Ideal Reactors (CSTR and PFR)

In chemical reaction engineering, reactors are vessels where chemical transformations occur. Two of the most foundational models for continuous, steady-state flow reactors are the Continuous Stirred-Tank Reactor (CSTR) and the Plug Flow Reactor (PFR).

These "ideal" models represent extremes of mixing behavior, allowing engineers to size reactors and predict conversion without complex computational fluid dynamics.

The Continuous Stirred-Tank Reactor (CSTR)

An ideal CSTR assumes perfect mixing. This means that the moment the reactants enter the tank, they are instantaneously and uniformly dispersed throughout the entire reactor volume.

Consequently, the temperature, concentration, and reaction rate inside the reactor are identical everywhere and match the conditions of the exit stream.

The design equation for a CSTR, derived from a steady-state mass balance (In - Out + Generation = 0), is:

$V = \frac{F_{A0} X}{-r_A}$

Where:

• $V$ = reactor volume [$\text{m}^3$]
• $F_{A0}$ = molar feed rate of reactant A [$\text{mol/s}$]
• $X$ = fractional conversion of reactant A [dimensionless, 0 to 1]
• $-r_A$ = reaction rate of A evaluated at the exit conditions [$\text{mol}/(\text{m}^3 \cdot \text{s})$]

The Plug Flow Reactor (PFR)

An ideal PFR is modeled as a cylindrical pipe with no radial mixing and no axial mixing (no back-mixing). The fluid flows through the reactor as a series of infinitely thin "plugs," each spending the exact same amount of time in the reactor.

Because the concentration of reactants decreases as the fluid travels down the tube, the reaction rate changes continuously along the length of the reactor.

The design equation for a PFR requires integration over the changing conversion:

$V = F_{A0} \int_0^X \frac{dX}{-r_A}$

Residence Time ($\tau$)

The space-time, or residence time ($\tau$), is the time necessary to process one reactor volume of fluid based on the entrance volumetric flow rate ($v_0$):

$\tau = \frac{V}{v_0}$

It provides a useful measure of how long, on average, a molecule remains inside the reactor.

Worked Example

Problem: A first-order liquid-phase reaction ($A \rightarrow \text{Products}$) with a rate constant $k = 0.5 \text{ min}^{-1}$ is carried out isothermally. The required conversion is 80% ($X = 0.8$). The initial concentration is $C_{A0} = 2 \text{ mol/L}$ and the volumetric flow rate is $v_0 = 10 \text{ L/min}$. Calculate the required volume for an ideal CSTR.

Solution:

1. Determine the exit concentration of A: $C_A = C_{A0}(1 - X) = 2(1 - 0.8) = 0.4 \text{ mol/L}$
2. Determine the reaction rate at exit conditions (first-order): $-r_A = k C_A = (0.5)(0.4) = 0.2 \text{ mol/(L}\cdot\text{min)}$
3. Determine the molar feed rate: $F_{A0} = C_{A0} v_0 = (2)(10) = 20 \text{ mol/min}$
4. Apply the CSTR design equation: $V = \frac{F_{A0} X}{-r_A} = \frac{20 \times 0.8}{0.2} = \frac{16}{0.2} = 80 \text{ L}$

The required volume for the CSTR is 80 liters.

Engineering Check

For reactions with typical kinetics (where rate decreases as reactant concentration drops), a PFR will always require less volume than a CSTR to achieve the same conversion. This is because the PFR operates at higher average concentrations (and thus higher average rates) throughout its length, whereas the CSTR operates entirely at the lowest concentration (the exit concentration). Real-world reactors (which suffer from dead zones, channeling, and short-circuiting) will always fall somewhere between these two ideal limits.