Simulation of Chromatographic Processes: Uptake Kinetics and

Lumped Body KineticsSandeep Ramesh Hadpe

Vijay MaranholkarSouhardya Roy

Modelling of Uptake Kinetics using Pore Diffusion Model

• Equation to solve:

• Nature of equation: • System of Partial Differential Equation (PDE). • 2 variables: c q • Equations: 1.

• Thus, it can not be solved directly.

• Solution: • Elimination of one of the variables• Substitution

• Reduced Equation:• Substitution:

• Elimination: q gets eliminated. • Reduced Equation: Single Partial Differential Equation. • Solver used: ‘pdepe’ solver for PDE in MatLab.

• Boundary Conditions:

• Initial Conditions:

• Bulk Fluid Concentration:• Conservation Equation:

• Valid in general.

• Analytical Form:• Valid for external film mass transfer.

• Both the conservation equation and the analytical form can be separately modelled using MatLab.

• There will be some difference between the two plots due to the extra constraint introduced in the analytical form. However, the trend should be same in both.

• The plots obtained can also be used to find the value of the parameter Film Mass Transfer coefficient kf directly using Curve Fitting in MatLab.

• This further can be used to determine De, i.e. Effective Pore Diffusion coefficient.

Conservative Equation Analytical Form

Time

C (Bulk Conc.)

Parameter EstimationResin R x 104 kf x 104

(Estimated)De x 108 (Estimated)

Resin S 75 2 1.245Resin T 45 3.3 1.1Resin K 75 1.06 2.25Resin A 35 4.5 1Resin M 75 2 1.24

35 45 75 750

0.5

1

1.5

2

2.5

3

3.5

4

4.5

54.5

3.3

2

1.061 1.1 1.24

2.25

Kf and De variation with Re

kf x 10^4(Estimated) De x 10^8 (Estimated)

Rx10^4

Modelling of Lumped Body Kinetics using Breakthrough Model

• Equation to Solve:

• Nature of Equation: • System of two Partial Differential Equations. • 2 variables: c q• Varied w.r.t: t x • Equations: 2.

• Solution using: ‘pdepe’ solver in MatLab.

• Substitution:

• Initial Conditions:

• Boundary Conditions:

• BC w.r.t q have to be designed for validating the form of the ‘pdepe’ solver.