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