Pellet equations

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 ... 

If intraparticle diffusion effects are important and an effectiveness factor ij is employed (as in equation 3.8) to correct the chemical kinetics observed in the absence of transport effects, then it is necessary to adopt a stepwise procedure for solution. First the pellet equations (such as 3.10) are solved in order to calculate t) for the entrance to the reactor and then the reactor equation (3.87) may be solved in finite difference form, thus providing a new value of Y at the next increment along the reactor. The whole procedure may then be repeated at successive increments along the reactor. 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

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 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 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 pellet equations are coupled to the following material and energy balances for the flowing gas... 

Thus for < = 0 the ring-shaped catalyst pellet becomes a cylindrical catalyst pellet. Equation 6.41 is illustrated in Figure 6.10. In this diagram 1 is plotted versus A lines with a constant geometry factor T are drawn. The four comers of the diagram represent... 

These single-pellet equations are then coupled to convective flow equations for olefins [Eq. 13] and paraffins [Eq. (14)] within interpellet voids in order to describe the overall reactor. 

The collocation methods can be shown to give rise to symmetric, positive definite coefficient matrices that is characterized with a acceptable condition number for diffusion dominated problems or other higher order even derivative terms. For convection dominated problems the collocation method produces non-symmetric coefficient matrices that are not positive definite and characterized with a large condition number. This method is thus frequently employed in reactor engineering solving problems containing second order derivatives of smooth functions. A t3q)ical example is the pellet equations in heterogeneous dispersion models. 

An example would be the dehydration of ethanol to ethylene and its dehydrogenation to acetaldehyde. If both reactions are first order, selectivity is unaffected by internal mass transport the ratio of the rates of reactions, 1 and 2 is k jkj at any position within the pellet. Equation (11-89) cannot be applied separately to the two reactions because of the common reactant A. The development of the effectiveness-factor function would require writing a differential equation analogous to Eq. (11-45) for the total consumption of A by both reactions. Hence k in Eq. (11-89) would be k- + k2 and Fp would be (Tp) -1- (rp)2- Such a development would shed no light on selectivity. 

Once the rate of reaction for the individual grain is known, the overall pellet equation can be written ... 

The particle-pellet equation is written in terms of a generalized effectiveness factors defined as... 

The external mass transfer in the pellet equation takes the form... 

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). 

For the simple unimolecular irreversible first order reaction and for spherical pellets, equation (5.97) can be written as. 

With regard to the incorporation of the catalyst pellet diffusional effects on the reactor behaviour, the pellet equation can be incorporated into the overall model equations without the introduction of the effectiveness factor concept discussed in chapter 5, or the effectiveness factor can be incorporated expressing the totality of the diffusional effects on the rate of reaction in the form of one number. The use of the effectiveness number concept taxes the simulation with some extra computational time, however it gives deeper insight into the system and the effect of diffusional resistances on the performance of the reactor. 

The details of the pellet equations are given in chapter 5, sections... 

Reactor design in the presence of internal diffusion involves simultaneous solution of the reactor and pellet equations (see Chapter 12). These are given in Table 20.9 along with the boundary conditions in both dimensional and nondimensional forms. 

Pellet equation for estimating e in the reactor equations (spherical pellet)... 

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

This equation allows us to determine the exit concentration of the jth cell, c (j), from the known inlet concentration c (j-l) and the solution of the single-pellet equations. It should be noted that the accumulation of mass in the interpellet space was neglected in Eq. (13), since its time constant is typically 80 times smaller than that of the intrapellet transients. The external mass transfer coefficient of CO, k, of Eqs. (7) and (13) was estimated from the de Acetis-Thodos correlation, as given in Smith (Ji). 

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... 

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