The differential equations of diffusion

During a diffusion process, e.g. the migration of an additive from a plastic into the atmosphere, a change in the concentration of the diffusing substance takes place at every location throughout the plastic. The mass flux caused by diffusion is represented by a vector quantity whereas the concentration c and its derivative of time t is a scalar quantity and is connected by the flux with help of the divergence operator. The following example serves to emphasize this relationship. 

In a body with any given shape, e.g. a piece of soap, there is an aroma compound which is initially uniformly distributed throughout the entire body. During storage without any packaging a decrease in concentration takes place due to diffusion into the atmosphere particularly in the outer layers of the soap. The resulting scalar concentration field with the levels Ci c2 c3 (Fig. 7-1) forms a gradient field that describes the external direction of the aroma compound flux. 

Now consider only a suitably small section of the soap in the form of a cube with side lengths of Ax, Ay, and Az (Fig. 7-2). 

The aroma compound will diffuse in as well as out of the cube because of its perpendicular side surface areas. Due to the greater decrease in the aroma near the soap s external surface, the flux out of the side of the cube closer to the surface is greater than the flux into the side of the cube that lies deeper in the soap. The difference between the aroma diffusing in and out will be positive which means one can consider the cube as an aroma source. As a consequence of the flux out of the cube, the concentration in the cube decreases with time. The concentration is also a function of time, c = c(x, y, z, t) and its decrease with time, i.e. the partial derivative —dc/dl in the cubic volume AV = Ax Ay Az, represents the net flux out of the cube and designated div/ the divergence of the flux. 

Mathematically the divergence is obtained as the sum of the differences between flux components in and out of the cube in the coordinate axis direction with respect to the cube s volume. Placing the coordinate axis parallel to the corners of the cube as a helpful construction (Fig. 7-2), then one can label the incoming flux component contributions through the side walls Ay Az at the location x with /X(x) and the outgoing component through the opposite side wall at x + Ax on the x-axis with Jx(x + Ax). When this is done in the same manner for the other components, then one gets  

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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 ... 

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). 

Diffusion in a sphere may be more common than that in a cylinder in the pharmaceutical sciences. The example we may think of is the dissolution of a spherical particle. Since convection is normally involved in solute particle dissolution in reality, the dissolution rate estimated by considering only diffusion often underestimates experimental values. Nevertheless, we use it as an example to illustrate the solution of the differential equations describing diffusion in the spherical coordinate system [1],... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

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. 

Derivations will be made of the differential equation of the pore and the expression for the rate of diffusion into the pore or the rate of reaction in the pore. The reaction is 2A B at constant pressure. 

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

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

A more sophisticated approach is to avoid the postulate of a shock and instead to state the differential equations of conservation of mass, momentum, and energy to include more properties of a real fluid. Including the effects of viscosity, heat conditions, and diffusion along with chem reaction gives eqs with a unique solution for given boundary conditions and so solves the determinacy problem. The boundary conditions are restricted by the assumption that the reaction begins and is completed with the region considered. 

Studying anew the differential equations of heat conduction, diffusion, gas motion and chemical kinetics under the conditions of a chemical reaction (flame) propagating in a tube, through a narrow slit or under similar conditions, using the methods of the theory of similarity we find the following dimensionless governing criteria ... 

With the help of these coefficients all the differential equations of the diffusion of the different substances and the equations of heat conduction for the chemical reaction take exactly the same form, with identical L and F in all the formulas ... 

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. 

It does not appear possible to consider in general form, discarding the assumptions of steady propagation of the regime with constant velocity, the differential equations of heat conduction and diffusion in a medium in which a chemical reaction is also running. 

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 Integration of the Differential Equation for Diffusion Potentials The Planck-Henderson Equation... 

Derivation of the Differential Equation Describing Diffusion and Reaction 741... 

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

However, as with the penetration theory analysis, the difference in magnitude of the mass and thermal diffusivities with cx 100 D, means that the heat transfer film is an order of magnitude thicker than the mass transfer film. This is depicted schematically in Fig. 8, The fall in temperature from T over the distance x is (if a = 100 D) about 0% of the overall interface excess temperature above the datum temperature T.. Furthermore, in considering the location of heat release oue to reaction in the mass transfer film, this is bound to be greatest closest to the interface, and this is especially the case when the reaction becomes fast. Therefore, two simplifications can be introduced as a result of this (i) the release of heat of reaction can be treated as am interfacial heat flux and (ii) the reaction can be assumed to take place at the interfacial temperature T. The differential equation for diffusion and reaction can therefore be written... 

The differential equation of Laplace (V c = 0) describes the formation of fractal distribution of solid matter that results from very different processes (Figure 7.5). Diffusion-limited aggregation (c is the concentration), electrogalvanic deposition (c is the electric potential), and viscous invasion (c is the local pressure) are three Laplacian processes that produce similar fractal distributions. They all imply a strong positive feedback and have the same mathematics. The first two are of significance for materials synthesis. 

D = Djh jc). Also, define two matrices of scalar functions hy A = (dj]c(r)) F = fjk T)). The differential equations of reactor kinetics for the multigroup diffusion model become (neglecting delayed neutrons)... 

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