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Making connections: different-sized pipes

For incompressible fluids, the density of the fluid remains constant throughout, no matter the flow rate or the size of the opening through which the fluid flows. We say that, to ensure continuity of flow, the amount of fluid that flows past any point is constant. That amount can be measured by either volume or mass.

Flow rate has units of volume/time (m 3 /s or L/s). Mass flow rate ( Δ m Δ t ) has units of mass/time (kg/s) and can be calculated from the flow rate by using the density:

m =   ρ V

The average mass flow rate can be found from the flow rate:

Δ m Δ t = m t =   ρ V t =   ˙ ρ Q = ρ A v

Suppose that crude oil with a density of 880 kg/m 3 is flowing through a pipe with a diameter of 55 cm and a speed of 1.8 m/s. Calculate the new speed of the crude oil when the pipe narrows to a new diameter of 31 cm, and calculate the mass flow rate in both sections of the pipe, assuming the density of the oil is constant throughout the pipe.

Solution: To calculate the new speed, we simply use the continuity equation.

Since the cross section of a pipe is a circle, the area of each cross section can be found as follows:

For the larger pipe:

A 1 =   π ( d 1 2 ) 2 =   π ( 0.275 ) 2 = 0.238 m 2

For the smaller pipe:

A 2 = π ( 0.155 ) 2 =   0.0755 m 2

So the larger part of the pipe ( A 1 ) has a cross-sectional area of 0.238 m 2 , and the smaller part of the pipe ( A 2 ) has a cross-sectional area of 0.0755 m 2 . The continuity equation tells us that the oil will flow faster through the portion of the pipe with the smaller cross-sectional area. Using the continuity equation, we get

A 1 v 1 =   A 2 v 2
v 2 =   ( A 1 A 2 ) v 1 =   ( 0.238 0.0755 ) ( 1.8 ) =   5.7   m / s

So we find that the oil is flowing at a speed of 1.8 m/s through the larger section of the pipe ( A 1 ), and it is flowing much faster (5.7 m/s) through the smaller section ( A 2 ).

The mass flow rate in both sections should be the same.

For the larger portion of the pipe:

( Δ m Δ t ) 1 =   ρ A 1 v 1 = ( 880 ) ( 0.238 ) ( 1.8 ) =   380   kg / s  

For the smaller portion of the pipe:

( Δ m Δ t ) 2 =   ρ A 2 v 2 = ( 880 ) ( 0.75538 ) ( 5.7 ) =   380   kg / s

And so mass is conserved throughout the pipe. Every second, 380 kg of oil flows out of the larger portion of the pipe, and 380 kg of oil flows into the smaller portion of the pipe.

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 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.

Questions & Answers

how can I read physics...am finding it difficult to understand...pls help
rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
David Reply
what is velocity
Jiti
the speed of something in a given direction.
Ju
what is a magnitude in physics
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Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
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What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
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omega= omega not +alpha t derivation
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u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
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the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
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if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
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define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
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why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
and he to multiply
Owofemi
stuff happens
Ju
God plays dice
Ju
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
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ximena Reply
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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