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CE 3372 Water Systems Design
Closed Conduit Hydraulics-I
Flow in Closed Conduits
• Diagram• Energy Equation• Head Loss Models
– Pipe loss– Fitting loss
• Moody Chart Problems• Direct Method (Jain equations)• Branched Systems• Looped System
Diagram
Suction Side
Lift Station
Discharge Side
Diagram
Mean Section Velocity
• In most engineering contexts, the mean section velocity is the ratio of the volumetric discharge and cross sectional area.
• The velocity distribution in a section is important in determining frictional losses in a conduit.
€
V =Q
A
Energy Equation
• The energy equation relates the total dynamic head at two points in a system, accounting for frictional losses and any added head from a pump.
Energy Equation
1
2
Head Loss Models
• Darcy-Weisbach• Hazen-Williams• Chezy-Mannings
Darcy-Weisbach• Frictional loss proportional to
– Length, Velocity^2
• Inversely proportional to– Cross sectional area
• Loss coefficient depends on– Reynolds number (fluid and flow properties)– Roughness height (pipe material properties)
Darcy-Weisbach• Frictional loss proportional to
– Length, Velocity^2
• Inversely proportional to– Cross sectional area
• Loss coefficient depends on– Reynolds number (fluid and flow properties)– Roughness height (pipe material properties)
Darcy-Weisbach• DW Head Loss Equation
• DW Equation, Discharge Form, CIRCULAR conduits
Hazen-Williams• Frictional loss proportional to
– Length, Velocity^(1.8)
• Inversely proportional to– Cross section area (as hydraulic radius)
• Loss coefficient depends on– Pipe material and finish
• WATER ONLY!
Hazen-Williams• HW Head Loss
• Discharge Form
Hydraulic Radius• HW is often presented as a velocity equation
using the hydraulic radius
• The hydraulic radius is the ratio of cross section flow area to wetted perimeter
Hydraulic Radius• For circular pipe, full flow (no free surface)
AREAAREA PERIMETERPERIMETER
D
Chezy-Manning
• Frictional loss proportional to– Length, Velocity^2
• Inversely proportional to – Cross section area (as hydraulic radius)
• Loss coefficient depends on– Material, finish
Chezy-Manning
• CM Head Loss
• Discharge form replaces V with Q/A
Fitting (Minor) Losses
• Fittings, joints, elbows, inlets, outlets cause additional head loss.
• Called “minor” loss not because of magnitude, but because they occur over short distances.
• Typical loss model is
Fitting (Minor) Losses
• The loss coefficients are tabulated for different kinds of fittings
Moody Chart
• Moody-Stanton chart is a tool to estimate the friction factor in the DW head loss model
• Used for the pipe loss component of friction
Examples
• Three “classical” examples using Moody Char– Head loss for given discharge, diameter, material– Discharge given head loss, diameter, material– Diameter given discharge, head loss, material
Direct (Jain) Equations
• An alternative to the Moody chart are regression equations that allow direct computation of discharge, diameter, or friction factor.
Branched System• Distribution networks are multi-path pipelines• One topological structure is branching
Branched System• Node
– Inflow = Outflow– Energy is unique value
• Links– Head loss along line
Branched System
Head loss in each pipe
Common head at the node
Branched System
Continuity at the node
Branched System
• 4 Equations, 4 unknowns• Non-linear so solve by
– Newton-Raphson/Quasi-Linearization
• Quadratic in unknown, so usually can find solution in just a few iterations
Looped System• Looped system is extension of branching
where one or more pipes rejoin at a different node.
Looped System• Nodes:
– Inflow = Outflow– Energy Unique
• Links– Head loss along pipe– Head loss in any loop is zero
LOOP
Examples
• Branched System• Loop System
Hydraulic Grade Line
• Hydraulic grade line is a plot along a conduit profile of the sum of elevation and pressure head at a location.
• It is where a free surface would exist if there were a piezometer installed in the pipeline
Energy Grade Line
• Hydraulic grade line is a plot along a conduit profile of the sum of elevation, pressure, and velocity head at a location.
HGL/EGL