Passive Mass Transport Processes in Cellular Membranes and their Biophysical Implications - Porous Media: Applications in Biological Systems and Biotechnology - page 298

used to analyze membrane phenomena. This formalism defines phenomenolog-
ical parameters describing transport properties of a membrane, and expresses
the fluxes of solvent and solute in terms of existing thermodynamic forces.
The derivation of the KK equations is well known but for completeness we
present its outline.

8.2.1 Derivation of Phenomenological KK Equations
For a thermodynamic system, the dissipation function ψ is defined as
ψ = δT =
k

X k J k (8.1)

where δ expresses entropy production d i S/dt in irreversible processes in the
system and T is the temperature. The function ψ is the sum of products of
thermodynamic stimuli X k and flux densities J k conjugated with them, and
is a measure of energy consumption in irreversible processes.
We consider a membrane system shown in Figure 8.1. A membrane M of
thickness ∆ x separates two nonelectrolytic solutions of different concentra-
tions. If the solutions are well mixed (e.g., by mechanical stirrers m), and the
volumes are suciently large, a stationary concentration profile develops on
the membrane (Table 8.1).
Assuming isothermal conditions, the dissipation function for an infinitesi-
mal membrane thickness element dx can be expressed as

n

ϕ = J i · grad ( − µ i ) (8.2)

i =1

where J i is the flux density for the i th solution component and µ i is its
chemical potential. Since the flux density is constant across the thickness of

M

J v

c 2 c 1

j s

μ ∆ x
i

P 2 μ i P 1

m m

FIGURE 8.1
Membrane system ( M is the membrane of thickness ∆ x; m is the mixing
devices; P 1 , P 2 is the pressures; c 1 , c 2 is the concentrations; µ i , µ ∆ x

is the

i

chemical potentials; J v is the volume flux; j s is the solute flux).