CS1022 Computer Programming & Principles Lecture 1.1 Introduction to Course.
LECTURE 1.1
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Transcript of LECTURE 1.1
9/9/2014
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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