Consider the in-line blending system of Fig. 18.1a. It is proposed that $w$ and $x$ be controlled using $a$ conventional multiloop control scheme, with $w_{A}$ and $w_{B}$ as the manipulated variables. Derive an expression for the RGA and recommend the best controller pairing for the following conditions: $w = 4 lb/min$ and $x = 0.4$ .

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Accepted Answer

Assuming perfect mixing, a process model can be derived from the following steady-state mass balances:

Total mass: [latex]w = w_{A} + w_{B} (18-41)[/latex]

Component A: [latex]xw = w_{A} (18-42)[/latex]

Substituting Eq. 18-41 into Eq. 18-42 and rearranging gives

[latex]x=\frac{w_{A} }{w_{A}+w_{B}}[/latex] (18-43)

The RGA for the blending system can be expressed as

[latex] w_{A} w_{B}[/latex]

?[latex]= \begin{matrix} w \ x \end{matrix} \begin{bmatrix} \lambda & 1-\lambda \ 1-\lambda & \lambda \end{bmatrix}[/latex]

Relative gain λ can be calculated from Eq. 18-34 after the four steady-state gains are calculated

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

[latex]K_{11} = \left(\frac{∂w}{∂w_{A} } ight) w_{B}=1[/latex] (18-44a)

[latex]K_{12} = \left(\frac{∂w}{∂w_{B} } ight) w_{A}=1[/latex] (18-44b)

[latex] K_{21} = \left(\frac{∂x }{∂w_{A}} ight) w_{B}=\frac{w_{B} }{(w_{A}+w_{B})^{2}}=\frac{1-x}{w}[/latex] (18-44c)

[latex]K_{22} = \left(\frac{∂x }{∂w_{B}} ight) w_{A}=\frac{-w_{A} }{(w_{A}+w_{B})^{2}}=\frac{-x}{w}[/latex] (18-44d)

Substituting into Eq. 18-34 gives [latex]λ = x[/latex]. Thus, the RGA is

[latex] w_{A} w_{B}[/latex]

?[latex]= \begin{matrix} w \ x \end{matrix} \begin{bmatrix} x & 1-x \ 1-x & x\end{bmatrix}[/latex]

Note that the recommended pairing depends on the desired product composition [latex]x[/latex]. For [latex]x = 0.4, w[/latex] should be paired with [latex]w_{B}[/latex] and [latex]x[/latex] with [latex]w_{A}[/latex] . Because all four relative gains are close to [latex]0.5[/latex], control loop interactions will be a serious problem. On the other hand, if [latex]x = 0.9, w[/latex] should be paired with [latex]w_{A}[/latex] and [latex]x[/latex] with [latex]w_{B}[/latex]. In this case, the control loop interactions will be small. Note that for both cases, total flow rate [latex]w[/latex] is controlled by the larger component flow rate, [latex]w_{A}[/latex] or [latex]w_{B}[/latex].

Question 18.3: Consider the in-line blending system of Fig. 18.1a. It is pr...

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