Macrosolute flux

Suppose that the solvated macrosolute radius is r,. Further, let the macrosolute concentration in the larger pores (tp > Tj) he equal to C, i.e. the macrosolute concentration at the feed solution-memhrane interface. On the other hand, the macrosolute concentration in smaller pores (tp < Tj) will he zero. The macrosolute flux through the membrane is therefore... 

An early study by Bixler and Rappe [88] showed that glass beads (up to 100 p,m size) added to a stirred cell UF of a macrosolute were able to significantly enhance flux. The mechanism was probably eddy formation and thinning of the concentration boundary layer by particle interaction. Similar effects were reported by Fane [89] who noted that enhancement required significantly supramicron particles and that smaller particles could in fact add to the deposit resistance. 

Practical considerations, however, require a compromise between the ideal goals and process economics. One major factor is the lack of reliable information and/or molecular weight distribution of macrosolutes. As a result, application specialists or process engineers typically recommend a pore diameter which is about 75% of the smallest particle size or a MWCO value of about 50-60% lower than the smallest macrosolute. The objective is to maximize flux without sacrificing solute retention below the set minimum requirements. 

In practice, however there could be differences between the observed and estimated flux. The mass transfer coefficient is strongly dependent on diffusion coefficient and boundary layer thickness. Under turbulent flow conditions particle shear effects induce hydrodynamic diffusion of particles. Thus, for microfiltration, shear-induced difflisivity values correlate better with the observed filtration rates compared to Brownian difflisivity calculations.Further, concentration polarization effeets are more reliably predicted for MF than UF due to the fact diat macrosolutes diffusivities in gels are much lower than the Brownian difflisivity of micron-sized particles. As a result, the predicted flux for ultrafiltration is much lower than observed, whereas observed flux for microfilters may be eloser to the predicted value. 

Fane et al. (1982) discussed the possibility of UF flux enhancement by particulates. It was found that rigid particles larger than 1 pm could enhance flux. Cohesive and compressible particles, even if large, would cause flux reduction. Milonjic et al. (1996) filtered hematite suspensions and found that increased pressure and stirring lead to a increased flux. Chudacek and Fane (1984) measured deposit layers of several pm on UF membrane by macrosolutes and silica colloids. 

The volume flux given above is for the pure solvent If there are solute (macrosolute) molecules in the feed solvent, the velocity of the solute molecule in the pore may or may not equal the local pore velocity of the solvent The latter is especially likely to be true for macrosolutes, macromolecules, proteins, etc. In general, the solute/ macrosolute molecule can also difflise along the pore, down its own concentration gradient The solute flux through the membrane is related to the solute flux through the pores, Af , as follows ... 

In the absence of any concentration polarization, and Cfi are equal to Cg and respectively. The extent of concentration polarization and its effects on the solvent flux and solute transport for porous membranes and macrosolutes/proteins can be quite severe (see Section 6.3.3). This model is often termed the combined diffusion-viscous flow model (Merten, 1966), and it can be used in ultrafiltration (see Sections 6.3.3.2 and 7.2.1.3). The relations between this and other models, such as the finely porous model, are considered in Soltanieh and Gill (1981). 

A quantitative analysis of separation in UF requires first a knowledge of the transport rates of the solvent and the macrosolutes through the UF membrane. When the macrosolute molecular weights are not high (>1000), the membrane pores may have dimensions in the range l-2nm the osmotic pressure of a concentrated solution of such macrosolutes will be significant with respect to the applied pressure difference, AP. The molar solvent flux under ideal conditions will be described by the flux expression (3.4.54) (Vilker etal., 1981) ... 

For larger membrane pores, and larger macrosolutes whose solutions have much smaller osmotic pressures, the membrane is sometimes called a microporous ultriffUter, the membrane pores are micropores/mesopores/large pores and the solvent volume flux (N zV ) is described by Darcy s law, expression (3.4.88) ... 

Figure 6.3.26. Ultrafiltration, (a) UF in a batch cell macrosolute concentration profile infeed side (b) Piston driven UF in a batch cell bulk flow parallel to the force, (c) Observed behavior ofsolvent flux vs. AP in macrosolute ultrafiltration. For an explanation of (l)-(4), see the text.

The result illustrated by (6.3.142b) is singularly important in UF. If in the feed solution, the mass-transfer coefficient kji of the macrosolute i (which is rejected (partially or totally) at the membrane surface) from the interface to the bulk liquid is not sufficiently large relative to the solv-(6.3.142b) ent flux through the membrane, the wall concentration. 

While Figure 6.3.26(c) illustrates how increasing polarization affects the solvent flux, relation (6.3.142b) may be rearranged to quantify how the macrosolute retention/ rejection/transmission is affected simultaneously (if the macrosolute is not rejected completely). Rearrange this equation as follows ... 

Suppose Cip = 0, i.e. the membrane effectively rejects/ retains the macrosolute i. Since C,gei is fixed for a particular macrosolute i in a given environment, one can increase the solvent flux only hy increasing ku- Although the solvent flux expressions (6.3.141a,h) suggest that an increase in AP win increase the solvent flux, no such increase takes place under the condition of gel polarization. 

Thus, knowing the solvent volume flux (IujI = A/j Fj) through the membrane and the macrosolute mass-transfer coefficient, ku, in the feed solution, the true macrosolute retention of the membrane, Rtnie> may be determined from the observed retention values, Robs- A plot of ((1 — Robs)/ flobs) against the observed ju l in a semilog plot will yield the value of R,n,e from the intercept ((1 — R me)//lttue))-From the slope of the line, one can also obtain an estimate of ifca in the system. 

The above analysis/description of solvent flux and macrosolute rejection/retention/ttansmission far an ultra-flllration membreme was carried out in the context of a pseudo steady state analysis in a batch cell (Figure 6.3.26 (a)). Back diffusion of the macrosolute from the feed solution-membrane interface to the bulk solution takes place by simple difflision against the small bulk flow parallel to the force direction. The resulting mass-transfer coefficients for macrosolutes will be quite small the solvent flux levels achievable will be quite low. For practically useful ultrafiltration rates, the mass-transfer coefficient is increased via different flow configurations with respect to the force. 

The magnitude of the right-hand side in parentheses is the wall flux of species i, the flux through the membrane. Since the membrane rejects the macrosolute completely, we get... 

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