Boundary Value Problems dispersion

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

Example of an Axial Dispersion Model. Linear and Nonlinear Two-point Boundary Value Problems (BVPs)... 

The axial dispersion model has led to the two-point boundary value problem (5.37) for uj from uj = ujstart = 0 to u> = uJend = 1- DEs are standardly solved by numerical integration over subintervals of the desired interval [uj start, we d. For more on the process of solving BVPs, see Section 1.2.4 or click on the Help line under the View icon on the MATLAB desktop, followed by a click on the Search tab in the Help window and searching for BVP . 

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. 

Inclusion of axial dispersion in the plug-flow model makes the model equations a boundary-value problem, so that conditions at both the reactor inlet and outlet need to be specified. The commonly used boundary conditions are the so-called Danckwerts type (12), although their origin goes back to Langmuir (13). 

The disadvantage of including axial dispersion is that an exit boundary condition must be specified, and in cases where an analytical solution is not available, a numerical boundary-value problem must be solved in the axial direction, rather than an initial-value problem. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

We have previously pointed out that the use of the dispersion model changes the reactor analysis from an initial-value (PFR) to a boundary-value problem. As a result, we should worry about the form of the boundary conditions to use for equation (5-77). This is illustrated in Figure 5.16, where several possible configurations of inlet and outlet conditions are shown. Hopefully, this is not to make a... 

The above dispersion relation exhibits the existence of curious waves, such that the group velocity is positive but the phase velocity is negative and the amplitude increases in the direction of the phase velocity. Thus, it is difficult to predict what waves are generated by imposing disturbance and how they propagate subsequently. Then, we return to the original differential equation and we consider the initial-boundary value problem, where a disturbance with a certain frequency is imposed at a location in the uniform flow and from an initial instant and to a specified direction. 

In cases of different dispersion coefficients (laminar flow), the analytical problem is close to intractable. For nonlinear reactions the axial dispersion model leads to a set of two-point boundary value problems which must be solved by an appropriate iterative numerical scheme. This is a great disadvantage of the model. We conclude that the axial dispersion model is cumbersome in reactor type calculations and should be abandoned. The reasons for this can be stated as follows. 

The dispersion model is at best only an approximate representation of physical reality, often in contradiction with selected physical evidence. For example, the dispersion model predicts the appearance of tracer upstream of the injection point, yet, all the experimental evidence in packed beds points to downstream spreading of the dye only. The boundary conditions of the dispersion model are often difficult to meet in practice. Considering the above discrepancies between the model and physical reality it seems hardly justified to do the elaborate boundary value problem calculations. This is especially true since for RTDs close to PFR micromixing effects cannot be pronounced. It seems more appropriate to relate the intensity of dispersion as given by eq. (37) to an equivalent number of tanks in series. 

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. 

Computer literacy is assumed, and there are many problems that require computer solution, particularly as one becomes involved with nonisothermal reactors, boundary-value dispersion problems, the more advanced fixed-bed problems, and interpretation of kinetic data. We have not tried to get into the software business here, in view of the continuing rapid evolution of various aspects of that field. We have yielded to the temptation in a couple of instances to suggest, in outline, some algorithms for specific problems, but in general this is left up to the reader. 

The problem to be solved in this paragraph is to determine the rate of spread of the chromatogram under the following conditions. The gas and liquid phases flow in the annular space between two coaxial cylinders of radii ro and r2, the interface being a cylinder with the same axis and radius rx (0 r0 < r < r2). Both phases may be in motion with linear velocity a function of radial distance from the axis, r, and the solute diffuses in both phases with a diffusion coefficient which may also be a function of r. At equilibrium the concentration of solute in the liquid, c2, is a constant multiple of that in the gas, ci(c2 = acj) and at any instant the rate of transfer across the interface is proportional to the distance from equilibrium there, i.e. the value of (c2 - aci). The dispersion of the solute is due to three processes (i) the combined effect of diffusion and convection in the gas phase, (ii) the finite rate of transfer at the interface, (iii) the combined effect of diffusion and convection in the liquid phase. In what follows the equations will often be in sets of five, labelled (a),..., (e) the differential equations expression the three processes (i), (ii) (iii) above are always (b), (c) and (d), respectively equations (a) and (e) represent the condition that there is no flow over the boundaries at r = r0 and r = r2. 

For convenience, the relevant dimensionless numbers for gas-particle flow derived in this section are collected in Table 1.1. In practice, one must choose appropriate values for U and L corresponding to a particular problem. For example, they may be determined by the inlet and/or boundary conditions. However, one case of particular interest is particles falling in an unbounded domain for which convenient choices are T = t/p and U = ul = Up - f/gl = Tp g (i.e. the settling velocity). For this case, there is no source term for p and so it relaxes to zero at steady state due to the drag. The disperse-phase Mach number thus becomes infinite. For settling problems, the particle Archimedes number (see Table 1.1) is often used in place of the Froude number. 

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