151300413 Fluid Flow Lecture Notes 9

8/12/2019 151300413 Fluid Flow Lecture Notes 9

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Lecture No. 9

MEASUREMENT OF FLUID FLOW

9.1 General types o lo! "eters

Obstruction meter. Fluid meters that belong to this type indicate a flowby a change in pressure.

Full bore meter (Venturi meter, orifice meter, flow nozzle)

Insertion meter (Pitot tube)

Variable area meter. An area meter is one in which the pressure drop isconstant and the reading is dependent upon a ariable flow area.

!otameter

Others" #agnetic field, orte$ shedding, turbine meter etc.

9.# $entur% "eter

Venturi meter is a full%bore obstruction meter with conerging and dierging

sections. It consists of a tube with a constricted throat which causes the

increase of fluid elocity at the e$pense of pressure. &he throat is followedby a gradually dierging section where the elocity is decreased with an

increase in pressure and slight friction losses. A schematic of Venturi meter

is shown in Figure '..

Figure '.. chematic of a Venturi meter

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$eloc%ty at t&e t&roat 'u#()

c + c

+ P g- -u +g /

= = ('.)

Mass lo! rate 'm0( an* +eloc%ty %n t&e "a%n l%ne 'u1()

+ + +m u A u A =0 ('.+)For incompressible fluid flow"

++ + +

Au u

A= ('.1)

where"

is the Venturi coefficient2

3+

diameter of throat 4

diameter of main line 4= 2

%P is the pressure drop, P5 P+2

is the density of the fluid2

/ is the differential head, ( P/ = )2

- is a correction factor.

I"portant notes)

If 6 7.+8, then .

Values of "

3 7.'9 for a well%designed orifice

7.'9 for 43 + to 9 inches

7.'' for 4: 9 inches

3 7.'9 for /erschel type with !e : +77,777

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Values of -"

For incompressible fluids (i.e., li;uids)" - 3

For compressible fluids (i.e., gases)"

+

+

+

+ +

P

PP

-P

P P

P P

= ('.)

where"p

.

=

ee also E,uat%on 1-#1and F%/ure 1-10of Perry

(*thed.)

&he permanent pressure loss across Venturi meter is dependent on

and the discharge cone angle () in the dierging section"

3 8 5 *o 7 to 8? of %P

: 8o 7 to 17? of %P

4isadantage" e$pensie compared to the other head meters.

9. Or%%ce "eter

A sharp% or s;uare%edged orifice is a clean%cut s;uare%edged hole with

straight walls perpendicular to the flat upstream face of a thin plate faced

crosswise of a channel.&he stream issuing from such an orifice attains its

minimum cross section (vena contracta) at a distance downstream of the

orifice which aries with .

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Figure '.+. chematic of a sharp%edged orifice

$eloc%ty at t&e or%%ce)

co o+ c

+ P g- -

u +g /

= = ('.8)

where" o is the orifice discharge coefficient

- is a correction factor.

@ther parameters are as defined in ection '.+.

I"portant notes)

Values of o3 f(!e, )"

+.87.+ 9

o 7.*8 7.8'8' 7.71+ 7.9 '.*

!e

('.)

=;uation '. has been plotted to gie F%/ure 9. below (or F%/ure

1-#-of Perry, *thed.)

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Figure '.1. oefficient of discharge for s;uare%edged circular (Figure7%+7 of Perry, *thed.)

Values of -"

For compressible fluid (i.e., gases)"

+

7. 7.18 P-

P

=

('.*)

ee also F%/ure 1-10of Perry (*thed.)

9.2 3%tot tu4e

A pitot tube is an insertion meter that measures local elocity (i.e., elocity at

a point). Applications include finding the elocity of a moing craft such as a

boat or an airplane. A similar deice, called Pitot static tube shown in Figure

'., is used to measure the elocity at different radial positions in a pipe.

*8

A

B


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Figure '.. Pitot tube with sidewall static tap (Fig. 7%8 Perry, *thed.)

$eloc%ty at A '%.e.5 !&ere t&e t%p %s locate*(5 uo)

o cu +g / ('.9)

where"

is the correction coefficient

I"portant notes)

Values of "

3 7.7 for simple Pitot tubes

3 7.'9 to for Pitot static tubes

For gases at elocities : +77 ft s%"

B Ao c

B B

P Pu +g

P

= ('.')

Cenerally, for compressible fluids"

c A Bo +

o

B

g P Pu

#a + #a ...

+

= ('.7)

For =;uations '.' and '.7, PAand PBare impact and static pressures,respectiely.

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If the Pitot static tube is inserted at the middle of a tube or pipe, theelocity measured is ma$imum (uo3 uma$). &o determine the aerageelocity in the tube or pipe, we use Figure '.8.

Figure '.8. Velocity ratio ersus !eynolds number for smooth circularpipes (Figure 7%* of Perry, *thed.)

9.6 Rota"eter

&he rotameter (Figure '.) consists of a solid floator plummetthat is free to

moe inside a gradually tapered ertical glass tube. &he fluid flows upwardand the flow rate is indicated by the e;uilibrium position reached by the floatwhich can be read from the adDacent scale usually etched on the glass tube.

Balance of forces at e;uilibrium (i.e., steady%state)"

4 C B f f f f f c c c

g g gF F F V V V

g g g = = ('.)

where"

F4 drag force resulting from form and s>in friction for flow aroundthe float

FB buoyant force acting so as to raise the float

FC graity force acting downward on a float

Vf olume of the float

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f density of the float

density of the fluid

Figure '.. chematic representation of a rotameter

=nergy balance between and + (see Figure '.)

+ ++ +

c

P P u uF 7

+g

= ('.+)where"

F summation of drag or friction losses

ontinuity e;uation

+ +m u u =0 ('.1)ombining e;uations '.+ and '.1

++ c+

+

Pu +g F

= ('.)

c +++

P

u +g

= ('.8)

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FB F

4

F

C

+

+

Flow


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4rag force (F4)

+4 f f f f F P P = ('.)where"

%Pf pressure drop acting on the top of the float

+f fraction of the ma$imum pressure drop (%P) that is not

recoered

ombining e;uations '., '.8 and '.

f f ! +

f +

+

+V g

u

=

('.*)

where"

!

f

= rotameter coefficient

f + (rotameter tapers gradually)

f f+ !

f +

+gVu

= ('.9)#ass flow rate

f f! +

f

+gVm

=0 ('.')

Important notes"

Proper design of rotameter float will ma>e !constant oer wide rangeof !e.

For a constant%density fluid in a single rotameter, the terms within thes;uare root symbol of e;uation '.9 are practically constant andindependent of flow rate2 hence,

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E ! +u ('.+7)

&he preceding e;uation shows that the elocity is e;ual to a constantmultiplied by the minimum cross section for flow (+).

4isadantage" e$pensie in large installations.

9.0 Notc&es an* We%rs

otch An opening in the side of the tan> or reseroir which e$tendsaboe the surface of the fluid. It is used to measure dischargeflow rate.

Geir A notch on a lager scale, usually found in riers. It maybesharp%crested but may also hae a substantial depth in thedirection of flow. It can be used to measure flow rate or raisewater leels.

T&e General We%r E,uat%on

Gith reference to Figure '.*,

Velocity through the strip" u +gh ('.+)

4ischarge through the strip" H Au b h +gh =H /

7 7

H b +gh h /

+

7

H +g bh dh ('.++)

Figure '.*. =lemental strip of flow through a notch

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Rectan/ular !e%r

Figure '.9. !ectangular weir

b 3 B

1/+ +

theo7

+H B +g h dh B +g/

1= = ('.+1)

1

+actual d

+H +g/

1= ('.+)

$notc& !e%r

Figure '.'. V%notch or triangular weir geometry

b + / h tan+= ('.+8)

/ +theo7

H + +gtan / h h dh+

=

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/1 8

+ +theo

7

+H + +gtan /h h

+ 8

= 8

+theo

9H +gtan /

8 +

= ('.+)8

+actual d

9H +gtan /

8 +

= ('.+*)

7UESTIONS 8 3ROLEMS)

. =$plain how a Pitot tube measures the speed of a boat.

+. =$plain the principle of ariable area meters.

1. A Pitot tube is inserted into the center of an air duct 7.9 m in diameter.

A pressure gage attached to the Pitot tube reads %P 3 ' m+. &he

Pitot tube coefficient is 7.'9+. alculate the mass flow rate of air, at a

temperature of 18o and a pressure of 7 >Pa.

. A Venturi meter with an entrance diameter of 7.1m and a throatdiameter of 7.+m is used to measure the olume of gas flowingthrough a pipe. &he discharge coefficient of the meter is 7.'.Assuming the specific weight of the gas to be constant at '.+ m 1,calculate the olume flowing when the pressure difference betweenthe entrance and the throat is measured as 7.7 m on a water J%tubemanometer.

8. A Venturi meter is used for measuring flow of water along a pipe. &hediameter of the Venturi throat is two%fifths the diameter of the pipe.

&he inlet and throat are connected by water%filled tubes to a mercuryJ%tube manometer. &he elocity of flow along the pipe is found to be

+.8 / ms, where / is the manometer reading in m of mercury.

4etermine the loss of head between inlet and throat of the Venturiwhen / is 7.' m. &he relatie density of mercury is 1..

. Brine (specific graity .9) is flowing though a 97%mm pipe at ama$imum rate of 7.78 m1s. In order to measure the flow rate, a

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sharp%edged orifice, connected to simple J%tube manometer is to be1'7 mm /g. Ghat size orifice should be installedK

*. A closed tan> has a 7.7+8%m diameter orifice in one of its erticalsides. &he tan> contains oil to a depth of 7. m aboe the centre ofthe orifice and the pressure in the air space aboe the oil is maintained

at 1*97 m+ aboe atmospheric. 4etermine the discharge from theorifice. &he coefficient of discharge of the orifice is 7. and therelatie density of oil is 7.'.

9. A Venturi meter is fitted in a horizontal pipe of 7.8 m diameter tomeasure the flow of water which may be anything up to +7 m1h. &hepressure head at the inlet for this flow is 9 m aboe atmospheric andthe pressure head at the throat must not be less than * m belowatmospheric. Between the inlet and the throat there is an estimatedfrictional loss of 7? of the difference in pressure head between thesepoints. alculate the minimum allowable diameter for the throat.

'. 4educe an e$pression for the discharge of water oer a right%angledsharp edged V%notch, gien that the coefficient of discharge is 7..

A rectangular tan> m by m has the same notch in one of itsshort ertical sides. 4etermine the time ta>en for the head, measuredfrom the bottom of the notch, to fall from 8 cm to *.8 cm.

7.4eelop a formula for the discharge oer a '7%degree V%notch weir interms of head aboe the bottom of the V.

A channel coneys 177 literssec of water. At the outlet endthere is a '7%degree V%notch weir for which the coefficient of dischargeis 7.89. At what distance aboe the bottom of the channel should the

weir be placed in order to ma>e the depth in the channel .17 mK Giththe weir in this position what is the depth of water in the channel whenthe flow is +77 literssecK

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151300413 Fluid Flow Lecture Notes 9

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