What is the Ideal Formula for Calculating Flow Rate of a Fluid?

[perkhouse]

I am trying to calculate the flow rate of a fluid with relatively low compressibility and a known viscosity (in centipoise) through a specific sized orifice under an exact known pressure.

My engineering background is in electronics, and although I had courses in physics and I actually had a course in hydraulics 20-something years ago, the memories dealing with fluids are long gone.

If someone could give me a formula I could easily plug into a spreadsheet, that would be ideal!

 

[stewartcs]

Check out this link...

https://www.physicsforums.com/showthread.php?t=185803

CS

 

[perkhouse]

CS,
Thanks for the reply, but I haven't used calculus since I took the course back in the early 80's. Isn't there a simpler formula for approximating the flow using only the 3 variables: viscosity, pressure, and orifice diameter?

Thanks again!

edit: Sorry, I read the wiki page; didn't see your screenshot of the spreadsheet. Of course, it looks like you know the flow rate and are solving for orifice diameter. Can your spreadsheet be easily modified to go the other way?

 

[perkhouse]

Ok, I got it working, but I didn't know what to use for the pressure drop, so I just cut my maximum pressure by 50% to at least get an estimate.

Another question though... What function does the inlet flow play? It's not in any of the calculations and I would assume (like electrical current) the flow will be the same at the inlet and outlet.

 

[stewartcs]

Well there are two different types of flow rates: Volumetric flow rate and mass flow rate. The mass flow rate is just the volumetric flow rate times the density of the fluid. The volumetric flow rate (Q) is equal to the cross-sectional area of the pipe (ID) times the velocity of the fluid.

The volumetric flow rate at the inlet may not be the same as the volumetric flow rate at the end of the pipe due to various things like for example the diameter, line restrictions, or elevation changes in the pipe system.

As far as orifices go, they are usually placed in the pipe to restrict the flow rate at the outlet (down-stream end). The inlet flow rate is important because orifices can only flow a certain amount of fluid depending mainly on their diameter. That is to say that they have a specific flow coefficient. If the inlet flow rate is too high for the orifice to handle, then the outlet flow rate will be reduced to whatever the max flow rate of the orifice is. In gravity feed tank systems it's not that big of a deal, however, in a pumping system it could pose a serious problem. If the over-pressurization devices (relief valve or pressure switch) do not relief the extra pressure fast enough the pipe could rupture under high pressure. The pressure builds if the inlet flow rate is higher than the out flow rate (remembering that we are assuming the orifice flow rate is less than the inlet flow rate in a pumping system).

So in that sense the inlet flow rate is important (in determining the orifice size).

Also, the pressure drop across the orifice is normally measured. The pressure drop is the actual amount of pressure lost due to the increase in velocity through the orifice. The velocity is increased due to the diameter becoming smaller. With an ideal fluid and pipe, Bernoulli's principle (continuity equation) gives the relation. I think that was what the link to the wiki page was if I remember correctly. Basically it’s a conservation of energy principle problem.

 

[perkhouse]

Okay, well that does make it a little clearer for me. Thanks!

Basically, what I have is a low-pressure (max 7 psi) system, so not much danger of the pipes rupturing. I will be placing a solenoid valve in the supply line after the pump. The pump and the solenoid will be powered at the same time. The orifice (aka valve) has an internal diameter approximately equal to 40% of the internal diameter of the pipe and I wanted to know what type of hit on the flow rate I might expect. From my nearest estimation (using your formulas), the minimum flow rate after the "orifice" is still about 10 times the maximum needed flow rate for the system, so I don't think I'll have any problems whatsoever, but I just wanted to be reasonably sure before I install it. After I install it, I'll check the pressure before and after the orifice just to see how the real compares with my estimate of 50% (3.5 psi) drop. If the drop is less than 3.5 psi, I won't worry about it, but if higher then I'll run the calculations again to make sure I still have a pretty good margin of safety.

Thanks for all your help!

 

[stewartcs]

Sounds like a plan.

Also, I inadvertently wrote “volumetric” flow rate instead of “mass” flow rate which is misleading.

The mass flow rate at the inlet and outlet may be different depending on the velocity of the fluid since it is a real fluid and not an ideal one. This is due to density changes at high velocity. It results in a choked flow condition.

The total volume of fluid that flows into the pipe (in a closed system) will be the same amount that flows out of the pipe, eventually. My intention was to show that (in an ideal system, i.e. perfectly incompressible fluid with 0 viscosity) the pressure or velocity of the fluid will change due to the various reason I had previously listed, and that the rate of flow in can be different than what flows out under certain circumstances (orifice flow for example). In the case of an orifice, you can have a restricted flow which limits your outlet flow rate to what ever the orifice can flow. That of course led to the pumping system analogy and burst pipes.

Anyway, just wanted to try and clear that up so it wasn't misleading.