Consider the schematic diagram of a controlled blending process shown in Fig. 8.1. The control objective is to control the mass fraction of the exit stream, x, by adjusting inlet flow rate, w2, using a feedback controller. The mass fractions of a key chemical component in the inlet streams, x1 and

Question

Consider the schematic diagram of a controlled blending process shown in Fig. $8.1$ . The control objective is to control the mass fraction of the exit stream, $x$ , by adjusting inlet flow rate, $w_{2}$ , using a feedback controller. The mass fractions of a key chemical component in the inlet streams, $x_{1}$ and $x_{2}$ , are constant, and mass flow rate $w_{1}$ is a disturbance variable. The liquid volume $V$ is constant. The composition sensor/transmitter (AT) and the current-to-pressure transducer $( I / P )$ are both direct-acting devices.

What is the minimum amount of information you would need in order to decide whether the feedback controller, AC, should be reverse-acting or direct-acting?

Accepted Answer

Two pieces of information are needed to specify controller action:

i) Is the control valve fail open or fail close

ii) Is [latex]x_{1}>x_{2}[/latex] or [latex]x_{1}

If [latex]x_{1}>x_{2}[/latex], then the mass balance is:

[latex]\begin{aligned}&x_{1} w_{1}+x_{2} w_{2}=x w=x\left(w_{1}+w_{2} ight) \&x_{1}>x_{2} \& herefore x_{1}=x_{2}+\Delta \&\left(x_{2}+\Delta ight) w_{1}+x_{2} w_{2}=x\left(w_{1}+w_{2} ight) \&x_{2} w_{1}+\Delta w_{1}+x_{2} w_{2}=x\left(w_{1}+w_{2} ight) \&x_{2}\left(w_{1}+w_{2} ight)+\Delta w_{1}=x\left(w_{1}+w_{2} ight) \&x=x_{2}+\frac{\Delta w_{1}}{\left(w_{1}+w_{2} ight)}\end{aligned}[/latex]

Since all the variables in the equation are positive, then [latex]x>x_{2} .[/latex] The only way to decrease [latex]x[/latex] is to increase [latex]w_{2}[/latex] (but [latex]x[/latex] can never be less than [latex]x_{2}[/latex] ). Therefore, [latex]w_{2}[/latex] should be increased when [latex]x[/latex] increases, in order to have [latex]x[/latex] decrease. If the control valve is fail open (air-to-close), then the composition controller output signal [latex]p[/latex] should decrease. Thus a reverse-acting controller should be selected. Conversely, for a fail close (airto-open) control valve, a direct-acting controller should be used.

If [latex]x_{1}

[latex]\begin{aligned}&x_{1} w_{1}+x_{2} w_{2}=x w=x\left(w_{1}+w_{2} ight) \&x_{1}

Since all the variables are positive, then [latex]x

Question 8.16: Consider the schematic diagram of a controlled blending proc...

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