Outer iteration

Outer iteration Design points x xl03, mol/1 QX108... 

After two outer iterations the following results are obtained. 

Modification of Describing Equations. In the simple K-value and enthalpy models just described, there are Ng+6 parameters, a, A, B, C, D, E and F, which have characteristics that make them excellent choices for the iteration variables of an outer iteration loop. This is in fact the essence of the inside-out concept. Its success, however, rests on the ability to transform and rearrange the describing equations to properly accommodate these variables. The result should be an efficient and well-behaved inner iteration loop in which values of the primitive variables - now regarded as dependent variables - are calculated. When the inside loop is converged, the actual K-value and enthalpy models can be employed to calculate new values of the simple model parameters. 

DEW P. The calculational scheme here is shown in Fig. 12.13. We read and store T and yk, along with appropriate constants. Since we can calculate neither the d> nor the yk, all values of each are set equal to unity. Values of P are found from the Antoine equation, and Eqs. (12.29) and (12.27) are then solved for P and xk. Evaluation of y now allows recalculation of P by Eq. (12.29). With this rather good estimate of P, we evaluate and enter an inner iteration loop that converges on values for xk and (yj. Subsequent recalculation of P by Eq. (12.29) leads to the outer iteration loop that establishes the final value of P. Since the xk calculated within the inner loop are not constrained to sum to... 

Witli tliese initial values of tlie y, and initial values of tlie Ki can be calculated by Eq. (14.22). The P and 4>r values are already available from tire preliminary DEW P and BUBL P calculations. Equations (14.19) and (14.20) now provide starting values of F and dF/dV for Newton s method as represented by Eq. (14.21). Repeated application of tlris equation leads to the value of V for which Eq. (14.19) is satisfied for the present estimates of A",. The remaining calculations serve to provide new estimates of y and 4> from which to reevaluate AT,. Tlris sequence of steps (an outer iteration) is repeated until there is no sigirificant change in results from one iteration to the next. After the first outer iteration, the values of V and (dF/dV) used to start Newton s method (an imrer iteration) are simply the... 

For unsteady flows the system of non-linear equations are linearized in the iteration process within each time step, since all the solvers are limited to linear systems. The iterative process is thus performed on two different levels. The solver iterations are performed on provisional linear systems with fixed coefficients and source terms until convergence. Then, the system coefficients and sources are updated based on the last provisional solution and a new linearized system is solved. This process is continued until the non-linear system is converged, meaning that two subsequent linear systems give the same solution within the accuracy of a prescribed criterion. A standard notation used for the different iterations within one time step is that the coefficient and source matrices are updated in the outer iterations, whereas the inner iterations are performed on provisionally linear systems with fixed coefficients. On each outer iteration, the equations solved are on the form ... 

We normally drop the time step index n - - 1 and introduced an outer iteration counter n. thus represents the current estimate of the solution At the beginning of each outer iteration, the two last terms on the right hand side of (12.159) are evaluated using the variables at the preceding outer iteration. 

The components of the momentum equation are usually solved sequentially, meaning that the components of the momentum equation are solved one by one. Since the pressure used in these iterations has been obtained from the previous outer iteration or time step, the velocities computed from (12.159) do not generally satisfy the discretized continuity equation. The predicted velocities do not satisfy the continuity equation, so the uf at iteration 1/ are not the final values of the velocity components. To enforce the continuity equation, the velocities need to be corrected. This is achieved by modifying the pressure field. 

After solving the Poisson equation for the pressure, by use of (12.164), the final velocity field at the new iteration, v t, is calculated from relations on the form (12.162). At this point, we have a velocity held which satisfies the continuity condition, but the velocity and pressure fields do not necessarily satisfy the momentum equations on the form (12.160). We begin another outer iteration and the process is continued until the velocity held which satisfies both the momentum and continuity equations is obtained. 

To obtain the solution at the new time step in the implicit method, several outer iterations are performed. If the time step is small, only a few outer iterations per time step are necessary. For steady problems, the time step may be infinite and the under-relaxation parameter acts like a pseudo-time step. 

The velocities obtained by solving the linearized momentum equations on the form (12.160), by use of the previous outer iteration values for the pressure and the density, do not satisfy the mass conservation equations (12.161). When the mass fluxes computed from these velocities and the previous outer iteration density (denoted by F ) are inserted into the discretized continuity equation, we obtain ... 

For one outer iteration, the temperature may be regarded as provisionally fixed, hence by use of the ideal gas law we may write ... 

Bove [16] proposed a different approach to solve the multi-fluid model equations in the in-house code FLOTRACS. To solve the unsteady multifluid model together with a population balance equation for the dispersed phases size distribution, a time splitting strategy was adopted for the population balance equation. The transport operator (convection) of the equation was solved separately from the source terms in the inner iteration loop. In this way the convection operator which coincides with the continuity equation can be employed constructing the pressure-correction equation. The population balance source terms were solved In a separate step as part of the outer iteration loop. The complete population balance equation solution provides the... 

This method starts off by fixing the temperature and pressure and iterating around the vapor fraction to calculate the equilibrium phase separation and compositions. The first step is an isothermal Hash calculation. If T and P are in fact the independent variables, the solution obtained in the first step is the desired solution. If either Tori and one more variable are specified, then another, outer iterative loop is required. The outer loop iterates around P or T (whichever is not fixed) until the other specified variable is satisfied. 

Figure 7.7 shows the convergence of the penalty function method. At each outer iteration r, the plotted objective functional value is the optimal with corresponding constraint violation quantified as q. It was 17.2 at the very beginning (corresponding to the initial controls) and dropped finally to 5.9 x 10 at the convergence, which was attained in 11 outer iterations. 

Figure 7.7 The intermediate, optimal objective functional I, penalty factor a, and the constraint norm q versus outer iterations...

The constraint violation in terms of q was 1182 at the very beginning with initial controls and converged finally to 1.1 x 10 in five outer iterations. Upon convergence, the final values of yi, y, and 2/3 were... 

With initial controls, the constraint violation in terms of q was 55.7, which reduced and converged to 3.3x10 in 10 outer iterations. At convergence, the optimal objective functional was —4.45, which corresponds to the final product concentration of 4.45 g/cm. The optimal final time reduced from 60 to 33.3 min. 

As foreshadowed in Section 9.7.1, the iterative strategy is based on an outer iteration on discrete local current densities. For given current densities, the coupled initial value problem for the channel fluxes and temperatures are solved by marching, with some terms handled implicitly. Details of the iterative strategy are given below. 

Interesting mathematical problems arise in the convergence of the outer iteration where it is desirable to use some procedure to accelerate the convergence. We can write the pth iterative solution of the multigroup equation in the form... 

As in the case of successive overrelaxation, the efficiency of the application of Chebyshev polynomials in accelerating the outer iterations depends upon the accurate estimation of the particular constant, 5, the dominance ratio for the matrix T. A practical numerical method for estimating <7 is given in [45]. 

Other methods for accelerating the convergence of the outer iterations exist, and are interesting in their own right. With the definition of the matrices in 3, we now write our discrete time independent eigenvalue problem in the matrix form... 

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