# What is the force needed to exert on the piston for a certain extrusion rate of a syringe tube?

I am trying to design a syringe pump for microfluidic extrusion uses. I have a 5cc syringe tube that I will use as the cartridge. We would be using it to extrude a variety of materials so the viscosity is uncertain. I want to know what's the best way to find out what the maximum linear force is needed from the piston to deliver a certain extrusion speed of certain materials.

I was thinking maybe doing an experiment with different materials in the cartridge would be the most straight forward but at this time no one is allowed back in the lab... Is there a good theoretical way to find a ballpark range of force that might be needed (so I can choose the right motor)? Maybe the Poiseuille Equation?

I am very new to this. Thanks for any advice!

• I'd start by trying to get a quantitative feel for the force required. Get some syringes from the nearest pharmacy, fill them with various fluids - maybe Golden Syrup (a viscous sugar syrup), etc., invert the syringe on a kitchen scales and press down on the body. Measure the force (N = kg x 9.81) and the time taken (from which you can calculate the rate). That may help you determine if you've got gross errors when you start to do the calculations. – Transistor May 25 '20 at 22:05
• F=ma to calculate force. Pressure in a fluid is different. – Rhodie May 31 '20 at 0:02
• The early sections of Faber (1995, Fluid dynamics for physicists, Cambridge University Press) use a syringe as an extended case study to survey the field of fluid mechanics. That's good news and bad news. It's good news in that there's an easy-to-get source where you can read up on the issue. It's bad news in that it indicates that almost any fluid-mechanical phenomenon could be going on in a syringe, so in order to get a straight answer to your question, you're going to have to specify a lot more detail about the geometry of the syringe, the fluid to be used, and the extrusion rate. – Daniel Hatton Oct 25 '20 at 22:48

I think you can use the Bernoulli equation $$p + 1/2\rho V^2 + \gamma z$$, which is constant along a streamline. Here $$p$$ is the pressure, $$\rho$$ the density of the fluid, $$V$$ velocity of the fluid, $$\gamma$$ the specific weight and $$z$$ the height/position.
Depending on how you place the syringe the problem will be different, lets assume you place it horizontally. Since you want to know the extrusion speed, it is the $$V$$ at the opening of the syring that is interesting. We set up the equation as follows: $$p_1 + 1/2\rho V_1^2 + \gamma z_1 = p_2 + 1/2\rho V_2^2 + \gamma z_2$$. Point 1 is inside the syringe and point 2 is just at the opening.
Since the syringe is horizontal, there is no height difference and we can place $$z_1 = z_2 = 0$$. Furthermore, the fluid inside the syringe, near the piston, is assumed to be at rest which gives $$V_1 = 0$$. This reduces the equation to: $$p_1 = p_2 + 1/2\rho V_2^2$$.
$$p_1$$ is known and can be calculated from: $$p_1 = F/A$$, where $$F$$ is the force and $$A$$ the area of the piston. $$\rho$$ is also known which is the density of the fluid. Now we can calculate $$V_2$$ as: $$V_2 = \sqrt{(p_1 - p_2)*2/\rho}$$.