Let's solve each part step-by-step: --- ## **Given Data Recap** - **Overall flow rate, F:** 100 L/h - **Initial enzyme concentration, C₀:** 1 g/L - **Centrifuge separation factor, C₁₁:** 5 - **Centrifuge flow ratio, L/F:** .6 (L = supernatant, F = feed) - **Membrane filter rejection:** 95% (only 5% passes through) --- ## **(a) Enzyme concentration \( C_L \) after centrifuge** Separation factor \( C_{11} = \frac{C_L}{C_} = 5 \) But the separation factor is typically defined as the ratio of the concentration in the concentrate to the feed. Here, let's assume \( C_L \) is the enzyme concentration in the supernatant **after centrifuge**. But usually, the separation factor (for removal) is: \[ C_{11} = \frac{C_}{C_L} \] So, \[ C_L = \frac{C_}{C_{11}} = \frac{1}{5} = .2\, \text{g/L} \] --- ## **(b) Permeate flow rate \( P \) and permeate enzyme concentration \( C_2 \)** ### **Flow rates:** - **After centrifuge, supernatant flow rate \( L \):** \[ L = (L/F) \times F = .6 \times 100 = 600\, \text{L/h} \] ### **Permeate flow rate \( P \):** Assume no volume loss in the membrane filter, so the permeate is the flow that passes through: \[ P = L = 600\, \text{L/h} \] ### **Permeate enzyme concentration \( C_2 \):** - **Membrane rejects 95% of enzyme:** \[ C_2 = (1 - \text{rejection}) \times C_L = .05 \times .2 = .01\, \text{g/L} \] --- ## **(c) Final flow rate and enzyme concentration after membrane filter** - **Final flow rate (permeate):** \( P = 600\, \text{L/h} \) - **Final enzyme concentration:** \( C_2 = .01\, \text{g/L} \) --- ## **(d) Overall percentage of enzyme separated from the solution** - **Initial amount of enzyme in feed:** \[ \text{Initial amount} = F \times C_ = 100 \times 1 = 100\, \text{g/h} \] - **Amount of enzyme in permeate:** \[ \text{Permeate enzyme} = P \times C_2 = 600 \times .01 = 6\, \text{g/h} \] - **Amount of enzyme separated:** \[ \text{Enzyme separated} = \text{Initial} - \text{Permeate} = 100 - 6 = 994\, \text{g/h} \] - **Percentage separated:** \[ \text{Percentage separated} = \frac{994}{100} \times 100\% = 99.4\% \] --- ## **Final Answers** a) The enzyme concentration after the centrifuge, \( C_L \), is **.2 g/L**. b) The permeate flow rate, \( P \), is **600 L/h**, and the permeate enzyme concentration, \( C_2 \), is **.01 g/L**. c) The final flow rate after the membrane filter is **600 L/h**, and the enzyme concentration is **.01 g/L**. d) The overall percentage of enzyme separated from the solution is **99.4%**.