Tearing Methods

A number of iterative procedures have been developed for solving Eqs. (13-18) and (13-19) simultaneously for V and T9. Frequently, and especially if the feed contains components of a narrow range of volatility, convergence is rapid for a tearing method in which a value of T9 is... 

Tridiagonal-Matrix Algorithm Both the BP and the SR equation-tearing methods compute hqnid-phase mole fractions in the same way by first developing linear matrix equations in a manner shown by Amundson andPontinen [Jnd. ng. Ch m., 50, 730 (1958)]. Equations (13-69) and (13-68) are combinedto eliminate yjj and yij + i (however, the vector yj stiU remains imphcitly in K j) ... 

Various procedures for solving Eqs. (13-149) to (13-161), ranging from a complete tearing method to solve the equations one at a time, as shown by Distefano, to a complete simultaneous method, have been studied. Regardless of the method used, the following considerations generally apply ... 

B. Tearing Methods for Large Scale Systems of Equations. 212... 

A correlation may be established between the concentration of oxidized lipids and the TEARS value, expressed as MDA equivalents, in uM units. Correction is due in some cases for the interference by dyes or other factors. For example, the presence of anthocyanins in red cabbage leaves or turbiditjf causes overestimation of lipid hydroperoxides in plant tissue by the TEARS method. TEARS was used to assert the level of endogenous peroxides in hypo- and hyperthyroidism, both conditions being characterized by low lipid and lipoprotein plasma levels and enhanced oxidative metabolism . In a procedure for determination of TEARS in edible oils, the sample is placed in a centrifuge at 12000 g before measuring at 532 nm (e = 1.56 x 10 M cm ) . A usual procedure for determination of TEARS in certain complex matrices involves steam distillation of the aldehydes responsible for the value, instead of extraction. In nitrite-cured meats, excess nitrite may cause nitrosation of MDA, thus interfering with distillation. To avoid this interference sulfanilamide is added, which is converted to a diazonium salt and... 

Tearing methods in which the equations are divided into groups... 

A great many tearing methods for solving the MESH equations have been proposed. According to Friday and Smith (1964), tearing methods may be analyzed in the following terms ... 

Most tearing methods are based on keeping all the equations of a given type together, e.g., the M equations or the H equations, for all the stages at once. 

Simultaneous Convergence Methods One drawback of some tearing methods is their relatively limited range of application. For example, the BP methods are more successful for distillation, and the SR-type methods are considered better for mixtures that exhibit a wide range of (pure-component) boiling points (see, however, our remarks above on modified BP and SR methods). Other possible drawbacks (at least in some cases) include the number of times physical properties must be evaluated (several times per outer loop iteration) if temperature- and composition-dependent physical properties are used. It is the physical properties calculations that generally dominate the computational cost of chemical process simulation problems. Other problems can arise if any of the iteration loops are hard to converge. 

Murphree efficiencies are easily incorporated within simultaneous convergence algorithms (something that is not always easy, or even possible, with some tearing methods). (As an aside, note that vaporization... 

A. The cut-and-try ( tear ) method. The recycle stream flows can be estimated and the calculations continued to the point where the recycle is calculated. The estimated flows are then compared with those calculated, and a better estimate is made. The procedure is continued until the difference between the estimated and the calculated flows is within an acceptable tolerance. 

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