A process control engineer has decided to install an automated shower control system in a bathroom of her mansion. The design calls for a system that can deliver 3 gal/min of water at 110 °F by mixing hot water at 140 °F, with colder water at 70 °F. Flow and temperature transmitters are available

Question

A process control engineer has decided to install an automated shower control system in a bathroom of her mansion. The design calls for a system that can deliver $3 gal / min$ of water at $110^{\circ} F$ by mixing hot water at $140^{\circ} F$ , with colder water at $70^{\circ} F$ . Flow and temperature transmitters are available along with control valves for adjusting the hot and cold water flow rates.

(a) Calculate the required flow rates of hot and cold water, assuming that the density and heat capacity of water are constant.

(b) Calculate the relative gain array for the system, and recommend a pairing of controlled and manipulated variables.

(a) Calculate the required flow rates of hot and cold water, assuming that the density and heat capacity of water are constant.

(b) Calculate the relative gain array for the system, and recommend a pairing of controlled and manipulated variables.

Accepted Answer

The process scheme is shown below

a) Steady state material balance:

[latex]q_{1}+q_{2}=q_{3}[/latex] (1)

Steady state energy balance:

[latex]q_{1} C\left(T_{1}-T_{r e f} ight)+q_{2} C\left(T_{2}-T_{r e f} ight)=q_{3} C\left(T_{3}-T_{r e f} ight)[/latex] (2)

By substituting (1) in (2) and solving:

[latex]\begin{aligned}&q_{1}=9 / 7 gpm \&q_{2}=12 / 7 gpm\end{aligned}[/latex]

b) The steady-state gain matrix [latex]K[/latex] must be calculated :

[latex]\left[\begin{array}{l}T_{3}^{\prime} \q_{3}^{\prime}\end{array} ight]=\left[\begin{array}{ll}K_{11} & K_{12} \K_{21} & K_{22}\end{array} ight]\left[\begin{array}{l}q_{1}^{\prime} \q_{2}^{\prime}\end{array} ight][/latex] (3)

From (1), it follows that [latex]K_{21}=K_{22}=1[/latex]. From (2),

[latex]q_{3} T_{3}=q_{1} T_{1}+q_{2} T_{2}[/latex] (4)

Substitute (1) and rearrange,

[latex]\begin{gathered}T_{3}=\frac{q_{1}}{q_{1}+q_{2}}\left(T_{1}+T_{2} ight) (5) \K_{11}=\left(\frac{\partial T_{3}}{\partial q_{1}} ight)_{q_{2}}=\left(T_{1}+T_{2} ight)\left[\frac{\left(q_{1}+q_{2} ight)-q_{1}}{\left(q_{1}+q_{2} ight)^{2}} ight]=\frac{\left(T_{1}+T_{2} ight) q_{2}}{\left(q_{1}+q_{2} ight)^{2}} \K_{12}=\left(\frac{\partial T_{3}}{\partial q_{2}} ight)_{q_{1}}=\left(T_{1}+T_{2} ight)\left[-\frac{q_{1}}{\left(q_{1}+q_{2} ight)^{2}} ight]\end{gathered}[/latex]

RGA analysis:

[latex]\lambda_{11}=\frac{1}{1-\frac{K_{12} K_{21}}{K_{11} K_{22}}}=\frac{1}{1-\left(-\frac{q_{1}}{q_{2}} ight)}=\frac{q_{2}}{q_{2}+q_{1}} \quad ightarrow \quad \lambda_{12}=1-\lambda_{11}=\frac{q_{1}}{q_{2}+q_{1}}[/latex]

Thus the RGA is,

[latex]\Lambda = \begin{matrix} T_{3} \ q_{3} \end{matrix} \overset{\begin{matrix} q_{1} & q_{2} \end{matrix} }{ \left\lgroup \begin{matrix} \frac{q_{2}}{q_{2}+q_{1}} & \frac{q_{1}}{q_{2}+q_{1}} \ \frac{q_{1}}{q_{2}+q_{1}} & \frac{q_{2}}{q_{2}+q_{1}} \end{matrix} ight group }[/latex]

Substitute numerical values for numerical conditions,

[latex]\Lambda = \begin{matrix} T_{3} \ q_{3} \end{matrix} \overset{\begin{matrix} q_{1} & q_{2} \end{matrix} }{ \left\lgroup \begin{matrix}\frac{4}{7} & \frac{3}{7} \ \frac{3}{7} & \frac{4}{7} \end{matrix} ight group }[/latex]

Pairing: [latex]T_3 - q_2 /q_3 - q_1[/latex]

Question 18.16: A process control engineer has decided to install an automat...

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