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Volume of a Pipeline

The attached document shows a pipeline of 24 inch diameter (approx. 600mm) buried 1 m below the ground. There is a water pipe which prevents the pipe from going horizontally and hence it has to follow one of two pathways ie. either along the dark blue 5 mm diameter curves and exit at the bottom or along the dotted red double 40 mm diameter curve and join the blue line at the bottom and then exit.

The big diagram is a cross-sectional view of the pipeline and view A is a view looking at the front of the pipeline where flow enters. View A is shown below with all the dimensions given.

The task of this posting is to determine the volume for the section below the ground for the two cases mentioned above ie. the green striped case and the red striped case. One can notice that in the case of the red stripe the trench will have to be dug deeper than in the case of the green stripe case. Note that the dimensions of the bends and other dimensions are not given for fun. Also state any assumptions necessary.

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Solution This solution is FREE courtesy of BrainMass!

See the attached jpg file.

All red bends have the same diameter of 40 mm
All blue bends have the same diameter of 5 mm
The drawing explains why the volumes are equal without any calculations, perhaps with a few words of explanations.
Therefore it is enough to calculate the volume for 0 mm bends.

A few words of proof:

Consider areas 1, 2, 3, and 4 between blue and red lines in the attached enhanced drawing.
These areas represent in fact volumes as similar sections can be drawn parallel to this section.
All the 4 areas have exactly the same area by construction as these are areas between 90 degree quarter-circles of radii 5 mm and 40 mm tangent to the sides of the square corners.

Areas 1 + 2 represent the volume of extra digging for red pipe as compared to blue pipe.
Areas 3 + 4 represent the volume of digging saved for red pipe as compared to blue pipe.
As the savings exactly compensate the extra digging, the volumes needed to dig for blue and red pipes are exactly the same.

The same proof goes for any two pipes of any diameter that can be situated in a similar ralative position as in the drawing.
Therefore, if we want to calculate the volume of earth needed to be dug, we can do it for a pipe with bends of 0 mm radius.

As digging is likely to be done in square sections similar to section A, the depth of digging on the left should be
1000 mm + 600 mm = 1600 mm
and on the right it would be
4000 mm + 600 mm = 4600 mm

Assuming 1m length for both the horizontal branches, we have to dig
1.6 m x 1 m = 1.6 m^2
on the left, and
4.6 m x (1 m + 0.6 m) = 7.36 m^2
on the right (adding the diameter of the vertical part of the pipe)

If the width of the dig is taken to be 1 m, the volume to dig is

(1.6 m^2 + 7.36 m^2) x 1 m = 8.96 m^3

Q:
What would happen if the top red curve does not exist and instead the flow followed the top blue curve and the bottom red curve in the 2nd case instead of following the 2 red curves?

A:
The volumes would still be the same as area 4 would exactly compensate area 2

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