Dynamic Optimization of Sequential Reactions

Computer Methods in Chemical Engineering

Problem Statement:

  • Reactant A is converted into a desired product B, which, in turn, is degraded into a useless form C. (Irreversible sequential reaction.) 
              k1        k2
    A --------> C ------> D

  where k1=k10exp(-E1/R/T)       E1= 75000J/mole   k10=109min-1
        k2=k20exp(-E2/R/T)       E2=125000J/mole   k20=2*1015min-1
    The following reaction kinetics apply. 
      dx1/dt = -k1*x1           I.C.: x1(0)=x10=1 mole/liter
  dx2/dt =  k1*x1 - k2*x2    I.C.: x2(0)=x20=0 mole/liter
  where x1=concentration of A
        x2=concentration of B
    1. Find the optimum constant temperature in a batch reactor to maximize x2 at tf=8min. Equivalently, find the temperature along the length of a tubular plug-flow reactor with a residence time of 8min such that we achieve the highest yield of B at the reactor exit.

      Solution:

      Note that although it takes several steps to find the level of B at the end of reaction, it is a plain function of temperature. Thus, this is a static optimization problem.

      • Static Optimization: Sequential Reaction: A-->B-->C, Isothermal, Optimize T

    2. Find the optimum temperature profile T(t) in a batch reactor. If you encounter trouble numerically with an infinite temperature, limit the temperature to T=[Tmin Tmin]=[250K 550K].

      Solution:

      • Dynamic Optimization: Sequential Reaction: A-->B-->C, Non-isothermal, Optimal Temperature Profile

    3. Because we can only heat/cool the reactor at a limited rate, how would you apply constraints on the limits of temperature gradient, i.e., force dT/dt to lie always within two limiting values? (The solution in the last part does not address this question. You figure it out.)
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    Computer Methods in Chemical Engineering -- Dynamic Optimization of Sequential Reactions
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