# Pressure loss in a syringe

I'm currently working on a problem which is really giving me some issues.

The problem concerns the force required to expel water from a syringe. We have a 20 ml syringe (which is $2\times10^{-5}$ meters cubed) with a diameter of 1 cm, full of water. The needle of the syringe is 40 mm in length and has a diameter of 0.2 mm. All of the water must be expelled from the syringe in 20 s. How much force must be applied to the syringe head to achieve this?

Ordinarily this is fine, but we have to include the pressure loss as a result of the friction in the needle. I'm using the Darcy–Weisbach equation to determine this. I calculated the speed the fluid needs to flow at by dividing the flow rate by the cross-sectional area of the needle. I've used a Moody chart to get $f_D$ as 0.046, and I'm using $\rho = 998.21$. I'm guessing the pressure loss in the needle is therefore $$0.046\times\frac{0.04}{0.002}\times\frac{998.21\times31.8^2}{2} = 4.64\,\mathrm{MPa}$$ Is that correct? In which case, how do I now get to the force from here?

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What is the relation between force and pressure? – Bernhard Mar 14 '13 at 6:35
I got a slightly different friction coefficient. But I used the fact that in case of a laminar flow the friction coefficient is equal to $64/Re$. And I also got quite a different speed (so no 31.8 m/s) how did you got this? And the rest of the syringe will also ad some drop in pressure, however a lot less than the needle (maybe check if this can be neglected?) and you could also add some transition factors due to the sudden transition from the wide diameter to the smaller one of the needle. – fibonatic Jun 4 '13 at 15:14
You really should add units to make it readable - it may even give you hints to the error. – Volker Siegel Apr 23 '14 at 19:51

$f=p\times a$
$f=p\times a + f_p$