Accelerated convergence for Newton‐type iterations applied to multicomponent distillation problems

Abstract

corresponds to the phase contours and Eq. lob to the x-axis so that the resulting plot conforms to the conventional Nichols chart. Figure 2 illustrates this from where it can be Seen that incorporation of Eqs. 6, lOa, and lob as above into the Nichols chart makes it so generalized and versatile as to permit both open-and closed-loop process identification. Thus, in addition to the usual M and a contours plotted on the A and 6 coordinates, Figure 2 also provides loci of constant A and 6 superimposed on the M and a coordinates. It may be noted from the chart that, while the M and A values coincide, the a and 6 values differ from each other by a constant factor of 180" as per Eqs. 10a and lob. Using Figure 2 the closedloop (or open-loop) identification is achieved by the usual procedure of entering the chart through A and 6 (or M and a) coordinates, and reading of M and a (or A and 6) contour values by interpolation. Any existing computer program describing the conventional Nichols chart may also be used for open-loop identification simply by incorporating Eqs. 6, 10a and lob into the program. NOTATION A = open-loop magnitude ratio C = controlled variable G(s) G* (s) = open-loop process transfer function = closed-loop process transfer function i = 4-1 M = closed-loop magnitude ratio R = set point W = frequency, radians/time Greek Letters a 6 = closed-loop phase angle, degrees = open-loop phase angle, degrees LITERATURE CITED Coughanowr, D. R., and L. B. Koppel, Process Systems Analysis and Deshpande, P. B., "Process Identification of Open-loop Unstable Systems," Koppel, L. B., "Sampled Data Control of Unstable Processes," Znd. Eng. Luyben, W. L., "Sample-Data Control of Second-Order Open-loop Un-Luyben, W. L., "Effect of Reaction Rate on the Open-Loop Stability of Luyben, W. L., and M. Melcic, "Consider Reactor Control Lags," Hydroc. Control, McGraw Hill, New York (1965). AfChE