General Form of the Solution

The design of fixed-bed ion exchangers shares a common theory with fixed-bed adsorbers, which are discussed in Chapter 17. In addition, Thomas(14) has developed a theory of fixed-bed ion exchange based on equation 18.21. It assumed that diffusional resistances are negligible. Though this is now known to be unlikely, the general form of the solutions proposed by Thomas may be used for film- and pellet-diffusion control. 

The form of the solution for one-dimensional diffusion is illustrated in Fig. 5.3. The solution c(x,t) is symmetric about x = 0 (i.e., c(x,t) = c(—x,t)). Because the flux at this location always vanishes, no material passes from one side of the plane to the other and therefore the two sides of the solution are independent. Thus the general form of the solution for the infinite domain is also valid for the semi-infinite domain (0 < x < oo) with an initial thin source of diffusant at x = 0. However, in the semi-infinite case, the initial thin source diffuses into one side rather than two and the concentration is therefore larger by a factor of two, so that... 

For the rectangular channels the diffusional deposition problem was solved only for an infinitely wide duct. Real channels have a finite ratio of the height to the width, bh/bw. If it is 1, the formula allows one to use the flow rate rather than the flow velocity. The general form of the solution for such channels with... 

The frequency dependencies of k and k" are plotted in Fig. 14.7 and are characteristic of typical dispersion curves experimentally observed for dielectrics. It is important to note that Eqs. (14.33) and (14.34) are only valid for a dilute gas, since it was implicitly assumed that the local field was identical to the applied field. To solve the problem more accurately for solids, the local rather than the applied field would have to considered in Eq. (14.29). Fortunately, doing this does not change the general forms of the solutions it only modifies the value of the resonance frequency coq (see App. 14A). 

All of the parameters in the Welsz modulus can be measured experimentally. The inequalities in Equations 38 are for effectiveness factors greater than 95% it is best to satisfy these inequalities by at least an order of magnitude (18). Given the general form of the solution to Equation 21 for Langmulr-Hlnshelwood kinetics, which are effectively zero order or greater (15), it seems reasonable to propose that the inequalities in Equation 38 provide a safe estimate of negligible substrate limitations for more complex kinetic expressions. 

As mentioned above, the obvious step is to solve equation (3.17) sequentially, i.e. start with Pj, then substitute it into the equation for Pj, solve it and so on. For the case of (3.17) this can be done analytically and it becomes clear there is a general form of the solution for P,- (see example 3.1). In more complex cases this cannot be done analytically but can still be carried out numerically in the style of Liu and Amundson [15]. The simple system (3.17) is dso amenable to a direct analytical solution via the Laplace transform [16], Pj... 

It is not difficult to continue this process, finding and solving the new ordinary differential equation in each interval from the solution in the previous interval and the condition that the solution be continuous everywhere. This, however, is a tedious process. What we need is either to see a pattern and deduce from it the general form of the solution for all s values or to use one of the other methods to be discussed below. 

Because there is about the energetic propagation, the form and the behavior of the total Poynting vector associated to the DS branches of the diffracted directions is natural to be analyzed. To this aim, the general form of the solutions of the waves propagated on the difiiacted directions associated to the DS branches will be firstly expressed in a form derived from the Laue s (5.104) and Bragg s (5.111) solution in the plane wave approximation the present discussion follows Birau and Putz (2000) and (Putz Lacr a, 2005) ... 

What is the general form of the solution map for problems in which you are given the mass of a reactant in a chemical reaction and asked to find the mass of the product that can be made from the given amount of reactant ... 

The self-consistency comes in because depends on p(z) through V(z). This set of equations can then be solved by a relaxation technique. For diblock copolymers and blends, additional distribution functions are necessary, but the general form of the solution is similar. Scheutjens and Fleer " were the first to exploit the numerical solution of the SCF equations on a lattice for polymers in solution using eq. (9.11). It is possible now to generate an extremely large number of conformations and study N as large as 100000. Their method has been widely applied to study adsorption of poly-... 

The general form of the solution of Eq. (3) with the boundary condition (19) is the following ... 

As an example we eonAsider the solution of Equation (2.80) with a value of a = 50, in this case the general form of the elemental stiffness equation is written as... 

The form of the solution for the bridging contribution is specific to the reinforcement and the crack geometry but is of the general form (35)... 

The general form of the anodic polarisation curve of the stainless steels in acid solutions as determined potentiostaticaiiy or potentiodynamically is shown in Fig. 3.14, curve ABCDE. If the cathodic curve of the system PQ intersects this curve at P between B and C only, the steel is passive and the film should heal even if damaged. This, then, represents a condition in which the steel can be used with safety. If, however, the cathodic curve P Q also intersects ED the passivity is unstable and any break in the film would lead to rapid metal solution, since the potential is now in the active region and the intersection at Q gives the stable corrosion potential and corrosion current. 

This difficulty is due to the fact that the form of the differential equation does not convey any information regarding the existence of cycles. However, the form of the solution does yield this information. Unfortunately, in entirely new problems we generally know the form of the equation and not that of its solution. 

The form of the solution of the dispersion equation (11.61) depends on the sign of the determinant D = q + Pl, i.e., on the values of the characteristic parameters g and P. The latter are determined by the physical properties of the liquid and its vapor, as well as the values of the Peclet number. This allows us to use g and P as some general characteristics of the problem considered here. 

The general form of the (RISM) integral equation appropriate for treating solute-solvent interactions which has been derived in the literature is given by " ... 

In general, the form of the solution to the dynamic model equations will be in the form... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

When the adsorbent molecides are not independent, we can no longer use the relation (D.2) for the GPF of the system. In this case, we must start from the GPF of the macroscopic system from which we can derive the general form of the BI for any concentration of the adsorbent molecule. The derivation is possible through the McMillan-Mayer theory of solution, but it is long and tedious, even for first-order deviations from an ideal solution. The reason is that, in the general case, the first-order deviations would depend on many second-virial coefficients [the analogue of the quantity B2(T) in Eq. (D.9)]. For each pair of occupancy states, say i and j, there will be a pair potential [/pp(R, i,j), and the corresponding second-virial coefficient... 

The general form of the decay of C2 is shown in Figure 7. This could probably reduce to only one line, as found in our results. It is not immediately obvious whether we are measuring 3 or A4. This will depend upon the magnitudes of a23 and a24. We must examine the behavior of this solution further. 

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