Engineer Analyzing Microsyringe Dynamics - Solutions for Velocity & Flow Rate

[cvoss_1228]

I am an engineer (MS) with no background in fluid dynamics. I am analyzing a microsyringe application for my company. Consider the following:

I have a tiny horizontal syringe (no gravity effect). Initially the plunger is fully pushed in, such that there is no air or liquid in the barrel, and liquid (water) fills the needle that is attached to the barrel. The needle has one end connected to water at atmospheric pressure. The needle is 2.5 mm long with a diameter of 0.25 mm. The barrel area is 5 mm^2. From this starting position the barrel is pulled out with a 40N constant force. There is 5N of friction opposing plunger motion. We can assume the plunger is zero mass.

The questions I need help answering are:
1. What is the solution for velocity of the plunger as a function of time?
2. What is the solution for flow rate through the needle as a function of time?
3. Is there a maximum flow rate and/or acceleration, and if so how is this calculated?

 

[Andy Resnick]

If you need an accurate answer, it's far easier to measure it.

However, for a ballpark estimate, you can try using Poiseuille flow: the pressure difference between the needle ends is (35/8) N/mm^2, and that will give you the volume flow, as well as the plunger velocity. You should at least calculate the Reynolds number to see if the flow is laminar (it may be in the needle, but it sure isn't as the plunger is filled).

 

[cvoss_1228]

...you can try using Poiseuille flow: the pressure difference between the needle ends is (35/8) N/mm^2,...

That pressure difference comes out to be 5000psi. Where did the 35/8 come from? Per your suggestion I did the Poiseuille calculation and got a pressure that varies linearly with volumetric flow rate. It comes out to be 3.8 psi (0.026N/mm^2) for 1 cm^3/s (1mL/s). Am I to conclude that the answer for maximum flow rate is that which produces pressure difference between the needle ends of 1atm ? That turns out to be about 4 cm^3/s. Would it be correct to assume that when the volumetric displacement rate of the plunger moving up the barrel exceeds the maximum flow rate of the needle, vacuum (or low pressure water vapor) will be created?

 

[ericgrau]

1. Complicated, but very brief anyway since the mass to be accelerated is so low. Soon you hit a constant flow; see #3. You should still verify that the system rapidly reaches the velocity in #3. Just use F=ma to find the initial acceleration.
2. Area x velocity, naturally.
3. Yes, when the pressure loss due to flow in the needle equals the pressure exerted on the plunger (force / area) then the flow rate and plunger velocity will be constant. You're already getting tips on figuring that out.

Also note that 5000 psi of suction is impossible, as this would make the absolute pressure inside the needle less than zero. The air above the water tank may only push the water with 14.7 psi plus the pressure from the height of the water. Instead, once the pressure drops in the syringe the water will cavitate: essentially, boil into vapor due to low pressure. Then the plunger will accelerate almost as fast as its mass allows; infinitely if it is massless. There would be a water vapor gap between the plunger and water decompressing and filling with more water vapor as needed. Conceptually, there is no longer a "connection" between the plunger and water tank.

So the actual pressure on the plunger may not exceed ~101 kPa (~14.7 psi) and probably a bit less than that to prevent cavitation. Or a little more than that if the water tank is deep, on the order of meters. 100 kPa x 5mm^2 = 0.5 N.

 

[Andy Resnick]

That pressure difference comes out to be 5000psi. Where did the 35/8 come from? Per your suggestion I did the Poiseuille calculation and got a pressure that varies linearly with volumetric flow rate. It comes out to be 3.8 psi (0.026N/mm^2) for 1 cm^3/s (1mL/s). Am I to conclude that the answer for maximum flow rate is that which produces pressure difference between the needle ends of 1atm ? That turns out to be about 4 cm^3/s. Would it be correct to assume that when the volumetric displacement rate of the plunger moving up the barrel exceeds the maximum flow rate of the needle, vacuum (or low pressure water vapor) will be created?

35/8 was a typo (sorry)= (40N - 5N)/5 mm^2. Your other statements are correct.

 

[cvoss_1228]

1. Complicated, but very brief anyway since the mass to be accelerated is so low. Soon you hit a constant flow; see #3. You should still verify that the system rapidly reaches the velocity in #3. Just use F=ma to find the initial acceleration.
2. Area x velocity, naturally.
3. Yes, when the pressure loss due to flow in the needle equals the pressure exerted on the plunger (force / area) then the flow rate and plunger velocity will be constant. You're already getting tips on figuring that out.

Also note that 5000 psi of suction is impossible, as this would make the absolute pressure inside the needle less than zero. The air above the water tank may only push the water with 14.7 psi plus the pressure from the height of the water. Instead, once the pressure drops in the syringe the water will cavitate: essentially, boil into vapor due to low pressure. Then the plunger will accelerate almost as fast as its mass allows; infinitely if it is massless. There would be a water vapor gap between the plunger and water decompressing and filling with more water vapor as needed. Conceptually, there is no longer a "connection" between the plunger and water tank.

So the actual pressure on the plunger may not exceed ~101 kPa (~14.7 psi) and probably a bit less than that to prevent cavitation. Or a little more than that if the water tank is deep, on the order of meters. 100 kPa x 5mm^2 = 0.5 N.

This is helpful. Since (40N-5N)>0.5N, I assume that the pressure will drop close to vacuum rather quickly, and gas (water vapor or other gas) will fill the part of the barrel volume that has not yet been occupied by water flowing in from the needle. I witnessed this exact effect on a regular syringe with a small needle. With a positive net force on the plunger, theoretically can I say that the plunger continues accelerating and never reaches constant velocity (Q#1)? For the flow rate through the needle, I conclude that the steady state flow rate is calculated through the Poiseuille eqn using a P=14.7psi, which comes out to 4mL/sec (Q#3), right? As for #2, the velocity of the flow in the needle as a function of time, how would that be computed? I am guessing that it is quite complicated given that there is a viscous force opposing motion that varies with velocity? What I am really interested in here is what is the "time constant" in getting up to the steady state flow rate through the needle.

 

[Danger]

I discovered the vacuum effect the hard way. Back in the good old days when people would give a 6-year-old things like syringes, explosives, etc. to play with, I tried drawing something up too fast. The plunger slipped out of my grasp, snapped back down, and shattered the barrel. I didn't get cut, but I was sufficiently taken aback that I was overly cautious with them until they started making them out of plastic rather than glass.

 

[cvoss_1228]

I discovered the vacuum effect the hard way. Back in the good old days when people would give a 6-year-old things like syringes, explosives, etc. to play with, I tried drawing something up too fast. The plunger slipped out of my grasp, snapped back down, and shattered the barrel. I didn't get cut, but I was sufficiently taken aback that I was overly cautious with them until they started making them out of plastic rather than glass.

Danger,
That's quite a story. I've read about the effect that you are talking about being due to huge releases of energy creating shock waves as a result of cavitation. Cavitation as I understand is the process of creation and destruction of bubbles due to low pressures. In your case, the shock waves may have shattered the syringe.

 

[Danger]

In your case, the shock waves may have shattered the syringe.

Had the circumstances been different, I would agree to that possibility. In actuality, though, it was the high-speed impact of a glass plunger into the bottom of a glass barrel, sort of like dropping a marble into a champagne flute.

 

[cvoss_1228]

What I am really interested in here is what is the "time constant" in getting up to the steady state flow rate through the needle.

I took a rough stab at this using simple F=ma neglecting viscosity / flow restriction. At a pressure difference of 14.7psi across the needle, the time to get to a fully restricted flow rate of 4mL/sec comes out to be 0.02 ms. This illustrates to me that the inertial mass of the fluid in the needle is really low compared to the force of the vacuum compared to atmospheric pressure, which is intuitive to me. My guess is that the 0.02ms shouldn't be more than an order of magnitude off when factoring in viscosity / flow restriction.