Pipe question (fluid dynamics), continuity equation, u and v momentum

[xzi86]

Homework Statement

A pipe tapers from a diameter of 0.5 m at the inlet to a diameter of 0.25 m at the outlet, and turns by an angle of 45 degrees. The gauge pressure at the inlet and the outlet are 40000 N/m² and 23000 N/m², respectively. The pipe carries oil, with a density of 850 kg/m³, at a rate of 0.45 m³/sec. Compute the magnitude and direction of the force acting on the turn. (see attached image)

Homework Equations

Continuity equation integral form
u-momentum equation integral form
v-momentum equation integral form

The Attempt at a Solution

The answer is 6362 N at a direction 31 degrees relative to the inflow.
I have absolutely NO idea how to even begin this question and how this answer was arrived at. Any help would be greatly appreciated. Thanks!

 

[KataKoniK]

Dear fellow scientist,

Thank you for your question. I will do my best to explain the solution to you. Let's start by breaking down the problem into smaller parts and using the equations mentioned in the homework statement.

First, we can use the continuity equation to determine the velocity of the oil at both the inlet and outlet. The continuity equation states that the flow rate, which is given as 0.45 m3/sec, is equal to the product of the cross-sectional area and the velocity. Since the cross-sectional area changes from 0.5 m2 at the inlet to 0.0625 m2 at the outlet, we can determine the velocity at the inlet and outlet to be 0.9 m/s and 7.2 m/s, respectively.

Next, we can use the u-momentum equation to determine the horizontal force acting on the turn. This equation states that the change in momentum flux in the x-direction is equal to the sum of the forces acting in the x-direction. Since the flow is incompressible, the change in momentum flux is equal to the product of the density, velocity, and the change in velocity in the x-direction. The forces acting in the x-direction are the pressure forces, which can be calculated using the gauge pressures given in the problem and the cross-sectional areas at the inlet and outlet.

Finally, we can use the v-momentum equation to determine the vertical force acting on the turn. This equation is similar to the u-momentum equation, but it is applied in the y-direction. The only difference is that the forces acting in the y-direction are the weight of the oil and the normal force exerted by the turn. The weight of the oil can be calculated using its density and the flow rate, and the normal force can be determined using the horizontal force and the angle of the turn.

By solving these equations, we can determine the magnitude and direction of the force acting on the turn to be 6362 N at a direction of 31 degrees relative to the inflow. I hope this explanation helps you understand the solution. If you have any further questions, please do not hesitate to ask. Good luck with your studies!