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POLYMERIZATION OF SILICIC ACID: CATALYTIC EFFECT OF FLUORIDE BY R. Iainingfluoride at pH 1.5 to 1.6.
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0 3 MILLIMOLAR AI203 FROM COMMERCIAL S I L I C A T E
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Fig. 5.-Effec,t of pH and fluoride ion concentration on sols containing 0.3 millmole per liter of A1203.
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Fig. S.--Effect of trace amounts of fluoride and aluminum on I .O molar Si02 sols made from pure sodium silicate.
June, 1952
MATHRMATICS OF ADSORPTIONIN BEDS
aluminum ion was examined a t lower fluoride concentrations, as shown in Fig. 7. This was carried out a t constant p H 1.5 to 1.6, where the effects are most readily noted. I t will IIC noted that the limit H represents the gel-time as catalyzed by hydroxyl ion alone. Thus, for a given concentration of fluoride ion, there is required a certain concentration of alriininum ion to inactivate the fluoridc. Extrapolation of tlw curve for 5.7 millimolar fluoride, to thc liniit H, givcs an A1,OJ concentration of 30 millimolar. This would correspond to about 5 moles of A1203 per fluoride equivalent, or an atomic ratio of A1:F = 10: 1 , a t this pH, in order to inhibit completely the catalytic effect of the fluoride ion. In the case of sols made from pure sodium silicate, the effect of trace amount,s of fluoride and aluminum ions is still more marked. I n Fig. 8, it is shown that a t pH 1.5, 1.8 millimoles of A1203 inhibit the catalytic effect of 0.1 millimole of fluoride.
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683
I n the absence of aluminum ions, traces of fluoride greatly accelerate gelling. Comparison of l>hesols from the two types of silicates cannot be made, however, because additional unknown trace impurities appear to lay a role. For example, a sol was made up from the purifiexsilicate to which aluminurn and fluoride was added to concentrations of 0.3 and 0.7 millimolar, corresponding t,o thc aluminum and fluoride contents of the sol made froin commercial silicate. However, the sol from commercial silicate gelled in around 75 hours at p H 1.5, while the sol from the “pure” silicate, with AI and F added, gelled in 30 hours. Additional unknown stabilizers are probably present in the commercial silicate.
Acknowledgment.-The author is grateful to Ralph E. Lawrence for analyses carried out in connection with this investigation.
MATHEMATICS OF ADSORPTION I N BEDS. T7. EFFECT OF INTRAPARTICLE DIFFUSION I N FLOW SYSTEMS I N FIXED BEDS BY PAUL R. KASTEN,LEONLAPIDUSAND NEALR. AMUNDSON Departntenl of Chemical Engineering, University of filinnesota, Minneapolis 14, Alinnesota Received J u l y 9, 1961
Equations have been derived for heat and mass transfer operations in fixed beds of solids through the interstices of which a fluid flows. The effect of intraparticle diffusion or conduction is considered as well as the resistance to transfer at the particle surface. This paper contains essentially a generalization of the results of Anselius,’ Schumann,*N u s ~ e l tWicke4 ,~ and Hougen and Marshall.6 The use of the derived equations may be limited because of the restrictive assumption of a linear isotherm for adsorption. The case of heat transfer should be more readily amenable to experimental verification. Although the functions involved in the solution are only exponential and Bessel the form of the solution may seem prohibitively complicated. However, the presence of the exponential factors should produce such rapid convergence that not more than a few terms should be needed in a calculation.
Introduction The problem of the transfer of heat or mass from a fluid to a granular bed through which it is flowing has been the subject of a voluminous literature both from the experimental and mathematical viewpoints. Early work in the field was concentrated on heat exchange because of the importance of heat regenerators, blast furnaces, and similar operations. In recent years the problem received further impetus by the widespread industrial use of selective adsorption schemes for difficult separations and of fixed and moving bed catalytic reactors, and further by the interest of chemists in the theory of chromatography. In general it has been assumed that the rate-determining step in these processes is the rate of mass or heat transfer from the fluid to the solid surface of the packed bed. In a few other cases, a.s in chromatography, it has been assumed that the flow of fluid through the bed is so slow there is either, virtually, an equilibrium established between the fluid and the solid a t each point of the bed or the rate determining step is the kinetic process of adsorption itself. These researches all have the common defect of neglecting the effect of intraparticle diffusion or conduction. This assumption is probably valid for very small particles but the experimental evidence is inconclusive. Two recent (1) A. Anzelius, Z. angew. Math. Mech., 6, 291 (1926). (2) T.E. W. Schumann, J. Franklin J n s t . , 208, 405 (1929). (3) W. Nusselt, Z. Ver. deut. Inn., 56, 2021 (1911); Tech. Mech. Theirno., 1, 417 (1930). (4) E. Wicke, Kolloid Z., 121 93, 129 (1940). ( 5 ) 0..4. Horigen and JV. R. Marshall, Ckem. Eng. Progress, 43, 197 ( 1W i ) ,
studies by Foster and Daniels6 and Eagleton’ have shown that in the cases of adsorption of nitrogen dioxide on silica gel and of water on activated alumina the diffusion of adsorbate inside the particle may be the rate-determining step. With these facts in mind the authors sought to to derive equations which would describe the pheiiomenon of heat or mass transfer from a fixed bed of solid to a flowing fluid stream. The discussion will be confined to adsorption although it will be shown what modifications need be made for heat transfer. It is assumed that the packed bed of solids consists of porous spheres of uniform diameter. The solid has adsorbent properties and therefore adsorbs solute from the stream of fluid flowing through the interstices. The solute must be transferred from the bulk of the fluid in the void volume among the particles by turbulence to the bounding film of fluid about a sphere. The solute diffuses through the film and also through the void volume of the particle and is then adsorbed on the internal surface of the adsorbent. Hence it is obvious that there are many rate processes occurring and that these occur in series. Which one, or ones, determine the overall rate depends probably on the system being investigated, and it would be more than coincidence if intraparticle diffusion never played a role. If one knows the initial condition of the system, along with flow rates, particle size, method of packing, adsorbent properties of the solid, etc., it wohld be desirable to have formulas which would predict (6) E. G . Foster and F. Daniels, Ind. Eng. Chem., 45, 986 (1951). (7) L. C. Eagleton, private communication.
