A Syringe is filled with a solution of viscosity n=1.75x10^-3 pa s. The plunger of syringe has cross sectional area 9 x 10^-5 m^2. The needle has length 30mm and internal radius 3 x 10^-4 m. Intravenous pressure in patient is 2.0 kPa. What force must be applied to plunger of the syringe to inject 2ml solution intravenously in 3 seconds?
To find the force required to inject the solution intravenously, we can use the equations related to fluid mechanics. Let's break down the problem into steps:
Step 1: Find the flow rate of the solution
To find the flow rate, we can use Poiseuille's Law, which relates the flow rate (Q) through a cylindrical tube to the pressure difference (∆P) and the viscosity (η) of the fluid, as well as the dimensions of the tube. The formula for flow rate is as follows:
Q = (π * ∆P * r^4) / (8 * η * L)
where:
Q = flow rate of the solution (in m^3/s)
∆P = pressure difference across the needle (in Pa)
r = radius of the needle (in meters)
η = viscosity of the solution (in Pa s)
L = length of the needle (in meters)
We need to convert the given 2 ml of the solution to m^3:
2 ml = 2 x 10^-6 m^3
Next, we need to convert the length of the needle from mm to meters:
Length of needle = 30 mm = 30 x 10^-3 m
Now we can substitute the given values into the formula to find the flow rate of the solution.
Step 2: Find the force required to inject the solution
The force required to inject the solution can be calculated using the following formula:
Force = Pressure x Area
where:
Force = force required to inject the solution (in N)
Pressure = intravenous pressure in the patient (in Pa)
Area = cross-sectional area of the syringe's plunger (in m^2)
We need to convert the given 2 ml of the solution to m^3:
2 ml = 2 x 10^-6 m^3
Next, we substitute the given values into the formula to find the force required to inject the solution.
Now, let's calculate the values step by step:
Step 1: Calculate the flow rate of the solution
Given:
∆P = 2.0 kPa = 2.0 x 10^3 Pa
r = 3 x 10^-4 m
η = 1.75 x 10^-3 Pa s
L = 30 x 10^-3 m
Using the formula:
Q = (π * ∆P * r^4) / (8 * η * L)
Q = (π * (2.0 x 10^3 Pa) * (3 x 10^-4 m)^4) / (8 * (1.75 x 10^-3 Pa s) * (30 x 10^-3 m))
Calculate Q using the given values.
Step 2: Calculate the force required to inject the solution
Given:
Pressure = 2.0 kPa = 2.0 x 10^3 Pa
Area = 9 x 10^-5 m^2
Using the formula:
Force = Pressure * Area
Substitute the given values into the formula and calculate the force required.
Note: Make sure to convert all units to SI units before performing calculations to ensure consistency and accuracy.
I hope this helps! Let me know if you have any further questions.