Liquid Chromatography

Chromatography is an important group of methods to separate closely related components of complex mixtures. The following example simulates the separation of species in High Performance Liquid Chromatography (HPLC). In this technique an injector introduces a sample as a zone in a liquid mobile phase. The mobile phase containing the sample zone is pumped through a column containing a solid stationary phase; Figure 1 shows a diagram of such an instrument.

Figure 1: Diagram of an HPLC system.

The mobile and stationary phases are chosen so that the samples are distributed to varying degrees between the two phases. Those components that strongly adsorb to the stationary phase move only slowly with the flow of the mobile phase, and those that are weakly adsorbed move more rapidly. As the sample zones progress through the column, the components are separated into discrete zones that are recognized by a detector, situated beyond the outlet of the column.

Model Definition

This model studies the separation of two species under conditions of nonlinear chromatography in a 1D geometry. The Transport of Diluted Species in Porous Media interface is used, with convection and adsorption in porous media accounted for.

The equation for analyte transport through a chromatographic column, with constant porosity, is computed by:

(1)

Here, ci is the concentration of component i (SI unit: mol/m3), ε is the porosity, ρ is the density of the media within the column (for the mix of liquid and solid matrix, SI unit: kg/m3), kP,i is an adsorption isotherm, and u is the volume average velocity of the fluid phase (SI unit: m/s). The second term on the right-hand side describes the mixing of the solutes, including mechanical mixing (dispersion) and molecular diffusion. The two last terms on the right-hand side are a reaction rate term and a fluid source term.

Equation 1 comes from the following derivation:

First, consider the dispersion of the chromatographic zone to be negligible as it progresses through the column. The mass transport equation will then take the form:

where S denotes the specific surface area of the particles in the column (SI unit: m2/kg), ρp denotes the density of the solid particles (SI unit: kg/m3), ε equals the column porosity, A gives the inner area of the column tube, ni equals the analyte concentration in the stationary phase of component i (SI unit: mol/m2), v describes the mobile phase flow (SI unit: m3/s), and ci equals the analyte concentration in the mobile phase of component i (SI unit: mol/m3).

Equation 1 is defined according to the ideal model for chromatography that assumes that the equilibrium for the analyte between the mobile and stationary phases is immediate, that is:

where cP,i is the concentration of the component adsorbed to the solid (moles per dry unit).

The mass transport equation for the ideal chromatography model therefore becomes:

The dispersion or band broadening of the analyte zone is a result of a great number of random processes that the analyte experiences (for example, inhomogeneous flow and diffusion in pores and the mobile phase). It is therefore possible to formally express the band broadening as a diffusion process with an effective diffusion constant, Deff. Thus, Deff is a measure of the chromatographic system’s efficiency for a particular analyte. This constant is closely related to the concept of the height equivalent of a theoretical plate, H, that is customarily used in chromatographic practice. It can be shown that:

where vzi is the migration velocity of the analyte zone through the column. A mass balance that includes the zone-dispersion term gives the following equation:

Here Φ = Sρ( 1 − ε )/ε denotes the phase ratio of the column (SI unit: m2/m3), vl = v/(εA) gives the linear velocity of the mobile phase in the column (SI unit: m/s), and Deff is the effective diffusion constant (SI unit: m2/s).

This first example covers two components. The adsorption isotherm for both components is assumed to follow a Langmuir adsorption isotherm, that is,

and

where Ki is the adsorption constant for component i (SI unit: m3/mol), and n0i is the monolayer capacity of the stationary phase for component i (SI unit: mol/m2).

Using an effective zone-dispersion term and the kP,i adsorption isotherm notation gives Equation 1 without the reaction rate and a fluid source terms:

Input Data

This example looks at the progress of the chromatographic zone within the column. The physical data for the column correspond to a 12 cm-by-4 mm inner diameter column filled with 5 μm porous particles. The rest of the input data appear in Table 1.

Table 1: input data.
name	Value
S	100 m2/g
ρp	2300 kg/m3
ε	0.6
vl	1.322 mm/s
Deff1	1⋅10-8 m2/s
Deff2	1⋅10-8 m2/s
K1	0.04 m3/mol
K2	0.05 m3/mol
n01	1⋅10-6 mol/m2
n02	5⋅10-7 mol/m2

The injector concentrations for the two components are described by a normal distribution and are set up with the help of a Gaussian pulse function with an amplitude of 1 (Figure 2).

Figure 2: Injection pulse with amplitude 1.

The maximum inlet concentrations are 1 mol/m3 and 10 mol/m3 for components 1 and 2, respectively.

Results and Discussion

Figure 3 shows the mobile concentration zones of component 1 at various times. Initially, the concentration is zero in the column and at approximately 5 s the whole component mixture has been injected through the leftmost boundary (the inlet). At approximatively 385 s, the first trace of component 1 exits the rightmost boundary (the outlet). The zones are nearly symmetrical and normally distributed, indicating that the solution is affected by an almost linear adsorption isotherm.

Figure 3: The concentration of component 1 in the mobile phase at various times.

In Figure 4, the zones for both of the components are displayed at various times. For the present conditions a clear separation of the two components occurs within the column; the component zones do not overlap as they reach the outlet. As an example, at t=300 s, component 2 is no longer present in the mobile phase of the column, while most of component 1 still remains. Here, it is shown that component 2 depends on a nonlinear adsorption isotherm, quickly obtaining an asymmetrical mobile concentration zone (cf. Figure 2).