1.Viscous Flow in Pipes

7/25/2019 1.Viscous Flow in Pipes

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VISCOUS FLOWIN PIPES

Dr. Azmahani Sadikin

Room : C16-101-09Off no : 07- 4537750

[email protected]
mailto:[email protected]:[email protected]


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WHY PIPES?

Have many application in engineering system

(particularly in fluid and thermal system). E.g : not only in water supply system also in human

body (blood vessel system), oil & gas industry, steampower plant, air-conditioning system, hydraulic system,

in car etc Pipes (circular x-section) = ducts (non-circular),

conduits, tubes (small circular pipes)

Q : Why study this topic?

To understand the flow characteristics in pipes viscousflow - friction - directly related to pressure drop andhead loss in pipes - the pressure drop is then used todetermine the pumping power requirement.


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General Characteristics of Pipe Flow

Assumptions:

The pipe is completely filled with fluid (if the pipe is not

full, it is called open channel and not possible to maintainpressure difference).

The conduit is round.

The fluid is incompressible.

Viscous fluid.


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Classification of Fluid Flow in Pipes

The fluid flow in pipes can be classified as laminar orturbulent.

This laminar or turbulent flow can be characterized byusing Reynolds number.

The laminar flow is characterized by smooth streamlinesand occur at low velocities or at Re < 2100.
http://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Laminar_flow


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While turbulent flow is characterized by velocityfluctuations and highly disordered motion (called eddies)and occur at high velocities or at Re > 4000.

The flow between 2100 < Re < 4000 is calledtransitional flow


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Reynolds Number, Re

The Reynolds number Re is a dimensionless numberthat gives a measure of the ratio of inertia forces toviscous forces.

The concept was introduced by George Gabriel Stokes in1851, but the Reynolds number is named after OsborneReynolds (18421912), who popularized its use in 1883.

Reynolds number is used to characterize different flowregimes whether it is laminar or turbulent flow.

The transition from laminar to turbulent flow depends on thegeometry, surface roughness, flow velocity, surface

temperature, and type of fluid, among other things.
http://en.wikipedia.org/wiki/Dimensionless_numberhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Inertial_forcehttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/George_Gabriel_Stokeshttp://en.wikipedia.org/wiki/Osborne_Reynoldshttp://en.wikipedia.org/wiki/Osborne_Reynoldshttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Turbulenthttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Osborne_Reynoldshttp://en.wikipedia.org/wiki/Osborne_Reynoldshttp://en.wikipedia.org/wiki/George_Gabriel_Stokeshttp://en.wikipedia.org/wiki/Viscoushttp://en.wikipedia.org/wiki/Inertial_forcehttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Dimensionless_number


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Reynolds Experiment


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Entrance Region and Fully DevelopedRegion

The region near where the flow enters the pipe is calledthe entrance region.

Here, the fluid typically enters the pipe with a nearly

uniform velocity profile at section (1).


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As the fluid moves through the pipe, viscous effects cause it tostick to the pipe wall (the no-slip condition).This is true whether the fluid is relatively inviscid air or a very

viscous oil.Thus, a boundary layer in which viscous effects areimportant is produced along the pipe wall such that the initial

velocity profile changes with distance along the pipe, x,

until the fluid reaches the end of the entrance length,section (2), beyond which the velocity profile does not varywith x.

The boundary layer has grown in thickness to completely fill thepipe.

The shape of the velocity profile in the pipe depends on whetherthe flow is laminar or turbulent, as does the length of theentrance region, . Typical entrance lengths are given by,


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Once the fluid reaches the end of the entrance region, section(2), the flow is simpler to describe because the velocity is

a function of only the distance from the pipe centerline, r,and independent of x.

This is true until the character of the pipe changes in someway, such as a change in diameter, or the fluid flows

through a bend, valve, or some other component atsection (3). The flow between (2) and (3) is termed fullydeveloped.

Beyond the interruption of the fully developed flow [at section(4)], the flow gradually begins its return to its fullydeveloped character [section (5)] and continues with thisprofile until the next pipe system component is reached[section (6)].


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Example : entrance length

Water flows through a 15 m pipe with 1.3 cm diameterat 20 l/min. What fraction of this pipe can beconsidered at entrance region?


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Pressure and Shear Stress

Fully developed steady flow in a constant diameter pipe maybe driven by gravity and/or pressure


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Fully Developed Laminar Flow

As is indicated in the previous section, the flow in long,straight, constant diameter sections of a pipe becomesfully developed. That is, the velocity profile is the same atany cross section of the pipe. Although this is truewhether the flow is laminar or turbulent, the details of the

velocity profile (and other flow properties) are quitedifferent for these two types of flow.

The knowledge of the velocity profile can lead directly to otheruseful information such as pressure drop, flowrate, head

loss, etc.3 methods could be used for this purpose :1. By applying F = ma to a fluid element2. From Navier-stokes equation3. From dimensional analysis


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Fully Developed Laminar Flow

By applying F=ma to a fluid element :

refer to derivation


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Shear stress distribution in Fully Developed Laminar Flow


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Average velocity :

Velocity at centerline (Umax) :

Flowrate: -> is called

Poiseuille law

Local velocity:

Pressure drop :


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For non-horizontal/inclined pipe :

The adjustment necessary to accountfor non-horizontal/inclined pipes, canbe easily included by replacing the

pressure drop,p, by the combinedeffect of pressure and gravity, p-l sin, where is the angle between thepipe and the horizontal.Exercise : From F=ma derive V and Q

for inclined pipe.


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Exercise : Laminar Flow

1. Using F=ma derive and proof that u = Vc [1 r2/R2]

2. Find velocity ratio u/Umax

3. For laminar flow in a round pipe of radius, R, at whatdistance from the centerline is the actual velocityequal to the average velocity.

4. In fully developed laminar flow in a circular pipe, thevelocity at R/2 (midway between the wall surface andthe centerline) is measured to be 6 m/s. Determinethe velocity at the center of the pipe.

5. The velocity profile in fully developed laminar flow in acircular pipe of inner radius R = 2 cm, in m/s, is givenby u(r) = 4(1- r2/R2). Determine the average andmaximum velocities in the pipe and the volume flow

rate.


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Example #1 : Laminar Flow


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Example #2 : Laminar Flow


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Consider a long section of pipe that

is initially filled with a fluid at rest.As the valve is opened to start theflow, the flow velocity and, hence,the Reynolds number increase fromzero (no flow) to their maximum

steady-state flow values. Assumethis transient process is slowenough so that unsteady effects arenegligible.

Transition form Laminar to Turbulent Flow

For an initial time period the Reynolds number is small enough forlaminar flow to occur. At some time the Reynolds number reaches 2100,and the flow begins its transition to turbulent conditions. Intermittentspots or bursts of turbulence appear. As the Reynolds number isincreased the entire flow field becomes turbulent. The flow remainsturbulent as long as the Reynolds number exceeds approximately 4000.


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Turbulent characteristic : random, chaotic, fluctuations and eddies.

Most flows encountered in engineering practice are turbulent.However, turbulent flow is a complex mechanism and the theory ofturbulent flow remains largely undeveloped.Therefore, we must rely on experiments and the empirical or semi-empirical correlations developed for various situations.

Fully Developed Turbulent Flow


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The experimental studies show that the shear stress in turbulent flowis much larger due to the turbulent fluctuations and the shear stress is

not merely proportional to the gradient of the time-average velocity.

Therefore, it is convenient to think of the turbulent shear stress as

consisting of two parts: the laminar component and the turbulentcomponent, or the total shear stress in turbulent flow can beexpressed as

where is the eddy or turbulent viscosity

where,

Turbulent Shear Stress

and


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However, in practice it is not easy to use and this eddy viscosity changesfrom one turbulent flow condition/point to another cannot be looked upin handbooks. Several semiempirical theories have been proposed to

determine approximate values of . For example, the turbulent processcould be viewed as the random transport of bundles of fluid particlesover a certain distance, the mixing length, from a region of onevelocity to another region of a different velocity. By the use of some adhoc assumptions and physical reasoning, it was concluded that the eddyviscosity was given by,

Thus, the turbulent shear stress is


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Viscous Sublayer Outer Turbulence Sublayer

Viscous shear stress is dominant Both viscous and turbulenceshear are important (although

turbulent shear is expected to besignificantly larger)

Random, fluctuating/eddying ofthe flow is essentially absent

Considerably mixing andrandomness to the flow

is an important parameter is not important

is not important is important


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Turbulent Velocity Profile

- much flatter than laminar profile.

- can be broken into three regionsi. the viscous sublayerii. the overlap regioniii. the outer turbulent layer

Unlike laminar flow, the expressions for thevelocity profile in a turbulent flow has beenobtained through the use of dimensionalanalysis, experimentation, andsemiempirical theoretical efforts.

An often-used correlation is the empirical power-law velocity profile

and


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The value of n can be obtain from graph below. However thetypical value of nis between 6 to 10.


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However, this power law cannot be valid nearthe wall (refer figure).

So, in the viscous sublayerthe velocity profilecan be written in dimensionless form

For the overlap region, the following expression has been proposed :

and


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..Formula from Cengel

(i)

(ii)


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Example #1 : Turbulent Flow


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Example #2 : Turbulent Flow

Exercise


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Pressure Drop and Head Loss


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Exercise : Pressure Drop and HeadLoss in Pipes

Water at 5 (= 1000 kg/m3 and = 1.519 x 10-3kg/m.s) is flowing steadily through a 0.3 cm diameter9 m long horizontal pipe at an average velocity of 0.9

m/s. Determine :a) the head loss

b) the pressure drop

c) the pumping power requirement to overcome the

pressure drop.


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LOSSES IN PIPES

Always describe as pressure drop or head loss. A quantity of interest in the analysis of pipe flow is the pressure

drop, Psince it is directly related to the power requirements ofthe pump to maintain flow.

Therefore, the analysis of losses in pipes is very useful in

estimating the pressure drop occurs. Besides the pipe size and material also the velocity in pipe, the

pipe components such as pipe fittings, valves, diffusers etc alsoaffect the flow patterns/conditions and this also contributed tothe losses.

When a head loss is considered, the steady-flow energyequation is expressed as


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Pressure Drop and Head LossIn practice, it is found convenient to express the pressure loss for

all types of fully developed internal flows (laminar or turbulent flowsetc).

The pressure loss and head loss for all types of internal flows(laminar or turbulent, in circular or noncircular pipes, smoothor rough surfaces) are expressed as

Where for

And f for turbulent can be obtain from Colebrook equation or Moodychart.


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TYPE OF LOSSESThere are 2 type of losses major losses and minor losses.

Major losses caused by fluid friction.

given by,

Minor losses - due to changes in the pipe cross section/ pipecomponents.

When all the loss coefficients are available, the total head loss in apiping system is determined from

If the entire piping system has a constant diameter, the totalhead loss reduces to


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MAJOR LOSSES

Major losses occur due to friction in pipe.

It depends on Reynolds no, surface roughness, lengthand diameter of pipe, and also the velocity in pipe.

Friction factor, f is depends on Reynolds no and surfaceroughness.

It can be obtained from the eqns. such as the Karman &Prandtl and Colebrook & White. But it is more easier from

Moody Chart.


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Surface Roughness,

Surface roughness of pipe is depends on pipe materialand how it been manufactured.

Different pipe material gives different value of surfaceroughness.

Rough pipe wall surface gives high value of surface

roughness and it will contribute larger losses.

While smooth pipe (i.e have lower surface roughness or

= 0) contribute lower losses.


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Surface roughness on rough and smooth wall


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G l t i l i M j L


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General steps in solving Major Lossesproblems.

1. Determine Re where Re = VD/.

If Re4000 (turbulent flow)

2. Determine surface roughness, and then relative roughness /D.

3. Obtain the value of friction factorf from Moody chart (base on Redan /D obtained before)

4. Calculate the losses head due tofriction hf.

2. Calculate friction factor f where ffor laminar,

f = 64/Re

3. Calculate the losses head due tofriction hf.

Note : f value only influenced by Re.no. and not by the value ofrelative roughness because the

pipe surface is smooth (i.e =0)


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Moody Chart


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MINOR LOSSES

Minor losses is due to changes in the pipe cross section. It is depends on the velocity in pipe and the geometry of

pipe components and this can be describe by the value ofloss coefficient KL.

Different shape and geometry of pipe component givesdifferent value of KL.

Sometimes minor losses can be a major losses forexample in short pipes where there are a suction pipe of a

pump with strainer and foot valves.


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KL for pipe entrance


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KL for pipe entrance (graph)


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KL for pipe exit


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KL for sudden contraction


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KL for sudden expansion


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Other method to calculate KL for sudden

expansion (by using the equation obtained fromsimple energy analysis)


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KL for typical diffuser


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KL for 90 bend


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KL

for pipe components


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PUMPING POWER REQUIREMENT

When a piping system involves a pump, the steady-flow energyequation is expressed as


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Common Types of Problems

In the design and analysis of piping systems that involve

the use of the Moody chart (or the Colebrookequation), we usually encounter three types ofproblems :

1. Determining the pressure drop (or head loss) when the

pipe length and diameter are given for a specified flowrate (or velocity).

2. Determining the flow rate when the pipe length anddiameter are given for a specified pressure drop (orhead loss).

3. Determining the pipe diameter when the pipe lengthand flow rate are given for a specified pressure drop (orhead loss).

l 1


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Example 1 :Water flows from basement (point 1) to the second floor ofbuilding through the copper pipe with diameter of 1.9 cm at flowrate 0.000756 m3/s and flows out from the faucet with diameterof 1.27 cm (point 2) as shown in Figure. With the viscosity ofwater, = 1.12 x 10-3 Ns/m2, calculate the head losses of thepipe system.


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Exercise : Final Exam Semester I

Session 2011/2012

b) A 80 percent efficient pump delivers water at 20C ( = 998.2

kg/m3 and = 1.002 x 10-3 Ns/m2) from one reservoir to anotherat 6 m higher. The piping system consists of 15 m of galvanized-iron 5-cm diameter pipe ( = 0.15 mm), a reentrant entrance (KL= 1.0), two screwed 90 long-radius elbows (KL = 0.41 each),and a screwed-open gate valve (KL = 0.16). What is the inputpower required in with a 6 well-designed conical expansion (KL =

0.3) added to the exit? The flow rate is 0.02 m3

/s.

(15 marks)

N i l C d it


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Noncircular ConduitsMost of the pipes used for engineering purposes are circular.However some of them are not circular in their cross section.

For noncircular pipes, the diameter in the previous relations can bereplaced by the hydraulic radius which defined as RH= A/P,whereA is the cross-sectional areaof the pipe (m2)and P is itswetted perimeter (m).

For circular pipe,

Reynolds no :

Relative roughness :

Head loss :

Replace hydraulic radius in Re, relative roughness and head loss given


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Example : Non-circular pipes

Air with density, = 1.221 kg/m3 and = 1.46 x 10-5 m2/sis forced through a 30.48 m long horizontal squareduct of 0.23 x 0.23 m at 0.708 m3/s. Find the pressuredrop if =0.0000914 m.


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EXERCISES


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Exercise : Laminar Flow in Horizontaland Inclined Pipes

Consider the fully developed flow of glycerin at 40Cthrough a 70 m long, 4 cm diameter, horizontal,circular pipe. If the flow velocity at the centerline is

measured to be 6 m/s, determine the velocity profileand the pressure difference across this 70 m longsection of the pipe, and the useful pumping powerrequired to maintain this flow. For the same usefulpumping power input, determine the percent increaseof the flow rate if the pipe is inclined 15 downwardand the percent decrease if it is inclined 15 upward.The pump is located outside of this pipe section.

T 1 S I S i 2011/12


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Test 1 Semester I Session 2011/12

QUESTION 1

(a) Using appropriate sketches, discuss the differences of velocityprofiles between laminar and turbulent flow in pipe. Provideexplainations of these patterns.

(6 marks)(b) For fully developed laminar pipe flow in a circular pipe, the velocity

profile is given by ,

where R is the inner radius of the pipe.

The 4 cm diameter pipe carries oil, with = 890 kg/m3 and =

0.07 kg/ms. The measured pressure drop per unit length is 72kPa/m; determine:

i. maximum velocity;ii. volume flowrate; andiii. shear stress at the point 1 cm from pipe wall.

(9 marks)

T t 1 S t I S i 2011/12


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Test 1 Semester I Session 2011/12

QUESTION 2

(a) A commercial steel pipe (equivalent roughness, = 0.045 mm) of80 mm diameter and 1000 metre long (horizontal pipe) is carryingwater at the flowrate, Q = 0.008 m3/s. Calculate loss of head, hf@ hL , if water flow in :

i. a rough pipe, orii. a smooth pipe (assumption)

(b) Determine the maximum diameter of pipe and loss of head if theflow is considered fully developed turbulent flow.

Assume , = 1000 kg/m3 and = 0.00015 kg/ms.

(15 marks)


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Final Exam Semester I Session 2011/2012

a) (i) In a pipe flow, what are the differences between uniform

velocity and uniform velocity profile?(ii) Using appropriate sketches show where each of themoccur.

(iii) Provide physical explanations on both phenomena above.(10 marks)

b) A 80 percent efficient pump delivers water at 20C ( = 998.2

kg/m3 and = 1.002 x 10-3 Ns/m2) from one reservoir to anotherat 6 m higher. The piping system consists of 15 m of galvanized-iron 5-cm diameter pipe ( = 0.15 mm), a reentrant entrance (KL= 1.0), two screwed 90 long-radius elbows (K

L

= 0.41 each),

and a screwed-open gate valve (KL = 0.16). What is the inputpower required in with a 6 well-designed conical expansion (KL =

0.3) added to the exit? The flow rate is 0.02 m3/s.(15 marks)

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1.Viscous Flow in Pipes