A small artery has a length of 1.10×10−3m and a radius of 2.50×10−5m. If the pressure drop across the artery is 1.15 kPa, what is the flow rate through the artery? Assume that the temperature is 37 °C and the viscosity of whole blood is 2.084×10−3Pa·s.
To find the flow rate through the artery, we can apply the Poiseuille's Law equation, which relates the flow rate (Q) to the pressure drop (ΔP), viscosity (η), and the dimensions of the artery (length L and radius r):
Q = (ΔP * π * r^4) / (8 * η * L)
Given information:
Length of artery (L) = 1.10×10^-3 m
Radius of artery (r) = 2.50×10^-5 m
Pressure drop across the artery (ΔP) = 1.15 kPa = 1.15 × 10^3 Pa
Viscosity of whole blood (η) = 2.084×10^-3 Pa·s
Now, substitute these values into the equation:
Q = (1.15 × 10^3 Pa * π * (2.50×10^-5 m)^4) / (8 * 2.084×10^-3 Pa·s * 1.10×10^-3 m)
Let's calculate the flow rate:
Q ≈ 17.31 mm^3/s
Therefore, the flow rate through the artery is approximately 17.31 mm^3/s.
To determine the flow rate through the artery, you can use Poiseuille's law, which relates the flow rate to the pressure drop, radius, length, and viscosity of the fluid. Here's how you can calculate it step by step:
1. Convert the pressure drop to pascals (Pa):
1.15 kPa = 1.15 × 10^3 Pa
2. Calculate the flow rate using Poiseuille's law:
Flow rate (Q) = (π * ΔP * r^4) / (8 * η * L)
where:
- ΔP is the pressure drop (in Pa),
- r is the radius of the artery (in meters),
- η is the viscosity of the fluid (in Pa·s),
- L is the length of the artery (in meters), and
- π is a mathematical constant approximately equal to 3.14159.
Plugging in the given values:
Q = (π * 1.15 × 10^3 * (2.50 × 10^-5)^4) / (8 * 2.084 × 10^-3 * 1.10 × 10^-3)
3. Simplify the equation:
Q = (π * 1.15 × 10^3 * 2.50^4 * 10^-20) / (8 * 2.084 × 10^-9 * 1.10 × 10^-3)
4. Calculate the value:
Q ≈ 3.32 × 10^-8 m^3/s
So, the flow rate through the artery is approximately 3.32 × 10^-8 m^3/s.