Differential Equation of Diffusion

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

By changing the variables, the partial differential equation of diffusion has been turned into an ordinary differential equation... 

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. 

Noncompartmental models were introduced as models that allow for transport of material through regions of the body that are not necessarily well mixed or of uniform concentration [248]. For substances that are transported relatively slowly to their site of degradation, transformation, or excretion, so that the rate of diffusion limits their rate of removal from the system, the noncompartmental model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. A number of noncompartmental models deal with plasma time-concentration curves that are best described by power functions of time. 

The usual differential equation of diffusion processes is due to Pick... 

Differential equation of diffusion (2.7) then is discretized by using a doublelayered by time and triple-layered by spatial coordinate centered finite difference scheme ... 

Solution of the differential equation of diffusion (6-101) with boimdary conditions (6-102) gives a depth concentration profile of fluorine groups (KrF) within the krypton film ... 

The differential equation of diffusion in a thin isotropic sheet is built as follows ... 

By making the assumption that the variables of space x and of time t are separable, an attempt can be made to find a solution for the partial differential equations of diffusion, when the diffusivity is constant. 

As a final point, a continuous catalyst surface could have been adopted rather than a discrete one, and partial differential equations of diffusion and conduction could have been used rather than those shown here. A great deal of interest in recent years has been directed toward studies of such diffusion-reaction systems, prompted strongly by the fascinating spatial patterns of the now-famous Belousov-Zhabotinskii reaction [69,70]. There have been some applications of such studies to catalytic processes (by Pismen [71-73], Sheintuch [73-75] and Hlavacek [22,25,76]) but the possibilities have not been clearly delineated. Concentration and temperature patterns on catalytic surfaces analogous to the colored bands in the B-Z system might be expected. 

Equation (79) is the differential equation of diffusion (or mass transfor) in a moving flow. In it, besides the concentration, the flow velNavier-Stokes (20) to (22) and the equation of continuity (18). 

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