Modified Bernoulli equation
 Hunt_and_Annoy
 Posts: 404
 Joined: Sat Feb 12, 2005 1:00 pm
Re: Modified Bernoulli equation
Curse all of you who know calculus. I don't get it until junior year/ next year. But I see we're bringing math into soaking. Ben leads us into the next generation!
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
For the CPS equation you don't need Calculus. For the air pressure equation you needed to use a lot more, but you could do it with Calc. 1 knowledge if you weren't too intimidated by the variables.
You might want to start looking up some of the basics of Calculus right now to get ahead. It's really useful in every science and I'm convinced many people could learn it earlier than it's taught. The basics aren't hard to understand.
You might want to start looking up some of the basics of Calculus right now to get ahead. It's really useful in every science and I'm convinced many people could learn it earlier than it's taught. The basics aren't hard to understand.
Last edited by SSCBen on Sun Mar 23, 2008 3:04 am, edited 1 time in total.
 Silence
 Posts: 3825
 Joined: Sun Apr 09, 2006 9:01 pm
Re: Modified Bernoulli equation
I'll back that up  you can understand the concept fairly easily. Knowing how to do it is a bit harder, of course. But even then, it isn't too bad  it just relies on you knowing 10 years' worth of math first.
Basically, given a function f(x), calculus lets you do two things:
 get a derivative, which is a function that tells the slope of f(x) at any xvalue. (the rate of change)
 get an integral, which is a function that finds the sum of all values of f(x) up to any xvalue. (the area between the curve and the xaxis)
So that lets you do a few things. For example, if you're given a force function F(t), you can relate it to an acceleration function a(t) using F=m*a. The integral of a(t) is v(t) (because v=a*t). The integral of v(t) is x(t), where x is the position/distance from the origin (because x=v*t).
Basically, given a function f(x), calculus lets you do two things:
 get a derivative, which is a function that tells the slope of f(x) at any xvalue. (the rate of change)
 get an integral, which is a function that finds the sum of all values of f(x) up to any xvalue. (the area between the curve and the xaxis)
So that lets you do a few things. For example, if you're given a force function F(t), you can relate it to an acceleration function a(t) using F=m*a. The integral of a(t) is v(t) (because v=a*t). The integral of v(t) is x(t), where x is the position/distance from the origin (because x=v*t).
 Hunt_and_Annoy
 Posts: 404
 Joined: Sat Feb 12, 2005 1:00 pm
Re: Modified Bernoulli equation
Cool xD I've actually though of selfteaching calc, but been too lazy.
I see there's still a character limit.
I see there's still a character limit.
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
I had forgotten about that limit. I kept decreasing it until people stopped complaining. The character limit is 25 characters. I could lower it, but I'd hope that people can manage 25 characters. If you have a problem with it let me know.
 Hunt_and_Annoy
 Posts: 404
 Joined: Sat Feb 12, 2005 1:00 pm
Re: Modified Bernoulli equation
I don't xD But when I accidentally hit enter it complained at me. I think a limit is good.
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
After doing some reading, I realized that the normal Bernoulli equation doesn't work correctly for air pressure. The normal Bernoulli equation only works for steady flow. Since the flow drops in air pressure, I have to use the unsteady Bernoulli equation, which is different. I'm going to use that, add some things for minor loss from different shaped nozzles, bends, and changes in pipe size, and then add some different pressure dependent efficiency part. That should be even better.
Once I have all this figured out and checked I'll make an article detailing exactly what all this means and make a spreadsheet. This should really help people planning water guns because it will let them know within a certain range how their water gun will perform.
Once I have all this figured out and checked I'll make an article detailing exactly what all this means and make a spreadsheet. This should really help people planning water guns because it will let them know within a certain range how their water gun will perform.
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
You can ignore most of the above. Steady flow refers to something other than what it sounds like initially: http://en.wikipedia.org/wiki/Steady_flow
The minor loss stuff still should be useful however. Here's a page I found that lists a bunch of the coefficients: http://www.engineeringtoolbox.com/minor ... d_626.html
These might make our calculations a lot more accurate. We'd have to account for bends, the valve, the the nozzle shape, and the nozzle exit (which appears to be a uniform 1.0 coefficient for some reason).
The minor loss stuff still should be useful however. Here's a page I found that lists a bunch of the coefficients: http://www.engineeringtoolbox.com/minor ... d_626.html
These might make our calculations a lot more accurate. We'd have to account for bends, the valve, the the nozzle shape, and the nozzle exit (which appears to be a uniform 1.0 coefficient for some reason).
 Silence
 Posts: 3825
 Joined: Sun Apr 09, 2006 9:01 pm
Re: Modified Bernoulli equation
The "constant velocity" seems to apply to the shortterm  on the scale of turbulence, not on the scale of dropoff. I guess you figured that out, but where does that leave us? Is what you did earlier correct?
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
I didn't read that part. I guess I will have to redo it again.
After doing some math, friction seems negligible for water guns unless you have one that's hundreds of feet long. So to keep things simple I'm not going to add that too the equation, but I'll keep looking at other things.
After doing some math, friction seems negligible for water guns unless you have one that's hundreds of feet long. So to keep things simple I'm not going to add that too the equation, but I'll keep looking at other things.

 Posts: 12
 Joined: Fri Mar 28, 2008 1:08 am
Re: Modified Bernoulli equation
While pipe friction may be small in a typical water gun, fitting losses in valves, elbows, sudden contractions and sudden enlargements will be substantial. Tabulated values for equivalencies for standard valves and fittings are probably included in your text. For example, a swingtype check valve produces 340 times as much friction as an equivalent diameter of pipe.
Contractions and expansions are accounted in the Bernoulli Equation friction term (script F in most texts). Friction for these terms is: F = KV^2/2g, where K is an impirical constant called the resistance coefficient, and is dependant on the ratio of the pipe diameters involved. For enlargements, K = [1(d1^2/d2^2)]^2 (sorry for the ugly equation). For contractions, K is derived from experimental data. It is probably charted in your text.
I hope this helps.
Contractions and expansions are accounted in the Bernoulli Equation friction term (script F in most texts). Friction for these terms is: F = KV^2/2g, where K is an impirical constant called the resistance coefficient, and is dependant on the ratio of the pipe diameters involved. For enlargements, K = [1(d1^2/d2^2)]^2 (sorry for the ugly equation). For contractions, K is derived from experimental data. It is probably charted in your text.
I hope this helps.
Last edited by Silence on Wed Apr 02, 2008 8:47 pm, edited 1 time in total.
 Silence
 Posts: 3825
 Joined: Sun Apr 09, 2006 9:01 pm
Re: Modified Bernoulli equation
I'm just curious, how would you combine friction values calculated for individual parts of the system? ie, given forces f1, f2, and f3 in different parts of the water gun, how would you combine them into the final equation? I haven't taken fluid dynamics as I'm still in high school, so I'm a little lost here. Fluid dynamics feels a lot different than kinetics and kinematics.
Don't worry about the ugly equations...plain text isn't a very elegant medium. I write out the equations on paper anyway.
(By the way, I added a space between your paragraphs since it's difficult to indent. The forum software removes extra whitespace.)
Don't worry about the ugly equations...plain text isn't a very elegant medium. I write out the equations on paper anyway.
(By the way, I added a space between your paragraphs since it's difficult to indent. The forum software removes extra whitespace.)
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
Thanks tim jones, but I already had mentioned that equation a few posts up with a link. I didn't know about the approximation for pipe increases and decreases though, so that definitely will help the spreadsheet. Is that for smooth (i.e. conical) enlargements or sudden though? That would make a difference.
I'm wondering if the slight increase in area from a pipe to a fitting has a noticeable affect on flow now. That's worth investigating.
@SilentGuy: These equations calculate pressure loss (or head losssame thing but with a conversion) based upon certain things. They're usually based on the velocity at the certain point, so for each different point, you put the velocity in terms of the volume (here it would be v*A=Q=dV/dt so v=(dV/dt)/A). You can add all the different ones then and subtract them from the current pressure (or in the case of my equation, the pressure as a function of starting pressure, etc.).
I'm wondering if the slight increase in area from a pipe to a fitting has a noticeable affect on flow now. That's worth investigating.
@SilentGuy: These equations calculate pressure loss (or head losssame thing but with a conversion) based upon certain things. They're usually based on the velocity at the certain point, so for each different point, you put the velocity in terms of the volume (here it would be v*A=Q=dV/dt so v=(dV/dt)/A). You can add all the different ones then and subtract them from the current pressure (or in the case of my equation, the pressure as a function of starting pressure, etc.).

 Posts: 12
 Joined: Fri Mar 28, 2008 1:08 am
Re: Modified Bernoulli equation
@SilentGuy: Ben's post addresses accumulating fitting friction loss contributions in the coefficient method (the method described in his post's link), and for contractions and expansions. I'll elaborate here on how the fitting equivalent length method that I referenced handles accumulating contributions.
The friction term is calculated as a function of pipe length (delta x), inside diameter (D), and a friction factor (f). The friction factor is itself a function of pipe roughness (e) and the Reynolds number (Re). For laminar flow regions, the friction factor may be calculated directly from f = 16/Re. For turbulant flow, f is a complex function, usually found on a chart. Fitting head losses are accumulated in the delta x variable, but only for pipes of equal diameter, roughness and turbulence. Areas of the flow with significant differences in these factors would require a separate friction term.
@Ben: Sorry, I missed your link on my first read through this thread. Since I'm new around here, I've been skimming most threads in an attempt to get up to speed. I wanted to specifically address sudden expansions (as found at the barb fitting on a CPS gun) and sudden contractions (as found at the pressure chamber of an APH gun), which were not directly addressed.
The friction term is calculated as a function of pipe length (delta x), inside diameter (D), and a friction factor (f). The friction factor is itself a function of pipe roughness (e) and the Reynolds number (Re). For laminar flow regions, the friction factor may be calculated directly from f = 16/Re. For turbulant flow, f is a complex function, usually found on a chart. Fitting head losses are accumulated in the delta x variable, but only for pipes of equal diameter, roughness and turbulence. Areas of the flow with significant differences in these factors would require a separate friction term.
@Ben: Sorry, I missed your link on my first read through this thread. Since I'm new around here, I've been skimming most threads in an attempt to get up to speed. I wanted to specifically address sudden expansions (as found at the barb fitting on a CPS gun) and sudden contractions (as found at the pressure chamber of an APH gun), which were not directly addressed.
 SSCBen
 Posts: 6449
 Joined: Sat Mar 22, 2003 1:00 pm
Re: Modified Bernoulli equation
Hey, you were trying to help, and that's all that matters. Don't hesitate to post something useful even if you think it might have been posted before.
Later today after I finish some homework I'm going to take a look at the unsteady Bernoulli equation with minor loss taken in to account. I'll then compare against the data I have for Supercannon II. At that point a flat efficiency constant will hopefully work.
Later today after I finish some homework I'm going to take a look at the unsteady Bernoulli equation with minor loss taken in to account. I'll then compare against the data I have for Supercannon II. At that point a flat efficiency constant will hopefully work.