Calculate Flow Rate of a Syringe: Poiseuille's & Bernoulli's Laws

[zoom1]

I am trying to come up with a mathematical model so that, when the displacement of the plunger of a syringe is know, I can calculate the amount of a specific liquid in the barrel. Or the relationship between the speed of the plunger and flow rate at the tip of the needle (Again assuming that the properties of the liquid in the barrel is known).

I have very limited knowledge in fluid dynamics.

I made a quick research and found the Poiseuille's law for laminar flow

Flowrate= π.r⁴ (P-P₀) / 8.η.L

where r is the radius of the pipe or tube, P₀ is the fluid pressure at one end of the pipe, P is the fluid pressure at the other end of the pipe, η is the fluid's viscosity, and L is the length of the pipe or tube

P₀ will be atmospheric pressure if I assume I am injecting in air.
P will be the pressure on the syringe plunger
r is the radius of the barrel
η is the dynamic viscosity of the fluid
L is the length of the fluid inside the barrel.

The flow rate unit is m³/s when I use the SI units.

Also I have the Bernoulli's law to calculate the pressure difference between the syringe barrel, hub and needle, when their cross-section area is given.

So, I will be converting the Pressure at the tip of the needle (which is shown as pressure P2 on the illustration) to pressure at the barrel of the syringe to use for the parameter P₀. Am I correct at this point?

Also, therefore, the length I will be using for the parameter L is the dashed red line, which is the distance between the plunger and end of the barrel. This is my assumption, since I already convert the pressure at the tip of the needle to barrel of the syringe. Is this correct?

If I am all correct up until this point, my last question is the following;

I am assuming that I can exert enough pressure for the desired flow rate. How can I convert the P parameter into speed so that I can calculate the speed of the plunger for the desired flow rate?

 

[boneh3ad]

You seem to be dramatically overcomplicating this. If you know how fast the plunger is moving and the dimensions of the syringe, then you know the volume flow rate (and therefore mass flow rate for a liquid) by default. Since mass is conserved, that means the mass flow rate coming out of the needle has to be the same.

 

[zoom1]

You seem to be dramatically overcomplicating this. If you know how fast the plunger is moving and the dimensions of the syringe, then you know the volume flow rate (and therefore mass flow rate for a liquid) by default. Since mass is conserved, that means the mass flow rate coming out of the needle has to be the same.

So, what you are suggesting is considering the height of the liquid in the barrel and the radius of the plunger along with the length of the needle with its radius. I am ignoring the hub, since its the intermediate part.

Volume_barrel = Volume_needle + Volume_output

r_barrel² * dh_barrel/dt = r_needle² * dh_needle/dt

This is what I came up with.

 

[boneh3ad]

Not exactly. In the syringe, as you push the plunger, you are reducing the interior volume of the syringe (and therefore the mass of liquid that can fit in there) at a certain rate. If you already know how fast the plunger is moving, then you know how fast the volume/mass in the syringe is reducing. All of that reduction has to go somewhere, and that somewhere is through the needle. So in other words, you know the mass flow rate of the whole system already. You would just need the dimensions of the needle to work out average velocity and other similar values you might care about. You could use that value also to determine the pressure gradient using Poiseuille's law that you cited above, and from there you could get the actual velocity profile in the needle if you felt so inclined.

In essence, if you know that the diameter of the barrel is $d_{\mathrm{b}}$, the position of your plunger is $x$, where $x=0$ corresponds to the plunger being fully depressed, and the rate at which your plunger is moving, $\dot{x}$, then you know that the volume of your syringe, $V_{\mathrm{b}}$, is changing as $\dot{V}_{\mathrm{b}} = \frac{\pi}{4}d_{\mathrm{b}}^2\dot{x}$. You could also change that into mass flow rate, $\dot{m} = \rho \dot{V}_{\mathrm{b}}$, if you know the density. That's all you need to know. That volume/mass of liquid leaving the inside of the syringe at a known rate all has to be moving somewhere, and that somewhere is through the needle.

 

[zoom1]

Not exactly. In the syringe, as you push the plunger, you are reducing the interior volume of the syringe (and therefore the mass of liquid that can fit in there) at a certain rate. If you already know how fast the plunger is moving, then you know how fast the volume/mass in the syringe is reducing. All of that reduction has to go somewhere, and that somewhere is through the needle. So in other words, you know the mass flow rate of the whole system already. You would just need the dimensions of the needle to work out average velocity and other similar values you might care about. You could use that value also to determine the pressure gradient using Poiseuille's law that you cited above, and from there you could get the actual velocity profile in the needle if you felt so inclined.

In essence, if you know that the diameter of the barrel is $d_{\mathrm{b}}$, the position of your plunger is $x$, where $x=0$ corresponds to the plunger being fully depressed, and the rate at which your plunger is moving, $\dot{x}$, then you know that the volume of your syringe, $V_{\mathrm{b}}$, is changing as $\dot{V}_{\mathrm{b}} = \frac{\pi}{4}d_{\mathrm{b}}^2\dot{x}$. You could also change that into mass flow rate, $\dot{m} = \rho \dot{V}_{\mathrm{b}}$, if you know the density. That's all you need to know. That volume/mass of liquid leaving the inside of the syringe at a known rate all has to be moving somewhere, and that somewhere is through the needle.

Given
$\dot{V}_{\mathrm{b}} = \frac{\pi}{4}d_{\mathrm{b}}^2\dot{x}$

Then
$\dot{x}_{\mathrm{n}} = \frac{4\dot{V}_{\mathrm{b}}}{\pi d_{\mathrm{n}}^2}$

where subscript $n$ stands for needle.

Asking just to make it sure.

Thanks for the help, this is quite straightforward compared to complex equations I was mentioned before. However, I am still curious, what is the difference between this conversion of mass approach and Poiseuille's law? With Poiseuille's law I can get the required pressure on the plunger for a given flowrate. Is this the only difference or can I get more than this with Poiseuille's law.

Also, how would the actual velocity profile help me more?

 

[Chestermiller]

Yes, Poiseulle's law let's you calculate the pressure on the plunger. The actual velocity profile would not be of more help to you.