Diffusion with a first order reaction

Substituting the equation for —above into Equation (12-65), we form the differential equation describing diffusion with a first-order reaction in a catalyst bed ... 

Consider diffusion with a first order reaction in a semi-infinite plane ... 

Example 4.7. Unsteady State Diffusion with a First-Order Reaction... 

Consider diffusion with a first order isothermal reaction in a rectangular pellet.[ll] [8]. The governing equation and boundary conditions for concentration in dimensionless form are ... 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

Figure 4.10.17 Conversion in an ideal plug flow reactor (C= 1) and in a tubular reactor with laminar flow (negligible molecular diffusion) for a first-order reaction (Do = kr) approximations for laminar flow as given by Eq. (4.10.31) are also shown.

A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Pick s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, 1), the first order reaction rate constant k. the film thickness L and the concentration Cas of solute in a saturated solution. The reaction is initially carried our at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K7... 

FIGURE 8.1 Fraction unreacted versus dimensionless rate constant for a first-order reaction in various isothermal reactors. The case illustrated with diffusion is for = 0.1. 

Comparing this equation with Equation (8.34) shows that 3At/Y is the flat-plate counterpart of aAII - We thus seek a value for t/T below which diffusion has a negligible effect on the yield of a first-order reaction. 

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

According to equation 8.5-28, the nth-order surface reaction becomes a reaction for which the observed order is (n + l)/2. Thus, a zero-order surface reaction becomes one of order 1/2, a first-order reaction remains first-order, and second-order becomes order 3/2. This is the result if De is independent of concentration, as would be the case if Knud-sen diffusion predominated. If molecular diffusion predominates, for pure A, DecrcA, and the observed order becomes n/2,with corresponding results for particular orders of surface reaction (e.g., a first-order surface reaction is observed to have order 1/2). 

The rate of chemical attack will depend on the concentration according to the order of the reaction (i.e. in a zero-order reaction the rate is independent of concentration, in a first-order reaction the rate depends linearly on concentration, and in second-order reaction the rate depends on the square of concentration). Increasing the concentration, therefore, provides a means of acceleration. Remember, however, that chemical attack on plastics is a liquid-solid and not a liquid-liquid reaction, such that the reaction laws only hold if there is free movement of all chemical species with no limitations due to diffusion or transport and no barrier layers. Since this is rarely the case, temperature is preferred as a means of acceleration. 

A plug flow or tubular flow reactor is tubular in shape with a high length/diameter (1/d) ratio. In an ideal case (as in the case of an ideal gas, this only approached reality) flow is orderly with no axial diffusion and no difference in velocity of any members in the tube. Thus, the time a particular material remains within the tube is the same as that for any other material. We can derive relationships for such an ideal situation for a first-order reaction. One that relates extent of conversion with mean residence time, t, for free radical polymerizations is ... 

First consider a single cylindrical pore of length L, with reactant A diffusing into the pore, and reacting on the surface by a first-order reaction... 

Molar Volume Change. With decrease in fluid density (expansion) during reaction the increased outflow of molecules from the pores makes it harder for reactants to diffuse into the pore, hence lowering /. On the other hand, volumetric contraction results in a net molar flow into the pore, hence increasing For a first-order reaction Thiele (1939) found that this flow simply shifted the S versus Mj curve as shown in Fig. 18.5. 

Equation (171) turned out to be the same result that had been obtained using the simpler boundary conditions assuming no diffusion in the fore and after sections. In other words, the solution of Eq. (168) with the general boundary conditions gives the same result as with the simpler boundary conditions. Wehner and Wilhelm used their analytical solutions for a first order reaction to show that this indeed was true Eqs. (169)... 

The absorption of ozone from the gas occurred simultaneously with the reaction of the PAH inside the oil droplets. In order to prove that the mass transfer rates of ozone were not limiting in this case, the mass transfer gas/water was optimized and the influence of the mass transfer water/oil was studied by ozonating various oil/water-emulsions with defined oil droplet size distributions. No influence of the mean droplet diameter (1.2 15 pm) on the reaction rate of PAH was observed, consequently the chemical reaction was not controlled by mass transfer at the water/oil interface or diffusion inside the oil droplets. Therefore, a microkinetic description was possible by a first order reaction with regard to the PAH concentration (Kornmuller et al., 1997 a). The effects of pH variation and addition of scavengers indicated a selective direct reaction mechanism of PAH inside the oil droplets... 

The pure compound rate constants were measured with 20-28 mesh catalyst particles and reflect intrinsic rates (—i.e., rates free from diffusion effects). Estimated pore diffusion thresholds are shown for 1/8-inch and 1/16-inch catalyst sizes. These curves show the approximate reaction rate constants above which pore diffusion effects may be observed for these two catalyst sizes. These thresholds were calculated using pore diffusion theory for first-order reactions (18). Effective diffusivities were estimated using the Wilke-Chang correlation (19) and applying a tortuosity of 4.0. The pure compound data were obtained by G. E. Langlois and co-workers in our laboratories. Product yields and suggested reaction mechanisms for hydrocracking many of these compounds have been published elsewhere (20-25). 

Newson (1975) was among the first to develop a pore plugging model of demetallation to predict catalyst life. By using the pore structure model of Wheeler (1951), the pellet was assumed to have N pores of identical length but with a specified distribution of pore radii. Metal deposition was assumed to be a first-order reaction over an outer fraction of the pore length and to have a uniform thickness. This model showed that the broadness of the size distribution had little effect on the catalyst life for the same average radii, but that increasing the radii from 45 to 65 A more than doubled the catalyst life. The restricted form of the diffusivity (see Section IV,B,5) was not employed in this model. 

For reactions with a first-order (or pseudo-first-order) rate constant of < 0.01s 1, little reaction will occur in the diffusion film. 

Here, we are concerned with intraparticle diffusion controlling a first-order readsorption reaction therefore, the Damkohler number (15), which reflects the ratio of reaction to diffusion rates within catalyst particles, is given by ... 

Example 9.3 Effectiveness factor for first-order irreversible reaction-diffusion system Consider a first-order reaction occurring on the pore walls of a catalyst with equimolar counter diffusion. Assume that isothermal conditions are maintained, and a catalyst with simple slab geometry is used (Figure 9.1). If the -coordinate is oriented from the centerline to the surface, the steady-state reaction diffusion equation for reaction A — B between reactant A and product B is... 

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.)... 

Diffusion factor y in cylindrical catalyst mass. The case of cylindrical channel filled with catalytic surfaces having one entrance for the reactant leads for a first-order reaction to... 

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