Optimization using numerical solutions

Since the recovery yield achieved rmder the experimental conditions giving the maximum production rate is of the order of 60%, a first-order approximation of the loading factor giving a recovery yield Y is Lf = 0.6L /Y. Then the column efficiency is adjusted to achieve the required recovery yield. This method has been adopted because the results of numerical optimization suggest that there is a quasi-linear relationship between the recovery 5deld and both the mobile phase velocity and the loading factor [7]. 

As we have seen, there are two simple objective fimctions for a numerical optimization the production rate and the specific production. More complex schemes 

The maximization of the production rate by numerical methods was first studied by Ghodbane and Gulochon [7], who used the equilibrimn-dispersive model to calculate the band profiles under a specific set of experimental conditions, the Knox [25] equation to relate the mobile phase flow velocity and the column efficiency, and a Simplex algorithm for the optimization. They discussed only the simultaneous optimization of two parameters, the mobile phase velocity and the sample size, or the particle size and the column length. Their main conclusions were that 

The optimum experimental conditions for maximum production rate of the first and the second component are quite different. 

When the production rate obtained with a given coliunn is optimized, the max-immn production rate is achieved at high mobile phase velocities. The production rate depends heavily on the column efficiency xmder analytical conditions most of this efficiency is traded off for a short cycle time. 

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Another possible way to improve the separation performance of SMB units is to change the feed concentration during each successive period. This operation mode, called the Modicon process, was suggested by Schramm et al. [72]. These authors optimized the process using numerical solutions of the equilibrium-dispersive model and compared the performance of the Modicon process with that of conventional SMB. They concluded that a cyclic modulation of the feed concentration allows a significant improvement of the separation performance The productivity and the product concentration can be increased and the specific solvent consumption decreased compared to those achieved with conventional SMB. 

Natarajan et al. [56] discussed the optimization of ion-exchange displacement separations of proteins by using numerical solutions of the soHd-fUm linear driving force model. The equilibrium isotherms of the protein-salt multicomponent... 

To exemplify both aspects of the formalism and for illustration purposes, we divide the present manuscript into two major parts. We start with calculations of trajectories using approximate solution of atomically detailed equations (approach B). We then proceed to derive the equations for the conditional probability from which a rate constant can be extracted. We end with a simple numerical example of trajectory optimization. More complex problems are (and will be) discussed elsewhere [7]. 

Computerized optimization using the three-parameter description of solvent interaction can facihtate the solvent blend formulation process because numerous possibihties can be examined quickly and easily and other properties can also be considered. This approach is based on the premise that solvent blends with the same solvency and other properties have the same performance characteristics. Eor many solutes, the lowest cost-effective solvent blends have solvency that is at the border between adequate and inadequate solvency. In practice, this usually means that a solvent blend should contain the maximum amount of hydrocarbon the solute can tolerate while still remaining soluble. 

Some problems in functional optimization can be solved analytically. A topic known as the calculus of variations is included in most courses in advanced calculus. It provides ground rules for optimizing integral functionals. The ground rules are necessary conditions analogous to the derivative conditions (i.e., df jdx = 0) used in the optimization of ordinary functions. In principle, they allow an exact solution but the solution may only be implicit or not in a useful form. For problems involving Arrhenius temperature dependence, a numerical solution will be needed sooner or later. 

On computational stability of iterative methods. Until recent years the iterative method with optimal set of Chebyshev s parameters was of little use in numerical solution of grid equations. This can be explained by real facts that various sequences turn out to be nonequivalent in computational procedures. 

This representation permits analytic calculations, as opposed to fiiUy numerical solutions [47,48] of the Hartree-Fock equation. Variational SCF methods using finite expansions [Eq. (2.14)] yield optimal analytic Hartree-Fock-Roothaan orbitals, and their corresponding eigenvalues, within the subspace spanned by the finite set of basis functions. 

The data shown in this section demonstrate that the simultaneous optimization of the solute geometry and the solvent polarization is possible and it provides the same results as the normal approach. In the case of CPCM it already performs better than the normal scheme, even with a simple optimization algorithm, and it will probably be the best choice when large molecules are studied (when the PCM matrices cannot be kept in memory). This functional can thus be directly used to perform MD simulations in solution without considering explicit solvent molecules but still taking into account the dynamics of the solvent. On the other hand, the DPCM functional presents numerical difficulties that must be studied and overcome in order to allow its use for dynamic simulations in solution. 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

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