Objective function stoichiometric

Note in Table 5.10 that many of the integrals are common to different kinetic models. This is specific to this reaction where all the stoichiometric coefficients are unity and the initial reaction mixture was equimolar. In other words, the change in the number of moles is the same for all components. Rather than determine the integrals analytically, they could have been determined numerically. Analytical integrals are simply more convenient if they can be obtained, especially if the model is to be fitted in a spreadsheet, rather than purpose-written software. The least squares fit varies the reaction rate constants to minimize the objective function ... 

The stoichiometric data is included in order to perform material balances in each unit operation. The second column of the stoichiometric data shows the amount of raw material required (tons) per unit mass (tons) of the overall output, i.e. s6 + si + s8. The third column shows the ratio of each byproduct (si and. S 8 ) to product (s6) in ton/ton product. The objective function is the maximisation of product (56) output. A 20% variation in processing times was assumed. 

By a stoichiometric objective function we mean one which is concerned only with the changes in concentration of the species. Thus, if for the moment we define cproduct streams, such an objective function is of the form... 

The objective function to be considered is of one of the simpler types, either stoichiometric or material. In the first case it is required to maximize the final extent of reaction, Ci in the second case this will also be required if the objective function is monotonically increasing with Cl. Here the interesting problem is to find the optimal policy subject to some restriction on the total holding time, say... 

We will continue the study of optimal problems with a stoichiometric objective function, for some of the tricks of the trade can be 78... 

Such a construction as this is only useful in the case of stoichiometric objective functions, but it does give a simple presentation of the complete optimal policy. We shall find this presentation useful in the next chapter. 

One important issue that still needs attention is the objective function. It is intuitively obvious that if a separation cost is not associated with it, we will usually end up getting near-complete separations of products, and hence complete conversions to the extent possible within stoichiometric constraints. Thus the AR in concentration space can easily be the entire stoichiometric space. Unfortunately, it is difficult to get an accurate representation for the separation cost, e.specially when sharp splits are not enforced. Here, we present a simple cost model by assuming that the variable cost of separation is determined by two factors, namely, the difficulty of separation and the mass flow rate through the separator. 

S Selectivity, stoichiometric matrix, objective function for parameter estimation ... 

Enforcing stoichiometric, capacity, and thermodynamic constraints simultaneously leads to the definition of a solution space that contains all feasible steady-state flux vectors. Within this set, one can find a particular steady-state metabolic flux vector that optimizes the network behavior toward achieving one or more goals (e g., maximize or minimize the production of certain metabolites). Mathematically speaking, an objective function has to be defined that needs to be minimized or maximized subject to the imposed constraints. Such optimization problems are typically solved via linear programming techniques. 

Specifically, SKM seeks to overcome several known deficiencies of stoichiometric analysis While stoichiometric analysis has proven immensely effective to address the functional capabilities of large metabolic networks, it fails for the most part to incorporate dynamic aspects into the description of the system. As one of its most profound shortcomings, the steady-state balance equation allows no conclusions about the stability or possible instability of a metabolic state, see also the brief discussion in Section V.C. The objectives and main requirements in devising an intermediate approach to metabolic modeling are as follows, a schematic summary is depicted in Fig. 25 ... 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

Another application of the analysis of the stoichiometric matrix is flux balance analysis (Edwards et al. 2002). Often the number of fluxes in the system exceeds the number variable metabolites making equation (3) an underdetermined set of linear equations, that is, many different combinations of fluxes are consistent with system steady state. One approach is to measure the fluxes that enter and exit the cell. Because intracellularly there are many redundant pathways, this does not enable one to determine all fluxes. Isotope labelling may help then (Wiechert 2002). Another approach to then find a smaller number of solutions is to postulate that the solution should satisfy an additional objective. This objective is taken to be associated with optimal functioning of the network, for instance maximization of some flux or combination... 

The objectives in this step are to assess thermodynamic feasibility for each of the stoichiometric equations and to list them in groups with the same chemical functions. For clarity many intermediate reactions have been omitted from the rearrangements below. Reactions with values greater than lOkcal moIe arc not included at this point. 

Therefore, BA can be seen as a productivity function that should be maximised by design. For example, the excess of reactant can give higher stoichiometric yield RY, but lower real balance yield BA. Increasing EAp to the theoretical limit of one is an objective of process design. This can be achieved by replacing steps involving unrecoverable auxiliary chemicals with operations where the recycle of materials is possible. 

If the selectivity of the MIP catalyst is the main objective, the partial poisoning of active centers might be a way to improve the performance of the system. The imprinting procedure generates a statistical distribution of selective and less selective reactions centers. Studies indicate that the least selective sites are the most reactive [27]. The reaction of an MIP catalyst with sub-stoichiometric amounts of a catalyst poison under kinetic control should, therefore, result in a less active but more selective MIP catalyst. As a poisoning reaction, the covalent modification of functional groups or the irreversible complexation of a metal center could be employed (Fig. 20). 

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