Engineering & Science
Levenspiel CSTR vs. PFR Reactor Volume Ratio (1st & 2nd Order)
Calculate the isothermal CSTR-to-Plug-Flow-Reactor volume penalty V_CSTR / V_PFR across target conversion X (0 < X < 1) for first-order ((X / (1 - X)) / (-LN(1 - X))) and second-order (1 / (1 - X)) kinetics.
See why 90% conversion of a 2nd-order reaction needs a 10x larger CSTR than a tubular PFR!
Octave Levenspiel’s famous reciprocal-rate plots (1 / -r_A vs. conversion X) reveal the central design trade-off in chemical reaction engineering: a Plug-Flow Reactor (PFR) integrates the area UNDER the 1 / -r_A curve as concentration gradually falls down the tube, whereas a Perfectly Mixed CSTR immediately dilutes fresh feed down to the exit concentration C_A = C_A0 * (1 - X), sizing volume by the giant rectangle X / -r_A(exit)!
Dividing isothermal CSTR Damköhler space-time by PFR space-time eliminates rate constant k and feed concentration C_A0 completely, leaving pure analytical volume-penalty ratios: for 1st-Order kinetics, V_CSTR / V_PFR = (X / (1 - X)) / (-LN(1 - X)); for 2nd-Order kinetics, V_CSTR / V_PFR = (X / (1 - X)^2) / (X / (1 - X)) = 1 / (1 - X)! At X = 0.90 (90% conversion), a single CSTR must be 3.909x larger than a PFR for a 1st-order reaction and a staggering 10.00x larger for a 2nd-order reaction!
| Target Fractional Conversion X | 1st-Order Volume Ratio V_CSTR / V_PFR | 2nd-Order Volume Ratio 1 / (1 - X) | Reactor Selection Insight |
|---|---|---|---|
| Moderate Conversion (X = 0.50) | 1.443x Larger CSTR | 2.00x Larger CSTR | Manageable Single-Tank Penalty |
| High Conversion (X = 0.90) | 3.909x Larger CSTR | 10.00x Larger CSTR | Use PFR or 3–4 CSTRs in Series! |
| Boundary Singularity (X <= 0 or X >= 1) | Complete-Conversion Guard | "Invalid Conversion (0 < X < 1)" | Infinite Volume at X -> 1 |
1st-Order Isothermal Volume Ratio V_CSTR / V_PFR = (X / (1 - X)) / (-LN(1 - X))
Evaluates exact CSTR backmixing penalty for positive first-order irreversible reactions across 0 < X < 1.
=IF(OR(B2 <= 0, B2 >= 1), "Invalid Conversion (0 < X < 1)", ROUND((B2 / (1 - B2)) / (-LN(1 - B2)), 3))2nd-Order Isothermal Volume Ratio V_CSTR / V_PFR = 1 / (1 - X)
Evaluates exact bimolecular/second-order CSTR-to-PFR volume ratio.
=IF(OR(B2 <= 0, B2 >= 1), "Invalid Conversion (0 < X < 1)", ROUND(1 / (1 - B2), 2))Try the example
Use the original values or change them to test your understanding.