Water flows in a channel with uniform curvature ?. The channel has height ℎ = 5 mm and width ? = 190 mm (normal to the drawing plane). The curved part of the channel has a length of ? = 100 mm in the ?-direction. You can assume that the channel height is very small compared to the curvature radius. Neglect effects of gravity. a) Assuming that the pressure is uniform on each of the channel walls, determine the pressure difference between the upper and lower wall that is needed if the required net force on the two walls is 2000 N pointing downward (in the negative ?-direction). Use a sign convention that a positive pressure difference or gradient means the pressure at the higher ?-position is higher. b) Your team agree that, from experience, the average velocity of the fluid is ?= 53 m/s. Assuming that the velocity is uniform across the channel, and neglecting gravitational effects, determine the curvature ?required to generate this fo
Water flows in a channel with uniform curvature ?. The channel has height ℎ = 5 mm and width ? = 190 mm (normal to the drawing plane). The curved part of the channel has a length of ? = 100 mm in the ?-direction. You can assume that the channel height is very small compared to the curvature radius.
Neglect effects of gravity.
a) Assuming that the pressure is uniform on each of the channel walls, determine the pressure difference between the upper and lower wall that is needed if the required net force on the two walls is 2000 N pointing downward (in the negative ?-direction).
Use a sign convention that a positive pressure difference or gradient means the pressure at the higher ?-position is higher.
b) Your team agree that, from experience, the average velocity of the fluid is ?= 53 m/s. Assuming that the velocity is uniform across the channel, and neglecting gravitational effects, determine the curvature ?required to generate this force with this flow velocity.
Use a sign convention that a positive value indicates that the centre of the radius is above the channel, i.e. the middle of the channel is bent downwards. A negative radius indicates the middle of the channel is bent upwards.
Your team agree on a radius of ? = 100mm for the centreline of the channel. One of your team members points out that velocity is not uniform across the channel. To improve your analysis, assume that the flow in the channel can be split into three streamtubes, of height h/3, each having a constant velocity. The two streamtubes near the walls can be assumed to have average velocity ??=25m/s.
For each of the streamtubes, use the streamtube's centreline to determine its radius.
c) Determine the pressure gradient in the streamtube next to the upper wall.
Determine the pressure difference across the streamtube next to the upper wall.
d) Determine the velocity in the middle streamtube, if the overall flowrate is to be the same as in b)
Determine the pressure gradient in the streamtube in the middle.
Determine the pressure difference across the streamtube in the middle.
e) Determine the pressure gradient in the streamtube next to the lower wall.
Determine the pressure difference across the streamtube next to the lower wall.
f) Determine the overall force for this velocity profile.
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