76
Ind. Eng. Chem. Fundam., Vol.
18, No. 1, 1979
COMMUNICATIONS A Diffusion Model to Explain the Increase of Reaction Rate in Ultrasonic Cavitation An explanation for the observed increase in rates of ionic reactions occurring in ultrasonic cavitation is suggested on the basis of an eddy diffusion mechanism, and evidence to support the view is presented.
Introduction Application of high intensity ultrasounds to reacting systems is found to increase the specific rate constant in some systems. Although this effect of ultrasounds is attributed to a phenomenon known as cavitation which implies nucleation, growth, and subsequent collapse of gas bubbles or cavities by the action of sound waves, the precise role of cavitation in increasing the reaction constant is somewhat obscure. Fogler and Barnes (1968) have attributed the cause for the increase of reaction rate in the hydrolysis of methyl acetate to the high temperature reached within cavitation bubbles. The same system has also been studied by Chen and Kalback (1967) and Couppis and Klinzing (1974);they believe that the increase in the reaction rate is due to an increased frequency of collisions between ions caused by cavitation pressure gradient and temperature rise. In addition to high temperature and pressure rise within cavitation bubbles, cavitation in ultrasonic fields transforms a fairly large amount of acoustic energy into shock wave energy through the collapse of millions of tiny bubbles. The shock wave energy dissipates in the medium and in the process of energy dissipation the shock waves could influence the diffusion coefficient of reactants to increase the rate constant in diffusion-controlled systems. The acid-catalyzed hydrolysis in the methyl acetatewater system is a fast reaction controlled by the diffusion (Benson, 1968) of reacting ions in the solvent. It is therefore most likely that the rate constants in this and other similar systems are enhanced by ultrasonic cavitation because of its effect on the diffusion process. This hypothesis is detailed and supported by the published results in this communication. R a t e Constant i n Diffusion Controlled Reactions The specific rate constant of diffusion-controlled reactions can be obtained from the Debye equation (Moelwyn-Hughes, 1971) which, for dilute solutions with constant integral charges on ions and at a constant temperature, may be written in the form KR = k'R (Dl + D2) (1) in which D1 and D2 are molecular diffusion coefficients of reacting ions with radii rl and r2, so that R = rl + r2. In the presence of a flow situation the diffusion coefficient due to the flow can be added with the molecular values in eq 1 to obtain the true rate constant provided characteristics of the flow satisfy certain conditions. For an isotropic turbulent flow Batchelor (1959) has deduced a characteristic wave number 114
kB=($)
0019-7874/79/1018-0076$01.00/0
which sets the upper limit above which the eddy diffusion of scalar quantities by small scale eddies becomes weaker than molecular diffusion. In a flow with Nsc >> 1 this characteristic wave number is larger than the Kolmogoroff wave number, defined by kK
=
(
:)'I4
(3)
Corresponding to this number the eddy length scale is l / k K . Since the eddy length scales in the range of wave number smaller than k K are usually very large compared with most reacting ions in solutions the diffusion effect of the eddies below wave number hK should have little influence on the collision frequency of ions. The effect of eddy diffusion on the rate constant of reacting ions is therefore expected to fall in the wave number range hK
