Distillation MESH stage balances (tridiagonal)

A three-stage slice of a distillation column at fixed K-values and constant molal overflow — the component material balances form the classic tridiagonal system the inside-out inner loop solves.

Governing equation

$x_j$

Liquid mole fraction of the light component on stage j — the unknowns.

$L$

Liquid molal overflow (mol s⁻¹), constant.

$V$

Vapor molal overflow (mol s⁻¹), constant.

$K$

Equilibrium ratio y/x (fixed in the inner loop).

$f_j$

Feed / boundary source on stage j (reflux on top, feed on the middle stage).

Derivation

1. 1
   $$L_{j-1} x_{j-1} + V_{j+1} y_{j+1} + f_j = L_j x_j + V_j y_j$$

   Component material balance on stage j: liquid down from above plus vapor up from below plus feed equals liquid and vapor leaving.

2. 2
   $$y_j = K_j x_j$$

   Equilibrium closes the vapor composition; in the inside-out inner loop the K-values are frozen, making the balances linear in x.

3. 3
   $$L\,x_{j-1} + V K\,x_{j+1} - (L + V K)\,x_j + f_j = 0$$

   Constant molal overflow makes L and V stage-independent; each row couples only to its neighbours, so the system is tridiagonal (the Thomas algorithm).

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

• Seader, Henley & Roper, Separation Process Principles, 3rd ed., Ch. 10.
• Boston, J.F. & Sullivan, S.L. (1974). A new class of solution methods for multicomponent, multistage separation processes. Can. J. Chem. Eng. 52.