Iterative calculation, example

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

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. 

FIG. 13-46 Comp arison of the assumed and calculated profiles from the first column iteration in Example 4 with the final computer solution,... 

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,. 

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. 

Calculations for adiabatic flow require the use of equations 6.63 and 6.64. For example, if the upstream conditions P and Vi are known, and G and d, are specified, C can be calculated from equation 6.64 then V2 from equation 6.63. Substituting this value of V2 in equation 6.64 gives P2- If the logarithmic term is not negligible, an iterative calculation will be needed to determine V2 from equation 6.63. 

The problem is best understood by considering an example. One of the most common iterative calculations is a vapor-liquid equilibrium bubblepoint calculation. 

The iterative calculations start by assuming a guess for AT, (it is necessary to assume a value for Xj and -for example the values previously obtained in the dew- and bubble-points evaluation). With these values of AT, the eq. (2.3-12) is solved iteratively for Z. With this value ofZ, using equations (2.3-8) and (2.3-9) new values of vapour- and liquid-phase compositions are calculated and new values of Kt calculated for the next iteration. The iterations stop when satisfactory convergence is reached. 

The preceding results may be converted to the basis of the new product flow rate. Mass-balance equations are linear, so that if the product flow rate is doubled, for example, all flow rates in the process are doubled. In this example then, the new flow rates are obtained by multiplying the previously calculated values by the ratio of the new-to-previous product-stream flow rates (880/99.99) = 8.801. This ratio can be calculated in the scratch pad (say location eh), the new CO feed rate, 880.1, entered into a 3, and the iterative calculations repeated until convergence is attained. The iteration could be avoided, however, if each mass balance had been entered as shown above, but multiplied by a coefficient e3 (the scratch-pad location). Initially, e3 would have been set equal to 1 and the calculations would have proceeded just as shown above. The user then enters 88Q/c8 for location e3, repeats the calculation, and obtains the final flow rates without iteration. 

Isotope ratio measurements were employed for the quantification of analytical data using the isotope dilution strategy. For example, isotope dilution analysis was developed by Sanz-Medel s group for the determination of selenomethionine in Se enriched yeast material by HPLC-ICP-MS using a Se-enriched selenomethionine spike obtained by growing yeast on a Se rich culture medium. For Cr(III)/Cr(VI) determination in yeast, Caruso et al employed the double spike species specific isotope dilution technique measured by HPLC-ICP-MS. The isotope pattern deconvolution approach applied in this work delivers a more intuitive and elegant solution to an otherwise complex data analysis without the need for iterative calculations as widely practised in double spike isotope dilution. The results are in exact agreement with the conventional isotope dilution calculations. ... 

Solution. The operating pressure corresponds to a dew-point condition for the vapor distillate composition. The composition of the reflux corresponds to the liquid in equilibrium with the vapor distillate at its dew point. As shown in Example 4.8, propylene (P) and 1-butene (B) form an ideal solution. Ideal X-values are plotted for lOO E in Fig. 4.4. The method of false position can be used to perform the iterative calculations by rewriting (7-21) in the form... 

All the topics are illustrated with examples that are closely related to practical process simulation problems. At the end of each chapter, additional calculation examples are given to enable the reader to extend his comprehension. An introduction to a larger number of problems can be found in Qiapter 15. These problems and their solutions can be downloaded from the site www.ddbst.com. The problems partially require the use of the software Mathcad and the Dortmund Data Bank Software Package - Explorer Version. Both packages can be downloaded from the Internet. The DDBSP Explorer Version is free to use, whereas Mathcad is only available for free during a tryout period of 30 days. Mathcad files enable the users to perform the iterative calculations themselves and get a feeling for their complexity. Often, typical pitfalls in process simulation are covered in the examples, which should be helpful for the reader to avoid them in advance. 

The underlying principles of the bootstrap approach are easily understood, and the iterative calculations are simple, for example as a macro written for Minitab (see Bibliography). The most important applications in analytical practice are likely to be more complex situations than the one in the above example. Suggested uses have included the estimation of between-laboratory precision in collaborative trials (see Chapter 4), and in the determination of the best model to use in multivariate calibration (see Chapter 8). 

Once the frequency sweep has been carried out, the software iteratively calculates the values of the elements in the model. For the example shown in Figure 2.24, it finds ... 

Once a functional form for e( ) has been chosen depending on a given number of parameters, the latter can be determined by means of iterative calculations to minimize the differences between each R and the corresponding measured response. The limit of this method is that the a priori choice of the form of e(A,) and the functional relationship R between response and exposure is critical for the correct determination of the action spectrum. FFowever, the use of multivariate statistical analysis (such as, for example, principal component analysis) allows more detailed BWFs to be obtained. ... 

This is the dew point temperature problem, which can most easily be solved through the use of iterative calculations. The process is relatively straightforward and similar to the previous example, except that we don t know the initial temperature. So we complete the following steps ... 

At higher pressures can be substantially different from unity even at high temperatures, as we will see with the case of the ammonia synthesis (Example 15.7). In such cases can be estimated with an equation of state. This requires an iterative calculation which is initialized by setting k =... 

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