Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
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A Dual-Series Solution for the Effectiveness Factor of Partially Wetted Catalysts in Trickle-Bed Reactors Patrick L. Mills and Milorad P. Dudukovie" Chemical Reaction Engineering Laboratory, Department of Chemical Engineering, Washington University, St. Louis, Missouri 63 130
A model is formulated which accounts for the effect of partial external wetting on the catalyst effectiveness factor in trickle-beds for an isothermal irreversible, first-order reaction with respect to the nonvolatile liquid reactant. The boundary conditions give rise to an unconventional type of dual-series equations which are solved by a novel application of the least-square method to yield the catalyst effectiveness for slab, cylindrical, and spherical shapes. The combined effects of fractional pore filling and partial external wetting on the catalyst effectiveness for slab geometry is also briefly examined. The computed results allow a better understanding of the effect of particle scale incomplete contacting on trickle-bed performance. The utility of the developed mathematical method for other situations arising in trickle-bed reactors is outlined. The method is also applicable to a number of mixed-boundary value problems arising in chemical engineering.
Introduction Proper interpretation of global rate data obtained from laboratory scale, pilot plant, or commercial size trickle-bed reactors is perplexing due to the unavailability of sound theory and correlations which allow an assessment of the various rate limitations often present in this type of reactor. Of particular importance is the need to ascertain the magnitude of the diffusional intrusion which is present in the case of porous catalysts and to determine the effect of incomplete catalyst, wetting on the catalyst effectiveness factor (Mears, 1974; DudukoviE, 1977; DudukoviE and Mills, 1978; Ramachandran and Smith, 1978). One may distinguish, in principle, between reactor scale and particle scale incomplete contacting in trickle beds. Any gross liquid flow maldistribution resulting from poor distributor design, unusual reactor height-to-diameter ratio, or reactor-to-particle diameter ratio may cause sections of the packing to be almost completely dry giving rise to reactor scale incomplete contacting. However, there exists strong experimental evidence that a t certain conditions catalyst pellets are incompletely wetted in a reactor consisting of a single string of spheres (Davidson et al., 1959; Satterfield et al., 1969), as well as in a packed column (Onda et al., 1967) even when perfect global liquid distribution is achieved. In such situations of particle scale incomplete contacting the usual expressions for the catalyst effectiveness factor are invalid. The catalyst utilization is then clearly a function of particle scale incomplete wetting, reaction type, liquid reactant volatility, and solubility of the gaseous reactants in the reacting liquid species. As a consequence of the preceding complications the assignment of a catalyst effectiveness factor to date has essentially been performed incorrectly since the standard available expressions for the effectiveness factor as a function of the Thiele modulus were used. One objective of this study was to assess the effects of incomplete external catalyst wetting and fractional pore filling on the effectiveness factor in trickle-bed reactors. This evaluation allowed a better understanding of the complex phenomena involved and offered a clear quantitative explanation of the effects of particle scale partial wetting on catalyst utilization. This investigation also allowed a comparison of previously proposed approximate expressions for the effectiveness factor (Mears, 1974; DudukoviE, 1977) to those of a more rigorous, exact solution. This suggested 0019-7874/79/1018-0139$01 .OO/O
certain restrictions which should be observed when utilizing the previous expressions. For reasons of simplicity, only a case of first-order reaction with respect to a nonvolatile liquid reactant is treated. This situation is more applicable to reactions typically encountered in petroleum processing operations where it can be assumed that the gaseous reactant is present in large excess. Recently, Ramachandran and Smith (1978) presented the solution for the case of first-order reaction with respect to the gas reactant. The mathematical method developed in this paper can be successfully used to compute the effectiveness factor in such situations also. The other objective of this study was to develop an efficient and reasonably accurate computational method for the solution of dual series arising in mixed-boundary value problems of the tl e treated in this paper. The method developed is genei d and can be successfully used to solve other mixed-boundary value problems in chemical engineering (Mills and DudukoviE, 1979). General Equations The general equation describing diffusion and reaction of a liquid reactant in a porous catalyst pellet is given by the following dimensionless form for an isothermal, irreversible first-order reaction confined to the liquid filled region of the pellet C2u - a 2 u = O
in
V,
(1)
At the boundary separating the wetted interior region of the pellet from the actively wetted exterior the mass transfer rate across the boundary layer of the liquid to the solid surface must equal the diffusion transport into the pores of the pellet. This requires that the boundary condition -1_ au + u = 1 onaVw Bi, au be satisfied a t the actively wetted liquid-solid interface. The remaining boundary of the wetted portion of the pellet separates the wetted region of the pellet from the dry interior and dry exterior. Therefore, the following condition should be satisfied everywhere on this boundary -1_au + u = e onaVD (3) Bid au 0 1979
American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
The explicit appearance of the parameter e in eq 3 indicates that there may exist a finite but small concentration of liquid reactant in the gas phase surrounding the dry portions of the pellet due possibly to volatilization of the liquid reactant. In hydrodesulfurization or hydrodenitrogenation reactions, for example, the degree of volatilization is usually not significant (Henry and Gilbert, 1973). The assumption that the kinetic rate can be described adequately by one which is first order with respect to liquid reactant is suggested by Satterfield (1975) for many reactions typical of the petroleum refining industry. This is supported by the experimental evidence of Frye and Mosby (1967),who examined the hydrodesulfurization of selected sulfur compounds. Other studies, however, have indicated that when a range of reactivities for the sulfur containing species exists that second-order kinetics prevails after lumping is performed (Weekman, 1969). An alternative approach was recently suggested by Yitzhaki and Aharoni (1977). It was shown that the summation of the first-order reactions gives an observed higher reaction that is best modeled by parallel reaction kinetics rather than second-order expressions. For simplicity reasons, the second-order expression was not considered. I t should be pointed out that eq 1-3 also describe a partially externally wetted pellet with diffusion and first-order reaction with respect to the gaseous reactant when liquid totally fills the interior as treated by Ramachandran and Smith (1978). The parameter Biwthen contains the gas-liquid and liquid-solid mass transfer coefficients while Bid is dependent on the gas-solid mass transfer coefficient. By addition of another equation V2u - J2u = O in v d (la) one could describe the problem of the reaction occurring in both liquid and gas phases and thus treat the case of a volatile liquid reactant. This is the situation that frequently arises in hydrogenation of chemicals (Sedricks and Kenny, 1973; Satterfield and Ozel, 1973; Germain et al., 1973). However, the phenomenon of partial liquid pore fill-up under reaction conditions is not well understood a t present in spite of the excellent pioneering work of Kesten and Sangiovanni (1971) on the dynamics of such a situation and Rony (1968) in the related area of supported liquid phase catalysis. From the previous discussion it is apparent that the boundary conditions specified earlier, although general in nature, may be simplified in the case of reactions typical of the petroleum industry. In this case the reactants are usually relatively nonvolatile at the operating temperatures and pressures. Of equal significance is that operations are usually in the regime free of major mass transfer limitations. Thus for a typical petroleum industry reaction the Robin conditions eq 2 and 3 may be simplified to the Dirichlet and Neumann conditions u =1 onaVw (4) and au - = 0 onaVD (5) av The quantity of interest is the catalyst effectiveness which is defined to be the ratio of the diffusion hindered rate in a partially liquid filled, incompletely externally wetted catalyst to that rate which would exist if the pellet were completely wetted both on the interior and exterior with reactant whose concentration was identical with that in the bulk
01 0
Figure 1. Illustration of the partial externally wetted slab catalyst.
where V is defined to contain both wetted and nonwetted regions. The Porous Slab Before proceeding to the more practical catalyst geometries commonly utilized in trickle-bed operations, it is instructive to consider the effect of partial external wetting on the effectiveness factor for a slab catalyst pellet whose pore filling is assumed to be complete (Figure 1). Such is expected to be the case for hydrocarbon feeds which in principle readily fill the interior pore structure. It is assumed that the pellet is finite in two dimensions and infinite in the third. The lines of symmetry are a t Ez = 0 and t1 = 1. At the outside surface t1 = 0 the slab is wetted for 0 5 E2 < Eo and dry for to< E2 I1. The surface t2= 1 is dry also. The mass balance of liquid reactant eq 1 now takes the form
where t1 and t2 are dimensionless n and y coordinates, respectively. It is further assumed that the reactant is nonvolatile and zero concentration gradient of reactant exists at all the dry boundaries and lines of symmetry. In the absence of mass transfer limitations, the Dirichlet type boundary condition eq 4 can be applied a t the wetted boundary. The complete set of boundary conditions is presented as t1=o u = l OItz
