A hypodermic syringe is attached to a needle that has an internal radius of 0.303 mm and a length of 3.02 cm. The needle is filled with a solution of viscosity 2.00 10-3 Pa · s; it is injected into a vein at a gauge pressure of 16.1 mm Hg. Ignore the extra pressure required to accelerate the fluid from the syringe into the entrance of the needle.
(a) What must the pressure of the fluid in the syringe be in order to inject the solution at a rate of 0.251 mL/s?
To find the pressure of the fluid in the syringe, we need to use the relationship between pressure and fluid velocity as described by the Hagen-Poiseuille equation.
The Hagen-Poiseuille equation states that the flow rate through a needle is given by:
Q = (πΔP r^4) / (8ηL)
Where:
Q is the flow rate,
ΔP is the pressure difference across the needle,
r is the radius of the needle,
η is the viscosity of the fluid, and
L is the length of the needle.
In this case, we are given the flow rate Q as 0.251 mL/s, which is equal to 0.251 x 10^-6 m^3 /s.
We are also given the radius of the needle r as 0.303 mm, which is equal to 0.303 x 10^-3 m.
The length of the needle L is given as 3.02 cm, which is equal to 3.02 x 10^-2 m.
The viscosity of the fluid η is given as 2.00 x 10^-3 Pa · s.
We can rearrange the equation to solve for ΔP:
ΔP = (8ηQL) / (πr^4)
Substituting the given values into the equation, we get:
ΔP = (8 x 2.00 x 10^-3 x 0.251 x 10^-6 x 3.02 x 10^-2) / (π x (0.303 x 10^-3)^4)
Evaluating the expression, we find:
ΔP ≈ 4.23 x 10^5 Pa
Now, to find the pressure of the fluid in the syringe, we need to add the gauge pressure of 16.1 mm Hg:
P_syringe = ΔP + P_gauge
Converting the gauge pressure to pascals:
P_gauge = (16.1 x 133.322) Pa
Evaluating the expression, we find:
P_syringe ≈ (4.23 x 10^5) + (16.1 x 133.322) Pa
Therefore, the pressure of the fluid in the syringe must be approximately equal to the calculated value.