Question

The human circulation system has approximately 1×109 capillary vessels. each vessel has a diameter of about 8 µm . assuming cardiac output is 5 l/min, determine the average velocity of blood flow through each capillary vessel.

Answer 1

It seems you mean 1x10^9 because there are a lot of capillaries indeed. The answer I got is v = 1.65788 * 10^9 meters / second

Given:
vessel diameter = D = 8um
r = D/2 = 4um
v = velocity of blood flow is equal to cardiac output / total capillary area

So, v = Q/A

Where:
Q = cardiac output = 5 liters / minute
A = total capillary area = pi(r^2) multiplied by the number of capillaries

Solve A = pi(r^2) x 10^9
A = pi(4um^2) x 10^9
= pi(16um^2) x 10^9 = 50265000000 um^2

Convert to meters. You will get 50265 m^2

Since velocity uses a meters/second unit format, you should convert Q to something more applicable.

Q = 5 Liters / minute
= 5 L/minute x 0.001 cubic meter/ 1L
= 0.005 m^3/minute

Combine them all and you will get:

v = Q/A = (0.005m^3 / min) / 50265 m^2 = 0.005 m^3 / (min)(50265 m^2)

Then convert minute to seconds, and cancel out m^2 with m^3.

v = 0.005m / 50265 minutes (1minute / 60 seconds) = 0.005m / [60(50265)] seconds

v = 0.00000000165788 meters / second
= 1.65788 * 10^9 meters / second