Approximate Analytical Solutions

Such an expansion will be referred to as the parabolic isotherm. The simplifying assumption made here is of a physical nature. It restricts the range of validity of 

Equation 10.14 is exact for a parabolic isotherm. However, it cannot be solved in closed form without some further simplifications. These simplifications will be of a mathematical nature, rendering the equation approximate. Several approaches are possible at this stage [15]. The physical (parabolic isotherm) and the mathematical (see below) simplifications combine to give an approximate solution. It is important to imderstand the difference between the two types of simplifications and their different consequences. 

Originally, Houghton [13] derived his equation with the assumption that the mass transfer kinetics is infinitely fast but that axial dispersion caimot be neglected. In view of the previous discussion (Section 10.1), we can extend the validity of the Houghton approach to the case of a finite rate of mass transfer, by lumping axial dispersion and mass transfer contributions into an apparent dispersion coefficient. 

Ignoring the term AC in the denominator of the second and third terms of Eq. 10.14 gives 

A solution of Eq. 10.17 can be derived using the Cole-Hopf transform. This solution gives the elution profile of a finite width pulse at the end of an infinitely long 

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The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Although only approximate analytical solutions to this partial differential equation have been available for x(s,D,r,t), accurate numerical solutions are now possible using finite element methods first introduced by Claverie and coworkers [46] and recently generalized to permit greater efficiency and stabihty [42,43] the algorithm SEDFIT [47] employs this procedure for obtaining the sedimentation coefficient distribution. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

Figure Al. a) Porosity distribution for a ID melt column (solid curve) assuming constant melt flux (see Spiegelman and Elliott 1993). Average porosity is shown as the dashed line, b) Emichment factors (a) calculated from the analytical solution (solid curves) and approximate analytical solution (dotted curves) for °Th and Ra. c) Emichment factors (a) calculated from the numerical solution of Spiegelman and Elliott (1993) for °Th and Ra. In these plots, depth (z) is non-dimensionalized. See text for explanation.

Figure A2. ( Ra/ °Th) and f °Th/ U) calculated from the analytical solution (solid light curves), approximate analytical solution (dotted light curves) and full numerical solution (solid dark curves). Horizontal curves represent constant maximum porosity ( x), while vertical curves represent constant upwelling rates (W ) in cm/yr. Selected contours are labeled. Contours range from 1-100 cm/a and 0.1-10% for upwelling velocity and maximum porosity, respectively. See text for explanation.

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

Equations 8.5-34 and -35 are nonlinearly coupled through T, since kA depends exponentially on T. The equations cannot therefore be treated independently, and there is no exact analytical solution for cA(r) and T(r). A numerical or approximate analytical solution results in tj expressed in terms of three dimensionless parameters ... 

Equations 9.2-28 and -29, in general, are coupled through equation 9.2-30, and analytical solutions may not exist (numerical solution may be required). The equations can be uncoupled only if the reaction is first-order or pseudo-first-order with respect to A, and exact analytical solutions are possible for reaction occurring in bulk hquid and liquid fdm together and in the liquid film only. For second-order kinetics with reaction occurring only in the liquid film, an approximate analytical solution is available. We develop these three cases in the rest of this section. 

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

An approximate analytical solution to this system has been proposed by van Krevelen and Hoftijzer (Eq. (27) and Fig. 45.3)) ... 

Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer <5, is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is ... 

In a recent paper, an approximate calculation was made of effects (b) to (d) above (19), using an approximate analytical solution for the diffusion problem, for the case where the reaction occurs readily over a short range of separation distances of the reactants. In the present report, we summarize the results of our recent calculations on a numerical solution of the same problem. A more complete description is given elsewhere (28). One additional modification made here to (19) is to ensure that the current available rate constant data at AG° = 0 (Appendix) are satisfied. 

An approximate analytical solution (see above) yields that Tc is nearly independent of the tunnel mode frequency. Experimentally, it is found that in H2-xDxSQ substantial variations in take place with variations in x [54]. This fact does not contradict the above results since the tunnel mode is renormalized through the coupling C as -/q (S ) - (S > /g4T). As C... 

For the case of nonlinear adsorption isotherms, no analytical solutions exist the mass balance equations must be integrated numerically to obtain the band profiles. Approximate analytical solutions are only possible for the cases where the solute concentration is low and accordingly, the deviation from linear isotherm is only minor. All the approximate analytical solutions utilize a parabolic adsorption isotherm q = aC( -bC). This constraint prevents us from drawing general conclusions regarding most of the important consequences of nonlinearity. 

When working with a computational transport code, there is httle reason to simplify equation (2.14) further. Our primary task, however, is to develop approximate analytical solutions to environmental transport problems, and we will normally be assuming that diffusion coefficient is not a function of position, or x, y, and z. We can also expand the convective transport terms with the chain rule of partial differentiation ... 

The exact solution of Ecj 1 is unknown. Only numerical or approximate analytical solutions are available. Solutions are obtained in terms dimensionless variables and parameters... 

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

An approximate analytical solution is presented for the RIS model of flexible polymer configurations. The method is applied to calculate and . 

A more tractable approach to shock wave propagation in water is that of Kirkwood and coworkers. For details of this rather involved analysis, the reader is referred to Ref 1, pp 29—33 and 104—106. The basic assumptions of this theory are that behind the shock front the entropy is constant, ie, ds=0, and that the conversion of the expl to its products occurs at constant volume. With these assumptions, it is then possible to get approximate analytical solutions of the equation of motion in terms of the enthalpy of the system... 

Koutecky obtained the numerical solution for the catalytic current at a DME by rigorous solution of the convective-diffusion equations by the expanding-plane model with =0 [199]. Subsequently, an approximate analytical solution was obtained which holds over the whole range of k j with an error of not more than 1% [188]. The equations are... 

An approximate analytical solution of the equations ate< l,l — a[Pg.467]

A further effect of D and A when they influence nuclear relaxation is that again an angle (if axial) or two angles (if rhombic) are needed which takes into account the location of the resonating nucleus within the D (A) tensor frame. The theoretical approach to the description of / i under these circumstances gives approximate analytical solutions [44], Numerical solutions [45,46], and a computer program (http //www.cerm.unifi.it) [47] are also available. 

The first contribution, which led to an approximate analytical solution, was then improved in further studies by Levart et al. [48]. For moderate blocked ratio values (0.5 < P< 1), ( blocked fraction of the surface), some contributions, such as that of Filinovsky [49], attempted to consider the convective terms for two-dimensional concentration fields but disregarded the diffusion terms in direction parallel to the electrode plane. 

An approximate analytical solution of Eq. (2.77) can be found, and, in fact, experimental data correspond to the condition ... 

Several useful approximate analytical solutions to Eq. (4.8) were developed. A well-known example is Higuchi s equation, based on a pseudosteady-state approach7 ... 

Lee, P. Diffusional release of a solute from a polymeric matrix Approximate analytical solutions. J. Membrane Sci. 7 255—275, 1980. 

An approximate analytical solution of Eq. (9.46) may be obtained by assuming that heat transport is confined to a thickness 8(t), verifying... 

For a second-order catalytic mechanism with an irreversible chemical reaction, an approximate analytical solution has been reported [86] ... 

We have so far been able to obtain exact explicit analytic solutions for (a) the case where only processes (i) and (ii) are significant, and (b) the case where only processes (ii) and (iii) are significant. We have also obtained an approximate analytic solution for the case where all three processes (i), (ii) and (iii) occur, but where the loss of radicals occurs predominantly by process (ii) rather than by prodess (iii). As a generalisation of case (a), we have obtained a general solution which covers the case where the parameters which characterise the processes (i) and (ii) are themselves time-dependent. The general solution to case (b) requires modification if processes of type (ii) do not occur. Complete solutions have been obtained for three special cases of (b), namely, decay from a Stockmayer-01Toole distribution of locus populations, decay from a Poisson distribution of locus populations, and decay from a homogeneous distribution of locus populations. 

We consider in this section some approximate analytical solutions to the electronic Schrodinger equation, in order to provide some basic insight into the energetics of the making and breaking of chemical bonds. Since most of the results are well known from quantum mechanics/chemistry, we will only present the key points and sometimes omit detailed proofs. 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

Approximate Analytical Solutions for Models of Three-Dimensional Electrodes by Adomian s Decomposition Method... 

Adomian s Decomposition Method is used to solve the model equations that are in the form of nonlinear differential equation(s) with boundary conditions.2,3 Approximate analytical solutions of the models are obtained. The approximate solutions are in the forms of algebraic expressions of infinite power series. In terms of the nonlinearities of the models, the first three to seven terms of the series are generally sufficient to meet the accuracy required in engineering applications. 

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