The CSTR Design Equation

A steady-state mole balance turns reactor sizing into one algebraic step.

reaction engineeringCSTRreactor design

⏱️ About 18 min

A CSTR is the simplest continuous reactor model: once you know the exit rate, you can size the tank.

💡

The big idea: For a steady-state CSTR, the entire reactor behaves as if it were at the exit conditions, giving an algebraic design equation.

🎯 By the end, you'll be able to

Write the steady-state mole balance for a CSTR
Use conversion to rewrite the balance in design form
Apply first-order kinetics to obtain a compact sizing formula
Compute CSTR volume from v0, k, and target conversion

📎 Helpful to know first

Batch Reactor Design

The CSTR assumption

A continuous stirred-tank reactor (CSTR) is assumed to be perfectly mixed. That means the concentration inside the tank is the same as the concentration in the outlet stream.

🔑 Perfect mixing means one rate value

Because the tank is uniform, the reaction rate used in the design equation is evaluated at the exit conditions. This is why a CSTR is often larger than a PFR for the same conversion when kinetics slow down with conversion.

Steady-state mole balance on A

At steady state, accumulation is zero. A mole balance on species A becomes:

in - out + generation = 0.

For consumption of A, the generation term is -r_AV, or equivalently V(-r_A) with (-r_A) positive.

\[ F_{A0} - F_A + (-r_A)V = 0 \quad \Rightarrow \quad V = \frac{F_{A0}X}{(-r_A)_{\text{exit}}} \]

CSTR design equation in terms of conversion X, with the rate evaluated at exit conditions.

First-order, constant-density simplification

For constant-density flow with first-order kinetics, (-r_A) = k C_A. If we define v₀ as the volumetric flow rate, then F_A0 = v₀C_A0. At the exit of an ideal CSTR, C_A = C_A0(1 - X).

\[ V = \frac{F_{A0}X}{k\,C_{A0}(1-X)} = \frac{v_0 X}{k(1-X)} \]

CSTR volume for a first-order reaction under constant-density conditions.

⚠️ Evaluate the rate at the exit

A common mistake is to plug in the inlet concentration for the rate. In a CSTR, the whole tank has the exit concentration, so you must use the exit value when computing (-r_A).

📝 Worked example: A first-order liquid reaction runs in an isothermal CSTR. The volumetric flow rate is v0 = 0.010 m3/s and k = 0.20 s-1. Find the required CSTR volume for X = 0.75.

Use the first-order CSTR sizing formula: V = v0 X / (k(1 - X)).
Compute 1 - X = 0.25.
Compute numerator v0 X = 0.010 × 0.75 = 0.00750 m^3/s.
Compute denominator k(1 - X) = 0.20 × 0.25 = 0.050 s^-1.
Compute V = 0.00750 / 0.050 = 0.150 m^3.

✓ 0.150 m^3

✏️ Practice: An isothermal CSTR carries out a first-order reaction with k = 0.50 s-1. The flow rate is v0 = 0.020 m3/s. What CSTR volume is needed for X = 0.60?

m^3

Solution

Use V = v0 X / (k(1 - X)).
Compute 1 - X = 0.40.
Compute v0 X = 0.020 × 0.60 = 0.0120 m^3/s.
Compute k(1 - X) = 0.50 × 0.40 = 0.20 s^-1.
Compute V = 0.0120 / 0.20 = 0.060 m^3.

Check your understanding

1. In the ideal CSTR model, the concentration in the tank is equal to:

Perfect mixing makes the tank uniform; the tank concentration equals the outlet concentration.

2. For a steady-state CSTR mole balance, which term is zero?

At steady state, accumulation is zero by definition.

3. For first-order kinetics and constant density, the CSTR volume for a fixed X scales with:

From V = v0 X / (k(1 - X)), for fixed X the scaling is V ∝ v0/k.

✅ Key takeaways

A steady-state CSTR has zero accumulation.
Perfect mixing means the rate is evaluated at exit conditions.
The CSTR design equation is algebraic: V = FA0 X / (-rA) at exit.
For first-order, constant density: V = v0 X / (k(1 - X)).

➡️ Next we derive the PFR design equation, where the rate changes continuously along the reactor.

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