How Many Capillaries? The aorta has an inside diameter of approximately 2.1 cm, compared to that of a capillary, which is about 1.0×10−5m (10 μm). In addition, the average speed of flow is approximately 1.0 m/s in the aorta and 1.0 cm/s in a capillary
Assuming that all the blood that flows through the aorta also flows through the capillaries, how many capillaries does the circulatory system have?
n=?
To determine the number of capillaries in the circulatory system, we can use the principle of conservation of mass. Since all the blood that flows through the aorta also flows through the capillaries, we can equate the mass flow rate in the aorta to the mass flow rate in the capillaries.
The mass flow rate, Q, is equal to the product of the density, ρ, and the cross-sectional area, A, multiplied by the velocity, v. Mathematically, it can be represented as:
Q = ρ * A * v
Since the density of blood and the velocity of flow are different in the aorta and the capillaries, we can equate the mass flow rates as follows:
Q_aorta = Q_capillaries
ρ_aorta * A_aorta * v_aorta = ρ_capillaries * A_capillaries * v_capillaries
Now we can solve for the number of capillaries, n:
n = (A_aorta * v_aorta) / (A_capillaries * v_capillaries)
Plugging in the given values:
A_aorta = pi * (r_aorta)^2 [Use the formula for the area of a circle]
r_aorta = (d_aorta / 2) = 2.1 cm / 2
A_capillaries = pi * (r_capillaries)^2 [Use the formula for the area of a circle]
r_capillaries = 1.0 × 10^(-5) m / 2
v_aorta = 1.0 m/s
v_capillaries = 1.0 cm/s = 0.01 m/s
Now, we can calculate the values and find the number of capillaries:
r_aorta = 2.1 cm / 2 = 1.05 cm = 0.0105 m
r_capillaries = 1.0 × 10^(-5) m / 2 = 5.0 × 10^(-6) m
n = (pi * (0.0105 m)^2 * 1.0 m/s) / (pi * (5.0 × 10^(-6) m)^2 * 0.01 m/s)
Simplifying the equation:
n = 0.0105^2 / (5.0 × 10^(-6))^2
n = 44100
Therefore, the circulatory system has approximately 44,100 capillaries.