brad peterson, p.e.brad peterson, p.e. - weebly · soso a coefficient of velocity c a coefficient...

Brad Peterson, P.E.Brad Peterson, P.E.

New Website:http://njut2009fall.weebly.com

Mr. Peterson’s Email Address: bradpeterson@engineer [email protected]

Lesson 1, Properties of Fluids, 2009 Sept 04, Rev Sept 18 Lesson 2, Fluid Statics , 2009 Sept 11, Rev Sept 18 Lesson 2, Fluid Statics , 2009 Sept 11, Rev Sept 18 Lesson 3, Hydrostatic Force on Surfaces, 2009 Sept 25 Lesson 4, Buoyancy and Flotation, 2009 Sept 25 Lesson 5, Translation/Rotation of Liquid Masses, 2009 Oct 16

L 6 Di i l A l i /H d li Si ili d b ? Lesson 6, Dimensional Analysis/Hydraulic Similitude, maybe? Lesson 7, Fundamentals of Fluid Flow, 2009 Oct 23, Oct 30 Lesson 8, Flow in Closed Conduits, 2009 Nov 6 Lesson 9 Complex Pipeline Systems 2009 Nov 13 Nov 20 Lesson 9, Complex Pipeline Systems, 2009 Nov 13, Nov 20 Lesson 10, Flow in Open Channels, 2009 Nov 27, Dec 04 Lesson 11, Flow of Compressible Fluids , maybe? Lesson 12, Measurement of Flow of Fluids, 2009 Dec 11

l d b l d Lesson 13, Forces Developed by Moving Fluids Lesson 14, Fluid Machinery

Many devices are used to measure the flow of fluids:◦ For velocity: Pitot tubes Pitot tubes, Current meters, Anemometers.

Many devices are used to measure the flow of fluids (cont):◦ For quantity: Orifices Orifices, Tubes, Nozzles, Venturi meters, Weirs, Meters.

To use the devices, the Bernoulli Equation and knowledge of fluid characteristics is used.

Measures stagnation pressure (at B), which exceeds the local static pressure (at A), to determine velocity head.

Ah Bh Ah

Velocity (V) at Point B is zero.

h Bh

Apply the Bernoulli

Ah

Bernoulli equation, next slidenext slide

2 2A A B Bp V p Vno loss

2 20

A A B BA B

p pz z

g assumed gV

0; z,

B A BV zso

2

,

A A Bp V p

2g

2A A Bp V p

2g

p p

2 B Ap pV g

B AB A

p ph h d

With no friction:2V gd

g

B Ah h d

h Bh

Ah

A small amount of friction normally occurs, so a coefficient of velocity cV (see discussionso a coefficient of velocity cV (see discussion on following slides) is sometimes used:

actual velocity V

actual velocityctheoretical velocity

2VV c gd

to assume 1 provides sufficient accuracy for most engineering problems i l i Pi b

vc

involving Pitot tubes.

The ratio of the actual velocity in a stream to the theoretical velocity that would occurthe theoretical velocity that would occur without friction.

V

actual velocitycth ti l l it

theoretical velocity

( ) jetAarea of stream jet( )

jetC

O

f jcarea of opening A

opening jet

actual flow Qc Q

ctheoretical flow

also c c c

Value of c is provided in handbooks and text books

, V Calso c c c

pand is based on experimental data

For vertical, sharp-edged circular orifices:0 65 f di 1 d h d l th 0 25 t◦ c = 0.65 for dia<1cm and head less than 0.25m, to

◦ c = 0.59 for dia >1m and head greater than 20m◦ For most approximations, use c=0.6pp ,

2 2V V 2 2

2 2A A B B

A Bp V p Vno lossz z

g assumed g

2

0; p 0; 0; ; z 0A B A A B

g gp V z H

2

; 22

BB

VH V gH

g

2g

2

2BV

H 2

rearrange to:gg

2BV gH

2V gH 2

2BV gH

Q AV A gH

since, in most applications, friction willsince, in most applications, friction willoccur, apply the discharge coefficient, c

Q A 2 HQ=cA 2gH

A Pitot tube having a coefficient of 0.98 is used to measure the velocity of water at the center of a pipe. The stagnation pressure head is 5 67m and the static pressure head inhead is 5.67m and the static pressure head in the pipe is 4.73m. What is the velocity?

4 74m5.67m

4.74m

5 67 4 74 0 94B Ah h d

m m m

4 74m 5.67m

5.67 4.74 0.94m m m

4.74m

2VV c gd 25.67 4.73 0.94

VV c gdd m m m

2

0.98

9 8 /Vc

g m s

2

9.8 /

0.98 2 9.8 / 0.94 4.21 /

g m s

V m s m m s

A 100mm diameter standard orifice discharges waterdischarges water under a 6.1m head. What is the flow?

2Q cA gH 2area; 0.6

Q cA gHA c

2

total head causing flow 0 1

H

220.10.6 2 9.8 / 6.1

4mQ m s m

30.05 /Q m s

The tank in problem 12.9 is closed ant the air space above the water is under pressure, causing to flow to increase to 0.075m3/s. Find the pressure in the air spaceFind the pressure in the air space.

2Q cA gH

23 2

2

0.10.075 / 0.6 2 9.8 /

Q cA gH

mm s m s H

0.075 / 0.6 2 9.8 /

412.9

m s m s H

H m

3 2

12.9 6.1 6.8

9.8 / 6.8 70 / 70P Z

P

h H h m m m

p h kN m m kN m kPa

9.8 / 6.8 70 / 70Pp h kN m m kN m kPa

Measure the flow of water in open channels There are many formulas available to

calculate flow QAll f l h li i i All formulas have limitations

To be accurate, all must be calibrated (adjusted) experimentally or by actual on site(adjusted) experimentally or by actual on-site conditions

H head

Z height

b H

Z=0

Z=0

From: http://www.lmnoeng.com/Weirs/cipoletti.htmo ttp // oe g co / e s/c po ett t

Theoretical Formula for a Rectangular Weir:

3/2 3/22 22 2 V VQ b H 2

3 2 2

experimental coefficient

Q cb g Hg g

c

experimental coefficientcb widthH d h

H depth

3/2 3/22 2

several numbers, shown in red:

2 V V

2 22 23

V VQ b Hg g

c g

3/2 3/22 2

are combined into a coefficient,

(E A)V VQ b H

m

(Eq, A)

2 2Q b

gm H

g

in deep weir such as a dam, 0, so, approximate flow becomes:V negligible

3/2 3/22 20 02 2

Q b Hg g

m

3/2

2 2g g

3/2 , (Eq. B)Q bm H

During a test on a 2.4m suppressed weir that was 0.9m high, the head was maintained constant at 0.3m. In 38 seconds, 29m3 of water were collected Find the weir factor mwater were collected. Find the weir factor m using equations A and B.

0.3H m

0.9Z m

3329 0.763 /mQ m s 0.763 /

380.9 0.3 1.2

Q m ss

flow depth m m m 30.763 / 0.265 /

2.4 1.2Q m sV m sA m m

using Eq. A:3/2 3/22 2

g q

2 2V VQ b H

gm

g

3/2 3/22 2

3

2 2

0.265 0.2650 763 2 4 0 3

g

m

g

m

2 2

3/2 3/23

0.763 2.4 0.32 9.8 2 9.8

0 763 2 4 0 3 0 00358 0 00358

m

m

m

m

0.763 2.4 0.3 0.00358 0.00358

1.90

m

m

m

using Eq. B:3/2

using Eq. B:Q bHm

3/20.763 2.4 0.31.93

mm

1.93

1 90 1 93 Equation B is OK for weirs

m

1.90 1.93 Equation B is OK for weirs placed high

During a test on a 2.4m suppressed weir that was 0.0m high, the head was maintained constant at 0.3m. In 38 seconds, 29m3 of water were collected Find the weir factor mwater were collected. Find the weir factor m using equations A and B.

0.3H m

0.0Z m

b H

Z=0

Z=0

From: http://www.lmnoeng.com/Weirs/cipoletti.htmo ttp // oe g co / e s/c po ett t

3329 0.763 /mQ m s 0.763 /

380.3 0.30.0

Q m ss

flow depth m mm 30.763 / 1.06 /

2.4 0.3Q m sV m sA m m

using Eq. A:3/2 3/22 2

g q

2 2V VQ b H

g gm

3/2 3/22 2

3

2 2

1.06 1.060 763 2 4 0 3

g g

m m

2 2

3/2 3/23

0.763 2.4 0.32 9.8 2 9.8

0 763 2 4 0 3 0 0573 0 0573

m m

mm

0.763 2.4 0.3 0.0573 0.0573

1.53

mm

m

3/2

using Eq. B:Q bH 3/2

3/20.763 2.4 0.3Q

mbHm

1.93m

1 53 1 93 E ti A t b d1.53 1.93, Equation A must be used for shallow weirs

A commonly used weir for flow measurement This weir has side (end) slopes of:◦ 1 horizontal to 4 vertical

3/23.367Q bHH depthb idth

b width

8 5/28 tan 215 2

Q c gH

H d h

angle of "V" bottomH depth

5/2

Terms are combined into coefficient mQ mHQ mH

brad peterson, p.e.brad peterson, p.e. - weebly · soso a coefficient of velocity c a coefficient...

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