122. Flow Rate and Its Relation to VelocityLearning Objectives
where $V$ is the volume and $t$ is the elapsed time. The SI unit for flow rate is ${\text{m}}^{3}\text{/s}$, but a number of other units for $Q$ are in common use. For example, the heart of a resting adult pumps blood at a rate of 5.00 liters per minute (L/min). Note that a Example 1: Calculating Volume from Flow Rate: The Heart Pumps a Lot of Blood in a LifetimeHow many cubic meters of blood does the heart pump in a 75year lifetime, assuming the average flow rate is 5.00 L/min? Strategy Time and flow rate $Q$ are given, and so the volume $V$ can be calculated from the definition of flow rate. Solution Solving $Q=V/t$ for volume gives $$V=\text{Qt}\text{.}$$Substituting known values yields $$\begin{array}{lll}V& =& \left(\frac{5\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{L}}{\text{1 min}}\right)(\text{75}\phantom{\rule{0.25em}{0ex}}\text{y})\left(\frac{1\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}}{{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{L}}\right)\left(5\text{.}\text{26}\times {\text{10}}^{5}\frac{\text{min}}{\text{y}}\right)\\ \text{}& =& 2\text{.}0\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{.}\end{array}$$Discussion This amount is about 200,000 tons of blood. For comparison, this value is equivalent to about 200 times the volume of water contained in a 6lane 50m lap pool. Flow rate and velocity are related, but quite different, physical quantities. To make the distinction clear, think about the flow rate of a river. The greater the velocity of the water, the greater the flow rate of the river. But flow rate also depends on the size of the river. A rapid mountain stream carries far less water than the Amazon River in Brazil, for example. The precise relationship between flow rate $Q$ and velocity $\overline{v}$ is $$Q=A\overline{v}\text{,}$$where $A$ is the crosssectional area and $\overline{v}$ is the average velocity. This equation seems logical enough. The relationship tells us that flow rate is directly proportional to both the magnitude of the average velocity (hereafter referred to as the speed) and the size of a river, pipe, or other conduit. The larger the conduit, the greater its crosssectional area. Figure 1 illustrates how this relationship is obtained. The shaded cylinder has a volume $$V=\text{Ad}\text{,}$$which flows past the point $\text{P}$ in a time $t$. Dividing both sides of this relationship by $t$ gives $$\frac{V}{t}=\frac{\text{Ad}}{t}\text{.}$$We note that $Q=V/t$ and the average speed is $\overline{v}=d/t$. Thus the equation becomes $Q=A\overline{v}$. Figure 2 shows an incompressible fluid flowing along a pipe of decreasing radius. Because the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. In this case, because the crosssectional area of the pipe decreases, the velocity must necessarily increase. This logic can be extended to say that the flow rate must be the same at all points along the pipe. In particular, for points 1 and 2, $$\begin{array}{c}{Q}_{1}={Q}_{2}\\ {A}_{1}{\overline{v}}_{1}={A}_{2}{\overline{v}}_{2}\end{array}\}\text{.}$$This is called the equation of continuity and is valid for any incompressible fluid. The consequences of the equation of continuity can be observed when water flows from a hose into a narrow spray nozzle: it emerges with a large speed—that is the purpose of the nozzle. Conversely, when a river empties into one end of a reservoir, the water slows considerably, perhaps picking up speed again when it leaves the other end of the reservoir. In other words, speed increases when crosssectional area decreases, and speed decreases when crosssectional area increases. Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, and so the equation must be applied with caution to gases if they are subjected to compression or expansion. Example 2: Calculating Fluid Speed: Speed Increases When a Tube NarrowsA nozzle with a radius of 0.250 cm is attached to a garden hose with a radius of 0.900 cm. The flow rate through hose and nozzle is 0.500 L/s. Calculate the speed of the water (a) in the hose and (b) in the nozzle. Strategy We can use the relationship between flow rate and speed to find both velocities. We will use the subscript 1 for the hose and 2 for the nozzle. Solution for (a) First, we solve $Q=A\overline{v}$ for ${v}_{1}$ and note that the crosssectional area is $A={\mathrm{\pi r}}^{2}$, yielding $${\overline{v}}_{1}=\frac{Q}{{A}_{1}}=\frac{Q}{{\mathrm{\pi r}}_{1}^{2}}\text{.}$$Substituting known values and making appropriate unit conversions yields $${\overline{v}}_{1}=\frac{(0\text{.}\text{500}\phantom{\rule{0.25em}{0ex}}\text{L/s})({\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}/\text{L})}{\pi (9\text{.}\text{00}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}{)}^{2}}=1\text{.}\text{96}\phantom{\rule{0.25em}{0ex}}\text{m/s}\text{.}$$Solution for (b) We could repeat this calculation to find the speed in the nozzle ${\overline{v}}_{2}$, but we will use the equation of continuity to give a somewhat different insight. Using the equation which states $${A}_{1}{\overline{v}}_{1}={A}_{2}{\overline{v}}_{2}\text{,}$$solving for ${\overline{v}}_{2}$ and substituting ${\mathrm{\pi r}}^{2}$ for the crosssectional area yields $${\overline{v}}_{2}=\frac{{A}_{1}}{{A}_{2}}{\overline{v}}_{1}=\frac{{\mathrm{\pi r}}_{1}^{2}}{{\mathrm{\pi r}}_{2}^{2}}{\overline{v}}_{1}=\frac{{r}_{{1}^{2}}}{{r}_{{2}^{2}}}{\overline{v}}_{1}\text{.}$$Substituting known values, $${\overline{v}}_{2}=\frac{(0\text{.}\text{900}\phantom{\rule{0.25em}{0ex}}\text{cm}{)}^{2}}{(0\text{.}\text{250}\phantom{\rule{0.25em}{0ex}}\text{cm}{)}^{2}}1\text{.}\text{96}\phantom{\rule{0.25em}{0ex}}\text{m/s}=\text{25}\text{.}\text{5 m/s}\text{.}$$Discussion A speed of 1.96 m/s is about right for water emerging from a nozzleless hose. The nozzle produces a considerably faster stream merely by constricting the flow to a narrower tube. The solution to the last part of the example shows that speed is inversely proportional to the square of the radius of the tube, making for large effects when radius varies. We can blow out a candle at quite a distance, for example, by pursing our lips, whereas blowing on a candle with our mouth wide open is quite ineffective. In many situations, including in the cardiovascular system, branching of the flow occurs. The blood is pumped from the heart into arteries that subdivide into smaller arteries (arterioles) which branch into very fine vessels called capillaries. In this situation, continuity of flow is maintained but it is the sum of the flow rates in each of the branches in any portion along the tube that is maintained. The equation of continuity in a more general form becomes $${n}_{1}{A}_{1}{\overline{v}}_{1}={n}_{2}{A}_{2}{\overline{v}}_{2}\text{,}$$where ${n}_{1}$ and ${n}_{2}$ are the number of branches in each of the sections along the tube. Example 3: Calculating Flow Speed and Vessel Diameter: Branching in the Cardiovascular SystemThe 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 $${n}_{2}=\frac{\left(1\right)\left(\pi \right){\left(\text{10}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(\mathrm{0.27\; m/s}\right)}{\left(\pi \right){\left(4.0\times {\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}\left(0.33\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)}=5.0\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{capillaries}.$$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 crosssectional 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. Section Summary
Conceptual QuestionsExercise 1What is the difference between flow rate and fluid velocity? How are they related? Exercise 2Many 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 crosssectional area through which it flows.) Exercise 3Identify some substances that are incompressible and some that are not. Problems & ExercisesExercise 1What 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? Show/Hide Solution Solution$\text{2.78}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$ Exercise 2The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to ${\text{cm}}^{3}\text{/s}$. (b) What is this rate in ${\text{m}}^{3}\text{/s}$? Exercise 3Blood is pumped from the heart at a rate of 5.0 L/min into the aorta (of radius 1.0 cm). Determine the speed of blood through the aorta. Show/Hide Solution Solution27 cm/s Exercise 4Blood is flowing through an artery of radius 2 mm at a rate of 40 cm/s. Determine the flow rate and the volume that passes through the artery in a period of 30 s. Exercise 5The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions (see Figure 3). On average the river has a flow rate of about 300,000 L/s. At the gorge, the river narrows to 20 m wide and averages 20 m deep. (a) What is the average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m? Show/Hide Solution Solution(a) 0.75 m/s (b) 0.13 m/s Exercise 6A major artery with a crosssectional area of $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$ branches into 18 smaller arteries, each with an average crosssectional area of $0\text{.}\text{400}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$. By what factor is the average velocity of the blood reduced when it passes into these branches? Exercise 7(a) As blood passes through the capillary bed in an organ, the capillaries join to form venules (small veins). If the blood speed increases by a factor of 4.00 and the total crosssectional area of the venules is $\text{10}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$, what is the total crosssectional area of the capillaries feeding these venules? (b) How many capillaries are involved if their average diameter is $10.0\phantom{\rule{0.25em}{0ex}}\mu \text{m}$? Show/Hide Solution Solution(a) $40.0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$ (b) $5\text{.}\text{09}\times {\text{10}}^{7}$ Exercise 8The human circulation system has approximately $1\times {\text{10}}^{9}$ capillary vessels. Each vessel has a diameter of about $8\phantom{\rule{0.25em}{0ex}}\mu \text{m}$. Assuming cardiac output is 5 L/min, determine the average velocity of blood flow through each capillary vessel. Exercise 9(a) Estimate the time it would take to fill a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take to fill if you could divert a moderate size river, flowing at $\text{5000}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/s}$, into it? Show/Hide Solution Solution(a) 22 h (b) 0.016 s Exercise 10The flow rate of blood through a $2\text{.}\text{00}\times {\text{10}}^{\text{\u20136}}\text{m}$ radius capillary is $3\text{.}\text{80}\times {\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$. (a) What is the speed of the blood flow? (This small speed allows time for diffusion of materials to and from the blood.) (b) Assuming all the blood in the body passes through capillaries, how many of them must there be to carry a total flow of $90\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$? (The large number obtained is an overestimate, but it is still reasonable.) Exercise 11(a) What is the fluid speed in a fire hose with a 9.00cm diameter carrying 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose? Show/Hide Solution Solution(a) 12.6 m/s (b) $0.0800\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/s}$ (c) No, independent of density. Exercise 12The main uptake air duct of a forced air gas heater is 0.300 m in diameter. What is the average speed of air in the duct if it carries a volume equal to that of the house’s interior every 15 min? The inside volume of the house is equivalent to a rectangular solid 13.0 m wide by 20.0 m long by 2.75 m high. Exercise 13Water is moving at a velocity of 2.00 m/s through a hose with an internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose’s nozzle is 15.0 m/s. What is the nozzle’s inside diameter? Show/Hide Solution Solution(a) 0.402 L/s (b) 0.584 cm Exercise 14Prove that the speed of an incompressible fluid through a constriction, such as in a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a largerdiameter region.) Exercise 15Water emerges straight down from a faucet with a 1.80cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is the flow rate in ${\text{cm}}^{3}\text{/s}$? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension. Show/Hide Solution Solution(a) $\text{127}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{\text{3}}\text{/s}$ (b) 0.890 cm Exercise 16Unreasonable Results A mountain stream is 10.0 m wide and averages 2.00 m in depth. During the spring runoff, the flow in the stream reaches $\text{100,000}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/s}$. (a) What is the average velocity of the stream under these conditions? (b) What is unreasonable about this velocity? (c) What is unreasonable or inconsistent about the premises?
