Optimal reactor profiles

Figure 8.6 shows the component costs combined to give a total cost which varies with both reactor conversion and recycle inert concentration. Each setting of the recycle inert concentration shows a cost profile with an optimal reactor conversion. As the recycle inert concentration is increased, the total cost initially decreases but then... 

Example 6.5 Find the optimal temperature profile, T z), that maximizes the concentration of component B in the competitive reaction sequence of Equation (6.1) for a piston flow reactor subject to the constraint that F=3h. 

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. 

As many other industries, the fine chemical industry is characterized by strong pressures to decrease the time-to-market. New methods for the early screening of chemical reaction kinetics are needed (Heinzle and Hungerbiihler, 1997). Based on the data elaborated, the digital simulation of the chemical reactors is possible. The design of optimal feeding profiles to maximize predefined profit functions and the related assessment of critical reactor behavior is thus possible, as seen in the simulation examples RUN and SELCONT. 

Kokossis and Floudas (1994) extended the MINLP approach so as to handle nonisothermal operation. The nonisothermal superstructure includes alternatives of temperature control for the reactors as well as options for directly or indirectly intercooled or interheated reactors. This approach can be applied to any homogeneous exothermic or endothermic reaction and the solution of the resulting MINLP model provides information about the optimal temperature profile, the type of temperature control, the feeding, recycling, and by-passing strategy, and the optimal type and size of the reactor units. 

Optimal feed profile for a second order reaction in a semi-batch reactor under safety constraints, Experimental study. Journal of Loss Prevention in the Process Industries, 12 (11), 485-93. 

In general, an objective function in the optimization problem can be chosen, depending on the nature of the problem. Here, two practical optimization problems related to batch operation maximization of product concentration in a fixed batch time and minimization of batch operation time given amount of desired product, are considered to determine an optimal reactor temperature profile. The first problem formulation is applied to a situation where we need to increase the amount of desired product while batch operation time is fixed. This is due to the limitation of complete production line in a sequential processing. However, in some circumstances, we need to reduce the duration of batch run to allow the operation of more runs per day. This requirement leads to the minimum time optimization problem. These problems can be described in details as follows. 

All simulation results given here are based on the optimization problem PI. The objective in the problem formulation is to find the optimal reactor temperature profile, such... 

N. Aziz, M.A. Hussain, I.M. Mujtaba, Performance of different types of controllers in tracking optimal temperature profiles in batch reactors, Comp. Chem. Eng. 24 (2000) 1069-1075. 

Fig, 10. Piecewise constant approximation of optimal reactor temperature profiles. 

Here, we consider two alternatives. First, we consider the sequential approach, where we optimize the reactor network with an optimal temperature profile, then integrate the maintenance of this optimal profile with the energy flows in the rest of the flowsheet. In the second case, we solve the above problem with the simultaneous formulation proposed in (PIO). 

Clearly, formulation (PI2) is an optimal control problem with differential equation constraints, where the y s, and the temperature are the control profiles. The solution to this model will give us the optimal separation profile along the reactor. It is clear that 7(a) models the effect of separation within the reactor network. If all the elements of the vector 7(0 ) are the same (which implies that there is no relative separation between the species in the reactor), the second term for the governing differential equation vanishes, since... 

The solution to (P12) gives us the optimal separation profile as a function of age within the reactor. However, except in the case of reactive phase equilibrium, the assumption of a continuous separation profile is not really required. Furthermore, a continuous separation profile may not be implementable in practice. To address this, we take advantage of the structure of a discretization procedure for the differential equation system. In this case, we choose orthogonal collocation on finite elements to discretize the above model. This results... 

This work presents the on-line level control of a batch reactor. The on-line strategy is required to accommodate the reaction rate disturbances which arise due to catalyst dosing uncertainties (catalyst mass and feeding time). It is concluded that the implemented shrinking horizon on-line optimization strategy is able to calculate the optimal temperature profile without causing swelling or sub-optimal operation. Additionally, it is concluded that, for this process, a closed-loop formulation of the model predictive controller is needed where an output feedback controller ensures the level is controlled within the discretization intervals. 

Fig. 9.1 Optimal temperature profiles for consecutive reactions and various lengths of reactor.

In this section we shall be concerned with more realistic models of tubular reactors. The isothermal reactor is obviously the simplest type, but it implies that either there are no large heat effects or that they can be completely dominated by temperature control. The reactor with an optimal temperature profile is clearly the most desirable, but this means that the rate of heat exchange can be regulated precisely at each point. Between these two extremes there is a range of designs about which something should be said. We shall not always solve the equations in detail but we shall try to show the important features of the behavior of the reactor by means of examples. 

To find the optimal temperature profile, the Pontryagin maximum principle can be used. However, in this case, the Pontryagin maximum principle results in a very simple optimization criterion, that is at each point along the length of the reactor find the temperature that maximizes the net rate of reaction at this point . Proof of the fact that the Pontryagin maximum principle in this case gives this simple criterion is given in the next section. 

The optimal temperature profiles obtained pose the practical problem of how to implement this optimal temperature policy. The optimal temperature profiles obtained suggest that the reactor should be operated with quite a high feed temperature followed by a very efficient cooling to decrease the temperature sharply at the first 15% of the reactor depth. The implementation of this optimal policy is technically difficult and expensive. A number of suboptimal policies were suggested and discussed (Elnashaie et ai, 1987b) to overcome the technical difficulties associated with the implementation of the absolute optimal policy. One of the main technical problems associated with optimal temperature is the very high temperature at the reactor inlet part which the catalyst may not be able to withstand. 

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