Question

Water flows horizontally in a pipe of diameter 7.7 cm at a rate of 0.429 m/s and a gauge pressure of 7.22x10³ pa. the fluid (still in the pipe) then descends a 4.65 m high hill, passes through a reducer and into a smaller 3.72 cm diameter pipe, and then flows horizontally again. determine the gauge pressure (in pa) in the lower section of pipe.

Answer 1

To determine the gauge pressure in the lower section of the pipe, calculations using Bernoulli's equation and the equations of fluid dynamics need to be performed. The gauge pressure in the lower section is approximately 7.15x10^3 Pa.

Step-by-step explanation:

To determine the gauge pressure in the lower section of the pipe, we need to consider the change in height and the change in diameter. Firstly, let's calculate the pressure at the top of the hill using the given information. The gauge pressure at the top of the hill can be calculated using the Bernoulli's equation: P1 + 1/2 * rho * V1^2 + rho * g * h1 = P2 + 1/2 * rho * V2^2 + rho * g * h2. Since the fluid is at rest at the top of the hill, V1 = 0.

Given that P1 = 7.22x10^3 Pa, h1 = 0 and h2 = 4.65 m, we can calculate P2. Next, we can calculate the gauge pressure in the lower section of the pipe using the change in diameter. The gauge pressure in the lower section can be calculated using the equation P2 = P3 + 1/2 * rho * V3^2, where P3 is the gauge pressure in the lower section and V3 is the velocity in the lower section.

Now, let's calculate P2. Rearranging the equation, we have P2 = P1 - rho * g * h2 = 7.22x10^3 - (1000 kg/m^3 * 9.8 m/s^2 * 4.65 m) = 7.22x10^3 - 45597 Pa.

To calculate the gauge pressure in the lower section, we need to find V3. In the horizontal pipes, the flow is steady, so the velocity remains constant. Therefore, V3 = V1 = 0.429 m/s. Plugging in the values, we have P2 = P3 + 1/2 * rho * V3^2. Rearranging the equation, we can solve for P3: P3 = P2 - 1/2 * rho * V3^2 = (7.22x10^3 - 45597 Pa) - (0.5 * 1000 kg/m^3 * (0.429 m/s)^2).

Calculating P3, we get P3 = 7.22x10^3 - 45597 Pa - 81 Pa. Therefore, the gauge pressure in the lower section of the pipe is approximately 7.15x10^3 Pa.