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Flow Measurement I

Hydromechanics VVR090

Need for Flow Measurements

Accurate record of discharge crucial in many applications.

Often required to estimate flow rate:

• to wastewater treatment plants

• to irrigation channels

• for water consumption in households/industry

• in rivers (evaluation of water resources, dilution potential)

• to hydropower plants (turbines; regulation of flow)

• to industrial processes

Purpose: to discuss a number of methodologies to measure the discharge in the field and laboratory.

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Flow Measurements

• stream gaging (rating curve)

• tracers

• measurement flumes

• weirs

• sluice gates

• point velocity measurements in a cross section

• turbine meters

• orifices, nozzles

Discharge Measurement Stations: Stage Recording

Stage: elevation of the water surface relative to a datum

Measurement method: Derive a functional relationship between stage and discharge (through calibration). Perform accurate measurements of the discharge at a number of different stages.

Rating curve

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Discharge Measurement Stations: Structures

Rectangular weirV-notch weir

Parshall flume Broad-crested weir

Criterion for Establishing Control Section for Discharge Measurement

Structure establishing the control section should:

• not create undesirable flow field disturbances

• have sufficient height to eliminate the effects of variable downstream conditions

• be designed so that small changes in discharge results in measurable change in the stage

• be stable and permanent (even under extreme conditions)

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Method of Discharge Measurement

1. Divide cross section into a number of subsections (N).

2. Determine average velocity in each subsection .

3. Summarize product between subsection area and average velocity:

1

N

i ii

Q u A=

=∑

iu

iA

Velocity might be measured with a propeller meter, electromagnetic current meter, or acoustic doppler velocimeter (ADV).

Propeller Meter

• measure uni-directional current

• sensitive to sediment

• not disturbed by vertical currents

• easy to operate

Water velocity proportional to the propeller rotational speed (calibration needed).

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Electromagnetic Current Meter

When the flowing water moves through a magnetic field a voltage is induced proportional to the velocity.

Acoustic Doppler Velocimeter (ADV)

A sound pulse is emitted and the doppler shift occurring when this pulse is reflected against particles in the water is related to the particle velocity (= water velocity).

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Measurement Strategy

Cross section is divided into a number of vertical sections (no section should contain more than 10% of the flow). 20-30 vertical sections are typically required. The number of measurements in a cross section should be limited to allow coverage during constant flow conditions.

Velocity profile in a subsection:

*2.5 lno

yu uy

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Average velocity is approximately obtained from:

(0.8 ) (0.2 )2

(0.4 )

u D u Du

u u D

+=

=

(see problem from tutorial)

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Integration Method to Obtain Flow

Numerical integration technique:

1 1

1 2 2

Ni i i i

i ii

b b b bQ u y − +

=

− −⎛ ⎞= +⎜ ⎟⎝ ⎠

∑

Trapezoidal method:

1/ 2 1/ 2

1

1 1

2

2 2

Ni i

i ii

i i i ii

d dQ u w

b b b bw

− +

=

− +

+⎛ ⎞= ⎜ ⎟⎝ ⎠

− −= +

∑

Structures for Flow Measurements

Structures may be placed in a channel to measure the flow rate (in some cases existing structures may be used to estimate the flow rate).

Examples of structures used for flow measurements:

• weirs

• critical-depth flumes

• culverts

• bridge piers

• dams

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Weirs

Types of weirs (classified according to shape):

• rectangular

• V-notch

• trapezoidal

• parabolic

• special type (e.g., Cipoletti, Sutro)

Distinguish between:

• Broad-crested

• Sharp-crested

Broad-Crested Weir

A structure with a horizontal crest above which the water pressure is considered hydrostatic.

Limits: 0.08 < H1/L < 0.50

(0.08 from limit for energy losses; 0.50 from curvature of the streamlines)

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Broad-Crested Weir

Discharge Formula for Rectangular Broad-Crested Weir

Apply Bernoulli equation between upstream section and the control section (critical depth occurs here).

h1H1

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Modified discharge formula for rectangular broad-crested weir (taking into account simplifying assumptions):

1/ 23/ 21

2 23 3D vQ C C g Th⎛ ⎞= ⎜ ⎟⎝ ⎠

Cv: velocity coefficient (correct for neglecting velocity head in the approach channel)

CD: discharge coefficient (correct for neglecting viscous effects, turbulence, non-uniform velocity distribution etc)

Velocity coefficient Cv is given by:

1

1v

HCh

φ⎛ ⎞

= ⎜ ⎟⎝ ⎠

f: power of the variable h in the discharge equation

H1 is normally not known => equation hard to use

Diagram over Cv

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Rectangular Broad-Crested Weir

Typical layout

Weir block placed in a rectangular channel perpendicular to the flow direction

Discharge Coefficient

CD assumed constant if:

1 1

1

0.08 0.33 and 0.35h hL h p

≤ ≤ ≤+

CD = 0.848 (basic discharge coefficient)

Outside the limits CD should be multiplied with a correction coefficient F.

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Triangular Broad-Crested Weir

Suitable for measuring a wide range of discharges.

Discharge formula for triangular broad-crested weir:

( )1/ 2

5/ 21

16 2 tan / 225 5D vQ C C g h⎛ ⎞= θ⎜ ⎟

⎝ ⎠

Discharge coefficient given by diagram:

(Cv from previous figure)

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Sharp-Crested Weir

Weir is considered sharp-crested if H1/L > 15.

Sufficient air should be supplied behind the nappe.

Sharp-Crested Weir

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Discharge formula for sharp-crested weir:

( )1

1/ 21

0

2 ( )h

eQ C g b z h z dz= −∫

Main assumptions in the derivation:

• height of water level above weir crest is h1

• velocity over the crest horizontal

• approach velocity can be neglected

Definition Sketch: Deriving Discharge Formula for Sharp-Crested Weirs

Derivation of discharge equation not based on the presence of critical section; integrate through the section at the weir crest. Employ Bernoulli equation to get velocity through this section.

h1

z

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Correct for assumptions made by introducing a discharge coefficient Ce.

( )1

1/ 21

0

2 ( )h

Q g b z h z dz= −∫

( )12u g h z= −

Velocity in control section (from Bernoulli equation):

Total flow over the weir:

( )1/ 2 3/ 21

2 23 eQ C g bh=

Rectangular sharp-crested weir:

Three weir types:

• fully contracted

• partially contracted

• full width

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Discharge coefficient:

1

1

1

1

/1 0.602 0.075 /0.8 0.597 0.045 /0.6 0.593 0.018 /0.4 0.591 0.0058 /

e

e

e

e

e

b T CC h pC h pC h pC h p

= +

= +

= += +

( ) ( )1/ 2 5/ 21

8 2 tan / 215 eQ C g h= θ

Triangular sharp-crested weir:

Two weir types:• fully contracted• partially contracted

Ce coefficient

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Discharge Equations for Weirs I

Discharge Equations for Weirs II

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Other Weirs

Trapezoidal control section.

Cipoletti weir

Sutro weir Discharge linearly proportional to total head (above some level)

Discharge formulas:

Cipoletti weir ( )1/ 2 3/ 21

2 23 D vQ C C g Th=

Sutro weir ( ) ( )1/ 212 / 2DQ C b ga h a= −

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Example: Sharp-Crested Weir

Treating the upper edge of the pipe as a sharp weir crest, estimate the flow rate when the water depth in the basin is 0.75 m.