Nonisothermal facilitated transport

Example 9.14 Nonisothermal facilitated transport An approximate analysis of facilitated transport based on the nonequilibrium thermodynamics approach is reported (Selegny et al., 1997) for the nonisothermal facilitated transport of boric acid by borate ions as carriers in anion exchange membranes within a reasonable range of chemical potential and temperature differences. A simple arrangement consists of a two-compartment system separated by a membrane. The compartments are maintained at different temperatures T] and T2, and the solutions in these compartments contain equal substrate concentrations. The resulting temperature gradient may induce the flow of the substrate besides the heat flow across the membrane. The direction of mass flow is controlled by the temperature gradient. 

A set of example chemical reactions for boric acid that take place in an anion exchange membrane is 

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become 

According to the Onsager reciprocal rules, we have LqB = LBq. If we replace the chemical potential in terms of the concentration (instead of activity), we transform Eqs. (a) and (b) 

To estimate the flow of borate, and assuming that the phenomenological coefficients are dependent on the average concentration (CBI + CBII)/2 linearly, we have 

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Under nonisothermal conditions, however, densification kinetics may not be diffusion controlled. If the temperature is being raised, the stress to move (and to generate) dislocations will steadily decrease. Material transport by dislocation motion will occur, and the extent of this transport may be dependent primarily on the final temperature rather than on the time of the heat treatment. In other words, as the temperature is increased, a steady procession of dislocations which were previously anchored will move. The facilitation of motion of dislocations by diffusion processes may be overshadowed by the contribution of the increasing temperature. 

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