Iterative calculation

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain... 

START ITERATIVE CALCULATION OF ACTUAL VAPCR COMPOSITION, ZKI)... 

A common iterative procedure is to solve the problem of interest by repeated calculations that do not initially give the correct answer but get closer to it as the calculation is repeated, perhaps many times. The approximate solution is said to converge on the correct solution. Although no human would be willing to repeat an iterative calculation thousands of times to converge on the right answer, the computer does, and, because of its speed, it often arrives at the answer in a reasonable amount of time. 

There is a great difference between various simulators (5) in terms of how easily and how well the hypothetical calculation units can be incorporated in the simulation. The trial-and-error calculations, which ate called iterative calculations, do not always converge for every flow sheet being simulated. Test problems can be devised to be tried with various simulators to see if the simulator will give a converged solution (11). Different simulators could take different numbers of iterations to converge and take different amounts of computet time on the same computet. 

X is calculated from Eq. (2-37) and Tf is calculated from Eq. (2-38). Iterative calculation maybe required. 

Shooting Methods The first method is one that utihzes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundaiy conditions. Consider the nonlinear boundaiy value problem... 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequahty constraints are present and all equahty constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objec tive func tion. Veiy few reahstic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

Example 2 Calculation of Kremser Method For the simple absorber specified in Fig. 13-44, a rigorous calculation procedure as described below gives results in Table 13-9. Values of were computed from component-product flow rates, and corresponding effective absorption and stripping factors were obtained by iterative calculations in using Eqs. (13-40) and (13-41) with N = 6. Use the Kremser method to estimate component-product rates if N is doubled to a value of 12. 

Equation (22-106) gives a permeate concentration as a function of the feed concentration at a stage cut, 0 = 0, To calculate permeate composition as a function of 0, the equation may be used iteratively if the permeate is unmixed, such as would apply in a test cell. The calculation for real devices must take into account the fact that the driving force is variable due to changes on both sides of the membrane, as partial pressure is a point function, nowhere constant. Using the same caveat, permeation rates may be calciilated component by component using Eq. (22-98) and permeance values. For any real device, both concentration and permeation require iterative calculations dependent on module geometiy. 

In order to calculate the effect of several passes an iterative calculation is needed using the initial profile at each pass to represent Cq. Clearly for the second pass, die concentration profile given in the right-hand side of tire above equation must be used. It is clear that tire partition constant of the impurity between the solid and liquid is the most significant parameter in tire success of zone refining. 

The design of any of the distillation processes discussed requires choosing an operating pressure, bottoms temperature, reflux condenser temperature and number of trays. This is normally done using any one of several commercially available process simulation programs which can perform the iterative calculations discussed in Chapter 6. 

Multicomponent distillation is by far the common requirement for process plants and refineries, rather than the simpler binary systems. There are many computer programs which have been developed to aid in accurately handling the many iterative calculations required when the system involves three to possibly ten individual components. In order to properly solve a multicomponent design, there should be both heat and material balance at every theoretical tray throughout the calculation. 

As the starting geometries for iterative calculation, we take all the possible structures in which bond lengths are distorted so that the set of displacement vectors may form a basis of an irreducible representation of the full symmetry group of a molecule. For example, with pentalene (I), there are 3, 2, 2 and 2 distinct bond distortions belonging respectively to a, b2 and representations of point group D21,. 

Figure 15, Iterative calculation of the instanton path. The labels 1-9 show gradual improvement of the instanton trajectory shape using the MP2/cc-pVDZ ab initio data. After switching to the CCSD(T)/(aug-)cc-pVDZ ab initio method, only two more steps needed to achieve convergence and obtain the final results. Taken from Ref. [104].

As in Example BSTILL, a column containing four theoretical plates and reboiler is assumed, together with constant volume conditions in the reflux drum. The liquid behaviour is, however, non-ideal for this water-methanol system. The objective of this example is to show the need for iterative calculations required for bubble point calculations in non-ideal distillation systems, and how this can be achieved with the use of simulation languages. 

Because the calculation of these residuals does not require any iterative calculations, the overall computational requirements are significantly less than for the explicit estimation method using Equation 14.15 and the explicit LS estimation method using Equations 14.16a and b (Englezos et al. 1990a). 

The principal use of this method is in the rating of an existing exchanger. It can be used to determine the performance of the exchanger when the heat transfer area and construction details are known. The method has an advantage over the use of the design procedure outlined above, as an unknown stream outlet temperature can be determined directly, without the need for iterative calculations. It makes use of plots of the exchanger effectiveness versus NTU. The effectiveness is the ratio of the actual rate of heat transfer, to the maximum possible rate. 

The design of vertical thermosyphon reboilers requires iterative calculations in which the exchanger needs to be divided into zones. The energy and pressure balances need to be performed simultaneously. Frank and Prickett17 performed a range of detailed simulations and presented the results graphically. This can be used as the basis of preliminary design. 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

Using Eqs. (5-42)-(5-46) in Section 5.3.2.2 with iterative calculations, the predicted CHF were compared with Columbia University data (Fighetti and Reddy, 1983). The comparison was made by examining the statistical results of critical power ratios (DNBRs), where... 

Equation (9.53) for the desired molecular field is nonlinear, typically solved iteratively. For this molecular-field approach to become practical, an alternative to this nonlinear iterative calculation is required. A natural idea is that a useful approximation to this molecular field might be extracted from simulations with available generic force fields. Then with a satisfactory molecular field in hand, the more ambitious quasichemical evaluation of the free energy can be addressed, presumably treating the actual binding interactions with chemical methods specifically. This is work currently in progress. 

Table III. Results of non-iterative calculations for Ni clusters (Ref. 46). Numbering of atoms as in Fig. 31.

Internal return rate. The internal return rate (IRR), also known as the discounted cash flow return rate, is the iteratively calculated discounting rate that would make the sum of the annual cash flows, discounted to the present, equal to zero. As shown in Figure 2, the IRR for Project Chem-A is 38.3%/yr. 

A feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. A more efficient way is to use an unfeasible path approach to solve the NLP problem however, many of these large-scale NLP methods are only efficient in solving problems with few degrees of freedom. A decoupled SQP method was proposed by Tjoa and Biegler (1991) that is based on a globally convergent SQP method. 

Iterative calculation. The solution of the M-estimator Eqs. (10.31) and (10.32) is obtained by means of an iterative procedure that, for iteration q, is made up of the following steps ... 

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