Reaction Engineering: Reactor Design

Reactor design sits at the point where chemistry becomes manufacturing. For a given set of reactions, engineers must choose a reactor type, predict how much reactant converts to product, and control how much of that product is the desired one when multiple reactions occur. The practical questions are straightforward: How big should the reactor be? What operating conditions achieve the target conversion? How do we maximize selectivity while meeting safety and cost constraints? The answers depend on kinetics, mixing, and how long molecules spend inside the vessel.

This article focuses on design equations and selection criteria for common reactor types, with emphasis on conversion, selectivity, and the role of residence time distribution (RTD) when real reactors deviate from ideal behavior.

Core performance measures: conversion, rate, and selectivity

Reactor sizing begins with the reaction rate law and a definition of performance.

Conversion of species A is the fraction consumed: $X_A=\frac{F_{A0}-F_A}{F_{A0}}$ for flow systems, or $X_A=\frac{N_{A0}-N_A}{N_{A0}}$ for batch.

For multiple reactions, conversion alone is not enough. Engineers track selectivity, which quantifies how effectively reactant forms the desired product rather than byproducts. A common definition is the ratio of desired product formation rate to undesired product formation rate, or on an overall basis: $S_{P/B}=\frac{\text{moles of desired product formed}}{\text{moles of byproduct formed}}$. In parallel and series networks, selectivity typically depends strongly on concentration, temperature, and mixing.

The design equations: batch, CSTR, and PFR

All ideal reactor design equations are expressions of a mole balance combined with kinetics. The difference is how concentration changes in time or along the reactor.

Batch reactor

A batch reactor is a closed vessel operated for a finite time. It is common in specialty chemicals, pharmaceuticals, and situations where flexibility matters.

For species A: $\frac{d{N}_A}{dt}=-r_{A}V$ or, in terms of concentration for constant volume: $\frac{d{C}_A}{dt}=-r_A$.

If kinetics are known, the time to reach a target conversion is obtained by integration. Batch operation is valuable for multi-step recipes, temperature programming, and handling slow reactions, but it does not provide steady continuous throughput without scheduling multiple vessels.

Continuous stirred-tank reactor (CSTR)

In an ideal CSTR, contents are perfectly mixed and the exit stream matches the reactor composition. The steady-state mole balance for A is: $F_{A0}-F_A+r_{A}V=0$, leading to the sizing equation: $V=\frac{F_{A0}-F_A}{-r_A}=\frac{F_{A0}{X}_A}{-r_A}$, where $r_A$ is evaluated at exit conditions.

This “rate at the exit” feature is central. For reactions where rate decreases sharply with conversion (for example, many positive-order reactions), a CSTR often requires a larger volume than a plug flow reactor to reach the same conversion. However, strong mixing can improve temperature control and can sometimes improve selectivity in networks where suppressing high concentrations of an intermediate is beneficial.

Plug flow reactor (PFR)

In an ideal PFR, fluid elements move through the reactor with no axial mixing, and composition changes continuously along the length. The differential design equation is: $\frac{d{F}_A}{dV}=r_A$, or in conversion form: $\frac{d{X}_A}{dV}=\frac{-r_A}{F_{A0}}$.

Integrating from inlet to outlet yields the required volume. The PFR tends to be more volume-efficient for many reactions because it maintains higher reactant concentrations near the inlet, where rates are highest. That same concentration profile can either help or hurt selectivity in multiple-reaction systems, depending on which reactions are favored at high concentration.

Multiple reactions: how reactor choice affects selectivity

When more than one reaction occurs, the “best” reactor is rarely determined by conversion alone. Two common patterns illustrate why.

Parallel reactions (A forms desired P and undesired B)

If A can react to both P and B, selectivity depends on how each rate scales with concentration. Suppose: $r_P=k_P{C}_A^m$ and $r_B=k_B{C}_A^n$. Then the instantaneous selectivity ratio scales as: $\frac{r_P}{r_B}\propto C_A^{m-n}$.

• If $m>n$, higher $C_A$ favors the desired path. A PFR (higher inlet concentrations) may give better selectivity than a CSTR.
• If $m<n$, lower $C_A$ favors the desired path. A CSTR (uniform, often lower concentration for a given conversion) can improve selectivity.

Temperature affects selectivity through the Arrhenius dependence of rate constants. If the undesired reaction has a higher activation energy, selectivity may worsen at higher temperatures even as conversion increases. Reactor choice and heat removal strategy become intertwined.

Series reactions (A → P → B)

When the desired product P is an intermediate that can further react to an undesired B, the key is controlling the residence time so P is formed but not over-processed. Batch operation can stop at the optimal time. In flow reactors, the concentration and residence time profiles matter:

• A PFR often allows better control of intermediate formation along the reactor length, but if residence time is too long, P will be consumed downstream.
• A CSTR tends to have more back-mixing, which can increase the exposure of P to conditions that convert it to B, potentially reducing yield of P.

In practice, intermediate maximization often leads to staged reactors (for example, multiple CSTRs in series) or a PFR with quench, separation, or side withdrawal, depending on the process constraints.

Residence time distribution (RTD): when “ideal” is not real

Real reactors deviate from ideal mixing assumptions due to channeling, dead zones, bypassing, or axial dispersion. RTD quantifies the distribution of times that fluid elements spend in the reactor, which can strongly affect conversion and selectivity, especially in multiple-reaction networks.

RTD is commonly measured using a tracer test. The normalized exit age distribution is denoted $E(t)$, and it satisfies: ${\int }_0^{\infty }E(t)dt=1$. The mean residence time is: $\bar{t}={\int }_0^{\infty }tE(t)dt$.

Ideal benchmarks:

• Ideal PFR: $E(t)$ is a sharp spike at $t=\tau$, where $\tau =V/{\nu }_0$ is the space time and ${\nu }_0$ is volumetric flow rate.
• Ideal CSTR: $E(t)=\frac{1}{\tau }{e}^{-t/\tau }$.

Why RTD matters:

• In a series reaction where overexposure destroys the desired intermediate, a “long tail” in RTD (some fluid staying much longer than average) can significantly reduce selectivity.
• For parallel reactions with different kinetics, back-mixing and dispersion can shift the effective concentration history and change product distribution.

RTD does not replace kinetics, but it connects hydrodynamics to performance. In design and troubleshooting, RTD helps distinguish whether poor selectivity is caused by chemistry (rate constants, temperature) or by flow non-idealities (maldistribution, stagnant regions).

Selection criteria: choosing between batch, CSTR, and PFR

Reactor selection is a balance of kinetics, selectivity, operability, and control.

When a batch reactor is a strong choice

• Flexible production and frequent changeovers.
• Tight control over reaction time for intermediate products.
• Complex recipes with staged additions, temperature ramps, or sampling-based endpoints.

When a CSTR is advantageous

• Excellent mixing and heat transfer for strongly exothermic reactions.
• Processes needing stable outlet composition and straightforward control.
• Selectivity benefits from lower reactant concentration or from damping concentration gradients.

CSTRs are also commonly arranged in series to approach plug-flow-like conversion while retaining mixing and heat removal benefits.

When a PFR is preferred

• High throughput continuous production with good volume efficiency.
• Reactions that benefit from high reactant concentration early in the reactor.
• Systems where minimizing back-mixing improves yield.

In practice, the “PFR” may be a tubular reactor with some axial dispersion; design often includes pressure drop constraints, heat exchange considerations, and catalyst placement if the reaction is heterogeneous.

Practical design workflow

A disciplined reactor design approach typically follows this sequence:

1. Define objectives: target conversion, selectivity, throughput, and constraints (temperature limits, safety, downstream separations).
2. Establish kinetics for all relevant reactions, including side reactions, ideally over the expected operating range.
3. Evaluate ideal reactor models (batch, CSTR, PFR) using the appropriate design equations to estimate size and performance.
4. Assess selectivity sensitivity to concentration and temperature, especially for parallel and series networks.
5. Account for non-ideal flow using RTD or dispersion concepts when scale-up or performance risk warrants it.
6. Iterate with heat and mass transfer realities, because temperature control and mixing can dominate real outcomes.

Reactor design is ultimately an exercise in aligning reaction kinetics