Diffusion counter-gradient

Complex strain flame-front regime. Where the flame fronfs are still lamella-like but thickened due to enhanced turbulent diffusivity. Scalar transport is expected to be counter-gradient in this regime. 

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. 

This counter-gradient behaviour, which would have required the diffusivity to be negative, was demonstrated unequivocally for scalars by Denmead and Bradley [149,... 

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 reservoirs are CSTRs. Fresh reactants flow continuously at a fixed flow rate into these feed reservoirs, where they are vigorously stirred to ensure uniform and constant feed conditions. The reactants diffuse into the gel from the two reservoirs and generate counter gradients of the various reactants. The reactants are distributed in the reservoirs A and B in such a way that the pattern-forming reaction cannot occur in the reservoirs, in contrast to the one-sided CFUR. 

The tlrermodynamic activity of nickel in the nickel oxide layer varies from unity in contact with tire metal phase, to 10 in contact with the gaseous atmosphere at 950 K. The sulphur partial pressure as S2(g) is of the order of 10 ° in the gas phase, and about 10 in nickel sulphide in contact with nickel. It therefore appears that the process involves tire uphill pumping of sulphur across this potential gradient. This cannot occur by the counter-migration of oxygen and sulphur since the mobile species in tire oxide is the nickel ion, and the diffusion coefficient aird solubility of sulphur in the oxide are both vety low. 

An ion-selective electrode contains a semipermeable membrane in contact with a reference solution on one side and a sample solution on the other side. The membrane will be permeable to either cations or anions and the transport of counter ions will be restricted by the membrane, and thus a separation of charge occurs at the interface. This is the Donnan potential (Fig. 5 a) and contains the analytically useful information. A concentration gradient will promote diffusion of ions within the membrane. If the ionic mobilities vary greatly, a charge separation occurs (Fig. 5 b) giving rise to what is called a diffusion potential. 

The more volatile constituent is transferred under the action of a concentration gradient from the liquid to the interface, where it evaporates and then diffuses into the vapour stream. The less volatile component is transferred in the opposite direction and, if the molar latent heats of the components are the same, equimolecular counter-diffusion takes place. 

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. 

Needless to say, the assumption of plug flow is not always appropriate. In plug flow we assume that the convective flow, i. e., the flow at velocity qjAt = v that is caused by a compressor or pump, is dominating any other transport mode. In practice this is not always so and dispersion of mass and heat, driven by concentration and temperature gradients are sometimes significant enough to need to be included in the model. We will discuss such a model in detail, not only because of its importance, but also because the techniques used to handle the ensuing boundary value differential equations are similar to those used for other diffusion-reaction problems such as catalyst pellets, as well as for counter-current processes. 

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