274
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
Low Polarization in Laminar Ultrafiltration of Macromolecular Solutions Woon-Fong Leung and Ronald F. Probstein” Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
Ultrafiltration of macromolecular solutions in steady, plane laminar channel flow is studied under conditions of low polarization. The ultrafiltrate flux-pressure behavior is explained by an osmotic pressure model in which both the osmotic pressure and diffusivity are taken to be concentration dependent. A theoretical solution of the mass transfer problem for the flux through the membrane is obtained by a boundary layer integral method. The accuracy of the integral solution is checked for the limiting case of a linear osmotic pressure-concentration relationship and constant diffusivity by comparing with Brian’s finite difference solution. The ultrafiltration of bovine serum albumin at pH 4.7 is studied experimentally and the measured ultrafiltrate fluxes are found to agree very well with the theoretically calculated values using Vilker’s nonlinear osmotic pressure data and a diffusivity relation obtained by linearly interpolating the diffusivity values between the gel and dilute concentration limits. A further confirmation of the results is found in the good agreement obtained between the integral solution predictions and the early and previously unexplained flux measurements of Huffman.
Introduction Ultrafiltration is a pressure-driven membrane separation process which can be used on macromolecular solutions. Under hydrostatic pressures applied across the membrane, typically in the range of 0.7 X lo5 to 7 X lo5Pa (- 10-100 psig), the solvent is forced through the membrane and the macromolecules are retained. Rejection is usually close to complete. The accumulation of the macromolecules at the membrane surface, termed concentration polarization, gives rise to an osmotic pressure resistance to flow across the membrane that reduces the effective driving pressure. For this reason the solution is usually circulated past the membrane in order to “mix up” the flow and reduce the solute concentration at the membrane surface. The solution is then removed from the ultrafilter in a concentrated form. No matter what the nature of the flow past the membrane, there is always some accumulation of macromolecules at the surface. As a consequence of the resulting polarization, an increase in applied pressure results in less than a proportional increase in permeate flux. This polarization effect increases with pressure and the flux curve becomes more nonlinear. At sufficiently high pressure, the flux approaches a limiting value which, in steady state, is independent of membrane permeability and pressure. This behavior is illustrated in Figure 1. Michaels (1968) has postulated that at sufficiently high concentration, gelation takes place and a gel layer forms on the membrane which limits the flux through the membrane. This high polarization gel model has been exploited by Shen and Probstein (1977) and Probstein et al. (1978) and has yielded accurate predictions of the limiting flux in laminar flow. It is the purpose here to derive predictions for the ultrafiltrate flux in laminar flow for the low polarization regime and to compare with experimental results using bovine serum albumin, a protein solution. The basis of the low polarization model is that increasing resistance to permeation through the membrane at higher pressure is a consequence of increased osmotic pressure associated with high solution concentrations. For dilute macromolecular solutions the osmotic pressure is normally small since it is proportional to the solute concentration and inversely proportional to the molecular weight of the solute. However, for polymer solutions, such as protein, with concentrations above about 00 19-7874/79/10 18-0274$01.OO/O
1wt % the osmotic pressure no longer follows the linear Van’t Hoff law and the second- and third-order virial terms become important (Goldsmith et al., 1971; Vilker, 1975). This is graphically illustrated in Figure 2, where we have plotted the osmotic pressure of bovine serum albumin solution as a function of its weight percent concentration. The data are from the experiments of Vilker (1975). It can be seen that at 40 g % and pH 7.4 the osmotic pressure is about 2.8 X lo5 Pa (-40 psi) compared with an applied transmembrane pressure range for ultrafiltration of 0.7 X lo5 to 7 X lo5Pa, with an upper limit of 4 X lo5 P a more common. Because of the relatively low diffusivity of large molcmz/s) concentration polarization effects are ecules (more pronounced in ultrafiltration than in reverse osmosis where the solute diffusivities are 10 to 100 times larger. Following Michaels’ (1968) idea, we therefore assume that the principal resistance to flux transport through the membrane at low pressure operation is due to an increase in osmotic pressure associated with concentration polarization at the membrane. The amount of flux transmitted through the membrane is then given by the relation (e.g., Merten, 1963) u, = A[AP - dc,)l (1)
where u, is the local flux corresponding to the osmotic pressure, T , of the solution evaluated at the wall concentration c,. For the solvent alone the flux is linear in the applied transmembrane pressure difference, Ap,with the coefficient of proportionality, A, equal to the membrane permeability coefficient (see Figure 1). We have neglected here the osmotic pressure of the permeate since, as noted above, the macromolecule rejection is nearly complete. Once the concentration at the membrane surface (the wall) is known then the flux-pressure curve can be specified. The determination of this concentration is a mass transfer problem. Kozinski and Lightfoot (1972) examined low polarization ultrafiltration in a rotating disk geometry. They numerically integrated the one-dimensional concentration diffusion equation subject to the osmotic pressure boundary condition and found that concentration polarization could account for experimentally observed reductions in perlheate flux when ultrafiltering bovine serum albumin solutions. In what follows we analyze by an
0 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
i -/
,
/
# S o l #v t l o no
v
e
-
275
In the present paper we solve the mass transfer problem by an approximate boundary layer integral method. The first step in the procedure is to integrate eq 2 across the concentration boundary layer and apply the boundary conditions of eq 4 and 5 a t the membrane and edge of the layer, respectively. The resulting differential-integra1 equation expressed in dimensionless form is
r‘Im nVelocity e daby n d Flow Intern01 Channel Design1
A p p l i e d P r e s ~ u r eA, p
Figure 1. Schematic of ultrafiltrate flux-pressure behavior.
where
Albumin Concentrotion lg/100cm3)
Figure 2. Osmotic pressure of bovine serum albumin in 0.15 M saline water (Vilker, 1975).
integral method the two-dimensional concentration polarization problem for a steady, plane laminar channel flow with perfectly rejecting membranes constituting the parallel channel walls, and compare the results with experiment. Governing Equations The velocity profile in the ultrafiltration channel is assumed to be fully developed or fully “mixed” and the concentration polarization layer to be thin compared to the channel width and growing. This is justified by the ) Peclet number very high Schmidt number ( v m / D mand based on the channel length (UL/D,). Here, v,, D,, and U are the bulk kinematic viscosity, diffusivity, and velocity, respectively. The diffusion equation governing the developing concentration profile c(x,y)for a constant solution density and a concentration-dependent diffusion coefficient D ( c ) may be written
where in accord with the boundary layer approximation, diffusion along the channel has been neglected in comparison with transverse diffusion. Here, x and y are distances along and transverse to the channel and u and u are the corresponding velocity components. The appropriate boundary conditions are: a t the channel inlet a specified bulk feed concentration
x=o; c=c, (3) a t the membrane surface complete rejection of the macromolecules
(4) at the edge of the diffusion layer the concentration taking on the bulk value and the concentration gradient vanishing (5)
with h the half-channel height. Equation 6 is an exact relation for the boundary conditions specified. The basis of the approximate Karman integral method used here is to assume the behavior of the velocity and concentration profiles across the diffusion layer with the axial behavior expressed in parametric form, and then to integrate out the j j dependence. This leads to an ordinary differential equation in terms of the parameters 8, ow, and E,, all of which are implicit functions o f f . Integral Method Solution (Parabolic Velocity Profile) Doshi et al. (1971) have carried out a related integral method analysis for a variable property concentration boundary layer in a reverse osmosis laminar channel flow. We adopt their description of the velocity profiles, but neglect the concentration dependence of the viscosity, based on the results of Kozinski and Lightfoot (1972) which show this dependence to have little effect even in the limit of high polarization. In this case, the behavior of the velocity profiles is not coupled directly to that of the concentration profile. For the developing concentration layer, the velocity profiles are expressed as the truncated series
where S($ is the fraction of permeate that has been ultrafiltered from the channel inlet to the point a. This fraction is given by
with D, C 0 and with 0 IS(a) I1. Generally, S will be quite small compared to unity in a developing layer. For the concentration profile we assume the parabolic form
[ (;) + (;)
c=l+(cw-l) 1-2
=
+
...I
(10)
with F , = E,(%) and F = a(%). This equation satisfies the boundary conditions at the edge of the diffusion layer specified by eq 5. Substituting the profiles of eq 8 and 10 into the integral relation of eq 6 and integrating we find
276
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
where y = ,C - 1 is termed the degree of polarization. Substituting the concentration profile eq 10 into the complete rejection boundary condition at the membrane, eq 4, gives
where D, = D,/D,. Finally, we rewrite the basic membrane relation eq 1 in the dimensionless form D, = & - b?i (13) where we define
Equations 11to 13 specify the three unknown parameters y(%),S(%), and ow(%). Some characteristics of the solution behavior may be seen by noting that 6
