The blending system in Fig. 13.3 has a bypass stream that allows a fraction f of inlet stream w2 to bypass the stirred tank. It is proposed that product composition x be controlled by adjusting f via the control valve. Analyze the feasibility of this control scheme by considering its steady-state

Question

The blending system in Figure below has a bypass stream that allows a fraction f of inlet stream [latex]w_{2}[/latex] to bypass the stirred tank. It is proposed that product composition x be controlled by adjusting f via the control valve. Analyze the feasibility of this control scheme by considering its steady-state and dynamic characteristics. In your analysis, assume that [latex]x_{1}[/latex] is the principal disturbance variable and that [latex]x_{2}, w_{1},\space and \space w_{2} [/latex] are constant. Variations in liquid volume V can be neglected because [latex]w_{2} ≪ w_{1}[/latex].

Accepted Answer

The dynamic characteristics of the proposed control scheme are quite favorable because product composition x responds rapidly to changes in the bypass flow rate. In order to evaluate the steady-state characteristics, consider a component balance over the entire system:
[latex]w_{1}\bar{x}_{1}+w_{2}x_{2}=w\bar{x}[/latex] (1)
Solving for the controlled variable gives,
[latex]\bar{x}=\frac{w_{1}\bar{x}_{1}+w_{2}x_{2}}{w} [/latex] (2)
Thus [latex]\bar{x}[/latex] depends on disturbance variable [latex]\bar{x}_{1}[/latex] and four constants ([latex]w_{1}, w_{2}, x_{2},\space and \space w)[/latex], but it does not depend on bypass fraction, f. Thus, it is not possible to compensate for sustained disturbances in [latex]x_{1}[/latex] by adjusting f. For this reason, the proposed control scheme is not feasible.
Because f does not appear in Equation (2), the steady-state gain between x and f is zero. Thus the bypass flow rate can be adjusted, but it does not provide a control degree of freedom. However, if [latex]w_{2}[/latex] could also be adjusted, manipulating both f and [latex]w_{2}[/latex] could produce excellent control of the product composition.

Question 13.3: The blending system in Fig. 13.3 has a bypass stream that al...

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