Sequential solutions

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.

Successive Ionic Layer Adsorption and Reaction (SILAR) and Related Sequential Solution-Phase Deposition Techniques... 

Several review articles have been published about SILAR-grown films.4-7 The SILAR technique, including its advantages and disanvantages and the equipment employed, is presented in Section 8.2. Materials that have been prepared by SILAR are reviewed in Section 8.3. Short descriptions of the related ILGAR, ECALE, and other sequential solution-phase techniques follow in Sections 8.4- 8.6. 

SILAR AND RELATED SEQUENTIAL SOLUTION-PHASE DEPOSITION TECHNIQUES... 

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

Other Sequential Solution-Phase Deposition Techniques 270... 

Equality constraints. The Eduljee correlation involves two parameters Rm, the minimum reflux ratio, and Nm, the equivalent number of stages to accomplish the separation at total reflux. His operating equations relate N, a, XF, XD, and XB (see Table El2.4A for notation) all of which have known values except XB as listed in Table E12.4A. Once R is specified, you can find XB by sequential solution of the three following equations. 

From a computational viewpoint, the presence of recycle streams is one of the impediments in the sequential solution of a flowsheeting problem. Without recycle streams, the flow of information would proceed in a forward direction, and the cal-culational sequence for the modules could easily be determined from the precedence order analysis outlined earlier. With recycle streams present, large groups of modules have to be solved simultaneously, defeating the concept of a sequential solution module by module. For example, in Figure 15.8, you cannot make a material balance on the reactor without knowing the information in stream S6, but you have to carry out the computations for the cooler module first to evaluate S6, which in turn depends on the separator module, which in turn depends on the reactor module. Partitioning identifies those collections of modules that have to be solved simultaneously (termed maximal cyclical subsystems, loops, or irreducible nets). 

To execute a sequential solution for a set of modules, you have to tear certain streams. Tearing in connection with modular flowsheeting involves decoupling the interconnections between the modules so that sequential information flow can take place. Tearing is required because of the loops of information created by recycle... 

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. 

Besides the electrochemical flow cells just mentioned, other hardware for sequential solution-deposition scenarios has been developed to form... 

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. 

One feasible network would correspond to the cold streams Cl, C8, and C9 diverted to suitable jacketed reactor compartments, as the simple network in Fig. 14 shows. The hot streams not shown in this network are matched directly with cooling water (CW), and the amount of steam used here is very small. Note that this network would require the same minimum utility consumption predicted by the solution of (PIO). It can be inferred that the network in Fig. 14 is equally suitable for both the simultaneous and sequential solutions. In fact, Balakrishna and Biegler (1993) showed that, for exothermic systems in which the reactor temperature is the highest process temperature, the pinch point is known a priori as the highest reactor temperature (in this case, the feed temperature) and the inequality constraints in (PIO), Qh 2h () ). F G P. can be replaced by a simple energy balance constraint. This greatly reduces the computational effort to solve (PIO). 

Suppose that instead of a simultaneous solution of the equations, a sequential solution is wanted. In what order should the equations be solved Partition the equations so that a sequential solution can be executed. Lump together blocks of equations that still have to be solved simultaneously. 

Step 1 One way to solve this problem is to do a sequential solution. You start with the feed, and solve the mass balance for the mixer, labeled MIXR. Since you do not yet know the amount of A in stream S5, assume it is zero and go on. Then there is one mole of A fed to the reactor, 40 percent of it reacts, and the unreacted part is recycled into stream S5. 

Sequential solution of flow, then concentration for efficiency Restart from simpler model to improve convergence and efficiency Plotting in 3D geometries ... 

The block of inter-linked columns offers robust simulation of a combination of complex distillation columns, as heat-integrated columns, air separation system, absorber/stripper devices, extractive distillation with solvent recycle, fractionator/quench tower, etc. Because sequential solution of inter-linked columns could arise convergence problems, a more robust solution is obtained by the simultaneous solution of the assembly of modelling equations of different columns. 

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