in the bath was measured using a hydrophone; the nominal root mean square (rms) value was found to be 1 bar. This corresponds to a peak amplitude of 1.41 bar. The predicted value based on the power rating (250 W) and the exposed area of bath is very close to this, being 1.38 bar. The latter value was used for quantitative predictions. It should be noted that, since the glass culture tubes used had small thickness compared to the acoustic wavelength a\ 25 kHz, there is very little attenuation of amplitude across the glass tube wall. Experimental conditions As mentioned earlier, the reaction vessel was fixed in the bath always at the same location, The volume of the reacting solution was fixed as 10ml so that the level of the solution in the vessel remained the same from experiment to experiment. These precautions were necessary to obtain reproducible results as it was found that cavitation pattern depended on the location of reaction vessel as well as the liquid level in it. Thirty minutes of sonication increased the bath temperature by 6°C. Bath water was changed after 15 min of sonication to maintain the bath temperature nearly constant (maximum variation was 3°C) so as to reduce the effect of its variation on the reaction behaviour. These experimental conditions were maintained throughout the work.

Modelling of a batch sonochemicalreactor Experimental procedure In the present work, experiments were carried out with different KI concentrations (1.0, 2.5, 5.0, 10.0, 15.0,20.0 and 25.0% w/v) in aqueous solutions under a variety of atmospheres: nitrogen, air and oxygen. Experiments were also carried out with different compositions of argon-oxygen atmospheres but with 1 M (16.6%) solution of KI in water. A few experiments were carried out under nitrogen atmosphere with the addition of ammonium molybdate to the aqueous solution. A saturated solution of the desired gas was prepared by boiling 100 ml of distilled water for an hour and passing the desired gas continuously through it, for about half an hour, until it reached the ambient temperature. Potassium iodide was dissolved in this solution. These solutions were sonicated for different times and rates of liberation of iodine were determined as described below. During the sonication, the gas atmosphere was maintained by passing the corresponding gas continuously over the solution in the culture tube. Method of analysis The iodine liberated during the entire experiment is very meagre and hence a spectrometer (UV-2100, SHIMADZU) was chosen for analysing the iodine. The peak at 354 nm was monitored. Calibration was done as follows. 0.01 N iodine solution was prepared by dissolving the corresponding amount of resublimed iodine in a solution with the particular concentration of KI for which calibration had to be done. The concentration of iodine in this standard solution was determined by titrating against sodium thiosulphate which was standardized against 0.1 N potassium iodate solution. The standard iodine solution was diluted with the same KI solution to obtain iodine solutions of different known concentrations. These solutions were used to calibrate the instrument for that particular KI concentration. The iodine liberated upon sonication with a particular concentration of KI solution was estimated by using the corresponding unsonicated KI solution as blank. THE MODEL

Physical description of the model The oxidation of iodide ion in the aqueous phase was taken as the system to be modelled. The solution of KI is expected to have a large number of nuclei, i.e. cavitation-generating spots. These could be small bubbles already present or arising out of collapsed cavitation bubbles or these could be gas entrapped in the crevices at the reactor wall. When the liquid medium is subjected to ultrasound, the nuclei respond differently depending on their initial size and the amplitude and frequency of the sonic field. This aspect is fully discussed in Flynn (1964); however, more recent studies of Arakeri and Chakraborty (1990) have shown that at frequencies lower than 100 kHz transient cavities dominate. Transient cavities are those which grow and collapse more or less in one acoustic

819

cycle, with the collapse phase being extremely violent. One criterion, which can be used to designate a cavity as transient is the condition that during the collapse phase the bubble wall velocity should exceed the velocity of sound in the liquid medium. There are additional questions in studying the cavity motions of this type, e.g. whether the thermodynamic path of the cavity contents can be considered to be isothermal or adiabatic or a combination of the two. Similarly, there is the question whether free evaporation and condensation of the surrounding solvent can be allowed during the entire history of the cavity motion. If one wants to include all the thermophysical aspects of the cavity motion, the problem is quite formidable. As shown in a recent study (Gaitan et al., 1992), there are severe limitations, using even the most complex models, in terms of predicting the end conditions in a cavity during the collapse phase. In view of this, in the present work, a simple model suggested by Flynn (1964) has been used. In this model, the entire growth period and the initial phase of the collapse are assumed to be isothermal; whereas the later stage of the collapse is considered to be adiabatic. This transition from isothermal to the adiabatic during the collapse phase is taken to occur when the internal gas pressure is equal to the vapour pressure of the liquid at its bulk temperature. The adiabatic collapse phase is assumed to end when the bubble wall velocity reaches the sonic velocity in the liquid medium. It is not clearly established what happens to such a cavity finally; it probably disintegrates mixing its contents into the bulk of the liquid. During the adiabatic phase of the bubble collapse, temperature and pressure inside the bubble increase due to compression, yielding extreme conditions. This can result in free radical formation from the water vapour and oxygen, if it is present in the dissolved gas. Although the formation of various radicals is the result of kinetics of various reactions among the available species, the present model assumes that thermodynamic equilibrium is attained at the end of the collapse phase. The equilibrium compositions of free radicals can, therefore, be evaluated at the extreme conditions of the bubble collapse from the thermodynamic properties of the constituents. The various radicals released into the medium undergo several reactions, one of them being the oxidation of iodide ion. In the literature, it has been suggested that these reactive radicals move into a thin shell of a few nanometres around the bubble where the reaction occurs, but this picture assumes that the bubble collapses symmetrically. However, recent findings (Lauterborn and Bolle, 1975) indicate that the bubble collapses asymmetrically giving rise to microjets which cause intense local mixing. Further, the collapsing bubble may fragment during collapse giving rise to one or more microreactors where the mixing is intense. The number and lifetimes of such microreactors are not possible to evaluate with our present state of understanding. As the number of cavities collapsing per cm3 is of the order of 10’ per second, there are a large number of microreactors forming

D. V.

880

PRASAD NAIDU

and dissipating simultaneously, resulting in comparatively high rates of mixing in the whole reactor. In view of the rapidity of the reaction, it is quite possible that diffusion plays a significant role. However, in the absence of precise information about the microreactors and the diffusional aspects associated with them, we have used a lumped approach considering the whole reactor as well mixed. The released radicals can, therefore, be considered as perfectly mixed in the bulk liquid. Thus, the various components of the proposed model involve the following:

(1) growth and collapse of the cavity,

et al.

This equation is known as the Rayleigh-Plesset tion and is given by Plesset (1949):

Here, R is the instantaneous radius of the bubble, fi is the bubble wall velocity, ii is the bubble wall acceleration, PL is the pressure in the liquid at the bubble interface, P, is the time-varying pressure field imposed by the sound field, and pL is the liquid density. The pressure just outside the bubble surface and the pressure inside the bubble are related by P,(R)

(2) evaluation of temperature and pressure at the end of collapse, (3) calculation of equilibrium compositions at collapse conditions, (4) material balance equations for various species in the liquid phase of the reactor. Each of these components has to be analysed to obtain the composite model of the reactor in which sonochemical reaction is proceeding. Growth and collapse of cavities The following assumptions are made for the growth and collapse of cavities.

(1) The intensity of the sound field is assumed to be

= Pi - 2a/R.

P, = PgO(RO/R)3”. The value whereas it C, and C,, (2) and (3)

(4)

(5)

(6)

(7) (8)

The expansion and collapse of a bubble is governed by the bubble dynamics equation. The continuity and momentum balance equations in spherical coordinates can be used to obtain the governing equation for radially symmetric cavity motion in inviscid liquids.

(3)

of c( is unity for an isothermal process, is equal to y. the ratio of the specific heats for an adiabatic process. Substituting eqs into eq. (l), we obtain

($y” - P-(t)].

constant.

(3)

(2)

Here Pi is the internal pressure of the bubble and is equal to PO f P, where P, is the partial pressure due to gas content, and P, is the partial pressure due to vapour content. Surface tension is given by u. The gas pressure inside the bubble changes as the bubble radius changes. As the total amount of gas in the bubble remains constant, the bubble radius and the gas pressure are, therefore, related by

(2) The bulk temperature itself is assumed to be

constant to correspond to the experimental conditions maintained. The gas and vapour present in the bubble are assumed to behave ideally. The expansion phase and the initial part of collapse phase of the bubble are assumed to be isothermal, whereas the end phase of the collapse is assumed to occur adiabatically. The partial pressure of the vapour inside the bubble is assumed to be equal to the vapour pressure of the liquid corresponding to the bulk liquid temperature during the isothermal periods. The transition from isothermal to adiabatic conditions in the collapse phase is assumed to occur when partial pressures of gas and vapour in the bubble equalize. It should be noted that beyond this point the cavity is assumed to be a closed system, on which work is being done. The end conditions of cavity collapse are taken to be the point when the bubble wall velocity is equal to the sonic velocity in the liquid medium. The liquid is assumed to be incompressible, and viscous forces are neglected.

equa-

(4) The external pressure, i.e. the time-varying pressure field P,, is obtained from the characteristics of the applied ultrasonic radiation. It is a sinusoidal function of time and can be written as P, = Pb - P,sinwt

(5)

where Pb is the atmospheric pressure in the present case and P,, is the pressure amplitude of the ultrasonic field. P. has to be obtained from measurement of intensity and has been discussed already. We obtain after substituting eq. (5) for P, in eq. (4) and rearranging

(Pb - P,sinot) R where dR dt=

1

3 Ii2 2R

R.

(6)

(7)

As mentioned earlier, the entire growth phase and the initial part of the collapse phase are isothermal. During this phase, eqs (6) and (7) have to be integrated with initial conditions

t = 0,

R = R.

and

2 =0

(8)

Modelting

of a

batch sonochemicalreactor

by setting a = 1 and P. = P,. The collapse phase becomes adiabatic once the gas pressure P, is equal to P.; let R2 be the radius of the bubble at which this transition occurs. The value of R, itself is easily calculated in terms of R. from eq. (3) as indicated below: P, = P, = Peo(R,/RZ)3.

- (Pb - P,sinwt) R

1

(11)

Temperature and pressure at the end of collapse The temperature and pressure at the collapse of the bubble can be calculated from the bubble size at collapse R,, using the adiabatic law ‘)

(12)

and P, = P,(R,/R/)3Y.

(13)

P2 is the pressure inside the bubble when R = R,, and is equal to the sum of the partial pressures of gas and vapour. However, at Rz, these are equal to each other. Thus P2 = 2P,. Further, T2 = T,, with T, being the bulk liquid temperature. Thus both temperature and pressure at the end of bubble collapse can he calculated from the radius, R,. As no vapour exchange occurs during the collapse phase, the total number of moles present in the bubble when R = R2 and at the time of collapse need not be equal because of chemical reactions occurring during CES 49:6-F

P

(14)

RLlT

where R, is the universal gas constant. The total moles of gas species at the time of collapse are also given by eq. (14) since P, = Pg at the beginning of the adiabatic phase of the collapse period. In the case of mixture of gases, the individual components were calculated using the known gas composition. This mixture of vapour and gases was used to compute the amounts of radicals formed by equilibration at T, and P,. Several physical properties appear in the above equations. They were obtained as follows. The nature of the dissolved gas affects y. Computations for a gas and vapour mixture were made by using an average y value given by

(10)

where the initial nuclei are assumed to be existing at atmospheric pressure. Later, we shall discuss how the initial cavity size was chosen.

T, = T2(R2/R,)3”-

4

n,=-nR:-i 3

3 d2 2 R’

This equation was solved for R < R2 until the surface velocity of the bubble reached the velocity of sound. At this point the bubble was assumed to collapse. Let R, denote the radius of the cavity at this point. Thus, R, could be found by solving the eqs (6) and (7) in the isothermal phase (d > 0 and R > R, when d < 0) and eqs (7) and (10) in the adiabatic phase (R < R2 when ri =Z0) with initial conditions (8). It should be mentioned here that the initial gas pressure PO0 is related to the initial cavity size by P, + P,, = Pb + 2u/R0

this phase. In the present case, the change in moles due to reactions was found to be negligible (~3%) and, hence, the number of moles was assumed to remain constant during collapse, and could be calculated from conditions obtaining at R = R1, by using the ideal gas law. Thus

(9)

The values of P,, the partial pressure of the vapour in the bubble, are different during the isothermal and adiabatic phases. Whereas it is equal to the saturation vapour pressure during the isothermal phase, its value changes continuously during the adiabatic portion of the collapse phase. In view of our assumption of the bubble being a closed system, neither heat nor mass exchange is permitted between the bubble and the liquid for R =g R,. Thus, pressures and volumes of the bubble during this phase were related using the adiabatic gas laws. The bubble dynamics equation therefore gets modified to

881

Equations (6) and (10) involve the saturation vapour pressure of liquid at the sonication temperature. As the KI concentration increases, the vapour pressure exerted by water is lowered. Horvath (1985) has given the following expression for the IL-H20 aqueous system: d=

+

[ln(l

- x) - 1.1517x + 0.0148x1’2

+ 3.204x’.’ - 11.44x2].

(16)

Here

CB

x=CB + 55.51’ Ce is the concentration of KI in mol/l, x is the mole fraction of ICI in the KI-H,O mixture, and 4 is the osmotic coefficient, given by -1000

(18)

ln (~.PH,o) dJ= - 36C, [ 1

where PHzO is the vapour pressure of pure water at 298” K, the condition of experimentation and was taken to be 3168 N/m2 (Perry, 1963). Another property playing an important role is the surface tension of the KI solution. In the KI-HI0 system, an increase in the KI concentration increases the surface tension. The surface tension of the aqueous KI solution was measured by a stalagmometer and was empirically correlated as (T_~~_~_,= 0.07272 + E

,

Cs.

(19)

Calculations of equilibrium compositions Owing to the collapse, the contents inside the cavity are subjected to extreme conditions resulting in

D. V.

882

PRASAD

a large number of reactions.

The equilibrium compositions can be calculated using free-energy minimization algorithms. One such program, SOLGASMIX (Ericksson, 1975) was used here to compute the equilibrium compositions. The program is capable of dealing with multicomponent and multiequilibria situations. It needs the C, vs temperature relationship, entropy and heat of formation of all components. The program further needs as input the species expected to be present. The data required for various components were obtained from the JANAF tables (1964). The program produces as output the equilibrium compositions and fugacities of various species for any given temperature, pressure and input compositions of mixture of gases and vapour. As mentioned earlier, the species expected to be present in the bubble at equilibrium are H,O, gases constituting the atmosphere, and the following radicals and atoms: 0, H, bH and Hb,. Thus, these species were listed as expected products. The program was fed with T,, P, and the composition of the cavity calculated from the bubble dynamics equation to obtain the composition of the equilibrium mixture which is released into the liquid phase.

NAIDU et al.

atmosphere like nitrogen, iodine is the only product formed. Under this condition, all the bH radicals are consumed to produce solely iodine. As mentioned earlier, the nuclei could be gas pockets present in crevices in the walls, the fragments of the collapsed cavities, etc. The number of nuclei growing and collapsing is then expected to reach steady state quickly after the sonication begins. Let n be the number of cavities collapsing per unit volume per unit time. Let Ci be the number of moles of the ith type of radical released by each cavity when it collapses. Thus, nc represents the rate of release of the ith radical species into solution. The material balance equations for the various components can in general be written as: Hydroxyl radicals: dC., ~ = nCA - kzCa - kl CACB. dt

Iodine: dCc dt = BlklC,CB Hydroperoxy

Material balance equations

During sonication of KI solutions, KI does not enter into the bubble as it has negligible volatility. The various radical species formed in the bubble are H, dH and H’Oz, etc. These species mix with the liquid phase when the bubble fragments at the end of collapse, and undergo further reactions. Thus, radicals mixed with the liquid react either with the ions present in the liquid medium or combine with themselves to yield various products. Hart and Henglein (1985) give the various possible reactions which could take place in the liquid phase as 2KI + 2dHtl-

2KOH + I2

dH + dHrZ-H,O, ZH’O, + 2KIH’O, + tiO,-

Ir* 2KOH k3 H202

20 + 21P + 2K+ + 2Hz0-%Iz

(A) (W

+ I2 + O2 +O,

(C) (D)

+ O2 + 2KOH + Hz.

(E)

Thus, the products formed in these reactions are I2 and H,Oz. If sonication is carried out in the presence of the catalyst, ammonium molybdate, H20L also can rapidly oxidize iodide ion to yield iodine: H202

+ 2KI --* 2KOH + 12.

Thus, in the presence of the catalyst

(F)

and an inert

Gv

+ B&C,&

+ P&&&E.

(21)

radicals:

dCI, = ttG - k3Cf, - k4CDCB. dt

(22)

Oxygen atoms: dCEznc dt

E - ksC&e.

(23)

The above equations are to be solved to obtain the concentration of various species as a function of time.

RESULTS

AND Dl!3CUSSION

The model eqs (20)-(23) show an interesting feature. As mentioned before, the rate at which radicals are being released into the liquid phase or nc reaches a steady state soon after sonication begins. Thus, it is possible that radical concentrations reach a quasisteady state; i.e. dC,/dt = 0. If that happens, and if CB is sufficiently large, the rate of liberation of iodine also reaches a quasisteady state. Thus, the equations have the capacity to evlain one of the key features observed by Hart and Henglein (1985). The model equations (20)-(23) can be solved to find the rate of production of iodine. The solution requires the rate constants appearing in the equations as well as the rates at which the various radicals are being released into the liquid phase. Calculation of the latter requires n, the number of bubbles collapsing per unit time per unit volume, and G, the amount of each radical species present in each bubble. The radical species contained in each bubble at the time of collapse can be calculated using SOLGASMIX once the temperature, pressure and gas composition in the bubble at that time are known. These quantities have to be obtained

Modelling of a batch sonochemical reactor from the solution of the bubble dynamics equation which requires Pa and RO as inputs. We first discuss how all these parameters were fixed.

Model parameters Amplitude of the pressurefield. The solution of the bubble dynamics equation to obtain T, and P, requires the amplitude of the sound field. The pressure amplitude of the sound waves was estimated as indicated earlier and was taken to be 1.38 x lo5 N/m’. Initial cavity size. The initial cavity sizes are very difficult to estimate. It is usually treated as a parameter in the literature on cavitation, but is expected to be greater than the size of the critical nucleus. Assuming uniform pressure in the surrounding medium and that mechanical equilibrium prevails, the size of a critical nucleus that grows explosively at a liquid pressure of Pu: is given by (Young, 1989) P. - PLc = 4tr/3Rc.

(24)

Thus, the smallest bubble that can grow corresponds to the lowest pressure reached in the system; the latter is given by P,,, - Pa and is equal to -0.38 x\105 N/m’ in the present case. While R, is the radius of the critical nucleus at PLc, R. is the radius of the critical nucleus at P.,,, and is related to R, by the following equation:

In the present study, RO was found to vary from 1.05 to l.lOpm for solutions with different KI concentrations, yielding negligible variation. Flynn (1964) has shown that the maximum radius, R,.,, that a cavitation bubble reaches is a sensitive function of RO. if its value is close to the critical value. Gradually as R. is increased, the dependence of R,,. on R. is reduced. Thus, in the present study, R,, was fixed at 2 pm. Moreover, cavities of this size reached the collapse condition within one cycle at the frequency employed. Number of bubbles collapsing. The number of bubbles collapsing per unit time per unit volume, n, depends only on the operating conditions of the sonicator, i.e. the power input, frequency and type of the reaction vessel. It is not expected to depend on the nature of dissolved gas and the KI concentration. However, there is no precise way of calculating n. Hence, information from some experiments has to be utilized to determine this. Under inert atmosphere, equilibrium calculations showed that the amounts of 0 and HO, contained in a collapsing bubble are negligible compared to the amount of dH. Moreover, all the dH radicals generated yield iodine if ammonium molybdate is present in the liquid phase. These facts suggest that the experiments carried out with ammonium molybdate under nitrogen atmosphere can be used to obtain n as described below.

883

The value of n so determined was used for all other conditions since the number of bubbles collapsing depends only on the operating conditions. For these experiments, we need to consider only dH production and reaction with iodide ion since H202 formed by recombination is rapidly utilized for oxidation of iodide ion under the influence of the catalyst. Thus, for this case, the pertinent equations are

dC1

-=nnC,-kk,CACB dt dCc dt = B,k,C,C,.

(214

These equations indicate that a quasisteady state in the rate of production of iodine can be reached when nc = k, CACB = 0.5 dCJdt. Thus, the rate of production of iodine observed in these experiments was used to obtain n, which is discussed next. Figure 1 shows a plot of the radius of the cavitating bubble vs time obtained from the solution of the bubble dynamics equations. It can be seen that the collapse is very rapid. The final temperature and pressure reached in this case were computed to be 2064 K and 78 atm. Table 1 shows the details of the results from SOLGASMIX program. It can be seen that hydroxyl radicals are the dominant radical species. Figure 2 shows the basic observations of the amount of I2 liberated against time. It can be seen that quasisteady rates have been reached very quickly. The rate of liberation of iodine was calculated from these data to be 2.945 x 10m8 mol/l s, and using this rate, the value of n was fixed as 2.6445 x 10” i/1 s. It may also be noted that at such small rates of liberation of iodine, the amount of KI consumed for the formation of iodine is negligible. Thus, for the typical reaction times used, the concentration of ICI is assumed to be constant.

,,a

5

Fig. 1. Radius of bubble during growth and collapse under sinusoidal pressure wave (initial cavity size = 2.0ym, P, = 1.38 x lo5 N/m’ and frequency = 25,ooO Hz).

D. V.

884

PRASAD

NAIDV

et a1.

Table 1. Equilibrium compositions of a bubble at collapse conditions (concentration of KI = 15%, T, = 2064 K, PI = 78.3 bar) Component

Moles of input mixture 1.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0

N* 02 H-20 OH 0 Hb, HZ

7~ q CEE_EC

Equilibrium composition (mol)

-LLCCG *0111 *tttt -a e!ZcWS’

1.00

1.0478x 9.9631 x 9.6692 x 5.007 x 7.5287 x 2.5725 x 2.016x 3.213~

10-s 10-l lo+ 10-b lo-’ 1O-3 1O-5 lo-*

1.0

(3

l.Oa

KI KI 5.0~ KI 10.0% KI 15.0% KI 20.0~ KI 25.0~ KI

2.B

2 s 0.5

I

-

0.0

0

e /

/

mo

O*O 0

P

,/

/

Fig. 3. Iodine liberated as a function of time during irradiation of KI solutions under air atmosphere.

5

Timel~min)

Fig. 2. Iodine generated as a function of time during ir-

Air

atmosphere

radiation of 15% KI under N, atmosphere with ammonium molybdate.

Reaction rate constants. The model equations (20)-(23) indicate that a quasisteady state can be reached if the rate of generation of radicals equals the rate of their consumption by various reactions, provided the iodide ion consumed is negligible. The latter is generally true since the reactions were carried out only for short periods. Under such quasisteady conditions, dCJdt becomes zero for all i and,dCJdt attains a constant value since CB is very nearly constant. Thus, the equations become 0 = nCA - k2C: - klCACB (2OB)

-

0 = nC,, - k,C;

0 =

nG

-

- k4CDCB k5CECB.

(2W (23~

These equations can be solved for CA, CD and CE, and used to calculate the rate of production of I2 as

Fig. 4. Comparison of predicted and observed rates of iodine liberation under air atmosphere.

The data obtained under air atmosphere with seven concentrations of KI were used to evaluate the various rate constants. Figure 3 shows the basic observations on amounts of iodine liberated as a function of time with KI concentration as a parameter. Once again, it can be seen that a quasisteady state has been attained quickly and that the rate of iodine liberation is constant. Figure 4 shows the data for the rate of IZ liberation vs KI concentration as points. From these data, and from eq. (26), a non-linear parameter estimation program (COSTAT) was used to determine the ratios kf/k, and kt/k,. The value of k2 was reported by Hart and Anbar (1970) to be 0.6 x 10” dm3/gmol s. The value of k3 was adjusted such that quasisteady state was reached within 3 min. The values of the rate constants are shown in Fig. 4. The calculations made on the basis of these rate constants are shown as a line. The values of the rate

Modelling of a batch sonochemical reactor Table 2. Amount of hydroxyl radicals

at collapse conditions

885

(mol/mol of HzO)

Atmosphere %KI 1.0 2.5 5.0 10.0 15.0 20.0 25.0

(w/v)

5.921 6.241 6.805 9.114 9.669 1.130 1.326

x x x X x X x

1o-4 10m4 1O-4 10-a lo-* 10-S 10-s

constants as well as the n value were used to predict the rate of iodine production in all the other experiments. As mentioned earlier, the temperature rose by about 3°C during each experiment, and we have neglected the effect of this rise on the rate constants. It may be noted from the data that the rate of liberation of iodine increases with KI concentration. This could be due to two factors. Firstly, the amounts of hydroxyl radicals contained in collapsing bubbles increase with KI concentration. Calculations from the model, shown in Table 2, do confirm this. Secondly, as the concentration of iodide ion is increased, the consumption of radicals by iodide ion becomes more effective than by the recombination of radicals. Thus, the rate of iodine liberation should increase with increased KI concentration. With different atmospheres, the same model equations can be used. Only the value of c changes from one atmosphere to another because the nature of the dissolved gas affects the extreme conditions generated, which in turn affects the equilibrium compositions of the contents of the gas phase. Model predictions: effect of KI concentration and dissolved gas The major parameters which can influence the rate of I2 generation are the KI concentration and the nature of the gas dissolved in the solution. To study the effect of KI concentration on the rates of I2 liberation, experiments were conducted with nitrogen and oxygen atmospheres. The values of y for both the gases are the same and hence they are identical so far as temperature and pressure reached at collapse. The effect of dissolved gas was studied by using an atmosphere of argon-oxygen mixtures of different compositions.

Oxygen

Air

Nitrogen 2.453 2.558 2.743 3.467 3.636 4.122 4.684

x X x x x x x

3.624 3.780 4.052 5.123 5.373 6.092 6.922

10-s 10-s 1O-J lo-’ 10-s 1O-3 lo- 3

x X x x X x x

lo-’ lo-’ lo-’ lo-’ lo-’ 1O-3 10 - s

%Kl

Fig. 5. Collapse conditions under N,, air and Oz

atmo-

4 “0

Nitmgtrn atmosphere

l l

- . 0-lm 0

5 Cont.

10 of

..

l

15 KI(%

Experimental Model 20

25

30

W/V)

Comparison of model predictions and observed rates of liberation of iodine under N2 atmosphere.

Fig. 6.

Nitrogen atmosphere. Experiments were conducted with different KI concentrations (1,2.5, 5.0,10.0, 15.0, 20.0 and 25.0%) under nitrogen atmosphere. The pressures and temperatures reached at collapse conditions are shown in Fig. 5. It is seen that with increasing KI concentration, both collapse temperature and pressure rise, probably because of the decrease in vapour pressure and increase in surface tension values. In the experiments, the amount of iodine liberated was measured as a function of time. Here too a linear

relationship between the amount liberated and time of sonicatian was observed. Experimentally measured rates for various KI concentrations are shown in Fig. 6. It should be noted that for any KI concentration the rates with Nz are lower than those with air atmosphere. This is because of the presence of oxygen, which leads to formation of greater amounts of dH radicals during bubble collapse under air atmosphere.

886

D. V. PRASAD NAIDU er

Table 2 shows the amount of dH radicals obtained with each gas atmosphere and KI concentration. It should be mentioned that in all these cases the amounts of dH are greater by an order of magnitude compared to other radical species. It can be observed from Table 2 that amounts of hydroxyl radicals obtained increase with Kl concentration; but the proportionality is less than linear. As the rate of liberation of iodine is proportional to the hydroxyl radical content of the bubbles, their dependence on KI concentration should be analogous. This is confirmed from the comparison of the experimental rates observed under Nz atmosphere with those calculated from the model presented in Fig. 6. It is seen that the predictions are in agreement without any adjustable parameters at lower concentrations, giving confidence in the model developed. However, there is a systematic deviation at higher concentrations. Oxygen atmosphere. Experiments under oxygen atmosphere were conducted, and again the rates of I2 liberation were found to remain constant with time. The observed rates of iodine liberation are shown in Fig. 7 and it is seen that they increase with KI concentration. The reasons for this have already been discussed. It is seen that the rates under oxygen atmosphere are higher than those observed with both N1 and air because of higher amounts of the dH radicals generated (see Table 2). The rates of I, formation were calculated and have been compared with observations in Fig. 7. It is seen that the model is able to explain the trends in the experimental results satisfactorily, although the predictions are consistently higher than observations. Atmospheres ofAr-O2 mixtures. Hart and Henglein (1985) observed a maximum in the rate of liberation of iodine as the oxygen content of Ar-0, gas atmosphere was increased. Prediction of this behaviour would provide a more stringent test of the model. Thus, in this set of experiments, 1 M KI solution was sonicated using different compositions of Ar-O2 mix-

al.

It was found that the amount of I2 generated linearly with time for each composition. The rate of liberation of iodine is thus constant. Here the final temperatures reached for different compositions are different due to the variation of y values. These are shown in Table 3. It is seen that for pure argon the temperatures attained are far higher ( r 2SOO°C) than for nitrogen, air or oxygen. When the oxygen concentration increases, the temperatures and pressures fall, the lowest corresponding to the case of pure oxygen. This is entirely due to the fact that y for argon is greater than that for oxygen. The different temperatures and pressures achieved during bubble collapse as well as the differences in the oxygen content in the gas gave rise to different equilibrium compositions in the bubble. These are shown in Table 3 and it is seen that the hydroxyl radicals generated reach a maximum concentration at around 10% oxygen content. Higher temperatures and higher oxygen contents favour formation of hydroxyl radicals. Thus, the hytures.

varies

Cont. Fig.

of Kl(%ww/v)

7. Comparison of model predictions and observed rates of iodine liberation under 0, atmosphere.

Table 3. Conditions of collapse for Ar-OZ mixture [concentration of KI = 16.6% (1 M)]

AT-O, (% V) 100-O ,98-2 96-4 94-6 92-8 90-10 88-12 80-20 70-30 60-40 50-50 40-60 30-70 20-80 10-90 0~100

Tr (K) 3172 3139 3107 3075 3044 3014 2954 2870 2739 2607 2499 2400 2304 2224 2146 2073

Pr (bar) 121.4 120.2 118.0 117.7 116.5 115.5 114.2 109.9 104.8 99.8 95.7 91.9 88.4 85.1 82.1 79.4

dH

(mol/mol

4.87 5.081 5.237 5.303 5.309 5.266 5.2041 4.6199 3.769 2.8754 2.3255 1.7102 1.3036 9.8426 7.428 5.575

H,O)

x lo-* X lo- * x lo- * X 10-z X 10-Z x lo-’ x lo-* x 1O-2 x 1O-2 x lo- 3 x IO-’ x lo-’ x lo-* x 1O-3 x 1O-3 X IO--’

887

Modelling of a batch sonochemical reactor

may need modifications for other systems. Thus, the present model provides a framework, which can be used as a starting point for developing more rigorous models. Acknowledgements-We acknowledge the contribution made by Mr. S. Maitra in the early stages of model formulation and the contribution made by Ms. M. Jhansi towards implementation of SOLGASMIX. We acknowledge Prof. A. G. Menon, ISU for permission to use the SOLGASMIX

program originally developed by Prof G. Ericksson. NOTATION

concentration

O2 composition

(*v)

Fig. 8 Comparisons of rates under Ar-OZ atmospheres.

droxyl radical content passes through a maximum at an intermediate composition. The present experimental results with an Ar-0, atmosphere are shown in Fig. 8. It is seen that the rate of iodine liberation passes through a maximum at around 10% oxygen, which agrees remarkably well with the composition where the maximum in dH generation is reached, as shown by the equilibrium calculations. The model predictions are also shown in Fig. 8. The agreement is satisfactory. It should be pointed out that this confirms the qualitative reasoning offered earlier by Hart and Henglein (1985). While doing similar experiments, they observed the peak to be at a composition of 30% 02-70% Ar. The difference in the composition where the peak is located is most probably due to the differences in equipment, frequency and the size of nuclei existing in their reactor. The predictions made for Ar-O2 atmospheres with R. = 2.5 ,um are also shown as a dashed line in Fig. 8. It can be seen that the predictions are still satisfactory. Even though results for other atmospheres and KI concentrations computed with R. = 2.5 pm are not shown, the trends are the same. It should be noted here that when R. is changed the extreme conditions reached in the bubble do change, but the nature of n adjusts to maintain the predictions relatively unaltered. CONCLUSIONS

The present work shows that, in spite of the uncertainties in the initial nuclei size and the heat and mass transport occurring during bubble collapse, it is possible to predict quantitatively the formation of Iz from a KI solution, when the solution is sonicated. The reasonable predictions made by the model also lend a degree of confidence in considering the bubble contents to be in thermodynamic equilibrium and the reactor contents to be well mixed. Although these assumptions have worked for the present system, they

CP G

ki n

P p2 P, P. pb Pf p*

P 90 P HZ0 pi PL

PLC

PS

P”

?i

R RCI R2

t T2 T/ xi

of ith species, gmol/l

moles of ith type of radicals/atoms in a single bubble, gmol specific heat at constant pressure, Cal/ gmol “C specific heat at constant volume, cal/ gmol “C i = 1, 2, 3, 4, 5, constants reaction rate l/gmol s number of bubbles collapsed per unit volume per unit time, l/l s number of moles of water vapour present in a single bubble at R2 pressure field, N/m’ pressure inside bubble when R = RI, N/m’ time-varying pressure field, N/m2 pressure amplitude of ultrasonic field, N/m2 atmospheric pressure, N/m” collapse pressure when R = R,, N/m2 partial pressure of the gas inside the bubble, N/m2 initial gas pressure inside the bubble, N/m2 vapour pressure of pure water at 298 K, N/m2 total pressure inside the bubble, N/m2 pressure in the liquid just at the bubble surface, N/m2 pressure corresponding to the radius of critical nucleus, N/m* saturation vapour pressure of the liquid, N/m2 partial pressure of vapour in the bubble, N/m2 ( = dC,/dt) rate of reaction, gmol/l s radius of the bubble at any time, m radius of the initial nucleus, m radius of the bubble at the beginning of the adiabatic collapse, m radius of critical nucleus, m collapse radius of the bubble, m ( = 8.314 J/mol K) universal gas constant surface velocity of the bubble, m/s acceleration of the surface of the bubble, m/s2 time, s temperature inside bubble when R = R,, K collapse temperature when R = R,, K moIe fraction of species i

D. V. PRASAD NAIDU etal.

888 Greek ; Y Pr. u d

letters isotropic coefficient stoichiometric coefficient ratio of specific heats density of the liquid, kg/m’ surface tension of the liquid, N/m osmotic coefficient for electrolytic solution

0

angular frequency, rad/s

Subscripts A B

O-H

c D

KI I2 H-0,

E

0

REFERENCES

Alippi, A., Cataldo, F. and Galbato, A., 1992, Cavitation in sonochemistry: decomposition of carbon tetrachioride in aqueous solution of potassium iodide. Ultrasonics 30, 148151. Arakeri, V. H. and Chakraborty, S., 1990, Studies towards potential use of ultrasonics in hydrodynamic cavitation control. Current Science 59, 13261333. Cum, G., Galli, G. and Gallo, R., 1992, Role of frequency in the ultrasonic activation of chemical reactions. Ultrasonics 30, 267-270. Eriksson, G., 1975, Thermodynamic studies of high temperature equilibria. XII SOLGASMIX, a computer program

for calculation of equilibrium composition in multiphase systems. Chem. Ser. 8, lM)-103. Flynn, H. G., 1964, Physics of acoustic cavitation in liquids, in Physicul Acoustics (Edited by W. P. Mason), pp. 57-172. Academic Press, New York. Gaitan, D. F., Crum, L. A., Church, C. C. and Roy, R. A., 1992, Sonoluminesccnce and bubble dynamics for a single, stable, cavitation bubble. J. acousf. Sot. Am. 91, 31663183. Hart. E. J. and Henelein. A.. 1985. Free radical and free atom reactions in the gonolysis of a&teous iodide and fonnate solutions. .I. phys. Chem. 89, 4342-4347. Hart, E. J. and Anbar, M., 1970, The Hydrared Electron, p. 230. Wiley, New York. Horvath. A. L.. 1985. A Handbook of Aaueous Electrolvte , Soluri& Ellis Horwood, Chichester. * JANAF Thermochemical Tables. 1986, 3rd Edition. American Chemical Society, New York. Lauterborn, W. and Bolle, H., 1975, Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72(2), 391-399. Ley, S. V. and Low, C. M. R., 1989, Ultrasound in Synthesis. Springer, New York. Lorimer. J. P. and Mason, T. J., 1987, Sonochemistry. Part l-the physical aspects. Chem. Sot. Rev. 16,239-274. Perry, J. H. (Editor), 1963, Chemical Engineers’ Handbook, 4th Edition, McGraw-Hill, New York. Plesset. M. S.. 1949. The dvnamics of cavitation bubbles. Trans. ASME., J. &l. M&h. 71, 277-282. Suslick, K. S., 1989, The chemical effects of ultrasound. Scient. Am. 260(2), 62-68. Suslick, K. S., 1990, Sonochemistry. Science 247, 1439-1445. Weissler, A., 1959, The formation of hydrogen peroxide by ultrasonic waves: free radicals. J. Am. them. Sot. 81, 1077-1085. Young, R. F., 1989, Cavitation. McGraw-Hill, New York.