10.8 Flow rate and its relation to velocity  (Page 3/6)

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${n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\text{,}$

where ${n}_{1}$ and ${n}_{2}$ are the number of branches in each of the sections along the tube.

Calculating flow speed and vessel diameter: branching in the cardiovascular system

The aorta is the principal blood vessel through which blood leaves the heart in order to circulate around the body. (a) Calculate the average speed of the blood in the aorta if the flow rate is 5.0 L/min. The aorta has a radius of 10 mm. (b) Blood also flows through smaller blood vessels known as capillaries. When the rate of blood flow in the aorta is 5.0 L/min, the speed of blood in the capillaries is about 0.33 mm/s. Given that the average diameter of a capillary is $8.0\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ , calculate the number of capillaries in the blood circulatory system.

Strategy

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.

Solution for (a)

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

$\overline{v}=\frac{\left(5.0\phantom{\rule{0.25em}{0ex}}\text{L/min}\right)\left({\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/L}\right)\left(1\phantom{\rule{0.25em}{0ex}}\text{min/}\text{60}\phantom{\rule{0.25em}{0ex}}\text{s}\right)}{\pi {\left(0\text{.}\text{010 m}\right)}^{2}}=0\text{.}\text{27}\phantom{\rule{0.25em}{0ex}}\text{m/s}.$

Solution for (b)

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

${n}_{2}=\frac{\left(1\right)\left(\pi \right){\left(\text{10}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(0.27 m/s\right)}{\left(\pi \right){\left(4.0×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(0.33×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)}=5.0×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{capillaries}.$

Discussion

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 $\text{200}×{\text{10}}^{6}$ per 1 kg of muscle. For 20 kg of muscle, this amounts to about $4×{\text{10}}^{9}$ capillaries.

Section summary

• Flow rate $Q$ is defined to be the volume $V$ flowing past a point in time $t$ , or $Q=\frac{V}{t}$ where $V$ is volume and $t$ is time.
• The SI unit of volume is ${\text{m}}^{3}$ .
• Another common unit is the liter (L), which is ${\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$ .
• Flow rate and velocity are related by $Q=A\overline{v}$ where $A$ is the cross-sectional area of the flow and $\overline{v}$ is its average velocity.
• For incompressible fluids, flow rate at various points is constant. That is,
$\left(\begin{array}{c}{Q}_{1}={Q}_{2}\\ {A}_{1}{\overline{v}}_{1}={A}_{2}{\overline{v}}_{2}\\ {n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\end{array}}\text{.}$

Conceptual questions

What is the difference between flow rate and fluid velocity? How are they related?

Many figures in the text show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which it flows.)

Identify some substances that are incompressible and some that are not.

Problems&Exercises

What is the average flow rate in ${\text{cm}}^{3}\text{/s}$ of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

$\text{2.78}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$

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