The staged system model for a three-stage absorber is presented in Eqs. 2-73–2-75, which are in state-space form. A numerical example of the absorber model suggested byWong and Luus1 has the following parameters: H = 75.72 lb, L = 40.8 lb/min, G = 66.7 lb/min, a = 0.72, and b = 0.0. Using the

Question

The staged system model for a three-stage absorber is presented in Eqs. 2-73-2-75, which are in state-space form. A numerical example of the absorber model suggested by Wong and Luus ${ }^{1}$ has the following parameters: $H=75.72 lb , L=40.8 lb / min , G=66.7 lb / min , a=0.72$ , and $b=0.0$ . Using the MATLAB function $s s 2 t f$ , calculate the three transfer functions $\left(Y_{1}^{\prime} / Y_{f}^{\prime}, Y_{2}^{\prime} / Y_{f}^{\prime}, Y_{3}^{\prime} / Y_{f}^{\prime}\right)$ for the three state variables and the feed composition deviation $Y_{f}^{\prime}$ as the input.

$\tau \frac{d x_{1}}{d t}=K\left(y_{f}-b\right)-(1+S) x_{1}+x_{2}$ (2-73)

$\tau \frac{d x_{3}}{d t}=\delta x_{2}-(1+S) x_{3}+x_{f}$ (2-75)

[latex] au \frac{d x_{1}}{d t}=K\left(y_{f}-b ight)-(1+\S) x_{1}+x_{2}[/latex] (2-73)

[latex] au \frac{d x_{3}}{d t}=\delta x_{2}-(1+\S) x_{3}+x_{f}[/latex] (2-75)

Accepted Answer

The system equations are:

[latex]\begin{aligned}&A_{1} \frac{d h_{1}^{\prime}}{d t}=q_{i}^{\prime}-\frac{1}{R_{1}} h_{1}^{\prime} \quad, \quad q_{1}^{\prime}=\frac{1}{R_{1}} h_{1}^{\prime} \&A_{2} \frac{d h_{2}^{\prime}}{d t}=\frac{1}{R_{1}} h_{1}^{\prime}-\frac{1}{R_{2}} h_{2}^{\prime} \quad, \quad q_{2}^{\prime}=\frac{1}{R_{2}} h_{2}^{\prime}\end{aligned}[/latex]

Using a state space representation,

[latex]\begin{gathered}\dot{x}=A x+B u \y=C x+D u \ ext { where } \quad x=\left[\begin{array}{l}h_{1}^{\prime} \h_{2}^{\prime}\end{array} ight] \quad, \quad u=q_{i} \quad ext { and } \quad y=q_{2}^{\prime}\end{gathered}[/latex]

then,

[latex]\begin{aligned}\left[\begin{array}{c}\frac{d h_{1}^{\prime}}{d t} \\frac{d h_{2}^{\prime}}{d t}\end{array} ight] &=\left[\begin{array}{cc}-\frac{1}{R_{1} A_{1}} & 0 \\frac{1}{R_{1} A_{1}} & -\frac{1}{R_{2} A_{2}}\end{array} ight]\left[\begin{array}{l}h_{1}^{\prime} \h_{2}^{\prime}\end{array} ight]+\left[\begin{array}{c}\frac{1}{A_{1}} \0\end{array} ight] q_{i}^{\prime} \q_{2}^{\prime} &=\left[\begin{array}{cc}0 & \frac{1}{R_{2}}\end{array} ight]\left[\begin{array}{c}h_{1}^{\prime} \h_{2}^{\prime}\end{array} ight]\end{aligned}[/latex]

Therefore,

[latex]A =\left[\begin{array}{cc}-\frac{1}{R_{1} A_{1}} & 0 \\frac{1}{R_{1} A_{1}} & -\frac{1}{R_{2} A_{2}}\end{array} ight] \quad, \quad B =\left[\begin{array}{c}\frac{1}{A_{1}} \0\end{array} ight], \quad C =\left[\begin{array}{ll}0 & \frac{1}{R_{2}}\end{array} ight], \quad E =0[/latex]

[latex]{ }^{1}[/latex] Wong, K. T., and R. Luus, Model Reduction of High-order Multistage Systems by the Method of Orthogonal Collocation, Can.J. Chem. Eng. 58, 382 (1980).

Question 6.25: The staged system model for a three-stage absorber is presen...

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