## Flow rate and pressure measurement through recoil

- SSCBen
**Posts:**6449**Joined:**Sat Mar 22, 2003 1:00 pm

### Flow rate and pressure measurement through recoil

A few days ago I had the idea to measure flow rate as a function of time by measuring the recoil as a function of time. This method is advantageous because no special modifications to the water gun will be necessary--a water gun can be used as is. Another advantage is that the flow will not be affected in any way by the measurement unlike some sort of propeller based system. The last advantage is that it's reasonably cheap and not difficult to make with some basic electronics.

After a little derivation (that I might post later) I found that if the water gun is fixed such that it is essentially immobile the volumetric flow rate is a function of the recoil force, the nozzle diameter, the density of the fluid, and the contraction coefficient. The contraction coefficient likely is new to most people here. The stream diameter is not the same as the nozzle orifice diameter in non-ideal cases. The contraction coefficient is the ratio of the stream area to the orifice area. Conical nozzles are near ideal so they have a contraction coefficient of 1. Sharp orifices (like drilled endcap nozzles) have a contraction coefficient of about 0.61.

Q(t) = (d / 2) * sqrt(pi * Fr(t) * Cc / rho)

where

Q is the volumetric flow rate

d is the nozzle orifice diameter

pi is pi

Fr is the recoil force and is a function of time

Cc is the contraction coefficient

rho is the density of the fluid

Interestingly, the gauge pressure of the fluid as a function of time can be found with an even simpler derivation. The force of a pressure differential is simply the pressure differential multiplied by the area. This force here is the recoil force and the area is the nozzle orifice area. Rearrange the equation and you get deltaP(t) = Fr(t) * 4 / pi * d^2. (Edit: This was corrected)

This method also allows for the shot duration to be measured with high precision. I intend to use a piezo electric element to measure the force and the element has a very fast response. In fact, I intend to use a piezo electric element in a homemade pressure transducer to measure pressure as a function of time in a pneumatic gun, where the time from the valve opening to the projectile leaving the barrel is on the order of milliseconds.

Cc can also be calculated if the total water shot over the duration of the shot is known. As that volume isn't difficult to measure because it's the difference between what was put in and what was left, finding Cc isn't too difficult either, though, an iterative method like a Newton-Raphson loop likely is necessary. 0.61's a good starting point for someone not interested in this.

So, in summary, a recoil force measurement allows you to measure flow rate and pressure as a function of time as well as shot duration.

I'll be experimenting with my pressure transducer circuits over the next few weeks and eventually I'll post a full circuit and setup procedure for this.

After a little derivation (that I might post later) I found that if the water gun is fixed such that it is essentially immobile the volumetric flow rate is a function of the recoil force, the nozzle diameter, the density of the fluid, and the contraction coefficient. The contraction coefficient likely is new to most people here. The stream diameter is not the same as the nozzle orifice diameter in non-ideal cases. The contraction coefficient is the ratio of the stream area to the orifice area. Conical nozzles are near ideal so they have a contraction coefficient of 1. Sharp orifices (like drilled endcap nozzles) have a contraction coefficient of about 0.61.

Q(t) = (d / 2) * sqrt(pi * Fr(t) * Cc / rho)

where

Q is the volumetric flow rate

d is the nozzle orifice diameter

pi is pi

Fr is the recoil force and is a function of time

Cc is the contraction coefficient

rho is the density of the fluid

Interestingly, the gauge pressure of the fluid as a function of time can be found with an even simpler derivation. The force of a pressure differential is simply the pressure differential multiplied by the area. This force here is the recoil force and the area is the nozzle orifice area. Rearrange the equation and you get deltaP(t) = Fr(t) * 4 / pi * d^2. (Edit: This was corrected)

This method also allows for the shot duration to be measured with high precision. I intend to use a piezo electric element to measure the force and the element has a very fast response. In fact, I intend to use a piezo electric element in a homemade pressure transducer to measure pressure as a function of time in a pneumatic gun, where the time from the valve opening to the projectile leaving the barrel is on the order of milliseconds.

Cc can also be calculated if the total water shot over the duration of the shot is known. As that volume isn't difficult to measure because it's the difference between what was put in and what was left, finding Cc isn't too difficult either, though, an iterative method like a Newton-Raphson loop likely is necessary. 0.61's a good starting point for someone not interested in this.

So, in summary, a recoil force measurement allows you to measure flow rate and pressure as a function of time as well as shot duration.

I'll be experimenting with my pressure transducer circuits over the next few weeks and eventually I'll post a full circuit and setup procedure for this.

Last edited by SSCBen on Thu Jun 04, 2009 8:28 pm, edited 1 time in total.

- Silence
**Posts:**3825**Joined:**Sun Apr 09, 2006 9:01 pm

### Re: Flow rate and pressure measurement through recoil

Sounds good.

I'm a little confused because you don't use the coefficient of contraction when calculating dP/dt. The CC itself makes sense.

Waiting to see how things go with the piezoelectric sensor. What'll you use for logging the results?

I'm a little confused because you don't use the coefficient of contraction when calculating dP/dt. The CC itself makes sense.

Waiting to see how things go with the piezoelectric sensor. What'll you use for logging the results?

- cantab
**Posts:**1492**Joined:**Fri Oct 19, 2007 1:35 pm

### Re: Flow rate and pressure measurement through recoil

It'll work, but it sounds rather complicated.

I work on Windows. My toolbox is Linux.

Arsenal:

Super Soaker: XP215, 2xXP220, Liquidator, Aquashock Secret Strike M(odded), Arctic Blast M, CPS1200, CPS2100, SC Power Pak, 3l aquapack, 1.5l aquapack

Water Warriors: Jet, Sting Ray M, Shark, Argon M, Tiger Shark, PulseMaster

Others: Waterbolt, The Blaster, Storm 500, Shield Blaster 2000, generic PR gun, generic backpack piston pumper (broken), 3l garden sprayer M, 10l water carrier:

Arsenal:

Super Soaker: XP215, 2xXP220, Liquidator, Aquashock Secret Strike M(odded), Arctic Blast M, CPS1200, CPS2100, SC Power Pak, 3l aquapack, 1.5l aquapack

Water Warriors: Jet, Sting Ray M, Shark, Argon M, Tiger Shark, PulseMaster

Others: Waterbolt, The Blaster, Storm 500, Shield Blaster 2000, generic PR gun, generic backpack piston pumper (broken), 3l garden sprayer M, 10l water carrier:

- SSCBen
**Posts:**6449**Joined:**Sat Mar 22, 2003 1:00 pm

### Re: Flow rate and pressure measurement through recoil

I have thought of a potential issue with this setup in that opening a ball valve would affect the measurement. This I don't consider to be too big of an issue because the opening time would be only a small fraction of the total time when the recoil would be low. If the recoil is high then the force from opening the valve likely is negligible.

This page has an image that might be helpful in understanding this: http://www.usbr.gov/pmts/hydraulics_lab ... 02_08.html

I'm going to use the microphone input of my computer. This assumes I can make a filter to at least approximately undo the high pass filter at the microphone input. If not then I'll look into making something that uses a USB or serial port.

Edit: It'll be a while before I make any pressure transducer or this. People at Spudfiles are having an interesting discussion of my planned pressure transducer circuit and how to best design it.

dP/dt? I don't calculate that anywhere. If you mean deltaP(t) (the pressure differential), the area is the area of the hole where the pressure differential is, not the stream area, so the Cc is not necessary.Silence wrote:Sounds good.

I'm a little confused because you don't use the coefficient of contraction when calculating dP/dt. The CC itself makes sense.

Waiting to see how things go with the piezoelectric sensor. What'll you use for logging the results?

This page has an image that might be helpful in understanding this: http://www.usbr.gov/pmts/hydraulics_lab ... 02_08.html

I'm going to use the microphone input of my computer. This assumes I can make a filter to at least approximately undo the high pass filter at the microphone input. If not then I'll look into making something that uses a USB or serial port.

It'll definitely be complicated, but once it's made and I establish a procedure to analyze the data, the results should be well worth the effort (or at least reasonably interesting enough to justify the effort).It'll work, but it sounds rather complicated.

Edit: It'll be a while before I make any pressure transducer or this. People at Spudfiles are having an interesting discussion of my planned pressure transducer circuit and how to best design it.

Last edited by SSCBen on Thu Jun 04, 2009 7:25 pm, edited 1 time in total.

- Silence
**Posts:**3825**Joined:**Sun Apr 09, 2006 9:01 pm

### Re: Flow rate and pressure measurement through recoil

Hmm, I guess I misunderstood the pressure part.

Overall I'm a little confused as to how you got the flow equation. In your version the area of the cross section (π r²) is found inside the square root...but in the page you linked to, it's found outside the square root. Similarly, you have the Cc inside the square root, when it seems like it ought to be outside it, or at least squared within.

The dimensions work for both, though.

Overall I'm a little confused as to how you got the flow equation. In your version the area of the cross section (π r²) is found inside the square root...but in the page you linked to, it's found outside the square root. Similarly, you have the Cc inside the square root, when it seems like it ought to be outside it, or at least squared within.

The dimensions work for both, though.

- Drenchenator
**Posts:**807**Joined:**Fri Jun 18, 2004 12:00 pm

### Re: Flow rate and pressure measurement through recoil

From what I know about the flow equations, the area of the cross section was in there as .25 pi d^2, and parts of this canceled out from the square root. Simple enough to understand.

The circuit actually is not complicated at all, at least after taking a circuit theory class (I was in the same one with Ben). Just never thought of doing with this with it though!

I really can't wait until Ben gets this done. It would be one of the more interesting things we've done. I've been interested in seeing how flow and pressure vary with time in a water gun for a long time, and never realized we could measure the two at a high rate.

The circuit actually is not complicated at all, at least after taking a circuit theory class (I was in the same one with Ben). Just never thought of doing with this with it though!

I really can't wait until Ben gets this done. It would be one of the more interesting things we've done. I've been interested in seeing how flow and pressure vary with time in a water gun for a long time, and never realized we could measure the two at a high rate.

The Drenchenator, also known as Lt. Col. Drench.

- Silence
**Posts:**3825**Joined:**Sun Apr 09, 2006 9:01 pm

### Re: Flow rate and pressure measurement through recoil

I just don't understand why the square root was applied to the area in the first place.

Here's a simplified version of Ben's equation: Q = sqrt(A * F / rho)

And from the page he linked to: Q = A * sqrt(2 * g * h)

I'm looking forward to these measurements, too. Technically, finding flow and pressure over time is easy if you know pressure as a function of the volume of water inside the water gun. With air pressure that can be found through Boyle's law, but with stretched rubber, we only have a few rules of thumb. Maybe P(V) can be found for CPS with these result.

Beyond that, the device may be more useful with Nerf, where pressure and flow aren't as uniform or predictable throughout the system.

Here's a simplified version of Ben's equation: Q = sqrt(A * F / rho)

And from the page he linked to: Q = A * sqrt(2 * g * h)

I'm looking forward to these measurements, too. Technically, finding flow and pressure over time is easy if you know pressure as a function of the volume of water inside the water gun. With air pressure that can be found through Boyle's law, but with stretched rubber, we only have a few rules of thumb. Maybe P(V) can be found for CPS with these result.

Beyond that, the device may be more useful with Nerf, where pressure and flow aren't as uniform or predictable throughout the system.

- SSCBen
**Posts:**6449**Joined:**Sat Mar 22, 2003 1:00 pm

### Re: Flow rate and pressure measurement through recoil

Hmm... I'm not completely sure about the accuracy of this derivation so I'll wait to learn some more about this. I did some research and on NASA's educational pages they used a different thrust equation that takes thrust as the combination of the pressure differential and momentum effects. My equation is derived from the momentum effects only as I had assumed the two would be equal based on something I read at Spudtech by D_Hall.

Either way, we'll have to wait a while while acceptable DIY computer instrumentation inputs are developed. Go figure.

Either way, we'll have to wait a while while acceptable DIY computer instrumentation inputs are developed. Go figure.