Calculating Water Flow Speed at Points A and B: Pipe Branching Equations

[redshift]

I've struggled with this since this morning with hardly any progress.
"There is a level pipe filled with water flowing through it. Branching off from this pipe are two pipes going perpendicularly upward, with a cross sectional area Sa, Sb, respectively, and there being a difference in water height of h. Find the water flow speed at points A and B."

Since no figures are given, I guess the answer is an equation. I also assume A and B are the points in the main pipe where the branches are.

Since the pressure should (i think) be equal in both branch pipes, even if their diameters are not the same, could A = [Sa(h)]/(Sb-Sa) and B = [Sb(h)]/(Sb-Sa)?

Regards

 

[modmans2ndcoming]

shouldn't the presser in the pipe that goes higher be less due to the need to overcome gravity for a longer period of time?

 

[M98Ranger]

Thank you for your question. Calculating water flow speed at points A and B in a branching pipe system can be a challenging task, but with the proper equations and understanding of fluid dynamics, it can be achieved. To start, we need to consider a few key factors that will affect the water flow speed at points A and B: the cross-sectional areas of each pipe (Sa and Sb), the difference in water height (h), and the pressure within the pipes.

First, we can use the Bernoulli's equation to determine the pressure at points A and B. This equation states that the total pressure at any point in a fluid system is equal to the sum of the static pressure, dynamic pressure, and hydrostatic pressure. In this case, the dynamic pressure (which is caused by the flow of water) will be the same at both points A and B, as they are at the same elevation in the main pipe. Therefore, we can simplify the equation to:

P + ρgh = constant

Where P is the static pressure, ρ is the density of water, g is the acceleration due to gravity, and h is the height of the water column.

Next, we can use the continuity equation, which states that the mass flow rate at any point in a system is constant. This means that the mass flow rate at point A must be equal to the mass flow rate at point B. We can express this as:

ρAv = ρBvB

Where A and B represent the cross-sectional areas at points A and B, and v and vB represent the water flow speed at those points.

Now, we can combine these equations to solve for the water flow speed at points A and B. First, we can rearrange the continuity equation to solve for vB:

vB = (Av)/(B)

Then, we can substitute this into the Bernoulli's equation:

P + ρgh = (ρAv)/(B)

Solving for v, we get:

v = √[(2gh)/(1-(A/B)^2)]

Therefore, the water flow speed at points A and B can be calculated using this equation. It is important to note that this equation assumes that the flow of water is steady and incompressible, and that there are no losses due to friction or other factors in the system.

I hope this explanation helps you understand the process for calculating water flow speed at points A and B in a branching pipe