Kinetic Theory

7.3. Kinetic Theory#

7.3.1. Rate Theory Models#

In contrast to the stochastic models, there are models derived from the following differential equation known as the transport-dispersive equation (or the convection-diffusion equation).

\[ \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} = D \frac{\partial^2 C}{\partial x^2} - k(C - C^*) \]

where:

  • \(C\) is the solute concentration in the mobile phase,

  • \(C^*\) is the concentration in the stationary phase,

  • \(u\) is the linear velocity of the mobile phase,

  • \(D\) is the axial dispersion coefficient,

  • \(k\) is the rate constant for exchange between phases,

  • \(x\) is the position along the column,

  • \(t\) is time.

These models are often classified into the following three according to the levels of simplification.

Model

Mass Transfer Kinetics

Axial Dispersion

Phase Equilibrium

GRM

Explicit, detailed

Yes

Non-instantaneous

LKM

Lumped, simplified

Yes

Non-instantaneous

EDM

Neglected (instantaneous)

Yes

Instantaneous

Simply in set inclusion notation:

\[ GRM \supseteq LKM \supseteq EDM \]

7.3.2. Equilibrium Dispersive Model#

Rate theory models are usually computed numerically because the differential equation can be solved analytically only in special cases. However, for EDM, [URMQ21] solved it analytically under applicable conditions, which we use as an additional constraint to improve the decomposition.

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