Let's tackle **Question 2: Liquid Depth Calculation** step by step. --- ## **Step 1: Understand the Problem** We need to **calculate the liquid depth (broth height, \( H_L \))** in the pilot tank, given: - Tank diameter, \( D = 1 \) m - Tank height, \( H = 2 \) m - Loading factor = 64% - The tank has a protruding section on top, height \( h_a = \frac{1}{4}D \) - The rest of the tank is cylindrical --- ## **Step 2: Calculate Protruding Section Height (\( h_a \))** Given: \[ h_a = \frac{1}{4} D = \frac{1}{4} \times 1 = .25 \text{ m} \] --- ## **Step 3: Calculate Total Working (Broth) Volume** The **loading factor** indicates the fraction of the tank's total volume that is filled with liquid (broth): \[ \text{Loading factor} = \frac{\text{Working volume}}{\text{Nominal (total) volume}} \] ### **Calculate Nominal (Total) Volume:** The tank is a cylinder: \[ V_{\text{nominal}} = \pi \left(\frac{D}{2}\right)^2 H \] Plug in the values: \[ V_{\text{nominal}} = \pi \left(\frac{1}{2}\right)^2 \times 2 = \pi \times .25 \times 2 = \pi \times .5 \approx 1.57 \text{ m}^3 \] ### **Calculate Working Volume:** \[ \text{Working volume} = \text{Loading factor} \times V_{\text{nominal}} \] \[ \text{Working volume} = .64 \times 1.57 \approx 1.01 \text{ m}^3 \] --- ## **Step 4: Calculate Broth Height (\( H_L \))** **The working volume is the volume of liquid (broth) in the tank.** The liquid fills the cylindrical section up to height \( H_L \) (not the entire tank height, since some headspace is needed for gas, foaming, etc.). \[ \text{Working volume} = \pi \left(\frac{D}{2}\right)^2 H_L \] Solve for \( H_L \): \[ H_L = \frac{\text{Working volume}}{\pi \left(\frac{D}{2}\right)^2} \] Plug the numbers in: \[ H_L = \frac{1.01}{\pi \left(\frac{1}{2}\right)^2} \] \[ = \frac{1.01}{\pi \times .25} \] \[ = \frac{1.01}{.785} \] \[ \approx 1.29 \text{ m} \] --- ## **Step 5: Check That \( H_L \) Fits in the Tank** The tank height is 2 m, with a protruding section of \( h_a = .25 \) m, so the cylindrical section is \( H - h_a = 2 - .25 = 1.75 \) m. Since \( H_L = 1.29 \) m is less than 1.75 m, the calculation is valid. --- ## **Final Answer** ### **Liquid Depth (Broth Height):** \[ \boxed{H_L \approx 1.29 \text{ m}} \] --- ### **Summary Table** | Parameter | Value | |----------------|-------------| | Tank diameter | 1 m | | Tank height | 2 m | | Protruding top | .25 m | | Loading factor | 64% | | Working volume | 1.01 m³ | | Broth height | 1.29 m | --- **If you need the justification for the stirred-tank reactor (Question 1), let me know!**