Determining Water Velocity and Pressure Difference in a Pipe

What is the velocity at points 1 and 2 in a pipe where water flows at a rate of 0.10 m^3/s, with different diameters at each point, and point 2 being open to the atmosphere?

To determine the velocity at points 1 and 2 in the pipe, we can use the equation A1v1 = A2v2 = Q, where A is the cross-sectional area and v is the velocity. Given that the flow rate Q is 0.10 m^3/s, and the diameters at points 1 and 2 are 0.4 m and 0.20 m respectively, we can calculate the velocities.

Calculating Velocities:

At point 1:

A1v1 = A2v2

0.1 m^3/s = v1 * π(0.4 m)^2

v1 = 0.8 m/s

At point 2:

0.1 m^3/s = v2 * π(0.2 m)^2

v2 = 3.2 m/s

Therefore, the velocity at point 1 is 0.8 m/s and at point 2 is 3.2 m/s.

Calculating Pressure Difference:

Next, we can use Bernoulli's equation to determine the pressure difference between the ends of the pipe:

P1 - P2 = 0.5 * ρ(v2^2 - v1^2) + ρgh

Given that the density ρ is 1000 kg/m^3, gravitational acceleration g is 9.8 m/s^2, and the height difference h is 3.0 m, we can substitute the values to find the pressure difference.

P1 - P2 = 0.5 * 1000 kg/m^3 * ((3.2 m/s)^2 - (0.8 m/s)^2) + 1000 kg/m^3 * 9.8 m/s^2 * 3.0 m

P1 - P2 = 34,200 N/m^2

Therefore, the pressure difference between point 1 and point 2 in the pipe is 34,200 N/m^2.

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