# 22.1 Flow rate and its relation to velocity  (Page 4/6)

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The 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 ${\text{m}}^{3}\text{/s}$ ?

Blood 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.

27 cm/s

Blood 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.

The Huka Falls on the Waikato River is one of New Zealand’s most visited natural tourist attractions (see [link] ). 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?

(a) 0.75 m/s

(b) 0.13 m/s

A major artery with a cross-sectional area of $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$ branches into 18 smaller arteries, each with an average cross-sectional 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?

(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 cross-sectional area of the venules is $\text{10}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$ , what is the total cross-sectional area of the capillaries feeding these venules? (b) How many capillaries are involved if their average diameter is $10.0\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ ?

(a) $40.0\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$

(b) $5\text{.}\text{09}×{\text{10}}^{7}$

The human circulation system has approximately $1×{\text{10}}^{9}$ 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.

(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?

(a) 22 h

(b) 0.016 s

The flow rate of blood through a $2\text{.}\text{00}×{\text{10}}^{\text{–6}}\text{-m}$ -radius capillary is $3\text{.}\text{80}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}\text{/s}$ . (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.)

(a) What is the fluid speed in a fire hose with a 9.00-cm 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?

(a) 12.6 m/s

(b) $0.0800\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}\text{/s}$

(c) No, independent of density.

The 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.

Water 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?

(a) 0.402 L/s

(b) 0.584 cm

Prove 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 larger-diameter region.)

Water emerges straight down from a faucet with a 1.80-cm 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.

(a) $\text{127}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{\text{3}}\text{/s}$

(b) 0.890 cm

Unreasonable 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?

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