Batch Reactors perfectly stirred : reaction conversion
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1. Reaction conversion
2. Batch reactor : reaction speed as a function of conversion
3. Interactive Reaction Kinetics & Conversion Calculator
4. Conversion in case of multiple reactions
It can be interesting to define a conversion rate in order to express the reaction speed. The conversion is based on the limiting reactant in a reaction.
1. Reaction conversion
What is the conversion in a chemical reaction ?
The reaction conversion (CSTR reaction conversion) is a measure of the progress of the reaction referring to the limiting reactant. The reaction will indeed not be able to go further once one of the reactants is consumed. The conversion rate of the reaction, based on the limiting reactant (named A hereafter), can then be defined as \(X_A\) by the following equation:
\[n_A = n_{A,0} \cdot (1 + \nu_A \cdot X_A)\]
With:
\(n_A\) = quantity of the limiting reactant A at time t (mol)
\(n_{A,0}\) = quantity of the limiting reactant A at \(t=0\) (mol)
\(\nu_A\) = stoichiometric coefficient associated with the limiting
reactant A in the reaction considered. As we refer to a reactant,
\(\nu_A < 0\)
\(X_A\) = conversion rate relatively to the limiting reactant A
At \(t=0\) : \(X_A = 0\)
At \(t = \text{end of reaction}\) : \(X_A =
-1/\nu_A\)
2. Batch reactor : reaction speed as a function
of conversion
The mass balance in a batch reactor, perfectly stirred and isothermal, shows that the reaction speed of a reactant A is:
\[r_A = \frac{1}{V \cdot \nu_A} \cdot \frac{dn_A}{dt}\]
For the limiting reactant A, the quantity of material at an instant \(t\) is:
\[n_A = n_{A,0} \cdot (1 + \nu_A \cdot X_A)\]
Thus the reaction rate can be expressed as a function of the conversion:
\[r_A = \frac{n_{A,0}}{V} \cdot \frac{dX_A}{dt}\]
When the reactor has a constant volume, we can use concentration (mol/L or mol/m\(^3\)) instead of quantity of material (mol), which is very often more practical. The reaction speed as a function of the conversion becomes:
\[r_A = [A]_0 \cdot \frac{dX_A}{dt}\]
Batch Reactor Kinetics & Conversion Calculator
Inputs
Calculated Outputs
Initial Limiting Reactant (\(n_{A,0}\)): 1000.0 mol
Remaining Reactant (\(n_A\)): 200.0 mol
Concentration at Target (\([A]_t\)): 0.400 mol/L
Initial Reaction Rate (\(r_{A,0}\)): 0.010 mol/(L·s)
Reaction Rate at Target (\(r_{A,t}\)): 0.002 mol/(L·s)
Calculated Batch Time (\(t\)):
Time Required: 321.9 seconds (5.36 min)
Conversion Progress: 80%
💡 Batch Reactor Engineering Rules of Thumb
- Conversion Economics: Sizing optimization typically targets 85% to 95% conversion. Attempting to reach 99%+ conversion exponentially increases reaction residence times and decreases overall unit productivity.
- Heat Transfer Limits: High conversion targets of highly exothermic reactions can yield high adiabatic temperature rises. Reactor cooling profiles must match the maximum kinetic rate (which occurs at \(t=0\) when \(X_A = 0\)).
- Gas-Phase Volume Variance: The constant-volume assumption safely holds for liquid-phase systems. If gas-phase batch operations change total moles, use the expansion factor \(\epsilon_A\) to model density variations.
- Mass Transfer Limitations: Verify reactor impeller power inputs. Viscosity shifts during high-conversion polymerization reactions can dramatically disrupt perfect stirring, creating hot spots.
3. Conversion in case of multiple reactions
When the limiting reactant is involved in multiple reactions, the relations above can be generalized the following way:
\[n_A = n_{A,0} + n_{A,0} \cdot \sum_{i=1}^{R} \nu_{A,i} \cdot X_{A,i}\]
If the reactor is at constant volume, we also have:
\[r_i = [A]_0 \cdot \frac{dX_{A,i}}{dt}\]