Dilution model, carrier transport

Widdas s quantitative model of the simple carrier was able to explain a number of earlier observations and to make predictions about what would be observed in more complex experiments on membrane transport. Thus it was a highly productive scientific insight. One of the earlier, apparently anomalous, results that the theory explained was the dramatic fall of membrane permeability found for solutes which were rapidly transported as solute concentration was increased. For example, in the human red blood cell, Wilbrandt and colleagues had previously measured a permeability constant for glucose which was 1000 times higher in dilute solutions of glucose than it was in a concentrated solution. This phenomenon, subsequently called saturation kinetics, is formally equivalent to the fall, as substrate concentration increases, in the proportion of substrate converted to product by a limited amount of an enzyme. 

In this problem, we calculate the time-averaged axial transport rate for the special case of an oscillating pressure-driven flow in a straight wall channel with a cross-sectional shape that we denote as S and a wall that we denote as B. We consider a two-component system consisting of a carrier gas and a second dilute species whose concentration we denote as c. The latter is assumed to have a uniform gradient in the axial direction that we denote as ft, i.e., dc/dz = j. We assume that this second component is sufficiently dilute that the velocity field is wholly determined by the carrier gas, which is modeled as an incompressible, Newtonian fluid with a density p and viscosity //. The pressure gradient is assumed to take the form... 

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