Let's break down the solution step by step based on the **batch growth data** and the reactor configuration question: --- ## **Step 1: Analyze the Batch Growth Data** Let's summarize what the data tells us: | Time (h) | X (g/l) | P (g/l) | dX/dt (g/l·h) | dP/dt (g/l·h) | |----------|---------|---------|---------------|---------------| | | .3 | <.01 | — | — | | 3 | 1. | <.01 | .30 | — | | 6 | 2.3 | <.01 | .55 | — | | 8 | 4. | .010 | 1. | .005 | | 9 | 5.1 | .025 | 1.3 | .010 | | 10 | 6.5 | .060 | 1.4 | .045 | | 10.5 | 7. | .10 | 1.4 | — | | 11 | 7.4 | .10 | .60 | .059 | | 12 | 7.7 | .17 | .20 | .072 | | 13 | 7.8 | .26 | .02 | .105 | | 14 | — | .36 | — | .130 | | 15 | 8. | .47 | ~ | .087 | | 16 | 8. | .54 | ~ | .042 | | 17 | — | .58 | — | .021 | | 18 | — | .60 | — | .005 | Where: - **X** = Biomass concentration - **P** = Product concentration - **dX/dt** = Biomass production rate - **dP/dt** = Product formation rate --- ## **Step 2: Find Maximum Product Formation Rate** From the table, the **maximum dP/dt** is: - At 14 h: **dP/dt = .130 g/l·h** --- ## **Step 3: Understand the Reactor Configuration Question** You have **three tanks**: - 900 L - 600 L - 300 L **Feed flow rate = 100 L/h** You are to **configure the tanks (in series)** to maximize product formation. --- ## **Step 4: Reactor Theory Consideration** **For product formation:** - Maximum product formation rate (dP/dt) occurs after the maximum biomass growth rate. - Ideally, you want most of the flow to be at the **maximum dP/dt** stage. **CSTRs in Series:** - **First tank:** High substrate, mostly growth phase. - **Last tank(s):** Lower substrate, higher product formation. **To maximize product, you want:** 1. **First tank**: Large enough for growth, leading into the production phase. 2. **Subsequent tanks**: Hold cells in conditions with high product formation rate. --- ## **Step 5: Calculate Residence Time** **Total volume = 900 + 600 + 300 = 180 L** **Total residence time** = Total volume / Flow rate = 180 L / 100 L/h = **18 hours** This matches the batch time data (up to 18 h). --- ## **Step 6: Assign Volumes to Maximize Product Formation** - **Growth phase** (max dX/dt): ~ to 10.5 h - **Production phase** (max dP/dt): 10.5 h onward (peak at 14 h) **Fraction of total time at max dP/dt:** - Peak dP/dt at 14 h (out of 18 h total): \( \frac{14}{18} \approx .78 \) You want the **largest tank(s) at the stage with highest dP/dt**. ### **Best Configuration:** - Use the **smallest tank for the growth phase** (early phase, up to ~10.5 h). - Use the **largest tanks for product formation** (late phase, after 10.5 h). #### **Specific Assignment:** - **First tank (growth phase):** 300 L Residence time = 300 L / 100 L/h = **3 h** - **Second tank (transition):** 600 L Residence time = 600 L / 100 L/h = **6 h** - **Third tank (production phase):** 900 L Residence time = 900 L / 100 L/h = **9 h** So, from entry to exit: - -3 h: 1st tank - 3-9 h: 2nd tank - 9-18 h: 3rd tank (this covers the peak dP/dt period!) --- ## **Step 7: Final Answer** ### **To maximize product formation:** **Configuration:** **Feed → 300 L tank → 600 L tank → 900 L tank → Product** - This setup ensures that most of the cell mass spends the longest time (9 h) in the stage where product formation rate is highest. --- ## **Summary Table** | Tank Volume (L) | Residence Time (h) | Stage | |-----------------|--------------------|-------------------| | 300 | 3 | Growth | | 600 | 6 | Transition | | 900 | 9 | Product formation | --- ### **Final Answer** > **Arrange the tanks in order: 300 L → 600 L → 900 L (in series), with a flow rate of 100 L/h. This maximizes the time spent at the highest product formation rate, thus maximizing overall product formation.**