Diffusion equation dimensionless

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

Proceeding as in the previous section, the appropriate dimensionless reaction-diffusion equation can be written in terms of the wavefront coordinates as... 

If both reactions (11.1) and (11.30) occur, the dimensionless reaction-diffusion equation for the concentration of B in terms of the travelling-wave coordinate z can be written as... 

The equations were transformed into dimensionless form and solved by numerical methods. Solutions of the diffusion equations (7 or 13) were obtained by the Crank-Nicholson method (9) while Equation 2 was solved by a forward finite difference scheme. The theoretical breakthrough curves were obtained in terms of the following dimensionless variables... 

This equation is of exactly the same form as the dimensionless convective-diffusion equation at the RDE (p. ). Furthermore, in dimensionless... 

Since the form of the dimensionless convective-diffusion equation for tube and channel electrodes is exactly the same as for rotating electrodes, we can immediately conclude that the steady-state collection efficiency, N0, under conditions of uniform surface concentration at the generator electrode (which corresponds to the limiting current at the generator or to any point on a reversible wave) is, once again... 

The dimensionless distance parameter may be found by substituting the definition of Ax into Equation 20.7. In the case of semi-infinite linear diffusion, Equation 20.17 is used to define Ax this substitution yields... 

In equation 17, Pes (Pes = VqL/D) is the Peclet number for mass transfer, as defined in Table II, and scales the importance of convective solute transport by the velocity in the bulk (scaled by V0) to diffusion. The dimensionless solute balance at the melt-crystal interface is the generalization of equation 3 for a steadily solidifying interface dD v... 

Solutions of the diffusion equation inevitably involve the dimensionless parameter Dtla2 in such a way that diffusive redistribution becomes significant as this parameter approaches unity. Here a is some characteristic dimension of the diffusive region. In the case where the medium is homogeneous, without a microscopic substructure (e.g., a glass or a liquid), a is the macroscopic dimension. In rocks that consist of numerous mineral grains, the relevant a is usually the individual grain size rather than the macroscopic dimension. 

This scale is now used. The three variables c, x and t are rendered dimensionless by the normalisations in (2.25) and applying these to (2.33) results in the new dimensionless diffusion equation... 

The model geometries are shown in Figure 43, and the basic dimensionless promoter (O ) reaction-diffusion equation, governing both phenomena, is... 

Now, if in this period, both evaporation from the granules and the internal equilibrium of water vapor in the bed are rapid compared to the transfer of moisture over the bed-stream interface, then the diffusion equation can be solved (23). Using the dimensionless parameter... 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

We have already noted that mass transfer in a liquid is almost always characterized by large values of the Peclet number (the Peclet number for mass transfer involves the product of the Schmidt number and Reynolds number instead of the Prandtl number and Reynolds number) and that the dimensionless form of the convection-diffusion equation governing transport of a single solute through a solvent is still (9-7), with 6 now being a dimensionless solute concentration. For transfer of a solute from a bubble or drop into a liquid that previously contained no solute, the concentration 6 at large distances from the bubble or drop will satisfy the condition... 

Equation (1) is the relevant time-dependent diffusion equation expressed in the form of dimensionless variables ... 

For each w satisfying condition (4.4.27), the leading terms of the asymptotic expansions for Eqs. (4.4.26) and (4.4.28) with the same boundary conditions coincide in the inner and outer regions. Therefore, as Pe - 0, in the diffusion equation one can replace the actual fluid velocity field v by w. This fact allows one to use the results presented later on in Section 4.11. Namely, as w we take the velocity field for the potential flow of ideal fluid past the cylinder. This approximation yields an error of the order of Pe in the inner expansion. By retaining only the leading terms in (4.11.15), we obtain the dimensionless diffusion flux at small Peclet numbers in the form... 

In the particle-centered spherical coordinate system (R, 6, dimensionless variables comprises the convective diffusion equation... 

The dimensionless species concentration distribution (C) is described by the time-dependent convection-diffusion equation ... 

Simultaneous solution of the convective diffusion equations for mass and heat must be done numerically in all but trivial cases. The solutions can be based on dimensioned variables like z and T, and this has the advantage of keeping the physics of the problem close at hand. However, the solutions are then quite specific and must be repeated whenever a design or operating variable is changed. Somewhat more general solutions, while still numerical, can be obtained through the judicious use of dimensionless variables, dimensionless parameters, and dimensionless functions. Table 8.1 defines a number of such variables. 

In this equation D (m2.s 1) represents the radon diffusivity, X the radioactive decay constant (s 1), C (Bq.m3) the radon concentration in the pore space, R (Bq.kg1) the radium concentration in the material, p (kg.m3) the bulk density of the dry material, E (dimensionless) the radon emanation power coefficient for the pore spaces, s (dimensionless) the total porosity and 0 (dimensionless) the moisture. The solution of the diffusion equation for an homogeneous medium represents the flux release from the waste material to the surface, Jt (Bq.m 2.s ). For a system without cover we obtain (Rogers, 1984) ... 

The dimensionless steady-state convective-diffusion equation requiring solution is... 

In 1968, Dyer, Gettins, and Molyneux (28) studied Ba2+ isotopic ion exchange in BaA over the temperature range 19° to 65 °C. They did not show plots of dimensionless time as a function of time but stated that these plots were linear and, therefore, obey the simple diffusion equation. There is some discrepancy between this work and the work of Hoinkis and Levi at the Hahn-Meitner Institute (35, 37) that is described above, in which they state that in their study of Ba2+ isotopic ion exchange, linear plots of dimensionless time vs. time are not obtained and fit the data by an equation for a two-step process. 

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