What assumptions does this batch reactor design calculator make?

This calculator assumes a perfectly mixed, isothermal, constant-volume batch reactor with a single irreversible reaction A → products. The rate law is taken as rA = −k CA^n with reaction orders n = 0, 1, or 2. No mass transfer limitations, side reactions, or heat effects are included.

How do I size a batch reactor for a target conversion?

To size a batch reactor, choose the reaction order and rate constant k, specify the initial concentration CA0 and desired conversion X, then compute the required reaction time t using the integrated rate expression. The reactor volume is then V = F_A0 · t / (CA0 · X), where F_A0 is the molar feed rate of A. This tool automates these steps for zero, first, and second-order reactions.

Can I use this tool for non-isothermal or variable-volume batch reactors?

Not directly. Non-isothermal or variable-volume batch reactors typically require solving coupled differential equations for energy and mass balances, often numerically. This calculator is intended for quick conceptual design and homework-level problems under isothermal, constant-volume assumptions.

What units should I use for the rate constant k?

The units of k depend on the reaction order. For zero-order, k has units of concentration/time (e.g., mol·L⁻¹·s⁻¹). For first-order, k has units of 1/time (e.g., s⁻¹). For second-order, k has units of 1/(concentration·time) (e.g., L·mol⁻¹·s⁻¹). Make sure your k units are consistent with the concentration and time units you choose.

Batch Reactor Design Calculator (Isothermal, Constant Volume)

Batch reactor design theory (isothermal, constant volume)

A batch reactor is a closed vessel where reactants are charged at the beginning of the batch, allowed to react for a specified time, and then discharged. For a single irreversible reaction A → products in an isothermal, constant-volume batch reactor, the mole balance on A is:

\[ \frac{dN_A}{dt} = r_A V \] \[ N_A = C_A V,\quad r_A = -k C_A^n \] \[ \Rightarrow \frac{dC_A}{dt} = -k C_A^n \]

Conversion definition

Conversion of A is defined as:

\[ X = \frac{N_{A0} - N_A}{N_{A0}} = 1 - \frac{C_A}{C_{A0}} \quad\Rightarrow\quad C_A = C_{A0}(1 - X) \]

Integrated rate expressions

Integrating the rate law from \(t = 0\) (where \(X = 0\)) to time \(t\) (conversion \(X\)) gives the design equations used in this calculator.

Zero-order reaction (n = 0)

\[ \frac{dC_A}{dt} = -k \quad\Rightarrow\quad C_A = C_{A0} - kt \] \[ X = 1 - \frac{C_A}{C_{A0}} = \frac{kt}{C_{A0}} \] \[ t = \frac{X C_{A0}}{k} \]

First-order reaction (n = 1)

\[ \frac{dC_A}{dt} = -k C_A \quad\Rightarrow\quad \int_{C_{A0}}^{C_A} \frac{dC_A}{C_A} = -k \int_0^t dt \] \[ \ln\left(\frac{C_A}{C_{A0}}\right) = -kt \quad\Rightarrow\quad C_A = C_{A0} e^{-kt} \] \[ X = 1 - e^{-kt} \quad\Rightarrow\quad t = -\frac{1}{k}\ln(1 - X) \]

Second-order reaction (n = 2)

\[ \frac{dC_A}{dt} = -k C_A^2 \quad\Rightarrow\quad \int_{C_{A0}}^{C_A} \frac{dC_A}{C_A^2} = -k \int_0^t dt \] \[ -\left(\frac{1}{C_A} - \frac{1}{C_{A0}}\right) = -kt \quad\Rightarrow\quad \frac{1}{C_A} = \frac{1}{C_{A0}} + kt \] \[ C_A = \frac{C_{A0}}{1 + k C_{A0} t} \quad\Rightarrow\quad X = 1 - \frac{1}{1 + k C_{A0} t} \] \[ t = \frac{X}{k C_{A0} (1 - X)} \]

Relating batch time to reactor volume

For conceptual sizing, you often know the desired production rate (moles of A processed per unit time) and the batch cycle time. If the molar feed rate of A is \(F_{A0}\) (e.g., mol/s) and each batch runs for time \(t\) to reach conversion \(X\), then the moles of A consumed per batch are:

\[ N_{A,\text{consumed}} = C_{A0} V X \]

To match a desired average throughput, you can rearrange to estimate the required reactor volume:

\[ V = \frac{F_{A0} \, t}{C_{A0} X} \]

This calculator uses this relationship when you provide both \(F_{A0}\) and the computed batch time \(t\).

Worked example

Suppose you have a first-order liquid-phase reaction with \(k = 0.2\ \text{s}^{-1}\), initial concentration \(C_{A0} = 1.0\ \text{mol·L}^{-1}\), and you want 90% conversion (\(X = 0.9\)).

Choose first-order kinetics and enter \(k = 0.2\ \text{s}^{-1}\), \(C_{A0} = 1.0\ \text{mol/L}\), \(X = 0.9\).
The design equation for first-order is \(t = -\ln(1 - X)/k\).
Compute \(t = -\ln(0.1)/0.2 \approx 11.51\ \text{s}\).
The concentration at this time is \(C_A = C_{A0}(1 - X) = 0.1\ \text{mol/L}\).

If you also specify a molar feed rate \(F_{A0} = 10\ \text{mol/s}\), the required reactor volume to process this feed in a single batch is:

\[ V = \frac{F_{A0} t}{C_{A0} X} = \frac{10 \times 11.51}{1.0 \times 0.9} \approx 128\ \text{L} \]

Frequently asked questions

What assumptions does this calculator make?

The tool assumes:

Perfect mixing (spatially uniform concentration and temperature).
Isothermal operation (temperature constant in time).
Constant liquid volume (no significant density or volume change).
Single irreversible reaction A → products with simple power-law kinetics.

How do I choose the correct reaction order?

The reaction order should come from kinetic experiments or literature data. Plotting \(\ln C_A\) vs. \(t\) (first-order), \(1/C_A\) vs. \(t\) (second-order), or \(C_A\) vs. \(t\) (zero-order) and checking linearity is a common approach.

Can I use this for gas-phase or non-ideal systems?

For gas-phase reactions with significant pressure or volume changes, or for non-ideal mixtures, you should work in terms of partial pressures or activities and may need a more detailed model. This calculator is best suited for liquid-phase or pseudo-liquid systems where the constant-volume assumption is reasonable.

How can I extend this to non-isothermal design?

Non-isothermal batch reactor design couples the mole balance with an energy balance, often requiring numerical integration of ODEs. Typical steps include specifying heat of reaction, heat transfer coefficient, jacket temperature, and solving for \(T(t)\) and \(C_A(t)\) simultaneously using tools like MATLAB, Python, or process simulators.

Batch Reactor Design Calculator

Batch Reactor Design Inputs

Results

Key outputs

Conversion vs. time (sample points)