Parabolic flux distribution

Parabolic Flux Distribution p = it. Yovanovich [131-133] reported the solution for the parabolic flux distribution corresponding to p = V2. 

General Expression for Flux Distribution of the Form (1 - u2)1. Yovanovich [130] chose the general flux function f(u) = (1 - u2y with parameter p, which gives (1) the isoflux contact when p = 0, (2) the equivalent isothermal strip when p = -Vi, and (3) the parabolic flux distribution when x = Vi to develop another general solution ... 

Parabolic Flux Distribution p = i6. The parabolic flux distribution p = Vi gives... 

The true isothermal strip solution and the equivalent isothermal flux solution predict values of the dimensionless spreading resistance that are in close agreement provided e < 0.4. The parabolic flux distribution gives the greatest values of the spreading resistance, followed by the isoflux values, which are greater than the values for the isothermal strip. 

As discussed previously, several solar photoreactor geometries can be reduced to cylindrical glass tubes externally illuminated by different types of reflectors, like parabolic troughs, CPC, V-grooves, or without reflector, directly illuminated by the sun. In this section the general solution of the PI approximation for this t)q5e of photo reactors is reported. This general solution is applicable to any particular reactor if the flux distribution impinging on the wall of the tubular reaction space is known. 

The exponentially parabolic Treanor distribution function, which provides a significant overpopulation of the highly vibrationally excited states, was illustrated in Fig. 3-3. To analyze the quite complicated W flux (3-122), it can be divided into linear and non-linear components ... 

Figure 18 Flux concentration distribution of a 90° rim angle parabolic trough solar concentrator. Adapted from Arancibia-Bulnes and Cuevas (2004), with permission from Elsevier.

There are two distinct contributions to the flux. The initial 3-correlated contribution, which gives rise to the transition state rate, and a retarded backflow j t) associated with third-body collisions. The temporal characteristics of the flux can be determined from the phase space distribution function R, t R 0), R(0)) which, for the inverted parabolic potential, is ... 

The Taylor dispersion problem is closely related to that discussed in the previous section, but also differs from it in some important fundamental respects. In the preceding problem, we assumed that the fluid was initially at a constant temperature upstream of z = 0 and that there was a constant heat flux into (or out of) the tube for all z > 0. In that case, the system has a steady-state temperature distribution at large times, and it was that steady-state problem that we solved. In the present case, there is no steady state. If the velocity were uniform across the tube instead of having the parabolic form (3 220), the temperature pulse that is initially at z = 0 would simply propagate downstream with the uniform velocity of the fluid, gradually spreading in the axial direction because of the action of heat conduction (i.e., the diffusion of heat). After a time /, the pulse would have moved downstream by a distance Uf, and the temperature pulse would have spread out over a distance of 0(s/(K tt)). Even in this simple case, there is clearly no steady state. The temperature distribution continues to evolve for all time.21... 

In Fig. 2, a numerical example of the angular distribution of emission power radiated from the point source is presented for the case of a two-dimensional square lattice of air holes in a polymer. The energy flux is strongly anisotropic, showing a relatively small intensity in all directions except along directions associated with parabolic points of iso-frequency surface, where the intensity tends to infinity. Predictions of asymptotic analysis on far-field radiation pattern (2) are substantiated with FTDT calculations, revealing a reasonable agreement (Fig. 2, left). 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

The simulated profiles of adsorbate, methylene chloride, at different times are given in Fig. 6.11, in which the development of the concentration profiles in the column with time is seen. In comparison with Fig. 6.1, it is found that the simulation is closely similar. Yet after careful comparison, the shape of concentration distribution in the adsorption section (represented by the red brackets) is somewhat different. The parabolic shape of purge gas concentration distribution is more obvious by using standard Reynolds mass flux model due to better simulation near the column wall. 

---