<< Chapter < Page Chapter >> Page >
n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 , size 12{n rSub { size 8{1} } A rSub { size 8{1} } {overline {v rSub { size 8{1} } }} =n rSub { size 8{2} } A rSub { size 8{2} } {overline {v rSub { size 8{2} } }} } {}

where n 1 size 12{n rSub { size 8{1} } } {} and n 2 size 12{n rSub { size 8{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 μ m , calculate the number of capillaries in the blood circulatory system.

Strategy

We can use Q = A v ¯ size 12{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 v ¯ size 12{Q=A {overline {v}} } {} or v ¯ = Q πr 2 size 12{ {overline {v}} = { {Q} over {πr rSup { size 8{2} } } } } {} for a cylindrical vessel.

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

v ¯ = 5.0 L/min 10 3 m 3 /L 1 min/ 60 s π 0 . 010 m 2 = 0 . 27 m/s . size 12{ { bar {v}}= { { left (5 "." 0`"L/min" right ) left ("10" rSup { size 8{ - 3} } `m rSup { size 8{3} } "/L" right ) left (1`"min/""60"`s right )} over {π left (0 "." "010 m" right ) rSup { size 8{2} } } } =0 "." "27"`"m/s"} {}

Solution for (b)

Using n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 1 size 12{n rSub { size 8{1} } A rSub { size 8{1} } {overline {v rSub { size 8{1} } }} =n rSub { size 8{2} } A rSub { size 8{2} } {overline {v rSub { size 8{2} } }} } {} , assigning the subscript 1 to the aorta and 2 to the capillaries, and solving for n 2 size 12{n rSub { size 8{2} } } {} (the number of capillaries) gives n 2 = n 1 A 1 v ¯ 1 A 2 v ¯ 2 . Converting all quantities to units of meters and seconds and substituting into the equation above gives

n 2 = 1 π 10 × 10 3 m 2 0.27 m/s π 4.0 × 10 6 m 2 0.33 × 10 3 m/s = 5.0 × 10 9 capillaries . size 12{n rSub { size 8{2} } = { { left (1 right ) left (π right ) left ("10" times "10" rSup { size 8{ - 3} } " m" right ) rSup { size 8{2} } left (0 "." "27"" m/s" right )} over { left (π right ) left (4 "." 0 times "10" rSup { size 8{ - 6} } " m" right ) rSup { size 8{2} } left (0 "." "33" times "10" rSup { size 8{ - 3} } " m/s" right )} } =5 "." 0 times "10" rSup { size 8{9} } " 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 mm 3 size 12{"mm" rSup { size 8{3} } } {} , or about 200 × 10 6 size 12{"200" times "10" rSup { size 8{6} } } {} per 1 kg of muscle. For 20 kg of muscle, this amounts to about 4 × 10 9 size 12{4 times "10" rSup { size 8{9} } } {} capillaries.

Section summary

  • Flow rate Q size 12{Q} {} is defined to be the volume V size 12{V} {} flowing past a point in time t size 12{t} {} , or Q = V t size 12{Q= { {V} over {t} } } {} where V size 12{V} {} is volume and t size 12{t} {} is time.
  • The SI unit of volume is m 3 size 12{m rSup { size 8{3} } } {} .
  • Another common unit is the liter (L), which is 10 3 m 3 size 12{"10" rSup { size 8{ - 3} } `m rSup { size 8{3} } } {} .
  • Flow rate and velocity are related by Q = A v ¯ size 12{Q=A {overline {v}} } {} where A size 12{A} {} is the cross-sectional area of the flow and v ¯ size 12{ {overline {v}} } {} is its average velocity.
  • For incompressible fluids, flow rate at various points is constant. That is,
    Q 1 = Q 2 A 1 v ¯ 1 = A 2 v ¯ 2 n 1 A 1 v ¯ 1 = n 2 A 2 v ¯ 2 . size 12{ left none matrix { Q rSub { size 8{1} } =Q rSub { size 8{2} } {} ##A rSub { size 8{1} } {overline {v}} rSub { size 8{1} } =A rSub { size 8{2} } {overline {v}} rSub { size 8{2} } {} ## n rSub { size 8{1} } A rSub { size 8{1} } {overline {v}} rSub { size 8{1} } =n rSub { size 8{2} } A rSub { size 8{2} } {overline {v}} rSub { size 8{2} }} right rbrace "." } {}

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 cm 3 /s size 12{"cm" rSup { size 8{3} } "/s"} {} of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?

2.78 cm 3 /s size 12{"cm" rSup { size 8{3} } "/s"} {}

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Physics 105: adventures in physics. OpenStax CNX. Dec 02, 2015 Download for free at http://legacy.cnx.org/content/col11916/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics 105: adventures in physics' conversation and receive update notifications?

Ask