Question

Average blood velocity is 11cm/s in the human ascending aorta. The average diameter of the aorta is 2.1cm.

a) How much blood passes through the aorta each day in gallons?

b) How does this compare to the total quantity of blood you have in your body?

c) If 45% of the aorta becomes clogged, what is the new blood velocity?

Answer 1

A) Ve= 11cm/s (velocity)

d= 2.1cm (diameter) or r=d/2=1.05cm (radius)

Flow rate, Q= Ve * A (where, A= cross sectional area of Aorta)

To calculate the cross sectional area of aorta, we assume it to be a cylinder and the area to be calculated as circular, so we apply the formula to calculate area of circle, i.e.,

A= πr_{^{​​​​2}}

A= 3.14 * 1.05 * 1.05 = 3.46 cm^{​​​​​2} = 3.5 cm^{​​​​​2}

Now we cam easily calculate the flow rate, Q= Ve * A

Q= 11 * 3.5 = 38.5 cm^{​​​​​3} per sec

This means that 38.5 cm^{​​​​​3} blood is passing through the aorta at any given second.

And we can convert this unit of cm^{​​​​​3}/sec into gallon/day by multiplying the value to 22.82

And we get Q= 38.5 * 22.82 = 878.7 gallons per day.

This implies, that in a day, the volume of blood flowing through the cross sectional area of aorta is 878.7 gallons.

B) It is interesting to note that it is quite large volume as compared to the total volume of blood, which is 1.2 to 1.5 gallons. This is because our heart beats 100,000 times each day, and as it pumps small portions of blood each time, it accounts for such a large volume flowing through its cross section. This volume, normally can be upto 2000 gallons per day.

C) Modifying the above formula of flow velocity,

Q=Ve*A

=Ve*πr^{​​​​​2}

Note that Q is directly proportional the square of the radius of aorta. So, if the radius is decreased by 45%, the flow rate will decrease, but most importantly, the velocity will increase. Because the velocity is inversely proportional to the radius.

Ve=Q/πr^{​​​​​2}

For example, if the radius of cross sectional of aorta is halved, the new velocity will be 4 times the previous. Here, in the question, the area is approximately decreased to half (since the aorta is clogged 45%), so the new velocity will be slightly less than 4 times.

Ve~1/(1/2*1/2) (if we assume Q to be constant)

Ve~2*2~4 times