(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 35 cm/s.
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?
cm/s
To calculate the mass flow rate (in grams per second) of blood in the aorta, you need to use the formula:
Mass flow rate = density × cross-sectional area × flow speed
(a) Mass flow rate in the aorta:
Given:
Density (ρ) = 1.0 g/cm³
Cross-sectional area (A) = 2.0 cm²
Flow speed (v) = 35 cm/s
Using the formula, substitute the values:
Mass flow rate = 1.0 g/cm³ × 2.0 cm² × 35 cm/s
First, convert the flow speed from cm/s to m/s since the density is in g/cm³:
Flow speed (v) = 35 cm/s = 0.35 m/s (1 cm = 0.01 m)
Now, calculate the mass flow rate:
Mass flow rate = 1.0 g/cm³ × 2.0 cm² × 0.35 m/s
Next, convert cm² to m²:
Cross-sectional area (A) = 2.0 cm² = 0.0002 m² (1 cm² = 0.0001 m²)
Mass flow rate = 1.0 g/cm³ × 0.0002 m² × 0.35 m/s
Multiply the values:
Mass flow rate = 0.00007 g/s or 7.0 × 10⁻⁵ g/s
Therefore, the mass flow rate in the aorta is 7.0 × 10⁻⁵ g/s.
(b) To find the flow speed in the capillaries, you can use the continuity equation, which states that the product of cross-sectional area and flow speed is constant in a closed system.
Continuity equation:
A₁v₁ = A₂v₂
Given:
A₁ = 2.0 cm² (cross-sectional area of the aorta)
v₁ = 35 cm/s (flow speed in the aorta)
A₂ = 3.0 × 10³ cm² (combined cross-sectional area of the capillaries)
Using the continuity equation, we can solve for v₂:
A₁v₁ = A₂v₂
Rearranging the equation:
v₂ = (A₁v₁) / A₂
Substituting the values:
v₂ = (2.0 cm² × 35 cm/s) / (3.0 × 10³ cm²)
Now, convert the values to consistent units:
v₂ = (2.0 cm² × 0.35 m/s) / (3.0 × 10³ cm²)
= (0.007 m²/s) / (30000 cm²)
Next, convert cm² to m²:
v₂ = (0.007 m²/s) / (30000 × 10⁻⁴ m²)
= (0.007 m²/s) / (3 m²/s)
Dividing the values:
v₂ = 0.00233 m/s
Therefore, the flow speed in the capillaries is approximately 0.00233 m/s or 0.233 cm/s.