78
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
istence of an optimum sound intensity above which the rate of reaction decreases rather than increasing. As indicated in eq 9, AK increases with the energy t released by the collapse of the cavitation bubbles. A bubble will collapse if the collapse time is equal to or shorter than one-half the period of oscillation of the sound wave. With an increase of sound intensity or applied voltage to the sound transducer, the cavitation bubble size becomes larger, increasing the collapse time and reducing the chance of collapse at a fixed frequency. A reduction in the number of bubbles that collapse will result in a lesser amount of energy release. On the other hand, larger drops release greater energy after their collapse; the released energy is therefore optimum a t the applied voltage which gives the largest bubble size with collapse time equal to one-half the period of sound oscillation. Equation 14 is considered valid when V is within the voltage corresponding to the optimum energy release.
Nomenclature
C, C1 = dimensionless constants C2 = constant in eq 7 C3 = constant in eq 9 C4 = constant in eq 14 D = molecular diffusion coefficient, cm2 s-l De = eddy diffusion coefficient, cm2 s-l E = energy released through cavitation k = wave number kB = Batchelor characteristic wave number kK = Kolmogoroff characteristic wave number K'R = reaction rate constant in cavitation fields KR = specific rate constant, mo1-ls-l k 1 = constant in eq 1 AK = increase in specific rate constant Nsc = Schmidt number, u/D
rl, r2 = radius of reacting ions R = radius of the sphere of diffusion Ti = period of an eddy, s U = mean eddy velocity, cm s-l U , = eddy velocity, cm s-l V = transducer voltage, V V , = transducer voltage just before the onset of cavitation Greek Letters t
= energy dissipation rate, cm2
u
= kinematic viscosity, cm2 s-l
X = eddy length, cm
Literature Cited Barren, E., Porter, C., J. Am. Chem. Soc., 63, 3434 (1941). Batchelor, G. K., J. Fluid Mech., 5 , 113 (1959). Benson, D., "Mechanisms of Inorganic Reactkms in SduUofts", p 204, W a w H i Y , London, 1968. Chen, J. W., Kalback, W. M., Ind. Eng. Chem. Fundam., 6 , 175 (1967). Couppis, E. C., Klinzing, G. E., A1Ch.E J., 20, 485 (1974). Fogler, S . . Barnes, D., Ind. Eng. Chem. Fundam., 7 , 222 (1968). Hinze, J. O., "Turbulence", p 165, McGraw-Hill, New York, N.Y., 1959. Levich, V. G., "F'hysim-chemical Hydrodynamics", p 215 RenticeHall, Englewocd Cliffs, N.J., 1962. Moelwyn-Hughes, E. A., "The Chemical Statics and Kinetics of Solutions", p 99, Academic Press, New York, N.Y., 1971. Plesset, M. S., "Bubble Dynamics and Cavitation Erosion", in L. Bjorno, Ed., "Proc. Symp. Finite Amplitude Wave effects in Fluids, Copenhagen", IPC Science and Technology Press, Surrey, England, 1971. Shinnar, R., Church, J. M., Ind. Eng. Chem., 52, 523 (1960). Sirotyuk, M. G., "Energy b h n c e of Sound Field in Cavitation" in L. D. Rozenberg, Ed., "High Intensity Ultrasonic Fields", Plenum Press, New York, N.Y., 1971.
Department of Chemical Engineering T h e Polytechnic of Wales Pontypridd, Mid Glamorgan CF37 1D L Wales, United Kingdom
M. Seraj-ud Doulah
Received for review August 15, 1977 Accepted September 29, 1978
CORRESPONDENCE Correspondence on Analysis of Gas Flow in Fluidized Bed Reactors negative gas flow and then duplicate Potter's equations without even introducing the concept of bubble wake and solid circulation! This shows the immense power of contradiction. Allow one and you can prove anything.
Equation 1of Bukur's paper 11978) assumes that all gas in excess of minimum fluidization goes through the fluidized bed as bubbles alone, while eq 2 assumes that these fast rising bubbles are accompanied by clouds which take up some of the dense phase. These represent two distinctly different physical pictures which if used together lead to a logical contradiction. Potter et al. and Kunii and Levenspiel explained gas reversal in fluidized beds as a consequence of bubble wake action and solid circulation. But by cleverly manipulating this one contradiction Bukur was able to come up with
Department of Chemical Engineering Oregon State University Corvallis, Oregon 97331
Professor Levenspiel's correspondence which concerns my recent paper actually represents a critique of Partridge and Rowe's (196613) model. Equations 1 and 2, as well as eq 3 , 7 , and 10 of my paper, are the basic equations of this model and have been used by many investigators up to the present time. Professor Levenspiel claims now that Partridge and Rowe's model is based on a logical contradiction, which is embodied in eq 1 and 2.
Equation 1 does not assume that all gas in excess of minimum fluidization goes through the bed as bubbles alone as stated by Professor Levenspiel. It represents a general statement which gives the physical description of the bed, and it says that the total gas flow is divided between the bubbles a n d associated clouds and the interstitial gas. It is essentially a material balance equation and no assumptions are made about the actual gas flow
Literature Cited Bukur, D.
B.,Ind.
Eng. Chem. Fundam.. 17, 120 (1978).
Octave Levenspiel
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
rates. Equation 2 specifies the flow rate of gas which passes through the bed in form of bubbles alone (visible bubble flow rate). This expression was derived by Partridge and Rowe (1966,a),and their analysis produced the result in agreement with one of the basic postulates of the simple two-phase theory regarding the bubble flow rate. However, in the simple two-phase theory one doesn’t take into account the existence of bubble clouds as is done in Partridge and Rowe’s model (1966a,b). There are no logical contradictions in this development, and Partridge and Rowe are not using two distinctly different pictures of the bed. However, there is some evidence, both theoretical and experimental, that eq 2 tends to overestimate the visible bubble flow rate. A theoretical analysis of Lockett e t al. (1967) predicts smal1e.r values of GB than eq 2 . Experimental measurements of Grace and Harrison (1969), Geldart (1971), and Chavarie and Grace (1975a,b) indicate that the visible bubble flow rate is smaller than eq 2 predicts. Equation 20 of my paper, which is proposed for
79
the visible bubble flow rate, is in qualitative agreement with these experimental results. It seems that Professor Levenspiel is not familiar with Potter’s analysis (1971) of Partridge and Rowe’s model, which was cited in my paper. Potter obtained eq 9 of my paper, with f , being substituted by eq 10 without introducing concepts of bubble wake and solid circulation.
Literature Cited Chavarie, C., Grace, J. R., Ind. Eng. Chem. Fundam., 14, 75 (1975a). Chavarie, C., Grace, J. R., Ind. Eng. Chem. Fundarn., 14, 79 (1975b). Geldart, D., Ph.D. Dissertation, University of Bradford, U.K., 1971. Grace, J. R., Harrison, D., Chem. Eng. Sci., 24, 497 (1969). Lockett, J. J., Davidson, J. F., Harrison, D., Chem. Eng. Sci.. 22 1059 (1967). Partridge, B. A., Rowe, P. N., Trans. Inst. Chem. Eng., 44, T349 (1966a). Partridge, B. A., Rowe, P. N., Trans. Inst. Chem. Eng., 44, T335 (1966b). Potter, 0. E., “Fluidization”, J. F. Davidson and D. Harrison, Ed., 332-333, Academic Press, London, 1971
Faculty of Technology University of Novi Sad 21000 Novi Sad, Yugoslavia
CORRECTION
In the Correspondence, “Catalytic Oxidation of Phenol in Aqueous Solution over Copper Oxide”, by A. I. Njiribeako, R. R. Hudgins, and P. L. Silveston [Ind.Eng. Chem. Fundam., 17, 234 (1978)], the reply by Professor Katzer is correct: the value of 148 g/L is a thousandfold too high. It should be 148 mg/L.
Dragomir B. Bukur
