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Unformatted text preview: 7.6 Multiple Reactions 395 sumed, the increase in the rate constant is more significant and the product r = kc A is larger inside the pellet. Because the effectiveness factor compares the actual rate in the pellet to the rate at the surface conditions, it is possible for the effectiveness factor to exceed unity in a nonisothermal pellet, which we see in Figure 7.19. A second striking feature of the nonisothermal pellet is that multi- ple steady states are possible. Consider the case Φ = . 01, β = . 4 and γ = 30 shown in Figure 7.19. The effectiveness factor has three possi- ble values for this case. We show in Figures 7.20 and 7.21 the solution to Equation 7.74 for this case. The three temperature and concentra- tion profiles correspond to an ignited steady state (C), an extinguished steady state (A), and an unstable intermediate steady state (B). As we showed in Chapter 6, whether we achieve the ignited or extinguished steady state in the pellet depends on how the reactor is started. Aris provides further discussion of these cases and shows that many steady- state solutions are possible in some cases [3, p. 51]. For realistic values of the catalyst thermal conductivity, however, the pellet can often be considered isothermal and the energy balance can be neglected . Multiple steady-state solutions in the particle may still occur in prac- tice, however, if there is a large external heat transfer resistance. 7.6 Multiple Reactions As the next step up in complexity, we consider the case of multiple reactions. Some analytical solutions are available for simple cases with multiple reactions, and Aris provides a comprehensive list , but the scope of these is limited. We focus on numerical computation as a general method for these problems. Indeed, we find that even numeri- cal solution of some of these problems is challenging for two reasons. First, steep concentration profiles often occur for realistic parameter values, and we wish to compute these profiles accurately. It is not un- usual for species concentrations to change by 10 orders of magnitude within the pellet for realistic reaction and diffusion rates. Second, we are solving boundary-value problems because the boundary conditions are provided at the center and exterior surface of the pellet. Boundary- value problems (BVPs) are generally much more difficult to solve than initial-value problems (IVPs). A detailed description of numerical methods for this problem is out of place here. We use the collocation method, which is described in more detail in Appendix A. The next example involves five species, 396 Fixed-Bed Catalytic Reactors γ = 30 β = . 4 Φ = . 01 A B C c r 3 2.5 2 1.5 1 0.5 1.2 1 0.8 0.6 0.4 0.2 Figure 7.20: Dimensionless concentration versus radius for the non- isothermal spherical pellet: lower (A), unstable middle (B), and upper (C) steady states....
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