The relative gain array for a refinery distillation column associated with a hydrocracker discussed by Nisenfeld and Schultz (1971) is given by u1 u2 u3 u4 ? = y1 y2 y3 y4 [ 0.931 0.150 0.080 -0.164 -0.011 -0.429 0.286 1.154 -0.135 3.314 -0.270 -1.910 0.215 -2.030 0.900 1.919 ] The four controlled

Question

The relative gain array for a refinery distillation column associated with a hydrocracker discussed by Nisenfeld and Schultz (1971) is given by

[latex] u_{1} u_{2} u_{3} u_{4} [/latex]

?= [latex] \begin{matrix} y_{1} \ y_{2} \ y_{3} \ y_{4} \end{matrix} \left [ \begin{matrix} 0.931 & 0.150 & 0.080 & -0.164 \ -0.011 & -0.429 & 0.286 & 1.154 \ -0.135 & 3.314 & -0.270 &-1.910 \ 0.215 & -2.030 & 0.900 & 1.919 \end{matrix} ight ][/latex] (18-45)

The four controlled variables are the compositions of the top and bottom product streams [latex](y_{1}, y_{2})[/latex] and the two side streams [latex](y_{3}, y_{4})[/latex]. The manipulated variables are the four flow rates numbered from the top of the column; for example, the top flow rate is [latex]u_{1}[/latex]. Find the recommended pairing using the RGA.

Accepted Answer

To determine the recommended controller pairs, we identify the positive relative gains that are closest to one in each row and column. From the rows, it is apparent that the recommended pairings are [latex]y_{1}- u_{1}, y_{2}-u_{4}, y_{3}-u_{2}[/latex], and [latex]y_{4}-u_{3}[/latex]. Note that this pairing assigns [latex]u_{2} to y_{3}[/latex] rather than to [latex]y_{1}[/latex], even though its relative gain of 3.314 is farther from one. This choice is required because pairing any other manipulated variable with [latex]y_{3}[/latex] corresponds to a negative relative gain, which is undesirable. The [latex]y_{1}- u_{1}[/latex] and [latex] y_{4}-u_{3}[/latex] relative gains are close to one, so these selections are straightforward.

Question 18.4: The relative gain array for a refinery distillation column a...

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