### Flow rate and pressure measurement through recoil

Posted:

**Wed Jun 03, 2009 10:55 pm**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.