1 1
ABSTRACT
ABSTRACT
The
The expeexperirimenment t was was condconductucted ed in in ordorder er to to demdemonsonstratrate te BerBernoulnoulli`li`s s TheTheoreorem m toto investigate validity of Bernoulli`s
investigate validity of Bernoulli`s Theorem when applied to the steady flow of water inTheorem when applied to the steady flow of water in tapered duct and to measure the flow rates and both static and total pressure heads in a tapered duct and to measure the flow rates and both static and total pressure heads in a rigid convergent or divergent tube of geometry for a range of steady flow rates. In fluid rigid convergent or divergent tube of geometry for a range of steady flow rates. In fluid dynamics, Bernoulli principles states that for an viscid flow, an increase in the speed of dynamics, Bernoulli principles states that for an viscid flow, an increase in the speed of the fluid occurs silmutaneously with a decrease in pressure or a decrease in the fluid the fluid occurs silmutaneously with a decrease in pressure or a decrease in the fluid potential energy
potential energy . In . In order to order to demonstrate Bernoulli`s Theorem demonstrate Bernoulli`s Theorem , model , model FM!" BernoulliFM!" Bernoulli app
apparaaratus test is tus test is useused d in in thithis s expeexperimriment .The ent .The watwater er floflow w ratrate e is measuris measure e by by usiusingng volumetric method . The time to collect #$ water in the tan% was measured. $astly ,the volumetric method . The time to collect #$ water in the tan% was measured. $astly ,the flow rate , velocity , and by using continuity e&uation to find the velocities and find the flow rate , velocity , and by using continuity e&uation to find the velocities and find the difference of the velocities were calculated using the data of the results and from the data difference of the velocities were calculated using the data of the results and from the data given .Based on result ta%en it has been analysed that the velocity of the fluid is increase given .Based on result ta%en it has been analysed that the velocity of the fluid is increase when it flowing from wider to narrower tube regardless the type of flow and pressure when it flowing from wider to narrower tube regardless the type of flow and pressure different. The velocity is increase as
different. The velocity is increase as the pressure different is increase for all the pressure different is increase for all types of flow.types of flow. The
The velvelociocitieties s difdifferferent ent is is a a posposititive ive valvalue ue thathat t shoshown wn thithis s expeexperimriment ent is is valvalid id for for bernoulli`s e&uation.
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INTRODUCTION
INTRODUCTION
Bernoulli’s
Bernoulli’s theorem, theorem, inin fuidfuid dynamics is the relation among the pressure, dynamics is the relation among the pressure, ve
velolocicityty, , anand d elelevevatatioion n in in a a momoviving ng fufuid id whwhicich h arare e liliququid id or or gagas, s, ththee compressibility and viscosity which is internal riction o which are negligible compressibility and viscosity which is internal riction o which are negligible and the
and the fowfow o which is o which is steady, or laminar. (aniel Bernoulli ,1!"#$ .steady, or laminar. (aniel Bernoulli ,1!"#$ .
Bernoulli%s principle is named ater the &wiss scientist
Bernoulli%s principle is named ater the &wiss scientist aniel Bernoullianiel Bernoulli who published his principle in his boo' ydrodynamica
who published his principle in his boo' ydrodynamica in 1!"#. )n the 1!**s,in 1!"#. )n the 1!**s, aniel Bernoulli
aniel Bernoulli investigated the orces pres investigated the orces present in ent in a movinga moving fuidfuid The BritishThe British eng
engineineer er 'sb'sbornorne e (ey(eynoldnolds s ververify ify the the exiexistestence nce of of thethese se lamlaminainar r , , tratransinsilatlation ion andand turbulent flow regimes by in)ecting some dye strea%s into the flow in a glass pipe . turbulent flow regimes by in)ecting some dye strea%s into the flow in a glass pipe . +am
+aminainar r fow fow is is chacharacracterterieied d by by smosmooth oth strstreameamlinlines es and and highighly hly ororderdereded motion . -hen the fows is laminar , the dye strea' orms a straight and motion . -hen the fows is laminar , the dye strea' orms a straight and smooth line at low velocities and when the fow become turbulent , it has a smooth line at low velocities and when the fow become turbulent , it has a bu
bursrst t o o fucfuctutuatatioion n in in ththe e trtranansisititiononal al reregigime me anand d iiq q aag g rarapipidldly y anandd randomly. (./engel ,2**0$
randomly. (./engel ,2**0$
In fluid
In fluid dynamdynamics, Bernoulli princiics, Bernoulli principles states that ples states that for an for an visciviscid d flowflow, an , an increincrease inase in the speed of the fluid occurs silmutaneously with a decrease in pressure or a decrease in the speed of the fluid occurs silmutaneously with a decrease in pressure or a decrease in tthe he flfluiuid d popottenentitial al enenerergy gy .* .* ++..enengegel l , , !-!---// .Be.Bernornoullulli’s i’s ririncinciple ple cacan n bebe de
demomonsnstrtratated ed by by the the BeBernrnououlllli i eqequauatitionon. . hhe e BeBernrnououlllli i eqequauatition on is is anan appr
appro3o3imate imate relarelation tion betwbetween een prepressurssure, e, velocvelocity, ity, and and elevaelevation.tion.-hile -hile thethe /ontinuity equation relates the speed o a fuid that moving through a pipe to /ontinuity equation relates the speed o a fuid that moving through a pipe to
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states that the sum o the pressure, the potential, and 'inetic energy per unit states that the sum o the pressure, the potential, and 'inetic energy per unit volume is equal to a 53ed constant at any
volume is equal to a 53ed constant at any point o a fuid.point o a fuid.
Bernoulli%s principle can be e3plained in terms o the law o conservation Bernoulli%s principle can be e3plained in terms o the law o conservation o ener
o energygy. . he wor' donhe wor' done e by corby correrespospondinding volung volumes in the mes in the widwider ander and narrower pipes will be e3pressed by the product o the pressure and the narrower pipes will be e3pressed by the product o the pressure and the volume. &ince the speed is greater in the narrower pipe, the 'inetic energy o volume. &ince the speed is greater in the narrower pipe, the 'inetic energy o tha
that t volvolume ume is is grgreateaterer.h.hen, en, by by the the law o law o coconsernservatvation ion o o eneenergrgy, y, thithiss increase in 'inetic energy must be balanced by a decrease in the pressure4 increase in 'inetic energy must be balanced by a decrease in the pressure4 volume product, or, since the volumes are equal, by a decrease in pressure volume product, or, since the volumes are equal, by a decrease in pressure ..
Bernoulli6s heorem emonstration 7odel 8 97 2: apparatus consists o a Bernoulli6s heorem emonstration 7odel 8 97 2: apparatus consists o a cl
clasassisicacal l ;;ententuri uri mamade de o o clcleaear r acacryrylilic c . . seseriries es o o wawall ll tatappppingings s alalloloww measu
measuremerement nt o o the the statistatic c prepressure distribssure distribution along ution along the convergithe converging ng duct ,duct , while a total head tube is provided to traverse along the centre line o the while a total head tube is provided to traverse along the centre line o the ttesest t sesecctitioon n . . hehese se tatapppipinngs gs aarre e coconnnnececteted d to to a a mamanonomemeteter r bbanan'' in
incocorprpororatatining g a a mamaninioold ld wiwith th aiair r blbleeeed d vavalvlve. e. rresessusuririaatition on o o ththee manometers is acilitated by a hand pump.
manometers is acilitated by a hand pump.
his
his unit unit has has been been designed designed to to be be used used with with a a ydraulics ydraulics Bench Bench oror students to study the characteristics o fow through both converging and students to study the characteristics o fow through both converging and diverging sections . uring the e3periment , water is ed through a hose diverging sections . uring the e3periment , water is ed through a hose connector and students may control the fow rate o the water by ad<usting o connector and students may control the fow rate o the water by ad<usting o a fow regulator valve at the outlet o the
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applied as well as in other circumstances where the theorem is not su=cient applied as well as in other circumstances where the theorem is not su=cient to describe the fuid behavior.
to describe the fuid behavior.
AIM AIM
he ob<ective
he ob<ective o this o this e3periement is to e3periement is to demonstrate Bernoulli6s heorem.demonstrate Bernoulli6s heorem. Besid
Beside to e to invesinvestigattigate the validity o the Bernoe the validity o the Bernoulli equaulli equation when appltion when applied toied to the steady fow o water in a tapered duct and to 5nd the time ta'en to the steady fow o water in a tapered duct and to 5nd the time ta'en to co
collellect ct "+ "+ o o wawaterter, , the the volvolumetumetric ric fow fow ratrates es o o the the watwater,er,the the prpressessureure di>erence at all manometer tube
di>erence at all manometer tube which is static head, velocity, dynamic headwhich is static head, velocity, dynamic head and also the total head.
and also the total head.
THEORY THEORY
he
he bernoulli bernoulli equation equation is is an an approappro3imate 3imate relation relation between between pressure pressure ,, velocity , and elevation and is valid in a regions o steady , incompressible velocity , and elevation and is valid in a regions o steady , incompressible fow , where net
fow , where net rictional orces are negligible.( ./engrictional orces are negligible.( ./engel , 2**0 el , 2**0 $$ he
he well4'nown well4'nown Bernoulli Bernoulli equation equation is is derived derived under under the the ollowingollowing assumptions which are the liquid is incompressible and non4viscous. Beside assumptions which are the liquid is incompressible and non4viscous. Beside th
the e fofow w is is ststeaeady dy anand d ththe e vevelolocicity ty o o ththe e liliququid id is is leless ss ththan an ththe e crcrititicicalal velocity or the liquid. lso ,there is no loss o energy due to riction. hen , velocity or the liquid. lso ,there is no loss o energy due to riction. hen , this is e3pressed with the
this is e3pressed with the ollowing equation8ollowing equation8 ’’
Bernou
Bernoulli0lli0s s e&uatie&uation on showsshows th
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&uantities are constant. Therefore the three terms must be interchangeable so that, for &uantities are constant. Therefore the three terms must be interchangeable so that, for example, if in a hori1ontal system the velocity head is increased then the pressure head example, if in a hori1ontal system the velocity head is increased then the pressure head must decrease.
must decrease.
p 2 Fluid static pressure at the cross secti p 2 Fluid static pressure at the cross sectionon
3 2 4ensity of the flowing fluid 3 2 4ensity of the flowing fluid g 2 +cceleration due to gravity g 2 +cceleration due to gravity v 2 Mean velocity of fluid flow at
v 2 Mean velocity of fluid flow at the cross sectionthe cross section 1 2 5levation head of
1 2 5levation head of the center at the cross section with respect to the center at the cross section with respect to a datuma datum h6 2 Total *stagnation/ head
h6 2 Total *stagnation/ head
+ccording to the Bernoulli7
+ccording to the Bernoulli7s theorem of fls theorem of fluid uid flow through a pipe, the total head h6flow through a pipe, the total head h6 at any cross section is constant. In a real flow due to friction and other imperfections, as at any cross section is constant. In a real flow due to friction and other imperfections, as well as
well as measumeasurement uncertairement uncertaintiesnties, , the results will deviate the results will deviate from the from the theortheoretical ones. Inetical ones. In experimental setup, the centerline of all the cross sections are considering lie on the same experimental setup, the centerline of all the cross sections are considering lie on the same hori1ontal plane *which may choose as the datum, 1 2 -, and thus, all the 817 values are hori1ontal plane *which may choose as the datum, 1 2 -, and thus, all the 817 values are 1eros so that the above e&uation reduces to
1eros so that the above e&uation reduces to
This represents the total head This represents the total head at a cross section.
at a cross section.
The static pressure is that pressure which would be measured by an instrument moving The static pressure is that pressure which would be measured by an instrument moving with the flow. 9owever, such a measurement is rather difficult to ma%e in a practical with the flow. 9owever, such a measurement is rather difficult to ma%e in a practical situation. There was no pressure variation normal to straight streamlines. This fact ma%es situation. There was no pressure variation normal to straight streamlines. This fact ma%es it possible to measure the static pressure in a flowing fluid using a wall pressure tapping, it possible to measure the static pressure in a flowing fluid using a wall pressure tapping, placed in
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pressure tap is a small hole, drilled carefully in the wall, with its axis perpendicular pressure tap is a small hole, drilled carefully in the wall, with its axis perpendicular to theto the
sur
surfacface. e. If the If the holhole e is is perperpenpendicdiculaular r to to the duct the duct walwall l and and frefree e frofrom m burburrsrs, , accaccurauratete measurements of static pressure can be made by connecting the tap to a suitable pressure measurements of static pressure can be made by connecting the tap to a suitable pressure measuring instrument.
measuring instrument.
9
9iigguurre e ::88 mea
measursuremeement nt o o static pressure static pressure In a fluid stream far In a fluid stream far ffrroom m a a wwaallll, , oor r
where streamlines are curved, accurate static pressure measurements can be made by where streamlines are curved, accurate static pressure measurements can be made by careful use of a static pressure probe, shown in Figure "*b/. :uch probes must be careful use of a static pressure probe, shown in Figure "*b/. :uch probes must be designed so that the measuring holes are place correctly with respect to the probe tip and designed so that the measuring holes are place correctly with respect to the probe tip and stem to avoid erroneous results. In use, the measuring section must be aligned with the stem to avoid erroneous results. In use, the measuring section must be aligned with the loc
local al flflow ow dirdirectectionion. . :ta:tatitic c prepressssure ure proprobes or bes or any any varvarietiety y of of forforms ms are are avaavailailableble commercially in si1es as small as ;.< mm *;=; in./ in diameter. The stagnation pressure commercially in si1es as small as ;.< mm *;=; in./ in diameter. The stagnation pressure is obtained when a flowing fluid is decelerated to 1ero speed by a frictionless process. In is obtained when a flowing fluid is decelerated to 1ero speed by a frictionless process. In incom
incompresspressible flow, the Bernoullible flow, the Bernoulli i 5&uati5&uation on can can be used be used to relate to relate changechanges s in speed in speed andand pressure along a streamline for such a process. >eglecting elevation dif
pressure along a streamline for such a process. >eglecting elevation differences, soferences, so
If the static pressure is p at a point in the flow where the speed is v, then the stagnation If the static pressure is p at a point in the flow where the speed is v, then the stagnation pressure, ?o, where the stagnation speed, @
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Thus, if the stagnation pressure and the static pressure could be measured at a point, this Thus, if the stagnation pressure and the static pressure could be measured at a point, this would give the local flow speed.
would give the local flow speed.
9igure ? 8 9igure ? 8 silmutaneous
silmutaneous m
meaeassurureemement nt o o s
sttaaggnnaattiioon n aanndd static pressure
static pressure
In Figure <*a/,
In Figure <*a/, the static pressure corresponding to the static pressure corresponding to point + point + is read from the is read from the wall staticwall static pressure tap.
pressure tap. The stagnation The stagnation pressure is pressure is measured directly measured directly at + by the at + by the total head total head tube, astube, as shown. The stem of the total head tube is placed downstream from the measurement shown. The stem of the total head tube is placed downstream from the measurement
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There are factors can cause for pressure to vary along the pipe such as friction from There are factors can cause for pressure to vary along the pipe such as friction from the pipe7
the pipe7s s inninner er sursurfacface, e, the diamethe diameter of ter of the pipe if the pipe if it is it is smsmall the all the prepressssure is ure is lowlower er because
because the the velocity velocity is increased is increased there there are are bernoulli`s theory bernoulli`s theory , , density of density of the fluid the fluid in thein the pipe,
pipe, the the height height of of the the pipe pipe at at which which the the pipe pipe stands stands or or the the height height at at which which the the flowflow through i.e. gravity, and turbulence of the fluid. *:olte&,!-;"/
through i.e. gravity, and turbulence of the fluid. *:olte&,!-;"/
he
he venturi
venturi meter
meter consists
consists o
o a
a venturi
venturi tube
tube and
and di>er
di>erential
ential
pressure gauge. he venturi tube has a converging portion, a throat
pressure gauge. he venturi tube has a converging portion, a throat
and a diverging portion as shown in the 5gure below. he unction o
and a diverging portion as shown in the 5gure below. he unction o
the converging portion is to increase the velocity o the and lower its
the converging portion is to increase the velocity o the and lower its
static pressure. pressure di>erence between inlet and throat is thus
static pressure. pressure di>erence between inlet and throat is thus
dev
develop
eloped,
ed, whi
which
ch pr
pressu
essure
re di>
di>ere
erence
nce is
is cor
correl
related
ated with the
with the rate o
rate o
discharge. he diverging cone to change the area o the stream bac' to
discharge. he diverging cone to change the area o the stream bac' to
the entrance area and convert velocity head into pressure head.
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)deally, )deally,
owever, in owever, in
the case o real fuid fow, the fow rate will be e3pected to be less than that the case o real fuid fow, the fow rate will be e3pected to be less than that given by equation (".1#$ because o rictional e>ects and consequent head given by equation (".1#$ because o rictional e>ects and consequent head loss between inlet and
loss between inlet and throat. hereore,throat. hereore,
APPARATUS AND MATERIAL APPARATUS AND MATERIAL
h
h e e apap papa rratat uus s usus ed ed in in thth is is e3e3 pepe riri meme nnt t arare e BeBe rnrn ouou llll i’i’ s s prpr iincnc ipip lele appa
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1.
1. 7an7anomeometer ter ubeubes s 2. 2. est est &ec&ectiotionn ".
". --atater er )n)nlelet t :. :. nnioionsns ?. ir
?. ir Bleed Bleed &crew &crew 0. 0. ischarge ischarge ;;alvealve !.
!. Eland Eland nut nut #.ypodermic #.ypodermic probeprobe @
@.. dd<u<uststaablble e eeetet
9
9igure8 igure8 BernoulliBernoulli e3perimental e3perimental apparatus apparatus 5gure8 5gure8 stop watch stop watch 9 9iigguurre e 88 venturi meter venturi meter and the scale and the scale
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PROCEDURE
PROCEDURE
Ceneral :tartAup Ceneral :tartAup
The bernoullis theorem 4emonstration *model FM !"/ is supplied ready for use and only The bernoullis theorem 4emonstration *model FM !"/ is supplied ready for use and only re&uires connection to the 9ydraulic Bench *model FM ;;- / as follows
re&uires connection to the 9ydraulic Bench *model FM ;;- / as follows
;. The clear acrylic test section is ensure have been installed with the converging section ;. The clear acrylic test section is ensure have been installed with the converging section upstream . The unions have to tighten *hand tight only/. The total pressure probe have upstream . The unions have to tighten *hand tight only/. The total pressure probe have been withdrawn fully before releasing the couplings to dismantel the test sect
been withdrawn fully before releasing the couplings to dismantel the test section.ion.
!. The apparatus must be located on the flat top of the bench . !. The apparatus must be located on the flat top of the bench .
#. + spiri
#. + spirit level has been t level has been attached to baseboard and attached to baseboard and the feet must be ad)ust to level unit othe feet must be ad)ust to level unit onn top of the bench .
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E. @; was opened gradually and allow piping to fill with water until all air has been E. @; was opened gradually and allow piping to fill with water until all air has been expelled from the system .
expelled from the system .
;-. Trapped bubbles in the glass tube or plastic transfer tube have to chec%. The trapped ;-. Trapped bubbles in the glass tube or plastic transfer tube have to chec%. The trapped bubbles was removed from the system for better accuracy
bubbles was removed from the system for better accuracy..
>ote to remove the air bubbles , bleed the air out as follow >ote to remove the air bubbles , bleed the air out as follow
i.Cet a pen or screw driver to press the air bleed valve at the top right side of manometer i.Cet a pen or screw driver to press the air bleed valve at the top right side of manometer board .
board .
ii.+ir bleed valve was pressed lightly to allow fluid and
ii.+ir bleed valve was pressed lightly to allow fluid and trapped air to escape out.trapped air to escape out.
;;. Dater will flow into the venturi and discharge into the collection tan% of hydraulic ;;. Dater will flow into the venturi and discharge into the collection tan% of hydraulic bench at that time.
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indicated on its respective manometers . The total head pressure along the venture tube indicated on its respective manometers . The total head pressure along the venture tube can be measured by traversing the hypodermic tube.
can be measured by traversing the hypodermic tube.
>ote
>ote The The manometer manometer tube tube connected connected to to the the tapping tapping ad)acent ad)acent to to the the outlet outlet flow flow controlcontrol valve is used as a datum when setting up e&uivalent conditions for flow through test valve is used as a datum when setting up e&uivalent conditions for flow through test section.
section.
;<.The volumetric tan% with a stop watch
;<.The volumetric tan% with a stop watch was used to measure the actual flow of was used to measure the actual flow of water .water .
Bernoulli`s Theorem 4emonstration Bernoulli`s Theorem 4emonstration
;. the general start up procedures was
;. the general start up procedures was performed.performed.
!. +ll manometer tubing was connected properly to the corresponding pressure taps and !. +ll manometer tubing was connected properly to the corresponding pressure taps and are air bubble free.
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hypode
hypodermic tube was
rmic tube was held again
held against the fow o fuid or
st the fow o fuid orcing it to a stop
cing it to a stop
(
(er
ero
o ve
velo
locit
city$.
y$. he
he re
read
ading
ing in
in ma
mamo
momet
meter
er F
F me
measu
asure
res
s <u
<ust
st the
the
pressure tap , which does not obstruct the fow, thus measuring the
pressure tap , which does not obstruct the fow, thus measuring the
fow static pressure .
fow static pressure .
0. &tep ? was repeated or others cross section (FB ,F/, F, FA, F9$
0. &tep ? was repeated or others cross section (FB ,F/, F, FA, F9$
!. &tep " to 0 was repeated with three other decreasing fow rates by
!. &tep " to 0 was repeated with three other decreasing fow rates by
regulating the venturi discharge valve .
regulating the venturi discharge valve .
#. he velocity,;
#. he velocity,;
iBiBis calculated by using the Bernoulli6s equation where
is calculated by using the Bernoulli6s equation where
;
;
iBiBD
D
@.
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RESULT
RESULT
FIRST READING FIRST READING @olume *$/ @olume *$/ 33+verage Time *min/
+verage Time *min/ 0.280.28
Flow rate *$?M/
Flow rate *$?M/ 10.7110.71
r
rososs s sesectctiion on JsJsining g beberrnonoulullli i e&e&uauattiion on JJsising ng cocontntininuiuity ty e&e&uauattioionn 4i4iffffererenencece
ii h6 2 h h6 2 h99 *mm/ *mm/ h hii *mm/ *mm/ @ @iBiB22 ++ii2 2 @@icic 2 2 K Kavav= += +ii @ @iBiB2 @2 @icic *m=s/ *m=s/
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Start Free Trial Cancel Anytime. ;;GG-- ;;##-- --....HHEE -..-- ---!!--;; --..HHE E --..--- -4 4 ;;HH ;;<<;; --..<<HH -..-- ---##;;"" --..<<GG --..--;; 5 5 ;; ;;<<"" --..""HH -..-- ---##HH-- --..""GG --..--;; F F ;;<< ;;<<GG --..""-- -..-- ---<<##;; --..##"" --..-- SECOND READING SECOND READING
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Start Free Trial Cancel Anytime. B B ;;HHGG ;;-- --..GG## -..-- ---## --..""GG --..!! ;;HH## EEEE ;;..!!HH --..---!!--;; --..HH --..""!! 4 4 ;;GGEE ;;##HH --..HH-- -..-- ---##;;"" --..<<<< --..!!<< 5 5 ;;GG<< ;;"""" --..;; -..-- ---##HH-- --..""<< --..;; F F ;;GG"" ;;<<"" --..##EE -..-- ---<<##;; --..##!! --..--GG THIRD READING THIRD READING @olume *$/ @olume *$/ 33
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