CSTR mole balance (2nd-order)

The steady-state mole balance on a continuous stirred-tank reactor with a second-order rate law — outlet concentration from a single nonlinear balance.

Governing equation

$C$

Outlet (= tank) concentration (mol m⁻³) — the unknown.

$\dot V$

Volumetric flow rate (m³ s⁻¹).

$C_0$

Inlet concentration (mol m⁻³).

$k$

Second-order rate constant (m³ mol⁻¹ s⁻¹).

$V$

Reactor volume (m³).

Derivation

1. 1
   $$\frac{dN}{dt} = \dot V C_0 - \dot V C + r_A V = 0$$

   Mole balance on the well-mixed tank at steady state: in − out + generation = 0.

2. 2
   $$r_A = -k\,C^2$$

   Second-order rate law; the tank is at the outlet composition because it is perfectly mixed.

3. 3
   $$F(C) = \dot V (C_0 - C) - k\,C^2 V = 0$$

   A single nonlinear (quadratic) equation for the outlet concentration.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Fogler, Elements of Chemical Reaction Engineering, 5th ed., Ch. 2 & 5.