1_Introduction to Pipe Flow

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Introduction to Pipe Flow

1.0 Definitions

IDEAL FLUID

The actual flow pattern within a fluid is usually complex and difficult to model

mathematically. However theory can !e simplified considera!ly !y the assumption that

the fluid is ideal. In context of Fluid "echanics #ideal$ has a specific meanin%.

An ideal fluid &li'uid or %as( is one which has the followin% properties)

* incompressi!ility &i.e. does not chan%e volume no matter what pressure is applied

and thus it has a constant density(

* +ero viscosity &i.e. does not experience friction resistance to flow(* +ero surface tension

* does not chan%e phase

,ote that %ases and vapours are compressi!le and can only !e considered as ideal fluidswhen the flow velocity is very low. -ases can !e treated as perfect in which case the

perfect %as e'uations are applica!le.

TEAD/ FL01

A steady flow is one in which the properties of the fluid mass flow rates and heat transfer

rates do not chan%e with respect to time.

Unsteady flow is one in which chan%es are present with respect to time.

U,IF02" FL01

Uniform flow ta3es place such that the properties are the same at all points within the

control volume at any %iven instant. For example hi%h velocity flow some way from theentrance down a lon% strai%ht pipe may !e re%arded as uniform.

Figure 1 Uniform Flow

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta 7


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If :a; :!; :c pa; p!; pcetc at the same instant then the flow is uniform. If the

properties vary from place to place within the control volume at a %iven instant then the

flow is non*uniform.

0,E*DI"E,I0,AL FL01

In one*dimensional &7D( flow it is assumed that all properties are uniform over any plane

perpendicular to the flow direction.


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Figure 3 Actual velocity profile and equivalent 1D profile

2.0 Pipe Systems

The transport of a fluid &li'uid or %as( in a closed conduit &commonly called a pipeif it is

of round section or a duct if it is not round( is extremely important in our daily

operations. uch applications include)

* lar%e pipelines for transportin% oil and natural %as across countries* natural systems of #pipes$ in the !ody that carry !lood

* water pipes in our homes

* water distri!ution system that delivers water to our houses

* hoses and pipes in transport vehicles that transport fuel and hydraulic to variouscomponents and machines

* air conditionin% ducts that transport heated8cooled8humidified8dehumidified air

across different parts of !uildin%s

Althou%h all of these systems are different the fluid mechanics principles %overnin% thefluid motions are common.

ome of the !asic components of a typical pipe system are shown in Fi%ure >.?. They

include pipes &perhaps of different diameters( the various fittin%s used to connect the

individual pipes to form the desired system the flowrate control devices &valves( andpumps or tur!ines that add or extract ener%y to8from the fluid.



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Even the most simple pipe systems are actually 'uite complex when they are viewed in

terms of ri%orous analytical considerations. In this chapter we shall only consider the

simplest pipe flow topics coverin% laminar flow in lon% strai%ht constant diameterpipes.

source: www.powermat.com/pools

Figure 4 Typical Pipe System (in this case for solar/gas water heating of a swimming pool

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta ?
http://www.powermat.com/poolshttp://www.powermat.com/pools


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3.0 General Characteristics of Pipe Flow

For all flows involved in this chapter we assume that the pipe is completely filled withthe fluid !ein% transported as shown in Fi%ure >.@&a(. Thus we will not consider a

concrete pipe throu%h which rainwater is flowin% without completely fillin% the pipe as

shown in Fi%ure >.@&!(. uch flows are referred to open*channel flows.

The difference !etween open*channel flow and the pipe flow is in the fundamental

mechanism that drives the flow. For open*channel flow %ravity alone is the drivin% force the water flows down a hill. For pipe flow %ravity may !e important &the pipe need not

!e hori+ontal( !ut the main drivin% force is li3ely to !e a pressure %radient alon% the

pipe. If the pipe is not full it is not possi!le to maintain this pressurep1p2.

Figure 5

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aminar and !ur"ulent Flow

There are three types of flow. For flow at relatively low velocities it is found that thepathlines tend to !e smooth. At hi%h velocities pathlines tend to !e very irre%ular and

cross over another.

0s!ourne 2eynolds a British scientist and mathematician &7C?5 775( carried out a

series of simple experiments to illustrate and distin%uish !etween the two types of flow

usin% the simple apparatus show in Fi%ure >.. If water runs throu%h a pipe of diameter dwith and avera%e velocity : the followin% characteristics are o!served !y inectin%

neutrally !uoyant dye as shown)

* for small flowrates the dye strea3 will remain as well*defined as it flows alon% withonly sli%ht !lurrin% due to molecular diffusion of the dye into the surroundin% water.

In this case we have laminar flow.

* for a somewhat lar%er #intermediate flowrate$ the dye strea3 fluctuates in time andspace and intermittent !ursts of irre%ular !ehavior appear alon% the strea3. This is

transitional flow.

* for lar%e flowrates the dye strea3 almost immediately !ecomes !lurred and spreads

across the pipe in a random fashion. In this case we have tur!ulent flow.

Figure 6 Simple pparatus use! to o"ser#e laminar$ transitional an! tur"ulent flow.

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta


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Source:http://pen.physik.uni-kl.de

Figure % Flow #isuali&ation of laminar (a'$ transitional ("' an! tur"ulent (c' flow usingeynol!s e)periment

Figure * Time !epen!ence of flui! #elocity at a gi#en point

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta >


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For a smooth pipe the type of flow depends on the 2eynolds ,um!er which is a

dimensionless num!er %iven !y

Vd=2e

where V avera%e velocity of flow fluid density

d internal pipe diameter

fluid viscosity

The flow is laminar transitional or tur!ulent provided the 2eynolds ,um!er is #smallenou%h$ #intermediate$ or #lar%e enou%h$. ,ote that it is not only the fluid velocity that

determines the character of the flow its density viscosity and pipe si+e are of e'ual

importance. These parameters com!ine to produce the 2eynolds num!er.

In case a pipe that is not smooth &mainly due to corrosion scalin% or surface treatment(

the type of flow will also depend on the rou%hness of the internal surface apart from the2eynolds ,um!er.

The critical 2eynolds ,um!er is defined as the 2e ,o at which transition from laminar to

tur!ulent will occur. It is often very difficult to define exactly the critical 2e ,o.However for %eneral en%ineerin% purposes the followin% values are appropriate for a

round pipe)

* flow is laminar if 2e is less than approx 5766

* flow is tur!ulent if 2e is %reater than approx ?666

* flow is transitional in !etween approx 5766 and ?666

#ntrance $e%ion and Fully De&eloped Flow

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta C


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Any fluid flowin% in a pipe had to enter the pipe at some location. The re%ion of flow

near where the fluid enters the pipe is termed the entrance re%ion and is illustrated inFi%ure >.. It may !e the first few centimeters of a pipe connected to a tan3 or the initial

section of a lon% run of air conditionin% duct from a chiller.

As shown in Fi%. >. the fluid typically enters the pipe with a nearly uniform velocity

profile. As the fluid moves throu%h the pipe viscous effects cause it to stic3 to the pipe

wall &the no*slip !oundary condition(. A !oundary layer &in which viscous effects areimportant( is produced alon% the pipe wall such that the initial velocity profile chan%es

with distance alon% the pipe until the fluid reaches the end of the entrance len%th !eyond

which the velocity profile does not chan%e with distance alon% the pipe.

The shape of the velocity profile in the pipe depends on whether the flow is laminar or

tur!ulent as does the len%th of the entrance re%ion le.

The dimensionless entrance length led in smooth pipes is dependent on the 2eynoldsnum!er. Typical entry len%ths are %iven !y

2e6E.6=d

le for laminar flow

E7

2e?.?=d

le for tur!ulent flow

Figure + ,ntrance region an! fully !e#elope! flow in a pipe

'.0 (iscous Flow in Pipes )aminar flow only*

4 Dr. In%. Tonio ant 56675879 Department of "echanical En%ineerin% University of "alta


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Gonsider a cylindrical element of radius rin pipe in which a fluid is flow under laminar

conditions &see Fi%. >.76(. Let the velocity at this radius !e v.

Figure - Flow in a pipe

The forces actin% are

&7( Force causin% motion due to pressure difference

( ) 5557 rprpp ==

where p; pressure drop across len%th!

&5( :iscous dra% on cylindrical surface

; area viscous shear stress

= r!5

dr

dvr! = 5 when we have laminar flow only

ince flow is steady there is no resultant force. Thus

65

5=+

dr

dvr!rp

dr!

prdv

5

=

Inte%ratin%



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"!

prdr

!

prv +

=

= ??

55

Apply !oundary conditions) at the wall i.e. at r # d2 v; 6.

!

pdA7E

5

=

Thus the velocity distri!ution !ecomes

= 5

5

??r

d

!

pv

The a!ove e'uation shows that the velocity distri!ution is a pipe for laminar flow has the

shape of a para!oloid as shown !elow)

To find the flowrate $&m98s()

Gonsider an annular element in Fi%. >.76.

Flowrate throu%h element ; d ; area velocity

rrrd

!

prr$

== 5

5

??

55

Inte%ratin% the total dischar%e $is %iven !y

rrrd

!

prr$

d

==

5

6

5

5

??

55

!

pd$

75C

?=



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"ean velocity

!

pd

d

$

A

$V

95?

5

5

====

Thus the pressure drop pis %iven !y

5

95

d

!Vp

=

The a!ove e'uation %ives the pressure drop in a pipe due to friction. ,ote that the

pressure drop is lar%er for

* fluids of a hi%her viscosity

* hi%her flow

* velocities* lon%er pipes

* pipes with a smaller diameter.

The pressure drop is usually expressed in the form of loss of head hf. The Darcy formula

havin% the followin% form is usually used for the loss of head)

g

V

m

f!hf

5

5

=

where mis the hydraulic radius which is d%for a pipe.


1_Introduction to Pipe Flow

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