Explicit solution

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

In principle, Chen, given the flux relations there is no difficulty in constructing differencial equations to describe the behavior of a catalyst pellet in steady or unsteady states. In practice, however, this simple procedure is obstructed by the implicit nature of the flux relations, since an explicit solution of usefully compact form is obtainable only for binary mixtures- In steady states this impasse is avoided by using certain, relations between Che flux vectors which are associated with the stoichiometry of Che chemical reaction or reactions taking place in the pellet, and the major part of Chapter 11 is concerned with the derivation, application and limitations of these stoichiometric relations. Fortunately they permit practicable solution procedures to be constructed regardless of the number of substances in the reaction mixture, provided there are only one or two stoichiomeCrically independent chemical reactions. 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

For more than three components extremely heavy algebra is generated in attempting to solve the implicit flux relations, and in general no usefully compact explicit solution is obtained. However, there are two interesting special cases in which explicit flux relations can be obtained with an arbitrary nutr er of components in the mixture. Neither would be expected to correspond accurately with physical situations of practical interest, but they may provide useful qualitative, or semi-quantitative pointers to the behavior of more accurate models. 

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

T-Jhile the stoichiometric relations have rendered the above problem tractable by permitting an explicit solution of the dusty gas model flux relations, it should be pointed out that they do not lead to equally radical simplifications with all flux models. In the case of the Feng and Stewart models [49- for example, Che total flux of species r is formed by in-... 

Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-and-error enters, however, because (GSi)r and are mild functions of Tg and related Te, respectively, and aprehminary guess of Tg is necessaiy. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. 

On page 235-241 is the explicit solution used in Excel format to make studies, or mathematical experiments, of any desired and possible nature. The same organization is used here as in previous Excel applications. Column A is the name of the variable, the same as in the FORTRAN program. Column B is the corresponding notation and Column C is the calculation scheme. This holds until line 24. From line 27 the intermediate calculation steps are in coded form. This agrees with the notation used toward the end of the FORTRAN listing. An exception is at the A, B, and C constants for the final quadratic equation. The expression for B was too long that we had to cut it in two. Therefore, after the expression for A, another forD is included that is then included in B. 

We have seen that a kinetic scheme does not have to be very complex before explicit solutions for concentrations as functions of time become difficult or impossible to obtain. Even with those complex schemes for which solutions are possible, the... 

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

EXAMPLE Consider an m x n lattice, where rn is even. Prom the explicit solution to the dimer problem [boll79], given by... 

This system of i + 2 equations is nonlinear, and for this reason probably has not received attention in the least-squares method (207). We are able to give an explicit solution (163) for the particular case when Xy = xj and m,- = m for all values of i that is, when all reactions of the series are studied at a set of temperatures, not necessarily equidistant, but the same for all reactions. Let us introduce... 

If the explicit solution cannot be used or appears impractical, we have to return to the general formulation of the problem, given at the beginning of the last section, and search for a solution without any simplifying assumptions. The system of normal equations (34) can be solved numerically in the following simple way (164). Let us choose an arbitrary value x(= T ) and search for the optimum ordinate of the point of intersection y(= log k) and optimum values of slopes bj to give the least residual sum of squares Sx (i.e., the least possible with a fixed value of x). From the first and third equations of the set eq. (34), we get... 

Because of the terms Ir-RAl and Ir-rT explicit solutions to Eq. 3 carmot be obtained in position space. In such cases approximate solutions are usually expressed as truncated linear combinations of basis functions (LCAO expressions). In spite of its successes, the LCAO approximation experiences various difficulties (truncation limits, nature of the basis functions, etc.) hard to estimate and which are not entirely controllable [51]. 

Combining the two equations then gives an explicit solution for concentration Y1 and hence also Xj... 

Hybrid MPC-MD schemes may be constructed where the mesoscopic dynamics of the bath is coupled to the molecular dynamics of solute species without introducing explicit solute-bath intermolecular forces. In such a hybrid scheme, between multiparticle collision events at times x, solute particles propagate by Newton s equations of motion in the absence of solvent forces. In order to couple solute and bath particles, the solute particles are included in the multiparticle collision step [40]. The above equations describe the dynamics provided the interaction potential is replaced by Vj(rJVs) and interactions between solute and bath particles are neglected. This type of hybrid MD-MPC dynamics also satisfies the conservation laws and preserves phase space volumes. Since bath particles can penetrate solute particles, specific structural solute-bath effects cannot be treated by this rule. However, simulations may be more efficient since the solute-solvent forces do not have to be computed. 

The explicit solution of Eq. (27), which uses a Fourier transform or a bilateral Laplace transform, is described in detail in Ref. 38. Its eigenvalues and eigenvectors are determined by the nonlinear eigenvalue equation... 

And again, since any matrix multiplied by its inverse is a unit matrix, this provides us with the explicit solution for b, which was to be determined. 

The broadest class of models, phenomenological models, account explicitly for individual phenomena such as swelling, diffusion, and degradation by incorporation of the requisite transport, continuity, and reaction equations. This class of models is useful only if it can be accurately parameterized. As phenomena are added to the model, the number of parameters increases, hopefully improving the model s accuracy, but also requiring additional experiments to determine the additional parameters. These models are also typically characterized by implicit mean-field approximations in most cases, and model equations are usually formulated such that explicit solutions may be obtained. Examples from the literature are briefly outlined below. 

There are, however, obvious limitations. It is not possible to make a very small spherical electrode, because the leads that connect it to the circuit must be even much smaller lest they disturb the spherical geometry. Small disc or ring electrodes are more practicable, and have similar properties, but the mathematics becomes involved. Still, numerical and approximate explicit solutions for the current due to an electrochemical reaction at such electrodes have been obtained, and can be used for the evaluation of experimental data. In practice, ring electrodes with a radius of the order of 1 fxm can be fabricated, and rate constants of the order of a few cm s 1 be measured by recording currents in the steady state. The rate constants are obtained numerically by comparing the actual current with the diffusion-limited current. 

These second-order nonlinear differential equations have no explicit solution but can be solved numerically. The limiting substrate for the biofilm transformations is the one that penetrates the shortest distance into the biofilm. Equations (2.26) and (2.27) are, thereby, reduced to an equation corresponding to Equation (2.20). If the limiting substrate cannot be identified, approximations based on Equation (2.25) can be developed. 

For a first order reaction, m = 1, the catalyst effectiveness f) is independent of As, so that after elimination of A and As the explicit solution for the rate is... 

Explicit solutions for r can be made for some values of m. When m... 

Because IB = B, an explicit solution for B results. Note that the order of multiplication is critical because of the lack of commutation. Postmultiplication of both sides of Equation (A. 10) by A-1 is allowable but does not lead to a solution for B. 

This book provides a quantitative treatment for a variety of geochemical problems involving mass balance, equilibrium, dynamics, and transport. Numerous applications from igneous and sedimentary environments are presented in the form of problems and their explicit solutions. 

W. Wiechert, M. Mollney, N. Isermann, M. Wurzel, and A. A. de Graaf, Bidirectional reaction steps in metabolic networks. Part III Explicit solution and analysis of isotopomer labeling systems. Biotechnol. Bioeng. 66, 69 85 (1999). 

The equivalent to the law of mass action in equilibria are the sets of differential equations in kinetics. They are defined by the chemical reaction scheme. Again, there are explicit solutions for very simple models but most other models lead to sets of differential equations that need to be integrated numerically. Matlab supplies an extensive collection of functions for... 

Chapter 4, Model-Based Analyses, is essentially an introduction into least-squares fitting. It is crucial to clearly distinguish between linear and nonlinear least-squares fitting linear problems have explicit solutions while non-linear problems need to be solved iteratively. Linear regression forms the base for just about everything and thus requires particular consideration. 

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