Formula & Calculator
Plug Flow Reactor Volume (First-Order Reaction)
Sizes a plug flow reactor needed to achieve a target conversion for a first-order irreversible reaction at constant density.
Interpretation
PFR volume for first‑order: V = (F_A0 / k) · ln(1/(1−X)). Example: F_A0=10 mol/min, k=0.5 min⁻¹, X=0.8 → V ≈ 32.2 L.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| V | Reactor volume | m3 |
| F_A0 | Molar feed rate of reactant A | mol/s |
| k | First-order rate constant | 1/s |
| X | Target fractional conversion |
What it means
For an isothermal plug flow reactor (PFR) with a first‑order reaction A → products, the design equation gives the reactor volume V required to achieve a conversion X. The equation is V = (F_A0 / k) ln(1/(1 − X)), where F_A0 is the inlet molar flow of A, k is the rate constant, and X is the conversion. This expression is derived from the mole balance and the rate law. It shows that as conversion approaches 1, the required volume goes to infinity because the reaction rate becomes very slow at low concentrations. In practice, PFRs are used for high‑throughput, continuous production. The equation is also applicable to other reaction orders with different integral forms. It is essential for sizing tubular reactors, which are common in the petrochemical and polymer industries. Understanding the design equation for PFRs is a core competency in chemical reaction engineering.
Worked example
PFR Volume – Two Examples
Real‑World| Parameter | Value |
|---|---|
| F_A0 | 10 mol/s |
| k | 0.1 s⁻¹ |
| X | 0.8 |
| Parameter | Value |
|---|---|
| F_A0 | 5 |
| k | 0.05 |
| X | 0.9 |
Common mistakes
- First‑order reaction: The rate law must be −r_A = k C_A. If the order is different, the integral changes.
- Constant density: This equation assumes constant volumetric flow (ρ constant). For gas‑phase with pressure drop, adjust.
- Molar flow F_A0: Inlet molar flow of reactant A, in mol/s or kmol/h – consistent with k and V.
- Rate constant k: Units depend on reaction order; for first order, s⁻¹ (if using C in mol/m³).
- Conversion X: Must be between 0 and 1; at X=1, volume is infinite.
Applications
The design equation for a plug flow reactor (PFR) with first‑order reaction is V = (F_A0 / k) · ln(1/(1−X)). This relationship allows engineers to calculate the required reactor volume for a given conversion, or the conversion achievable with a given volume. PFRs are characterised by no axial mixing and are often used for fast reactions and high‑temperature processes. This equation assumes constant density and isothermal operation. Engineers apply it to design tubular reactors, catalytic cracking units, and polymerisation systems. By understanding the PFR equation, professionals can optimise reactor dimensions, select appropriate operating conditions, and integrate reactors with other process units to achieve overall process objectives.
- Design of tubular reactors for gas‑phase and liquid‑phase reactions
- Sizing of catalytic reactors (e.g., ammonia synthesis, cracking)
- Optimisation of reaction conversion with respect to volume and temperature
- Scale‑up of PFRs from laboratory data
- Comparison with CSTR performance for given kinetics
Frequently Asked Questions
For a first‑order reaction with constant density, the required volume is V = (F_A0 / k) · ln(1/(1 – X)). This is derived from the PFR mole balance: dX/dV = –r_A/F_A0 = k·C_A0·(1–X)/F_A0, integrated.
- First‑order reaction: –r_A = k·C_A.
- Constant density (no volumetric change).
- Isothermal operation (k constant).
- Ideal plug flow (no axial mixing).
- Applying it to reactions of different order.
- Using it when the density changes (gas‑phase with mole change).
- Ignoring the effect of temperature on k.
- Using outlet concentration instead of inlet in the integration.
As X approaches 1, ln(1/(1–X)) grows rapidly, so the volume increases asymptotically. High conversion requires disproportionately large reactors.
Volume is inversely proportional to k. A higher k (faster reaction) reduces the required volume.
Using F_A0 = C_A0·v₀, the equation becomes V = (v₀ / k) · ln(1/(1–X)). This is the familiar form for a first‑order PFR.
For a first‑order reaction, the PFR volume is always less than the CSTR volume for the same conversion (because the PFR operates at higher concentrations). The ratio V_PFR/V_CSTR = ln(1/(1–X)) / X.
Preliminary sizing of tubular reactors for first‑order reactions, often used in environmental engineering (e.g., UV disinfection) and polymerisation.