Upload
sata-ajjam
View
244
Download
1
Embed Size (px)
Citation preview
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 1/51
Chapter hree
Evaluationof TransferCoefficients
Engineering orrelations
Sincemostengineering roblems o not have heorctical olutions, largeportionof en-gineering nalysiss concerned
ith expeimentalnformation, hich s usually xpressedin lemlsof engineeringonelations. hese onelations.owever.,re imited o a specificgeomeffy,quipmentonfigumtion,oundary onditions,ndsubstance.s a result, heval-uesobfainedromcoffelations renotexact nd t is possibleo oblain wo differcnt nswersfiomtwo different offelationsor thesameproblem.herefore,neshouldkeepn mindthattheuseofa correlationntroducesnerror n lheorderof +25./..
Engineeringorelationsarcgiven n termsof dimensionlesslrmbers. orexample,hecorrelatioassed o dete nine he riction actor, eat ransfer oefficient,ndnasskansfercoefficient regenerally xpressedn the orm
I : I (Re)
Nu: Nu(Re,Pr)Sh: S1t1B"'"'
ln thischaptetsome f theavailableorrelationsor momentum,nergy, ndmassrans,po in different eomefieswi]l bepresented.mphasis ill beplaced nthecalculationsfforce or rateofwork),heat ansfer ate,andmass ransfertte under.steadyonrJitions.
Transfercoeffcents6
REFERENCE EMPERATURENDCONCENTRATION
Theevaluationfthedimensionlessumben hatappearn thecorrelationequireshephysical propefiies
f the luid to be knownor estimated.hese roperties,uchasdensity ndviscosity, epend n tempefaturcnd/or oncentration.emperaturend concenfuation,ntheotherhand, aryas a iunctionof position. wo commonly sed eferenceempemturesandconcenfiationsrc the bulk temperaturer conce ratiofiand heJtlm emperaturer
3.1.1 BulkTemperaturendConcentration
For flow insidepipes, hebulk emperaturer concentrationt a particularocationn thepipe s the averageemperaturcr concenradonf the luid were horoughly ixed, some-times alled hemirrg cup emperaturer concentration.hebrlk temperaturend hebulk
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 2/51
47 ChemicatEngineegprocesses
concentration are denoted by 16 and c6, respectively, and are deflned by
| [ , ,raerb:u#-
lJ^*oo
[["'oond
"= #-
JJ^,"0o(3-l)
(3-2)
\rhrreu, rs hec-omponenl[ velocir)n hedirectlonf mean oq.ror.ne case t flowpast odiesmmersednan nfinite luid, heburk emperaturendburk
:oncentrationecomeheree heamempeftru."no ."""J*'n "oii""i*',ir"
.?#1,"o,
3.1.2 Film Tempe,atureand Concentration
Thefln tenpe ratu re TL andthe il n conrcntrarior, c/, are defined asthe arithmefic averageof the bulk andsurface alues. .e..
(l-3)
fa:r- l
Tf:: - :- :- - and cl=
For flow oversubmerged bjects
n",=3=11s9L . V
where ubscript@ epresentsheconditions t hesualacer thewall.
3.2 FLOWPASTA FLATPLATE
Let usconsidera flat plate suspendedn a uniform streamof velocity r,@andtemperalure?€_as hownn Figue 3.1.Th; Iength f theptaten thedirection f flow s l, and rswidttris Iry.The.localvaluesof rhefriction factor, 'he l**f,""_U"r,-"ra
ifr"'Sln"i*rJi,i".*,aregiven n Table .1 or both aminar nd urbulentlow condirions.i,. ,"r_ il, ]. ,n"Reynoldsumber ased nthedistance, anddefined v
(3-4)
Theexpressionor the friction factorun(canebrainednaryricary;,r,"""i",r"',,'iiiilJIil.lx":Tir;,ft:rr,:;l,fl,.i
Elil*,1
Table3.1.,he otutvatLes . he t.ictionactoi the NLsse.t Jnbe,,ano nesheMoodnLrmberforiowovera iarpta,€
Shr
(D)
(E)
(F)
0.664Re;l /2 (A) o.Otoro.,
0.332Re,1/2prr/3 G) 0.0296R#5pfr/3
0.332Re.1/2scr/3 (q 0.0296R#5scr/3
Rer < 500,000 5 x 1 0 5 < R e r < 1 0 7
0.6 Pr<60 0 . 6 < S c < 3 0 0 0
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 3/51
T.arsferoeffiients48
tho irst oobtain h;ssolution singa mathematicalechniquealled hesimilaritysolunonor themethodof combinationof variables.Note rhatEqs. B) and (C) in Tabte3.1 canbeobtainedlom Eq.(A) by using he ChiltonColbumanalogy. inceanalytical olutions reimpossibleor turbulentlow'Eq. D) n Table .1 s obtainedxperimen;ly.Theuseofrhisequationn theChilton-Colbumnalogy ieldsEqs. E) and F).
The avemgevaluesof the ftiction factor, he Nusseltnumber,andthe Sherwood umbercanbe obtained rom the ocal valuesby theapplicationof the meanvalue heorem.n manvcases,owever,hebansitionrom aminaro turbulentow will occuron theplate. n rh;case,both the laminar 4nd turbulent low regionsmust be taken nto account; calcuiuunstheaveEgevalues.For example, f the transition akesplaceat r., where0 < r" < l,, thentheaveragetiction factor is givenby
Change f variable rom r to Rex reducesEq.(3-5)to
rn *| l/-"'rr,r,,.an",/*"r1r,,,,an",]
r<e14e
Substitutionf Eqs. A) and D) n Table .I inroEq.(3-6)gives
l.328Re:/2 o.o74ReX/5
t f / r . t L 1; l l ( f ^ tu . , t ^ I \ f ) to ,bJx lL L J 1 J t , J
( l-5)
(3-6)
whereRec, heRelnolds numberat thepoint of transition,andRe., the Relnolds numberbased n the engthof the plate,aredefinedby
, ". 0.074(J )= - i ;+
R.;-
TakingRe.= 500,000esultsn
1743
(3-9)
(3-i )
(3-7)
(3-8)
(3 -11)
(3-12)
. -. 0.074
n"l/t Rer,
The average aluesof thefriction factof theNusseltnumber,and the Sherwoodnumbercanbe calculatedn a similarway or a varietyof flow conditions. heresults resiven n
Table .2. n these orrelationsllphysical roperties usrbeevaluaredt the ilm t;mpgra-ture.
Once heavengevaluesof theNusseltandSherwood umbers redetermined,he aveRgevalues f theheatandmassfansfer oefncientsrecalculatedrom
. , . (Nu)k
(sh)D,4
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 4/51
p
9€e
008
: 9 a
€
H
E
399
aG
E6
€
.e
g
;P seE
5
E
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 5/51
TransferoeffcienlsE0
On the other hand, the rate of momentumhansfer, .e., the drag force, the rate of heat
transfetand herate of massansfer of species4 fromone sideof theplateare calculateals
/ t _ \FD= twLt
\ ie i l i ) \ f )
Q: QrL) lh)lrtu- r@
nA = (WL)(kc)lcA_ cA_)
(3-13)
(3-14)
(3-15)
Engineeringproblemsassociatedwith the flow of a fluid over a flat plate areclassifiedasfollows:
. Calculate he ffansferrate;giventhephysicalproperties,he velocity of the fluid, andthedimensionsf theplate.
. Calculate he lengthof theplate n thedirectionof ffow; given he physical properties,the velocityof the fluid, and hetransfer aie.
. Calculaiehe luidvelocity; iven hedimensionsf theplate, he ransferate,and hephysicalpropertiesof the fluid.
Example .1 Waterat 20.C flowsovera 2 m long latplatewith a velocityof 3 m/s. Thewidth of theplate s 1 m. Calculate he drag orceon one sideof theplate.
Solution
Physicalprcperties
fo r wre ra l20 .c (2sJ * , ln :a9oke /mr - - -l r
= 1001 l0 6 kg /m
Assumption
1. Steady-stateonditionsplevail.
Analysis
To determinewhich conelationto use for calculating he averaBeriction factor (/), wemust irst determinehe Reynoldsnumber:
^ Lu*p f2)13|994)
- u l 00 l 10 -6
Thercfore,both aminarand turbulent low rcgionsexiston theplate.TheuseofEq. (D) 1nTable3 2gives he friction factor as
. ". 0.074 1143 0.074 1743
n" ] t Re r 16 lgb ) r 6^ l 0o -- ' "
Thedrag orcecan henbe calcularedromEq. 3-13)as
/ l , \ Tr ^ ' rFD- twLt l ,ouLl r t - r t ,2 ' l (9q9r{3 l l { l t0 i )_27N
\ z / l 2 I
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 6/51
51ChemicalngineedngDcesses
Example3.2 AL at a temperature f 250c ffowsover a 30 cm wide electricresistancelatplateheaterwitha velocityof t3 m/s.Theheater issipatesnergynto h; ;;;;;;;;,","",r?.te f n30 W rr2. How long must theheaterbe in thedirectio*n f flo]' to, tn" .*u""temperatureot to exceed155 C?
Solution
Physical properties
The ilm tempemtures (25+155)/2=90.C.
l u=21 .95x10 -6 .m2 l sForairat90 C (363K) and1atm: ltr=30.58 x l0 3Wm.K
I
lPr = 0.704Assumptions
1 Steady-stateonditionsprevail.2. Both laminaralrd urbulent low regionsexistover he plate.
Analysis
lhe average onvectionheat transfercoefficientcanbe calculated rom Newton,s aw ofcooling s
@t#r*= A+:2rw/m2Todetermine hichcorelation o use,t is necessaryo calculateheReynolds umber.HowevetheReynoldsumberannot edeterminedpriorisincehe enj*, of m. t
"ur.,sunknow0_herefore.rjal-and_errorrocedue,no"tUe""A.Since,eas_.omeJaiiotftrarrunarnd ubulentlow egionsxist verhehearetheuse fEq. E) nTable .2glves
, \lNu)= tj:1: __ 00J7R"j 5
szrler. l
(21 )L I r , r r , , l / 5 |ru . )d lu - -
I 1 r , . , ISimplificationfEq. 2)yields
F(L) = L - 1.99 a/5+ l.l3 =0
(D
(3)
(4)
The lengthof the heatercan be determinedrom Eq. (3) by usingone of the numencalmethodsor roor indinggiven n Section .7.2 n appenaix . mJiterution ."t
"_"giu"o
by Eq. (A.7-25)s expresseds
0.02Lk tF (Lk_)
F(1 .01L* -t t - F t0 .99L r )Assuming al5^:1,, a starting aluecan be estrmateds L,=l.l4l.The iterations regivenn the ablebelow:
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 7/51
Transfercoefiicients 52
Lp0I
23
r.1411.249
1.252|.252
Thus, he engthof theplaie s approximately .25m. Now it is necessaryo check hevalidityof thesecond ssumption:
(1.25)(13)
Example3.3 A waterstorage ank open o the atmosphercs 12 m in length and6 m rnwidth. The water and the surounding air are at a tempemtureof 25.C, and the relativehumidityof the air s 607,. f rhewind blowsara velocity f 2 m/s along he ongsideofthe tank, what s the steady ateof water ossdue o evaporationrom thesurface?
Solution
Physical properti€s
Forairat25 C (298K): r,:15.54 x 10 6m2/s
Diffusioncoefficientof water("4) in air (6) ar 25.C (298K) :
r )aR r t /2 r . roc, l .2rDar r2o8 (DABr r ,1 l- :
I _ 12 .88 tO 5 r { i j| _2 .79 , tO-5m2 /<
\ J t J l
TheSchmidt umbers
=1.4 v 105 =+ Checks21.95 10 6
v 15.54 10-6
Dta 2 .19x10 5
For water t 25'C (298K): Pr.r': 0.03165 ar
Assumptions
l. Steady-stateonditionsrevail.2. Idealgasbehavior
Analysis
Todeterminewhichcorrelation o use,we must irst calculate heReynoldsnumber:
ne1 3e = .-(*111
- - r 's+' o'u t J . J 4 X t ( ' "
Since oth aminar td turbulent onditionsxist, heuseofEq. (F) n Table .2gives
(sh) (0.037 e1/5 871) c'/3 [0.037(1.s4106;a/5 s71]19.56;r/:2996
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 8/51
53 Chemicalngineednsrccesses
Therefote,heaveragemass ransfercoefficients
,0 " , {Sh)Dea_ r2000 l |2 .79t0 - . )t
- - - -t 2
- -465x l0 Jm/s
Thenumberof molesof H2O("4)evaporatedn unit time is
ne: A{k)k'f, _ ca@iil: A(k)(c"ft o.6c,ft):o.4Ak"t ";tThe aturadononcentradonf $aLer,xr. s
. - , -r e _ 001165
'A -Rr
:$314;1onQ5 + 2:7,
= 128 x 10-r kmol/m3
Hence,he ateof water osss
rit: ieMe =0.4A\k) 4tMe
: (0.4)(12 6)(4.65 l0 3)(1.28x 10-3)(18)(3600)l l . l kslh
3.3FLOWPASTA SINGLESPHERE
Consider single pheremmersedn an nfinite luid.Wemayconsiderwo exactly qulva_lentcases: i) thespheres sta$ant, rhe luid flowsoverthesphere, ll *re nolJ i.'stig1lant,thespheremoves
hrcughthefluid.According to Newton's second aw of motion, thebalanceof forcesactingon a single
:ri".'::t1,]:,:-":.1'A:jerDp. atrinsnastasnanttuidwithaconstantenijnal elocity,r. ls exprcsseon the onn
Cravitational orce= Buoyancy+ Dragforce (3-16)
l rD r r \ t "D3 " \ l nD2 , t t | . t\ i lpPg_ ( u Jrsr \ i ) \ i t , , i ) r
B_t l l
where p andp representhedensitiesfthe pafljcleand luid, espectively.n the iteraNre,thefriction facror / is also catled he lrag cifficrznr and s denotedby ir. sirnpiii"",""ofEq. (3-17) ives
t . , , - 4 g D p l p p _ p l' " _ t - -p (3_18)
Equafion3-18) anbe earrangedn dimenstonlessb.mas
^ 4/Rei =
JAr (3-19)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 9/51
Transiercoefiicients 54
where he Reynoldsnumber,Rep, and he Archimedesnumber,Ar, aredefinedby
D3pgP@r P)
fi
I : - K e p < l
18 .5l= * 2<Rep <500
tleii-
f : 0 .44 500 (Rep<2x105
Engineering roblemsassociated ith the motionof sphericalparticlesn fluids areclassifiedas ollows:
. Calculatehe erminal elocity, r given hev scosiiy f flu d, . and hepa ticle diameter,Dp.
. Calculateheparticle iameter, p; given heviscosity f the luid,p, and he eminalvelocity,,.
. Calculatehe luidviscosity,.;given heparticle iametetDp, and he eminalvelocity' v
The dimculty in theseprcblemsarises rom rhe act that the ricrion factor / in Eq.(3-19) sa complex unction of theReynoldsnumberand he Reynoldsnumbercannotbe detetmined
3.3.1 FrictionFactorCorrelations
For flow of a sphere hrough a stagnant luid, Lapple ard Shepherd 1940)presentedherexperimentalata n the brmof / versus ep.Theirdata anbe approximateds
0.413
(3-20)
(3-21)
(3-22)
(3,23)
(3-24)
( l-25)
Equations 3-22) and (3-24) are generally eferred to as Stokes' aw and Newton's aw,
respectively.In recent ears, fforts ave een irectedo obtain single omprehensivequationor he
friction actor hatcoversheentire ange f Rep.TuftonandLevenspiel1986) roposedhefollowing five-constant quation,whichcofielates he experimeltaldata or Rep ( 2 x 105:
1: 11r 1o.rr :n.orut t )KeP I * 16,300Re roe
3.3,1.1 Solutionso theengineeringroblems Solutionso theengineeringroblems e-scribedabovecannow be sunmatizedas ollows:
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 10/51
55 Chemicalngineenngrocesses
I Calculaterr; giyen p and Dp
SubstitutionfBq.
(3-25)ntoEq.(3-19) ives
Ar= t8(Rep0.173Ref65?)-j 31n4-' I + 16,300Re;loe
(3-26)
SinceEq. (3-26)expresseshe Archimedesnumberas a functionof theReynoldsnumoelcalculationof the terminalvelocity for a givenparticlediameterandfluid viscosityrequiresan iterativesolution.To circumvent his problem, t is necessaryo expresshe Reynoldsnumberas a function of the Archimedesnumber.The following explicit expressionelatingtheArchimedesnumber o the Reynoldsnumber s proposed y Turtonana btart 1Dt; r.
ReP #(l +o.o579Aro4r2)214 (3-27)
Theprocedue o calculatehe erminal elocitys as ollows:
a) CalculateheArchimedesumbertomEq.(3-21),b) SubstituteheArchimedesumber into Eq. (3-27)anddeiermine heReynoldsnum_
bet,
c) OnceheReynolds umbers determined,he erminal elocity anbe calculatedtomme equalon
,rlRep(3-28)
PDeExample 3.4 Calculate he velocitiesat which a drop of water,5 mm in diameter,wouldfall in air at 20'C and hesame izeair bubblewould;se throughwater t 20"C.
Solution
Physical properti€s
Forwarer r20'c (29JK.J: p-- oqg e/']
l / - l 00 l l0 6 kg /m"
Forairat20'C (293K):
Analysis
ln=1.zulks/nf[ r r= 18 .17 10okg/ms
Water droplet falling in air
To determile the teminal velocity of water, t is necessaryo calculate he Archimedesnumber singEq.(3-21):
A r_u 'pqp tp p t
_(5x l0 ) ' j ( q .8 ) r.2047) tqqg-.2047,
u- €l? ' fo-tr--4 46x to"
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 11/51
Transfermefftcients 56
TheRe),iolds umbers calculatedromEq. 3-27):
A rR e P= 1 " 1 - 0 . 0 5 7 q r 04 1 2) i 2 1 4
4.46x 106:fflr +o.os'ts(4.46t06\0t2l-t ta:3slr
Hence,he erminal elocity s
4 Ree ( l 8 . l 7x l 0 b r ( J581 )i / :
pDp-
( t2o47 l (5x 0 f= rudm/ \
Air bubble rising in water
In thiscase,heArchimedesumbers
, D'ogprpp pt (5 l0-rrr (9.8)gos) .2047 sqg)----n- -ilnoa' F-- Izrq ru
Theminussign ndicates hatthe motion of the bubble s in tie direction opposite o gravity,i.e., t is rising.TheReynolds umber nd he erminal elocity re
ArRec=
rr--r- 0.0579 jo4 l i 12 4
1.219 n6 .. ^ ̂ _-= _ f * :L t +0 .0s?q r.2 ts . , t 0b )0 .4lr 2 r ,_ t825
/ 1Rep t l00 l ^ l0 6) (1825) - -_o =7o,
:rsogrts to lt : o ; m/'s
I CalculateDp; givenprand u;
In thiscase, q.(3-19)mustbe reanangeduch hat heparticlediameters eliminated.fbothsidos f Eq.(3-19)aredivided y Re3p,he esulrs
JRt = Y (3'2e)
where , which s ndependentf Dp, is a dimensionlessumber efined y
y : s@e )p (3-:lo)3 ptri
SubstitutionfEq. (3-25)ntoEq.(3-29) ields
v - j1( t - o . ; : n"oosr ' *04 lJ - t3-Jt )
R"," ' 'Rep 16.300 e;o o
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 12/51
57 Chemicalnqineennsbcesses
SinceEq. (3-31)expresses as a functionof theReynolds umber, alculation f thepaticle diameteror agiven erminalvelocityand luid viscosity equiresan t.rutiu" .olut on.
Tocircumvenrhisproblem.he ollowing xplicir xprelsionelaring to theRe\notdsnumbersproposedy To\un ndAL$ahinl992)as
E(v)(3-32\
t 6 y t 3 / 2 0 y 6 / 1 )) n / n
vlryexp(:s 19?W -'#P) (3-31)
The procedureo calculatehe pafticlediameters as ollows:
a) Calculate fromEq. 3-30),b) Substituie inroEqs. 3-32)and 3-33) ndderermine ep,c) Once he Reynoldsnumber s determined,he particlediametercanbe calculatedtom
theequation
whereV(f) is givenby
& ReP
Example3.5 A gravity settlingchambers oneof the diverse angeof equipmentused oremoveparticulatesolidsfrom gassl€ams. In a settlingchamber,heeniering gasstreamencounters largeandabrupt ncreasen cross_sectionalreaas shown n the figurebeiowAs a resultof the sharpdecreasen the gasvelocity,he solidparticles eftlelown with
gravity.In practice,the gasvelocity through he chambershouldbe kept below 3 m/s toprcvent he re-enftainment fthe settledDarticles.
(l-34)
Sphedcaldustparticleshaving a densityol 2200kg/m3 are to be separatedrom an arstreamat a temperafiue f 25oC.Determine he diameterof the smallestparticle hat canberemovedn a settling hamber m long,2 m wide,and m high.
Solution
Physical properti€s
l r :1 .184skc/rn3l l l . : 18.41 lO-bkg/m.s
F_r _ -
For airat 25"C (298K) :
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 13/51
Tlansfercoefticients 58
Analysis
For the minimumparticlesize hat can be removedwith 1007.efficiency, he time requiredfor rhisparticte o fall a distance/ must be equal o thetime required o move hisparticle
hodzontally distance , i.e.,
H L . . /H \t -u ' : ( ' ) - - } ' ' - t ' l \Z /
where o) representsheaverageas elocityn thesettling hamber aking (u) 3 m/s,
the settling velocity of thepafiiclescan be calculated s
/ 1 \o1 (3){; }:0.a3 rnls
\ r /
Thevalue f y is calculatedrom Eq. 3-30) s
f^ .- 0.052 0.007 0.0001q'l ^, .= e \pLr 'r '{4J4r ' l i * . i4 t t ,
-t+u31n -
z+)
. . 4 g tpp p tp 4 {q .8 ' t2200 - 1845 , (18 .410 6 ,, - ,t - l
p-, ; ' .- r
' r ' rsas; 'o""-* ' "
Substitutionf thevalue f / intoEq. 3-33) ives
/ o.09 0.007 0.00019, {y )=exp \3 . t5+yt a I yn
_ _1 t r)
Therefore, he Reynoldsnumberand heparticlediameterare
^ v { } ) ZA .3n" " =
Gy . , ' . l 0 y "u , t ,-
l 6 \4 ;2 , t : , 2oA i4 )o , t1 t1 'n -z ) )
^ u Rer ( 18 .41 l0 6 t t2 .55 r^ , . , ^ -o ,_
' prr (1.1845)(0.43)
I Calculate ; given p and u,
In this case, q. (3-19)mustbe reanangedo hat he luid viscosity anbe eliminated.f
bothsides f Eq.(3-19)aredivided y Rezp,he esult s
f : x (3-35)
whereX, which s ndependentf&, is a dimensionlessumber efined y
Y : eDe (PP- -P) (3 -36)"3 pu?
SubstitutionfEq. (3-25)ntoEq.(3-35)gives
x_ j1{ t+o.ptn.o.osr1 0.411 __ \ j_J t )Kep l * l6 .300Re l ' "
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 14/51
59Chemic€lngineeing rocesses
SincoEq.(3-37) xFesses asa function ftheReynoldsumber, alculation fthe fluidviscosity or agiventeminal velocityandparliclediametercquiresan terativesolution.lbcircumventhis problem, he ollowing explicitexpressionelat;g X to the Reynoldsnumberrsproposedy Tosun ndAktanin 1992):
nee 2 j r t + t2oX o l t ti t t t X >0 .5 (3-3 )
The procedureo calculatehefluid viscosity s as ollows:
a) Calculare fromEq. l-16r,
b) Substitute intoEq.(3-39)anddetermineheReynoldsumber,
c) Once he Reynoldsnumbers determined,he luid viscositycanbe calculatedtom lneequatron
(3-3e)p \ P
Exl''.nph3.6 One way of measudngluid viscositys to usea fallingball viscometernwhicha sphericalall of knowndensitys droppedntoa fiuid,filted r;duaredylinder ndthe ime of fall for the ball for a specifieddistances recoraled.
A sphericalball, 5^mm n diameter,hasa alensiry f 1000kg/m3. Ir falls througha liquidof density 10kg/m3 ar 25.C and ravels
distance f l0 cim n f.f rnin.OetJnJne reviscosity f the iquid.
Solution
Theteminal velocityof thespheres
Distance l0 ^ l0 2:9 .26 ' t j - a m/sTime
Thevalueof X is calculatedrom Eq.(3-36) s
(1.8x60)
(9.8)(5 10 3)(1000 910) 4 8 D p \ p p - p )
3 pu? 3 (910)(9.26 l0 a)zSubstitutionf thevalueof X intoEq.(3-38) ivesheReynoldsumber s
R"" ?l + rlox-'o,u)n"t Ll, * ,20(is36)-20/11)4/tt3.2x ro 3
Hence, leviscosity f the luid s
Dp \g (5 l0 r ) (9 .26x t0 4 ) t9 l/ - f t : r - i= - l l2(g/ms
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 15/51
TEnsiercoefficients 60
3.3.1.2 Deviationsfrom idealbehavior It should e noted harEqs. 3_19) nd 3_25)areonly valid for a singlesphericalparticlefalling in an unboundedluid. The Dresence f
container allsandotherparticles swell asanydeviationsrom sphericalhape ffect neterminal elocityof particles. or example, s a resultof rheupflowof disphc;dfluid n asuspensionf uniformparticles,hesettling elocity fparticlesn suspensions slower hanthe erminal elocity fa singleparticlefthesame ize. hemostgeneralmpirical quationrelating hesettling velocity to thevolume ractionof pafticles,a), s givenby
ur(suspension)(3-40)
where he exponent dependson the Reynoldsnumberbasedon the teminal velocityof aparticle n an unboundedluid. In the iterature,valuesof r are cpofiedas
(3-41)
Particleshape s another actoraffecting eminal velocity.Thetenninalvelocityof a non_)phericalrnicle< e\. han hcr fa sphericalne ya facror frr 'r?ri . i ry,. . . ; . ,
u, single phere)
ur(non pherical)
Nu:2 + 0.6Ref;2prr/3
All propetiesn Eq.(3-44)mustbeevaluatedt he ilm temperature.
[4.65 5.00 Rep< 2": lz.soz.es 5oo Rep 2x 105
Sphedcity s definedas the ratio of thesuface areaof a spherehavingthe samevolumeasthe non-spherical article o the actualsurfaceareaof theparticle.
3.3.2 HeatTranslerCorrelations
Whena spherc s immelsedn an nfinite stagnantluid, the analyticalsolution or steadvs[a[econducrionr po.siblelnd he esullserpressedn he orm
u,(spherical)(3-42)
(3-43)
In thecase ffluid motion,hecontributionf theconvective echanism ustbe ncludedin Eq. 3-43).Cor.elationsor including onvectiveeat ransfer reas ollows:
Ranz-Marshall correlation
RanzandMarshall 1952) rcposedhe ollowingcorrelationor constant urfaceempcra_ture:
(3-44)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 16/51
61Chemicalngneerins rccesses
Whitak€r correlation
Whitaker (1972) consideredheat tansfer fromthe sphere o be a result of two parallelprocesses ccuring simultaneously. e assumedhat the aminarand urbulentcontriburrons
areadditiveandproposedhe ollowing equation:
Nu:2+ (0.4Rev2+ 0 06Re213)pp.10r*l1,)1/4 (3-45)
A11 ropertiesexcept ,.u shouldbe evaluated t fe. Equation 4.3-30) s valid for
3.5 Rep ?.6x lOa 0.71 pr ( 380 t.O< LL6lt .u 3.2
3.3.2.1 Calculntion f theheat ransfer ate Once heaverage ear ansfer oefficientsestimated y usingcorelations, the ate of heat ransferreds calculatedas
A: (1,D2)lhl lr. -r* l i l ' -46)
Example3.7 An instrument s enclosed n a protectivesphericalshell,5 cm in diame@r,and submergedn a river to measureheconcentrationsf pollutants.The temperature ndthevelocityof lhe riverare 10'C and 1.2m/s, respectively.oprevent nydamageo theinsfument as a resultof the ow river temperatue, he surface emperatures keptconstantat 32'C by installing lectrical eatersn theprotectivehell.Calculateheelectrical owerdissipated nder steadyconditions.
Solution
Physicalproperties
Forwater t 10'C (283K):
Forwaterat 32 C (305K). p :7 69 x 10 6kg/m.s
Analysis
System: rotectivehell
Understeady onditions, he electricalpowerdissipaied s equal o the mteof heat oss romthe shellsuface to the river The rate of heat oss s givenby
A: (" Dzp)De, _ r*)
To determiner), it is necessaryo calculateheRe),noldsumber
Dpuap (5 x l0-2)(1.2)(1000)
l,:
r:o+,. o_o
Ir: 1000e/mr-
fr: 1304 l0 oks/ms
It :587 x l0-3w/tn.K
Ik= 9 .32
(1 )
=4 .6x l ja (2 )
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 17/51
TErsier coeffrcients62
TheWhitakeronelation,q. 3-45),Bives
Nu 2+ (0.4Rey, 0 O6Re2/3)p'40L_ p;t /4
Nu:2+[0.4(4.6x l0a)r/2 0.06(4.6"
toa12/311s.32,10.+
/ 1304 10-6\ /a> . t - t :456 (3 )
\769 ,106 /
The verage eat ran\fer oefficients
. . / k \ / 58 / . t0 ' \( f t )=Nu l̂ l=(as6r l - - , l_sJsJw/m.K t4)\ uP / \ ) ^ ru )
Therefore,he rute of heat oss s calculatedrom Eq.(l ) as
a =[16 x 10 r)2](5353)(32- 10):925 w (5)
3,3.3 MassTJansfer orrelations
Whena spheres ilnmelsed n an nfinite stagnantluid, theanalyticalsolution or steadystatediffusionspossible2nd he esult sexpressedn the orm
sh: 2 (3-41)
Inthecase ffluidmotion, hecontdbutionfconvection ustbe akenntoconsidemtion.Corelations for convectivemass ransferare as ollows:
Ranz-Marshall coftelation
For constant urfacecompositionand ow masshansfer ates,Eq. (3-44)may beapplied omasshansferproblemssimply by replacilg Nu andpr with Sh and Sc, respectively,.e
sh:2+0.6Ref;2scr/3 (3-cx)
Equation3-48)s valid or
2<Rep<200 0.6<Sc<2.7
Fmsslingcorrelation
Frossling1938) roposedhe ollowing ofelation:
sh:2 + 0.552Re /2Sc1/3 Q_4s)
Equation3-49)s valid or
2<Rep<800 0.6<Sc<2.7
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 18/51
Steinberger-Tfeybalcorrelation
Thecofielation_originallyroposedy SteinbergerndTrcybal 1960)nciudes corecriontermfor naturalconvection.The lack of experimentaldata,however.makes his ter; v€rydifncult o calculaten mostcases. heeffeciof natualconvectionecomesegligibl;whentheReynoldsnumber s high, and heSteinberger_Treybalorrelation educesJ
"
63 Chemical lrqineeringrocesses
SteinbergerndTreybal 1960)modifiedheFrossling orelationas
which s vaiid or
sh : 2+ 0.552Re0r53cl/3
1500 Rep< 12,000 0.6< Sc< 1.85
Forwater 6) at 25oC(298K):
TheSchmidt umbers
(l-50)
(3-s2)
Sh= 0.347Re962cr/3 (3-5)Equation3-51)s recommendedor iquidswhen
2000<Rep<16,900
3,3.3,1-,Calculationofthe mass ransfer ate Once heaveragemassransfercoefficients
estimatedy using orrelations,hemteof mass f species4 tra-nsferreds catculateJ:r"
,ha = (r D2p)k)lc t. - u)Me
Example3.8 A solidsphere fbenzoic cid (p: 126.7g/m3)wirha diameter f 12mmis droppednto a longcylindricai ank iiledwith purewirer at 25.C. It tfreneignfof tfretank s3 m,detemineheamount fbenzoic ciddissolvedrom hesphere henlt eachesthebottom f the ank.Thesaturationolubility f benzoic cid n waier s 3.412 s/m3.
Solution
Physical properties
Ia= 1000s/m3
I u:892 x 10-6 s/m.s
IDo" : t .z t x lo em2 ls
pD AB
892x 0 o
(1000)(1.2110-e)
Assumptions
1. Initialaccelerarioneriod s negligible nd hesphercinstantaneously.
reachests terminal elocity
2. Diameter f thespherc oes otchange pprcciabty.hus, heReynolds umber ndthe erminal elocity emain onstant.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 19/51
Tnansfer oeffcients 64
3. Steady-stateonditions revail.4. Physicalpropetiesof waterdo not changeas a result of mass mnsfer.
Analysis
To determine he erminal velocityof the benzoicacid sphere,t is necessaryo calculateheArchimedesumber singEq. 3-21):
, D t rgp rpp p ' r l 2x I 0 r r J {9 .8 ) ( l 000 r i l 2b7 -1000 ] - ^ . ^A- - - -n- =1892 t ; - - r 'odx
ru
TheReynolds umbers calculatedrcm Eq. 3-27):
nep= $1 t . 0 .057qAru4 l2J2 r ' 1Ius AR tn6
- J U o - r uJ l r n n s ? q r s ^ R Y I n o r 0 4 l 2 l l 2 l 4 - 1 n s A
l 8
Hence, he teminal velocify is
, ,_ ,uRep _ '892 , l 0 o ' {40 t -0 ,_O. r_ r , ,'
pDp (1000) (12l0 i )
Since he benzoicacid sphere alls the distance f 3m with a velocity of 0.3 m/s, the alling
time s
Di\rance 3 _/__ =_ : l us
Time U.J
TheSherwood umbers calculatediom the Steinbergerreybal orrelation, q. 3-5-,,as
sh = 0.347Reg62c1/30.34'7(4056)a2('73't)1/3= 54t
Theaverage ass ransfer oef6cients
L .2 l l 0 - 't * . . t s l ( " r '4 I=rsa l r l __t- ) .46xlu 'm/s'\Dr , / \ t2"10- ' /
The mte of transferof benzoicacid(species"4)
to water s calcuiatedby using Eq. (3-52):
tnA= (t D2p)\k,)(cA,, cA-lMA: (n D2p)\k,l@t, pe-)
= [z(12 r0 t)']ts.+e"
to sy1:.+tzoy= s.+: t0-8 gTs
TheamounrfbenToiccid issol\edn l0 s s
MA = th^t = (8.43 10 8)(10): 8.43 l0-7kg
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 20/51
65 Chem'€lEngineeringtoesses
Verification of assumption# 2
The nitial massof the benzoicacid sphere, l4, is
f t r l 2 \ l 0 - l ) l lM^= t 1 f l 267 )_ . t46 , t0 ,kBL6r
Thepercentdedease n the massof t}le sphere s givenby
/ 8.43 l0-7 \r_ 11 . t00 0 .014qa\ 1 .146 l0 7
Therefore,he assumed onstancy f ,p and u, is ustified.
3-4 FLOWNORMAL O A SINGLECYLINDER
3.4.1 FriclionFactorCorrelations
For cross low over an nfinitely long circularcylindet LappleandShephed(1940)presentedtheir experimentaldata n the form of / versusReD, the Reynoldsnumberbasedon thediameterof thecylinder Their datacan be apprcximated s
. 6 .18J = m ReD 2 (3-53)
K€n
f
: t.2 104 Re,<
1.5 105 (3-54)
The friction factor / in Eqs.(3-53)and(3-54) s basedon iheprojectedareaof a cylinder,i.e., diameter imes ength,and Re2 is definedby
"Du*P
xeo=- ; - t3 -55 ,
Tosun ndAk$ahin 1992)proposedhe ollowingsingleequationor rhe riction actorthatcovers he entiremnge of the Reynoldsnumber n theform
6 t R
1: "]|1r +0.:on/e)8/5
Reo 1.5 105 (3-56)KC;
Once he riction factor s determined, hedrag orce s calculatedrom
r ,=rnt( \p, ,*\ r (r_57)
Example3.9 A distillation colurnnhasan outsidediameierof 80 cm and a heightof 10 m.Calculatehedrag orceexerred y air on hecolumnf thewindspeeds 2.5 m/s.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 21/51
Transieroeff ients 66
Solution
Physicalproperties
Forairar25.c (298K,. l l :1 1845 e/m3
'{rr: 18.41 10 6kg/m.s
Assumption
1. AiI temperatures 25'C.
Anallsis
l .romEq. 3-551lheeynold\umbers
- - Du -p (0 .8(2 .5 t .184518 ,41l0o
r l zex lu -
TheuseofEq. (3-56)giveshe ricrion actoras
6 t R
r - ^"$ lr+o.J6Relq)8"R"l -
6 .18
d##+rtt* 0 36(t sx 105)5/el8/512
Therefore,he drag orce s calculated romEq, (3-57)as
l I . \ T t ^ lF D (DL , l,p t i l f : r 0 .8 l 0 ) l { L 84s (2 .5 ) t rl ( 1 .2- 35 .5
1 L 2 I
3.4.2 HeatTransferCorrelations
As statedn Section .3.2, he analytical olutionor steady-stateonductiontom a sphereto a stagnantmedium givesNu = 2. Therefore, he correlations or heat transfer n spheri_cal geometry equirethat Nu --+2 as Re-->0. In the caseof a single cylinder,howevel nosolutionor thecase f steady-stateonductionxists.Hence,t is rcquired hatNu -->0 asRe-+ 0. The following heat ransfercorelationsareavailablen this case:
Whitaker correlation
Whitaker(1972)proposeda conelation n the orm
Nu: (o.4Rey'?0.06Re 3)pP4@*/;1/a (:-sr)
in whichall propertiesexceprr. arc evaluared t 7t. Equarion 3-58) s valid for
1 .0 (Rep (1 .0x105 0 .67<p r<300 0 .25<pe lpo<5 .2
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 22/51
67 Chemicalngineeing rccesses
Tabte 3.3. Consrantsor Eq. (3_59)or rhe ckcurarcylindern cross low
ReD
l-40 0.75 0.440 1000 0.51 0.5
1x 1032 x i05 0.26 0.62 x ld I x 106 0.076 o.j
Zhukauskascorrelation
Thecorelationproposedy Zhukauskas1972)s givenby
Nu: CRe3pr"(pr_/pr.)r/a (3_59)
10.37 if pr < 10
10.36 i P r l 0
and hevaluesof C and m aregiven n Table4.3.All prcpertiesexceptpru shoulal e evalu,ated t f6 in Eq.(3-59).
Churchill-Bernstein corelation
Churchilland Bemstein
1977)proposed singlecomprehensivequationhat coversheentircmngeof ReD for which data areavailable,as well as for a wide rangeof h. Thjs
equations in the form
0.62Rej.2Prr [. / Re., r5,8l ' /5 ,1_60) | t , 10 .4 /p t ' ,3 l t / 4
I'
\ 282 .000 /I
, .
whereall propefiiesareevaluated t the film temperature. quation(3_60)s rccommendedwhen
ReDpr > 0.2
3,4,2.1 Calculation of the heat transferrate Once he average eat ransfercoefficlent sestimated y using corelations,the rateof heat ransferreds calculatedas
Q = QtDL\ \h|lru - T6 (J -61 )
Example3.10 Assume hat apenon can be approximated sa cylinderof 0.3 m diameterand 1.8m heightwith a surfaceempemturef 30.C. Calculateherateof heat oss romthebodywhile hispersons subjecredo a 4 m/s wind wirha remperatuef _ 10.C.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 23/51
TEnsfer@efiicients 68
Solution
Physical pmperties
The llm temperatures (30 l0) 2: lO"C
lu - t t . ' t . 106kg /m. .
Fo ra i ra r r0 ' c (263 ) . lu :12 4 I0 6m27s
l k=z l .zZ" t0 r w/m.K
ln: o.zz
Iu -14 .18xt0om2/sFora i ra r0 .Cr280K, :
l l -2a .96^t0 'W/m.K
IPr= 6 714
For irar o.c (30J 1' u- tt '"+ 10-6 gTm
lPr :0 .71
Assumption
1. Steady-stateonditionsrevail.
Analysis
The ate f heatossrom he ody an ecalculaiedromEq. 3_61):
a : Q,DL) \h tQura ( l )
Dererminationf {1,)nEq. 1) equiresheReynoldsumberobeknown. heReynoldsnumberst 7; and7/ are
a t /6 - - l 0oc o '^ -Du ' - r03 r l4 'ReD ---= =i)7;;fr
=e.65 roa
at rJ=to"c ReD:919=: {q3r(11- =8.46x 0a14 . t8^ l 0 6 - - '
Whitaker correlation
TheuseofEq. (3-58)givesheNusselt umber s
Nu = (0.4Rey?+ 0.06Re2/3) f.4Q1- p,\t /4
: [0.4 9.65 104)r/2 0.06(9.65 tg+,vr1,t.tr,or f1.6?rl0-6-)r/a
'\ t 8 .64 , t0 .6 /
, Hence,he average eat ransfercoefficients
^ - . / 23 .28 . l 0 r \(r ' ) Nu{i |, , , ,
- , r ,u , (__J_)=
t6.6w/n. .K
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 24/51
69Chemi€lEngineeringro@sses
Substitutionf thisresult ntoEq.(l) giveshemteof heat ossas
a=(nxo.3 x r .8) (16.6) [30(- 10) ] :1126Zhukauskascorrelation
ForReo= 9.65x 104 ndPr < 10,,? 0.3?,and romTable .3 heconstantsreC = 0.26
andm : 0.6.Hence.heuleof Eq. 3-5s) i les
Nu = 0.26Reg6ProJT r- / Pr@I 4
-0 .2b {q .65 ,00 ,00 ,6 .72 ,01 ' (9? ) 'o
: : : o\v . / . /
Thercfore, heaverage eat tansfercoefficientand he rate of heat oss rom the body are
, l \ t ) ' ^(h) Nu( : ) - (226,(jj i- -) : r7.sw/m':
\ t ) / \ 0 .3 /
Q=Qt x0 .3x 1 .8 ) (L? .5 ) [30 - ( 0 ) ] 1188
Churchill-Bemstein corrclation
TheuseofEq. (3-60)gives
o .b2Re l2 r r / r [ , R .^ r t ' t - l ot
N , , - n 1 _ u - t t t - - - - - - - - - - t I' - " " l l - {0 .4 /Pr '2r l r4L'
\282.000/I
^ . .0 .b2(8.46\ 0arr r0 . t t4r r l r f, 18.a6too\" - lo / ' ,^ ," .+l t+aAlc i t4 t r t r l
^L
t \,sr "ooo I
: ' "
The average eat ransfercoefficientand he mie of heaf oss tom the body are
t l r t / 24 .80 .10 r r _ . . . , . .( h )= Nu{; l - r o r t l - l - l 6 w /m- K
\u . / \ 0 .3 I
a: (r x 0.3 1.8)(16)[30(-r0)] 1086
Comment: The rate of heat ossFedicted by the Zhukauskas onelation s 97ogreater
than hat calculatedusing heChurchill-Bernstein orrelation.t is important o note that no
two conelationswill giveexactly he same esult.
3,4.3 MassTransferCorrelallons
Bedingfield ndDrew(1950) roposedhe ollowing ofielationor cross- ndparallel-flow
of gaseso the cylinder in which mass ransfer o or from the ends of the cylinder is not
considered:
sh = 0.281Rey2 co44Q-62)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 25/51
Transrer oeffi ienrs 70
Equation 3-62) s valid for
400< ReD< 25,000 0.6< Sc< 2.6
For iquidsthe conelationobrainedby Linton andSherwood1950)maybeused:
sh= 0.281ReB6Scr/3 (3_63)
Equation3-63)s valid or
400< ReD< 25,000 Sc<.3000
3.4.3,1 Calculationof themass ransferrav Once he averagemass ansfer coelficientsestimated y usingcorelations,the rateof massof species4 ffallsferreds calculatecl s
nA: @DL)(kc)lcA. cA_lMA e_64)
whereMe is themolecularweightof species4.
Example3.11 A cylindricat pipeof 5 cm outsidediameters covercdwith a thin layerofethanol. ir at 30.C flowsnomal to the pipewith a velocityof 3 m/s. Derermirc heaveragemass ransfercoefficient.
Solution
Physical propertiesDiffusioncoefncientof edranol ,4) in air (B) at 30.C (303K) is
/ l n 1 \ l / ' / r ^ r \ t / 2
tDea) ro r=rD,qa t r r ' l ( '# l - r t .4s \ l0 5 r ( i : ) - t . 38x l0 5m27s\ r r J / \ J t J l
Forairat30'C 303K): v:16 x 10 6m2ls
The Schmidtnumber s
s":r|=ffiffi=rroAssumptions
l Steady-stateonditionsprevail.2. Isothermalsystem.
Anal)sis
The Reynoldsnumber s
Du* (5> t0 , r ( j )^^__( (D - -=
16 \ l oo-e r l )
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 26/51
where n1 and ? are the mass lorr mte and the specific volume of the ffuid, respectively.Note hat he erm n parenthesesn rhe ight-hand ideof EJ. (3-65) s knownas he rdlIork in thermodynamicsr.oran ncompressibleluid, .e.,y : 1/p: constant, q. 3-65)simplifies o
w:QLP)
71 Chemical ngjneeringrocesses
Theuseof thecorelationproposedy BedingfieldndDreq Eq. 3-62), ives
sh = 0.281Rev2c0.410.281(g3:'rt/20.rcro4:29
Thercfore,he avemgemass ransfercoefficient s
( / , . ) -sh l? ' ) : ,2q, / | 18 tn- :\ D / 1s ' ro : J :8r lo 'm/s
3.5 FLOW N CIRCULAR IPES
Therate of work done,W, to pumpa fluid canbe determiled rom the expression
w=*o:*(l oar) (3-6s)
(3-66)
whereQ is thevolumetriclowrateof the luid.CombinationfEq. (3-66)wirhEq. 3.1 1l)glves
Fp luJ : QJLPI
| l l , \ ' ll tnDL) l ;p \uJ.l / l (u) et^p lL \ z / l
Exprcssinghe average elocity n termsof the volumetric low late
(3-67\
. . a, - '
n D2 4
^ ̂ , 32pLfQ'1
n2Ds
reduces q.(3-68) o
(3-68)
(3-69)
(3-70)
Engineering roblemsassociated ith pipeflow areclassifiedas ollows:
. Determinehepressurerop, APl, or thepumpsize, -iz; iven hevolumetriclow rate,8, thepipediameter,D, and hephysicalpropertiesof the fluid, p and -,.
. Determinehevolumetriclow rate,Q; given hepressurerop, A P , thepipediameter,D, and hephysical ropertiesfthe fluid,p andp.
. Determine hepipe diameter,D; giventhe volumetric flow rate,Q, thepressure rop,A Pl, and le physical ropertiesf the luid,p andp.
Jwork doneon thesystenrs considered ositive-
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 27/51
Transter oefficients 72
3.5.1 FrictionFactorCorelations
3Srl.l. Laminarfow correlatio For aminarlow n acircular ipe, .e.,Re D (ulp/ tr <2100,hesolution f theequatioN f changc rves
(3-71\
The riction actor appearingn Eqs. 3-70)and 3-71)s alsocalled heFanninpnc-tionfactor. lowever,his s not heonlydefinirionor / availablen the jterature.inothercommonlyuseddefinitionfor / is the Darcyt.tion factor, fD, vthich s four times argerthan he Fanning riction factor, .e.,fD : 4 mercfore, for laminar low
(3-72)
3.5.1.2 Turbulentfow correlatio, Since o theoreticalolution xists or turbulentlow,the riction factor s usuallydetermined.om the Moodycharl (Ig44) in which t is exoresseclasa unction f theReynolds umbetRe,and he elative ipewall oughness./D.'Moodypreparedhischartby using heequarionroposedy Colebrook1938)
- 16' Re
a^ - :-,
| . . / r l D l . 2 6 l j \- _ - 4 l o s l - t - l
"J
--\3.706s Re,///
where€ is thesurfacecughness f thepipewall in metels.
3.5.1,3 SoLutionso theengineering roblems
I. Laminar flow
Fof flow in apipe,tle Re),notds umber s definedby
- D luJ 4pQ
subsrirurionfEq. 3-7+ynto q.1:-zr; ila"
r pD
41t D
(3-73)
(3-74\
I Calculat€APl
SubstitutionfEq.
or t7; givenQ and D
(3-75) nroEq.(3-70) ives
128pLQ
P Q(3-7s)
(3-76)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 28/51
73 Chemical igineering rocesses
Thepumpsize anbe calculatedrom Eq.(3-66) s
I281LLQ2(3-77)
(r-78.)
I CalculateQ; givenIAP and,
RearrangementfEq. (3-76)gives
I CalculateD; givenQand API
Reanangementf Eq.(3-76)gives
^ t D 1 L P-
lz8ttL
^ 1 1281LLQ1t/a
\ raP l(3-7e)
II. Ttrbulent flow
I CalculateAPl or W; givenO and ,
For hegiven alues f Q andD, theReynolds umber anbedetemined singEq. 3-74).
However, hen he values f Re ande/D areknown,determinationf / ftom Eq.(3-73)requiresan terativeproceduesince appears n bothsidesofthe equation.To avoid terativesolutions,effortshavebeendirected o express he friction factor, /, as an explicit functionof theReynolds umbetRe,and he elative ipewall roughness,/D.
Grcgory ndFogarasi1985) omparedhepredictionsf the welve xplicit elations rthEq. 3-73)and ccommendedheuseof theconelation roposedy Chen 1979):
| . . / t lD 5 .0452 \_= 4 tog l_ _ r ^oa r / r .80 )\/ [ \ 3.7065 ne
--- -,/
/ /D \ ll oo8
r 7 .1490 to8o8 lA= l . l + l | ( l -R l r
\25491 \ Re /
Thus, in order to calculate he prcssuredrop using Eq. (3-80), the following procedureshould e ollowedhroughwhichan temtive olutions avoided:
a) CalculateheReynolds umberrom Eq. 3-74),b) Substirute e nto Eq.(3-80)anddetemine ,c) UseEq. (3-70) o find hepressurerop.Finally, hepumpsizecanbedeterminedy
usinBEq. 3-64).
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 29/51
Transfer oefiicients 74
Example 3.12 What s the requiredprcssure ropperunit length n order o pumpwaterat
a \olumetric low rateof 0.03ml/s at 20"C through commercial teel ipeG = 4.0 x10 ) m) 20cmin diameter?
Solution
Physicalprop€rties
nlForw.terat20'C 1293 r I
p: >v'tKc/l-
[ r : l 00 l l 0 6kg /m.s
Anal)sis
TheReynold\umbersdeterminedromEq. t-741 \
^ 4pQ (4 ) tqqq r (0 .03 jnpD n(1001 t0 6 ) (0 .2 )
Substiturionf thisvalue nroEqs. 3-81) nd 3-80)gives
, t E lD \ i l oo8 /7 . t4oo \o8o8 l
\2 .s4o1 \ Re ./
T (4 .o l0 5 /0 .2 )l l l@8 r 7 .1490 08n81
L z.s4e i I' \ rer . ro j /
| . . / t /D s .0452 . \- - 4 roc \ too5 - -n t 'o8o , l
- f (4 .6 l0 5 /0 .2 ) 5 .0452 " I- -4 roc l3J065
-l9 l , l0 r
o8(1 8 l0 ' ) l = ls 14
Hence,he riction actors
f : 436x103
Thus,Eq.(3-70) ives hepressurcropperunirpipe ength s
LP 32p lQ2 i J2 r (asq) r4 .3b0 1 ) (0 .03 )2
r-
7 pr:-
,rr,6215-+urarm
I CalculateQ; givenlAPl and D
In thiscase,earrangementfEq. (3-70) ives
f : ( : ) '\a)whereY is definedby
(3-82)
12D5 lLP l
32pL(3-83)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 30/51
(3-84)
Thus, heprocedureo calculate he volumetric low mte becomes:
a) Calculate fiorn Eq.(3-83),b) Substitute into Eq. (3-84)and determine he volumetric low mte.
Example 3,13 What is the volumeffic flow rate of water n m3/s at 20oC that can be de-livered through a commercialsteelpipe (s = 4.6 x 10 | m) 20 cm in diameterwhen theprcssureabopperunit lengthof thepipeis 40 Pa/m?
SolutionPhysicalproperties
for warerr20'c {29i , In=o9oks/m]' - ' " ' '[ a
- l 00 l 10 -6kB /ms
Analysis
Substitutionfthegivenvaluesnto Eq.(3-83) ields
. , l n ' o ' l tP l / : r ' r 0 .2 r5 r40 t
\
3zPL
Y
rl2)rgqo)
Hence, q. 3-84)gives hevolumetic low rateas
75 Chemical ngineeringrocesses
SubstitutionfEqs.(3-74)and 3-82)ntoEq. 3-73) ields
a:-4Yb8(1k+ 2)
q=-4Ybc(:1L+Y2)
= (4x1.ee10')r"clq-##S2 (1001 t0 6)(0.2
(999)(1.99 10 3)il:0.03 m3/s
I CalculateD; givenQ and lAPl
Swamee ndJain 1976) ndCheng ndTurton 1990) rcsentedxplicitequationso solveproblems f this ype.These quations,owever,reunnecessarilyomplex. simpler qua-
tioncanbe obtained y us ing heprocedureuggestedy Tosun ndAklahin (1993)as ol-lows.Equation 3-70)can be reaffangedn the olm
whereN is deflnedby
f : (DN)5 (3-85)
. , 1 n l r .e111/5^=\n;td )
(3-86)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 31/51
Transfer@efficienls 76
For turbulent floq the value of/
variesbetween0.00025and0.01925.Using an averagevalueof 0.01for / givesa relationshipbetweenD and N as
r04
,=
"
(3_87)
SubstitutionfEq.(3-85)nto he efr-handide fEq.(3-73), ndsubsriuionofEqs. 3-74), (3-87), nd = 0.01 nto he ight-handideof Eq. 3-73)give
o .s ta / t I r , r t t r \ - r l 5D:": : I l roelv +s.8?5(--r-_,1 o. l7rI ( ] -88)' " \ t L \P?1v ' l J | /
Theprccedurco calculate hepipediameterbecomes:a) Calculate fromEq.(3-86),b) SubstituteN into Eq. (3-88)and determinehepipediameter.
Example 3,14 Waterat 20"C is to bepumped hrougha commercialsteelpipe (r = 4.6 x10 ) m) at a volumetriclow rateof 0.03mr/s. Dete.minehe diameter f thepipe f theallowable ressureropperunit ength fpipe s 40 Pa/m.
Solution
Physical roperties
( ^^^ .
Forwaler r20'c ,rn., ,. I a-
9q9ke/ml- - -[ r - l 00 l ^ l 0 6kg /ms
Analysis
Equation3-86)gives
, , I n ) Lp l \ t / s I n1 \4q l t t t ^"-\ ir;Ld) :L-t"t""ortl - 'oe
Hence, q. 3-88) iveshepipediameters
o .s t , t t r i a \ ' l r , \' / '
D=;{lrogl,N+s.87s{-+; l o.r7r l" \ l L \ p v , / ) | /
0 5 1 4 t T'
-ff i ( lr"el,+.0ro-,,,r.0s,+'5811,,100rI0o,l-o.,t, l ' ). - . \ r l qqq ) (0 .01 r i 1 .69 ) "
1 )
:0 .2 m
3.5.2 HeatTianslerCorrelations
For heat ransfer n circular pipes,variouscorrelations avebeensuggestedepending n theflow conditions,.e., aminar r turbulent.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 32/51
77Chemicalng'neeringro@sses
3.5.2.1 Laminar fow coffelation For 7un\nar low heattransfer n a circular tube withconstant all emperalue, ieder ndTate 1936) roposedhe ollowing orelation:
Nu= L86[Repr(r/a))t/t*/,.)o.tn (3-89)
(3.e0)
Nu = 0.027Rea/5 rr/3(p./p,)0 la
in which all propertiesexceptp& are evaluatedat the mean bulk temperature.Equa_tion 3-89)s valid or
13< Re< 2030 0.48< Pr< 16,700 0.0044 p/ u.tu 9.j5
Theanalytical olutiono thisproblems onlypossibleor very ong ubes,.e.,Z/D -+ oo.In thiscaseheNusselt umberemains onstantt 3.66..
3.5,2,2 Turbulentow correlatiors The following correlationsapprcximatehephysicalsituation uitewell for thecases fconstantwall emperaturendconsiant all heat lux:
Dittus'Boelter correlalion
DittusandBoelter 1930) roposedhe ollowing onelarionn whichall physical ropertiesareevaluated t the meanbulk tempenture:
Nu = 0.023Re4/5 l
10.4 torheadng
t 0.3 for coolingThe Dittus-Boeltercorelation is valid when
0.7< Pr< 160 Re 10,000
Sieder-Thtecorrelation
Theconelation roposedy Sieder ndTare 1936)s
L/D > t0
(3-91)
in whichall propertiesexceptpu aieevaluated t the meanbulk temperaturc.Equation 3-91
) is valid or
0.7< Pr< 16,700 Re 10,000 LID >,10
Whitaker correlation
Theequation roposedy Whilaker 1q72)s
Nu = 0.0 15ReO 3Pro42(p/p, )0 14 (3-92\
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 33/51
TEnster oefiicients 8
in which the Pmndtlnumberdependences basedon the work of Fdendand Metzner 1958),
and he functionaldependence f p//ru is from SiederandTare 1936).Al1physicalproper-ties except&o areevaluated t themeanbulk temperature. heWritaker correlation s validfor
2300<Re<l x 105 0 .48<Pr<592 0 .44< tL l pu<25
3,5,2.3 Calculatio of theheat tansfer rute Once heaverage eat ransfer coefficient scalculatedrom corelations y usingEqs. 3-89){3-92), her the rateof energy ans-ferred s calculatedas
a:(rDL)\h)^TLtt, (3-93)
\\,hereLTLM, logarithmicmetlfl emperature iffelence, s deined by
(7.. Tt\,,, (T, - tn\.,,] / t , v_# ( i - q t )
, ^ l t I a - t b t n I'Ltr' - tb"" )
Example 3.15 Steam condensingon the outer surfaceof a thin walled circular tube of65 mm diametermaintains unifom surfaceemperaturef 100'C. Oil flows hough hetube at an average elocity of 1 m/s. Determine he engthof the tube n order to increaseoil temperaturerom 40'C to 60'C. Physical ropertiesf theoil areas ollows:
I t t : | 2 .4" ru " Kg /m
At50 'C : I r - , : 4 .28 105
m2lsI
l P r : 143
At 100'C:pr. 9.3x 10 3kg/m.s.
Solution
Assumptions
l. Steady-stateonditions rcvail.2. Physical rcpertiesemain onstanl.3. Changesnkineticandpotential neryiesrenegligible.
AnalysisSystem: il in thepipe
The nventorymte equation or massbecomes
Rate f massn= Rare f massout:r i :p{r,)(ftr2/4) (1)
On tle otherhand, he nventory ate equation or energy educeso
Rateof energy n : Rateof energyout (2)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 34/51
79Chemical ngineeringrccesses
The terms n Eq. (2) are expressed y
Rateof energyn: m d p(Tu, - Toi * t DL\hJLhu (3)Rateofenergyout=m ep96.,, -f*9 @)
SincehewaII emperatures constant,heexpressionor AZaM,Eq. 3-94), ecomes
m"r:W (s)tnl
'' -"'" I
\7, - ru ,,/
SubstitutionfEqs.(l), (3), (4)and 5) ntoEq. 2) gives
= \'):'9"^( =') turD 4 (h \ \ t - - t , , , )
Noting hatStH= (r)/((r)pdpl :11u7G.p4, Eq.(6)becomes
9 -1. . t n( 4,) - | RePrrnf' rb i
) a,D 4 SrH r.-r t . , , ) 4 Nu \/ , - r r , , , /
To determineNu (or (l')), firut the Reynoldsnumber must be calculated.The mean bulkiemperatures (40+ 60)/2= 50'C and heReynolds umbers
" .D ( r ' ) ( 65x l0 l , { i r - 1519
, , " Lamina rl owe = - :+ l g * to s
Since he low s aminar, q. 3-89)mustbe used,.e.,
Nu= L86[RePr(r/r )ft/t ,It,,to" (S)
SubstitutionfEq. (8) ntoEq.(7)yields
. . - r , 4 ? " , r ,t l l
2- - -Repr _ l l j ' ' r n l 3
- ', ,
D L r4)fl.861 \T. - k"", ) J
- - f ( t 2 .4 \ t0 r / s . J \ t0 r l 0 ra ,7 t00 40 t1 "2 ^ . ^ ^- rrsrs)rr43)L ioi; i .86;
' " l ro-oo/ l :2602
ThP nrhe lcnorh i ( rhPn
L = (2602)(65 tO-3; = tO9m
Example 3.16 Ab at 20"C entersa circularpipeof 1.5cm niemal diameterwith a velocityof 50 m/s. Steamcondensesn the outsideof thepipeso as o keep he surface emperatureof tie pipeat 150"C.
a) Calculate he ength of thepiperequired o inqease air tempemtwe o 90oC.b) Discuq"heelTecr f surfaceoughne\\ n he ength f rhepipe.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 35/51
Transfer@effdenls 80
Solution
Physicalproperties
Themean ulk emperatures (20+ 90)/2 55'C
Forair at 20'C (293K): p :1.204 t kglr.ll3
l r ,- t o .S , l 0 -6kg /m.o
Fo ra i ra t 5 'C 328 r : { r - l8 Jq l0 6 m ' / sI
I Pr 0.707
Forair at 150C (423K)'.p = 23.86x 10 6kg/m.s.
Analysisa) System: ir in thepipe
The nventory ate equation or mass cduceso
Rateof mass fair in: Rateof mass f air out =m (l)
Notethat for compressible luids like air both density andaverage elocity dependontemperature ndpressurc.Therefore,using he nlet conditions
_ f - ' o n r< '2 rt i 1 : t r D ' t l L ' \D ) ) , - , " ,i " ' " ;
' '1 , t . 2047 , { s0 , -.06 , l 0
' kg /s
" - L 4 J
In Foblems dealing with the flow of compressibleluids, it is customary o deflnemassvelocity,G, as
c : = p \ t ) (2 )
The advantage f usingG is the act that t remains onstant or steady low of comptess-ible fluids throlgh ductsof unifom cross section. n this case
G : (l .2M'/)(50)= 60.24 g/n2 s
The nventorymte equation or energy s written as
Rateof energy n : Rateof energyout (3)
Equations3)15) of Example .15arealsoapplicableo thisproblem. herefore, eget
a lRePr . /7 . - h ' \_=_ - l n t -| ( 4 )
D 4 Nu \ l - - | a , , )
TheNusselt umbern Eq. 4)canbedetemined nly f theReynolds umbers known.TheReynolds umbers calculateds
^ DG (0.015)(60.24)Re- - - - -45 ,636 + Turbu len ll ow
rl 19.80 t0-o
The valueof, depends n the correlationsas ollows:
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 36/51
8l Chemi@l ngineeing rccesses
Dittus-Boelter conelation
Subsdtutionf Eq. -90r nroEq. 4 give\
t _ Re02pp6 ln / , - rb . "\ _
(45 .03b )02 (0 .707 )0o , "150 - 20 \ _ .0 ,D 0.0s2 \r. - Tb",, 0.092 \ lso _ 90/
-" "
Therefore,he required ength s
a : (58 .3 ) (1 .5 ) :87m
Sieder-Tatecorrelation
Substiru nof Eq. l-al , inLo q. 4 gi\e.
t Reo2P1 'J r / /p . lo ' 4
, t Tu , -7o "D 0. 08 \r. - Tu,,
{45 .636 )02 {0 .70712 ' l4 .80 . 0 " \ -0 ro . / t50 -20 \-oro8 \: : : " - ro "/
'n \rso o6/=t* '
Thercfore, he required ength s
, : ( 49 .9 ) (1 .5 ) :75m
Whitaker correlation
SubstitutionfEq. (3-92)nroEq.(4) gives
L Reo17p ro58 { t / / - / d t -0 ) " ,1 r , - T0 , "
n- 0-06-\h-r^)
r45 .b jo )07r0 .?07)058/ t9 .80 r t0 o \ 0 ra .
/ t50 -20 \
0 .06 \23 .80 t0 b / " ' \ t50 q0 / -" '
Therefore,ie required engths
t: (67)(1.5) 101 ln
b) Note that Eq. (4) is also exprcssedn the orm
t : ''n1
'-ru") " ' 4SLH \7. - rb,,,/
Theuseof theChilton-Colbumnalogy,.e., /2: StnPr2/3,educes q.(5) o
L I p? /3 . I T , -h . " 1 r i 0 .702 ) ) r . / r 50 -20 \ 0 .1068
o=z rt ' \ r , -u,1=, r
'" \"0r7- ,
16)
The riction actor anbecalculatedrom heChen orelation,Eq.(3-80)
| / e/D 5.0452 \
r'T:-o.cl]ro6s
*" ,ocrJ
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 37/51
TEnsreroefficieris2
. / | l D \ t o o 8 / 7 . 1 4 9 0 o 8 o 8 lA _ t _ | f | _ l\ 2.s4e1/ \ Re ./
Forvaious aluesf€/D, thecalculatedaluesf f, LID and arcgiven s ollows:
;7D--j --z/D
0 0.0053 57.9 86.90.001 0.0061 50.3 '75.5
0.002 0.0067 45.8 68.70.003 0.0072 42.6 63.90.0M 0.0077 39.8 59.7
Commenb The increase n surface roughnessncreaseshe friction factor and hencepowerconsumption.On theotherhand, he ncrcasen surface oughnessauses n ncreasein the heathansfercoefficientwith a concomitantdecrcasen pipe length.
3.5.3 MassTransterCorrelaiions
Mass ransfer n cylindrical tubes s encounteredn a vadetyof operations, uch as wettedwallcolumns,everse smosis, ndcross-flow ltmfihation.As in thecase f heat lanster,masshansfercorrelationsdependon whether he low is laminaror turbulent.
3.5.3.1 Laminar low correlation For laminar low mass tansfer n a circular tube with aconstant allconcentration,nexprcssionnalogouso Eq. 3-89)s givenby
sh 1.86[Rec(D/.)] ' /3 (3-e5)
Equation3-95)s valid or
lxescqoryll/32 z
3.5.3.2 TurbulentfowcorrcIations
Gilliland-SherwoodcorrelationGillilandandSherwood1934) onelatedle experimenralesLiltsbrainedrom wettedwalcolumns n the fom
Sh:0 023Reo 3Sco44 (3-96)
which is valid for
2000 Re 35,000 0.6< Sc< 2.5
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 38/51
83 Chemical ngineeringro@sses
Linton-Shenvoodcorrelation
Thecorelation roposedy LintonandSherwood1950)s givenby
Sh= 0.023Reo3Scl/3 (3-97)
Equation3-97)s valid or
2000 Re< 70,000 0.6< Sc< 2500
3.5,3,3 CaLculation f the mass ransferrate Once heaveragemass tansfercoefficient scalculatedromcorelations ivenby Eqs. 3-95)-(3-97),len dre ateofmassofspecies,4 transferreds calculared)
rir=
QrDL)(k")(Lc1) yMa (3-98)where"M,{ is the molecular weightof species"4, and(Lca)'y, logaithmic me.rnconcen-tration difference,s delnedby
(3.ee)
Example ,17 A smooth ubewith an ntemaldiameter f 2.5cm s cast tom solidnaph-thalene.Pureair enters he tube at an average elocity of 9 m/s. If the average ir pressureis I atm and the temperature s 40'C, estimate he tube length required or the averagcconcenhation f naphthalene apor n the air to reach257a
of the saturation alue.Solution
Physical properties
Diffusion oefficient fnaphthalene,{) in ail (6) ar40 C (313K) is
/ l l l \ 1 2 / ? 1 1 \ l / 2rDaa t : r - r : Dq r ) roo (
* )- r0 .b2 ' l 0 5 ) {# ) :6 .61> l0 om2 ls
\ JUUI \JUU/
Forairat40 C (313K): ,:16.95 x 10 6m2/s
TheSchmidt umbers
16 .q5>0 -o ^ -_r r - tB=6;r ' ro=
-z)o
Assumptions
1. Steady tate onditionsrevail.2. Thesystems sothermal.
Analysis
System: ir in thenaphthaleneube
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 39/51
Tlansfer@efficients 84
If naphthalenes designated s species"4, then the rate equation or the conservationof\pecie ,4becomes
Rateof moles f,4 in: Rateofmolesof,4 out (1) :
The ems in Eq.(1)areexpressedy
Rareofmotesof -4 n = nDLlk)(Lc LM e)
Rare f mofes f "4out= e(cA).d: (n D2 4)lu)(cA)Nt (3)
Since heconcentrationt the wall s constant,he expressionor (AcA)LM,Eg. 3_gg),becomes
(Lca,1p=--- -@ L-
rnl tA'I
L,e. - @e).*)
SubstitutionfEqs. 2)-(4) nroEq.(1)gives
;--;#'"1 ?:]= +1_q,"rr-o.xr=ooz((,1))Notethat Eq. (5) canalsobe expressedn the orm
f -oo' : ( . ' - )-o.o?r(R=) (o), \ s rM / \ Sh , /
The valueof a depends n thecofielationsas ollows:
Chilton-Colburn analogy
SubstitutionfEq. (2-73)nroEq.(6)gives
f,=o.onlscrTheReynolds umbers
- D lu ) (2 .5 10 2 ) r9 )Ke : - : . : - - i :_ -13 ,214 _+ Tu lbu lenrl ow
The friction factor can be calculated rom the Chen conelation, Eq. (3_g0).Takingt /D , .0 ,
, I e /D \ r rm8 /7 . t490 \0 .808r 7 . t490 \o .8s8l" - \zs4si) +(
R. /- \ t r *1 - r '16Yro-r
I f 5 .04s2 - l
{ t=-o^eL__ tosf l . t6,0 J, l - } /_0.0072
(4)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 40/51
E5Chemicalngineeringrocesses
Hence q.(7)
becomesL
D
The requLed ength s then
(0.072)(2\2.sq2/3
0.0072
L = (3'7.4)(2.5) 93.5cm
Linton-Sherwood correlation
SubstirutionfEq. l q7)inLo q. 6)gives
L ^ . ^ ^ . ^
i : 3. Reo s.2/l - J. (1 .274r0t'12.56t2'- 2o.4D
The tube ength s
L: (29.4)(2.s)'73.s cm
3,5.4 Flow n Non-Circular ucts
The conelationsgivenfor the friction factor, heat ransfercoefficient,and mass ansfer co-efficient are only valid for ductsof circular cross section.Thesecorelations can be used orflow in non-circular ducts by introducing be conceptof hJdraulic equivalentdiatneter,Dh,definedby
/ Florr area \
' \WefledperimeFr/
The Reynolds number based on the hydraulic equivalent diameter is
^ Dt\a)p
so hatthefriction factor, basedon the hydmulic equivalentdiameter, s related o Re, in theform
':'(#)
(3-100)
(3 -101)
(3-ro2,
where Q dependson thegeomefiyof rhe system.Sinceg : I only for a circularpipe, fte
useof the hydraulic quivalent iameters not recommendedor laminar low (Birdead1..2002;Fahien, 983). he hydraulic quivalent iameteror vaiiousgeometdess shownnTable3.4.Example 3.18 Water flows at an average elocity of 5 m/s through a duct of equilateraltriangulardoss-sectionwith one side, a, being equal o 2 cm. Elecffic wLes are wrappedaround he oufer surfaceofthe duct to providea constantwall heat lux of 100Wcm2. ffthe nletwater empemtules 25'C and heduct ength s 1.5m, calculate:
a) Thepowerrequircd o pumpwater hrough he duct,b) The exit water emperatue,c) The average eat ransfercoefficient.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 41/51
TEnsier coefticients 86
Iable3.4. Thehyd€ulic quivalenliameterorvarious eometnes
II
T
I o997ke/nJ
-
l1= ssz lo-o slmI Cp= 4180 /ks.K
-L.{3
Solution
Physi@l properties
For waterat 25
C
(298K) :
Assumptions
l Steady-stateonditionsprevail.2. Changesn kinetic andpotentialenergies rc negligible.3. Variations n p aDdCp with temperature renegligible.
Analysis
System:Water n theduct
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 42/51
E7Chemical ngineedngrccesses
a) Thepower equired s calculaied rom Eq.(2-11)
w:root:lo,tr(lrr,r,)r' lr,r (r)L \ 2 . / J
The friction factor in Eq. (1) can be calculated iom the modifiedform of the Chencorelation,Eq.(3-80)
| / €/D 5.0452 \
v7--o 'os| \ r .zoo5-
*o'o to , t2 l
where
/ F / D \ I l o q s / 7 . l 4 9 o t o 8 q 8 lA: \Ls4s i ) l \R . , /
( 3 )
The hydraulicequivalentdiameterand he Re),nolds umberare
u,: l- 1= t.tss"^/3 \/3
Dh lu )o { 1 .155 l0 - r } r5 r {997 rRe/,-.-=
8tt x t0 o-::-:---- =64.548 -+ Tubulent low
Substitutionof thesevalues n0oEqs.(3) and(2) and raking€/D 0 give
/7 . t490 t08e81 7 .1490 \o8o8 l^=l. .n.,/
: l* . '* / -28xro"
| .. | 5.04s2 . r
" rT--4" tL-60.*8109{2.8. l0 ' ) l+ / - 0.0049
Hence,hepowerrequired s calculated rom Eq.(1) as
W Ir:rrz to- ' ] r1.stfroozrrs, ',o.o*n,,r, rr. ,t Lz l l
{ b) The nventory ateequation or mass s
Rateof massn : Raieof mass ut ,? : p (D)f @) (4)'\ 4 , /
f J3t2 x Io-2 21. a= (997){5) l - | = 0.86Je/sL4l -
The rvertory rateequation or energy educeso
Rateofenergy n: Rateofenergy ut (5)
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 43/51
Transfercoefticients 88
Theterms n Eq. (5) areexpressedy
Rateof energyn = m dp e6,^_ Ta) + e.. (6)
Rateof energyout: r/,epq6*, _ f,4 (j)
wherep, is the rateof heat ansfer to water rom the ateralsurfaces f the duct.Sub_sirul ion f Eqs.6rand 7) ntoEq. 5)give.
b"..rb*--rbt,+:+ 2s riI?11+r9o'- so"c'n( p (0.863)(4t80)
c' Themean ulk emperaLue( t25 50)/2 17.5.C. r his emperature
tr:628 x 10-3rym.K and pt:4.62
Theuseof dreDittus-Boeltercorelation,Eq.(3 9O),gives
Nu = 0.023Re75pro.a 0.023(64,54g)a/s4.62)o.a2g9
Therefore,heaverage eat mnsfercoefficients
/L \ / i 28x l0 l \ ^(h) Nu{ l : tzqsr l\ uh \ rJ5s . lo t l- 16 57 /mzK
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 44/51
89chemical ngineedngbceses
NOTATION
A area,rrta packingsudacearcaperunit volume'1/m
e p heatcapacityatconstantpressrtre, J/kg K
ci concentration f species, kmol/m3
D diameter,m
Dn hydraulicequivalentdiameterm
De particle diameter,m
D,la diffusion coefficient or system 4 B, m2/s
Fp drag orce.N
/ ftiction factor
G massvelocity,kg/m' sI acceleration f gravitY, n/s2
j s Chilton-Colbum-factor for heat ansfer
j a Chilton Colbum -factor fol massfadsfer
k thermalconductivity,W/m K
&. massransfercoefficient,m/s
, length,m
M mass, g
m masslow rate, g/s
M molecularweight,kg/kmol
i molar lowmt€,kmol/s
P pressule,Pa
C heat ransfer ate,W
Q volumetriclowrate,m3/s
4 heat lur, W/m2
? gasconstant, /mol K
7 iemperatue,'Cor K
tr umq s
V volume,m3
, velocity,m/s
o superficialvelocity,m/s
,r terminalvelocitY,m/s
/ wo*, J: width, m
li/ rateof work, w
x rectangular ooldinate,m
A difference
€ porosity
s surfaceoughness f thePiPe,m
p viscosity,kg/m s
l) kinematicviscosity,m2/s
p density,kglm3
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 45/51
Tlansfercoeflicients 90
Overlin€s
Bracket
Superscript
permoleperunitmass
\a) averagealueof.1
sat saturation
Subscripts
A, I speciesn bina.ryystems, bulkc transition rom laminar o turbulentcft characteristic
/ filmi speciesn multicomponent ystemsin inletLM log-mean
out outletpb packedbedu wall or sudaceoo free stream
Dimensionl€ssNumbers
Ar ArchimedesnumberPr PmnddnumberNu NrrsseltumberRe Reynoldsnumber
ReD Reynoldsnumberbasedon thediameterRet Reynoldsnumberbasedonthe hydmulicequivalentdiameterRe, Reynoldsnumberbasedon the engthRe, Reynoldsnumberbasedon the distarce_rSc SchmidtnumberSh Sherwood umberStn Stantonnumber or heat ransferSfu Santon number or masshansfer
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 46/51
91 ChemicalEiqineeg Pbcesses
REFERENC€S
Bedingfield.C.H. a.d T.B. Drew, I 950. Aralogy berween eai ransferand mlss transferA psychometric tudy,Ind.Eng-Chem. 2, 1164.
Bird, R.B., W.E.StewarrandE.N. Lightfoot. 2002,Transportphenomcna,2ndEd., Wiley, N€wyork.Blausius, ., 1908, renzschleienn Flussigkeitenit kleinerReibu.g, . AngewMatb.phys.56.1Cher,N.H..1979.An explicitequationor fdction actornpipe,Ind.Eng.Chem. und. 8 (3).296.Cheng, .X. dd R. Turron, 990.How ro calculareipe ize withourreradon, hen.EnB.9? No .),187Churchill,S.W, 1977,Friction factorequationspans ll fluid flow regimes,Chem.Eng.84 (Noe 7),9 I.Churchiu.S.W.ed M. Bemstein.19?7,A corclarjng e4uation or forcedconveciion iom galesand iqui.ls ro a
circrle cylinder n doss flow. J. HeatTransfea99, 300.Colebrook,C.F., 1938-9.Turbulen ow in pipeswjth paltculd referenceo fte transitionEAion betweenhe
smoolhnd cu8h ipe aqs. . l lsr .Civ. lEng. . ,J3.Dittus. F.W:and L.M.K. Boelter, 1930,Un;venity of Califomia publicationson Engineering,Vol. 2, p_ L4J.
Berkeley.Dwivedi.PN. and S.N. Upadhyay, 977.panicte-fllid massbansfer n fixed and luidizedbeds. nd. Ens. Chem.
Prce$ De..De\.16.157Ergun, ..1952, luidnow hrcu8h acked olumns,Chem.ng. tog.48,89.Fahien, .W, I 983,Fundanentalsf Transporthenomena,Mccraw,Hilt,Newyork.Frlend,WL andA.B. Metzner, I958, Tufbulenrheat mDsfernside ubesand treanatogyamongheatjmassjod
momentumrdsfer, AIChE Joornal4. 393.Frossling. .. 1938,Beitr ceophy .52,170.Gilliland.E.R.andT.K. Sherwood.934, iffusion f vaporsntoairstreans,nd_ ne.Chem. 6,516.GregoryC.A-mdM. Fogarasi.985,Altenrareo stddard ricrion actor quarion. il Gas .83. 120.I apple..F.and .B Shepherd.qao.cr l .utJr .onotpxni .elrajccro. ie\ .j , l Eng.chem 2. 05.Lirton, WH. a.d T.K. SheNood, 1950,Ma$ r.esfer from sotid shapeso water n streamlineand urbulenl low.
Chen.Eng.Prog. 6,258.
Moody,L.F.. 19lu, Frjction facton for pipeflow, Truns.ASME 66, 671.Ranz,W:E. d WR. Mdshail, I952,Evaporationromdrops par I Chen.Eng.piog.48,l4l.Sieder, .N.dd c.E. Tate, 936.Heathansfer ndpressureropofljquids n lubes,nd.Eng.Chen.28, 1429.Steinberger, .L. dd R.E. Treybal. 1960,Mass ransfer rom a solid sphere o a flowing liqujd stream.AIChE
Iornal 6,227Swmee,PK. andA.K.Jain.1976. xplicir quationsor pipenow problems,. Hydr.Div ASCE102,657_Touson2007,Modelingn Transporthenomen,2ndd..Etsevier cience Technolos/ ooks.Tosun.L a.d I. Ak$hin. 1992,Explicit exlre$lons for the ricrion facror.Unpubtishcdeport,Middle EastTech
Tosu. L andL Aktahin.1993, alculareiticalpjping damelers,hem. ng. 00 Mdch), r65.Forconectronssee lsoChen.Ens. 100 July).8.
Tunon.R. andN.N.Cldk. 1987,An explicir clarionshipo redict sphericalanicle eminatvetociixpowde.Technolo$ 1. 2l .
Tunon,R. and o. Levenspiel,1986,A sho]1ote on rhe dragcofielarion o. spheres, owderTechnolosy47. 83.Whitaker, .. 1972, orced onvecrionearransferorrelationsor now n pipes, ast larptates.ingte ytindeB,
single sphercs, nd or flow in packedbedsdd tubebundtes.AIChEJournal I 8. 361.Zhukauskas, .. 1972,Adveces in HeatTransfer,Vol_8: Heai Transfer ilJmTubes . CrossFloq Eds.J.p.Hart
nettandT.F. rvine. Ji. AcademicPrcss.New Yor^.
SUGGESTEDEFERENCESORFURTHER TUDY
Brodkey. .S.andH.C.Hcrshey,988,Truspod Phenomena:UnifiedApproach. ccraw HiU,Newyork.Hines,A.L-andR.N.Maddox, 985,MassTransfer-FundmentatsndAppiications.rcnriceHall,Englewood
Clifl'. \etr Jer"e)Incrcpem,F.P md D.P.Dewitt, 2002,FundmentalsofHeat andMassTmnsfer.5th Ed..Witev. Newyofk.
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 47/51
TEnsfercoefficienls 92
Middleman,S., 1998,An IntrodDction o Mass dd Heat T.ansfer_ prirciptes of Analysisand Design,witey,
Skelland,A.H.P.,1974,DitrurionatMassTrmsfer.wiley, Newyork.Wbitakei,S., 1976,ElemerraryHeat TransferAnalysis,pergmon press,Newyork.
PROBLEMS
3.1 A flat plareof length2 m and width 30cm s to beplacedparallel o anair stueam rateinperaruef25oC.Whichsideofrheplare,.e.,lengtb rwidth,should einthedirectionol tlowsoas o minimizehedrag orce f:
a) Thevelociry fair is 7 m/s,b) Thevelocity f air s 30 m/s.
(Ans\yer:a) Lengrh b) Widrh)
3,2 Air at atmosphericressurend200.C lowsatg m/s overa latplate150 m ong nthedircction fflow and70 cm wide.
:lE^slimitele rareol coolingof rheplareso as o keep he surface cmperaruret 30oC.
b) Calculatehe drag orceexertedon theplate
(Answer:a) 1589W b) 0.058N')
3.3.^Water t l5'C flowsat 0.15m/s overa flatplate m Iong n thedirection f tlowand0.3 m wide. If energy s transfered from the top and botto; sudacesof the platetothe flowing streamat a steadymte of 3500 W, determinehetemperature t Ae piate sur_face.
(Answer:35'C)
3.4 Fins areused o incrcaseheareaavailable or heat ransferbetweenmetalwalls andpoorlyconductingluidssuchasgases. simple ectangularin s shown elow
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 48/51
93ChemicalEnginees Processes
If oneassumes,
. I= I (1 ' on l y .
. No heat s Iost rom the end or fiom the edges,
. The avengeheathansfercoefficient, t), is constantand uniformover the entire
surfaceof the fin,. The thermalconductivityofthe fin, k, is constant,
. The emperatwefthe medium uroundinghe in, fo, is unifbrm,
. Thewall temperature,u, is constant,
ihe resulting steady-stateempemture istribution s givenby
r r* '""["( ' - ; ) ]T. -T -
If the rate of heat oss rom the fin is 478 W, determinehe aveEgeheat ransfercoeflicient
for the followingconditions:€ : l'75"C; T. :260"C; k : I05 w/m K; t :4 cm;
W:30cm; -8 :5mm.
(Answer: 400wm2.K)
3.5 Considerhercctangularin Biveon Problem .4. Oneof theproblems f practical
interests the determination f the optimumvaluesof B and , to maximize he heat ransfer
mte rom he in for afixedvolume, , andW.Show hat heoptimum imensionsreglven
by
t ,h \V2 r t t t / ky \ l lI a N L J o l , r ^ | |. r ,_ \ r ,wr ) , , , , , . , ,
3.6 Consider herectangularin given n Problem4.4. If a laminar flow regionexistsover
theplate,show hat the optimumvalue of w for the maximumheat ansfer rate1]om le
fin for a fixedvolume,,
and hickness, , is givenby
N1't'vrl
wherekl is the thermalconductivityof the ffujd
3.7 A thin aluminumln (k :205 W/m K) of length L 20 cm has wo endsattached
to twoparallelwallswith temperatures = 100'c and7, = 90'c asshownn the igure
belovr'' he in loses ealby convectiono theambient ir at 7€ : 30'c with an average
heat ransfercoeflicientof (ft) : 120Wrn2 K through he top andbottom surfacesheat
loss rom heedgesmaybeconsideredegligible).
rzvusn rsl(l)iln
2lhlL2
KB
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 49/51
TGnsfer oetrcients 4
F__ r.=,o ._- _l
Oneof yolu friendsassumeshat there s no ntemal genetationof energywithin the rnanctdetermineshe steadystate emperature istributionwithin the fin as
' r _T
7,,- T-,
in whichN andQ aredefined s
- 2Q sinhNz
and Q=
- tL l T t -T* \' - \ r , -r- l
2stnhN L
a) Show hat here s ndeed o nremal enemtionf energywithin he in.b) Determine he ocationand he valueof theminimLrmJmperaturewithin the fin.
(Answeri :0.1 cm,7:30.14.C)
3.8 ReworkExample4.8 by using he RanzMarshallcorrelarion, q. (4.333), .heFrossling orrelarion, q. (a.3-34), nd hemodifiedFrossling oneJation, q. (4.335).
Why do the resultingSherwood umbers iffer significantly rom 541?
3,9 In an experiment arried ur at 20 C,^aglasssphere f density2620kg/ml falls
rh rou8hca rbon re l rach lo r i de tp= l5q0kg /mrandg -q .58 \ l 0akg /m. \Ju i tha re rm i_nalvelocity f65 cm/s.Detelminehediameter fthesphere.
(Answer:21mm)
3.10.
A CO2_bubbles dsing n a glass f beer20cm rall.Estimatehe ime equiredor abubble5 mm in diameter o reach he top if theprope iesof CO2and beercan e taker asequal o thoseof air andwater, espectively.
(Answer:0.54s)
2lhl
KB
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 50/51
95 Chemicalnglneedngrccesses
3.11 Show hat he useof theDittus-Boelterotelation,Eq (4 5 26)' together ith the
Chilton-Colbumnalogy, q.3.5-12)' ields
/ :0.046 Re-o
which is a good power-lawapprcxination for the friction factor in smoothcircularpipes
Calculate for ne : 105,106and107using hisapgoximate quation ndcompalehe
values ith hose btained y lsing theChen orrelation' q (4 5-16)
3.12 For aminariow of an ncompressibleewtonianluid n a circular ipe'Eq (45_12)
indicateshatthepressure rop spr;portional to thevolumetric lowmte For fully tu$[lent
nor .fro* tttut ttt" pres.uredrop n apipe sproportionalo thesquare f thevolumetic flow
rate.
3.13 Determinehepower o pumpa fluid at a volumetriclowmte of 1 1x l0 3m3/s
throueh : cm aiameter orizontalmoothpipe 10m- ong Thephysical ropertiesf rie
fluid re girenasp = ol5 kg/mrand / = l q2- l0 i kg/m<
(Ans\ter: 10.4W)
3.14 Thepurpose f bloodpressuren thehuman ody s to pushblood o the issues f
the organis; s; that theycanperform their functions Each ime theheart beats' l pumps
out btloodnto thearteries. he bloodpressue eachests maximumvalue' e systolic
Dressure. hen heheall contractso pump the blood ln betweenbeats' heheart s at rest
i"J rft" Uto"a pressure alls to a mi;imum value,diastolicpressureAn averag€healthy
Dersonassystolic nddiastolic ressuresf 120and80 nrmHg' espectivelyhehuman
iody hasabout5.6 L of blood. i it takes 0 s for blood o circulatehroughouthebody'
estimatehepower outputof theheart
(Answer:3.73w)
3,15 Waters in isothermalurbulentlowat 20'C through horizontal ipeof circular
cross-sectionwithl0cminside.liametel,Thefollowingexpefimentalvaluesofvelocityalemeasured( a function l rcdialdictance':
Thevelocitydistributionsproposedn the olm
where -ax is themaximum elocityandR is themdiusof thepipe Calculatehepressure
dropper unit length of thePiPe
(Answer:12.3Pa/m)
":^*('-1)"'
, '(cm) I 0.5 2.5 3.5
u . rm/s)0 .394
7/27/2019 Chapter -3-Evaluation of Transfer Coefficients Engineering Operations
http://slidepdf.com/reader/full/chapter-3-evaluation-of-transfer-coefficients-engineering-operations 51/51
. l
Transfercoeffi ienis 96
3,16 In Example4.15, he ength o diametermtio is expresseds
I - ' t ^1t '' ro'1
D 4SrH \T" _ 16,,,,l
UseheChilton-Colburnnalogy,.e.,
andevaluatehevalueof L/D. ls it a realisricalue? fiy/why nor?
3.17 Waterat l0'C enten a circularpipeof intemaldiameter .5 cm with an aveugevelocityof 1.2m/s. Steam ondensesn the outside f thepipesoas o keep he surtacetemperature f thepipeat 82"C. If the engrhof thepipejs 5 m, determinehe outlettgm_peratureof water.
(Answen5l 'C)
3.18 Dry air at I atmpressueand50oC enterca circular pipeof 12 cm intemaldiameterwith an average elocity of l0 cm/s. The inner surfaceof the pipe is coatedwith a rtunabso$entmaterialsoakedwith waterat 20.C. If the ength of thepipeis 6 m, calculateheamountof watervaporcaried out of thepipeperhour.
(Answer:0.067 g/h)
z= StnPl"