To calculate the stroke length of the piston and the work done per stroke, we'll use the provided information.
2.3.1 Calculate the Stroke Length of the Piston
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Determine the cross-sectional area of the piston: The area \( A \) of a circle is given by the formula:
\[ A = \pi \left( \frac{d}{2} \right)^2 \]
where \( d \) is the diameter of the piston.
Here, the diameter \( d \) is 250 mm, which we need to convert to meters:
\[ d = 250 , \text{mm} = 0.250 , \text{m} \]
Now, we can calculate the area:
\[ A = \pi \left( \frac{0.250}{2} \right)^2 = \pi \left( 0.125 \right)^2 \approx \pi (0.015625) \approx 0.049087 , \text{m}^2 \]
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Determine the stroke length \( L \): We know the volume of water delivered per stroke is 12.5 liters, which we need to convert to cubic meters:
\[ 12.5 , \text{liters} = 0.0125 , \text{m}^3 \]
The volume \( V \) produced by the piston during each stroke is also given by the formula for the volume of a cylinder:
\[ V = A \cdot L \]
Therefore, we can rearrange this to solve for \( L \):
\[ L = \frac{V}{A} \]
Substituting in the values:
\[ L = \frac{0.0125 , \text{m}^3}{0.049087 , \text{m}^2} \approx 0.2546 , \text{m} = 254.6 , \text{mm} \]
So, the stroke length of the piston is approximately 254.6 mm.
2.3.2 Calculate the Work Done Per Stroke
The work done \( W \) is given by the formula:
\[ W = F \cdot d \]
where \( F \) is the effective force and \( d \) is the stroke length. We have:
- Effective force \( F = 27 , \text{kN} = 27000 , \text{N} \)
- Stroke length \( d = 0.2546 , \text{m} \)
Now substituting the values into the work equation:
\[ W = 27000 , \text{N} \cdot 0.2546 , \text{m} \approx 6864.2 , \text{J} \]
So, the work done per stroke is approximately 6864.2 Joules.