DEGENERACY AND DECOUPLING CONTROL OF THE STIRRED MIXING TANK SYSTEM. Let us consider the stirred mixing tank expanded with an additional heater as in Example 21.10. If we fix the heater at a constant power and only consider the hot and cold stream flowrates as inputs, the linearized model is given

Question

DEGENERACY AND DECOUPLING CONTROL OF THE STIRRED MIXING TANK SYSTEM.

Let us consider the stirred mixing tank expanded with an additional heater as in Example 21.10. If we fix the heater at a constant power and only consider the hot and cold stream flowrates as inputs, the linearized model is given by Eq. (22.95). When some specific values are introduced for the physical dimensions of the stirred mixing tank system:

[latex]G(s) = \begin{bmatrix}\frac{0.261}{43.5s + 1} & \frac{0.261}{43.5s + 1} \ \ \frac{0.01235 (T_{H} - T_{s})}{21.7s + 1} & \frac{0.01235 (T_{C} - T_{s})}{21.7s + 1} \end{bmatrix} [/latex] (22.95)

(These physical dimensions correspond to those for the stirred mixing tank system used for instruction at the University of Wisconsin-Madison.)
Investigate the possible degeneracy for this system under two conditions:

• Condition 1

Hot Stream Temperature: [latex]T_{H}[/latex] = 53°C.
Cold Stream Temperature: [latex]T_{C}[/latex] = 13°C
Operating steady-state tank temperature: [latex]T_{s}[/latex] = 33ºC

• Condition 2

Hot Stream Temperature: [latex]T_{H}[/latex] = 28.5°C .
Cold Stream Temperature: [latex]T_{C}[/latex] = 28°C
Operating steady-state tank temperature: [latex]T_{s}[/latex] = 33ºC

Observe that for Condition 2 the hot stream has cooled off substantially, and the cold stream has heated up significantly so that the desired steady-state is above both feed temperatures. This is only possible because of the auxiliary heater.

Accepted Answer

Under Condition 1, the steady-state gain matrix is given by:

[latex]K = \begin{bmatrix} 0.261 & 0.261 \ 0.247 & -0.247 \end{bmatrix} [/latex] (22.96)

whose determinant is - 0.129; the RGA for this system is obtained as:

[latex]\Lambda = \begin{bmatrix} 0.5 & 0.5 \ 0.5 & 0.5 \end{bmatrix} [/latex] (22.97)

and even though the indication is that there will be significant interactions, we observe that decoupling is very feasible in this case. (We will show later what is required to achieve decoupling under these conditions.) Note that because of the - 0.247 element, the rows and the columns of Eq. (22.96) are, in fact, not linearly dependent even though they may appear to be so.

The gain matrix under Condition 2 is:

[latex]K = \begin{bmatrix} 0.261 & 0.261 \ -0.0556 & -0.0618 \end{bmatrix} [/latex] (22.98)

and we observe immediately that if we rounded all the elements off to only the second decimal place, Column 1 will be identical to Column 2.

The determinant of this matrix is - 0.00162, and the RGA is:

[latex]\Lambda = \begin{bmatrix} 10 & -9 \ -9 & 10 \end{bmatrix} [/latex] (22.99)

and it is now clear that the system will be easier to decouple under Condition 1 than under Condition 2.
There is a physical reason for the increased. degeneracy under Condition 2: observe that the difference between the hot and cold stream temperatures is only half of a degree. The implication is therefore clear: we no longer have two input variables, we essentially have only one (since the hot stream is almost identical to the cold stream), and as a result we have lost one degree of freedom in controlling the process.
Even though it is not crystal clear from the discussion given above, note that the degeneracy will affect only our control of the temperature; in no way does it affect our control of the level.

Question 22.8: DEGENERACY AND DECOUPLING CONTROL OF THE STIRRED MIXING TANK...

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