Explicit, Exponential Difference Solutions

Central differences are applied to diffusion problems, and upwind differences are applied to convective problems, but most cases have both diffusion and convection. This conundrum led Spaulding (1972) to develop exponential differences, which combines both central and upwind differences in an analytical solution of steady, one-dimensional convection and diffusion. Consider a control volume of length Ax, in a flow fleld of velocity U, and transporting a compound, C, at steady state with a diffusion coefficient, D. Then, the governing equation inside of the control volume is a simphflcation of Equation (2.14)  

The particular solution is found by applying boundary condition 1 to equation (7.16)  

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EXAMPLE 7.5 Comparison of explicit and implicit exponential differences with the exact solution... 

The problem of Example 7.3 will again be solved with explicit and implicit exponential differences, and compared with the analytical solution, equation (E7.4.7). This solution is given in Figure E7.5.1. Note that the explicit solution is close to the analytical solution, but at a Courant number of 0.5, whereas the implicit solution could solve the problem with less accuracy at a Courant number of 5. In addition, the diffusion number of the explicit solution was 0.4, below the limit of Di < 0.5. The implicit solution does not need to meet this criteria and had Di = 4. 

Consider now the charge distribution of ionized centers in the depletion region. For comparison, two cases can illustrate the difference between crystalline and a-Si H. The two cases are for n-type semiconductors, one with a single band of donor levels and the other with a uniform density of donor levels throughout the band gap. These two cases are illustrated in Fig. 2. In these cases, Poisson s equation can be explicitly solved. The solutions yield parabolic bands for the case with a single donor band and an exponential behavior for the continuous uniform-state density. The field dependences of the two cases differ also. For the discrete level, a linear dependence results, whereas an exponential behavior is obtained in the uniform-state case. The most striking difference is in the density of ionized states, which is uniform... 

In the derivation of the regular solution model the vibrational contribution to the excess properties has been neglected. However, as a first approximation the vibrational contribution can be taken as independent of the interaction between the different atoms, and this contribution can be factored out of the exponential and taken into account explicitly. The partition function of the solution is then given as... 

This large increase in the solvent power of ethylene on compression to 100 bar cannot be attributed to a hydrostatic pressure effect on the vapor pressure of naphthalene, since the pressure effect is explicitly accounted for in the exponential term in equation 1.1. Instead, the large difference in experimental and calculated naphthalene solubility at high pressures is associated with the nonideal behavior of ethylene as it is compressed to liquid-like densities in its critical region. It is the strength of the interactions between solvent and solute molecules that fixes the solubility depending of course on whether the solvent molecules are in close enough proximity to interact. Chapters 3 and 5 delve more deeply into intermolecular interactions and solubility behavior. 

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