Size a continuous stirred-tank reactor (CSTR) and a plug-flow reactor (PFR) for a first-order, constant-density liquid-phase reaction A → products. Enter the rate constant, the target fractional conversion, and the volumetric flow, and compare the two reactor volumes and residence times side by side.
The first design question for any reactor is how big it must be to reach a target conversion. For a first-order, constant-density liquid-phase reaction A → products, the answer is a closed-form expression for both the continuous stirred-tank reactor (CSTR) and the plug-flow reactor (PFR). This calculator evaluates both, reports their residence times, and shows the volume penalty you pay for choosing perfect mixing over plug flow.
Fractional conversion X is the fraction of the limiting reactant A that has reacted: X = (F_A0 − F_A)/F_A0, where F_A0 is the molar feed rate of A and F_A is what leaves. X ranges from 0 (no reaction) to 1 (complete consumption). For a first-order reaction the rate of disappearance of A is −r_A = k·C_A, and with constant density C_A = C_A0(1 − X). Everything in reactor sizing flows from this definition.
A CSTR is perfectly mixed, so the whole tank sits at the exit conversion and the rate is evaluated at that low outlet concentration. The design equation V = F_A0·X / (−r_A) becomes, for first order, V = Q·X / [k(1 − X)], where Q is the volumetric flow. The space time is τ = V/Q = X / [k(1 − X)]. Because the rate is fixed at the slow exit value, the required volume grows sharply as X approaches 1.
A PFR has no back-mixing; concentration falls smoothly from inlet to outlet, so the reaction runs faster near the entrance. Integrating V = F_A0·∫dX/(−r_A) for first order gives V = (Q/k)·(−ln(1 − X)) and τ = −ln(1 − X)/k. The logarithm grows far more slowly than the X/(1−X) term of the CSTR, which is why the PFR needs less volume for the same conversion.
For any positive reaction order, a PFR achieves a given conversion in less volume than a single CSTR, because it always operates at a higher average concentration (and therefore higher average rate). The ratio V_CSTR/V_PFR widens dramatically at high conversion. Space time τ = V/Q is the mean residence time; its reciprocal, space velocity (SV = Q/V), measures throughput per unit reactor volume. Engineers trade these off: a series of CSTRs approaches PFR performance, while a CSTR is favored when good mixing, heat removal, or solids handling matters more than minimum volume.
A CSTR is perfectly mixed, so the entire reactor operates at the low exit concentration, where the reaction rate is slowest. A PFR has a concentration gradient and spends much of its length at higher concentration and faster rate. For the same first-order conversion, V_CSTR = Q·X/[k(1−X)] always exceeds V_PFR = (Q/k)·(−ln(1−X)), and the gap grows as conversion approaches 1.
Space time τ = V/Q is defined from the inlet volumetric flow and reactor volume. For a constant-density (liquid-phase) system with no volume change, space time equals the mean residence time the fluid actually spends in the reactor. For gas-phase reactions with changing moles or density, the two can differ and you must track the actual flow.
Both design equations blow up at X = 1: the CSTR term X/(1−X) and the PFR term −ln(1−X) both go to infinity. Physically, driving a reaction to 100% conversion requires infinite reactor volume because the rate falls to zero as the reactant is exhausted. Real designs target 90–99% and recover the rest by recycle or a second reactor.
The closed-form expressions here assume constant density, which is the standard liquid-phase case. Gas-phase reactions with a change in total moles cause the volumetric flow to vary along the reactor, adding an expansion factor ε to the integrals. For those, use the full Levenspiel design equations rather than these simplified results.
Replace the rate law: for second order, −r_A = k·C_A², and the integrals change accordingly (the PFR term becomes X/[k·C_A0·(1−X)] and the CSTR term grows even faster with X). This calculator is specialized to first order. For other orders, set up the design equation with the correct −r_A and integrate, or solve numerically.