Derivation of the dimensionless mole balance

Two equations (the component mole balance and the overall mole balance) govern the mass and mole relationship between the pre-reaction and post-reaction state:

where

Fin is the incoming molar flow in kmol/s

zin is the incoming overall molar composition

Fout is the outgoing molar flow in kmol/s

zout is the outgoing molar composition

In this first step of derivation, we still consider Ratec an extensive rate of reaction (in kmol/s) per component. When we sum up Ratec over all components, we get the overall change in moles in the considered system.

In the next step of the derivation, we insert the equation for the component mole balance into the equation for the overall mole balance. We then divide the resulting equation by Fin, assuming that Fin does not equal zero.

We then apply a symbolic substitution of to arrive at the recommended equation form for equilibrium reactions:

No equation ever explicitly calculates the value of dz; the equation system implicitly defines dz.

This approach has several notable benefits:

• Because we symbolically substitute in dz, there is no chance of a division by zero happening, since the reaction equations fully define dz instead of the feed rate.

• dz is independent of the molar feed flow and only depends on the molar feed composition. As Fin approaches zero, the magnitude of dz stays constant since the reaction now depends only on zin. This has beneficial numerical effects in that the solver can recover from an intermittent zero flow. Even with a specified flow of zero, the solver can find a solution if the composition is well defined.

• Even with zero flow, the equation system has better scaling properties, as both the reaction and phase equilibrium are independent of the total amount of material flowing through the system. This is especially desirable in modelling chemical plants since some design variants may require zero flow in some parts of the simulation.