Some General Linearization Rules

As can be seen, both model equations are non-linear. The question is now, how can we get an impression of the behavior of the outlet concentration ca when the reactor throughput changes. 

One way is to simulate both equations and to give a change in the flow F. However, this does not provide us with much insight. 

Another way is try to find an analytical solution for both equations. This is not a trivial matter, however. We shall therefore introduce linearization as a useful tool to get iusight into process behavior. 

Linearization is based on a Taylor series develop of a function around the operating point. Any arbitrary frmction f(x) can be approximated by  

The original function is now approximated by its tangent to the curve in the reference point. The result of Eqn. (6.10) can now be used in the general model differential equation  

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In some cases, empirical rules can also relate thermodynamic properties to crystal structures. One of the best-known cases is in the AB5 systems where the equilibrium pressure is linearly correlated to the cell volume. As the cell volume increases, the equilibrium plateau pressure decreases, following a InPn law [64]. However, some exceptions exist to this rule such as in LaPts where electronic effects make the smaller unit cell more stable [65[. Nevertheless, generally, for intermetallic compounds the stability of the hydride increases with the size of the interstices [66]. A limitation of this empirical rule is that comparison between different types of intermetallics is impossible. For example, the stabilities of AB2 alloys cannot be compared with those of AB5 alloys [43]. 

A strongly bonded polyhedron chain which occurs on a crystal face contains ligands which bond either to cations of the chain or both to cations of the chain and to species in the adjacent aqueous solution. Any anion on such a chain and the cations to which it is bonded form a termination. Polyhedron chains are generally linear and have a small number of cation-cp (anion) terminations per unit length. In general, it is the incident bond-valence at the anion of the (bare) termination that controls the reactivity of that termination. If the incident bond-valence at the anion already satisfies the valence-sum rule, pZa = 0 in Eq. (1) and there is no driving force for that anion to react with any component of the adjacent aqueous solution. Conversely, if the incident bond-valence at the anion is less than that required by the valence-sum rule, the anion will react with some component of the adjacent aqueous solution to accord with the valence-sum rule. 

A natural question to ask is whether the basic model can be modified in some way that would enable it to correctly learn the XOR function or, more generally, any other non-linearly-separable problem. The answer is a qualified yes in principle, all that needs to be done is to add more layers between what we have called the A-units and R-units. Doing so effectively generates more separation lines, which when combined can successfully separate out the desired regions of the plane. However, while Rosenblatt himself considered such variants, at the time of his original analysis (and for quite a few years after that see below) no appropriate learning rule was known. 

So far, Eq. (103) is general and free from any approximation. To determine B (r), it is necessary to integrate Eq. (103) over the charging parameter X and to make some assumptions with respect to the X dependence of the correlation functions. The usual rules are the linear dependence of the correlation functions on X and the unique functionality of the bridge function already mentioned. But, in that case, there is some arbitrariness on it and, as pointed out by Lee [72], a quadratic dependence could equally well be assumed. In a rigorous way, a X dependent correlation function, say T(r, X), has to express as P X)T(r), with the conditions P(X) > P(X = 0) = 0 and P(X) < P(X = 1) = 1. Unfortunately, P(X) remains unknown whatever the correlation function under consideration. So, the way a test particle does couple with the rest of the fluid is an open question. The author has assumed that P(X) = Xn, namely h(r, X) = Xnh(r) and c(r, X) = Xnc r), which corresponds to an extention of the Kjellander-Sarman... 

Alkynes are hydrocarbons with carbon-carbon triple bonds as their functional group. Alkyne names generally have the -yne suffix, although some of their common names (acetylene, for example) do not conform to this rule. The triple bond is linear, so there is no possibility of geometric (cis-trans) isomerism in alkynes. 

There is no general rule relating the nucleophilic reactivity of cyclic monomer and linear polymer repeating unit, it depends on the nature of heteroatom and the size of the ring which affects the electronic structure of heteroatom. It is a common practice to estimate the order of nucleophilicities on the basis of basicities. Although it is only partly justified, this procedure enables semiquantitative comparisons of known pKa values whereas no universal scale of nucleophilicity exists. Some typical values of pKa for cyclic compounds and their linear analogs are given in Table 8 [99,100],... 

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