Batch Reactor Theory
Introduction
A batch reactor is a closed system where reactants are charged, reaction proceeds over time, and products are discharged after a specified reaction time. Unlike continuous reactors, composition changes with time rather than position. Batch reactors are widely used in pharmaceutical, specialty chemical, and fine chemical industries.
Mass Balance
For a batch reactor, the mole balance for species A becomes:
dNA/dt = rA × V
For constant volume and using concentration (CA = NA/V):
dCA/dt = rA
In terms of conversion X (where CA = CA0(1-X)):
dX/dt = -rA / CA0
Reaction Kinetics
The simulator supports several reaction types:
First Order: A → B
-rA = k × CA
Rate proportional to reactant concentration
Second Order: 2A → B
-rA = k × CA²
Rate proportional to concentration squared
Second Order: A + B → C
-rA = k × CA × CB
Rate depends on both reactant concentrations
Reversible: A ⇌ B
-rA = kfCA - krCB
Equilibrium limited reaction
Arrhenius Equation
The temperature dependence of reaction rate constants follows the Arrhenius equation:
k = A × exp(-Ea / RT)
- A = Pre-exponential factor (frequency factor)
- Ea = Activation energy (J/mol)
- R = Gas constant (8.314 J/mol·K)
- T = Absolute temperature (K)
Energy Balance (Non-isothermal)
For non-isothermal operation, the energy balance must be solved simultaneously:
dT/dt = [(-ΔHrxn)(-rA) - UA(T-Tj)] / (ρCp)
- ΔHrxn = Heat of reaction (J/mol)
- UA = Overall heat transfer coefficient × Area (W/K)
- Tj = Jacket/coolant temperature (K)
- ρCp = Volumetric heat capacity (J/m³·K)
Exothermic reactions (ΔHrxn < 0) release heat and can lead to thermal runaway if cooling is insufficient.
Batch Time Calculation
For isothermal operation, batch time can be calculated analytically for simple kinetics:
First Order
t = (1/k) × ln(1/(1-X))
Second Order (2A → B)
t = X / (k × CA0 × (1-X))
References
- Fogler, H.S. "Elements of Chemical Reaction Engineering"
- Levenspiel, O. "Chemical Reaction Engineering"
- Smith, J.M. "Chemical Engineering Kinetics"