PFR Design Equation (First Order)

A plug-flow reactor treats the feed as a series of independent slugs sliding down a tube without mixing back or forward. Concentration therefore falls smoothly along the length instead of dropping to the outlet value the instant it enters, and the reaction runs fast at the front where the reagent is strong. The volume follows from integrating the rate along the tube, giving $V = (v_0/k)\ln[1/(1-X)]$.

Put it side by side with the stirred tank on the same duty and the difference is startling. For 5 L/s at $k = 0.01\ \text{s}^{-1}$ and 80% conversion, the CSTR needed 2000 L. The PFR needs $0.5 \times \ln 5 = 805\ \text{L}$, two and a half times less. Push to 99% conversion and the gap widens to more than twenty to one, because the CSTR volume goes as $X/(1-X)$ while the PFR only goes as a logarithm. For any reaction of positive order, plug flow always wins on volume, and the higher the conversion the more decisively.

A subtlety with real consequences: if you run the same commissioning test in the two reactor types and fit a rate constant, you get different answers unless you use the right model. A 500 L vessel on 1 L/s hitting 90% conversion implies $k = 0.018\ \text{s}^{-1}$ if you treat it as a CSTR and $k = 0.0046\ \text{s}^{-1}$ if you treat it as a PFR, a factor of four. Before trusting either number, run a tracer test and find out which flow model the vessel actually obeys.