Models.chem.Liquid Chromatography 1

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1 | L I Q U I D C H R O M A T O G R A P H Y

L i q u i d Ch r oma t o g r a ph y

Introduction

Chromatography is an important group of methods to separate closely related

components of complex mixtures. The following example simulates the separation 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.

Mobile phase

reservoir

Pump Injector Column Detector

Figure 1: Diagram of an HPLC system.

The mobile and stationary phases are chosen so that the sample components 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

migrate through the column, the components are separated into discrete zones that

are recognized in a detector, situated after the outlet of the column.

Model Definition

This model studies the separation of two species under conditions of nonlinear

chromatography, where the analyte concentration is high, and in a 1D geometry.

The governing equation for analyte transport through a chromatographic column,

with constant porosity, is

(1)

Here, ci is the concentration of component i (SI unit: mol/m3), ε is the porosity, ρb

is the bulk density of the column solid phase (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

ε ρ+ bkP i,( )t∂

∂ciu ∇ci⋅+ ∇ DD i, ετF i, DF i,+( ) ci∇[ ]⋅ Ri S+ +

i=




2 | L I Q U I D C H R O M A T O G R A P HY

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.

If you neglect the dispersion of the chromatographic zone during the migration, the

mass transport equation takes the form

where S denotes the specific surface area of the particles in the column (SI unit: m2/

kg), ρ 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 is 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).

The ideal model for chromatography, which have been applied in Equation 1, assumes

that the equilibrium for the analyte between the mobile and stationary phases is

immediate, that is:

where cP ,i is the concentration of 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 that is obtained during the

migration through the column 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 asa 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

Sρ 1 ε–( )

ε--------------------

∂ni

∂t--------⋅

∂ci

∂t-------+

v

ε A-------

∂ci

∂ x-------⋅–=

∂cP i,

∂t-------------

cP i,∂

ci∂------------

∂ci

∂t-------⋅ kP i,= =

1 Sρ 1 ε–( )

ε--------------------

dn

dc-------⋅+

∂ci

∂t-------⋅

v

ε A-------

∂ci

∂ x-------–=





where v zi 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/

(ε A) 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 the migration within the column of two components. The

adsorption isotherm for both components is assumed to follow a Langmuir adsorption

isotherm, that is,

and

where K i 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 gives the following 1D equation:

I N P U T D A T A

This example looks at the migration of a chromatographic zone migrating only withinthe column. The physical data for the column correspond to a 12 cm-by-4 mm inner

diameter column filled with 5 µm porous particles (εcolumn= 0.6) with a specific

surface area of 100 m2/g, which gives a column phase ratio, Φ, of 1.533·108 m−1.

Def f

Hv zi

2------------=

1 Φ dn

dc-------⋅+

∂ci

∂t-------⋅ vl

∂ci

∂ x------- Def f

∂2ci

∂ x2

----------+–=

ni

n0i K ici

1 K ici+

--------------------=

dni

dci

---------

n0i K i

1 K ici+( )2

----------------------------=

ε ρ+ bkP i,( )∂ci

∂t------- u

∂ci

∂ x-------+

∂∂ x------ Def f i,

∂ci

∂ x-------

=





Additional input data appear in Table 1.

NAME VALUE

vl 2.22 mm/s

Deff1 1⋅10-8 m2/s

Deff2 3⋅10-8 m2/s

K 1 0.04 m3/mol

n01 1⋅10-6 mol/m2

K 2 0.05 m3/mol

n02 5⋅10-7 mol/m2

The initial concentrations for the two components are described by the normal

distribution

where a equals 1·105 m−2, and the starting point at t = 0 is at x = 0.01 m. You firstsolve the model for the initial injector concentrations c01 = c02 = 0.1 mol/m3, which

corresponds to the linear regime for the adsorption isotherm. In a second stage, you

increase the injector concentrations to c01 = 1 mol/m3 and c02 = 10 mol/m3.

Results and Discussion

Figure 2 shows the zones of the two components at various times (0 s, 80 s, and 160 s)as they migrate within the column. In this example, the analyte concentration at t = 0

is low for both components (c0i = 0.1 mol/m3); the solution is in the linear domain

TABLE 1: INPUT DATA

c0i x( ) c0i e a x 0.01–( )

2–

⋅=





of the adsorption isotherm. This implies that the zones are symmetrical and normally

distributed.

Figure 2: The concentrations of components 1 (solid) and 2 (dashed) in the mobile phaseduring the migration through the column at times 0, 80, and 160 s for initial injectorconcentrations c 01 = c 02 = 0.1 mol/m 3 .

When the initial concentrations increase, c01= 1 mol/m3 and c02 = 10 mol/m

3, the

solution enters into the nonlinear domain of the adsorption isotherm, changing the

behavior radically (see Figure 3). A comparison of the zone for component 2 between





the two simulations clearly shows that both the zone width and form is strongly

influenced by migrating into the adsorption isotherm’s nonlinear domain.

Figure 3: The concentrations of components 1 (solid) and 2 (dashed) in the mobile phaseduring migration through the column at times 0 s, 80 s, and 160 s for initial injectorconcentrations c 01 = 1 mol/m 3 and c 02 = 10 mol/m 3 .

References

1. D. DeVault, “The Theory of Chromatography,” J. Am. Chemical Soc., vol. 65, pp.

532–540, 1943.

2. S. Golshan-Shirazi and G. Guiochon, “Analytical solution for the ideal model of

chromatography in the case of a Langmuir isotherm,” Analytical Chemistr y , vol. 60,

no. 21, pp. 2364–2374, 1988.

3. B. Lin and G. Guiochon, Modeling for Preparative Chromatography , Elsevier

Publishing, Amsterdam, 2003.

Application Library path: Chemical_Reaction_Engineering_Module/

Mixing_and_Separation/liquid_chromatography_1





Modeling Instructions

From the File menu, choose New.

N E W

1 In the New window, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizard window, click 1D.

2 In the Select physics tree, select Chemical Species Transport>Transport of Diluted

Species in Porous Media (tds).

3 Click Add.

4 In the Number of species text field, type 2.

5 Click Study.

6 In the Select study tree, select Preset Studies>Time Dependent.

7 Click Done.

G L O B A L D E F I N I T I O N S

Parameters

1 On the Home toolbar, click Parameters.

2 In the Settings window for Parameters, locate the Parameters section.

3 In the table, enter the following settings:

Name Expression Value Description

Phi 1.533e8[1/m] 1.533E8 1/m Phase ratio

v_l 2.22[mm/s] 0.00222 m/s Linear velocity, mobile

phase

D_eff1 1e-8[m^2/s] 1E-8 m²/s Effective diffusion

constant, component 1

D_eff2 1e-8[m^2/s] 1E-8 m²/s Effective diffusion

constant, component 2

c01 0.1[mol/m^3] 0.1 mol/m³ Initial injector

concentration,

component 1

c02 0.1[mol/m^3] 0.1 mol/m³ Initial injector

concentration,

component 2





G E O M E T R Y 1

Interval 1 (i1)

1 On the Geometry toolbar, click Interval.

2 In the Settings window for Interval, locate the Interval section.

3 In the Right endpoint text field, type 0.12.

4 Click the Build All Objects button.

5 Click the Zoom Extents button on the Graphics toolbar.

T R A N S P O R T O F D I L U T E D S P E C I E S I N P O R O U S M E D I A ( T D S )

To obtain a properly resolved solution, use quadratic elements.

1 In the Model Builder window’s toolbar, click the Show button and select Discretization

in the menu.

2 In the Model Builder window, under Component 1 (comp1) click Transport of Diluted

Species in Porous Media (tds).

3 In the Settings window for Transport of Diluted Species in Porous Media, click to

expand the Discretization section.

4 From the Concentration list, choose Quadratic.

K1 0.04[m^3/mol] 0.04 m³/mol Adsorption constant,

component 1

K2 0.05[m^3/mol] 0.05 m³/mol Adsorption constant,

component 2

n01 1e-6[mol/m^2] 1E-6 mol/m² Monolayer capacity,

component 1

n02 5e-7[mol/m^2] 5E-7 mol/m² Monolayer capacity,

component 2

a 1e5[1/m^2] 1E5 1/m² Normal-distribution

parameter

x0 10[mm] 0.01 m Starting point

epsp 0.6 0.6 Porosity

rho 2300[kg/m^3] 2300 kg/m³ Solid material density

S 1e5[m^2/kg] 1E5 m²/kg Particle specific

surface area

v_in 1.332[mm/s] 0.001332 m/s Linear velocity, mobile

phase






5 Locate the Transport Mechanisms section. Select the Adsorption in porous media check

box.

Porous Media Transport Properti es 1

1 In the Model Builder window, expand the Transport of Diluted Species in Porous Media

(tds) node, then click Porous Media Transport Properties 1.

2 In the Settings window for Porous Media Transport Properties, locate the Model

Inputs section.

3 Specify the u vector as

4 Locate the Matrix Properties section. From the εp list, choose User defined. In the

associated text field, type epsp.

5 From the ρ list, choose User defined. In the associated text field, type

rho*(1-epsp).

6 Locate the Diffusion section. In the DF ,c1 text field, type D_eff1.

7 In the DF ,c2 text field, type D_eff2.

8 Locate the Adsorption section. In the kL,c1 text field, type K1.

9 In the cP ,max,c1 text field, type S*n01.

10 In the kL,c2 text field, type K2.

11 In the cP ,max,c2 text field, type S*n02.

Initial Values 11 In the Model Builder window, under Component 1 (comp1)>Transport of Diluted

Species in Porous Media (tds) click Initial Values 1.

2 In the Settings window for Initial Values, locate the Initial Values section.

3 In the c1 text field, type c01*exp(-a*(x-x0)^2).

4 In the c2 text field, type c02*exp(-a*(x-x0)^2).

Inflow 1

1 On the Physics toolbar, click Boundaries and choose Inflow.

2 Select Boundary 1 only.

Outflow 1

1 On the Physics toolbar, click Boundaries and choose Outflow.

2 Select Boundary 2 only.

v_in x





M E S H 1

Size

1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and

choose Edge.

2 In the Settings window for Size, locate the Element Size section.

3 Click the Custom button.

4 Locate the Element Size Parameters section. In the Maximum element size text field,

type 5e-5.

5 Click the Build All button.

S T U D Y 1

Step 1: Time Dependent

1 In the Model Builder window, expand the Study 1 node, then click Step 1: Time

Dependent.

2 In the Settings window for Time Dependent, locate the Study Settings section.

3 In the Times text field, type range(0,10,180).

4 Select the Relative tolerance check box.

5 In the associated text field, type 1e-4.

Solution 1

1 On the Study toolbar, click Show Default Solver .

2 In the Model Builder window, expand the Study 1>Solver Configurations node.

3 In the Model Builder window, expand the Solution 1 node, then click Time-Dependent

Solver 1.

4 In the Settings window for Time-Dependent Solver, click to expand the Absolute

tolerance section.

5 Locate the Absolute Tolerance section. In the Tolerance text field, type 1e-5.

6 On the Study toolbar, click Compute.

R E S U L T S

Concentration (tds)

Follow these steps to reproduce the plot in Figure 2:

1D Plot Group 2

1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group.





2 In the Settings window for 1D Plot Group, locate the Data section.

3 From the Time selection list, choose From list.

4 In the Times (s) list, choose 0, 80, and 160.

5 On the 1D Plot Group 2 toolbar, click Line Graph.

6 In the Settings window for Line Graph, locate the Selection section.

7 From the Selection list, choose All domains.

8 Click Replace Expression in the upper-right corner of the x-axis data section. From

the menu, choose Geometry>Coordinate>x - x-coordinate.

9 Click to expand the Coloring and style section. Locate the Coloring and Style section.

Find the Line style subsection. From the Color list, choose Black.

10 Right-click Results>1D Plot Group 2>Line Graph 1 and choose Duplicate.

11 In the Settings window for Line Graph, click Replace Expression in the upper-right

corner of the y-axis data section. From the menu, choose Component 1>Transport of

Diluted Species in Porous Media>c2 - Concentration.

12Locate the

Coloring and Style section. Find the

Line style subsection. From the

Line list, choose Dashed.

13 On the 1D Plot Group 2 toolbar, click Plot.

G L O B A L D E F I N I T I O N S

Parameters

1 In the Model Builder window, under Global Definitions click Parameters.

2 In the Settings window for Parameters, locate the Parameters section.

3 In the table, enter the following settings:

S T U D Y 1

On the Home toolbar, click Compute.


c01 1[mol/m^3] 1 mol/m³ Initial injector

concentration,

component 1

c02 10[mol/m^3]10 mol/m³

Initial injector

concentration,

component 2





R E S U L T S

1D Plot Group 2

Compare the plot with that in Figure 3.

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Models.chem.Liquid Chromatography 1