Differential equations diffusion modeling, boundary value problems

The differential diffusion equations system to solve when a potential pulse E is applied and the corresponding boundary value problem (bvp) when the expanding plane model for the DME is considered are ... 

The Mean Transport Pore Model (MTPM) described diffusion and permeation the model (represented as a boundary value problem for a set of ordinary differential equations) are based on Maxwell-Stefan diffusion equation and Weber permeation law. Parameters of MTPM are material constants of the porous solid and, thus, do not dependent on conditions under which the transport proeesses take place. 

The general solution to the differential equation includes many possibilities the engineer needs to provide initial conditions to specify which solution is desired. If aU conditions are available at one point [as in Eq. (8.1)], then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equation becomes a two-point boundary value problem (see Chapter 9). Initial value problems as ordinary differential equations arise in the control of lumped parameter models, transient models of stirred tank reactors, and generally in models where there is no diffusion of the unknowns. 

The analytical solutions presented above are most of all derived on the basis of the very simple Henry isotherm or the more physically sensible Langmuir isotherm. Beside these analytical solutions a direct integration of the initial and boundary value problem of the diffusion-controlled model is possible. To do so differentials are replaced by differences. This approximation leads to linear equation systems for each time step which have to be solved. As... 

Application of numerical methods have been rather seldom in studies of adsorption kinetics from micellar solutions. The main difficulties are probably connected with the large number of independent parameters. The first work belongs to Miller [146]. Fainerman and Rakita also published numerical results of the solution of the boundary value problem (5.236), (5.237), (5.245) [85]. Recently Danov et al. proposed an original method for solving the boundary value problem for the diffusion of micelles and monomers [92]. The system of equations was reduced to a system of ordinary differential equations by using a model concentration profile in the bulk phase. The obtained results agree better with dynamic surface tensions of micellar solutions than equation (5.248). 

Considering chemical application problems, a large number of them yields mathematical models that consist of initial-value problems (IVPs) for ordinary differential equations (ODEs) or of initial-boundary-value problems (IBVPs) for partial differential equations (PDEs). Special problems of this kind, which we have treated, are diffusion-reaction processes in chemical kinetics (various polymerizations), polyreactions in microgravity environment (photoinitiated polymerization with laser beams) and drying procedures of hygroscopic porous media. 

Needless to say, the assumption of plug flow is not always appropriate. In plug flow we assume that the convective flow, i. e., the flow at velocity qjAt = v that is caused by a compressor or pump, is dominating any other transport mode. In practice this is not always so and dispersion of mass and heat, driven by concentration and temperature gradients are sometimes significant enough to need to be included in the model. We will discuss such a model in detail, not only because of its importance, but also because the techniques used to handle the ensuing boundary value differential equations are similar to those used for other diffusion-reaction problems such as catalyst pellets, as well as for counter-current processes. 

When the Fickian diffusion model is used many reaction-diffusion problems in porous catalyst pellet can be reduced to a two-point boundary value differential equation of the form of equation B.l. This is not a necessary condition for the application of this simple orthogonal collocation technique. The technique in principle, can be applied to any number of simultaneous two-point boundary value differential equations. 

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