Computer programming overflow

However, the total number of equilibrium stages N, N/N,n, or the external-reflux ratio can be substituted for one of these three specifications. It should be noted that the feed location is automatically specified as the optimum one this is assumed in the Underwood equations. The assumption of saturated reflux is also inherent in the Fenske and Underwood equations. An important limitation on the Underwood equations is the assumption of constant molar overflow. As discussed by Henley and Seader (op. cit.), this assumption can lead to a prediction of the minimum reflux that is considerably lower than the actual value. No such assumption is inherent in the Fenske equation. An exact calculational technique for minimum reflux is given by Tavana and Hansen [Jnd. E/ig. Chem. Process Des. Dev., 18, 154 (1979)]. A computer program for the FUG method is given by Chang [Hydrocarbon Process., 60(8), 79 (1980)]. The method is best applied to mixtures that form ideal or nearly ideal solutions. 

Binary systems of course can be handled by the computer programs devised for multicomponent mixtures that are mentioned later. Constant molal overflow cases are handled by binary computer programs such as the one used in Example 13.4 for the enriching section which employ repeated alternate application of material balance and equilibrium stage-by-stage. Methods also are available that employ closed form equations that can give desired results quickly for the special case of constant or suitable average relative volatility. 

Variations of the basic McCabe-Thiele method, such as incorporating tray efficiencies, nonconstant molal overflow, side streams, or a partial condenser, are often outlined in standard texts. These modifications usually are not justified modern computer programs can calculate complex column arrangements and nonideal systems considerably faster and more accurately than use of any hand-drawn diagram can. 

It is desirable to choose the relaxation parameter as large as possible, but there are stability criteria. The update can be viewed as a matrix operation, and it is essential that the matrix iterations lead to a decrease in the error, not an increase. This only occurs if the spectral properties of the update matrix or propagator are such that the magnitudes of all eigenvalues are less than one. If not, the errors in the solution explode and the computer program quickly generates a numerical overflow ... 

Every care must be taken when developing an algorithm or conducting a simulation to minimize these errors. Numerical errors not only impact the accuracy of the simulation results, they also affect the numerical stability of the algorithm. Accumulation of numerical errors may result in numerical overflow (e.g., when a variable becomes larger than the maximum value a computer can handle) and a program crash. 

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