The Catalyst Pellet Equations

The six reactions described in (7.151) to (7.156) with their rate equations (7.157) to (7.162) take place in the catalyst pellet. Reactants and products are diffusing simultaneously through the pores of the catalyst. In our development of the diffusion-reaction model equations we make the following assumptions. 

Mass transfer through the catalyst pellet occurs by diffusion and only ordinary molecular and Knudsen20 diffusion are considered. 

The gases diffusing through the pellet obey the ideal gas law. 

Viscous flow is negligible and the reactor conditions are isobaric. 

The concentration profiles are symmetric around the center of the pellet. 

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Now let us consider the possibility that there will be a significant temperature difference between the bulk fluid and the external surface of the catalyst pellet. Equation 12.5.6 indicates that the temperature and concentration gradients external to the particle are related as follows ... 

For given values of the parameters we can compute the effectiveness factor r) as follows By integrating the bulk-phase IVPs (7.5) to (7.8) we find xAB, xBB, and yB. Then the catalyst pellet equations (7.9) to (7.11) allow us to find y, xA and xB from which we can find ij by the formula... 

This equation is a. nonlinear DE that we can integrate in order to find yB(l). Recall that at each point on the graph of yB we must also calculate r/ from the catalyst pellet equations (7.9) to (7.11) and the effectiveness factor equation (7.12). Recall that we can easily calculate xab and x u u from yB using the formulas... 

The catalyst pellet boundary value differential equations (7.172) and (7.173) can be solved via MATLAB using bvp4c or bvp4cf singhouseqr as practiced in Chapter 5. The reactor model DEs (7.166) and (7.180) to (7.182) can be solved via MATLAB s standard IVP solvers ode.. . The reactor model equations and the catalyst pellet equations used to compute the effectiveness factors rjj are all coupled. 

The solution of the bulk phase equations (equations 6.142, 6.144) together with the catalyst pellet equation (equations 6.131, 6.132), gives the concentration (or conversion) and temperature profiles along the length of the catalyst bed. 

The solution of the bulk gas phase equation cannot be performed separately from the catalyst pellet equations. The procedure is as follows ... 

Solve the catalyst pellet equations (non-linear two point boundary value differential equation) at the entrance of the reactor and evaluate the effectiveness factor. 

The effectiveness factors at each point along the length of the reactors are calculated for the key components methane and carbon dioxide, using the dusty gas model and simplified models I and II. The catalyst equations resulting from the use of the dusty gas model are complicated two-point boundary value differential equations and are solved by global orthogonal collocation technique (Villadsen and Michelsen, 1978 Kaza and Jackson, 1979). The solution of the catalyst pellet equations of the simplified models 1 and 2 at each point... 

The solution of the reactor model differential equations (6.225-6.227, 6.210) simulates the molar flow rates, pressure drop and energy balance of the reactor while the solution of the catalyst pellet equations (6.216-6.219) provide the effectiveness factors t/ s for equations (6.225-6.227). 

Eor strong diffusion resistance within the catalyst pellet, Equation 2.190 can be simplified to ... 

This is a much simpler expression than that obtained for the catalyst pellet (Equation 4.20e) but differs from if by being a function of the external gas-phase concentration Cg. We can use the fact to set lower limits to the pollutant concentration Cg, which can be efficiently removed by biofiltration (i.e., E 1). To exceed that threshold, we must have... 

Equation (15) is derived under the assumption that the amount of adsorbed component transferred by flow or diffusion of the solid phase may be neglected. This assumption is clearly justified in cases of fixed-bed operation, and it is believed to be permissible in many cases of slurries or fluidized beds, since the absolute amount of adsorbed component will probably be quite low due to its low diffusivity in the interior of the catalyst pellet. The assumption can, however, be waived by including in Eq. (15) the appropriate diffusive and convective terms. 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

Similarly, the experimental surface area may be equated to the surface area of pores as given by the model. Now, the mass of the catalyst pellet is the product of the pellet volume, Vp, and its density, Pp, so the experimentally determined surface area is Pp VpSg where Sg is the BET... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

By using the theory of multicomponent first-order reaction and diffusion (Wei, 1962), the specific conservation equation for A and B in the catalyst pellet is expressed as... 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

Derive the material and energy-balance design equations, including the equations of the catalyst pellets to calculate the effectiveness factor rj. 

Since the cooling jacket has cocurrent flow, the model consists of the set of four coupled initial value differential equations (7.5) to (7.8). Note that the first three DEs (7.5) to (7.7) contain the variable catalyst effectiveness factor rj. Thus there are other equations to be solved at each point along the length 0 < / < Lt of the reactor tube, namely the equations for the catalyst pellet s effectiveness factor rj. 

The Catalyst Pellet Design Equations, Calculating r] along the Length of the Reactor... 

Next we try to simplify the catalyst pellet design equations (7.9) to (7.11) which are used to calculate r/ for all points along the length of the reactor. 

The characteristic length is the thickness of the equivalent slab used in the single catalyst pellet equation and it is defined as the thickness lc of the catalyst slab that gives the same external surface to volume ratio as the original pellet. For Raschig16 rings this is given by... 

The differential equations (7.164), (7.165), (7.166), and (7.168) form a pseudohomogene-ous model of the fixed-bed catalytic reactor. More accurately, in this pseudohomogeneous model, the effectiveness factors rji are assumed to be constantly equal to 1 and thus they can be included within the rates of reaction ki. Such a model is not very rigorous. Because it includes the effects of diffusion and conduction empirically in the catalyst pellet, it cannot be used reliably for other units. 

Equation (7.172) is the generalized equation for the flux of component i diffusing through the catalyst pellet, subject to the boundary condition Ni(0) = 0 at the center z = 0. The mass-balance equations and the dusty gas model equations provide necessary equations for predicting the diffusion through the catalyst pellet. The dusty gas model equations are... 

The concentration and temperature profiles are calculated from the above non-linear equations using the Broyden quasi-Newton method. The effectiveness factors for the catalyst pellet may be expressed as... 

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