00
Ind. Eng. Chem. Fundam. 1082, 21,9 9
flows. The pressure flow was dealt with in an early work by Fredrickson and Bird (1959).The drag flow is a simple problem and the velocity distribution can be easily developed or found in Middleman (1977)and Bird et al. (1977). Our work treated the combined case. The combined flow in fact exists only in the region with a proper combination of k, n, and p and with A > 0 in Tables I-IV of our work. Other cases in these tables with A = 0 belong to the drag flow which has simple velocity distribution as mentioned above. We should have clearly indicated this in our paper. All the entries in these tables with A = 0 hence have no practical meaning. The table and equation given by MacSporran do provide a convenient means for determining the boundary which separates the combined flow from the drag one. It must be noted, however, that approximate boundary values of (with an accuracy to the third decimal point) for A = 0 can be easily determined from our tables by simple interpolation. The other reason why we did not determine the accurate boundary values of /3 is that the actual combined flow like those in wire coating are unlikely to exist near the boundary and can be easily identified. Therefore, for practical reasons, knowledge of the exact boundary values
of 0 is not necessarily needed and the tables given in our work can provide a lot of information. One thing for which we must apologize is that there are three wrong entries in the tables which we overlooked. The value of A for k = 0, n = 0.1,and 0 = 0.14 (in Table 111) should read 0.13544 instead of 0.23544. The two entries in Table IV should read 0.83664 for n = 0.8 and 0 = 0.016 and 0.59876 for n = 1.0 and /3 = 0.1. Other than this, the numerical tables are fairly accurate. Literature Cited Bird, R. B.; Armstrong. R. C.; Hassager, 0. “Dynamics of Polymeric Liquids”; Wlley: New York. 1977. Fredrlckson, A. G.; Bird, R. E. I n d . Eng. Chem. 1958, 50, 347. Lin, S. H.; Hsu, C. C. Ind. fng. Chem. fundam. 1980, 79, 421. Middleman, S. ”Fundamentals of Polymer Processing”; McOraw-Hilt New York. 1977.
Dielectric Materials Laboratory General Electric Company Pittsfield, Massachusetts 01201 American Cyanamid Company Bound Brook, New Jersey 08805
Sheng H. Lin* Chung C. HBU
Comments on “Feasibility of Decoupllng In Conventionally Controlled Dlstlllatlon Columns” Sir: In a recent article, Weischedel and McAvoy (1980) concluded that simulated results support their claim that decoupling of high sensitivity distillation columns is not feasible. In fact, the results they show demonstrate only that in highly nonlinear systems a compensation matrix with constant gains can be inappropriate. According to eq 5 in their paper the compensation matrix for their system labelled, “column C”, is 1
1
+0.9(57s + l)2e-0.7s (50.56 t 1)’
+1.05(30.7s + l)2ee‘s (28.5s t 1 ) 2 1
1 1
Leaving aside the practicality of applying such a compensator this compensation matrix diagonalizesthe system matrix formed when multiplying by the transfer function matrix for column C. It is well known, Rosenbrock (1974), that a diagonalized system is decoupled. Therefore, the only possible causes of the interaction shown are: (1)a poor linear transfer function fit to the simulated step responses; and (2) that the process transfer function is strongly dependent on the size or direction of the step disturbance. In fact, the authors have shown that the latter is true for column C. The process gain array for this system differs by an order of magnitude when comparing the results of a step down to a step up in the manipulated variables L and V. The authors estimated the gains for the transfer function array of column C by averaging the gains from the two types of response tests (i.e., steps up and down). This is a common procedure for single input/output systems but is inappropriate here. An examination of the Nyquist Array of the transfer function 0 198-43 131821 102 1-0099$0 1.2510
matrix, using the gains from the upward step responses, shows that the system is not dominant but can easily be made dominant by applying the compensation matrix
[: :] When using the gains from the downward step responses, the Nyquist Array shows that the system is strongly dominant and requires no decoupling. For purposes of comparison to the case of the upward step above, the compensation matrix here would be the identity matrix
[: :] By averaging the gains from these two extreme cases the authors have merely assured that compensation derived by the relative gain method is always inappropriate. Thus, the only conclusions that should be drawn from this communication are (1)that the relative gain decoupling method can be inappropriate, and (2) that the method of averaging gains from upward and downward step response tests can be inappropriate in multiple input/ output systems. One should not draw the conclusion that decoupling of high sensitivity columns is infeasible. Literature Cited Rosenbrock. H. H. “Computer-Aided Control System Design”; Academlc hess: London, 1974. Welschedel. K.; McAvoy. T. J. I n d . Eng. Chem. Fundem. 1980. 79, 379.
Process Computer Applications Department Shell Canada Limited Toronto, Ontario, Canada 0 1982 American Chemical Soclety
Steven Treiber
