SIMPLIFIED DECOUPLER DESIGN FOR THE WOOD AND BERRY DISTILLATION COLUMN. Using the simplified decoupling approach, design a decoupler for the Wood and Berry distillation column whose transfer function model was given in Example 20.6, Eq. (20.26).

Question

SIMPLIFIED DECOUPLER DESIGN FOR THE WOOD AND BERRY DISTILLATION COLUMN.

Using the simplified decoupling approach, design a decoupler for the Wood and Berry distillation column whose transfer function model was given in Example 20.6, Eq. (20.26).

[latex]G(s) = \begin{bmatrix}\frac{12.8 e^{-s}}{16.7s + 1} & \frac{- 18.9 e^{-3s}}{21.0s + 1} \ \ \frac{6.6e^{-7s}}{10.9s + 1} & \frac{-19.4e^{-3s}}{14.4s + 1} \end{bmatrix} [/latex] (20.26)

Accepted Answer

Recalling the transfer function matrix for this process, we now obtain, with the aid of Eqs. (22.20) and (22.21), that the transfer functions for the simplified decouplers are:

[latex]g_{I_{1}} = - \frac{g_{12}}{g_{11}}[/latex] (22.20)

[latex]g_{I_{2}} = - \frac{g_{21}}{g_{22}}[/latex] (22.21)

[latex]g_{I_{1}} = - \frac{\frac{- 18.9 e^{-3s}}{21.0s + 1}}{\frac{12.8 e^{-s}}{16.7s + 1}}[/latex]

and

[latex]g_{I_{2}} = - \frac{\frac{6.6e^{-7s}}{10.9s + 1}}{\frac{-19.4e^{-3s}}{14.4s + 1}}[/latex]

which simplify to:

[latex]g_{I_{1}} = 1.48 \frac{(16.7s + 1)e^{-2s}}{(21.0s + 1) }[/latex] (22.33)

and

[latex]g_{I_{2}} = 0.34 \frac{(14.4s + 1)e^{-4s}}{(10.9s + 1) }[/latex] (22.34)

Note that these interaction compensator blocks each have the form of a lead/lag system with a gain term and a time-delay term.
In terms of actual implementation, using Eqs. (22.33) and (22.34) in Eq. (22.18) implies that the overhead reflux rate, [latex]u_{1}[/latex] will be determined from:

[latex]\left. \begin{matrix} u_{1} = v_{1} + g_{I_{1}} v_{2} \ u_{2} = v_{2} + g_{I_{2}} v_{1} \end{matrix} ight\} [/latex] (22.18)

[latex]u_{1} = v_{1} + \left[ 1.48 \frac{(16.7s + 1)e^{-2s}}{(21.0s + 1)} ight] v_{2}[/latex] (22.35)

where, as we may recall, [latex]v_{1}[/latex] and [latex]v_{2}[/latex] are, respectively, the outputs of the Loop 1 (overhead composition) controller and Loop 2 (bottoms composition) controller. The bottoms steam flowrate will be determined from:

[latex]u_{2} = v_{2} + \left[0.34 \frac{(14.4s + 1)e^{-4s}}{(10.9s + 1) } ight] v_{1}[/latex] (22.36)

Question 22.1: SIMPLIFIED DECOUPLER DESIGN FOR THE WOOD AND BERRY DISTILLAT...

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