An incompressible fluid is flowing through a pipe with a constriction. The pipe is on an incline with an angle of 30.0 degrees. The narrow section is 15.0 m from the wide section and the narrow section is lower than the wide section. The velocity of the fluid in the wide section of the pipe is 4.00 m/s and the velocity of the fluid in the narrow section of pipe is 9.00 m/s. The pressure of the fluid in the wide section is 250 kPa. What is the pressure in the narrow section of the pipe? (density of the fluid is 1,000 kg/m3)
Question
An incompressible fluid is flowing through a pipe with a constriction. The pipe is on an incline with an angle of 30.0 degrees. The narrow section is 15.0 m from the wide section and the narrow section is lower than the wide section. The velocity of the fluid in the wide section of the pipe is 4.00 m/s and the velocity of the fluid in the narrow section of pipe is 9.00 m/s. The pressure of the fluid in the wide section is 250 kPa. What is the pressure in the narrow section of the pipe? (density of the fluid is 1,000 kg/m3)
Solution
To solve this problem, we can use Bernoulli's equation, which states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant for an incompressible, non-viscous fluid in steady flow.
Bernoulli's equation is given by:
P1 + 1/2 * ρ * v1^2 + ρ * g * h1 = P2 + 1/2 * ρ * v2^2 + ρ * g * h2
where:
- P1 and P2 are the pressures at the wide and narrow sections of the pipe, respectively,
- v1 and v2 are the velocities of the fluid at the wide and narrow sections of the pipe, respectively,
- h1 and h2 are the heights of the wide and narrow sections of the pipe from a reference level, respectively,
- ρ is the density of the fluid,
- g is the acceleration due to gravity.
Given:
- P1 = 250 kPa = 250,000 Pa,
- v1 = 4.00 m/s,
- v2 = 9.00 m/s,
- h1 - h2 = 15.0 m * sin(30.0 degrees) = 7.5 m (since the narrow section is lower than the wide section),
- ρ = 1,000 kg/m^3,
- g = 9.81 m/s^2.
We can substitute these values into Bernoulli's equation to find P2:
P2 = P1 + 1/2 * ρ * v1^2 + ρ * g * h1 - 1/2 * ρ * v2^2 - ρ * g * h2 = 250,000 Pa + 1/2 * 1,000 kg/m^3 * (4.00 m/s)^2 + 1,000 kg/m^3 * 9.81 m/s^2 * 7.5 m - 1/2 * 1,000 kg/m^3 * (9.00 m/s)^2 - 1,000 kg/m^3 * 9.81 m/s^2 * 7.5 m = 250,000 Pa + 8,000 Pa + 73,575 Pa - 40,500 Pa - 73,575 Pa = 217,500 Pa or 217.5 kPa.
So, the pressure in the narrow section of the pipe is 217.5 kPa.
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