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Chapter 8: Flow in Pipes
Chapter 8: Flow in Pipes 2
OBJECTIVES
1. Have a deeper understanding of laminar and turbulent flow in pipes and the analysis of fully developed flow
2. Calculate the major and minor losses associated with pipe flow in piping networks and determine the pumping power requirements
3. Understand the different velocity and flow rate measurement techniques and learn their advantages and disadvantages
Chapter 8: Flow in Pipes 3
INTRODUCTION
Flowing real fluids exhibit viscous effects, they:Stick to solid surfacesHave stresses within their body
From earlier we saw relationship between shear stress and velocity gradient:
τ α du/dyThe shear,τ in a fluid is proportional to velocity gradient the rate of change of velocity across the flowFor a ‘Newtonian’ fluid we can write:
τ = μ du/dyWhere μ is coefficient of velocity (or simply viscosity)
Chapter 8: Flow in Pipes 4
LAMINAR & TURBULENT FLOW
Injecting a dye into the middle of flow in pipe, what would we expect to happen.
All this three would happen with different flow rates.
Chapter 8: Flow in Pipes 5
LAMINAR & TURBULENT FLOW
Laminar flow:Motion of the fluid particles is very orderly, all particles moving in straight lines parallel to the pipe wallsEncountered when highly viscous fluid such as oil flows in small pipes or narrow passage.
Turbulent flowMotion is, locally, completely random but the overall direction of flow is one wayIn practice, most flows are turbulentThe intense mixing fluid in turbulent flow as the result of rapid fluctuation enhance momentum transfer between fluid particles which increase the friction force on the surface and thus required pumping powerThe friction factor reached maximum when flow become fully turbulent
Chapter 8: Flow in Pipes 6
REYNOLDS NUMBER, Re
Critical Reynolds number (Recr) for flow in a round pipe
Re < 2300 laminar2300 ≤ Re ≤ 4000 transitional Re > 4000 turbulent
Note that these values are approximate.
For a given application, Recr depends upon
Pipe roughnessVibrationsUpstream fluctuations, disturbances (valves, elbows, etc. that may disturb the flow)
Definition of Reynolds number
Reynold number is a non-dimensional number.
Has no unit!
Chapter 8: Flow in Pipes 7
Laminar and Turbulent Flows
Chapter 8: Flow in Pipes 8
Laminar and Turbulent Flows
For non-round pipes, define the hydraulic diameter Dh = 4Ac/PAc = cross-section areaP = wetted perimeter
Example: open channelAc = 0.15 * 0.4 = 0.06m2
P = 0.15 + 0.15 + 0.5 = 0.8mDon’t count free surface, since it does not
contribute to friction along pipe walls!
Dh = 4Ac/P = 4*0.06/0.8 = 0.3mWhat does it mean? This channel flow is
equivalent to a round pipe of diameter 0.3m (approximately).
Chapter 8: Flow in Pipes 9
The Boundary Layer
Chapter 8: Flow in Pipes 10
Fully Developed Pipe Flow
Comparison of laminar and turbulent flowThere are some major differences between
laminar and turbulent fully developed pipe flows
LaminarCan solve exactly
Flow is steady
Velocity profile is parabolic
Pipe roughness not important
It turns out that Vavg = 1/2Umax and u(r)= 2Vavg(1 - r2/R2)
Chapter 8: Flow in Pipes 11
Fully Developed Pipe Flow
TurbulentCannot solve exactly (too complex)Flow is unsteady (3D swirling eddies), but it is steady in the meanMean velocity profile is fuller (shape more like a top-hat profile, with very sharp slope at the wall) Pipe roughness is very important
Vavg 85% of Umax (depends on Re a bit)No analytical solution, but there are some good semi-empirical expressions that approximate the velocity profile shape.
Instantaneousprofiles
Chapter 8: Flow in Pipes 12
Chapter 8: Flow in Pipes 13
Types of Fluid Flow Problems
In design and analysis of piping systems, 3 problem types are encountered
1. Determine p (or hL) given L, D, V (or flow rate)Can be solved directly using Moody chart and Colebrook equation
2. Determine V, given L, D, p3. Determine D, given L, p, V (or flow rate)
1. Types 2 and 3 are common engineering design problems, i.e., selection of pipe diameters to minimize construction and pumping costs
2. However, iterative approach required since both V and D are in the Reynolds number.
Chapter 8: Flow in Pipes 14
Major and Minor losses
Total Head Loss( hLT) = Major Loss (hL)+ Minor Loss (hLM)
g
V
D
LfhEquationsDarcy l 2
'2
Due to wall friction Due to sudden expansion, contraction, fittings etc
g
VKhlm 2
2
K is loss coefficient must be determined for each situation
For Short pipes with multiple fittings, the minor losses are no longer “minor”!!
Chapter 8: Flow in Pipes 15
Major loss
Physical problem is to relate pressure drop to fluid parameters and pipe geometry
Differential Pressure Gauge- measure ΔP
PipeD
V
L
ρ μ ε
),,,,,( DLVP
Using dimensional analysis we can show that
DD
LVD
V
P
,,
2
1 2
Chapter 8: Flow in Pipes 16
Friction factor
g
V
D
Lfh
g
P
VD
LfPor
VfD
LPie
D
VDffactorfrictiondefine
VD
VD
D
LP
D
VD
D
L
V
P
LL
L
2
2
1
2
1
,
2
1,
,
2
1
2
2
2
2
2
2
2
2
11
2
VL
DPf
g
V
D
Lfh
L
L
Chapter 8: Flow in Pipes 17
Friction Factor
For Laminar flow ( Re<2300) inside a horizontal pipe, friction factor is independent of the surface roughness.
Df
D
VDf
Re,,
asiprelationshfunctionalthederivecanwellyTheoretica
onlyfie .Re
• For Turbulent flow ( Re>4000) it is not possible to derive analytical expressions.
• Empirical expressions relating friction factor, Reynolds number and relative roughness are available in literature
Re
64f For Laminar flow
Chapter 8: Flow in Pipes 18
Minor Losses
Piping systems include fittings, valves, bends, elbows, tees, inlets, exits, enlargements, and contractions.
These components interrupt the smooth flow of fluid and cause additional losses because of flow separation and mixing
We introduce a relation for the minor losses associated with these components
• KL is the loss coefficient.
• Is different for each component.
• Is assumed to be independent of Re.
• Typically provided by manufacturer or generic table (e.g., Table 8-4 in text).
Chapter 8: Flow in Pipes 19
Minor Losses
Total head loss in a system is comprised of major losses (in the pipe sections) and the minor losses (in the components)
If the piping system has constant diameter
i pipe sections j components
Chapter 8: Flow in Pipes 20
Flow at pipe inlets and losses from fittings
Rounded inlet Sharp-edged inlet
Head loss for inlets, outlets, and fittings:
where K is a parameter that depends on the geometry.For a well-rounded inlet, K = 0.1, for abrupt inlet K = 0.5(much less resistance for rounded inlet).
Chapter 8: Flow in Pipes 21
Bends in pipes:
Sharp bends result in separation downstream of the bend.
The turbulence in the separation zone causes flow resistance.
Greater radius of bend reduces flow resistance.
Chapter 8: Flow in Pipes 22
Chapter 8: Flow in Pipes 23
Chapter 8: Flow in Pipes 24
Transition losses and grade lines
Head loss due to transitions (inlets, etc.) is distributed over some distance.Details are often quite complicated.
Approximation: Abrupt losses at a point.
Chapter 8: Flow in Pipes 25
Chapter 8: Flow in Pipes 26
Chapter 8: Flow in Pipes 27
Piping Networks and Pump Selection
Two general types of networks
Pipes in seriesVolume flow rate is constantHead loss is the summation of parts
Pipes in parallelVolume flow rate is the sum of the componentsPressure loss across all branches is the same
Chapter 8: Flow in Pipes 28
Piping Networks and Pump Selection
For parallel pipes, perform CV analysis between points A and B
Since p is the same for all branches, head loss in all branches is the same
Chapter 8: Flow in Pipes 29
Piping Networks and Pump Selection
Head loss relationship between branches allows the following ratios to be developed
Real pipe systems result in a system of non-linear equations. Very easy to solve with EES!Note: the analogy with electrical circuits should be obvious
Flow flow rate (VA) : current (I)Pressure gradient (p) : electrical potential (V)
Head loss (hL): resistance (R), however hL is very nonlinear
Chapter 8: Flow in Pipes 30
Piping Networks and Pump Selection
When a piping system involves pumps and/or turbines, pump and turbine head must be included in the energy equation
The useful head of the pump (hpump,u) or the head extracted by the turbine (hturbine,e), are functions of volume flow rate, i.e., they are not constants.Operating point of system is where the system is in balance, e.g., where pump head is equal to the head losses.
Chapter 8: Flow in Pipes 31
Pump and systems curves
Supply curve for hpump,u: determine experimentally by manufacturer. When using EES, it is easy to build in functional relationship for hpump,u.
System curve determined from analysis of fluid dynamics equations
Operating point is the intersection of supply and demand curves
If peak efficiency is far from operating point, pump is wrong for that application.
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