# Engineer Analyzing Microsyringe Dynamics - Solutions for Velocity & Flow Rate

• cvoss_1228
In summary, a tiny horizontal syringe with a fully pushed in plunger and filled with water is pulled out with a constant force of 40N. There is 5N of friction opposing the motion. The flow rate and plunger velocity will eventually reach a constant value when the pressure loss due to flow in the needle equals the pressure exerted on the plunger. However, it is important to verify that the system rapidly reaches this velocity and to calculate the initial acceleration using F=ma. It is also important to note that 5000 psi of suction is impossible and the actual pressure on the plunger may not exceed ~101 kPa to prevent cavitation.
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?

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).

Andy Resnick said:
...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?

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.

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cvoss_1228 said:
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.

ericgrau said:
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.

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 said:
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.

cvoss_1228 said:
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 said:
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.

## 1. What is the purpose of analyzing microsyringe dynamics?

The purpose of analyzing microsyringe dynamics is to understand the behavior of the syringe and the flow of fluids through it. This can help in developing more efficient and accurate microsyringes for various applications, such as drug delivery and laboratory experiments.

## 2. What is velocity and why is it important in microsyringe dynamics analysis?

Velocity refers to the speed at which a fluid is flowing through the microsyringe. It is an important factor in understanding the performance of the syringe and can affect the accuracy and precision of fluid delivery. By analyzing velocity, engineers can make necessary adjustments to improve the functionality of the microsyringe.

## 3. How does flow rate impact the function of a microsyringe?

Flow rate is the volume of fluid passing through the microsyringe per unit time. It is a critical factor in determining the efficiency and accuracy of fluid delivery. A higher flow rate can result in faster delivery but may also lead to issues such as air bubbles and inconsistent flow. Analyzing flow rate can help engineers optimize the design and operation of microsyringes.

## 4. What are the common challenges in analyzing microsyringe dynamics?

Some common challenges in analyzing microsyringe dynamics include dealing with small and delicate components, ensuring accurate measurements of fluid volume and flow, and taking into account external factors such as temperature and air pressure. Additionally, the complex nature of fluid flow can make analysis difficult and require advanced techniques and equipment.

## 5. How can solutions for velocity and flow rate be applied in real-world settings?

The solutions for velocity and flow rate derived from microsyringe dynamics analysis can be applied in various real-world settings, such as medical and laboratory settings. They can help in designing and manufacturing more efficient and accurate microsyringes for drug delivery, fluid dispensing, and other applications. They can also aid in research and development of new medical treatments and technologies.

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