Calculating Flow Speed and Vessel Diameter: Branching in the Cardiovascular System

We can use $Q=A\overline{v}$ to calculate the speed of flow in the aorta and then use the general form of the equation of continuity to calculate the number of capillaries as all of the other variables are known.

The flow rate is given by $Q=A\overline{v}$ or $\overline{v}=\frac{Q}{{\mathrm{\pi r}}^2}$ for a cylindrical vessel.

Substituting the known values (converted to units of meters and seconds) gives

Using $n_1{A}_1{\overline{v}}_1=n_2{A}_2{\overline{v}}_1$, assigning the subscript 1 to the aorta and 2 to the capillaries, and solving for $n_2$ (the number of capillaries) gives $n_2=\frac{n_1{A}_1{\overline{v}}_1}{A_2{\overline{v}}_2}$. Converting all quantities to units of meters and seconds and substituting into the equation above gives

Note that the speed of flow in the capillaries is considerably reduced relative to the speed in the aorta due to the significant increase in the total cross-sectional area at the capillaries. This low speed is to allow sufficient time for effective exchange to occur although it is equally important for the flow not to become stationary in order to avoid the possibility of clotting. Does this large number of capillaries in the body seem reasonable? In active muscle, one finds about 200 capillaries per ${\text{mm}}^3$, or about $\text{200}\times {\text{10}}^6$ per 1 kg of muscle. For 20 kg of muscle, this amounts to about $4\times {\text{10}}^9$ capillaries.