Iterative loops

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

Each iteration requires only one call of the thermodynamic liquid-liquid subroutine LILIK. The inner iteration loop requires no thermodynamic subroutine calls thus is uses extremely little computation effort. 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

Figure 2 Outline of the steps involved in the preparation of a force field for the inclusion of new molecules and optimization of the associated parameters. Iterative loops (1) over individual external terms, (11) over individual internal terms, (111) over the external and internal terms. In loop (IV) over the condensed phase simulations, both external terms and internal terms are included.

Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

Executable statements affecting the order in which the program instructions are executed include conditional (branching) statements, iterative (looping) statements, and statements which call subprogram units. 

A common feature in the models reviewed above was to calculate pressure and temperature distributions in a sequential procedure so that the interactions between temperature and other variables were ignored. It is therefore desirable to develop a numerical model that couples the solutions of pressure and temperature. The absence of such a model is mainly due to the excessive work required by the coupling computations and the difficulties in handling the numerical convergence problem. Wang et al. [27] combined the isothermal model proposed by Hu and Zhu [16,17] with the method proposed by Lai et al. for thermal analysis and presented a transient thermal mixed lubrication model. Pressure and temperature distributions are solved iteratively in a iterative loop so that the interactions between pressure and temperature can be examined. 

All of the methods for designing laser pulses to achieve a desired control of a molecular dynamical process require the solution of the time-dependent Schrodinger equation for the system interacting with the radiation field. Normally, this equation must be solved many times within an iterative loop. Different possible approaches to the solution of these equations are discussed in Section V. 

A slight alternative to the above NIPALS algorithm is replacing the iteration loop (line 8-14) by ... 

The highly interactive nature of the balance and equilibria equations for the distillation period are depicted in Fig. 3.66. An implicit, iterative algebraic loop is involved in the calculation of the boiling point temperature at each time interval. This involves guessing the temperature and calculating the sum of the partial pressures, or mole fractions. The condition required is that Zyi + yw = 1. The iterative loop for the bubble point calculation is represented by the five interconnected blocks in the lower right hand corner of Fig. 3.66. The model of Prenosil (1976) also included an efficiency term E for the steam heating, dependent on liquid depth L and bubble diameter D. 

In this equation, the superscript ( ) indicates that a term is computed based upon the most recent information, which complies with the ( + l)th time level when all iterative loops have converged. Further, the convective transport and viscous generation of fluctuating kinetic energy have been collected in the explicit term D. The iterative solution procedure for the granular energy equations continues until the convergence criteria... 

The function PLS ca librat ion, m is the PLS equivalent to PCR calibration, m. The iterative loop implements equations (5.73)-(5.77). The prognostic vector vprog is introduced in the next section. 

C TEMPERATURE ITERATION LOOP 100 LOOP=LOOP+l IF(LOOP.GT.100)THEN WRITE(6,1)... 

Once the logic tree structure appears to be consistent, the first of three quality assurance tests is applied by examining the overall logic tree structure for completeness. The logic in each branch of the tree should be tested to determine if it is necessary and sufficient. (Details and tips for testing the logic are discussed in Section 9.6.2.) If the tree appears to be complete, the next quality assurance test is initiated. If the tree is incomplete, then the fact or logic problem is identified and the entire process is repeated. This is called an iterative loop. ... 

If the scenario is completely disproved by the known accepted facts, the reasoning is documented and the scenario is disregarded. If the scenario needs additional data in order to be proven or disproved, then the iterative loop path is followed and additional information is gathered. Sometimes this new information is very specific, precise, and limited in scope. Examples of tasks initiated by this iterative loop include ... 

If the deductive process has stalled and no further progress seems possible or likely, then the iterative loop calls for application of inductive investigation methods such as a checklist or HAZOP. The inductive methods may also benefit from use of the fact/hypothesis matrix tool described in this chapter. 

If the incident investigation team is satisfied with the root causes identified, then the investigation proceeds to the recommendation stage. If a problem or some incompleteness is noted, then an iterative loop is followed. 

---