Fluid Dynamics - Pressure at the end of a syringe

[Sean Devereux]

Hello there Physics Forum!

This is my first post here. This problem has been boggling me all day and I'm need of help in piecing together what I've come up with so far.

I'm currently working on a personal project related to paint spraying. In this project I'm using a syringe & needle to feed the paint through a spraying nozzle that will atomise the paint. However, to record the results I retrieve (diameter/reach of spray for different needle diameters and plunger speeds), I'd like to know at what pressure the paint is at when leaving the syringe.

The syringe will be depressed at a constant velocity of around 5mm/s or 0.005m/s.
The area of the syringe and needle are 0.00055m² and 2.01x10^-6m² respectively.

From this I've worked out the velocity at the end of the needle using:

v₂ = v₁ A₁ / A₂ = 1.372m/s

and a flow rate out of the end of the needle by simply multiplying velocity by area to get 9.93 L/h.

I have also used Bernoulli's to find the change in pressure between the syringe and the needle:

ΔP = ½ ρ ( v₁² - v₂² )
ΔP = 12253 Pa (assuming density of water for now 1060kg/m³)

However from here I'm a little stuck.

To utilise the pressure change I'd first need to know the pressure in the syringe? This would have to be worked out using the force which is presumably some function of the velocity of the plunger and the viscosity of the liquid inside. Also, will the pressure change as the syringe is emptied?

I'm not sure whether I'm over complicating or under complicating this problem. Please help!

Note: I realize that once the liquid leaves the needle it's pressure will technically be atmospheric. I assume I'm looking for velocity pressure or something to that effect?

Many thanks,
Sean.

 

[cjl]

I'm a bit confused what you're looking for. Your final statement is correct - the pressure will simply be ambient after exiting the needle. Also, I suspect the true pressure in the syringe will be more than 12.2kPa above the pressure at the exit, since I doubt viscosity is negligible. If all you care about is exit pressure though, that's kind of irrelevant.

 

[Sean Devereux]

I guess I'm just looking for the pressure in the syringe. I've got the idea of 'water pressure' leaving the syringe in my head, but I guess that's essentially the pressure of the water in the syringe which then translates to a flow rate and velocity of water out of the needle tip.

So my question is, what is the force the plunger applies at stated velocity onto the liquid in the syringe? From this I'll get my pressure.

 

[cjl]

The pressure in the syringe is probably easiest to figure out just using the force and area of the plunger, but that probably won't really translate to the flow rate and velocity out of the needle unless you account for viscous effects too.

 

[Chestermiller]

As cjl indicates, you need to consider the viscous flow in the needle, which will dominate over anything you calculate from Bernoulli for this system. The relationship between pressure drop and flow rate for viscous flow is described by the Hagen Poiseulle equation. Google this. To do the calculation, you are going to need to know the viscosity of the paint, and it is not going to be anywhere as low as that of water. Look up some typical values in the literature or online, and, better yet, measure the viscosity of the paint in a commercially available viscometer. You are going to need this.

 

[CWatters]

Is pressure the best thing to reference your measurements against? Why not the flow rate or velocity? They are much easier to measure/calculate.