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Minimized Volume for Reactors in Series

Two isothermal chemical reactors in series are used to carry out the reaction . These reactors can be continuous stirred-tank reactors (CSTR) or plug flow reactors (PFR).
1. Choose which reactors are used and in which order; four reactor arrangements are possible: CSTR followed by PFR, PFR followed by CSTR, CSTR followed by CSTR, or just one PFR (which is the same as a PFR followed by a PFR).
2. Select the fractional conversion from the second reactor by changing the conversion exiting the first reactor (which corresponds to changing the size of the first reactor).
This determines the minimum total volume.
The rate of disappearance of the reactant depends on the reaction order (first or second order) or the rate expression (Langmuir–Hinshelwood, or L–H, mechanism). The L–H mechanism is a dual-site irreversible reaction , where corresponds to a surface site. The area on a plot of the ratio (molar feed flow rate/rate of reaction) versus fractional conversion (a Levenspiel plot) is proportional to the reactor volume. The PFR volume is proportional to the area under the curve (from inlet to outlet conversion), and the CSTR volume is the difference in conversion multiplied by the axis ratio at the reactor outlet.

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DETAILS

Reactor Volume
is the inlet molar flow rate of reactant .
is the rate of disappearance of reactant in mol/time.
, are the respective inlet and outlet reactor conversions.
The CSTR volume is .
The PFR volume is .
Reaction Types
Assuming constant reactor inlet flow rate ,
,
where is the rate constant,
.
For first order ,
and for second order: ,
and for Langmuir–Hinshelwood: .
Assuming and rewriting in terms of ,
.
Reference
[1] H. S. Fogler, Essentials of Chemical Reaction Engineering, 1st ed., Upper Saddle River, NJ: Pearson Education, 2010 pp. 33–82, 445.
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