Sequential solution method

Within the framework of commercial CFD codes where sequential solution methods are standard, as they need to solve a number of user-specified transport equations, the two potential equations must then be solved through innovative source term linearization. ... 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

As is true in the design of many separation techniques, the choice of specified design variables controls the choice of the design method. For the flash chamber, we can use either a sequential solution method or a simultaneous solution method. In the sequential procedure, we solve the mass balances and equilibrium relationships first and then solve the energy balances and enthalpy equations. In the simultaneous solution method, all equations must be solved at the same time. In both cases, we solve for flow rates, compositions, and temperatures before we size the flash drum. 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

Fakhr Eldin et al. [22] described a simple sequential spectrophotometric method for the assay of penicillamine. The method is based on the complex formed when the drug is reacted with Fe(III) solution in hydrochloric acid media. The deep blue colored drug Fe(III) complex is monitored at a maximum wavelength of 600 nm. 

TABLE 8.4. Films Grown by ILGAR, ECALE, and Other Sequential Solution-Phase Deposition Methods... 

Cereal proteins when classified by the Osborne sequential extraction method yield four different classes albumins, which are water soluble, globulins, which are soluble in salt solutions, prolamins, which are soluble in alcohol-water mixtures, and glutelins, which are soluble in dilute acid or alkali. Chen and Bushuk added a fifth fraction by dividing the glutelin into two fractions, one soluble in dilute (0.05 m) acetic acid and the other insoluble in this reagent.5... 

To determine the various species and their concentrations in soil, many selective, semiselective, and sequential extraction methods have been developed. Species associated with various components in soil can be extracted with varying effectiveness (see Chapters 11 and 12). Thus, metal cations that are in solution, exchangeable, weakly held, or associated with carbonate and with... 

The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

The older modular simulation mode, on the other hand, is more common in commerical applications. Here process equations are organized within their particular unit operation. Solution methods that apply to a particular unit operation solve the unit model and pass the resulting stream information to the next unit. Thus, the unit operation represents a procedure or module in the overall flowsheet calculation. These calculations continue from unit to unit, with recycle streams in the process updated and converged with new unit information. Consequently, the flow of information in the simulation systems is often analogous to the flow of material in the actual process. Unlike equation-oriented simulators, modular simulators solve smaller sets of equations, and the solution procedure can be tailored for the particular unit operation. However, because the equations are embedded within procedures, it becomes difficult to provide problem specifications where the information flow does not parallel that of the flowsheet. The earliest modular simulators (the sequential modular type) accommodated these specifications, as well as complex recycle loops, through inefficient iterative procedures. The more recent simultaneous modular simulators now have efficient convergence capabilities for handling multiple recycles and nonconventional problem specifications in a coordinated manner. 

Fig. 3 UV-vis spectra of a 2.0 xM G6 - OH[Pd55](Aun) solutions (n = 0-255) prepared by the sequential-loading method. The DENs were prepared using ascorbic acid as the reducing agent, b 0.8 iM solutions of G6 - OH[Pd55](Au255), prepared by the sequential-loading method, and G6-OH(Au3i0) and G6 - OHfPdssA ss) prepared by the co-complexation method. Reprinted with permission from J Am Chem Soc, 2004, 126, 15583-15591. Copyright 2004 American Chemical Society...

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

The obvious alternative to the sequential optimization methods is the use of an interpretive optimization method. In such a method a limited number of experiments is performed and the results are used to estimate (predict) the retention behaviour of all individual solutes as a function of the parameters considered during the optimization (retention surfaces). Knowledge of the retention surfaces is then used to calculate the response surface, which in turn is searched for the global optimum (see the description of interpretive methods in section 5.5). For programmed temperature GC the framework of such an interpretive method has been described by Grant and Hollis [614] and by Bartu [615]. 

The Newton/sparse matrix methods now used by electrical engineers have become the solution method of choice. Hutchison and his students at Cambridge were among the first chemical engineers to publish this approach, in the early 1970s. They used a quasi-linear model rather than a Newton one, but the ideas were really very similar. (It appears that the COPE flowsheeting system of Exxon was Newton based it existed in the mid-1960s but slowly evolved into a sequential modular system. One must assume the Newton method failed to compete.)... 

Now that the problem is formulated we turn our attention to solving the equations. One solution method that we could use is the sequential modular method. For this method, select one of the process units as the starting point for the calculation. Then, assume values for some of variables to reduce the degrees of freedom to zero for that rniit. Next, precede unit-by-unit through the flow sheet... 

To obtain a solution for the material and energy balances in a flowsheet by the sequential modular method, you must partition the flowsheet, select tear streams, nest the computations, and thus determine the computation sequence. 

The conditions and procedures of precipitation play a significant role in catalyst morphology, texture, pore structure, physical strength, and consequently the performance (activity, selectivity, and stability) of the catalyst. By merely changing the sequence of solution addition, catalyst components can be precipitated simultaneously or sequentially. The methods of precipitation most often used are constant pH coprecipitation, sequential precipitation, acid-to-base precipitation, and base-to-acid precipitation. 

For steady-state simulations, several solution methods have been used. The one most frequently used is the sequential approach. In this method, numerical modules are used that calculate the output stream of a process unit from the input streams coupled with any additional information that is required to uniquely define the performance of the process unit. The simulation treats one unit model at a time. For a simulation flow sheet consisting of interconnected units, the order... 

Solution methods for optimization problems that involve only continuous variables can be divided into two broad classes derivative-free methods (e.g., pattern search and stochastic search methods) and derivative-based methods (e.g., barrier function techniques and sequential quadratic programming). Because the optimization problems of concern in RTO are typically of reasonably large scale, must be solved on-line in relatively small amounts of time and derivative-free methods, and generally have much higher computational requirements than derivative-based methods, the solvers contained in most RTO systems use derivative-based techniques. Note that in these solvers the first derivatives are evaluated analytically and the second derivatives are approximated by various updating techniques (e.g., BFGS update). 

EXAMPLE WITH RECYCLE COMPARISON OF SEQUENTIAL AND SIMULTANEOUS SOLUTION METHODS... 

In addition, the overall content of TEs in extractable (mobile) forms can be estimated prior to SEP by a single-extraction procedure using electrolyte solutions, such as 1.0 mol HNO3 or 1.0 mol L HCl (Ladonin, 2002). If the contents of TEs obtained by the screening protocol are rather low, it might then be unreasonable to use a sequential extraction method. 

In another method, designated the sequential precipitation method, an aqueous solution of ruthenium trichloride is contacted with the ammoniacal hydrazine solution (3). The resulting precipitate is filtered out of solution and reslurried in water, after which a solution of copper nitrate is added to the slurry. On subsequent addition of ammoniacal hydrazine solution, further precipitation occurs in the presence of the original precipitate. The total precipitate is then dried and reduced in the same manner as the coprecipitated preparations. 

Several methods have been reported for the preparation of magnetic-nanoparticle-reinforced conductive-polymer nanocomposites, including the oxidation in solution method, the electrochemical method, sequential deposition approaches, the irradiation method, etc. 

Equation (7.18) represents a set of N nonlinear algebraic equations, which must be solved by a trial and error method such as Newton-Raphson or successive substitution for y . j (Appendix A). This is the characteristic difference between the implicit and explicit types of solution the easier explicit method allows sequential solution one at a time, while the implicit method requires simultaneous solutions of sets of equations hence, an iterative solution at a given time... 

Many researchers have suggested that ultrasonication frequencies ranging from 20 to 1000 kHz can help in the preparation of metal NPs from aqueous solution of metal salts under reducing condition [44-46], 200 kHz is the optimum ultrasound frequency for reduction of metal salt as suggested by Ashok kumar et al. [47]. Ultrasonication method is most widely used green method for the production of Au NPs-rGO nanocomposites. For example, Vinodgopal et al. synthesized Au NPs-rGO composites by simultaneous and sequential reduction of GO and Au (III) with an ultrasonic frequency of 200 kHz [23], In the simultaneous reduction process, the mixtures of aqueous suspension of GO and HAuCl were ultrasoni-cated for 5 h to produce Au NPs-rGO composites. In the sequential reduction method, first GO is reduced to rGO and then aqueous solution of HAuCL was added and ultrasonicated for 2 h to form Au NPs-rGO com-... 

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