LECTURE 1.1

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LEARNING OUTCOMES:

1. To apply Hagen Poisseuille equation on laminar flows in bounded system

2. To apply Darcy Weisbach equation on laminar and turbulent flows in bounded system

3. To determine the flow friction factor, f using Moody chart

4. To determine head losses in pipe flow due to friction, separation (sudden contraction & expansion) and pipe fittings

TOPIC 1 : REAL FLUID FLOW

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1.1 Laminar Flow through Circular Pipes

1.2 Turbulent Flow in Bounded Conditions

1.3 Determining Friction Factor, f

using Moody Chart

1.4 Different forms of the Darcy Equation

1.5 Pipe Problems

1.6 Separation losses in pipe flow

1.7 Equivalent Length

CHAPTER 1

STEADY FLOW IN PIPES

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Laminar flow through Circular Pipes

under Steady and Uniform conditions

LECTURE 1.1

STEADY FLOW IN PIPES

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Deals with analysis of pipe flow running under

STEADY and UNIFORM condition.

Pipes are analysed for both laminar and turbulent

flow.

Head loss due to friction - results from shear stresses

derived by both laminar and turbulent flow.

Head loss also results in flow separation.

INTRODUCTION

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STEADY FLOW

Fluid flow in which all the conditions at any one point are constant with respect

to time

UNIFORM FLOW

Fluid flow in which all the conditions at any one point are constant with respect

to space

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Can be analysed theoretically by applying

momentum equation to the element of flow within the

pipe

The relationship between the shear stress and

velocity gradient is utilised to obtain the velocity

distribution within the cross section

Laminar Flow in Circular Pipes under

Steady and Uniform Conditions

No slip condition used for a given fluid where the

velocity of the fluid in contact with solid boundary =

velocity of the solid boundary

Stationary pipe case the velocity will equal to

zero = the velocity of the fluid at the location where

it is in contact also equals to zero

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Laminar Flow in Circular Pipes

• Annulus of radius r with elemental

thickness, δr

• Annulus part of the fluid flowing in

the pipe with radius R

p+p

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Velocity distribution in pipe due to laminar flow

Fluid density, ρ

Momentum equation applied to the annular element and summing the forces and equating them to the rate of change of momentum

Laminar Flow in Circular Pipes

p+p

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Force in the direction of motion due to :

Pressure, p

Shear stress,

Weight of element, W

Force in opposite direction due to :

p +P

+

p+p

Laminar Flow in Circular Pipes

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PRESSURE

Force due to the static pressure, p (in the direction of

flow) = pressure multiplied by the cross sectional

area of the annulus.

p= F/A

F =p. A

Similarly force due to p + δp, (in the opposite

direction)

Laminar Flow in Circular Pipes

= (p+ (p/x). x) 2 r r

EQN 1

EQN 2

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Laminar Flow in Circular Pipes

SHEAR FORCE

• Shear force acting along the inner surface

of the annulus

Shear Force = A

EQN 3

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Laminar Flow in Circular Pipes

• Similarly the shear force acting on the

outer surface (in the opposite direction)

Shear Force = (+ ) A

= ( + (/r). r) . 2rx *Ignoring higher order terms

EQN 4

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Laminar Flow in Circular Pipes

WEIGHT

• Weight of the element is equal to the

Weight = density x volume x g

where g is the gravitational acceleration

W= mg = g

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W sin

Where W = .2r. r x. g

sin = - (z/x)

W sin = -2r rx. g (z/x)

EQN 5

p+p

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Laminar Flow in Circular Pipes

MOMENTUM EQUATION

• Applying momentum equation, sum of the forces in

the direction of flow, must be equal to zero since there

is no acceleration of steady flow

Sum up all the forces, F = 0

(EQN 1) - (EQN 2) + (EQN 3) – (EQN 4) + (EQN 5) = 0

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• Substituting W = .2r. r x. g and sin = - (z/x) and

cancelling out terms, will yield,

Laminar Flow in Circular Pipes

01

01

01

xPgzr

rr

x

zgr

rrx

P

x

zgr

rrx

P

Piezometric Pressure

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PIEZOMETRIC PRESSURE

• It is independent of r, thus expression can be integrated

with respect to r

• Integrating the equation,

where C is the constant of integration • Substituting values at the centreline where r = 0, C = 0

Laminar Flow in Circular Pipes

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SHEAR STRESS & VELOCITY GRADIENT RELATIONSHIP

• Circular pipes distances are measured from the centre (r),

modify the expression

Laminar Flow in Circular Pipes

EQN 6

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Laminar Flow in Circular Pipes

• Substitute equation 6 into

• Result in the following

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• Hence velocity gradient, du

• Velocity obtained by integrating the

expression

Where D is the constant of integration

Laminar Flow in Circular Pipes

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Applying the boundary condition at the pipe wall,

where r = R, velocity, u = 0 due to no slip condition

Laminar Flow in Circular Pipes

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Substituting into

results in the velocity distribution across the cross

section in circular pipe of radius R

Laminar Flow in Circular Pipes

Where,

u = velocity at a specific point in any cross section

R = internal radius of the pipe

r = distance of the point measured from the centre of

the pipe x- section

μ = dynamic viscosity

p = pressure

= density of the fluid flowing

g = gravitational acceleration

z = elevation

dx = incremental distance along the pipe

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Previous equation describes the variation of the

velocity across the pipe cross section

Velocity variation is parabolic in nature

The maximum velocity occur in the centre of the pipe

where r = 0

Hence, maximum velocity

Laminar Flow in Circular Pipes

Velocity profile in a circular pipe for laminar flow

MAXIMUM VELOCITY

Flow rate can be determined by integrating the

incremental flow, δQ through

– Annulus of radius, r

– Thickness δr across the flow from r = 0 or r = R

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Laminar Flow in Circular Pipes

FLOW RATE

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Laminar Flow in Circular Pipes

Q = v . A

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Flow Rate in a Circular Pipes for

Laminar Flow

EQN 7

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Average velocity, v can be computed by

v = Q/A

where A = cross sectional area

A = πR2 or πd2/4, d = diameter of pipe

Therefore, substitute

Q = Av in the equation 7

where A = πR2

Average velocity for Laminar Flow in

Circular Pipes

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Average velocity for Laminar Flow in

Circular Pipes

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Comparing

&

yields,

Average velocity in the pipe is equal to half of the

maximum velocity (occurs in the centre)

Relationship between Average

velocity and Maximum velocity for

Laminar Flow in Circular Pipes

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Substituting with

(the pressure drop per length of the pipe)

into equation 7. Thus,

Finding Pressure drop for Laminar

Flow in Circular Pipes

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Rearranging the equation to obtain Δp,

Thus,

also known as the Hagen Poisseuille equation

Hagen Poisseuille equation for

Laminar Flow in Circular Pipes

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Head loss due to friction,

Substitute the head loss due to friction in the Hagen

Poisseuille equation

Head Loss due to friction for

Laminar Flow in Circular Pipes

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EXAMPLE 1.1

Laminar Flow in Circular Pipes

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Solution for example 1.1

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Solution for example 1.1

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Solution for example 1.1

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Laminar Flow in Circular Pipes

EXAMPLE 1.2

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Solution for example 1.2

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Solution for example 1.2

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Solution for example 1.2

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Solution for example 1.2

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Solution for example 1.2

L= dx = incremental distance along the pipe

p = p

z if not given, assume 0

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Laminar Flow in Circular Pipes

EXAMPLE 1.3

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Solution for example 1.3

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Laminar Flow in Circular Pipes

EXAMPLE 1.4

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Solution for example 1.4

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Solution for example 1.4

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Solution for example 1.4