Blood is pumped from the heart at a rate of 5L/min into the aorta of radius 2.0cm. Assuming that the density of blood 1x103 kg/m3 .Determine the velocity of blood through
the aorta
Q=vA
Q=5L/min=8.3e-5 m^3/sec
A= pi*r^2=3.14*4*10^-4=0.001256 m2
v=Q/A
v=8.3e-5/0.001256 = 0.006m/sec = 0.6cm/sec
for rate 4.5 L/min and radius r = 1.1 cm
Given
rate of blood pumped from the heart is = 4.5 L/min
we know that 1 L = 1000 cm^3
the flow rate in cm^3/s , by heart is = 4.5*1000/60 cm^3/s = 75 cm^3/s -------------(H)
the area of aorta is A = pi*r^2 = pi*1.1^2 = 3.80132 cm^2
now the rate of flow is A*v ----------(A)
where A is area of aorta and v is speed of blood in aorta
equating (H) to (A) ( from equation of continuity A1*V1 = A2*V2)
75 = A*v
v = 75/A
v = 75/3.80132 cm/s
v = 19.7299885303 cm/s
To determine the velocity of blood through the aorta, we can use the principle of continuity. According to this principle, the flow rate of a fluid remains constant as it passes through different sections of a pipe.
The formula for flow rate is:
Flow Rate = Area × Velocity
In this case, the flow rate of blood is 5 L/min, which can be converted to m^3/s by dividing by 1000 (1 L = 0.001 m^3) and by 60 (1 min = 60 s):
Flow Rate = 5 L/min × (0.001 m^3/L) × (1 min/60 s) = 8.33 × 10^(-5) m^3/s
The area of the aorta can be calculated using the formula for the area of a circle:
Area = π × radius^2
Area = π × (0.02 m)^2 = 4 × 10^(-4) m^2
Now, we can rearrange the formula for flow rate to solve for velocity:
Velocity = Flow Rate / Area
Velocity = (8.33 × 10^(-5) m^3/s) / (4 × 10^(-4) m^2)
Simplifying the expression:
Velocity = 2.08 m/s
Therefore, the velocity of blood through the aorta is 2.08 m/s.
To determine the velocity of blood through the aorta, we can use the principle of conservation of mass.
The formula for the conservation of mass is:
A1v1 = A2v2
Where:
A1 and A2 are the cross-sectional areas of the aorta at two different points
v1 and v2 are the velocities of blood at those two points
We have the following information:
- The blood is pumped from the heart at a rate of 5 L/min, which is equivalent to 5/60 m3/s (since there are 60 seconds in a minute).
- The radius of the aorta is 2.0 cm, which is equivalent to 0.02 m.
- The density of blood is 1x103 kg/m3.
First, let's calculate the cross-sectional area of the aorta at the point where blood is pumped from the heart. We can use the formula for the area of a circle:
A1 = π * r1^2
= π * (0.02 m)^2
Next, let's calculate the velocity of blood at the point where it is pumped from the heart:
v1 = (5/60 m3/s) / A1
Now, we need to determine the cross-sectional area of the aorta. Since the aorta is a tube, its cross-sectional area is constant. Therefore, we can use the area calculated at the point where blood is pumped from the heart.
A2 = A1
Finally, we can calculate the velocity of blood through the aorta at any point:
v2 = (A1 * v1) / A2
Now, let's plug in the values and calculate the velocity:
A1 = π * (0.02 m)^2
v1 = (5/60 m3/s) / A1
A2 = A1
v2 = (A1 * v1) / A2
Once you calculate these values, you will have the velocity of blood through the aorta.