The Multibed Adiabatic Reactor

In discussing the preliminary design of fixed bed reactors in Sec. 11.3 we mentioned that adiabatic operation is frequently considered in industrial operation because of the simplicity of construction of the reactor. It was also mentioned why straight adiabatic operation may not always be feasible and examples of multibed adiabatic reactors were given. With such reactors the question is how the beds should be sized. Should they be designed to have equal ATs or is there some optimum in the AT s, therefore in the number of beds and catalyst distribution In Section 11.3. this problem was already discussed in a qualitative way. It is taken up in detail on the basis of an example drawn from SOj oxidation, an exothermic reversible reaction. To simplify somewhat it will be assumed, however, that no internal gradients occur inside the catalyst so that the effectiveness factor is one. 

A very convenient diagram for visualizing the problem of optimizing a multibed adiabatic reactor is the conversion versus temperature plot already encountered in Sec. 11.3, and drawn in Fig. 11.5.d-l for the SO oxidation based on the rate equation of Collina, Corbetta, and Cappelli [113] with an effectiveness factor of 1. (For further reading on this subject see [114] and [115].) This equation is 

The coefficients ki, K2, and Kj were determined by nonlinear regression on 59 experiments carried out in a temperature range 420 to 590°C. The partial pressures are converted into conversions by means of the formulas For 1 mol SO2 fed per hour, the molar flow rates in a section where the conversion is x is given in the left-hand column. The partial pressures are given in the right-hand column. 

The figure has been calculated for the following feed composition 7.8 mole % SO2 10.8 mole %02 81.4mole % inerts, atmospheric pression, a feed temperature of 37 C, a mean specific heat of 0.221 kcal/kg °C (0.925 kJ/kg K) and a (— AW) of 21,400 kcal/kmol (89,600 kJ/kmol). Cooling by means of a heat exchanger is represented by a parallel to the abscissa in this diagram. 

The calculations do not necessarily proceed according to the direction of the process flow. This is only so for a final-value problem (i.e., when the conditions at the exit of the reactor are fixed). For an initial-value problem, whereby the inlet conditions are fixed, the direction of computation for the optimization is opposite to that of the process stream. In what follows an initial-value problem is treated. First consider the last bed. No matter what the policy is before this bed the complete policy cannot be optimal when the last bed is not operating optimally for its feed. The specifications of the feed of the last bed are not known yet. Therefore, the optimal policy of the last bed has to be calculated for a whole set of possible inlet conditions of that bed. 

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Consider now two steps, the last two of the multibed adiabatic reactor. From Bellman s maximum principle it follows that the optimal policy of bed 1 is preserved. This time X2 and Ti have to be chosen in an optimal way to arrive at... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

The temperature-composition relation in such a multibed adiabatic reactor is illustrated in Fig. 11.3-4 for ammonia synthesis [3]. The F, curve in this diagram represents the equilibrium relation between composition and temperature. The maximum ammonia content that could be obtained in a single adiabatic bed with inlet conditions corresponding to A would be 14 mole % as indicated by point B, and this would theoretically require an infinite amount of catalyst. The five-bed quench converter corresponding to the reaction path ABCDEFGHIJ permits... 

This figure clearly illustrates that the range within which multiple steady states can occur is very narrow. It is true that, as Hlavacek and Hofmann calculated, the adiabatic temperature rise is sufficiently high in ammonia, methanol and oxo-synthesis and in ethylene, naphthalene, and o-xylene oxidation. None of the reactions are carried out in adiabatic reactors, however, although multibed adiabatic reactors are sometimes used. According to Beskov (mentioned in Hlavacek and Hofmann) in methanol synthesis the effect of axial mixing would have to be taken into account when Pe < 30. In industrial methanol synthesis reactors Pe is of the order of 600 and more. In ethylene oxidation Pe would have to be smaller than 200 for axial effective transport to be of some importance, but in industrial practice Pe exceeds 2500. Baddour et al. in their simulation of the TVA ammonia synthesis converter found that the axial diffusion of heat altered the steady-state temperature profile by less than 0.6°C. Therefore, the length of... 

More complicated problems for sequences of stirred tanks can be devised, but they follow the pattern of multibed adiabatic tubular reactors to which we now turn. 

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