Boundary value problems diffusion modeling

Beginning with the innovative work of Tsuji and Yamaoka [409,411], various counter-flow diffusion flames have been used experimentally both to determine extinction limits and flame structure [409]. In the Tsuji burner (see Fig. 17.5) fuel issues from a porous cylinder into an oncoming air stream. Along the stagnation streamline the flow may be modeled as a one-dimensional boundary-value problem with the strain rate specified as a parameter [104], In this formulation complex chemistry and transport is easily incorporated into the model. The chemistry largely takes place within a thin flame zone around the location of the stoichiometric mixture, within the boundary layer that forms around the cylinder. 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

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 ... 

In membrane transport, one-dimensional models are usually used. If the permeates move independently of one another and with the ideal interface permeability, the simple diffusion, described by Fick s law, across the membrane is given by the boundary value problem... 

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... 

Analytical solutions as presented above are based on the very simple Henry isotherm, while for the frequently applied Langmuir isotherm an approximate solution as a power series can be obtained. For any other, more sophisticated isotherm, an analytical solution does not exist. Thus, a direct integration of the initial and boundary value problem of the diffusion-controlled model is required. Using a difference scheme [63] numerical results can be obtained for any type of an adsorption isotherm. The following models rely on such numerical methods. 

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). 

Our model of diffusion and heterogeneous reaction in one of the capillary tubes illustrated in Figure 1.9 is given by the following boundary value problem ... 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

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. 

NUMERICAL METHODS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS MODELING DIFFUSION PROCESSES... 

In Sections II and III, we considered boundary value problems modeling the diffusion process for some substance in a homogeneous material. We have studied problems in which the unknown function takes a prescribed value on the boundary (in Section II), or the diffusion flux of the unknown function is given on the boundary (in Section III). The right-hand sides of the equations were supposed to be smooth. Boundary layers appear in these problems for small values of the parameter. 

NUMERICAL METHODS FOR PROBLEMS MODELING DIFFUSION PROCESSES 299 boundary value problem e-uniformly on every subdomain... 

Numerical Methods for Singularly Perturbed Boundary Value Problems Modeling Diffusion Processes... 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

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. 

---