Terminal velocity of fluid in pipe

[Brage Eidsvik]

Hello,

How do you calculate the terminal velocity of a fluid through a pipe.
My problem includes turbulent flow, very high reynold number, and water (incompresible).

 

[Chestermiller]

I assume you're not talking about the average velocity, and your focus in on determining the entrance length required for the flow to be fully developed. Correct?

 

[Brage Eidsvik]

In my problem I have a tube at an incline, empty of water. I then open a gate letting the water flow into the pipe. All design parameters like cross sections, length, incline are all known. I wish to know what is the velocity of the water when it is fully developed?

 

[Chestermiller]

In my problem I have a tube at an incline, empty of water. I then open a gate letting the water flow into the pipe. All design parameters like cross sections, length, incline are all known. I wish to know what is the velocity of the water when it is fully developed?

You want the steady state flow rate. What are your thoughts on this so far? What is your background in fluid mechanics?

 

[Brage Eidsvik]

Yes. I am guessing that the terminal velocity will be a product of gravity pulling the water down, and the walls and viscosity counteracting this. That will be gravity versus drag.
I am writing a bachelor thesis about designing a hydro power plant and have completed 1 bachelor level course in fluid dynamics. I am hoping for either a solution to the velocity, or an estimation I can use.

 

[Chestermiller]

Yes. I am guessing that the terminal velocity will be a product of gravity pulling the water down, and the walls and viscosity counteracting this. That will be gravity versus drag.
I am writing a bachelor thesis about designing a hydro power plant and have completed 1 bachelor level course in fluid dynamics. I am hoping for either a solution to the velocity, or an estimation I can use.

Physics Forums rules do not permit members to solve the problem for you. But we can help you by giving hints and asking leading questions.

In your course in fluid dynamics, I assume you learned about turbulent flow, Reynolds number, friction factor, Moody diagram. If the pipe were horizontal and you knew the flow rate, could you determine the pressure drop?

 

[russ_watters]

In my problem I have a tube at an incline, empty of water. I then open a gate letting the water flow into the pipe. All design parameters like cross sections, length, incline are all known. I wish to know what is the velocity of the water when it is fully developed?

This seems to me like an odd thing to try to calculate as pertains to a hydro power plant. Can you explain what the relevance is? Do you have a sketch of your design you can provide?

 

[Brage Eidsvik]

Physics Forums rules do not permit members to solve the problem for you. But we can help you by giving hints and asking leading questions.

In your course in fluid dynamics, I assume you learned about turbulent flow, Reynolds number, friction factor, Moody diagram. If the pipe were horizontal and you knew the flow rate, could you determine the pressure drop?

I have calculated major and minor losses in the pipe with an assumed flow rate. Used Reynolds number, friction factor, Moody diagram and geometric coefficients.

 

[Brage Eidsvik]

This seems to me like an odd thing to try to calculate as pertains to a hydro power plant. Can you explain what the relevance is? Do you have a sketch of your design you can provide?

I will try to provide a sketch when i can get a hold of a scanner. The relevance is 1) to determine the pressure loss in the pipe which is mostly dependent on the steady flow velocity in the pipe. 2) to choose the exit nozzle diameter to get the optimal exit velocity.

 

[Chestermiller]

I have calculated major and minor losses in the pipe with an assumed flow rate. Used Reynolds number, friction factor, Moody diagram and geometric coefficients.

So, it sound like you have it all worked out. What do you need our help for?

 

[Brage Eidsvik]

I can't assume the flow rate. What are the design factors that control the flow rate in a tube?

 

[Chestermiller]

Do you know the pressure drop? Do you know the elevation of one end of the tube above the other end? Is your problem that you know how to get the pressure drop if you are given the flow rate, but not the other way around? Or is your problem that one end of the tube is elevated, and you don't know how to include that? Of, something else?

 

[Brage Eidsvik]

My problem is that I do not know at what speed the flow will become steady. If i have an infinite water reservoir and a pipe empty of water. If I let water into the pipe and let water run through until it reaches a stable velocity. What is the velocity?

 

[russ_watters]

The relevance is 1) to determine the pressure loss in the pipe which is mostly dependent on the steady flow velocity in the pipe. 2) to choose the exit nozzle diameter to get the optimal exit velocity.

Ok, but I don't see how that relates to the concept of "terminal velocity". Pressure loss is something you want to minimize because any additional height beyond that which would provide terminal velocity is all lost energy. And when a flow is constricted by a nozzle, flow in the pipe isn't at terminal velocity. Terminal velocity is a concept that I don't think I've ever seen applied in a piping system.

Choosing the nozzle diameter would typically be based on the flow rate and pressure available. To that end:

If i have an infinite water reservoir and a pipe empty of water. If I let water into the pipe and let water run through until it reaches a stable velocity. What is the velocity?

I would start by choosing/calculating the velocity at the nozzle and work backwards to find a pipe diameter that minimizes the velocity and therefore the pressure loss. The velocity of a fluid in a pipe is generally chosen, not calculated.

 

[Chestermiller]

Ok, but I don't see how that relates to the concept of "terminal velocity". Pressure loss is something you want to minimize because any additional height beyond that which would provide terminal velocity is all lost energy. And when a flow is constricted by a nozzle, flow in the pipe isn't at terminal velocity. Terminal velocity is a concept that I don't think I've ever seen applied in a piping system.

Choosing the nozzle diameter would typically be based on the flow rate and pressure available.

I think what he means is "steady state velocity."

 

[russ_watters]

I think what he means is "steady state velocity."

I'm not so sure. This implies to me a flow rate limited by gravity vs friction:

My problem is that I do not know at what speed the flow will become steady. If i have an infinite water reservoir and a pipe empty of water. If I let water into the pipe and let water run through until it reaches a stable velocity. What is the velocity?

In any case, a gravity induced velocity is still a steady state velocity and regardless of what the determining factors are - if not gravity - I don't see what relevance of the question is. Generally, the answer to "what is the velocity in my pipe?" is "as low as you can get it without spending too much on oversized pipes". It isn't calculated on pure fluid dynamics considerations, it is calculated/chosen based on economics vs performance.

Not sure if I'm missing something or not...

[edit]
Reading up on the Hoover Dam, it seems that for the Pelton (kinetic energy) style turbines, they utilize pipes that gradually narrow for an ever-increasing velocity, which apparently is more efficient than a low velocity (low pipe friction loss) and a more rapid acceleration at the nozzle:

Before water from Lake Mead reaches the turbines, it enters the intake towers and then four gradually narrowing penstocks which funnel the water down towards the powerhouse. The intakes provide a maximum hydraulic head (water pressure) of 590 ft (180 m) as the water reaches a speed of about 85 mph (140 km/h).

https://en.wikipedia.org/wiki/Hoover_Dam#Power_plant_and_water_demands
Here's a link discussing hydro plant design considerations: https://www.ifc.org/wps/wcm/connect...c57143498e5/Hydropower_Report.pdf?MOD=AJPERES

Penstock diameter and velocity are designed within an optimization analysis in which head losses are considered against penstock costs, which are determined by material, length, and thickness.

A paper specifically about penstock design: http://article.sciencepublishinggroup.com/pdf/10.11648.j.ijepe.20150404.14.pdf

 

[Chestermiller]

I will try to provide a sketch when i can get a hold of a scanner. The relevance is 1) to determine the pressure loss in the pipe which is mostly dependent on the steady flow velocity in the pipe. 2) to choose the exit nozzle diameter to get the optimal exit velocity.

Still waiting for that sketch. Can you do one using Powerpoint, and then copy it using the Snipping Tool (Windows), and then upload it to this thread?

 

[Brage Eidsvik]

Sorry for taking so long to respond.

The hydroplant design considerations contain a lot of the information I was looking for, thank you. I haven't finished it yet, but so far it's good.
Where did you find it? Any suggestion of where I should start when looking for reports like these?

I uploaded a rough sketch of the problem as an attachment to this post. The absolute roughness is 0.09mm.

 

[Chestermiller]

Sorry for taking so long to respond.

The hydroplant design considerations contain a lot of the information I was looking for, thank you. I haven't finished it yet, but so far it's good.
Where did you find it? Any suggestion of where I should start when looking for reports like these?

I uploaded a rough sketch of the problem as an attachment to this post. The absolute roughness is 0.09mm.

Your figure explains it nicely. I'll get back with you a little later today.

Chet

 

[Chestermiller]

For the system you have shown, a problem like this would be solved using the macroscopic mechanical energy balance equation, which is an extension of the Bernoulli equation (that takes into account frictional energy losses in the straight pipe sections and in bends, junctions, nozzles, etc). The macroscopic mechanical energy balance equation is given by:

$$\Delta \left(\frac{1}{2}\rho v^2 + \rho g z + p\right)+\sum\left(\frac{1}{2}\rho v^2 f\frac{4L}{D}\right)+\sum\left(\frac{1}{2}\rho v^2 e_v\right)=0$$
where the first summation is carried out over all long straight sections of pipe, f is the Fanning friction factor in each pipe section (correlated as a function of Reynolds number in many sources), the second summation is carried out over all fittings, junctions, bends, etc, and $e_v$ is a friction loss factor for each kind of junction (tabulated in many sources, including Transport Phenomena by Bird, Stewart, and Lightfoot).

Are you familiar with the equation? If so, here is how to use it: Assume a volumetric flow rate for the water, substitute into the equation, and see how well the equation is satisfied. If the left side of the equation does not add up to zero, modify the volumetric flow rate, and try again. So we are dealing with an iterative procedure, until the left hand side of the equation adds up to zero. For the terms that involve the $\Delta$, you take the first location on the top of the reservoir and the second location at the discharge from the nozzle (so $\Delta p=0$).

I'll stop here and give you a chance to ask questions.

 

[Brage Eidsvik]

I am familiar with each part of the equation, but I have never solved it iteratively. I'll try to solve it and post the results. Thank you for the help.