Co-current crossflow microfiltration in a microchannel

Abstract

Steady state crossflow microfiltration (CMF) is an important and often necessary means of particle separation and concentration for both industrial and biomedical processes. The factors controlling the performance of CMF have been extensively reviewed. A major factor is transmembrane pressure (TMP). Because microchannels have small height, they tend to have high pressure gradients in the feed-flow direction. In the extreme, these gradients may even reverse the pressure across the membrane (inciting backflow). It is therefore desirable to compensate for the effect of feed-flow on the TMP, aiming at constant transmembrane pressure (cTMP) at a value which maximizes filtrate flux. This is especially critical during filtration of deformable particles (e.g. erythrocytes) through low intrinsic resistance membranes. Filtration flux is generally taken to be directly proportional to TMP, with pressure drop along the channel decreasing in the flow direction. A co-current flow of filtrate in a suitably designed filtrate collecting channel is shown to allow the TMP to remain constant and permit the sieving surface to perform optimally, permitting up to twice as much filtration over that of a naïve configuration. Manipulation of the filtrate channel may be even more beneficial if it prevents backflow that might otherwise occur at the end of a sufficiently long channel. Experiments with erythrocyte suspensions, reported here, validate these concepts.

Appendix

1.1 Nomenclature

B :

Half height of the channel (m)

L :

Channel Length (m)

W :

Channel width (m)

ΔP :

Pressure drop across the channel (Torr)

Q _f :

Volumetric flowrate of the permeate (i.e. Filtration rate) (cm³/min)

Q _m :

Volumetric flowrate in main channel (cm³/min)

TMP :

Transmembrane pressure (Torr)

J _f :

Filtrate flux (cm³/cm²-min)

P ₁ :

Inlet blood pressure (Torr)

P ₂ :

Outlet blood pressure (Torr)

P ₃ :

Inlet plasma pressure (Torr)

P ₄ :

Outlet plasma pressure (Torr)

Greek symbols

γ_w:

Nominal wall shear rate (1/s)

τ_w:

Wall shear stress (Pa)

μ :

Viscosity of media (Poise)

1.2 Supporting equations

1.2.1 Microchannel height

As in a previous study (Amar et al. 2018), the thickness of each microchannel varied each time the apparatus was assembled and had to be accurately calculated each time. By measuring the change in pressure for various through-flow rates for a fluid of known viscosity (i.e. microfiltered water), the thickness (2B) of the channel was calculated using the equation for laminar flow in a narrow slit solved for B (Bird 2007):

$$ 2B=2\times \sqrt[3]{\frac{3}{2}\frac{\mu L}{W}\times \frac{Q_m}{\Delta P}} $$

(2)

The flow was assumed to be Newtonian, laminar, and fully developed, where Q_m is the volumetric flow rate, ΔP is the difference in pressure between inlet and outlet, W and L are the width and length of the channel, respectively, and μ is the viscosity of the fluid (Bird 2007).

Since the heights above and below the microsieves were not equal, different flows above and below the sieves were required to create the same pressure drop. A simplified version of the slit flow equation (Bird 2007) was used:

$$ {Q}_{plasma}=\left(\frac{3}{2}\right){Q}_{blood}{\left(\frac{B_{plasma}}{B_{blood}}\right)}^3 $$

(3)

The factor 3/2 compensates for the viscosity difference between the two fluids.

1.2.2 Shear rate and shear stress at the wall

The shear stress exerted at the flow boundaries, τ_w can be calculated by balancing the shear force at the wall against the pressure gradient for a slit channel (Bird 2007).

$$ {\tau}_w=\frac{\mu 3{Q}_m}{2{B}^2W} $$

(4)

Shear rates ($\dot{\gamma}=\frac{dV}{dZ}$) at the wall are found as the shear stress (τ_w) divided by the viscosity:

$$ {\gamma}_w=\frac{\tau_w}{\mu }=\frac{3{Q}_m}{2{B}^2W} $$

(5)

where Q_m is the volumetric flow rate, B is the half thickness of the channel, W is the width of the channel, and μ is the viscosity of the fluid (Bird 2007).