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In an industrial chemical process, an irreversible liquid-phase reaction $A \rightarrow \text{Products}$ is carried out under steady-state and isothermal conditions. Reactant A is fed into the reactor with an inlet molar flow rate ( $F_{A0}$ ) of 20 mol/s. The engineering design specification requires a fractional conversion ( $X$ ) of reactant A equal to 0.8 (80%). Kinetic evaluations indicate that at the exit concentration corresponding to this conversion, the reaction rate of reactant A ( $-r_A$ ) is 0.2 mol/(m³·s).

Calculate the required volume of an ideal Continuous Stirred-Tank Reactor (CSTR) ( $V_{\text{CSTR}}$ ) to achieve this conversion, and perform an engineering comparative analysis with an ideal Plug Flow Reactor (PFR). (Note: You can modify the variables in the input data section)

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Click to view industrial reactor comparison table (CSTR vs. PFR)

Engineering comparison criteria between steady-state CSTR and PFR:

Feature	CSTR	PFR
Mixing Behavior	Perfect instantaneous mixing (complete back-mixing)	Plug flow with zero axial back-mixing
Concentration Profile ($C_A$)	Spatially uniform throughout, exactly equals exit value ($C_{A,\text{exit}}$)	Monotonically decreases along tube length from inlet to outlet
Required Volume ($V$)	Substantially larger ($V_{\text{CSTR}} > V_{\text{PFR}}$ for positive reaction orders)	Significantly smaller; higher volumetric efficiency
Thermal Control	Excellent temperature control; easy to install internal coils or jackets	Prone to axial temperature gradients and hot spots in exothermic reactions
Typical Applications	Liquid-phase reactions, viscous media, slurries, highly exothermic reactions	Gas-phase reactions, high-throughput commodity chemicals, compact plants

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Step 1: Calculate the volume of the ideal CSTR ( $V_{\text{CSTR}}$ ) (Derived from the steady-state mole balance on reactant A: $\text{In} - \text{Out} + \text{Generation} = 0$ , giving $F_{A0} - F_A + r_A V = 0$ . Substituting $F_A = F_{A0}(1 - X)$ yields the CSTR design equation)

💡 Dimensional Consistency Check: Analyzing the units of the substituted variables: $\frac{[\text{mol/s}] \times [1]}{[\text{mol}/(\text{m}^3\cdot\text{s})]} = \frac{\text{mol}}{\text{s}} \times \frac{\text{m}^3\cdot\text{s}}{\text{mol}} = \text{m}^3$ . The units of moles ( $\text{mol}$ ) and seconds ( $\text{s}$ ) cancel out completely, yielding cubic meters ( $\text{m}^3$ ), which is the correct SI unit of volume. Multiplying by $1,000\text{ L/m}^3$ gives $80,000\text{ L}$ .

Step 2: Comparative Analysis with an Ideal Plug Flow Reactor ( $V_{\text{PFR}}$ ) (Understanding how mixing hydrodynamics govern the required reactor size)

For an ideal Plug Flow Reactor (PFR), the differential mole balance yields the integral design equation:

$V_{\text{PFR}} = F_{A0} \int_0^X \frac{dX}{-r_A}$

Assuming first-order reaction kinetics ( $-r_A = k C_A = k C_{A0}(1 - X)$ ) in a constant-density system:

In a CSTR, the reaction rate everywhere inside the tank is evaluated at the exit conversion ( $X = 0.8$ ), giving $-r_A = 0.2\text{ mol}/(\text{m}^3\cdot\text{s})$ .
In a PFR, the reaction rate begins at the inlet ( $X = 0$ ) where $-r_A = \frac{0.2}{1 - 0.8} = 1.0\text{ mol}/(\text{m}^3\cdot\text{s})$ (5 times faster than in the CSTR!) and decreases gradually along the tube length.

Integrating the PFR design equation yields:

$V_{\text{PFR}} = \frac{F_{A0}}{k C_{A0}} \ln\left(\frac{1}{1 - X}\right) = \frac{F_{A0}(1 - X)}{-r_A} \ln\left(\frac{1}{1 - X}\right)$

Substituting the numerical values:

$V_{\text{PFR}} = \frac{(20)(1 - 0.8)}{0.2} \ln\left(\frac{1}{1 - 0.8}\right) = 20 \ln(5) \approx 32.19\text{ m}^3$

Comparing the two volumes reveals a ratio of $\frac{V_{\text{CSTR}}}{V_{\text{PFR}}} = \frac{80}{32.19} \approx 2.49$ . This confirms the foundational reaction engineering principle that for normal kinetics (positive reaction order), a PFR will always require significantly less volume than a CSTR to reach the same conversion.

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