Differential equations for diffusion

The diffusion of impurities or heat is governed by differential equations, that can be derived from the molecular models just described. We only 

Consider two layers in a solid a distance Ax apart (Fig. A.2). Let the concentration of impurity atoms be Ci and Cz in the two layers, so the concentration gradient is 

If Ax is chosen to be equal to the diffusion step length I then the numbers of impurity atoms per unit area in the layers are Cil and Czl, respectively. In a time interval At = /v, half of these will jump to the left and half to the right, so the net flow of atoms from layer 1 to layer 2 is 

A second differential equation is needed for the analysis of non-steady impurity or temperature distributions. It can be derived from Eq. (A.7) or (A.9) on making the assumption that D is independent of the concentration, or k is independent of the temperature. Pigure A.2 shows a solid divided into layers of thickness x. In the finite difference heat transfer 

In a time interval At, the increase in the thermal energy stored in layer i is the difference between the heat flows across the left- and right-hand boundaries. The calculation is made for unit cross-sectional area, and yields infinite differences from the equation 

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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]. 

In this Section so far, ADM is used to solve theoretical generalized models in the forms of ordinary differential equations). For diffusion-convection problems, the distributions along the axial direction of the packed bed electrode were neglected in certain cases, and mass transfer in the three dimensional electrodes were characterized by an average coefficient kh... 

The dependence of the kinetics on dimensionality is due to the physics of diffusion. This modifies the kinetic differential equations for diffusion-limited reactions, dimensionally restricted reactions, and reactions on fractal surfaces. All these chemical kinetic patterns may be described by power-law equations with time-invariant parameters like... 

This is the differential equation for diffusion in one dimension (called Pick s second lav). 

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

Diffusion of A within the porous pellet takes place. If the pores arc very large this may be the normal type of molecular diffusion, but if the pore radius is smaller than the mean free path, a molecule will hit the pore wall more often than it hits its fellows, and this is the Knudsen regime of diffusion. Both types of diffusion can be described by Fick s law in which the flux is proportional to the concentration gradient, and if the diffusion coefficient is not in some sense large there may be large variations in the concentration of A within the pellet. Let r denote position within the catalyst particle then the concentration of A within the particle is a(r), a function of that position, and obeys the partial differential equation for diffusion with a(r) = as when r is a position on the exterior surface of the particle. Clearly, this is a complicated matter and we shall seek ways of simplifying it in Sec. 

Fick s second law for unidirectional diffusion may be derived from his first law [81], and it provides the fundamental differential equation for diffusion of an isotropic medium (similar properties in all directions) ... 

In general, a theoretical description of non-stationary (transient) processes in any chemical FS (especially, of third type) is a very difficult mathematical problem. It consists in solution of system of differential equations for diffusion of the components in moving coordinate system shown in Fig. 1.1. These equations are highly non-linear ... 

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 Figures 4 and 5 report the water content kinetics for the samples prepared with water/diacetine mixtures and water respectively. Following Crank the diffusion of a penetrant in a swelling matrix can be analyzed similarly to that in a non-swelling system by assuming a proper measure of the penetration depth. The solution of the differential equation for diffusion gives >, for sorption and desorption in the initial stage ... 

The analytical integration of the differential equation for diffusion with chemical reaction in a catalyst particle is achievable just for first-order reactions. A generalized modulus has been proposed for extending the use of the // expression in Equations 2.61 and 2.64a to any type of rate expression [17], at least approximately. For irreversible nth order reactions, the generalized modulus for a sphere becomes dependent on the exterior surface concentration ... 

In the case of more complicated kinetic expressions like LHHW equations, the effectiveness factor rj can be determined by numerical integration of the differential equations for diffusion with chemical reaction. The complex kinetics of a number of reactions can practically be approximated by simpler power function expressions. 

Example 6.2 Solution of Parabolic Partial Differential Equation for Diffusion. 

To obtain the general rate expression, we can start from the governing differential equation for diffusion through the product layer with interfacial chemical reaction and external mass transport providing the boundary conditions. Since each step occurs in series and is independent of the others, we can make use of the results already obtained in the previous two sections. 

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