Optimal Sizing of Two CSTRs Connected in Series

We would like to optimally size the two CSTRs for a specified value of final conversion (XAf= Xa. Let Xi and X2 be the space times of the two CSTRs, and Xaj be the conversion in the first reactor. The design equations for the two CSTRs are 

For a fixed value of Xa2 = Af (final conversion specified), X j is calculafed such fhaf fhe fofal space time x = Xj + X2 is minimum. Applying fhe necessary condition for minimum, we have 

However, the analytical equation for Xai can be derived for a first-order reaction by applying the condition for optimalify (Equation 3.45). The rate equation for a first-order 

Graphical procedure for optimal sizing of two CSTRs in series. 

This implies that tj = X2. Thus, for a first-order reaction, equal volumes of CSTRs correspond to the total minimum volume required for any specified conversion. 

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