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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 ater the &wiss scientist

Bernoulli%s principle is named ater the &wiss scientist aniel Bernoullianiel 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 chacharacracterterieied 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 iiq q aag 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 ririncinciple ple cacan n bebe de

demomonsnstrtratated ed by by the the BeBernrnououlllli i eqequauatitionon. . hhe 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' donhe 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 mamaninioold ld wiwith th aiair r blbleeeed d vavalvlve. e. rresessusuririaatition 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 oror 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.( ./engrictional 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 ollowingollowing 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 equation8ollowing 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. hereore,throat. hereore,

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 :. :. nnioionsns ?. 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,;

iBiB

is calculated by using the Bernoulli6s equation where

is calculated by using the Bernoulli6s equation where

;

;

iBiB

D

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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References

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