(a) Calculate the mass flow rate (in grams per second) of blood (ρ = 1.0 g/cm3) in an aorta with a cross-sectional area of 2.0 cm2 if the flow speed is 33 cm/s.
Answer is in g/s
(b) Assume that the aorta branches to form a large number of capillaries with a combined cross-sectional area of 3.0 103 cm2. What is the flow speed in the capillaries?
Answer ins in cm/s
(a) To calculate the mass flow rate of blood in the aorta, we need to multiply the density of blood (ρ) by the cross-sectional area of the aorta (A) and the flow speed of blood (v).
The formula to calculate mass flow rate (ṁ) is:
ṁ = ρ * A * v
Substituting the given values:
ρ = 1.0 g/cm^3
A = 2.0 cm^2
v = 33 cm/s
ṁ = 1.0 g/cm^3 * 2.0 cm^2 * 33 cm/s
To simplify the units, we need to convert the cm^3 units to liters (since 1 liter = 1000 cm^3).
ṁ = 1.0 g/cm^3 * 2.0 cm^2 * 33 cm/s * (1 L/1000 cm^3)
Now, we can calculate the mass flow rate:
ṁ = 1.0 g/cm^3 * 2.0 cm^2 * 33 cm/s * (1 L/1000 cm^3)
ṁ = 0.066 g/s
Therefore, the mass flow rate of blood in the aorta is 0.066 g/s.
(b) To calculate the flow speed in the capillaries, we need to use the principle of continuity. The principle of continuity states that the total flow rate remains constant when the fluid flows through different cross-sectional areas. In this case, the total flow rate in the aorta is equal to the total flow rate in the capillaries.
The formula to calculate the flow speed in the capillaries (v_capillaries) is:
v_capillaries = (A_aorta * v_aorta) / A_capillaries
Substituting the given values:
A_aorta = 2.0 cm^2 (cross-sectional area of the aorta)
v_aorta = 33 cm/s (flow speed in the aorta)
A_capillaries = 3.0 * 10^3 cm^2 (combined cross-sectional area of the capillaries)
v_capillaries = (2.0 cm^2 * 33 cm/s) / (3.0 * 10^3 cm^2)
Now, we can calculate the flow speed in the capillaries:
v_capillaries = (2.0 cm^2 * 33 cm/s) / (3.0 * 10^3 cm^2)
v_capillaries ≈ 0.022 cm/s
Therefore, the flow speed in the capillaries is approximately 0.022 cm/s.