Plug Flow Reactor Volume (First-Order Reaction)

The term ln(1/(1−X)) comes from integrating the first-order rate law dC_A/dz = −k·C_A/v₀ along the reactor length. Because the reaction rate is proportional to concentration, as reactant is consumed the rate slows down. The last few percent of conversion require progressively more volume because the driving force (concentration of A) is almost gone. At X = 0.5, the term equals 0.693. At X = 0.9, it is 2.303. At X = 0.99, it is 4.605. At X = 0.999, it is 6.908. So going from 90% to 99% conversion doubles the reactor volume, and 99% to 99.9% adds another 50% — even though the incremental conversion is only 0.9 percentage points.

For first-order reactions, PFRs always require less volume than CSTRs at the same conversion, and the advantage grows with conversion. At X = 0.5, the PFR is about 1.4× smaller; at X = 0.9, about 3.9× smaller; at X = 0.99, about 21× smaller. So PFRs are preferred when high conversion is required, when the reaction is fast, and when continuous operation is feasible. CSTRs are preferred when you need tight temperature control (a CSTR operates at uniform conditions), when the reaction has a sharp optimum temperature, or when the feed contains solids or is prone to fouling (CSTRs are easier to clean). For autocatalytic reactions, a CSTR followed by a PFR often gives the smallest total volume.

Temperature enters through the rate constant k, which follows the Arrhenius equation: k = A·exp(−E_a/RT). Higher temperature means larger k, which means less reactor volume for the same conversion. For a typical activation energy of 70 kJ/mol, raising the temperature from 300 K to 310 K increases k by a factor of about 2.3, and thus reduces the required volume by the same factor. This is why industrial reactors often run at high temperature. The limits are side reactions (which may have different activation energies and become more prominent at high T), catalyst stability, and materials of construction.

Space time τ = V / v₀ is the reactor volume divided by the inlet volumetric flow rate, expressed in seconds (or sometimes hours). For a first-order reaction in an ideal PFR, τ = (1/k) · ln(1/(1−X)). Space time is a measure of how long, on average, a fluid element resides in the reactor. Its reciprocal is space velocity SV = 1/τ, which tells you how many reactor volumes of feed are processed per unit time. For industrial processes, SV is often quoted in h⁻¹: SV = 2 h⁻¹ means the reactor processes two reactor volumes per hour.

Yes. The equation V = (F_A0/k)·ln(1/(1−X)) assumes constant volumetric flow rate, which is valid for liquid-phase reactions or for gas-phase reactions with no change in total moles. For gas-phase reactions with changing total moles (e.g., A → 2B), the volumetric flow rate changes as conversion proceeds, and the design equation must be modified. For the reaction A → products with ε = change in moles per mole of A reacted, the design equation becomes: V = (F_A0/k) · ∫₀^X dX / [(1−X)(1+εX)]. This integral is evaluated analytically or numerically depending on ε. Advanced reactor design software handles these cases automatically.

For a first-order reaction, k has units of reciprocal time — s⁻¹ in SI, or min⁻¹, h⁻¹, or day⁻¹ in other conventions. The design equation V = (F_A0/k)·ln(1/(1−X)) requires k in s⁻¹ if F_A0 is in mol/s and V is to be in m³ (mol·s⁻¹ / s⁻¹ = mol). If you use k in min⁻¹, then F_A0 must be in mol/min for V in m³. Whatever time unit you choose for k, use the same for F_A0. The hint under the k field shows the half-life t_1/2 = ln(2)/k for reference — a useful sanity check on whether the value is physically reasonable.