ME3 21 Mea s ur em me entSys t em ms s Dr . Ma yhew S pr i ng2 0 1 4 – 20 1 5
Ex pe r i me nt a lDe t e r mi na t i onofa n Or i fic eDi s c har geCoeffic ffic i ent
0 4 / 2 0 / 2 0 1 5
T e a m4
Za c hLe hm ma a n J os e phAr t hur J os e phKa l t e nt ha l e r Andr e w Ni e mann
RoseHul manI ns t i t ut eofTechnol ogy
T e r r eHa ut e , I N4 7 8 0 3
I nt r oduc t i on Whe ndi s c us s i ngj unc t i o nso ro bj e c t st ha ti mpe deflo wi npi pi ngne t wo r k si ti so f t e nus e f ult o kno wt hef a c t o rb yw wh hi c ht he s eo bj e c t si mpe det heflo w.Thedi s c ha r g ec o e ffi ffic c i e nte s s e n t i a l l y de s c r i be st her e l a t i ons h i pbe t we e nt hei de a lflo wi nas y s t e ma ndt hea c t ua l ,r e s t r i c t e dflo w c a u s e db ya no bs t r uc t i o no fs o me mek i nd.Thust hego a l so ft hi se x pe r i me ntwe r et ode t e r mi net he di s c ha r g ec oe ffic ffic i e ntf ora no r i fic ea ndt ode v e l opar e a s o na bl eunc e r t a i nt ya ppr o x i ma t i ont ha t wi l lv er i f yt hea c c ur a c yoft hedi s c ha r g ec oe ffic ffic i e nt . Addi t i ona l l y , i twa soure xpe c t a t i ont o demo mons t r at et hatt hedi s char gecoeffi ffic ci entwasbet ween0and1, base duponevi dencef r om e xt e r na ls o ur c e s . T oa c c o mp mpl i s ht he s eg oa l s , t heda t ar e duc t i o ne qua t i o nwa sde t e r mi ne df o rt hedi s c ha r g e c o e ffic ffic i e nto ft heo r i fic ea ndt hepa r a me me t e r so ft hes y s t e m we r ea l t e r e da ndme a s ur e dwi t ht he a ppr opr i a t ei ns t r ume nt a t i o n.Ama no me me t e rwa sus e dt oobt a i nt hepr e s s ur edi ffe r e nt i a lo ft he flui da c r os st heo r i fic ea ndar o t a me me t e rme a s ur e dt heflo wr a t eo ft heflui d. Byut i l i z i ng t r a ns duc e ra c c ur a c i e sa ndr e s ol ut i onva l ue s , a ndb ya l t e r i ngt henumbe ro fda t at r i a l sc o l l e c t e d, t heunc e r t a i nt yi nt her e s ul t a ntwa sde t e r mi ne d.
Ex pe r i me nt a lF a c i l i t i e sa ndI ns t r ume nt a t i on Thee x pe r i me nt a ls y s t e m us e di nt hi se x pe r i me ntc o ns i s t e do fawa wa t e rt a nk ,apump,i n puta nd o ut putv al v e s , a ndt hene c e s s a r ypi pi ngt odi r e c tt heflo wo ft heflui db e i ngpumpe d, i nt hi sc a s e wa t e r ,t ot heor i fic ebe i ngt e s t e d.
c he ma ma t i co ft heEx pe r i me nt a lS ys t e m Fi gur e1 .S F i gur e1i sas c he ma ma t i co ft hee nt i r ee x pe r i me nt a la ppa r a t uswhi c hi nc l ude sat a nk ,t heor i fic ei n que s t i o n,ma no me me t e r ,r o t a me me t e r ,c o nt r o lv al v e s , a ndt hea ppr o pr i a t epi pi ng .Ke yi nde t e r mi ni ng t hedi s c h a r g ec o e ffic ffic i e nto ft heor i fic ea r et her o t a me me t e ra ndt hema ma no me me t e r .
Sc hema mat i coft heManomet er F i g ur e2 . Theme meas ur andsi nt hi ss yst em ar eobt ai nedus i ngt hema manomet erandt her ot ame met er .The r o t a me me t e rme a s ur e st hev ol ume t r i cflo wr a t eo ft hewa t e rf r o m 2t o20l i t e r sp e rm mi i nut e .The ma no me me t e rme a s ur e st hea mo mo unto fpr e s s u r edi ffe r e nc ea c r o s st heo r i fic eb ydi s pl a y i ngwa wa t e r he i ghtf r o mo f0t o4 4 0mi l l i me t e r sa ss ho wn wni nF i gur e2 . Thema manome met erandr ot ame met erwer eut i l i zedi nt hi se xper i mentt omeas ur et woflui dhei ght s a ndt heflui dflo wr a t et ode t e r mi net hedi s c ha r g ec oe ffic ffic i e nto ft heor i fic e . F i gur e3i sac r o s s s e c t i o nv i e wo ft heor i fic edi s pl a y i ngflo w di r e c t i o na nda r e a si mpa c t i ngflui dflo w.
F i g ur e3 . Or Or i fic eSchem ma at i c
Oft hef a c t or sc ont r i but i ngt ot hedi s c ha r g ec oe ffic ffic i e nt , r a t e
´ ∀
C d
,va r i a bl e si nc l ude :v ol ume t r i cflo w
, ma no me me t e rh e i ghtb e f o r ea nda f t e rt heor i fic e , h1 and h2 r e s pe c t i v el y .Cons t a nt
va l ue sf ort hi se x pe r i me nti nc l ude :t heor i fic ea r e a , A 2 ,t hepi pear e a , A 1 , t hede ns i t yof t heflui d, ρ ,a ndt hegr a vi t a t i ona la c c el e r a t i onoft heEa r t h, g . Thema mas sflow r at esand pr e s s ur e so ut l i ne di nF i gur e3a r er e l a t e dt ot hema ma no me me t e rhe i ght sa n dv ol ume t r i cflo wr a t e sa s de s c r i be di nt hef ol l o wi wi ngs e c t i o n. Thefir s tma no me me t e rh e i ghtr e c o r de d, h1 i st a ke n ups t r e a mo ft heo r i fic ea n dt hes e c o ndr e a di ng ,
h2
i st a ke ndo wn wns t r e a mo ft heo r i fic e . The s e
he i ght sa r eout put so ft hema ma no me me t e ra tpo s i t i o ns6a nd7onF i gur e2 ,whi l et hea c t ua l me a s ur a ndi st hepr e s s ur eb e f o r ea n da f t e rt heo r i fic e . Ho we we v e r ,t hea c t ua lma no me me t e rr e a di ngs c a nb et r e a t e da sme me a s ur a ndsc o ns i de r i ngt he ya r eus e ddi r e c t l yi no urd at ar e duc t i o ne qua t i o n. I nor de rt oc o l l e c tt hene c e s s a r yi nf or ma ma t i ont ode t e r mi mi ne C d , t hepumpwast urnedont o a l l o wt hewa t e rt opa s st hr o ug ht hes y s t e m. m. Wi t hb ot hv a l v e sf ul l yope n ,t hepumpwa sr un unt i la l lo ft hea i rha de s c a pe d.S omeo ft hewa t e ra nda i rwa sd r a i ne df r o mt hema ma no me me t e rs o t ha tt heflui dl e v e l swe r ei nt hec e n t e ro ft hema ma no me me t e rr e a do uta ndnobub bl e sr e ma ma i ne d. Onc ea l loft he s ec r i t e r i awe we r eme me t ,t hee xpe r i me nt a lt r i a l swe r ec onduc t e d. Byo pe ni ngo rc l o s i ngt hei nputv al v e ,t heflo wr a t eo ft hewa t e rwa si nc r e a s e do rde c r e a s e d.T o c o l l e c tda t a ,t hev al v ewo wo ul db es e tt oas p e c i ficp os i t i o na ndt heflo wr a t ewo ul db eob s e r v e do n t her ot ame met erwhi l et hehei ght swer er ec or dedf r om t hema manome met er .Comp mpar i ngt heal i gnment o ft hev e r yt o po ft hefloa ti nt her o t a me me t e rwi t ht heme me a s ur i ngma r ki ngswa sc r uc i a lt ot he a c c ur a c yo ft hi sme me a s ur e me me nt .S pe c i fict ooure x pe r i me nti st hec a l i br a t i onoft her o t a me me t e rt o a c hi e v ei mpr o ve da c c ur a c y . Thi sc a l i br a t i o ni so ut l i ne di nAppe ndi xB. Wi t ht heflow benchoper at i ng,onedat um wascol l ec t edatfiveuni queflow r at esi nar andom or de r .Wer a ndomi z e dt heor de rofda t ac o l l e c t i onmi ni mi z e dt hepo t e nt i a le ffe c t so fhys t e r e s i s o nr a ndo m unc e r t a i nt y . Af t e rc o l l e c t i ngt hefiv edi ffe r e ntflo wr a t e sa ndc o r r e s p ondi ngv a l ue so f h1 and h2 , t he r a ndo m unc e r t a i nt yi nt hedi s c h a r g ec o e ffic ffic i e ntwa sde t e r mi ne d.Byc o mp mpa r i ngt hi sv a l uewi t h t hes ys t e ma ma t i cunc e r t a i nt yi nt hedi s c ha r g ec oe ffic ffic i e nt ,
C d
, onec oul ddet er mi nehow many
t r i a l swe r ene c e s s a r yt opr e v entt her a ndom unc e r t a i nt yf r om be i ngal a r g ec o nt r i but i ngf a c t or
t ot het o t a lunc e r t a i nt y . Wede t e r mi ne dt ha tt her a ndom unc e r t a i nt ys h oul db eo fe qua lo rl e s s e r ma gni t udet ha nt hes y s t e ma ma t i cu nc e r t a i nt y .Onc et hes t a nda r dde vi a t i o ni n
C d
was
de t e r mi ne df r o mt hefiv et r i a l s ,i twa su s e di nt her a ndo m unc e r t a i nt ye q ua t i o na ndwa s c o mp mpa r e dt ot hes y s t e ma ma t i cunc e r t a i nt y . I twa sde t e r mi mi ne dt ha t2 0t r i a l swe r ene c e s s a r yt o pr oduc ear a ndo m unc e r t a i nt yt ha twa sc o mp mpa r a bl et ot hes y s t e ma ma t i cunc e r t a i nt y . De t a i l so nt he ma t he ma ma t i c sr e qui r e dt oob t a i nt hes y s t e ma ma t i ca ndr a ndom unc e r t a i nt i e sa r eout l i ne di nt he f o l l o wi wi ngs e c t i o na nds e n s o ri nf or ma t i oni sdi s p l a y e di nAppe ndi xC.
Dat aReduc t i onandUnc er t ai nt yAnal ysi s Theda t ar e duc t i o ne qua t i o nf o rt hedi s c ha r g ei sde r i v e df r o mt hec o ns e r v at i o no fma ma s sa tt he ´ 1 ,i o r i fic e . Theflui dflo w be f or et heor i fic e ,m se qua lt ot heflui dflo wt hr ought heo r i fic e , m ´2
, ass hown wnby, m ´ 1=m´ 2 .
(1)
F r o m flui dpr i nc i pl e s , i ti sk no wn wnt ha t
´ = ρVA m ρVA = ρ ´∀
(2)
,
Thede ns i t yoft heflui di s ρ ,Vi st heve l oc i t yoft heflui da tt hegi v enl oc a t i on,A i st hec r o s s s e c t i o na la r e at ha tt heflui dpa s s e st hr ough,a nd
´ ∀
i st hev ol ume t r i cflo wr a t eoft heflui d.
As s umi ngt hede ns i t yi ss t i l lc o ns t a nta ndut i l i z i nge qua t i ons1and2 ,i ti sde t e r mi mi ne dt ha t , V 2=V 1
A1 A2
(3)
F r o m Be r no ul l i ’ se q ua t i o n, P1+
1 1 2 2 ρ V 1= P 2+ ρ V 2 2 2
,
wes o l v e df o rt hec ha ng ei npr e s s ur eo ve rt heor i fic e ;
(4)
( )
2
A 2 1− ] A 1 1
2
P1− P 2= ρ V 2 ¿
(5)
.
2
S i mpl i f y i ngt hi sweno w ha v eane we x pr e s s i o nf o rVa tt heo r i fic e ,
V 2=
√
2 ( P1− P2 )
( )
2
A 1 ] ρ [ 1 − A2
.
(6)
Thei de a lflo wr wr a t et hr o ug ht hes y s t e mi s
´ =V A ∀ ideal 2 2
(7)
Andt hedi s c h ar g ec oe ffic ffic i e nte s s e nt i a l l yi sar a t i obe t we e nt hea c t ua lflo wr a t e ,whi c hi s me a s ur e d,a ndt hei de a lflo wr a t e . Thi sr e l a t i o ns h i pc a nbede s c r i b e da s C d=
´ ∀ actual ´ ideal ∀
=
´mactual m ´ ideal .
(8)
Ut i l i z i nge qua t i ons6 , 7 ,a nd8 ,wea r r i v ea t
´= m
C d A 2
√
2
A 1 ) 1 −( A2
√ 2 ρ ( P − P ) 1
2
(9) .
Kno wi wi ngt ha tt hepr e s s ur ei sme a s ur e db yt hema ma no me me t e r , t hepr e s s u r ec a nbes i mpl i fie dt o , P= ρgh
,
(10)
wher e h i st hehe i g hti nt hema ma no me me t e ra nd g i st hegr a vi t at i onala cc e l e r a t i onoft he Ea r t h.Ut i l i z i nge qua t i ons2 ,9 ,a nd1 0 ,wear r i v ea tt heda t ar e duc t i one qua t i on;
h1− h2
(11)
A2 √ 2 g ( ¿ )
´ ∀ C d=
√
2
A 2 ) 1 −( A 1
.
¿
T ode t e r mi mi net het o t a lunc e r t a i nt yi nt hedi s c ha r g ec oe ffic ffic i e nt ,
w C
d
,t hes ys t e ma ma t i c
unc e r t a i nt ya ndt her a ndo m unc e r t a i nt yo ft hedi s c h a r g ec o e ffic ffic i e nta r er o ots um s q ua r e dt o produce, 2
2
2
w C = wC ,sys + wC ,rand d
d
C d
Thes y s t e ma ma t i cunc e r t a i nt yi n
(12)
d
,
w C ,sys d
.
, i sde pe nde ntupont hes y s t e ma ma t i cunc e r t a i nt i e s
oft heme meas ur andsas 2
2
2
∂ C d ∂ C ∂C w C ,sys=( ) w2h 2, sys+( d ) w2h 1, sys +( ´ d ) w 2∀´ ,sys ∂ h2 ∂ h1 ∂∀
(13)
2
d
wher e
w h , sys sys 1
,
i st hes ys t e ma ma t i cunc e r t a i nt yi nt hehe i ghta f t e rt heor i fic e ,
s y s t e ma ma t i cunc e r t a i nt yi nt hehe i ghtbe f or et heor i fic e , a nd
w ∀´ ,sys
w h ,sys 2
i st he
i st hes ys t e ma ma t i c
unc e r t a i nt yi nt hevol ume t r i cflo wr a t e .Theunc e r t a i nt i e soft hear e asf a c t ori nt ot hi sa na l ys i sa s we l le x c e p tt ha tt heunc e r t a i nt i e swe r eno tgi v e n, no rwa sa n yi nf or ma t i o nt ha twoul da l l o w us t ode t e r mi mi net heunc e r t a i nt y .I tc a nbeas s ur e dt ha tt heac c ur a c i e soft he s eva l ue si se x c e pt i ona l . A A Thes e ns i t i vi t yc oe ffic ffic i e nt so ft heor i fic ea r e a , 2 , a ndt hepi peor i fic e , 1 , a r er o ughl y −3
2 x 10
−4
and 1.35 x 10
r e s pe c t i ve l y .Thes e ns i t i vi t yoft hegr a vi t at i onalac c el e r at i on g −5
i se v ens ma ma l l e r ,be i ng 2.9 x 10
.Whe nt he s es ma ma l ls e n s i t i vi t yc o e ffic ffic i e nt sa r ec o mb mbi ne dwi t h
t hel o wv al ue so func e r t a i nt yi na c c u r a c yw we ede t e r mi ne dt ha tt hea r e a s ,a swe l la st he gr a vi t a t i ona la c c e l e r a t i ondono ts i gni fic a nt l yi mpa c tunc e r t a i nt ya na l ys i sa nda r ea s s ume dt obe a bs o l ut ev al ue s . Abe t t e rde t e r mi mi na t i onoft heunc e r t a i nt yc a nbede t e r mi mi ne di ft heunc e r t a i nt y i nt hea r e a si sa c c o unt e df or . T ode t e r mi mi net heunc e r t a i nt i e si nt hehe i ght s , t hea c c ur a c ya ndr e a da bi l i t yo ft hema ma nome t e r , h h a r eut i l i z e di nt hef ol l o wi wi ngwa wa y , a ndc a nbea ppl i e dt obo t h 1 and 2 , 2
2
2
w h,sys =wmanometer manometer , accuracy + w manometer manometer ,readability
(14) .
Theflo wr a t eunc e r t a i nt ywa wa sd e t e r mi ne df r o mt her o t a me me t e rc a l i br a t i o na sd e s c r i b e di n Appe ndi xB. Theda t ar e duc t i one qua t i onf ort hi sc a l i br a t i oni s
´= ∀
∆∀ ∆ t
(15) ,
wher e ∆∀ i st hev o l umec ha ng ei nt het a nka nd ∆ t i st hec h a ng ei nt i medur i ngwhi c h t hev ol umec ha ng eo c c ur r e d.F r o me qua t i o n1 5 ,t hes ys t e ma ma t i cunc e r t a i nt yoft heflo wr a t e be c ome s w
( )
∂ ´∀ 2 2 sightt glas glasss , acc acc + w sightglass, read (¿ ¿ sigh )+ ( wstopwatc read stopwatch, h, acc + wstopwatc stopwatch, h, read read ) ∂ ∆ t 2
2
2
2
w ∀´ ,sys=(
∂ ´∀
∂∆ ∀
2
(16) .
)¿
Thes u bs c r i pt s i ne qua t i on1 6r e f e rt opr o pe r t i e soft hes i ghtgl a s so nt he s i gh tg l a s sands t o pwa t c h t a nka nds t opwa t c hr e s pe c t i v e l y . The i ra c c ur a c i e sa ndr e a da bi l i t i e sde t e r mi mi net hene w unc e r t a i nt yi nv ol ume t r i cflo wr a t es i nc et he ywe r eus e dt oc a l i br a t et her e a di ngsf r om t he r o t a me me t e r . Ther andom unc er t ai nt yi n C d
i sde t e r mi mi ne df r om t henumbe ro ft r i a l s , n,t her e l a t e dt -
di s t r i but i onva l ue , t n−1 , a ndt hes a mp mpl es t a nda r dde v i a t i onoft he C d va l ue sf ort hegi v e n numbe ro ft r i a l s ,
w C ,rand = d
S C
d
us i ngt hee qua t i o n,
t n−1 S C
d
√ n
(17)
.
Theunc e r t a i nt i e sf ort hea c c ur a c ya ndr e a da bi l i t i e so ft het r a ns duc e r sa r egi v e ni nAppe ndi xC a ndt her e s ul t i ngunc e r t a i nt i e sa swe l la st hepa r t i a lde r i va t i v esoft heda t ar e duc t i one qua t i on i ne qua t i on1 1ar egi v e ni nT abl e1 . Table 1. Va Variable riable
impact on the overall
C d uncertaint
!arameter
"nit#
∀
mm
∆ t
#ec
h2
h1
mm
mm
∂ ´∀ ∂∀
3.521 × 10
−2
3.88 × 10
∂ ´∀ ∂ ∆ t
−7.33 × 10 3
3.36 × 10
141
∂ Cd ∂ h1
-3.06×10-4
1.17 × 10
194
∂ Cd ∂ h2
3.06×10-4
1.17 × 10
29.3
3
mm sec
h,sys
mm
Cd, total total
6
´ ∀´ ,sys
Cd, rand rand
!artial
3
%en#itivit &oe'' i
Ter erm m i
$pical Value (mean)
7
6
−6
−6
4
1.25 × 10
2
1.12 × 10
0.001851
0.020
3
Rot a me me t e rc a l i br a t i onwa se s s e nt i a la si t sunc e r t a i nt yc a us e d
´∀ ,sys
5
mm
t obe 1.179 x 10 sec
whi c hi sa no r de ro fma gni t udegr e a t e rt ha nt hec a l i br a t e dunc e r t a i nt yi nflo wr a t ea ndt hr e e or de r sofma gni t udegr e a t e rt ha nt heunc e r t a i nt yc a us e db yt hehe i ght s . F ort he s er e a s ons , t he r o t a me me t e rwa sc a l i br a t e dwi t ht hea v ai l a bl et a nka nds t opwa t c h.Unc e r t a i nt i e sf ora r e a sa nd gr a vi t ya r eno tpr o vi de da st he ywe we r ede e me me dt oha v eane gl i gi bl ee ffe c to nt heo ve r a l l unc e r t a i nt y .
,
Re s ul t sa ndDi s c us s i o n Theme me a nr e s ul tf ort heo r i fic edi s c h ar g ec oe ffic ffic i e ntwi t ha s s o c i a t e dunc e r t a i nt yi s C d=0.564 ! 0.020
F i g ur e4s ho ws wst her e l a t i o ns hi pb e t we e nt hedi s c h a r g ec o e ffic ffic i e nta ndt heflui dv ol ume t r i cflo w r a t easwe l la st heunc e r t a i nt ya s s o c i a t e dwi t he a c hva l ueus i ngt her o t a me me t e rc a l i br a t i on o ut l i ne di nAppe ndi xB. 0.6 0.6
0.58
0.56
Di sc har geCoeffic ffici ent( )
Ex pe r i me nt a lda t a
0.54
Re l a t i veunc e r t a i nt y 0.52
Not e :unc e r t a i nt yi sl a r ge r a tl o we we rflo wr a t e s 0.5 0.5
9
10
11
12
13
14
15
Vol umet r i cFl ow Rat e( L/ mi n)
Di s c ha r g ec o e ffi ffic c i e ntwi t hunc e r t a i nt yofa no r i fic ea tv ar yi ngv ol ume t r i cflo wr a t e s F i g ur e4 . Fi gur e4r eflect st heme meanoff ourmeas ur eme ment sateac hoft heflow r at esmeas ur ed.Comp mpl et e da t ai sa v a i l a bl ei nAppe ndi xD. Whe ndi s c us s i ngflui ddy na mi mi c si ti se s s e nt i a lt ome me nt i o nRe y no l dsnumb e r ,whi c hi sano ndi me ns i o na lt e r mt ha tc h ar a c t e r i z e st heflo w oft heflui d.Ther e l a t i o ns h i pbe t we e nRe y no l ds
numbe ra nddi s c ha r g ec oe ffic ffic i e ntc a nbes e e ni nF i gur e5t oe xhi bi tane a r pa r a bo l i cb e ha vi or whe nt r a ns i t i o ni ngi nt ohi g he rRe y no l dsn umb e r sf r o mt he1 0t ot he2 0 0 0r a ng e .F i g ur eX comp mpar esourmeas ur edval uesofdi s char gecoeffi ffic ci entandReynol dsnumber st ot hoseofan A 2 or i fic epl a t ewi wi t has i mi l a r A 1
r a t i o .Ourd a t ar e fle c t sas ma ma l lr a ng eofRe y no l dsn umb e r sa s
c o mp mpa r e dt ot her e f e r e nc eda t a . 0.75
0.7
0.65
Di s c har g eCo e ffic ffic i ent( ) 0.6
0.55
0.5 10
100
1000
10000
Reynol dsNumber( ) Thi sr e l a t i ons h i pi sb a s e dont e s t sdoneonas i mi l a ro r i fic epl a t et ot heoneus e di nour F i g ur e5 . s c h a r g eCo e ffic i e n tP e r f o r ma nc eo fVe n t u r i ,St a nd a r dCo nc e nt r i cOr i fic e e x pe r i me nt a t i on.Source:Di Pl a t e ,VCo n ea ndWe We d g eF l o wMe t e r sa tSma l lRe y no l d sNumb e r s , Aut ho r e db yCo l t e rL
Hol l i ngs h ea d,Ut a hS t a t eUni v e r s i t y . F i gur e5s ho ws wst ha to urda t ai nc r e a s e sf o rt het hr e el o we we rRe y no l dsnumbe rt r i a l s . Whe n c o mp mpa r e dt ot her e f e r e n c eda t af oras i mi l a ro r i fic epl a t e ,t hi st r e nddoe sn ota gr e e .Weha v eno j us t i fic a t i ona st owhyt hi si s .Thi swoul ds ugge s tt ha tf orourt r i al s , t hedi s c ha r gec oe ffic i e nti s notc ons t a nt .Ourr e s ul t showe ve r ,i ndi c a t et h atf oro urt r i a l s , di s c ha r g ec oe ffic ffic i e nti sc ons t a nt .
46
44
42
40
TubeHe Hei ghtDi ffer enc e( mm)
38
36
34
32
30 75
95
115
135
155
175
195
215
Vol umet r i cFl ow Rat e2( L2/ mi n2)
Thevol umet r i cflow r at es quar eswi t ht hedi ffer encei nmanome met ert ubehei ght F i g ur e6 . Fi gur e6s hows wshow t hema manome met ert ubehei ghtdi ffer encec hangeswi t ht hes quar eoft he v ol ume t r i cflo wr a t e .I fe qua t i on1 1i ss qua r e da ndr e a r r a ng e d,o nea r r i v e sa t
´ ∀
2
2
2
=C d
A 2 2 g ( h1− h2)
( )
(1 −
A 2 A1
2
)
(18)
,
whi c hde s c r i be st her e l a t i ons hi pi nFi gur e6. Thi sr e l a t i ons hi p,whi c hi sl i ne ar ,s t r o ngl ys ugge s t s C t ha tt hedi s c ha r g ec o e ffi ffic c i e nt d ,i sc o ns t a nt .I no t he rwor ds , s i nc et hevol ume t r i cflo wr a t e s qua r e da ndt hema ma no me me t e rhe i g htdi ffe r e nc ev a r ypr o po r t i o na t e l yt oo nea no t he r ,t he ne v e r y o t he re l e me me nti nt hi se q ua t i onmus tbec o ns t a nt .Thi se v i de nc es uppor t so urr e s ul t a nt , be i ngas i ngl eva l ueata l lflow r at e sandi smor ec onc l us i vet ha nt heRe ynol dsnumbe r c om mp par i s on.
C d
,
Conc l us i ons Thi se x pe r i me ntme a s ur e dt her e s u l t a ntdi s c ha r g ec oe ffic ffic i e nta ndt her e l a t e dunc e r t a i nt yo fa n o r i fic epl a t e , a nds u c c e s s f ul l yde mo mo ns t r a t e dt ha tt hi sr e s ul t a ntwa sb e t we e n0and1 ,a swe e x pe c t e dduet opr e v i ouse x pe r i me nt a t i o na ndr e s e a r c hc o nduc t e dont het o pi c . Wes uc c e s s f ul l y pr o ve dt hi swi t hi na ndunc e r t a i nt yo f ! 0.020 . I ti sr e c o mm mme nde dt ha ts i mi l a re x p e r i me nt sus ea ni n putr a ng ef o rt hev ol ume t r i cflo wr a t e be t we e n4a nd20l i t e r spe rmi nut e .Theme a s ur e me ntofvol ume t r i cflow r at ewast hel ar ge s t s o ur c eo func e r t a i nt yi noura na l ys i s . I ti sr e c o mm mme nde dt oi mpr o v es e n s o ra c c ur a c yby c a l i br a t i ngt her o t a me me t e rpr i ort ous e . Onc et hec a l i br a t i oni ss uc c e s s f ul l ye x ec ut e dus i ngt he wa t e rt a nka nds t opwa t c h ,t heunc e r t a i nt ydr a ma ma t i c a l l yde c r e a s e s .
Appendi xA-Pr i nc i pl esofOper at i on-Or i fic eFl owm wmet er Thema ma nome t e rc ons i s t so fas e tofgl a s st ube st ha tf ol l o w hydr os t a t i cpr i nc i pl e st opr o vi dea r e a douto fp r e s s ur e ,di s p l a y e di nwa t e rc o l umnhe i ght .Thi sa l l o ws wst heus e rde t e r mi net he pr e s s ur edi ffe r e nc ef o l l o wi wi nge qua t i o n1 0 .Thet ub e sa r ec o nne c t e da tdi ffe r e ntpo i nt si nt he flo ws ws t r e a m, m,s h owni nF i gur e7 ,i no r de rt ode t e r mi net hepr e s s ur edi ffe r e nc ea c r o s st heo r i fic e pl at e .
As s emb mbl yphot ogr aph F i g ur e7 . Whe npo we we ri si n put t e d,t hepumps e ndswa t e ro uto ft heho l di ngt a nka tac o ns t a ntr a t e t hr o ug ht hes y s t e m. m. Ai ra c c umul a t e si ns i det hes y s t e mb e c a us es o me meo ft hepi pe sa n dv a l v e si n t hes e t upa r ee x pos e dt ot hea i r .Thee x c e s sa i ri sr e l e a s e df r om t hes y s t e mb ya dj us t i ngt he ma no me me t e rv a l v ema ma r k e do nF i g ur e4 .Onc et he r ei sn oe x c e s sa i rr e ma ma i ni ngi nt hes y s t e m, m,t he s y s t e mi ss t a b i l i z e da nda c c ur a t er e a di ng sc a nbema ma deo nt hema ma no me me t e ra ndt her o t a me me t e r . Theme me ni s c uso ft hewa wa t e rl e v e li sr e a ddi r e c t l yf r om t hehe i ghtg ag ea t t a c he dt ot hema ma no me me t e r . I nde t e r mi ni ngt hedi s c h a r g ec o e ffic ffic i e nt ,t ub e snumb e r e d6and7we r eus e df o ra n a l y s i sb e c a us e t he yar ec o nne c t e di mme di a t e l ybe f o r ea nda f t e rt heor i fic e , a ss ho wn wni nF i gur e7 .Thepr e s s ur e di ffe r e nc ea c r o s st heor i fic ei spr opor t i ona lt ot heflo wr a t e .Ther o t a me me t e rc ont a i nsame me t a l l i c floa tt ha t ,onc et hes ys t e mi ss t a bi l i z ed,r e ma ma i nsa tac ons t a ntl e ve lwi t hi nt hede vi c e .Thi s de vi c ei spr e c a l i br a t e di nt hes ys t e mt odi s pl a ya nout putofl i t e r sp ermi mi nut e .Ther o t a me me t e ri s t hes e c o nds e n s o ru s e di ne x pe r i me nt a t i o n.I tpr oduc e st hefina lme a s ur a nd,flo wr a t e , us e df o r t heor i gi na lDRE,e qua t i o n1 1 .
Appe ndi xB-Pr i nc i pl e so fOpe r a t i o n–Ro t a me me t e rCa l i br a t i o n Ther o t a me me t e rwa sc a l i br a t e da si twa st hehi ghe s ts o ur c eo func e r t a i nt yi nt heme me a s ur e me me nt s y s t e m. m.Thi sc a l i br a t i ona l l o we we df oramo mor epr e c i s edi s c h ar gec o e ffic ffic i e ntt ober e por t e d.Thi s c a l i br a t i onc ons i s t e doft hewa t e rt a nk ,s t opwa t c h, a ndr ot a me me t e r .Thes i ghtgl a s st ha tdi s pl a ys t hewa wa t e rt a nkv ol umei so ut l i ne di nF i gur e8 .
F i g ur e8 .Ta nks i ghtgl a s sus e di nr o t a me me t e rc a l i br a t i on
T obe gi n,a ni ni t i a lflo wr a t ewa sr e c o r de df r o mt heout puto ft her o t a me me t e r .The nt hei nput v al v ewa wa sc l os e ds ot ha tt het a nkwo woul dfil l .Ani ni t i a lflui dl e v e lwa sr e c o r de da da nda f t e ra s pe c i fica mo mo unto ft i mee l a ps e dt hefina lv ol umewa sr e c o r de d.Thedi ffe r e nc ei nt he s et i me s a ndt hedi ffe r e nc ei nv ol umeme me a s ur e me me nt swa sus e di ne q ua t i o n1 5t ode t e r mi net hec a l i b r a t e d flo wr a t e .Thec a l i br a t e dflo wr a t e swe r eu s e di nt hes y s t e ma ma t i cunc e r t a i nt yc a l c ul a t i onoft he v ol ume t r i cflo wr a t e . F i g ur e9i st hec o mp mpa r i s o no ft her o t a me me t e rflo wr a t e sa ndt hemo mo r ea c c u r a t e l yde t e r mi ne d c a l i br a t e dflo wr a t e sf oundf r om t het a nkfil l i ngc a l i br a t i on.
15
14
13
Cal i br at edFl ow Rat e( L/ mi n)
12
11
10
9 10
11
12
13
14
15
16
Rot ame met erFl ow Rat e( L/ mi n)
Ro t a me me t e rCa l i br a t i onc ur v e F i g ur e9 . Thec a l i br a t i one qua t i onf ort her o t a me me t e ri s Calibrated Calibrated "low #ate=0.9426∗ #otameter "low #ate + 0.413 ,
(19)
whi c hi su s e dt oge ne r a t emo mo r ea c c u r a t eflo wr a t e sf r o mt her o t a me me t e rflo wr a t e sa ndul t i ma t e l y l e a dst omo r ea c c u r a t edi s c ha r g ec o e ffic ffic i e nt s . Theunc e r t a i nt yi nt hevo l ume t r i cflo wr a t e sf o r t hec a l i br a t e dflo wr a t e si se q ua t i on1 6i ns t e a doft her o t a me me t e re q ui v al e ntofe qua t i on1 4 . Sug ge s t i o nsf o rF ut ur eEx pe r i me nt a t i o nEffo r t s
Ca l c ul a t i ngva l ue sf o rdi s c ha r g ec o e ffi ffic c i e nt sc a nbes i mpl i fie da nde x pe di t e db yus i ngt he r o t a me me t e ro nt heor i fic ebe nc h,a ndno tt i mi ngt a nkfil l i ngs . F ors i mpl i c i t yac a l i br a t i ono ft he r o t a me me t e rh asb e e nc o mp mpl e t e di nt hi sa ppe ndi xyi e l di ngt hec a l i br a t i o ne qua t i o n1 9 .F ut ur e e x pe r i me nt sc a nbec o nduc t e dus i ngt hi sc a l i br a t i on, buti ti ss ugg es t e dt ha tflo wr a t e sa b ov e4 l i t e r spe rmi mi nut ea r em me e a s ur e d. Ourr e s ul t sa r er e por t e di nFi gur e1 0.
0.6 0.6
0.58
0.56
Di sc har geCoeffic ffici ent( ) Ex pe r i me nt a lda t a 0.54
0.52
Unc e r t a i nt yi s l a r g e ra tl o we we rflo w r a t e s
0.5 0.5 9
10
11
12
13
14
15
Rot ame met erVol umet r i cFl ow Rat e( L/ mi n)
Di s c ha r g ec o e ffi ffic c i e nta ndunc e r t a i nt ybe f or ec a l i br a t i ngt her o t a me me t e r F i g ur e1 0 . Thel a r g e runc e r t a i nt ya tl o we we rflo wr a t ei sduet ot hegr e a t e rs ys t e ma ma t i cunc e r t a i nt yoft he t r a ns duc e r si nt he i rl o we we ro pe r a t i onr a nge s . I ti ss ugge s t e dt ha tflo wr a t e sgr e a t e rt ha n4l i t e r s pe rmi mi nut ea r eus e dt omi t i g at et hi si s s ue .
Appendi xC-Syst ema mat i cUnc er t ai nt i esf ort heMe Meas ur ands Ta bl e2pr o vi de sal i s tofa l ls o ur c e so fs e ns o runc e r t a i nt i e sa ppa r e nti noure xpe r i me nt a t i on. Ther e s o l ut i o nsf o rt hema ma no me me t e ra ndr o t a me me t e rwe r et a k e ndi r e c t l yf r o mt he i rr e a do ut sa nd t he i rr e a da bi l i t i e sa nda c c ur a c i e sa r eha l foft he i rr e s ol ut i ons . Thet a nkwa sf oundt oha vea n a c c ur a c yo fa ne i ght hofal i t e ra na nde qua lr e a da bi l i t y . Ther e a da bi l i t yoft hes t opwa t c hwa s ha l fo ft her e s o l ut i o nb utt hea c c ur a c ywa wa sma ma deaq ua r t e rf r o mj udg e me me ntb a s e do nt he di ffic ul t yi nope r a t i ngt hes t opwa t c ha tt hec o r r e c tt i me . Ta bl e2 . S e ns o rAc c ur a c i e sa ndRe a da bi l i t i e s %en#or
e#olution
eaabilit
*ccurac
+anometer (mm)
5
2.5
2.5
otameter (-min)
1
0.5
0.5
$an Volume ()
0.25
0.125
0.125
%top/atch $ime (#)
0.01
0.005
0.25
Appendi xD –Exper i ment alDat a T a bl e3c o nt a i nst heme me a s ur a ndsa n dt her e s u l t a nt C d c a l c ul a t e df r om e qua t i on1 1f ort he f ourt r i a l sa tfiv es e pa r a t ev ol ume t r i cflo wr a t e s . Theflo wr a t e sdi s pl a y e da r et her o t a me me t e r r e a di ngspo s tc a l i br a t i o nus i nge qua t i on1 9 .Theme me a no ft het we nt yv a l ue so f C d i st heva l ue r e por t e df ort hi sor i fic e .
S ys t e m me a s ur a ndsa n dc a l c ul a t e dr e s u l t a ntf o rr a ndomi z e dt r i a l s Ta bl e3 .
´ V 10 15 12 11 14 13 10 15 12 11 14 13 10 15 12 11 14 13 10 15 12 11 14 13
[ ] $ min
h1 [ mm ]
165 148 157 161 150 154 165 150 158 161 152 154 166 149 157 162 152 153 165 150 158 160 153 153
h2 [ mm ]
203 225 207 204 217 211 203 226 208 205 219 211 202 225 207 204 218 211 201 226 208 204 219 211
∆ h [ mm ]
38 77 50 43 67 57 38 76 50 44 67 57 36 76 50 42 66 58 36 76 50 44 66 58
−¿
C d ¿
0. 544821 0. 565923 0. 565895 0. 561193 0. 567411 0. 572592 0. 544821 0. 569634 0. 565895 0. 55478 0. 567411 0. 572592 0. 55975 0. 569634 0. 565895 0. 567835 0. 571694 0. 567634 0. 55975 0. 569634 0. 565895 0. 55478 0. 571694 0. 567634
T out i l i z et hev al ue sa b ov ea nde qua t i on1 1t ode t e r mi mi net hedi s c ha r g ec o e ffi ffic c i e nt s , t hea r e a s , de ns i t y , a ndgr a v i t a t i o na la c c e l e r a t i o nmus tbekno wn wn. The s ev a l ue sa r epr e s e n t e db e l o wi n T a bl e4 .
Ta bl e4 . Ex pe r i me nt a lv a r i a bl e st ha ta r epr e s u me me dc o ns t a nt
Cons t ant 2
Or i fic eAr e a , A 2 ( mm )
792
2
Pi peAr e a , A 1 ( mm ) De ns i t yofWa t e r ,
ρ (
314 %g m
3
)
Gr a vi a t i ona lAc c e l e r a t i on, 2
m g( ) sec
1000 9 . 8 1
