Network equations solution

Attack by alkali solution, hydrofluoric acid and phosphoric acid A common feature of these corrosive agents is their ability to disrupt the network. Equation 18.1 shows the nature of the attack in alkaline solution where unlimited numbers of OH ions are available. This process is not encumbered by the formation of porous layers and the amount of leached matter is linearly dependent on time. Consequently the extent of attack by strong alkali is usually far greater than either acid or water attack. 

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

However, if the atoms are not related by symmetry, the normal rules break down. The homoionic N-N bond in the hydrazinium ion is an electron pair bond, but one in which N1 contributes 1.25 and N2 0.75 electrons. How can we apply the bond valence model in such cases where no solution to the network equations is possible One approach is to isolate the non-bipartite portion of the graph into a complex pseudo-atom. Thus in the hydrazinium ion the homoionic bond and its two terminating N atoms are treated as a single pseudo-anion which forms six bonds with a valence sum equal to the formal charge of —4. 

Stiff differential equations Differential equations with widely varying rate constants. Like neural networks, their solution depends upon careful selection of step sizes. 

An example of the solution of the network equations for the bond graph of CaCrFj is shown in Figure lO.lOd. 

Classical approach to solving this problem consists in assuming analytical model of the approximative function, for example taking the form of algebraic polynomial. In general it is going to be a certain vector function X t, C), where C is a vector of parameters sought after. The C vector is set upon all discrete observations in the network by solution of equations... 

We now consider elemental potentials of general form, for which the evaluation of the elemental deformation functions is essential to the solution of the network equations. 

Bidispersed Particles For particles of radius Cp comprising adsorptive subparticles of radius r, that define a macropore network, conservation equations are needed to describe transport both within the macropores and within the subparticles and are given in Table 16-11, item D. Detailed equations and solutions for a hnear isotherm are given in Ruthven (gen. refs., p. 183) and Ruckenstein et al. [Chem. Eng. Sci., 26, 1306 (1971)]. The solution for a linear isotherm with no external resistance and an infinite fluid volume is ... 

When the flow pattern is known, conversion in a known network and flow pattern is evaluated from appropriate material and energy balances. For first-order irreversible isothermal reactions, the conversion equation can be obtained from the R sfer function by replacing. s with the specific rate k. Thus, if G(.s) = C/Cq = 1/(1 -i- t.s), then C/Cq = 1/(1 -i-kt). Complete knowledge of a network enables incorporation of energy balances into the solution, whereas the RTD approach cannot do that. 

These equations allow either to predict the swelling degree (w = l/(p) as a function of external conditions or to calculate the network parameters from the correlation between the theoretical and experimental dependencies w(q) or w(p) [22, 102], An example of such a correlation is given in Fig. 2 and 5. As can be seen, theoretical predictions are in good agreement with experimental data. However, when the outer solution contains multivalent cations, only a semi-quantitative agreement is attained. 

The simultaneous determination of a great number of constants is a serious disadvantage of this procedure, since it considerably reduces the reliability of the solution. Experimental results can in some, not too complex cases be described well by means of several different sets of equations or of constants. An example would be the study of Wajc et al. (14) who worked up the data of Germain and Blanchard (15) on the isomerization of cyclohexene to methylcyclopentenes under the assumption of a very simple mechanism, or the simulation of the course of the simplest consecutive catalytic reaction A — B —> C, performed by Thomas et al. (16) (Fig. 1). If one studies the kinetics of the coupled system as a whole, one cannot, as a rule, follow and express quantitatively mutually influencing single reactions. Furthermore, a reaction path which at first sight is less probable and has not been therefore considered in the original reaction network can be easily overlooked. 

One approach to extend such theories to more complex media is network theory. This approach utihzes solutions for transport in single pores, usually in one dimension, and couples these solutions through a network of nodes to mimic the general structure of the porous media [341], The complete set of equations for aU pores and nodes is then solved to determine overall transport behavior. Such models are computationally intense and are somewhat heuristic in nature. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

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