Porous spherical catalyst particle

We start with an ideal, porous, spherical catalyst particle of radius R. The catalyst is isothermal and we consider a reaction involving a single reactant. Diffusion is described macroscopically by the first and second laws of Pick, stating that... 

Consider an isothermal porous spherical catalyst particle of radius R in which a single, irreversible first-order reaction takes place at steady state (Figure 2.11). Taking a spherical shell of thickness Ar at a radius r from the center, the steady-state component mass balance over a differential shell of volume 4nr Ar includes diffusion of reactant into and out of the control volume in the radial direction as well as reaction on the inner surface of the particle ... 

Figure 2.11 Reactant concentration and temperature profiles inside a porous spherical catalyst particle with exothermic reaction occurring at steady state. (Source Roberts [23]. Reproduced with permission of John Wiley Sons.)...

Figure 9-6 Profiles of the concentration of reactant A (Ca) and the temperature (7) inside a porous spherical catalyst particle in which an exothermic reaction is taking place at steady state. 7(0) is the temperature at the center of the particle (r = 0) and Ca(0) is the concentration at the center of the particle.

A model of the actual immobilization process with intact spherical catalyst particles was developed using the experimentally determined binding kinetics (48). The system was treated as a group of porous spheres suspended in a well-mixed solution of heparinase. The enzyme diffused through the porous network, where it reacted with the surface cyanate esters to produce the bound enzyme. 

Reaction rates inside a spherical isothermal porous catalyst particle depend on die rate of diffusion and the kinetics. When tiie reaction is fast compared to the diffusion, e.g. at higher temperatures, the reactants will be consumed near the surface and not aU the catalyst will be effectively used. The ratio of the actual reaction rate tiiroughout the particle to the reaction rate without any diffusion limitation is a good measure of how effectively the catalyst is used. This is called the effectivenesss factor, ri. Consider calculating the effectiveness factor for a spherical catalyst particle for a first-order reaction at the conditions specified below. The governing equation and boimdary conditions are given in the dimensionless form, and a first-order reaction is assumed ... 

Reactants must diffuse through the network of pores of a catalyst particle to reach the internal area, and the products must diffuse back. The optimum porosity of a catalyst particle is deterrnined by tradeoffs making the pores smaller increases the surface area and thereby increases the activity of the catalyst, but this gain is offset by the increased resistance to transport in the smaller pores increasing the pore volume to create larger pores for faster transport is compensated by a loss of physical strength. A simple quantitative development (46—48) follows for a first-order, isothermal, irreversible catalytic reaction in a spherical, porous catalyst particle. 

The porous nature of the catalyst particle gives rise to the possible development of significant gradients of both concentration and temperature across the particle, because of the resistance to diffusion of material and heat transfer, respectively. The situation is illustrated schematically in Figure 8.9 for a spherical or cylindrical (viewed end-on) particle of radius R. The gradients on the left represent those of cA, say, for A(g) + product(s), and those on the right are for temperature T the gradients in each case, however, are symmetrical with respect to the centerline axis of the particle. 

Figure 8.9 Concentration (cA) and temperature (Z ) gradients (schematic) in a porous catalyst particle (spherical or end-on cylindrical)...

The reactor, in which the gas phase will be virtually pure hydrogen, will operate under a pressure of 2 MN/m2 (20 bar). The catalyst will consist of porous spherical particles 3 mm in diameter, and the voidage, that is the fraction of bed occupied by gas plus liquid, will be 0.4. The diameter of the bed will be such that the superficial liquid velocity will be 0.002 m/s. The concentration of the aniline in the liquid feed will be 0.055 kmol/m3. 

Catalyst-supporting materials are used to immobilize catalysts and to eliminate separation processes. The reasons to use a catalyst support include (1) to increase the surface area of the catalyst so the reactant can contact the active species easily due to a higher per unit mass of active ingredients (2) to stabilize the catalyst against agglomeration and coalescence (fuse or unite), usually referred to as a thermal stabilization (3) to decrease the density of the catalyst and (4) to eliminate the separation of catalysts from products. Catalyst-supporting materials are frequently porous, which means that most of the active catalysts are located inside the physical boundary of the catalyst particles. These materials include granular, powder, colloidal, coprecipitated, extruded, pelleted, and spherical materials. Three solids widely used as catalyst supports are activated carbon, silica gel, and alumina ... 

The value of Do used to calculate the data in Table IV was obtained from direct measurement of the diffusion of hydrogen in the catalyst by the porous plug method (1, p. 189, method b). The value used was for the spherical beads of cogelled silica-alumina cracking catalyst used in the experiments to be reported here. The catalyst contained 10% AI2O3 by weight and had a surface area of 350 m.2/g. The value of the effective diffusivity of the catalyst particle for H2 at 27°C. (DHj) was found to be 7 X 10-3 cm.2/sec. The value of the effective diffusivity of the catalyst particle for cumene Dc, at reaction temperature was calculated from this measured hydrogen diffusivity by the equation... 

Effectiveness factors for a first-order reaction in a spherical, nonisothermal catalysts pellet. (Reprinted from R B. Weisz and J. S. Hicks, The Behavior of Porous Catalyst Particles in View of Internal Mass and Heat Diffusion Effects, Chem. Eng. Sci., 17 (1962) 265, copyright 1962, with permission from Elsevier Science.)... 

It is further assumed that B is not present in the feed oil, i.c. Cno =0 and that the reactions are taking place in spherical particles of porous material and are controlled by effective diffusion coefficients Da = Da = D. (The actual catalyst particles used in the experiments were short cylinders). 

Unfortunately, most enzymes do not obey simple Michaelis-Menten kinetics. Substrate and product inhibition, presence of more than one substrate and product, or coupled enzyme reactions in multi-enzyme systems require much more complicated rate equations. Gaseous or solid substrates or enzymes bound in immobilized cells need additional transport barriers to be taken into consideration. Instead of porous spherical particles, other geometries of catalyst particles can be apphed in stirred tanks, plug-flow reactors and others which need some modified treatment of diffusional restrictions and reaction technology. 

A bed of catalyst consisting of 200 g spherical egg-shell catalysts was employed in the fixed bed reactor. The catalyst bed was diluted by shattered steatite particles (0.9 mm < d < 1.6 mm) in a mass ratio 1 1 to obtain a plug flow system. The catalyst used throughout the study was prepared by coating spherical steatite particles of 4-5 mm diameter with a porous oxidic layer. The egg-shell catalyst contained 20 weight % active component, the thickness of the shell being 215 xm. The oxidic catalyst consisted mainly of Mo, V and Cu, its preparation has been described elsewhere [10]. 

In our efforts to verify deductively our modified capillary condensation theory we found two commercial porous bodies in which spherical elementary particles are arranged in a definite pattern. Therefore, if the radius of the elementary particles and their packing are given, a whole model of pore structure is clearly available for these specimens. By using these specimens as a catalyst or a catalyst carrier a series of investigations was carried out on catalytic activity in relation to the pore structure. 

This autoxidation property of carbons leads to a continuous loss of catalyst. When spherical carbon particles of 30 to 100 p,m were used in the oxidation with air of aqueous cyclohexanone solutions at 393 K in a trickle-bed reactor, a weak loss of carbon was observed after four weeks, and the originally smooth particles appeared rough and porous in scanning electron microscopy (SEM) [165]. The catalysts used with a nitrogen content of < 1% had been prepared from nitrogen-containing phenol-formaldehyde resin [166]. In this reaction cyclohexanone is oxidized to adipic acid and other dicarboxylic acids. 2-Hydroxycyclohexanone seems to be an intermediate. A carbon loss of several percent was also reported for other wet-air oxidation reactions of pollutants, mainly of phenol [167-169]. 

One of the-most familiar of catalytic chemicd reactors is the fixed-bed reactor. Here the catalyst particles remain in a fixed position and the reacting gases move through the catalyst bed. The catalyst particles are usually porous pellets, either cylindrical or spherical, ranging from in. or more in diameter. The ph3rsical structure of these pellets is usually such that the internal pore surface is infinitely greater than the actual pellet surface. Thus, the actual contact surface present is independent of pellet... 

The methods used for modeling-supported PTC systems are all based on the standard equations developed for porous catalysts in heterogeneous catalysis (Chapter 6). These are expressed in terms of an overall effectiveness factor that accounts both for the mass transfer resistances outside the supported catalyst particles (film diffusion resistance, expressed as a Biot number) and within them (intraparticle diffusional resistance, expressed in terms of a Thiele modnlns). Then, for any given solid shape, the catalytic effectiveness factor can be derived as a function of the Thiele modulus A. Thus, for a spherical support solid, we have... 

In order to obtain "good" fluidization in gas phase processes, spherical particles are preferred with a diameter in the range of 0.05 to 0.2 mm, with a relatively low density. Catalysts for catalytic cracking of petroleum (and a number of similar processes) are usually made of very porous silica- alumina particles with a diameter of about 0.075 mm. 

Figure 4.6. Schematic drawing of a spherical porous catalyst particle with radius Rp indicating concentration decreases going from the bulk (Co) to the surface (Cg) and then through the pores (C).

In the PDU, the 1.5 mm extrudate catalyst particles, which are cylinders 1.5 mm in diameter and 4.5 mm in length, are diluted with an equal volume of approximately spherical, non-porous inert particles, 1.0 mm in diameter. 

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