CSTR / PFR Reactor Sizing (First-Order Reaction)
Finds the volume an ideal continuous reactor needs to reach a target conversion for a first-order reaction. Choose a continuous stirred-tank reactor (CSTR) or a plug-flow reactor (PFR). Process engineers and students use it for a first sizing, and to see how much smaller a tubular reactor can be than a stirred tank for the same duty.
Equations
For a first-order reaction, −rA = k·CA, at constant density and temperature, the design equations are [1]:
CSTR: V = Q·X / (k·(1 − X))
PFR: V = (Q/k)·ln(1/(1 − X))
V is the reactor volume, Q the volumetric flow rate through it, k the first-order rate constant in 1/s, and X the fraction of the feed reactant converted, from 0 to 1. The space time τ = V/Q is the mean time the fluid spends in the reactor. A CSTR runs entirely at the low outlet concentration, so it reacts more slowly and needs more volume than a PFR, whose concentration falls gradually along its length.
Limits and common mistakes
- The equations hold only for first-order, irreversible kinetics. Second-order or reversible reactions need other forms.
- Density must stay constant. That suits most liquids. A gas-phase reaction that changes the number of moles needs the expansion factor ε [1].
- The reactor is isothermal. k changes strongly with temperature, so use k at the operating temperature.
- Enter X as a fraction: 0.9, not 90. The volume grows without limit as X approaches 1.
- Check the time unit of k. A value quoted per minute must be divided by 60 before you enter it here.
- Real stirred tanks and tubes fall between the two ideals. Add a margin, and for a tube check that the flow is close to plug flow.
Worked example
A liquid feed of 5 L/s undergoes a first-order reaction with k = 0.004 s−1 at the operating temperature. The plant needs 90 % conversion. How big must a stirred tank be, and how big a tubular reactor?
- Choose Reactor Type CSTR (ideal, continuous stirred-tank).
- Enter Volumetric Flow Rate (Q) 5 L/s, First-Order Rate Constant (k) 0.004 and Target Conversion (X) 0.9.
- Press Solve. The Solution shows:
Required Reactor Volume: 11.25 m³
Required Reactor Volume: 11,250 L - Choose Reactor Type PFR (ideal, plug flow) and press Solve again. The Solution shows:
Required Reactor Volume: 2.87823 m³
Required Reactor Volume: 2,878.23 L
Check by hand: Q/k = 0.005/0.004 = 1.25 m³. The CSTR needs 1.25 × 0.9/0.1 = 11.25 m³, and the PFR 1.25 × ln 10 = 2.878 m³. At 90 % conversion the stirred tank is almost four times the size of the tube.
