Exponential Difference Solutions

With an implicit scheme applied to exponential differences, equation (7.24) becomes 

Let s assume that we have zero-order and first-order source/sink terms (the most common case). Then, 

Equation (7.30) is solved with a tridiagonal matrix algorithm as described in Patankar (1980). First reform equation (7.29) into 

to solve for the concentration profile at t = n + l, equation (7.31) is rewritten as 

At the far end of the domain, i = I, a constant gradient is normally a good assumption, where 

---

Exponential differences Flux between control volumes is given by a weighted mean of conditions in both control volumes, as determined by the one-dimensional, steady flux solution. 

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) ... 

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. 

The stretched exponentials described above are characterized by scaling parameters oi and V. In different solutions a assumes a substantial range of values, while v varies over a considerably narrower range. 

The amplitude of the exponential differs for each species Q (eq) is the concentration of species A at equilibrium. The are solutions of the... 

V-halTf hpITf This will be satisfied by any value in the negative half plane, i.e. by any value of ha<0. Thus for an exponentially decaying solution, the trapezoidal algorithm will be stable for any desired step size, similar to the backwards difference algorithm. 

The above approximation, however, is valid only for dilute solutions and with assemblies of molecules of similar structure. In the event that concentration is high where intemiolecular interactions are very strong, or the system contains a less defined morphology, a different data analysis approach must be taken. One such approach was derived by Debye et al [21]. They have shown tliat for a random two-phase system with sharp boundaries, the correlation fiinction may carry an exponential fomi. 

When considering the construction of exactly symmetric schemes, we are obstructed by the requirement to find exactly symmetric approximations to exp(—ir/f/(2fi,)). But it is known [10], that the usual stepsize control mechanism destroys the reversibility of the discrete solution. Since we are applying this mechanism, we now may use approximations to exp —iTH/ 2h)) which are not precisely symmetric, i.e., we are free to take advantage of the superior efficiency of iterative methods for evaluating the matrix exponential. In the following, we will compare three different approaches. 

It is a property of this family of differential equations that the sum or difference of two solutions is a solution and that a constant (including the constant i = / ) times a solution is also a solution. This accounts for the acceptability of forms like A (t) = Acoscot, where the constant A is an amplitude factor governing the maximum excursion of the mass away from its equilibrium position. The exponential form comes from Euler s equation... 

The two states viz. disappearing mass of solute and number of erystals formed not only show opposing trends in time but also differ largely in magnitude. These attributes of the system together with the exponential form of nueleation kineties neeessitate very fine diseretization, whieh, eoupled with the faet that the differential equation eannot be solved off-line, would result in a very large and highly... 

FIG. 18 Scaling plot of L t) from the numerical solution of the rate equations for quenches to different equilibrium states [64] from an initial exponential MWD with L = 2. The final mean lengths are given in the legend. The inset shows the original L t) vs t data. Here = (0.33Lqo). ... 

Assuming fj, < 1/2, this solution implies a monotonic approach to equilibrium with time. From a purely statistical point of view, this is certainly correct the difference in number between the two different balls decreases exponentially toward a state in which neither color is preferred. In this sense, the solution is consistent with the spirit of Boltzman s H-theorem, expressing as it does the idea of motion towards disorder. But the equation is also very clearly wrong. It is wrong because it is obviously inconsistent with the fundamental properties of the system it violates both the system s reversibility and periodicity. While we know that the system eventually returns to its initial state, for example, this possibility is precluded by equation 8.142. As we now show, the problem rests with equation 8.141, which must be given a statistical interpretation. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

---