Reconciliation of inaccurate process measurements is an important problem in process control that can be solved using optimization techniques. The flow rates of streams B and C have been measured three times during the current shift (shown in Fig. E19.7). Some errors in the measurement devices

Question

Reconciliation of inaccurate process measurements is an important problem in process control that can be solved using optimization techniques. The flow rates of streams B and [latex]C[/latex] have been measured three times during the current shift (shown in Fig. E19.7). Some errors in the measurement devices exist. Assuming steady-state operation [latex]\left(w_{A}=\right.[/latex] constant [latex])[/latex], find the optimal value of [latex]w_{A}[/latex] (flow rate in [latex]kg / h[/latex] ) that minimizes the sum of the squares of the errors for the material balance, [latex]w_{A}+w_{C}=w_{B}[/latex] using an analytical solution.

Accepted Answer

The objective is to minimize the sum of the squares of the errors for the material balance, that is,

[latex]\min E=\left(w_{ A }+11.3-92.1 ight)^{2}+\left(w_{ A }+10.9-94.2 ight)^{2}+\left(w_{ A }+11.6-93.6 ight)^{2}[/latex]

Subject to [latex]w_{A} \geq 0[/latex]

Solve analytically,

[latex]\begin{aligned}\frac{d E}{d w_{A}}=& 0=2\left(w_{A}+11.3-92.1 ight)+2\left(w_{A}+10.9-94.2 ight) \&+2\left(w_{A}+11.6-93.6 ight)\end{aligned}[/latex]

Solving for [latex]w_{A \ldots} \quad w_{A}{ }^{o p t}=82.0 Kg / hr[/latex]

Check for minimum,

[latex]\frac{d^{2} E}{d w_{A}^{2}}=2+2+2=6>0[/latex], hence minimum

Question 19.7: Reconciliation of inaccurate process measurements is an impo...

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