NORA-Lec #1 VISCOUS FLOW IN PIPES_Published.pdf

8/19/2019 NORA-Lec #1 VISCOUS FLOW IN PIPES_Published.pdf

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VISCOUS FLOWIN PIPES

Dr. Norasikin Mat Isa

Room : C16-101-05

Off no : 07- 4537721

[email protected]

[email protected]

mailto:[email protected]:[email protected]:[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, steam power

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 and

head loss in pipes -› the pressure drop is then used to

determine 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 tomaintain pressure 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 or

turbulent. This laminar or turbulent flow can be characterised by

using Reynolds number.

The laminar flow is characterized by smooth

streamlines and occur at low velocities or at Re <

2100.


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While turbulent flow is characterized by velocity

fluctuations and highly disordered motion (callededdies) and occur at high velocities or at Re > 4000.

The flow between 2100 < Re < 4000 is called

transitional flow


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Reynolds Number, Re

The Reynolds number Re is a dimensionless number that givesa measure of the ratio of inertia forces to viscous forces.

The concept was introduced by George Gabriel Stokes in 1851,but the Reynolds number is named after Osborne Reynolds

(1842 –1912), who popularized its use in 1883.

Reynolds number is used to characterize different flow regimes

whether it is laminar or turbulent flow.The transition from laminar to turbulent flow depends on the

geometry, surface roughness, flow velocity, surface

temperature, and type of fluid, among other things.

http://en.wikipedia.org/wiki/Osborne_Reynolds


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Reynold’s Experiment


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Reynold’s Results

Reynolds Demonstration

For low velocities, the dye filamentwould pass straight down the tube

As the velocity was increased, acritical value was achieved and at thisvalue, the stream of dye began towaver

Further increase in velocity made thefluctuations more intense until thedye was no longer a distinct andunbroken thread, but quite suddenlymixed more or less completely with

the water


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Laminar Flow

In the first kind of flow, the particles of fluid are

moving entirely in straight lines even though thevelocity along each line may not be the same. Since

the fluid may be construed to be moving in layers or

laminar, this type of flow is referred to as Laminar

Flow.


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Turbulent Flow

The second type of flow is called Turbulent Flow and

the paths of fluid particles are no longer orderly butrandom in nature. For such flows, average propertiessuch as mean velocity are used for description. Thecharacteristics of a turbulent flow depend on its

environment and turbulent motion is consideredirregular on a small scale.


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The Reynolds Number

As laminar and turbulent flows are wholly different,

some criterion for distinction is required. Transition

from laminar to turbulent flow depends on:

Flow Velocity, u

Fluid Viscosity, μ

Pipe Diameter, D

Reynolds derived a dimensionless number whichrepresented the ratio of the magnitude of the

inertial forces in the fluid to the viscous forces.

uDuDRe


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The ratio is known as the

Reynolds Number, Re

and is a fundamental

characteristic of flow in

which inertial and viscous

forces are present. viscosity

velocity

parameter length

density

Re

u

D

uD

The Reynolds Number


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For flows in which inertia and viscous forces are the most

significant, Reynolds Number is the parameter used to

compare experimental observations.

High Reynolds Number Inertia Forces dominate

Low Reynolds Number Viscous Forces dominate

For flows with geometric similarity, the same Reynoldsnumber describes the flow regimes.

The Reynolds Number


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Entrance Region and Fully Developed Region

• The region near where the flow enters the pipe is

called the entrance region.• Consider a flow entering a pipe.

• Let us think of the entering flow being uniform, so

inviscid.

Flow at the entrance to a pipe


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As soon as the flow 'hits' the pipe many changes take place.

The most important of these is that viscosity imposes itself

on the flow and the "No Slip" condition at the wall of the pipe comes into effect.

Consequently the velocity components are each zero on the

wall, ie., u = v = 0.

The flow adjacent to the wall decelerates continuously.

We have a layer close to the body where the velocity buildsup slowly from zero at wall to a uniform velocity

towards the center of the pipe. This layer is what is

called the Boundary Layer.


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Viscous effects are dominant within the boundary layer.

Outside of this layer is the inviscid core where viscous

effects are negligible or absent.

The boundary layer is not a static phenomenon; it isdynamic. it grows meaning that its thickness increases

as we move downstream.

From Fig. 5, it is seen that the boundary layer from the walls

grows to such an extent that they all merge on the

centreline of the pipe.

Once this takes place, inviscid core terminates and the flow

is all viscous. The flow is now called a Fully DevelopedFlow.

The velocity profile becomes parabolic.


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Once the flow is fully developed the velocity profile does

not vary in the flow direction.

In fact in this region the pressure gradient and the shear

stress in the flow are in balance.

The length of the pipe between the start and the point

where the fully developed flow begins is called the EntranceLength.

Denoted by , the entrance length is a function of the

Reynolds Number of the flow.

In general,


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Entrance Region and Fully Developed Region


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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 thepipe centerline, r , and independent of x .

This is true until the character of the pipe changes in some

way, 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

fully developed .

Beyond the interruption of the fully developed flow [at

section (4)], the flow gradually begins its return to itsfully developed character [section (5)] and continues

with this profile until the next pipe system component

is reached [section (6)].


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Example : Entrance Length

Water flows through a 15m pipe with 1.3cm

diameter at 20 l/min. What fraction of this pipe

can be considered at entrance region?


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Pressure Along The Pipe

• It is easy to visualise that the forces acting upon the pipe

flow are inertial, viscous force due to shear and the

pressure forces.

• Let us ignore gravity, i.e., let the pipe be horizontal.

• When the flow is fully developed the pressure gradientand shear forces balance each other and the flow

continues with a constant velocity profile. The pressure

gradient remains constant.

• In the entrance region the fluid is decelerating. A balance

is achieved with inertia, pressure and shear forces.

• The pressure gradient is not constant in this part of the

flow and in fact, it decreases as shown in Fig.6


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Pressure along the pipe

Fig. 6: Flow at the entrance to a pipe


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Pressure and Shear Stress

Fully developed steady flow in a constant diameter pipemay be 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

at any cross section of the pipe. Although this is true

whether 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

other useful 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 element

2. From Navier-stokes equation

3. 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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Average velocity :

Velocity at centerline (Umax) :

Flowrate: -> is called

Poiseuille law

Local velocity:

Pressure drop :


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Flow Properties Equation Remarks

Entrance Length,

l e /D

l e /D = 0.06 Re

l e /D = 4.4 (Re)1/6

Laminar flow

Turbulent flow

Pressure drop

per unit length

p/l = 2 /r Valid for both laminar

and turbulent flow

Shear stress = 2 w r/D Valid for both laminarand turbulent flow

Pressure drop p = 4l w /D Valid for both laminar

and turbulent flow

Velocity profile ur = V c 1 – (2r/D) 2 Laminar flow

Average velocity V = (π R2 V c /2)/ πR2

V= V c /2

V = pD2 /32µl

Laminar flow

Flowrate Q = πD4 p/128µl Laminar flow

Summary - Flow properties for horizontal 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 what distancefrom the centerline is the actual velocity equal to the average

velocity.

4. In fully developed laminar flow in a circular pipe, the velocity atR/2 (midway between the wall surface and the centerline) is

measured to be 6 m/s. Determine the velocity at the center of

the pipe.

5. The velocity profile in fully developed laminar flow in a circularpipe of inner radius R = 2 cm, in m/s, is given by u(r) = 4(1-

r2/R2). Determine the average and maximum velocities in the

pipe and the volume flow rate.

h l/ l d


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For non-horizontal/inclined pipe :

The adjustment necessary to account

for non-horizontal/inclined pipes, can

be easily included by replacing the

pressure drop, Δp, by the combined

effect of pressure and gravity, Δ p-γ l sin

Ө, where Ө is the angle between the

pipe and the horizontal.

Exercise : From F=ma derive V and Q

for inclined pipe.


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Example #1 : Laminar Flow

Oil at 20°C (ρ = 888 kg/m

3

and µ = 0.800kg/m · s) is flowing steadily through a 5-

cm-diameter 40-m-long pipe (Figure). The

pressure at the pipe inlet and outlet are

measured to be 745 and 97 kPa,respectively. Determine the flow rate of oil

through the pipe assuming the pipe is (a)

horizontal, (b) inclined 15° upward, (c)

inclined 15° downward. Also verify that theflow through the pipe is laminar.


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Example #2 : Laminar Flow

An oil with a viscosity of µ = 0.40 N.s/m2 and density ρ = 900 kg/m3

flows in a pipe of diameter D = 0.020 m. (a) What pressure drop, p1 - p2,is needed to produce a flowrate of Q = 20. x 10-5 m3/s if the pipe is

horizontal with x 1 = 0 and x 2 = 10m? (b) How steep a hill, ϴ, must the

pipe be on if the oil is to flow through the pipe at the same rate as in

part (a), but with p1

= p2

T iti f L i t T b l t Fl


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Consider a long section of pipe that is

initially filled with a fluid at rest. Asthe valve is opened to start the flow,

the flow velocity and, hence, the

Reynolds number increase from zero

(no flow) to their maximum steady-

state flow values. Assume this

transient process is slow enough so

that unsteady effects are negligible.

Transition form Laminar to Turbulent Flow

For an initial time period the Reynolds number is small enough for laminarflow to occur. At some time the Reynolds number reaches 2100, and the flow

begins its transition to turbulent conditions. Intermittent spots or bursts of

turbulence appear. As the Reynolds number is increased the entire flow field

becomes turbulent. The flow remains turbulent as long as the Reynolds

number exceeds approximately 4000.


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T b l t Sh St


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The experimental studies show that the shear stress in turbulent flow is 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 turbulent component, or the total shear

stress in turbulent flow can be expressed as

where η is the eddy or turbulent viscosity

where,

Turbulent Shear Stress

and

dy

ud

turbulent lamtotal

dr

ud lam

y

uvuturbulent

''

However, in practice it is not easy to use and this eddy viscosity changes from one


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However, in practice it is not easy to use and this eddy viscosity changes from one

turbulent flow condition/point to another – cannot be looked up in handbooks.

Several semiempirical theories have been proposed to determine approximate

values of η . For example, the turbulent process could be viewed as the randomtransport of bundles of fluid particles over a certain distance, the mixing length,

from a region of one velocity to another region of a different velocity. By the use of

some ad hoc assumptions and

physical reasoning, it was concluded that the eddy

viscosity was given by,

Thus, the turbulent shear stress is

dy

ud m

2

2

2

dy

ud mturbulent


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turbulent lamtotal


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Turbulent Velocity Profile

- much flatter than laminar profile.

- can be broken into three regions

i. the viscous sublayer

ii. the overlap region

iii. the outer turbulent layer

Unlike laminar flow, the expressions for the velocity

profile in a turbulent flow has been obtained

through the use of dimensional analysis,

experimentation, and semiempirical 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 the typical

value of n is between 6 to 10.


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However this power law cannot be valid near the


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However, this power law cannot be valid near the

wall (refer figure).

So, in the viscous sublayer the velocity profile can be

written in dimensionless form

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

and

Formula from Cengel


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..Formula from Cengel…

(i)

(ii)


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1. Colebrook equation (explicitly): **

Darcy friction factor for turbulent flow

Friction factor f for turbulent can be obtain through

2. Colebrook equation (implicitly):

3. Moody chart (also generated by Colebrook equation).


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Moody chart


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Example #1 : Turbulent Flow

(a) For laminar flow, determine at what radial location you would

place a Pitot tube if it is to measure the average velocity in thepipe. (b) Repeat part (a) for turbulent flow with Re= 10 000

E l #2 T b l t Fl


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Example #2 : Turbulent Flow

Water at 20°C ( ρ = 998 kg/m

3

and ν = 1.004 x 10-6

m2

/s) flows througha horizontal pipe of 0.1-m diameter with a flowrate of Q = 4 x 10-2 m3

/s and a pressure gradient of 2.59 kP/m. (a) Determine the

approximate thickness of the viscous sublayer. (b) Determine the

approximate centreline velocity, V c , (c) Determine the ratio of the

turbulent to laminar shear stress, τturb / τlam , at a point midway

between the centreline and the pipe wall (i.e., at r = 0.025m)

E i


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Air under standard conditions flows through a 4.0-mm-diameterdrawn tubing with an average velocity of V = 50 m/s For such

conditions the flow would normally be turbulent. However, if

precautions are taken to eliminate disturbances to the flow (the

entrance to the tube is very smooth, the air is dust free, the tube

does not vibrate, etc.), it may be possible to maintain laminar flow.

(a) Determine the pressure drop in a 0.1-m section of the tube if the

flow is laminar. (b) Repeat the calculations if the flow is turbulent.

Exercise

Pressure Drop and Head Loss


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Exercise : Pressure Drop and Head Loss in Pipes

Water at 5º ( ρ = 1000 kg/m3 and μ = 1.519 x 10-3 kg/m.s) is

flowing steadily through a 0.3 cm diameter 9 m long horizontal

pipe at an average velocity of 0.9 m/s. Determine :

a) the head lossb) the pressure drop

c) the pumping power requirement to overcome the pressure

drop.

LOSSES IN PIPES


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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, ∆P since it is directly related to the powerrequirements of the pump to maintain flow.

• Therefore, the analysis of losses in pipes is very useful inestimating 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 also affect the flow patterns/conditions and

this also contributed to the losses.

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



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In practice, it is found convenient to express the pressure loss for all types

of fully developed internal flows (laminar or turbulent flows etc).

The pressure loss and head loss for all types of internal flows (laminar or

turbulent, in circular or noncircular pipes, smooth or rough surfaces) are

expressed as

Where for

And f for turbulent can be obtain from Colebrook equation or Moody

chart.


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TYPE OF LOSSES

There 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 a pipingsystem 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, length and

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 material

and how it been manufactured.

• Different pipe material gives different value of surface

roughness.• Rough pipe wall surface gives high value of surface

roughness and it will contribute larger losses.

•

While smooth pipe (i.e have lower surface roughnessor = 0) contribute lower losses.


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Surface roughness on rough and smooth wall


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General steps in solving Major Losses


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General steps in solving Major Lossesproblems.

1. Determine Re where Re = VD/µ.

If Re4000 (turbulent flow)

2. Determine surface roughness, and then relative roughness

/D.

3. Obtain the value of friction

factor f from Moody chart

(base on Re dan /D obtainedbefore)

4. Calculate the losses head due to

friction hf .

2. Calculate friction factor f where f for laminar,

f = 64/Re

3. Calculate the losses head due to

friction hf .

Note : f value only influenced by Re.

no. and not by the value of

relative roughness because the

pipe surface is smooth (i.e = 0)


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Moody Chart

MINOR LOSSES


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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 of loss

coefficient KL.

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

• Sometimes minor losses can be a major losses for example 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

Other method to calculate KL for sudden


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Other method to calculate KL for sudden

expansion (by using the equation obtained

from simple 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-flowenergy equation is expressed as

Common Types of Problems


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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

Colebrook equation), we usually encounter three

types of problems :

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

the pipe length and diameter are given for aspecified flow rate (or velocity).

2. Determining the flow rate when the pipe length and

diameter are given for a specified pressure drop (or

head loss).

3. Determining the pipe diameter when the pipe length

and flow rate are given for a specified pressure drop

(or head loss).

Example 1 :


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pWater flows from basement (point 1) to the second floor of building

through the copper pipe with diameter of 1.9 cm at flow rate 0.000756

m

3

/s and flows out from the faucet with diameter of 1.27 cm (point 2)as shown in Figure. With the viscosity of water, µ = 1.12 x 10-3 Ns/m2,

calculate the head losses of the pipe system.


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Exercise : Final Exam Semester I Session


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Exercise : Final Exam Semester I Session

2011/2012

b) A 80 percent efficient pump delivers water at 20°C (ρ

= 998.2 kg/m3 and μ = 1.002 x 10-3 Ns/m2)

from one reservoir to another at 6 m higher. The

piping system consists of 15 m of galvanized- iron 5-

cm diameter pipe (ε = 0.15 mm), a reentrantentrance (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 input power

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)

Noncircular Conduits


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Most 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, where A is

the cross-sectional area of the pipe (m2 ) and P is its wetted 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,ρ = 1.221 kg/m3 and ν = 1.46 x 10-5

m2/s is forced through a 30.48 m long horizontal square

duct of 0.23 x 0.23 m at 0.708 m3/s. Find the pressure

drop if ε=0.0000914 m.


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EXERCISES


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Exercise : Laminar Flow in Horizontal and

Inclined Pipes

Consider the fully developed flow of glycerin at 40ºC

through 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 profile and the pressuredifference across this 70 m long section of the pipe, and

the useful pumping power required to maintain this flow.

For the same useful pumping power input, determine the

percent increase of the flow rate if the pipe is inclined 15ºdownward and the percent decrease if it is inclined 15º

upward. The pump is located outside of this pipe section.

Test 1 Semester I Session 2011/12


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QUESTION 1

(a) Using appropriate sketches, discuss the differences of velocity profilesbetween laminar and turbulent flow in pipe. Provide explainations 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 ρ = 890 kg/m3 and μ = 0.07

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

i. maximum velocity;

ii. volume flowrate; and

iii. shear stress at the point 1 cm from pipe wall.

(9 marks)



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QUESTION 2

(a) A commercial steel pipe (equivalent roughness, ε = 0.045 mm) of

80 mm diameter and 1000 metre long (horizontal pipe) is carrying

water 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 the flow

is considered fully developed turbulent flow.

Assume , ρ = 1000 kg/m3

and μ = 0.00015 kg/ms.

(15 marks)

Final Exam Semester I Session 2011/2012


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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 them

occur.

(iii) Provide physical explanations on both phenomena above.

(10 marks)

b) A 80 percent efficient pump delivers water at 20°C (ρ = 998.2 kg/m3 and

μ = 1.002 x 10-3 Ns/m2) from one reservoir to another at 6 m higher. The

piping system consists of 15 m of galvanized- iron 5-cm diameter pipe (ε =

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 input power 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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Viscous Sublayer Outer Turbulence Sublayer


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Viscous Sublayer Outer Turbulence Sublayer

Viscous shear stress is dominant Both viscous and turbulence shear

are important (although turbulent

shear is expected to be significantlylarger)

Random, fluctuating/eddying of

the flow is essentially absent

Considerably mixing and

randomness to the flow

μ is an important parameter μ is not important

ρ is not important ρ is important

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NORA-Lec #1 VISCOUS FLOW IN PIPES_Published.pdf