Equations, derivations & Jacobians

Every equation the solver works with, derived from first principles and shown with its exact Jacobian. The derivatives aren't hand-typed — they're computed by the same forward-mode automatic-differentiation engine the solver uses, and every symbolic derivative on these pages is checked against it in our test suite. See the math, not just the answer.

Phase equilibrium

Rachford-Rice flash

The vapor-fraction objective every isothermal two-phase flash solves — one nonlinear equation in the vapor fraction ψ, monotonic so Newton always converges.

Derivation & Jacobian

Bubble-point temperature (Raoult + Antoine)

At the bubble point the vapor mole fractions sum to one; with Raoult's law and Antoine vapor pressures this is one nonlinear equation in temperature.

Derivation & Jacobian

Heat transfer

Heat exchanger duty (LMTD)

The log-mean-temperature-difference rating equation, and the sharply nonlinear sensitivity of duty to the terminal temperature approach.

Derivation & Jacobian

Material & energy balance

Adiabatic mixer energy balance

Two streams mix adiabatically; the outlet temperature is the enthalpy-weighted average of the inlets — the linear building block of every energy balance.

Derivation & Jacobian

Reaction

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.

Derivation & Jacobian

Separation

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.

Derivation & Jacobian

Hydraulics

Centrifugal pump head curve

A centrifugal pump's developed pressure follows its quadratic head-vs-flow curve — the flow-dependent pressure rise the pressure-driven solver uses.

Derivation & Jacobian

Equations, derivations & Jacobians

Phase equilibrium

Rachford-Rice flash

Bubble-point temperature (Raoult + Antoine)

Heat transfer

Heat exchanger duty (LMTD)

Material & energy balance

Adiabatic mixer energy balance

Reaction

CSTR mole balance (2nd-order)

Separation

Distillation MESH stage balances (tridiagonal)

Hydraulics

Centrifugal pump head curve

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