Fluid mechanics conservation of momentum problem

[ehilge]

Homework Statement

The problem involves fluid flow through a 180 degree bend in a pipe. I am trying to find out the force applied on the flanges connecting the bend to straight pieces of pipe. I am given information on the diameter of the pipe, the length of the pipe, weight flow rate, and pressure at both the inlet and outlet of the 'U'

Homework Equations

The Reynolds transport theorem using linear momentum.

The Attempt at a Solution

I need help figuring out how to approach the problem. A control volume can be defined surrounding the 'U' and cutting through the pipe at the flanges. Since there is only one inlet, one outlet, and no change in volume, the mass flow rate in must be the same as out. The area of the pipe is the same on each side, so it follows that the velocity of the fluid must be the same on each side. But continuing with this logic, the force on each flange would be the same in equal and opposite directions using the Reynolds transport theorem. And I don't see how that can be plausible considering the type of information in the problem statement and it appears that some energy is lost in the pipe. So my question is, where is the flaw in my logic?

 

[RTW69]

Use the linear momentum equation to find the flange reaction force. Looks like everything is in the x-direction. Is P_inlet=P_outlet? Velocities and unit vectors are in the same so angle between them is 0.

 

[ehilge]

Use the linear momentum equation to find the flange reaction force. Looks like everything is in the x-direction. Is P_inlet=P_outlet? Velocities and unit vectors are in the same so angle between them is 0.

I'm sorry, I failed to specify that, P_in > P_out which I guess is really the kicker here because that would imply energy is lost in the pipe somewhere, but I'm not sure how to work that in.

 

[ehilge]

if anyone is curious, I got the problem resolved today. The mass flow rate and velocity are indeed the same, and gravity accounts for the difference in pressure. I didn't realize what the orientation of the pipe was.

 

[DeeNos]

Your approach is on the right track, but there are a few key things to consider in this problem. First, the conservation of momentum principle states that the total momentum of a system remains constant unless acted upon by an external force. In this case, the external force is the pressure difference between the inlet and outlet of the pipe.

To solve for the force applied on the flanges, you will need to use the Bernoulli's equation, which relates the pressure, velocity, and elevation of a fluid at different points in a pipe. This will allow you to calculate the pressure difference between the inlet and outlet, which will then be used to calculate the force on the flanges.

Additionally, it is important to consider the shape of the pipe bend. A 180 degree bend will introduce some turbulence and friction in the flow, causing some energy to be lost. This is known as head loss and can be calculated using the Darcy-Weisbach equation.

Overall, your approach using a control volume and the continuity equation is correct, but you will need to incorporate the Bernoulli's equation and account for head loss to solve for the force on the flanges. I would recommend reviewing these equations and their applications in fluid mechanics to fully understand the problem and its solution.