Page 348 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN009K-419 July 19, 2001 20:57

Membranes, Synthetic, Applications 283

can be done, a commonly used characteristic of pores is as described later. First, however, an additional issue re-
their hydraulic diameter defined in Eq. (4). lated to complex flow fields within the pores of the mem-
brane should be considered. When the dimensions of the
d h = 4[total pore volume/total internal pore area] (4)
pores and those of the suspended solute are similar, such
The Carman–Kozeny equation for flux through such a as in the case of nanofiltration (NF) a process called “hin-
porous medium is dered transport” occurs (Deen, 1987).
2
(n A ˆv A ) o = εd (16ηk o k t L)
p = k h
p, (5)
h b. Complex effects.
where k o is a shape factor that accounts for the various i. Hindered transport. Intuition correctly predicts
possible pore shapes that result from packing of differ- complete rejection of a monodisperse solute from a
ent shaped particles comprising the selective layer. The solvent “A” by a membrane with a uniform pore size
complex flow path length is accounted for by introducing smaller than the solute “B” but much larger than the
2
another parameter, k t = (L e /L) , which is usually called solvent “A.” Rejection “R” of component “2” is defined
the “tortuosity.” Note, however, that the term “tortuosity” as R = 1 − C downstream /C upstream , where the concentrations
is also used by some authors to equal L e /L, so care must refer to solute B downstream and upstream, respectively.
be used when referring to the term tortuosity. The term, Again, in this case solvent flux can be described in terms
k h , is referred to as the hydraulic permeability (Coulson, of Eq. (5).
1949). Clearly, in this case, if
p is taken as the driving Surprisingly, intuition fails to predict the behavior of
force term in Eq. (1), the resistance term in Eq. (1) is a the same solute and solvent in a membrane with a uni-
complex function of the selective layer of the membrane form pore size larger than both the solvent and solute.
and the fluid properties. For packed beds of particles re- The expectation that such a membrane will provide no re-
sulting in effective pathways of average length, L e , the jection of the solute has been refuted repeatedly. Indeed,
so-called Kozeny constant, k o k t , is near 5 (Carman, 1937; careful experiments indicate that partial rejection of the
Leenaars and Burggraaf, 1985; Coulson, 1949). Accepting solute occurs even when the solute is considerably smaller
thisvalueallowsonetodeterminek h frompuresolventflux (say 1/10th as large as the pore size) (Miller, 1992; Deen,
vs pressure drop. For complex membrane morphologies, 1987; Ho and Sirkar, 1992; Happel and Brenner, 1965).
such an idealization is theoretically questionable. Nev- The extent of rejection increases monotonically to the total
ertheless, this approach provides a useful characterization rejection limit as the solute size approaches the pore size.
parameter for the effective diameter of the pores in the thin These effects arise both from entropic suppression of par-
region of the membrane that controls flux. To achieve high titioning and from augmented hydrodynamic resistance to
fluxes, membranes are often asymmetric with a thin selec- transport through the fine pores. Thus, in this case, for a
tive region supported on an open-cell support material. A porous membrane, thermodynamic partitioning can play
flow-based characterization of k h , is especially valuable a role in the physical chemical processes of transport.
for these complex media. Such characterizations avoid For a solute of finite dimensions, a decrease in solute
the need to measure the detailed values from Eq. (5) in the entropy occurs upon partitioning from an unbounded ex-
thin flux-determining layer at the membrane surface, but it ternal solution into a confined pore space. The decrease in
should be remembered that it is a highly simplistic view of entropy results in a lower solute concentration in the pore
reality. Despite its limitations, the Carman–Kozeny equa- compared to the external solution to allow equalization
tion remains popular, since more realistic descriptions of of chemical potential with the solute in the external fluid.
practical membranes are encumbered by the lack of ability For cylindrical pores, d h = d p , and the partition coefficient,
to characterize the actual detailed morphology of the K i = (C i ) internal /(C i ) external , between the internal and exter-
2
membrane. nal solutions is K i = (1 − λ) , where λ = d s /d p the ratio
The volumetric flux of a binary mixture of solvent and of solute to pore diameter (Happel and Brenner, 1965).
solute, j v , through a membrane can be expressed as Even for solutes 50% as large as the pore diameter, this
factorequals 0.25, yielding a fourfold reduction inconcen-
j v = n 1 ˆv A + n 2 ˆv B , (6)
tration within the pore. As λ approaches unity, K i drops
where n A ˆv A is the solvent flux in the presence of the so- tremendously. Related expressions exist for other geome-
lute and n B and ˆv B are the mass flux and partial specific tries, and the trends are similar (Happel and Brenner,
volume of the solute, respectively. For dilute solute con- 1965).
centrations j v ∼ n A ˆv A , and under dilute ideal conditions, Hindered transport of a solute moving within a contin-
j v ∼ n A ˆv A ∼ (n A ˆv A ) o [see Eq. (5))] (Miller, 1992). Com- uum of solvent in a small pore can be analyzed in terms of
plications due to so-called “concentration polarization” classical hydrodynamics (Deen, 1987). The penetrant-to-
and “fouling” often invalidate this simple approximation, pore size ratio (λ) and the position of the penetrant within a