How Much Pressure Is Needed to Inject Water at 1.5g/s Through a Tiny Needle?

[SoccaCrazy24]

A patient is given an injection with a hypodermic needle 3.2 cm long and 0.25 mm in diameter. Assuming the solution being injected has the same density and viscosity as water at 20°C, find the pressure difference needed to inject the solution at the rate of 1.5 g/s.

ok I have the equation V = [(P1-P2)(pi*r^4)]/(8*n*L)
(P1-P2)= pressure difference
r= .000125 m
n= .0010055
L= .032 m
I do not know how I am suppose to transform 1.5g/s to m/s because I am unsure of how to transfer grams to meters... once i get that I can find the difference in pressure...

In a similar problem with diameter being .28mm the answer came out to be 320kPa... help?

 

[mezarashi]

Pressure energy is converted into kinetic energy.
To convert volume flow into velocity, you can consider:

$$\dot{V}= A\cdot v$$

 

[quicknote]

I would first like to commend you on your use of the correct equation for fluid flow rate. It seems like you have a good understanding of the concept.

To address your question, you are correct in assuming that the units of grams per second (g/s) need to be converted to meters per second (m/s) in order to use the equation. In this case, we can use the density of water (1000 kg/m^3) to convert the mass flow rate to volume flow rate.

So, 1.5 g/s = (1.5/1000) kg/s = (1.5/1000)/1000 m^3/s = 1.5/1000000 m^3/s = 1.5*10^-6 m^3/s

Now, we can plug this value into the equation along with the other given values to find the pressure difference:

V = [(P1-P2)(pi*r^4)]/(8*n*L)

1.5*10^-6 m^3/s = [(P1-P2)(pi*(0.000125)^4)]/(8*0.0010055*0.032)

Solving for (P1-P2), we get:

(P1-P2) = (1.5*10^-6*8*0.0010055*0.032)/(pi*(0.000125)^4)

(P1-P2) = 39.5 Pa

Therefore, the pressure difference needed to inject the solution at the rate of 1.5 g/s is 39.5 Pa.

In the similar problem you mentioned with a diameter of 0.28 mm, the answer of 320 kPa seems reasonable. It is important to note that the pressure difference is highly dependent on the diameter of the needle, which is why the difference is much larger in that case.

I hope this helps clarify the concept for you. Keep up the good work!