What Is a Reactor Sizing & Scale-Up Calculator?

A Reactor Sizing & Scale-Up Calculator is a chemical engineering tool used to estimate how large a reactor should be, how long a reaction should run, how much catalyst is needed, how much heat must be removed, and how a laboratory or pilot reactor may be translated into a larger production reactor. Reactor sizing is a core part of chemical process design because the reactor often determines conversion, selectivity, safety, energy demand, residence time, and plant economics.

This calculator combines common design equations for CSTR, PFR, batch reactors, packed-bed reactors, heat removal, residence time, Arrhenius temperature scale-up, geometric scale-up, and mixing power. It is designed for students, teachers, chemical engineering learners, process development teams, and early-stage design screening. It helps users connect reaction kinetics with equipment size and operating conditions.

The most important concept behind reactor sizing is the design equation. For a steady-flow reactor, the molar feed rate of reactant A, target conversion, and rate of disappearance of A determine the volume or catalyst weight. If the rate is fast, the required reactor size is smaller. If the rate is slow, if conversion is high, or if concentration falls strongly with conversion, the required size increases. Reactor type matters because a CSTR operates at the exit concentration throughout the tank, while a PFR experiences a concentration gradient along the reactor length.

Scale-up is more complex than simply multiplying volume. A reaction that behaves safely and selectively in a small flask may generate heat too quickly in a large vessel. Mixing time may increase. Heat transfer area per unit volume may decrease. Gas-liquid transfer can become limiting. Catalyst beds may develop pressure drop, hot spots, or channeling. Agitation power, impeller tip speed, Reynolds number, and power per unit volume are all used as scale-up criteria, but none is universally correct for every process.

This tool therefore includes both sizing and scale-up modules. It estimates reactor volume, batch time, residence time, heat release, cooling area, adiabatic temperature rise, full-scale tank dimensions, impeller diameter, mixing speed, Reynolds number, and agitation power. Results should be treated as transparent engineering estimates, not final design authority.

How to Use This Reactor Sizing & Scale-Up Calculator

Use the CSTR / PFR tab when you know feed molar flow, feed concentration, kinetic rate constant, reaction order, and target conversion. Select CSTR if the reactor is a well-mixed continuous tank. Select PFR if the reactor behaves like plug flow with concentration changing along the reactor path. You can use a power-law rate model \(-r_A=kC_A^n\) or enter a known rate directly.

Use the Batch Time tab when you want the time required to reach a target conversion in a constant-volume batch reactor. Enter initial concentration, rate constant, reaction order, and conversion. The calculator includes a loading or cleaning time field to estimate total cycle time.

Use the Packed Bed tab when the rate is expressed per mass of catalyst. The calculator estimates catalyst weight, approximate bed volume, and bed height from bulk density and bed diameter. Use Residence Time to convert between reactor volume, volumetric flow, residence time, and space velocity.

Use Heat Removal for heat duty, cooling area, overall heat transfer coefficient, driving temperature difference, or adiabatic temperature rise. Use Temperature Scale-Up to estimate how a rate constant changes when temperature changes. Use Geometric Scale-Up to estimate full-scale dimensions from pilot scale. Use Mixing Power to compare constant power per volume, constant tip speed, constant Reynolds number, or manual full-scale speed.

Reactor Sizing Formulas

The CSTR design equation is:

The PFR design equation is:

A constant-density power-law rate model is:

Batch reactor time for constant volume is:

Residence time and space velocity are:

Packed-bed catalyst weight can be estimated by:

Heat duty from reaction is:

Cooling area is estimated from:

Adiabatic temperature rise is approximated by:

The two-temperature Arrhenius equation is:

Agitation power is commonly estimated with:

CSTR and PFR Sizing

A CSTR is a continuous stirred tank reactor. It is modeled as perfectly mixed, so the reactor contents have the same composition as the outlet stream. This makes the CSTR design equation simple, but it also means the entire reactor operates at the lower exit concentration when conversion is high. For positive-order reactions, that lower concentration gives a lower rate, so a CSTR can require more volume than a PFR for the same feed and conversion.

A PFR is a plug flow reactor. Fluid elements move through the reactor with minimal axial mixing. Concentration changes continuously along the reactor length. Near the inlet, reactant concentration is high and reaction rate can be high. Near the outlet, concentration is lower and rate decreases. The PFR design equation integrates the reciprocal rate from zero conversion to the target conversion.

For first-order liquid-phase reactions, analytical formulas exist. For more general order, the calculator performs a numerical integration for PFR sizing. This makes it more flexible for zero-order, first-order, second-order, fractional-order, and noninteger power-law rate models. For gas-phase reactions, an expansion factor \(\varepsilon\) can adjust concentration with conversion using a simplified relation.

Batch Reactor Time

A batch reactor is charged with reactants, operated for a period of time, and then emptied. It is common in pharmaceuticals, specialty chemicals, fermentation, polymerization, pilot testing, and multiproduct facilities. Batch operation is flexible but includes nonproductive time for charging, heating, cooling, cleaning, sampling, and discharging.

The batch design equation relates time to conversion through the reaction rate. In a constant-volume liquid batch reactor, concentration is often written as \(C_A=C_{A0}(1-X)\). The time to reach conversion \(X\) depends strongly on reaction order. First-order reactions have a logarithmic conversion-time relation. Second-order reactions become very slow near high conversion because concentration becomes small. This is why pushing conversion from 95% to 99% can require much more time than pushing from 50% to 80%.

Cycle time is not just reaction time. In production, total cycle time includes preparation, charging, temperature ramping, hold time, cooling, sampling, transfer, and cleaning. A reactor with fast chemistry can still have low productivity if handling time is large.

Packed Bed and Catalyst Sizing

A packed-bed reactor contains solid catalyst particles while reactants flow through the bed. The design variable is often catalyst weight rather than liquid volume. The packed-bed design equation uses a rate per mass of catalyst, written as \(-r_A'\). If the catalyst rate is known and roughly constant across the bed, catalyst weight can be estimated by \(W=F_{A0}X/(-r_A')\). If the rate changes strongly with conversion, temperature, pressure, or concentration, integration is required.

After catalyst weight is estimated, bed volume can be approximated using bulk catalyst density. Bed height follows from bed volume and cross-sectional area. Real packed-bed design requires pressure drop analysis, catalyst particle size, void fraction, heat transfer, hot spot risk, flow distribution, channeling risk, deactivation, regeneration, and mechanical support design.

Heat Removal and Adiabatic Temperature Rise

Reactor heat management is often more important than reactor volume. Exothermic reactions release heat. If heat is not removed fast enough, temperature rises, which may increase the reaction rate and release heat even faster. This feedback can create runaway risk. Endothermic reactions absorb heat and may require heating jackets, coils, fired heaters, or heat-transfer media.

The heat duty from reaction is estimated by \(Q=(-\Delta H_{rxn})F_{A0}X\). For an exothermic reaction, \(\Delta H_{rxn}\) is negative, so heat release is positive. Cooling area is estimated from \(Q=UA\Delta T_{lm}\). The required area grows when heat duty is large, when \(U\) is small, or when the temperature driving force is small.

The adiabatic temperature rise estimates how much temperature could rise if no heat were removed. It is a screening tool for safety and process feasibility. A large adiabatic temperature rise signals that temperature control, feed dilution, staged addition, external cooling, or alternative reactor configuration may be necessary.

Scale-Up, Mixing, and Agitation Power

Reactor scale-up is not simply geometry. When a vessel becomes larger, volume increases faster than surface area. This reduces heat-transfer area per unit volume and can make cooling harder. Mixing time can increase. Impeller power demand changes. Gas dispersion, suspension of solids, emulsification, and mass transfer may change. Therefore, engineers choose a scale-up criterion based on what controls the process.

Common agitation scale-up criteria include constant impeller tip speed, constant power per unit volume, and constant Reynolds number. Constant tip speed helps limit shear. Constant power per volume helps preserve turbulence intensity. Constant Reynolds number preserves dynamic similarity in some flows but may not preserve mixing time or gas dispersion. For many turbulent stirred-tank systems, constant \(P/V\) is a practical starting point, but sensitive biological or polymer systems may require shear constraints.

Agitation power is often estimated by \(P=N_p\rho N^3D_i^5\). This relation shows why impeller diameter and speed matter greatly. Small changes in speed can significantly change power. Full-scale power should always be checked against motor capacity, mechanical design, shaft torque, heat input, baffle design, and process limits.

Reactor Sizing Worked Examples

Example 1: CSTR sizing. If \(F_{A0}=2.5\) mol/s, target conversion is \(X=0.8\), and the exit rate is \((-r_A)=0.8\) mol/m³·s, then:

Example 2: First-order batch time. If \(k=0.003\,s^{-1}\) and \(X=0.85\), then:

Example 3: Heat duty. If \(\Delta H_{rxn}=-75000\) J/mol, \(F_{A0}=2.5\) mol/s, and \(X=0.8\), then:

Example 4: Mixing power. If \(N_p=5\), \(\rho=1000\), \(N=4\) rev/s, and \(D_i=0.13\) m, then:

Common Reactor Scale-Up Mistakes

The first common mistake is scaling volume without checking heat removal. A full-scale reactor has less surface area per unit volume than a small vessel, so cooling can become limiting. The second mistake is assuming kinetics measured in a flask automatically apply to a plant reactor. Mixing, mass transfer, catalyst wetting, temperature gradients, and impurity effects can change the observed rate.

The third mistake is using CSTR and PFR equations interchangeably. A CSTR operates at exit composition, while a PFR has a concentration profile. The fourth mistake is ignoring high-conversion rate slowdown. The fifth mistake is scaling agitation speed directly. Constant rpm rarely gives similar mixing across scales. The sixth mistake is ignoring safety. Exothermic reactions require calorimetry, relief design, interlocks, and hazard review before scale-up.