Viscous Flow in Pipes

FLUID MECHANICS

VISCOUS FLOW IN PIPE

Learning Outcomes

Characterize flows in pipes.

Explain laminar and turbulent flows and examine their differences

calculate losses in various segments of pipes

Upon the completion of this lecture, you will beable to:

EE038-3.5-3 FLUID MECHANICS VISCOUS FLOW IN PIPES 2

calculate losses in various segments of pipes

apply appropriate equations and principles to analyze a variety of

pipe flow situations.

4.1 Flow in Pipe

Pipe is a closed conduit through which fluid may betransported

Behaviour of fluid in pipe has wide variety of reallife applications ranging from large man-made pipesuch as the 800-miles Alaskan pipeline that carries


crude oil in such as a long distance to the naturalpipes that transports blood throughout humanbodies into and out of their lungs.

In fact numerous applications exist such as housingpipeline network, machinery pipes etc.

4.1 Flow in Pipe

Knowing there are this variety of applications, it isextremely important to understand thecharacteristic behaviour of moving fluid throughpipe.

Pipe fluid carrierFittings connectors for a desired configuration


Fittings connectors for a desired configuration of pipes

Pump as energy adderValve flowrate controller

4.2 General Characteristics of Pipe Flow

Conduits are not necessarily round cross-section but most ofthe common one such as water pipes, hydraulic hoses,and other conduits that are designed to withstand aconsiderable pressure difference across their wallswithout undue distortion of their shape.


Typical conduits of noncircular cross section includeheating and air conditioning ducts that are often ofrectangular cross section.

Normally the pressure difference between the inside andoutside of these ducts is relatively small. Hence, most of thebasic principles involved are independent of the cross-sectional shape, unless otherwise specified.


We consider a pipe completed filled with fluid as shownbelow (a).

We will look at the types of flow such as laminar, transitionalturbulent.


(a) Pipe flow (b) Open channel flow

Osborne Reynolds (1842 1912), a British scientist and mathematicianclassified these flows by using a simple apparatus below

He injected dye into a pipe in which water flowed due to gravity. Theentrance region of the pipe is depicted below.

Neutrally buoyant dye is injected, as shown, into the flowing fluid of



Neutrally buoyant dye is injected, as shown, into the flowing fluid ofgiven velocity V and pipe diameter D and a streaklike dye shape isformed (only in small enough flowrate)

VDRe



VDRe


Reynolds Number Flow

Re < 2300 Laminar


2300 < Re O 4000 Transitional

Re > 4000 Turbulent

Example 4.1

a. Evaluate the minimum time required to fillthe 1-liter bottle as shown if the flow is (i)Laminar (ii) Turbulent

b. Repeat a for 140oF temperature of thewater.


4.3 Fully Developed Flow

Fully developed flow

Each fluid particle moves at a constant axial velocity along a streamline The velocity profile u(r) remains unchanged in the flow direction. There is no motion in the radial direction. The velocity component in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady and fully developed.


4.3 Pressure Distribution


4.4 Laminar Flow

Laminar flow exhibits parallel streamlines

Between parallel surfaces, it can be considered that laminar flow is made

up of parallel layers that do not mix up at low average velocity

The Reynolds number indicates that a flow can be laminar, translational or

turbulent as a function of velocity, pipe diameter and viscosity

VDRe


There must be the critical velocity, diameter and viscosity

VDRe

4.4 Laminar Flow

Fully developed Laminar Flow


4.4 Laminar Flow

Fully developed Laminar Flow


Shear stress distribution within the fluid in a pipe (laminar or turbulentflow) and typical velocity profiles

4.4 Laminar Flow

rL

P

2

Shear Stress as a function of r

R

r

L

pRrv

2

2

14

)(

Velocity as a function of r

1

2

Average Velocity

L

pD

L

pRv

22

ave328

4

Volumetric Flow rate or Flow rate

pRpD

Q44

5


RL 4

L

pRv

2

max4

Maximum Velocity at r = 0

3

LL

Q8128

5

Head Loss

2g

v

D

Lfh

gD

Lvh

2

2

L

L

326

7

Hagen Poiseuille Equation

Darcy Energy Loss Equation

Example 4.2

A 200-m-long pipe made of 3-cm-diamter copper is used to

transport water at 5 liters / second (L/s) and at 4oC having viscosity

of 1.5028x10-3 kg/m.s. Determine the

a) average velocity of the water

b) friction factor

c) pressure drop


d) head loss

e) required pump power to overcome the head loss.

Example 4.3

Gasoline of density 680 kg/m3 and viscosity of 3.1x10-4 N.s/m2 flows

is to be transported in a smooth pipe of 40-mm diameter at a rate of

0.001m3/s. determine the ratio of turbulence and laminar head

losses to avoid the turbulence to occur.


4.5 Laminar vs Turbulent Flows

The fundamental difference between the laminar and turbulent flows is

that the laminar flow does not depend on the pipe wall surface

roughness thus the friction factor is constant (64/Re) regardless of the

relative roughness /D.


The turbulent flow is dependent on both density and the surface pipe

wall surface roughness. Thus the equation 7

2g

v

D

Lfh

2

L Re64 )D/(Re,f

(laminar) (Turbulent)

4.6 Minor and Major Losses

When fluid flows through pipe resistance against it exists in

various forms, thus there are pressure drops along the length of

the pipe.

This pressure drop is termed as loss. The loss is divided into two categories

namely minor and major losses


Minor Losses these losses occur in the following smooth flow interruptions

a. Inlets or exits

b. Sudden enlargement and contraction in a pipe.

c. Bends in a pipe.

d. Any other source of restriction such as pipe fittings and valves.


When fluid flows through pipe resistance against it exists in

various forms, thus there are pressure drops along the length of

the pipe.

This pressure drop is termed as loss. The loss is divided into two categories

namely minor and major losses


Major Losses these losses are due to the shearing resistance on the pipe

wall surface. Therefore, equation 7 defines the Major Loss or simply

is the Major Head Loss

and the total head loss in the pipe should be the

Minor Head Losses + Major Lead Losses

2g

v

D

Lfh

2

L


Minor losses are usually expressed in terms of loss coefficient

which is also called resistance coefficient where hL is the

additional irreversible head loss in the piping system caused by insertion of

the component and is defined as

g2/V

hK

2L

L

g

Ph LL

2

VKP

2

LL


ghL

2

KP LL

Minor losses are also expressed in terms of the equivalent length Lequiv

Where f is the friction factor and D is the diameter of the pipe that contains the

component.


Lequiv

2equiv

2

LL Kf

DL

g2

V

D

Lf

g2

VKh


The head loss caused by the component is equivalent to the head loss

caused by a section of the pipe whose length is Lequiv. This is simply

accounted for additional length for the pipe.

Minor Head Loss

g2

VKh

2

LL8

The total head losses therefore for a given pipe length with any of the

components (valves, elbows, etc) is the combination of the equation 7 and 8

i.e.


hhh minormajortotal ,L,L,L


For constant diameter

2g

V

2g

V

D

Lfh

2

j2

i

i

iitotal

jj,L

i,L K 9

2g

V K

D

Lf h

2

Ltotal

,L 10


Sharp-Edged Exit



Loss Coefficient for sudden contraction



Loss Coefficient for sudden expansion



Loss Coefficient for Round Edge



PIPE INLETS

Reentrant: KL = 0.80

T

Example 4.4

Water at 10C flows from a large reservoir to a smaller one

through a 5-cm diameter cast iron piping system, as shown in

Figure below. Determine the elevation z1 for a flow rate of 6 L/s.


FLUID MECHANICSLearning Outcomes

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Viscous Flow in Pipes