COMSOL 6.3 - Separation Through Dialysis

Dialysis is a widely used separation method. An example is hemodialysis, where membranes are used as artificial kidneys for people with renal failure. Other applications include the recovery of caustic colloidal hemicellulose during viscose manufacturing as well as the removal of alcohol from beer (Ref. 1).

In the dialysis process specific components are preferentially transported through a membrane. The process is diffusion-driven, that is, components diffuse through a membrane due to concentration differences between the dialysate and the permeate sides of the membrane. Separation between solutes is achieved as a result of the different diffusion rates across the membrane, which arise from differences in molecular size and solubility.

This example examines a process aimed at lowering the concentration of a contaminant component in an aqueous product stream. The dialysis equipment is made of a hollow fiber module, where a large number of hollow fibers act as the membrane. It focuses on the transport of the contaminant in the hollow fiber and through its wall.

Figure 1 shows a diagram of the hollow fiber assembly within a dialysis module where the dialysate flows inside while the permeate flows on the outside in a co-current manner. The contaminant is transported through the fiber walls to the permeate side. Species with a higher molecular weight are retained in the dialysate side, due to their low solubility and diffusivity through the membrane.

Figure 1: The hollow fiber assembly in a dialysis module.

Model Definition

This example models a piece of hollow fiber through which the dialysate flows with a fully developed laminar parabolic velocity profile. The fiber is surrounded by a permeate that flows laminarly in the same direction as the dialysate. The dialysate, the permeate, and the membrane are all examined in the results. The model domain is shown in Figure 2. Here, the angular gradients are considered negligible, so an axisymmetric approximation can be used.

Figure 2: Illustrations of the hollow fiber setup with the dialysate and permeate, and of the model domain.

You can draw a hexagonal-shaped unit cell of the fiber assembly as in Figure 3:

Figure 3: Hexagonal-shaped unit cell of the fiber assembly.

As a simplification, the hexagon is approximated as a circle in the model.

The contaminant is transported by diffusion and convection within the two liquids, whereas diffusion is the only transport mechanism through the membrane. The mass transport is modeled with the Transport of Diluted Species interface. To analyze the convective flux, the Laminar Flow interface is used, assuming that the flow is laminar.

The contaminant must dissolve into the membrane in order to be transported through it. The interface conditions between the liquids and the membrane are described by the dimensionless partition coefficient K:

(1)

where ci denotes the concentration of the contaminant (SI unit: mol/m3). The subscripts and superscripts describe the location in the dialysis fiber as displayed in Figure 4. This figure also shows a schematic concentration profile.

Figure 4: The concentration profile across the membrane (see Equation 1). Note that there are discontinuities in the concentration profile at the membrane boundaries.

To obtain a well-posed problem, an appropriate set of boundary conditions must be defined. Figure 5 displays the boundaries that need to be accounted for. Note that the concentration is discontinuous at the membrane-liquid interfaces; boundary conditions need to be set on both sides of the interface. Equation 1 is implemented at each of the two boundaries using a Partition Condition node. In addition to prescribing the concentration ratio, this node ensures that the mass flux is identical on both sides of the interface.

Figure 5: The boundaries accounted for in the model.

Danckwerts’ inflow conditions are set at the inlet to the dialysate and permeate. At the outlet, the convective contribution to the mass transport is assumed to be much larger than the diffusive contribution and is modeled by setting outflow conditions. Symmetry applies at the leftmost boundary for this axisymmetric model geometry, and no flux is set at the membrane edges and the rightmost boundary, since no species pass these.

Summary of Input Data

The input data are listed in the table:


Property	Value	Description
D	10-9 m2/s	Diffusion coefficient, liquids
Dm	10-9 m2/s	Diffusion coefficient, membrane
Rhf	0.2 mm	Inner radius, hollow fiber
Lm	0.28 mm	Thickness, membrane
Lpc	0.7 mm	Width, concentric permeate channel
H	21 mm	Length, fiber
Uave_dia	0.5 mm/s	Average velocity, dialysate
Uave_per	0.8 mm/s	Average velocity, permeate
K	0.7	Partition coefficient
c0	1 M	Inlet concentration, dialysate

Results and Discussion

The surface plot in Figure 6 visualizes the concentration distribution throughout the three model domains in 3D: the dialysate liquid inside the hollow fiber (nearest the center), the membrane, and the permeate liquid outside the hollow fiber. As the plot shows, the concentration inside the hollow fiber decreases markedly over the first 10 mm from the inlet. The contaminant transport, starting from the dialysate inlet, is shown in detail in Figure 7. In this figure the radial direction has been rescaled for clarity. Streamlines of the total flux of the contaminant is plotted, with arrows separated by a fixed time interval. Observing the number of arrows inside the membrane, it is evident that the time for the transport across the membrane increases with the downstream distance. This occurs since the driving force, the concentration difference between the dialysate and the permeate side, decreases with the downstream direction.

Figure 6: Concentration in the three domains using a revolution dataset.

Figure 7: Contaminant development using a scaled radial direction.The streamlines (black) show the total flux with arrows positioned using a fixed time interval. The arrow line (gray) shows the stationary velocity field.

Concentration jumps arise at the boundaries between the domains. This is shown in Figure 8 where the concentration profile at the middle of the fiber length is plotted along the radius of the model geometry.

Figure 8: Concentration across the three domains at the middle of the fiber length and at the outlet.

References

1. M. Mulder, Basic Principles of Membrane Technology, 2nd ed., Kluwer Academic Publishers, 1998.

2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons, 1960.

Chemical_Reaction_Engineering_Module/Mixing_and_Separation/dialysis

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Root

Load parameters from a text-file.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file dialysis_parameters.txt.

Draw the geometry and make selections.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type Rhf.

In the text field, type H.

Click

Click the

button in the toolbar.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type Lm.

In the text field, type H.

Locate the section. In the text field, type Rhf.

Click

Click the

button in the toolbar.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type Lpc.

In the text field, type H.

Locate the section. In the text field, type Rhf+Lm.

Click

Click the

button in the toolbar.

Form Union (fin)

In the window, click .

In the window for , click

Dialysate and Permeate

In the toolbar, click

and choose .

In the window for , type Dialysate and Permeate in the text field.

On the object , select Domains 1 and 3 only.

Membrane

In the toolbar, click

and choose .

In the window for , type Membrane in the text field.

On the object , select Domain 2 only.

Model transport by convection and diffusion in the dialysate and permeate, and diffusion in the membrane.

Transport of Diluted Species (tds)

Transport Properties - Dialysate and Permeate

In the window, under > click .

In the window for , type Transport Properties - Dialysate and Permeate in the text field.

Locate the section. From the u list, choose .

Locate the section. In the Dc text field, type D.

Transport Properties - Membrane

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , type Transport Properties - Membrane in the text field.

Locate the section. In the Dc text field, type Dm.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the c text field, type c0_dia.

Initial Values 2

Right-click > > and choose .

Select Domain 3 only.

In the window for , locate the section.

In the c text field, type c0_per.

Inflow 1

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the c0,c text field, type c0_dia.

Locate the section. From the list, choose .

Inflow 2

In the toolbar, click

and choose .

Select Boundary 8 only.

In the window for , locate the section.

In the c0,c text field, type c0_per.

Locate the section. From the list, choose .

Outflow 1

In the toolbar, click

and choose .

Select Boundaries 3 and 9 only.

Add a feature to implement the concentration ratio at the dialysate-membrane interface.

Partition Condition 1

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

In the Kc text field, type K.

Note that up and downside for the concentration are indicated in the . The arrow, situated on the selected boundary, points from the downside into the upside. In this case let the membrane side correspond to the upside to achieve the desired condition.

Add another feature to implement the concentration ratio at the permeate-membrane interface.

Partition Condition 2

In the toolbar, click