To determine the long-run average capacity of the system, we analyze the process by breaking it into its distinct phases and calculating the overall output over a repeating production cycle. Here's how:

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1. Key Points of the Process

• Input to the fermentation tank: 5,000 litres per day.
• Fermentation tank capacity: 10,000 litres.
• Fermentation time: 30 days.
• Output from the fermentation tank to the filter: 1,200 litres per day.
• Delay before sake begins flowing out of the filter: 10 days.
• Output from the filter: 1,200 litres per day.
• Cleaning times:
  • Fermentation tank: 3 days after being emptied.
  • Filter: 5 days after being emptied.

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2. Breakdown of the Cycle

Phase 1: Filling the fermentation tank

• Raw materials are added at 5,000 litres per day.
• To fill the tank to its capacity of 10,000 litres, it takes: $\text{Time to fill the tank} = \frac{\text{Tank capacity}}{\text{Input rate}} = \frac{10,000}{5,000} = 2 \text{ days.}$

Phase 2: Fermentation

• The fermentation process lasts for 30 days.

Phase 3: Emptying the fermentation tank

• The sake flows from the fermentation tank to the filter at 1,200 litres per day.
• To empty the entire tank (10,000 litres): $\text{Time to empty the tank} = \frac{\text{Tank capacity}}{\text{Output rate}} = \frac{10,000}{1,200} \approx 8.33 \text{ days.}$

Phase 4: Cleaning the fermentation tank

• The fermentation tank is cleaned for 3 days.

Phase 5: Filtering and final output

• Filtering starts as soon as sake enters the filter. However, sake only begins flowing out of the filter after a 10-day delay.
• Since sake flows into and out of the filter at the same rate (1,200 litres per day), the filter will empty 10,000 litres in: $\text{Time to empty the filter} = \frac{\text{Filter capacity}}{\text{Output rate}} = \frac{10,000}{1,200} \approx 8.33 \text{ days.}$
• After the filter is empty, it undergoes 5 days of cleaning.

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3. Total Cycle Time

Now, we calculate the total time for one complete cycle:

1. Filling the fermentation tank: 2 days.
2. Fermentation: 30 days.
3. Emptying the fermentation tank: 8.33 days.
4. Cleaning the fermentation tank: 3 days.
5. Emptying the filter (includes the 10-day delay): $10 \, \text{(delay)} + 8.33 \, \text{(emptying time)} = 18.33 \, \text{days.}$
6. Cleaning the filter: 5 days.

$\text{Total cycle time} = 2 + 30 + 8.33 + 3 + 18.33 + 5 = 66.66 \, \text{days.}$

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4. Total Output in One Cycle

The total output of sake in one cycle is the amount that flows through the filter, which is 10,000 litres.

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5. Long-Run Average Capacity

The long-run average capacity is the total output divided by the total cycle time: $\text{Long-run average capacity} = \frac{\text{Total output}}{\text{Total cycle time}} = \frac{10,000}{66.66} \approx 150 \, \text{litres per day.}$

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Final Answer

The long-run average capacity of this system is 150 litres per day.

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Do you want details on any part of this explanation or have other questions?

Here are five related questions to expand your understanding:

1. How does the cleaning time affect the system’s long-run average capacity?
2. What happens to the capacity if the fermentation time is reduced?
3. How would an increase in the filter’s output rate affect the long-run average capacity?
4. What are the implications of increasing the fermentation tank’s capacity?
5. How does the 10-day delay in filtering impact the overall efficiency of the system?

Tip: When analyzing process-based systems, always focus on identifying the bottleneck as it controls the overall throughput.