Calculating flow rate of a syringe

I wanted to know how would I go about calculating the flow rate of a syringe with a metal tip that is dispensing water using a pressurized air.

I have had a look on internet about this and on this form and I though that laminar flow equation might be the solution for this. $$\text{Flowrate}=\frac{\pi r^4(P-P_0)}{8\eta l}$$

From this equation the variable I can obtain are as follows:

• $r = \text{radius of the metal tip}$
• $\eta = \text{viscosity of water}$
• $L = \text{length of the metal tip}$

However, I don't quite understand how can I get $P$ and $P_0$

Since I'm supplying a certain pressure from the top of the syringe, this could be $P$ and the pressure that comes out of the syringe metal tip could be $P_0$. If what I said is right, how would I calculate the pressure at the tip?

Thank You

I have a follow up question for this:

If I have $P$(compressed air) - $P_0$(atmospheric pressure) and the $P_0$ is higher than $P$, then the value will be negative. So in this case would I do $P_0$ - $P$? I'm asking this as my pressure will range from 0Bar to 5bar and when it's 0.04Bar, atmospheric pressure is higher than the compressed air. what do I do in this case?

• Assuming your volumetric flow rate for laminar flow is correct (you didn't provide a reference) then $P_0$ is simply atmospheric pressure and $P$ would be the pressure of the compressed air. Make sure you check the Reynolds Number though... – Gert Nov 26 '15 at 15:44
• the reference is physics.stackexchange.com/questions/22978/… - would you say it's fine – Satvir Singh Nov 26 '15 at 15:47
• Yes, the formula is indeed correct: it's a reworked Darcy-Weisbach equation. – Gert Nov 26 '15 at 15:49
• Just another thing, would the amount of liquid in the syringe left affect results. I mean for example if I have xL in syringe and I dispense at 5Bar and amount goes lower and then I dispense again at 5Bar. Would the be affected. – Satvir Singh Nov 26 '15 at 15:53
• No. Water is basically incompressible. – Gert Nov 26 '15 at 15:54