Diffusion similarity

In each form of attack, solute concentration differences arise primarily by diffusion-related processes. As a consequence, stagnant conditions may promote attack, since concentration gradients near affected areas are reduced by flow and these concentration gradients supply the energy that drives diffusion. Similarly, high concentrations of dissolved species increase attack. Elevated temperature usually stimulates attack by increasing both diffusion and reaction rates. 

Total pressure will only change in a flnid if shaft work or work of extraneous forces are introduced. Therefore, total pressure would increase in the impeller of a compressor or pnmp it wonld remain constant in the diffuser. Similarly, total pressure would decrease in the turbine impeller but would remain constant in the nozzles. 

Both diffusion and convection are modes of mass transfer. Typically, large-scale mass transfer is accomplished by convection, and small-scale mass transfer is accomplished by diffusion. Similarly, large-scale heat transfer in the Earth is through convection (mantle convection), and small-scale heat transfer is through heat conduction (e.g., through the lithosphere). To treat the complicated convection pattern and diffusion requires a large computational effort. Some simple problems can be treated analytically. 

There is an increase in temperature as heat removal slows, along with a corresponding increase in the reaction rate. This phenomenon is known as the autoacceleration or Trommsdorff effect and can lead to catastrophic results if not properly controlled. Even with snccessfnl control of the reaction, it is difficult to remove the traces of remaining monomer from the polymer due to decreased diffusion. Similarly, it is difficult to get the reactions to proceed to completion due to limited monomer mobility. 

Gohrbandt s data for camphor spheres (40, 97) afford comparison of rates with diffusion controlling and with heat transfer controlling. Extrapolation to low temperatures of the heat transfer portion indicates sufficient heat transfer but inadequate diffusion. Similarly, extrapolation to high temperatures of the diffusion portion indicates sufficient diffusional driving force but inadequate heat transfer to maintain the surface temperature. 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

The polymer dissolution flux c/HP may be given by Fick s first law of diffusion (similar to Eq. 5.1) ... 

The value of /cro depends on the frequency of diffusion jumps, and thus on the rate of diffusion, similar to ka. This model was used by Rabinovich [42] as a basis in rate constant calculations of diffusion-controlled reactions its variant for the reactions of large and small molecules was used by Allen and Patrick [13]. Another very well known, and actually the first approach, is the treatment of this problem by Smoluchowski and his successors [41, 44-48]. 

Generally an Arrhenius (exponential) type of relation represents the diffusion coefficient as a fimction of the temperature, with AQa the activation energy of diffusion. Similarly the parameters b and K (9.16) can be expressed with Arrhenius functions with Qa the (isosteric) heat of adsorption. Consequently is also activated with a total apparent activation energy of (Qa AQa). For chemisorption AQa has about the same value as Qa [1]. For physical adsorption the value of AQa is < (0.5-0.66)Qa. Since the surface flux is small at very low temperature as well as very high temperature there must be a maximum. The possibility of observing this maximum depends on the relative magnitudes of Qa and AQa-... 

Our discussion so far has considered the impurities to be fixed in the ice lattice but this is obviously not exactly true for they can move slowly by solid-state diffusion. Similarly individual water molecules migrate in a self-diffusion process which can be followed by using isotopically labelled molecules. The structure and mass of these are very little different from those of ordinary water molecules, so that a study of their diffusion gives information about self-diffusion in the ice crystal. The only case where this is not obviously true is for the isotopes of hydrogen, where the mass ratio to the proton is considerable, but the nature of the experimental results enables us to sidestep this difficulty. 

Often a periodic vortical structure typical of EOl bifurcates from a time-dependent quiescent conduction state, prior to the onset of steady state. In this case, the width of the depletion front at the charge-selective interface first develops in time with the typical diffusion similarity scaling, /i. Subsequently, this evolution slows down, with some typical width selected, which further develops at a rate much slower than For a review of these time-dependent aspects of EOl, the reader is referred to [9] and references therein. 

This is called the Graham law of diffusion, similar in form to the Graham law of effusion eq. (7.4-42) obtained earlier for the free molecular flow (Knudsen flow). 

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