(Solved): Sodium hydroxide is diluted with de-ionized water in a perfectly agitated continuous flow tank. Th ...

Sodium hydroxide is diluted with de-ionized water in a perfectly agitated continuous flow tank. The concentration of sodium hydroxide in the tank can be assumed to be uniform at any given time. A sodium hydroxide mass balance gives: \[ \frac{d(V c)}{d t}=V \frac{d c}{d t}=q_{i} c_{i}-\left(q_{w}+q_{i}\right) c \] Where \( \mathrm{c} \) and \( \mathrm{c}_{\mathrm{i}} \) are concentrations(mass/volume) and \( \mathrm{q}_{\mathrm{w}} \) and \( \mathrm{q}_{\mathrm{i}} \) are volumetric flowrates (volume/time). Answer the following questions based on this system; 1.1. What are the input and output variables? [3] 1.2. Derive the transfer function from the given ODE. [15] 1.3. Write expressions for the Time constant and the Process Gain. [7] 1.4. If the inlet sodium hydroxide concentration changes from \( 4.4 \mathrm{~kg} / \mathrm{m} 3 \) to \( 6.3 \mathrm{~kg} / \mathrm{m} 3 \) derive an expression relating the output variable to time. [10] Data: For Question 1.4, assume the tank volume \( V=10 \mathrm{~m}^{3}, q_{w}=2 \mathrm{~m}^{3} / \mathrm{min} \) and \( q_{i}=2.8 \mathrm{~m}^{3} / \mathrm{min} \).