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**The Annapolis Valley Apple Puree Pipeline**

**Problem Statement:**

Why ship apples by truck from Kentville to Halifax? Consider the following issues:
- Hauling
- Loading into containers
- Squashing
- Spoilage
- Worry about bugs and worms
- Unloading
- Trucking
- Handling
- Labour

Instead, why not mechanically peel the apples in Kentville and pump them as a puree by pipeline straight to Halifax, just 120 km away? After that, add a touch of vegetable glue to reform the apples into any desired shape. What an exciting opportunity for creative apple reshaping! However, before plunging into apple reformation, let us see if this proposal is economical.

**Task:**

Ignoring capital costs in building the pipeline, estimate the pumping cost for such an operation. Provide costs in terms of $yr\(^{-1}\) and $kg\(^{-1}\) of shipped apples.

**Data:**
- Kentville is 220 ft above sea level.
- Pipe size is 4-in sch. 40 stainless steel.
- There will need to be 220 standard 90° elbows, 20 gate valves (normally fully open), and 100 unions.
- Assume pump and motor are 65% efficient overall.
- Electricity costs $0.12 kWh\(^{-1}\).
- Mean apple puree velocity is 1.2 m s\(^{-1}\).
- Density of the apple puree is 1015 kg m\(^{-3}\).
- Rheological characteristics of apple puree are \(\tau = 5.0 \dot{\gamma}^{0.5}\) (in SI units).

Answer :

Final answer:

The pumping cost can be calculated using Bernoulli's equation and considering the pressure drop due to friction, unions, elbows, and gate valves. The annual cost of electricity can be found by multiplying total power requirement by yearly hours and electricity cost. The cost per kilogram can be found by dividing the annual cost by the total mass of apple puree moved in a year.

Explanation:

The cost of pumping the apple puree can be calculated with respect to the power requirement of the pipeline. This can be determined using Bernoulli's equation and the equation for the pressure drop in a pipe due to friction, unions, elbows and gate valves. In Bernoulli's equation, we take into account the elevation change, the pressure change, and the change in velocity of the fluid. While calculating the frictional losses, we have to consider the length of the pipeline, the number of unions, elbows and gate valves.

Once the total power requirement is calculated, we can then calculate the annual cost for electricity. Given that the electricity cost is $0.12 per kWh, the pump and motor are 65% efficient, we can find the cost per year. E.g. if the result was 1000kW, then the annual cost would be 1000kW * 8760 hours/year * $0.12/kWh. This gives a cost in $/year.

To get the cost in $/kg, we divide the annual cost by the total mass of apples pureed in a year. That is, if we know the pipelines flow rate in kg/s, we can calculate the total mass moved in a year as Flow rate * number of seconds in a year, and then divide the total annual cost by this number to get the cost in $/kg.

In conclusion, this idea is creative and opens exciting possibilities for apple reshaping. However, the economical viability of this project can be assessed by calculating the pumping costs as explained above.

Learn more about Pumping Cost Calculation here:

https://brainly.com/question/32544924

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