A conventional multiloop control scheme consisting of two PI controllers is to be used to control the product compositions xD and xB of the distillation column shown in Fig. 18.1b. The manipulated variables are the reflux flow rate R and the steam flow rate to the reboiler S. Experimental data for

Question

A conventional multiloop control scheme consisting of two PI controllers is to be used to control the product compositions [latex]x_{D}[/latex] and [latex]x_{B}[/latex] of the distillation column shown in Fig. 18.1b. The manipulated variables are the reflux flow rate [latex]R[/latex] and the steam flow rate to the reboiler [latex]S[/latex]. Experimental data for a number of steady-state conditions are summarized below. Use this information to do the following:

(a) Calculate the RGA and determine the recommended pairing between controlled and manipulated variables.

(b) Does this pairing seem appropriate from dynamic considerations? Justify your answer.

Table E18.6

Run	R (lb/min)	S (lb/min)	[latex]x_{D}[/latex]	[latex]x_{B}[/latex]
1	125	22	0.97	0.04
2	150	22	0.95	0.05
3	175	22	0.93	0.06
4	150	20	0.94	0.06
5	150	24	0.96	0.04

Accepted Answer

i) Calculate the steady-state gains as

[latex]K_{11}=\left(\frac{\Delta X_{D}}{\Delta R} ight)_{S}=\frac{0.97-0.93}{(125-175) lb / min }=-8 imes 10^{-4} min / lb[/latex]

[latex]\begin{aligned}&K_{12}=\left(\frac{\Delta X_{D}}{\Delta S} ight)_{R}=\frac{0.96-0.94}{(24-20) lb / min }=+5 imes 10^{-3} min / lb \&K_{21}=\left(\frac{\Delta X_{B}}{\Delta R} ight)_{S}=\frac{0.06-0.04}{(175-125) lb / min }=+4 imes 10^{-4} min / lb\end{aligned}[/latex]

[latex]K_{22}=\left(\frac{\Delta X_{B}}{\Delta S} ight)_{R}=\frac{0.04-0.06}{(24-20) l b / min }=-5 imes 10^{-3} min / l b[/latex]

Substituting into Eq. 18-34,

[latex]\lambda_{11}=\frac{1}{1-\frac{K_{12} K_{21}}{K_{11} K_{22}}}[/latex] (18-34)

[latex]\lambda=\frac{1}{1-\frac{\left(5 * 10^{-3} ight)\left(4 * 10^{-4} ight)}{\left(-8 * 10^{-4} ight)\left(-5 * 10^{-3} ight)}}=2[/latex]

Thus the RGA is

[latex]^{x_D\ }_{\ x_B} \overset{R\qquad S}{\begin{pmatrix} 2 & -1 \ -1 & 2 \end{pmatrix} } [/latex]

Pairing for positive relative gains requires [latex]X_{D}-R, X_{B}-S[/latex].

ii) This pairing seems appropriate from dynamic considerations as well; because of the lag in the column, [latex]R[/latex] affects [latex]X_{D}[/latex] sooner than [latex]X_{B}[/latex], and [latex]S[/latex] affects [latex]X_{B}[/latex] sooner that [latex]X_{D}[/latex]

Question 18.6: A conventional multiloop control scheme consisting of two PI...

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