The Heart of the Process
The reactor is where raw materials become products — it is the unit that justifies the entire plant. Designing it requires two ingredients: kinetics (how fast the reaction goes) and a reactor model (how the fluid flows and mixes). Together they tell you how big the reactor must be to reach a target conversion, and how to maximize the desired product.
Rate Laws and Reaction Order
The rate of reaction describes how quickly reactants are consumed. For a reaction A + B → products, the rate law often takes the form:
−rA = k [A]a [B]b
The exponents a and b are the reaction orders with respect to each species; their sum is the overall order. Crucially, orders are found experimentally and are not generally the stoichiometric coefficients. A first-order reaction's rate is proportional to concentration; a zero-order reaction proceeds at a constant rate independent of concentration.
The Arrhenius Equation
The rate constant k depends sharply on temperature through the Arrhenius equation:
k = A · exp(−Ea / RT)
where A is the pre-exponential factor, Ea is the activation energy, R the gas constant, and T absolute temperature. Because T sits inside an exponential, modest temperature increases produce large rate increases — the familiar rule of thumb that rate roughly doubles per 10 °C. Plotting ln k against 1/T yields a straight line whose slope gives Ea, a standard way to extract activation energy from data.
Conversion
Conversion (X) is the fraction of the limiting reactant consumed:
X = (moles reacted) / (moles fed)
Conversion is the natural design variable: reactor design equations express the volume or time needed to reach a chosen X. Higher conversion is usually desirable but follows diminishing returns — the last few percent can require disproportionately large volume because the rate falls as reactant is depleted.
The Three Ideal Reactors
Reactor design starts from three idealized models, each with its own design equation derived from a mole balance:
| Reactor | Mode | Mixing | Design equation |
|---|---|---|---|
| Batch | Closed, no flow | Well mixed | t = NA0 ∫ dX / (−rA V) |
| CSTR | Continuous | Perfectly mixed | V = FA0 X / (−rA)exit |
| PFR | Continuous | No back-mixing | V = FA0 ∫ dX / (−rA) |
- Batch reactor: charge, react for a time, discharge. Flexible and common for small-volume, high-value products (pharmaceuticals, specialty chemicals).
- CSTR: a continuously fed, perfectly stirred tank. The whole vessel sits at the outlet composition, so the reaction proceeds at the (low) exit rate. Robust, easy to control temperature, and good for liquid-phase reactions.
- PFR: a tube in which fluid moves as a plug with no axial mixing. Concentration falls smoothly from inlet to outlet, so the reaction sees a range of rates.
CSTR vs. PFR Sizing
For a normal positive-order reaction, comparing the two design equations shows the PFR needs less volume for the same conversion. The reason is intuitive: the CSTR operates entirely at the exit (lowest) concentration and therefore the slowest rate, while the PFR experiences high concentrations and fast rates near its inlet. Graphically, the CSTR volume is a rectangle on a Levenspiel plot (1/−rA versus X), whereas the PFR volume is the smaller area under the curve.
The reverse holds for autocatalytic or some negative-order reactions, where a CSTR can be smaller. And CSTRs in series approach PFR behavior — an infinite number of CSTRs in series is mathematically equivalent to a single PFR.
Selectivity and Yield
Real chemistry rarely produces only the desired product; competing and consecutive reactions form byproducts. Two metrics matter:
- Selectivity: moles of desired product per mole of undesired product.
- Yield: moles of desired product per mole of reactant fed.
Reactor choice strongly influences selectivity. If the desired reaction is higher order than the side reaction, high concentration (favoring a PFR or batch) improves selectivity. If the side reaction is higher order, low concentration (favoring a CSTR) helps. Temperature is equally powerful: when the desired reaction has higher activation energy, higher temperature improves selectivity, and vice versa. Tuning concentration, temperature, and residence time to maximize selectivity is often worth more than squeezing out the last few points of conversion.
From Kinetics to Hardware
A typical design path: gather rate data and fit a rate law, extract k and Ea via Arrhenius, choose a target conversion and reactor type based on phase and selectivity needs, apply the appropriate design equation to size the volume, then add heat-transfer surface to manage the heat of reaction. The same fundamentals scale from a lab flask to a 100 m³ industrial reactor.