Temperature nonisothermal reactors

Omoleye, J. A., Adesina, A. A., and Udegbunam, E. O., Optimal design of nonisothermal reactors Derivation of equations for the rate-temperature conversion profile and the optimum temperature progression for a general class of reversible reactions, Chem. Eng. Comm., Vol. 79, pp. 95-107, 1989. 

The axial dispersion model is readily extended to nonisothermal reactors. The turbulent mixing that leads to flat concentration profiles will also give flat temperature profiles. An expression for the axial dispersion of heat can be written in direct analogy to Equation (9.14) ... 

Nonisothermal reactors. Nonisothermal operation brings additional complexity to the superstructure approach1112. In the first instance, the optimum temperature... 

Nonisothermal reactors with adiabatic beds. Optimization of the temperature profile described above assumes that heat can be added or removed wherever required and at whatever rate required so that the optimal temperature profile can be achieved. A superstructure can be set up to examine design options involving adiabatic reaction sections. Figure 7.12 shows a superstructure for a reactor with adiabatic sections912 that allows heat to be transferred indirectly or directly through intermediate feed injection. 

For nonisothermal reactors the key questions that the reactor designer must answer are (1) How can one relate the temperature of the reacting system to the degree of conversion that has been accomplished and (2) How does this temperature influence the subsequent performance of the system In responding to these questions the chemical engineer must use two basic tools—the material balance and the energy balance. The bulk of this chapter deals with these topics. Some stability and selectivity considerations are also treated. 

In the previous chapter we showed how nonisothermal reactors can exhibit much more complex behavior than isothermal reactors. This occurs basically because k(T) is strongly temperature dependent Only a single steacfy state is possible in the PFTR, but the CSTR, although (or because) it is described by algebraic equations, can exhibit even more interesting (and potentially even more dangerous) behavior. 

The state of a system is rigorously defined through the state variables of the system. The state variables of any system are chosen according to the nature of the system. The state of a boiler, for example, can be described by temperature and pressure, that of a heat exchanger by temperature, the state of a nonisothermal reactor by the concentration of the different components and their temperature, and the state of a bioreactor by the substrate concentrations and pH. 

Consider a nonisothermal reactor. The concentration C and temperature T change with time within the reactor may be described by the following equations ... 

A continuous emulsion polymerization is characterized by (i) the continual addition of monpmer, surfactant, and initiator to the reaction vessel and (ii) the withdrawal of a steady stream of reactor fiuid. Several such reaction vessels may be set up in series with one another so that the effluent from the first feeds the second, etc. Any species in such a system remains in the reaction vessel for a finite time, termed the residence time. Such reactor vessels may be either completely or incompletely mixed and operate under either isothermal or nonisothermal conditions. The incompletely mixed, nonisothermal reactor is characterized by the existence of spatial concentration and temperature gradients in the reactor. 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

For nonisothermal reactors, one of the reactor design equations, the energy transfer equation (see above), and an expression for the rate in terms of concentration and temperature must be solved simultaneously to give the conversion as a function of time. Note that the equations may be interdependent each can contain terms that depend on the other equation(s). These equations, except for simple systems, are usually too complex for analytical treatment. 

In the following analysis, both the solid and gas phases are assumed to move in countercurrent plug flow so that both concentrations (and temperature in nonisothermal reactors) vary with axial position. The material balance with boundary condition for the gas phase reactant is... 

In an endothermic reaction the temperature decreases as the conversion increases, unless energy is added to the system in excess of that absorbed by the reaction. Both the reduction in concentration of reactants, due to increasing conversion, and the reduction in temperature cause the rate to fall (Fig. 5-1 <2). Hence the conversion in a nonisothermal reactor will normally be less than that for an isothermal one. Adding energy to reduce the temperature drop will limit the reduction in conversion. If the... 

Nonisotfaermal reactor with isothermal cooling jacket. A coolant at constant temperature cooling jacket is added to the previous example to examine the perfomiancc of a nonisothermal reactor. In thi.s model, the boundary condition for the energy balance at the radial boundary is changed from the thermal insulation boundary condition to a heat flux boundary condition. 

Nonisothermal reactor with variable coolant temperature. This e.xamplc extends the third example by including the energy balance on the coolant in the cooling Jacket as the temperature of the coolant varie.s along the reactor. 

Single-Bed, Nonisothermal Catalysts. In an attempt to circumvent the undesirable formation of hydrogen sulfide in the presence of water vapor, a nonisothermal reactor was constructed by placing 536 g of Jamaican red mud catalyst in a 2-cm diameter 96%-silica tube. The catalyst-filled tube was inserted into the bottom half of the furnace. This resulted in a 15-cm uniform temperature hot zone and a 25-cm zone with temperatures gradually decreasing to about 100 °C at the lower reactor exit. The inlet gas consisted of 17% water vapor, 5.8% carbon monoxide, and 3.0% sulfur dioxide, and 74.2% helium. Figure 5 shows the dependence of the exhaust gas analysis on the hot-zone temperature of the Jamaican red mud catalyst. No sulfur dioxide was removed at hot-zone temperatures lower than 240 °C. At 250 °C, some sulfur dioxide was removed, and small quantities of hydrogen sulfide were formed. Above 300°C, more than 80% of the sulfur dioxide and virtually all of the carbon monoxide... 

Figure 4.25 Typical temperature and partial pressure profiles for nonisothermal reactor operation with the highly exothermic naphthalene oxidation reaction. (After R.J. Van Welsenaere and G.F. Froment, Chem. Eng. Sci., 25, 1503, with permission of Pergamon Press, Ltd., London, (1970).]...

A reminder neither of the two heat-transport models considered in this section account for thermal conduction in the bed. If gradients are substantial, this can be a miserable assumption as shown by the discussion of nonisothermal reactors in Chapter 7. It is probably fair to say that if any conduction effects are included, axial or radial, the problem formulation is not complicated very much (just the addition of a second derivative term with respect to temperature), but the solution to the problem becomes a numerical one and sufficiently complicated to be a subject... 

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