For structure-dependent diffusion, it should also be recalled that Z)eff = f(DAB). Therefore, the diffusivity of a substance in solids is a function of concentration, structural parameters, and other system characteristics.

Diffusivity in Nonideal Systems

An experimental apparatus for measuring the diffusivity in nonideal systems is shown in Figure 6. Two chambers of equal volume, 1 and 2, are separated by a porous membrane and contain a true solution of a food system, whose concentration is slightly higher in 1. Both bulk concentra-

Rotating magnet bar

Rotating magnet bar

Diffusion Food Processing
Figure 6. Diagram of a diffusion cell for determination of diffusivity in nonideal systems.

tions are maintained uniform by agitation (rotating magnet), and diffusion takes place only through the membrane. Under quasi-steady-state conditions, the amount of A diffusing through the membrane in a small time interval can be found by taking the mass balance in chambers 1 and 2 separately. For chamber 1:

For chamber 2:

Combining equations 72 and 73, rearranging, and integrating gives lnl

2 ePA YV

where V is the volume of each chamber; (CA1 — CA2)0 and (CAl — CA2)t are the concentration differences between the two chambers at the initial and subsequent times, respectively; t is time; Y is the thickness of the membrane; e is the effective porous fraction of the membrane; and f> is a cell constant (ft = 2 e/YV) that can be determined by experimenting with a substance of known diffusivity.

The diffusivity of a nonideal system DA is different from that of an ideal system £>ab- The relationship between DA and Z)^ is unpredictable and can be determined only by experiment. Furthermore, when binding reactions are present, the diffusivity (denoted by Db) is different from DA and Dab. A semitheoretical equation has been presented (8) to estimate the diffusivity DA of a globular-type protein in solution without binding action between the protein and the solute A as

where Cp is the protein concentration in kg of protein/m3. An equation that relates DA, Dab, and Db during diffusion of sodium caprylate in bovine serum albumin solution was reported (9) as

where a is the so-called diffusivity reduction shape factor for the protein, <pp is the volume fraction of protein in solution, Cb is the concentration of protein-bound solute A in g/m3, and C is the total concentration of solute A in g/m3.

Overall Mass Transfer Coefficient

In a two-phase system, the mass transfer rate can be written in terms of the overall mass transfer coefficient and the bulk concentrations or partial pressures of both phases

where Kg and Ki are the overall mass transfer coefficients for the gas and liquid phases, respectively; P% is the partial pressure of A at equilibrium with the bulk concentration of the liquid; and C% is the concentration of A at equilibrium with the bulk partial pressure of the gas phase, as shown in Figure 3. For dilute solutions, Henry's law

can be employed, where H is Henry's constant. It is also possible to write

Using equations 79 and 80, equation 77 becomes

Na = Kg(PA ~ PAi) + KgH(CAl - CA) (81) If these terms are rearranged, the following is obtained

By combining equations 52,53, and 82, the final expression is obtained

Hence, the overall mass transfer coefficient can be evaluated from the film coefficients. In a similar manner, an expression for Kt can be derived

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