Simultaneous solution method

The simultaneous solution method solved this example in four iterations. The Peng-Robinson equation of state was used to estimate K values and enthalpy... 

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

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

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. 

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

Since a stable steady state is sought, the method of false transients could be used for the simultaneous solution of Equations (5.29) and (5.31). However, the ease of solving Equation (5.29) for makes the algebraic approach simpler. Whichever method is used, a value for UAext pQCp is assumed and then a value for Text is found that gives 413 K as the single steady state. Some results are... 

The numerical techniques of Chapter 8 can be used for the simultaneous solution of Equation (9.3) and as many versions of Equation (9.1) as are necessary. The methods are unchanged except for the discretization stability criterion and the wall boundary condition. When the velocity profile is flat, the stability criterion is most demanding when at the centerline ... 

A generalized partial differential equation solver which handles simultaneous parabolic, one dimensional elliptic, ordinary and integral equations and uses B-splines with an adaptive grid was written to solve the model. Further details on the model and solution method can be found in Reference 14. 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

Modem tables of thermod5mamic data are made self-consistent either by methods of iteration or by computer-assisted simultaneous solutions [1]. The value of AH for any chemical reaction that can be obtained from two or more thermodynamic cycles should have the same value from each of those cycles if the data are obtained from the same tables. Any thermodynamic calculation, therefore, should be carried out using data from a single database whenever possible. Thus, in listing some values of enthalpies of formation below, we provide values from different databases in separate tables. 

The simultaneous solution of the equations for ai, 02, and K will yield an a versus X curve if all the underlying parameters were known. To this end, Futerko and Hsing fitted the numerical solutions of these simultaneous equations to the experimental points on the above-discussed water vapor uptake isotherms of Hinatsu et al. This determined the best fit values of x and X was first assumed to be constant, and in improved calculations, y was assumed to have a linear dependence on 02, which slightly improved the results in terms of estimated data fitting errors. The authors also describe methods for deriving the temperature dependences of x and K using the experimental data of other workers. 

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. 

Despite its success, the embedded model approach still requires repeated solution of the process model (and sensitivities). For large processes or for processes that require the solution of rigorous underlying procedures, this approach can become expensive. Moreover, for stiff or otherwise difficult systems, this approach is only as reliable as the ODE solver. The embedded model approach also offers only indirect ways of handling time-dependent constraints. Finally, the optimal solution of this approach is only as good as its control variable parameterization, which often can only be improved by a priori information about the specific problem. Consequently, we now consider the simultaneous approach to (16) as an alternative to solution methods for (17). 

Instead, the simultaneous method can be extended to select adaptively a sufficient number of finite elements. Here, we note that even if we set any element length to zero, the collocation equations and the continuity equations are still satisfied. Thus, any number of zero length (or dummy) elements can be added without changing the control or state profiles, or the solution to the NLP. Vasantharajan and Biegler (1990) take advantage of this important property and propose an adaptive element addition approach embedded within the simultaneous solution strategy. 

For simultaneous solution of (16), however, the equivalent set of DAEs (and the problem index) changes over the time domain as different constraints are active. Therefore, reformulation strategies cannot be applied since the active sets are unknown a priori. Instead, we need to determine a maximum index for (16) and apply a suitable discretization, if it exists. Moreover, BDF and other linear multistep methods are also not appropriate for (16), since they are not self-starting. Therefore, implicit Runge-Kutta (IRK) methods, including orthogonal collocation, need to be considered. 

Cuthrell, J. E., and Biegler, L. T., Simultaneous optimization and solution methods for batch reactor control profiles, Comp, and Chem. Eng. 13(1/2), 49-62 (1989). 

The simultaneous solution of eqns. (72) and (79) when h is not zero is generally achieved by a numerical method which considers small increments in reactor volume and then iterates the calculation of the resulting temperature and fractional conversion in a manner similar to that described for Sect. 2.5.3 for a batch reactor. Cooper and Jeffreys [3] give an illustrative example, together with a computer flow diagram, for calculating the reactor volume. 

Das and Haider described a simultaneous spectrophotometrie method for the analysis of binary dosage form mixtures of diloxanide furoate with metronidazole or with tinidazole [19], Powdered tablets or suspension, equivalent to 50 mg of the drug substances, were dissolved in 50 mL of dimethylformamide with shaking. After 15 minutes, the solution was diluted to 100 mL with water and filtered. A 1 mL portion of the filtrate was diluted to 50 mL with water, and the absorbance of the resulting solution measured at 320 and 262 nm for metronidazole and diloxanide furoate simultaneously. Alternatively, readings were taken at 318 and 262 nm for the simultaneous determination of tinidazole and diloxanide furoate. Recoveries were reported to be quantitative. 

So far this approach is analogous to most of the simultaneous optimisation methods. However, the optimisation is not continued by preselecting desired values for any criterion to construct contour plots (Figure 4.13 and 4.14), or to search for acceptable solutions [29]. 

While the number of independent variables is arbitrary in our definitions, it makes a tremendous difference in computations. Simultaneous solution of n equations and minimization in n dimensions are much more difficult than in one dimension. The main difference between one and several dimensions is that in one dimension it is possible to "bracket" a root or a local minimum point between some bracketing values, and then to tighten the interval of uncertainty. This gives rise to special algorithms, and hence the solution of a single equation and minimization in one variable will be discussed separately from the multidimensional methods. 

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. 

Requirements to use embedded solution techniques include giving accurate function and first derivative evaluations for Newton-based methods. Embedded convergence needs to be tight to ensure the results are accurate and precise. However, this can yield longer solution times and make a simultaneous solution approach preferred. Embedded solutions provide a way of reducing the model variables exposed to the overall solution. The modeler should ensure the eliminated... 

On the basis of the above mentioned workup procedure a new simultaneous resolution method was developed using two moles of racemic tetrahydrofuroic acid (3b) instead of 12 for decomposition of complex 11 (Scheme 6, L= 8). In a methyl alcohol/water system the (S)-3b isomer built up a new crystalline coordination complex (13) while two moles of (R)-8 liberated and one mole of (R)-3b remained in the solution. [25]... 

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