Counter-gradient flux

The shortcoming of such descriptions of exchange is that diffusion is not a local phenomenon, as implied by these formulations, and eddy sizes most important to exchange within the canopy can be many times larger than scales associated with the distribution of sources and sinks of heat, water vapour, etc. As an obvious example, gradient diffusion would not permit a secondary maximum in the wind profile in an extensive canopy, because such a profile would require a counter-gradient flux of momentum. 

However, as discussed in chap 1.2.7, the gradient-diffusion models can fail because counter-gradient (or up>-gradient) transport may occur in certain occasions [15, 85], hence a full second-order closure for the scalar flux (1.468) can be a more accurate but costly alternative (e.g., [2, 78]). 

The second component of the vacancy flux is related to the temperature gradient. In Stark [40] it was shown that, in the presence of a temperature gradient in the crystal, there arise an atom flux and a counter vacancy flux, which can be written in the form ... 

If a membrane separates two solutions with mixtures of counterions — in which each counter-ion is present only on one side of the membrane — and the same co-ion, we meet with a so-called multi-ionic system. These are also treated by F. Helfferich (53, 55) (ref. 55, page 327). An explicit solution of the flux equations in this case is obtained if the flow of co-ions is neglected and if all the counter-ions possess the same valency. Gradients of activity coefficients in the membrane and convection are also neglected. Diffusion coefficients and concentration of active groups are considered to be constant. It is assumed that there is equilibrium between the salt solution and the membrane surface on either side of the membrane. 

Consider now the situation when a counter ion concentration gradient that exactly balances the metal ion concentration gradient is established, so no flux of either ion across the membrane occurs. Under this condition, [MR ]o(m) = [MR ]f(ffl) and RH "./h, = [RH] (m), producing the expression... 

Mass transfer in catalysis proceeds under non-equilibrium conditions with at least two molecular species (the reactant and product molecules) involved [4, 5], Under steady state conditions, the flux of the product molecules out of the catalyst particle is stoi-chiometrically equivalent (but in the opposite direction) to the flux of the entering reactant species. The process of diffusion of two different molecular species with concentration gradients opposed to each other is called counter diffusion, and if the stoichiometry is 1 1 we have equimolar counter diffusion. The situation is then similar to that considered in the case of self-or tracer diffusion, the only difference being that now two different molecular species are involved. Tracer diffusion may be considered, therefore, as equimolar counter diffusion of two identical species. 

The driving force for mass transfer is the concentration difference or, more correctly, the concentration gradient of counter-ions between the two boundaries of the film. If theoretical refinements are ignored for the time being, the momentary flux equation for ion A may be written in terms of Tick s first law ... 

Of course, the applied field affects the transport of all ionic species. Since the concentration gradient of hydrogen ions runs counter to that of hgh, an increase in the flux of positively charged hormone from x = 0 to... 

Due to the concentration gradient imposed across the pellet, a counter-current mass transfer will occur. Let be the steady state flux of the component A and Nr be that of the component B. If the total pressure is maintained constant throughout the system, the mechanism for mass transfer through the particle is due to the combined molecular and Knudsen diffusions. The steady state flux of the component A for the case of a cylindrical capillary is (Chapters 7 and 8) ... 

The buffer reaction above can have a profound effect upon net CO2 transport, and does so via a mechanism which is perhaps more complex than it might appear at first glance. Clearly, because of the coupling via hydrogen ions, the dissociation of the weak acid affects the extents of formation of carbonate and bicarbonate ions. In moderately alkaline solutions, the OO2 flux is enhanced by that of bicarbonate, and diminished by the counter-transport of carbonate ion. The bicarbonate concentration gradient exceeds that of carbonate, and that difference is enhanced by the presence of buffer (23). Thus, simple inorganic weak acids have a positive effect upon CO2 transport in alkaline solutions, as illustrated in Figure 1, and confirmed in a later study (24). 

Diffusional transport of matter, as described in Chapter 7, can be analyzed in terms of the flux of atoms or equivalently in terms of the counterflow of vacancies. In the early development of sintering theory, the approach based on the counter-flow of vacancies driven by a vacancy concentration gradient was used predominantly. We will adopt this approach in the present section, but later in the chapter we will outline a more general approach based on the flux of atoms driven by a... 

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