Continuous Stirred Tank Reactor (CSTR) Calculator

Verified design equations for first‑order reactions. Compute volume, conversion, residence time, and Damköhler number with real‑time unit conversion.

Design equation: \( V = \frac{v_0 X}{k(1-X)} \)    \( X = \frac{k\tau}{1+k\tau} \)    with \( \tau = V/v_0 \)

k=0.1 1/s, X=0.5 → V=1.67 L
V=150 L, k=0.05 → X≈0.882
X=0.8, v₀=20 L/min → V=40 L
V=50 L, k=0.5 → X≈0.714

Understanding the CSTR – In Depth

A Continuous Stirred Tank Reactor (CSTR) is the idealization of a perfectly mixed reactor. The contents are uniform in composition and temperature, and the outlet stream has the same properties as the reactor contents.

1. Fundamental Assumptions

  • Perfect mixing – no spatial gradients; the reactor behaves as a single point.
  • Steady state – accumulation terms are zero (d/dt = 0).
  • Constant density (liquid phase) – volumetric flow rate in = out.
  • First‑order, irreversible reaction A → products with rate \( r_A = -k C_A \).

Under these assumptions, the mole balance on component A gives:

\( F_{A0} - F_A + r_A V = 0 \)

\( v_0 C_{A0} - v_0 C_A - k C_A V = 0 \)

\( C_A = C_{A0}(1-X) \) → \( v_0 C_{A0} X = k C_{A0} (1-X) V \)

\( \boxed{ V = \frac{v_0 X}{k(1-X)} } \)   or   \( \boxed{ X = \frac{k\tau}{1+k\tau} } \) with \( \tau = V/v_0 \)

2. Key Dimensionless Group – Damköhler Number (Da)

Da = kτ compares the characteristic reaction rate (1/k) to the residence time τ. It directly determines conversion:

  • Da ≪ 1 (reaction‑limited): X ≈ Da → conversion increases linearly with volume or k.
  • Da ≫ 1 (flow‑limited): X → 1 (complete conversion).
  • Da = 1 → X = 0.5 (50% conversion).

In the graph above, you see how X approaches 100% as volume (and thus Da) increases.

3. Comparison with Other Ideal Reactors

For the same first‑order reaction and equal volume, a Plug Flow Reactor (PFR) always gives higher conversion than a CSTR because the concentration is always higher in the PFR. The CSTR operates at the lowest concentration (outlet value), so it requires a larger volume for the same conversion. This is a key design trade‑off.

4. Thermal Effects and Energy Balance

Real CSTRs are often non‑isothermal. For exothermic reactions, the heat generated must be removed to avoid runaway. The energy balance couples with the mole balance and can lead to multiple steady states (ignition/extinction). Our calculator assumes isothermal operation, which is valid for many liquid‑phase systems or when cooling is effective.

5. Typical Rate Constants (First‑Order Approximation)

The rate constant k depends strongly on temperature via the Arrhenius law: \( k = A e^{-E_a/(RT)} \). Below are illustrative values at moderate temperatures (for learning purposes).

Reaction type k (1/s) at ~25°C k (1/s) at ~60°C
Hydrolysis of esters 10⁻⁴ – 10⁻³ 10⁻³ – 10⁻²
Enzyme catalysis 10² – 10⁴ higher but denatures
Decomposition of azo compounds 10⁻⁵ – 10⁻⁴ 10⁻⁴ – 10⁻³

6. Practical Applications

  • Wastewater treatment: Activated sludge processes are often modeled as CSTRs.
  • Polymerisation: Continuous stirred tanks are used for emulsion polymerisation.
  • Pharmaceuticals: Many liquid‑phase synthesis steps are carried out in CSTRs to maintain uniform quality.
  • Bio‑reactors: Fermenters for ethanol or antibiotics approximate CSTR behaviour.

Calculator validation: Tested with known values (e.g., v₀=10 L/min, k=0.1 1/s, X=0.5 → V = 1.667 L). The tool returns exactly that. Unit conversions are correctly applied: 1 L/min = 1.66667×10⁻⁵ m³/s, 1 1/min = 1/60 1/s, etc.

Frequently Asked Questions

This calculator is specifically for first‑order kinetics. For other orders (e.g., second‑order), the design equation changes. However, the Damköhler number concept can be extended. You can approximate by using an effective rate constant at the outlet concentration, but precise design requires numerical solution.

For gas‑phase reactions with mole change, volumetric flow rate is not constant. The design equation becomes more complex, often requiring expansion factors. This tool assumes constant density (liquid phase).

For consistency in the graph axis, volume is always displayed in cubic metres. However, your input and the main result are shown in the unit you selected (litres or m³).