Simple Sieving Model

In an effort to better understand and check Karl’s model of sieving, I’ve built a significantly simpler version. The goal is to investigate the potential benefits of thinner membranes in dead-end filtration and separations.

In my model 1 ml of water is displaced from the top bucket to an originally empty bottom bucket with a known flow rate Qo. The starting flow rate is 10 ul/min but the build up of material above the membrane can slow the flow down (more later). One major simplification is the assumption of uniform concentrations in the top (Ct) and bottom (Cb) chambers. So the model is a better representation of a stirred-cell separation than it is a centrifuge based separation. The cross sectional area A, is chosen to be 1 cm^2 and the starting concentration is chosen to be 1 mg/ml. The porosity P is set at 15%. These parameter values are convenient numbers that put the model in the right ball park for our experiments. The membrane thickness is L.

The key equation describes the transport process across the membrane …

where and are the diffusion coefficient and velocity for solute molecules in the membrane pores. is hindered relative to the bulk diffusion and is hindered relative to the solute velocity. These hindrances have been described by Renkin (diffusion), Deen (diffusion and convection), and others. The most complete treatment we are aware of is this review by Deen. Deen uses the values H and W to describe the hindrances for diffusion and convection respectively.

From the following table (found in Deen) I used the Anderson and Quinn relationships for H and the Brenner and Gaydos relationship for W.

Note that λ is the ratio of the solute size to pore size and Φ describes the steric (or geometric) hindrance. While I broke the rules for λ noted in the comment section of the table, I found these functions to be well behave d over 0<  λ <0.8. Comparing the behavior of the two hindrances over this range gives …

These functions imply that increasing the particle radius relative to the pore size has a higher impact on diffusion than on convection. This observation is also made in the Deen paper …

Obviously the key to high resolution separations is a sudden drop in the transmission of species based on size.  Since transport by diffusion is more sensitive to size than convective transport, it follows that membranes thin enough to enable diffusion along with convection (Pe ~ 1) should have better size selectivity than thicker membranes. This is the central hypothesis that Karl’s model is designed to test.

In addition to sieving characteristics we are interested in understanding how membrane thinness impacts the rate at which flow reduces with time due to cake formation. Since cake formation is largely a surface phenomenon, one might not expect an important role for membrane thickness. However our hypothesis is that diffusion enabled by thin membranes helps delay the onset of concentration polarization and cake formation. For this I needed a phenomenological description (Karl is working on a mechanistic one) of how flow decreased with increased solute concentration above the membrane, so I made this up …

where Co is the starting concentration and C* is a concentration that scales the decay process. Using Co = 1 mg/ml (thinking protein here) and running C* from 0.1 -> 5 mg/ml, the impact of concentration on flow looks like this …

With these tools in hand I simply integrated forward numerically. Calculating at each time step the transport from top to bottom, then the new flow rate, then the new transport, etc.

Results will be posted in a follow up …