Consider the feedback control system for the stirred-tank blending process shown in Fig. 11.1 and the following step test. The controller was initially in the manual mode, and then its output was suddenly changed from 30% to 43%. The resulting process reaction curve is shown in Fig. 12.16. Thus,

Question

Consider the feedback control system for the stirred-tank blending process shown in first Figure below and the following step test. The controller was initially in the manual mode, and then its output was suddenly changed from 30% to 43%. The resulting process reaction curve is shown in second Figure below. Thus,after the step change occurred at t = 0, the measured exit composition changed from 35% to 55% (expressed as a percentage of the measurement span), which is equivalent to the mole fraction changing from 0.10 to 0.30. Determine an appropriate process model for [latex]G riangleq G_{IP}G_{v}G_{p}G_{m}[/latex].

Accepted Answer

A block diagram for the closed-loop system is shown in first Figure below. This block diagram is similar to second Figure below , but the feedback loop has been opened between the controller and the current-to-pressure (I/P) transducer. An FOPTD model can be developed from the process reaction curve in second Figure above using the graphical method of Section 7.2. The tangent line through the inflection point intersects the horizontal lines for the initial and final composition values at 1.07 min and 7.00 min, respectively. The slope of the line is
[latex]S=(\frac{55\%-35\%}{7.00-1.07min} )=3.37\%/min[/latex]
The model parameters can be calculated as
[latex]K=\frac{\Delta x_{m}}{\Delta p}=\frac{55\%-35\%}{43\%-30\%} =1.54 [/latex] (dimensionless)
[latex] heta= 1.07 min[/latex]
[latex] au = 7.00 − 1.07 min = 5.93 min[/latex]
The apparent time delay of 1.07 min is subtracted from the intercept value of 7.00 min for the [latex] au[/latex] calculation.
The resulting empirical process model can be expressed as
[latex]\frac{X^{\prime}_{m}(s)}{P^{\prime}(s)}=G(s)=\frac{1.54e^{-1.07s}}{5.93s+1} [/latex]
Example 12.5 provided a comparison of PI controller settings for this model that were calculated using different tuning relations.

Question 12.9: Consider the feedback control system for the stirred-tank bl...

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