Blood takes about 1.55 s to pass through a 2.00 mm long capillary. If the diameter of the capillary is 5.00 μm and the pressure drop is 2.65 kPa, calculate the viscosity of blood. Assume laminar flow.
To calculate the viscosity of blood, we can use Poiseuille's law, which relates the flow rate of a fluid through a tube to the pressure drop and the properties of the tube.
According to Poiseuille's law, the flow rate (Q) through a cylindrical tube (such as a capillary) can be given by:
Q = (π * ΔP * r^4) / (8 * η * L)
Where:
- Q is the flow rate
- ΔP is the pressure drop across the tube
- r is the radius of the tube (half the diameter)
- η is the viscosity of the fluid
- L is the length of the tube
Given:
- ΔP = 2.65 kPa (convert to Pa by multiplying by 1000)
- r = 5.00 μm = 5.00 x 10^-6 m
- L = 2.00 mm = 2.00 x 10^-3 m
- Q = Volume / Time (Volume of blood = π * r^2 * L)
Step 1: Calculate the flow rate (Q)
Since we are given the time (1.55 s) it takes for the blood to pass through the capillary, we need to find the volume of blood passing through during that time. Using the volume formula for a cylinder, we get:
Volume = π * r^2 * L
Volume = π * (5.00 x 10^-6 m)^2 * (2.00 x 10^-3 m)
Now, let's calculate the flow rate using the formula for flow rate:
Q = Volume / Time
Step 2: Solve for η
Rearrange the equation to solve for η:
Q = (π * ΔP * r^4) / (8 * η * L)
η = (π * ΔP * r^4) / (8 * Q * L)
Substitute the given values into the equation and solve for η:
η = (π * (2.65 x 10^3 Pa) * (5.00 x 10^-6 m)^4) / (8 * Q * (2.00 x 10^-3 m))
Calculate the value of η.
To calculate the viscosity of blood, we can use Poiseuille's law, which relates the flow rate of a fluid through a cylindrical tube to the pressure drop across the tube. The formula is as follows:
Q = (pi * (r^4) * (P))/ (8 * η * L)
Where:
Q = Flow rate of the fluid
π = Pi, approximately 3.14159
r = Radius of the capillary (diameter / 2)
P = Pressure drop across the capillary
η = Viscosity of the fluid
L = Length of the capillary
We are given the following information:
- Length of the capillary, L = 2.00 mm = 0.002 m
- Diameter of the capillary, d = 5.00 μm = 5.00 x 10^-6 m
- Pressure drop across the capillary, P = 2.65 kPa = 2.65 x 10^3 Pa
- Time taken for blood to pass through the capillary, t = 1.55 s
First, let's convert the radius, diameter, and time to SI units:
- Radius, r = (diameter / 2) = (5.00 x 10^-6 m) / 2 = 2.50 x 10^-6 m
- Time, t = 1.55 s
Now, we can rearrange Poiseuille's equation to solve for the viscosity, η:
η = (pi * (r^4) * (P))/(8 * Q * L)
We need to find the flow rate, Q. The flow rate can be calculated by dividing the distance traveled by blood, L, by the time taken, t.
Q = L / t
Substituting the known values:
Q = (0.002 m) / (1.55 s) = 0.00129 m/s
Now, we can substitute the values for Q, P, r, and L into the equation for viscosity:
η = (pi * (2.50 x 10^-6 m)^4 * (2.65 x 10^3 Pa))/(8 * 0.00129 m/s * 0.002 m)
Evaluating the equation will give us the viscosity of blood.