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Flow Measurement II

Hydromechanics VVR090

Flow Measurements in Flumes

Use of flumes might be preferrable to measure flow rate compared to weirs.

Disadvantage of weir installations:

• results in relatively high head losses

• ”dead zone” created upstream (risk of sedimentation)

Measurement principle in a flume: contraction in width that generate a critical section

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Parshall Flume

Similar to a Venturi meter used in pipe flow.

Critical flow occurs in the flume throat followed by a hydraulic jump downstream.

Theoretical Analysis of Flow Rate

2 2

2

32 2 2

23 2

a ca c c

ac a

u uH y yg g

uy Hg

+ = + =

⎛ ⎞= +⎜ ⎟

⎝ ⎠

Energy equation from upstream point to critical section:

Continuity equation gives the flow rate:3/ 22

3/ 23/ 2

23 2

23

ac c c c a

a

uQ Wu y W gy y W g Hg

gWH

⎛ ⎞⎛ ⎞= = = +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

⎛ ⎞≈ ⎜ ⎟⎝ ⎠

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Empirical Discharge Relationships for Parshall Flumes

General equation:

Flow rate depends on flume width (W) and upstream water depth (Ha).

BaQ AWH=

Empirical discharge equations

Discharge from the flume should occur under free conditions.

⇒ A hydraulic jump develops and no effects from the downstream water level (Hb).

However, under some conditions the jump become submerged and Hb influences the discharge.

Submergence of the jump occurs if:

0.6 0.8b

a

HH

≥ −

(depending on width)

The main effect of the submergence is that the flow rate is reduced.

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Submerged flow:

''

BaQ AWH

Q CQ=

=

(a correction factor C is applied)

Figures to estimate flow rate for submerged conditions.

Q depends of Ha and Hb/Ha.

Flow Measurements in Culverts

For the case of flow in a culvert through an embankment measurements of the flow rate may be carried out.

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Culvert Flow

Flow through culvert depends on:

• upstream and downstream water level

• geometric properties of the culvert

• energy losses through the culvert

entrance losses

exit losses

frictional losses

The discharge through the culvert is determined from the continuity and energy equations (from an upstream to a downstream location).

Flow through the culvert divided into six cases.

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Culvert Flow Characteristics I

Classification based on the relationship between head (y1) and tailwater (y4) heights.

Culvert Flow Characteristics II

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Culvert Flow Characteristics III

Venturimeters

Involves a constriction in the flow. The constriction produces an accelerated flow and a fall in the hydraulic grade line (pressure) directly related to the flow rate.

Pressure measured at section 1 and 2

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Employ energy equation between 1 and 2 + continuity equation:

In practice, introduce a coefficient Cv to take into account frictional effects:

( )2 1 2

1 222 1

21 /

A p pQ g z zA A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠−

( )2 1 2

1 222 1

21 /

vC A p pQ g z zA A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠−

Nozzles

Often used to create jets and streams, but also for fluid measurements (flow nozzle). A flow nozzle is essentially a venturimeter without the diffuser cone.

Cv as a function of Re

Same equation applies as for the Venturimeter.

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Orifices

Used for many purposes in engineering, including measuring the flow rate.

A difference compared to nozzles is that the minimum flow section does not occur at the orifice but some distance downstream (in vena contracta).

Flow rate is given by:

( )1 2

1 2221

21 /

v c

c

C C A p pQ g z zC A A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠−

Area in vena contracta is:2 cA C A=

Flow through an orifice is often written as:

( )

1 21 2

221

2

1 /v c

c

p pQ CA g z z

C CCC A A

⎛ ⎞= + − −⎜ ⎟γ γ⎝ ⎠

=−

Hard to locate the downstream measurement instrument in vena contracta. Instrument located at a fixed distance downstream from the orifice.

Empirically determined values for C as a function of Reavailable.

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Submerged Orifice

Discharge from one large reservoir to another.

1/ 0

v c

A AC C C

≈→ ≈

Energy equation from upstream reservoir to section 2:

( )

22

1 2

2 1 2

2

2

Vh hg

V g h h

= +

= −

(hydrostatic pressure distribution in section 2)

For a real fluid:

( )2 1 22vV C g h h= −

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Flow rate:

( ) ( )2 2 1 2 1 22 2c vQ A V C C A g h h CA g h h= = − = −

Discharge to the atmosphere:

2 2c vQ C C A gh CA gh= =

Orifice Coefficients

Re number dependenceGeometry dependence

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Sluice Gate

Special case of orifice flow: only contraction on the top of the jet.

Pressure in vena contracta is assumed to be hydrostatic.

Energy equation from section 1 to 2 (ideal fluid):

2 21 2

1 22 2V Vy y

g g+ = +

Continuity equation: 21 2

1

yV Vy

=

Substitute in:

( )( )2 1 22

2 1

1 21 /

V g y yy y

= −−

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Adopt to a real fluid:

( )( )1 22

2 1

21 /

v cC C Aq g y yy y

= −−