porousmedia 97 Flowlhrough
Chapter four
FlowThroughPorousMedia
DESCRIPTION OF POROUSMEDIA By a "porous medium" is meant a solid, or a collection of solid particles, with sufficientopen spacein or around the particlesto enablea fluid to pass through or around them. There are va ous conceptualways of describinga porous mgdium. One conceptis a continuoussolid body with poresin it, suchas a brick or a block of saldstone. Such a medium is referredto as consolidated,and the poresmay be ulconnected("closedce11,"or impermeable)or connected ("open cell," or permeable).Anodrer conceptis a collection (or "pile") of solid particlesin a packed bed, where the fluid can passthrough the voids between the particles. This is relerrcd to as unconsolidated. A schematic representdtionis shown in Fig. 4-1. Either of these concepts may be valid, dependingupon the specificmedium under consideration,and both have been usedas the basisfor developingthe equationsthat desc be fluid flow behaviorwithin the medium.In practice,porous mediamay mnge from a "tight" oil bea ng rock formation to a packed column containing relatively large packing elementsand large void spaces. The pileof 50lidparticle, concepti. u'eful for eitherconsolidated or unconsolidatedmedia as a basis for analyzing the flow process,because many consolidatedmedia are actually made up of individual particlesthat
98 ChemielEngineeng Processes
a
T_ F-
L-I
I
\a)
-- j#i (b) Porous media- (a) Consolidated medjum;(b) unconsolidated
FGUBE4-l medium.
are just stuck together(e.g. sandstone). One of the key propertiesof a porousmediumis the porositys or void ftaction,which is def,nedby Totalvolume- Volumeof solids Totalvolume ,
asolirl
^v.in\
AA
(4-l )
wherelsoudis the areaof the solidphasein a crosssectionof areal. We alsodistinguishbetweenthe velocityof approach,or the ..superficial" velocityof the fluid, vs = Q/A
@_2)
and the "interstitial" velocity,whichis the actualvelocitywithin the pores or voids, Q _V" eAe
(4-3 )
4.1. HydraulicDiameter Becausethe fluid in a porous medium follows a tortuous path through cha[nels of varying size and shape,one method of describingthe flow
Flowlhmughporcusmedia 99
behaviorin the poresis to considerthe flow path asa "noncircularconduit." Thii rcqlirc;r rn .ri:':opriate definitionof the hydraulicdiameter: a, WE
, AiL WOL
Flowvolume Internalweltedsurfacearea
€ x Bedvolume (No. of panicles)(Surface area/Panicle)
(4-4 )
The medium,with overall dimensions,4I, is assumedto be made up of a collectionol indi\idualparticlesand may be eilhercon.olidatedor unconsolidated.The numberofparticles in the mediumcan be expressed as No.particles:
(BedvolumeXFraction ofsolidsin bed) Volume/Particle (BedvolumeXl- e)
(4s )
Volume/Particle Substitutionof this into Eq. (4-4) leadsto
,r: o, -r-(*)
(4'6 )
where a. : (particle surfacearea)/(particlevolume). If the particles are sphericalwith diameter,/, then as:6/d. Thus, for a medium composed ol uniform spherical particles, ^ "n
2de l(t - e)
(4-1 )
If the particlesare not spherical,the parameterI may be replacedby
(4-8 )
d:tl'ds-6las , herc lt is lhe spher[ciryfactor, defined by Surfacearcaof a spherewith samevolume asthe pafiicle Surfaceareaof the particle
(4-e)
and 4 is the diameterof a spherewith the samevolumeas the particle.
4.2.PorousMediumFrictionFactor The expressions for the hydraulicdiameterand the superficialvelocitycan beincorporatedinto the delinitionof the friction factorto givean equivalent exDression for the Dorousmediumfriction factor: '" -
et
etd€
t 4 L t D , t \ v i z/ 2 t - ) l t l
tlvl
-
e1tle3 )Ltl
tv!
(4-10)
100Chemi€lEngineerins ftocesses
l"fcst referencesuse Eq. (4-10) without the numericalfactor 11 _??s th3 deflnitionof the porousmediumfriction factor, i.e., ( 4 - 1 1)
Llt - €)v:
4.3 Porous Medium Fleynolds Number In like fashion,the hydraulicdiameterand the superficialvelocitycan be introducedinto the definitionof the Reynoldsnurnberto give ^,
DrVip
r vR e - - : ; . - : : : : 1t
2dtV,p J\t - t)1t
2dv,p . Jll - €Jl,
(4-t) \
Here again, lhe usual porousmediumReynoldsnumberis definedby Eq. (4-12) without the numericalfactor (2/3):
dv"p
^,
(4-13)
r r R e . P M: : - _ . lr-t)lL
4.4.FRICTION LOSSIN POFOUSMEDIA A. LaminarFlow By analogywilh laminarflow in a tube,thefrictionfactorin lanrinarflow wouldbe 16 or (4-r4) /" : /PM: A/R"
However,this expressionassumes that the total resistance to flow is due to the sheardeformation of the fluid, as in a uniform pipe. In reality the resistance is a resultof both shearand stretching(extersional)deformation as the fluid movesthroughthe nonuniformconvergingdivergingflow cross sectionwithin the pores.The "stretchi[g resistance"is the product of the extension(stretch)rate and the extensionalviscosity.The extensionrate in porousmedia is of the sameorder as the shearrate, and the extensional viscosityfor a Newtodan fluid is three timesthe shearviscosity.Thus, in praclicea valueof 150-180insteadof 72 is in closeragreementwith observationsat low Reynoldsnumbers,i.e.,
180
for 1{q",pM< l0
(4-15)
This is known as the Blake Koze y equationand, as noted, appliesfor NR",PM < 10.
porousmedia 101 Flowlhrough
B. Turbulent Flow At high Reynoldsnumbers(high turbule[celevels),the flow is dominatedby inertial forcesand "wail roughness,"as ill pipe flow. The porous medium can be consideredan "extremelyrough" conduit, \'/ith e/d - 1. Thus, the flow at a sufficieltly high Reynoldslumber should be fully turbulent and the friction factor should be constallt.This has beenconfirmedby observations, with the value of the constantequal to approximately1.75: .fna: l.'75
for NR",pM> 1000
(4-16)
This is known as the Bftke-Plumnel equation and, as noted, appliesfor > 1000. iy'a",p11 C. All Reynolds Numbers An expressionthat adequately.eprcsentsthe porous medium friction factor over ail valuesof Reynoldsnumber is
6 n ,= r . 7 5 +rv; q
(4-t7 )
Re.PM
This equatior with a valueof 150insteadof 180i,scalledthe Ergun equation and is simply the sum of Eqs (4- 15) and (4-16). (The more recent referenceslavor the value of 180, which is also more conservative.) Obviously, for 1{p..p14 < 10 the fi$t term is small relative to the second, and the Ergun equation reducesto the Blake-Kozeny equation. Likewise, for Np",p11> 1000 the first term is much larger than the secold, and the equation reducesto the Burke-Plummereqration. If the definitions of /pM and NRe.pMare itrserted irto the Ergun equation,the resultingexpressionfor the irictioral energyloss (dissipation) per unit massof fluid in the medium is
I -,".|L+ 4r(1. rso .,' = r.7sr';tf :u)'/ d\E'/ d'€'p
(4-18)
102Chemjcel Engineer ng Processes
4.5. PERMEABILITY The "permeability" of a porous medium (-Q is definedas the proportionality constantthat relatesthe flow rate through the nedium to the pressure drop, the cross-sectionalarea, the fluid viscosity, and net Ilow length through tho medium:
(4-1e) This oquationdefinesthe permeability(-&1J and is known as -Dar.]]r /dx,.The most common unit for the penneabilityis the "darcy," which is dellnedas the flow ratein cm'/s that resultswhena plessuredrop ol I atm is appliedto a porous medium that is 1 cm' in cross-sectional arca and 1 cm long, lbr a fluid with viscosityof I cP. It shouldbe evidentthat the dimensionsof the dar^cyare L2, and^ the conversion lactors are (approximately) 10-3 cm'f darcy= 10 " ft'/darcy. The flow propertiesoftight, crudeoil bearing, rock fornations are often describedin penneabilityunits of millidarcies. lf the Blake Kozeny equationfor lan'inar flow is usedto describethe frilrtion loss,which is then equatedto AP/p from the Bernoulliequation,the resultineexoressionfor the flow rate is
^ s:
-LPAI
d|€t
\
(4-20 )
r-.r U8o{1=/
By cornparisonofEqs. (4- 19)and(4- 20),itis evidentthat the pemeability is identical to thc tcnn in bracketsin Eq. (4-20), which shows how the pemeability is relatcd to the equivalentparticle size and porosity of the medium. SinceEq. (4-20) appliesonly for laminar flow, it is cvident that the permeabilityhas no meaningunder turbulent flow conditions. 4.6. MULTIDIMENSIONAL FLOW Flow ir1 a porous medium in two or three dimensionsis important in situationssuch as the production of crude oil from reservoirformations. Thus,it is ofintercst to considerthis situdtionbriefly and to point out some characteristics of the governingcquations. Consider the flow of an incompressiblefluid through a two-dimensiolal porous medium,as illustratedin Fig. 4-2. Assumingthat the kinetic energychangers negligibleand that the flow is laminar as characterizedby Darcv's law. the Bernoulli equationbecomes
-
lLP \ \-+c^z)=et=
or
^(e\ = -i!!d \.p /
Kp
tlV,L
Kp
(4-21)
Flowthro!ghporousmedla103
Equation(4- 22) wherethe densitycancelsout if the fluid is incompressible. can be appliedin both the x and ] directions,by taking l, : Arc for the )t directionand Z = Al for the I direction:
F-ax
---+l
T I Ay.
Two-dimensional flowin a porousmedium.
FrcuRE 4-2
a.o
ttv,
ao
K
aia
and
Ao
pV, ao
AJ,
K
(4-24 )
A!
If Eq. (4-23 ) is differentiatedwith respectto r. and Eq. (4-24) is differand -Kto be entiatedwith respectto y and the resultsare added,assuming,& ger we consranr,
#o
do
aF*;F:-K\
I tav^*. i.vj\
a'
o!):"
^
fluid, the termin parentheses is zeroas a resultofthe For an incompressible conservationof mass(e.g.,the microscopiccontiluity equation).Equation (4-25) can be genefdli/ed ro rhreedimen\ion..is
v2o=o
(4-26)
which is called the Laplaceequation.TlTesolution ol this equation,along with appropriate boundary conditions, determinesthe potential (e.9., pressure)distribulionwithin the medium.The derivativesof this potential then determinethe velocitydistributionin the nedium [e.g.,Eqs. (4 23) and (4- 24)1.The Laplace equation thus governsthe three-dimensional (potential)flow of an inviscidfluid. Note that the Laplac€equationfollows fron Eq. (4-25) for eitheran inconpressible viscousfluid, by virtue of the continuity equation,or for any flow with negligibleviscosityeffects(e.g.,
104Chemical Engineering Proe$es
compressible flow outsidethe boundarylaye. neara solid boundary).It is interestingthat the sameequationgovernsboth of theseextremecases. The Laplac€ equation also appliesto the distribution of electrical potentialand curent flow in an electdcallyconductingmediumas well as the temperaturcdistribution and heat flow in a thermally conducting medium. For example,if O =+ E,y =+ i, ar:d p/K =+ re. where /e is the electricalresistivity(/e : RA/Ax), Eq. (4- 22) becomesOhm'sla\,:
ff:
-,",,, v2E:0, and *-#:,
(4-21 )
Also, with O =+ T, V =+ q, and 1(/p + t, wherek is the thermalconduclivity, the sameequationsgovemthe flow ofheat in a thermallyconducting medium(e.g.,Fouier's lat'))l
*t:
-Io,
vzr:0,
cnd Y*Y:o ox tr))
(4-28 )
By making use of theseanalogies,cleclricalanalogmodelscan be con, structedthat can be usedto deter.dnethe pressureand flow distribution in a porousmediumfrom measurements ofvoltageand cu ent distribution in a conductingmedium,for example.The processbecomesmore complex, however,whcn the local permeabilityvaries with position within the medium,which is often the case. 4.7.HEATTBANSFER IN PACKED SEDS For heatandmasstransferthrougha stationaryor streaniinefluid ro a singlespherical particle, it hasbeensholvnn)CorlsonI1l. Vol L Ch.9.rhattheheatandmassrransler coefficienfs rcachlimitinglow valuesgivenby:
@.2e) whercNu'(: h.l/k) attd Sh'(=hDrllDJ are the Nusseltand Sherwoodnumberswirh respectto the fluid, respectively. kamers [2] has shown that, for conditions of forced convecrion. rhe heat transfer coefficient can be reDresenledbv:
N u ' : 2 . o + 1 3 P r a $ + o . 6 6 P rsor R esf
(4.30)
whereft?i is the parlicleReynoldsnumberr.dp/p basedon the supedcial velociryr. of the fluid, and Pr is &e PrandtlnumberCr"p/*. This expressionhas been obfained on the basis of experim€ntalresulis obtained with fluids of Prandtl numbersranging ftom 0.7 to 380. For natumlconvection,Ranz andMarshall[3] havegiven: N u' : 2.0+ O.6Prt/1 er'tla where Gr' is the Grashof number -
(4.31)
porousmedia 105 FJowthrough
Resultsfor packedbedsarc much more diflicult to oblain becauserhe ddving lorce cannotbe measuredvery rcddily, Gupta and Thodus .ll suggestthat rhe j-faclor for heat transfer,jr , forms thc most satisfactorybasisof corrclalionfor experimental resultsand havcproposedthat: ei6 = 2.06Re",a t75
(4.12)
where:e is the voidage of the bed, '. j', - S! P; atd S/': Stanlon numberi/Crpr.. The j-faclors for heatandnass transfer,j/, andjl. arefound!o be equal,andrherefore equaion4.28 can also be usedfor the calculationol masstransferrates. Reproduciblccorrelarionsfor the heat transfcrcoefficientberweena fiuid fiowing througha packedbedandthe cylindricalwall ofthe coniainerarevery diflicult ro obtainThe mafu difficulty is that a wide rangcof packingconditionscan occur in the viciniry of the walls.However,the rcsultsquoledby Ze|z ard Orhmerfjl suggestthat: Nu a Relt oe
(4.33)
It may be noredthat in this exprcssionthe Nusseltnumberwith respect10the tubewall lr'r is rclatedto the Reynoldsnumberwith respccL to the paticle Rel..
4.8.PACKEDCOLUMNS
V
Since packed columnsconsistof shapedparriclescontainedwirhin a column, their behaviourwill in many ways be sinilar io that of packedbeds which have already beenconsidered. Thereare,however,scvenl inlpofiantdifferences which makefie direct applicationof thc cquationsfor pressuregradientdifncult.Fircr.rhe sizeol rhe packing elementsin the columnwill generallybe very muchlargerandthe Reynoldsnumberwill lhereforebe suchlhal the flow is turbulent.Secondly,the packingelemenrs will normally be hollow, and thereforehave a large amountof internalsurfacewhich will offer a higherflow reslstance than their extemalsudace.The shapestoo arc speciallydesigned to producegood masslransfcrcharacteristics wi$ relativelysmall pressuregradienrs. Althoughsomeofthc generalprinciplesalreadydiscussed canbe usedro predicrpressurc gradientas a functionof flowrale.it is necessary to rcly heavilyon the literatureissued by the manufacturercof the packings. In general,packedrowersareused1brbnnginglwo phasesin contactwilh oneanother and lbcrc will be strcnginieractionberweenthe lluids. Nonnally ore of rhe fluids will preferentially wct lhe packingandwill flow asa 6lm overils surface:the secondfluid then passes throughtheremainingvolumeolthe colunn. With gas(or vapour)-liquidsystems, the liquid will normallybe the weuingfluid and the gasor vapourwill rise throughrhc columnmakingclosecontacrwith the down llowingliquid andhavinglirde directcontac! with the packingelements. An exampleof thc liquid gassysiemis an absorpfion process whcrea solublegasis scrubbedfrom a mixturc of gasesby meansof a liquid, as shown in Figure4.3. In a packedcolumn usedfor distillation,the morc volatile componeni
106Chemi€lEngineeng Proc€sses
of, say, a binary mixture is progressively rransfered ro the vapour phase and the less volatile condensesout in the liquid.
In order to obtain a good rate of transfer per unit volume of the tower, a packins is selec(edwhich \rill promoler high interfacialareabetweenlhe tlvo phasesand a igh degreeof turbulence in the fluids. Usually inqeased aroaand turhilence are achieved; the expenseof indeased capital cost and,/orpressurediop, and a balance must be maale belweenthesefactors when arriving at an economic design.
Liquidin
Uquid Fignre 4.3, Pacled absorptioncolunn
4.&1.GeneraldescriDtion The constuction of packedtowers is relatively straighdorward.The shell of the colunn may be constructedfrom metal, ceramics,glass,or plasticsmaterial, or from metal with a
porousmedia 107 Flowthrou9h
corosion-resislant lining.The columnshouldbc mounlcdtruly verticallyto helpunifom liquid dislribulion.Detailedinfomation on the nechanicaldesignandmountingofindustrial scalecolumn she11s is giveD by BRoWNELL and Youngt6l, MoLyNELxtit and in BS 5500t8l Thc bed of packingrestson a supportplatewhich shouldbe designedto haveat least 75 per cent ftee areafor the passageof the gas so as to offer as low a resislanceas possible.The simplestsupporiis a grid wilh relaiivelywidciy spaccdbarson which a few layersof largeRaschigorpartitionringsarc stacked.One suchanangement is shown in Figure4.4. The gas injectionplate describedbyl-ev sli6hn in Figure,l.5 is passageways designedto provideseparaie for gasandliquid so thatthey nccdnot vie for passage though the sameopening.This is achievedby providingthe gasinlctsto the bed at a Dointabovethe level at which liouid leavesthe bed-
Figure.1.,1. Grid bar suppofsfor packedto$es
FiguF 4.5.
The gasnrjectionplatc(z')
At the top of the packedbed a liquid distributorof suitabledesignprovideslor the uniform irrigation of the packing which is necessaryfor satisfactoryoperation.Four
108Ch€mical Efgineer ngProcesses
examplesof different disrriburors are shown in Figure a.6 tl0l asfollows:
, and may be described
(a) A simpleorificerypewhich givesvery finedistribulionthoughil mustbe correctly sizedfor a particularduty and shouldnot be usedwherethereis anv risk of rhe hole' plugCing (b) The notchedchimneylype of disrjburor,which hasa good rangeof flexibitityfor the mediumand uppcrflowrates,andis nor proneto blockage (c) Thc notchedtrough disftiburorwhich is spcciallysuitablefor rhe targersizesof fower,and,becauseof its largefie€ area,il is alsosujtablefor rhe highergas.ales (d) The perforated ring type of distributorfor usewith absomtioncolunns wherehish ga. rdres d relairel) s'nall. quiJ rle\ arccn(oLnrerec. This rlpe ir c.pe.,aii1 suitablewherepressure lossmustbe minimiscd.For ihe largcrsizeof tower,where installationtbroughmanholcsis necessary. it lnay be madeup in flangedsecrions:
FiguE 4.6 Typesofliquid distriburo/ro)
Uniform liquid l'low is essentialif the bes! useis io be madeof the packingand,lf the loweris high, re-disributingplaresarenecessary. Thescplaiesarencededat intervais
porous Flowthmugh media109
of about2i 3 column diametersfor Raschigrings and about 5-10 column diameters for Pall rings,but are usualiynot morethan6 m apartllll.A "hold-down"plateis often placed at the top of a packedcolumn to minimise movemenland breakageof the packing causedby surgesin ffowrates.The gas inlet shouldalso be designedfor uniform flow over the cross-sectionand the gas exit should be separatefrom the liquid inlet. Further detailson intemalfittingsaregiven by Lsva[g]. Columns for both absorption and distillation vary in diameter from about 25 mm for small laboratory purposesto over 4.5 m for large industrial operations;these industrial rangingftom colunmsmay be 30 m or morein height.Columnsmay operateat pressures high vacuumto high pressure, the optimumpressurcdcpendingon both the chemicaland the physical propertiesof the system.
4.8.2.Packings Packingscan be dividedinlo four main classes-broken solids,shapedpackings,grids, and structuredpackings. Broken solids are the cheapestform and are used h sizesfrom about10 mm to 100mm accordingto the size of the column.Althoughthey frequendy form a good corosion-resistant material they are not as satisfactory as shapedpackings either in regard to liquid flow or 10 effective surface ollered for transfer. The packing should be of as unifom size as possibleso as to producea bed of uniform characteristics with a desired voidage. areRaschigrings,Pallrings.Lessingdngs,andBerl The mostcommonlyusedpackings saddles.Newer packings include Nutter rings, lntalox and Intalox metal saddles,Hy-Pak. and Mini rings and, becauseof their high perfonnancecharacterislicsand low pressue drop,thesepackingsnow accountfor a largeshareof themarket.Commonlyusedpacking elementsare illustrated in Figue 4.7. Most of these packings are available in a vr'ide plastics,carbon,andsometimes rubber, rangeof materialssuchasceramics,metals,g1ass, Ceramic packings are resistantto conosion and comparalively cheap,but are heavy and may require a strongerpacking support and foundalions.The smaller mefal rings are also availabie made from wire mcsh, ard tbesegrve much-improvedtransfercharacterisiicsin smallcolumns. A non-poroussolid should be usedif thereis any risk oi crystal fo.mation in the pores whenthe packingdries,as this can give dse to seriousdamageto the packingelements. However,someplasticsare not very goodbecausethey are not wettedby many liquids. Channelling.that is non uniform distribution of liquid acrossthe colunm cross-section,is muchlessmarkedwith shapedpackings,andthei resislance to flow is much1ess. Shaped packingsalso give a more effective surfaceper unit volume becausesurfacecontactsare reducedto a minimum and the film llow is much improved comparedwith broken solids. On the olher hand, the shapedpackings are more expensive,particularly when small sizes are used.The voidageobtainablewjth thesepackingsvariesfton about 0.45 to 0.95. Ring packings are either dumped into a tower, dropped in small quanriries, or may be individually stackedif 75 mm or larger in size. To obtain high and uniform voidagc and to preventbeatage, i! is often found better lo dump the packingsinto a tower ful] of iiquid. Stackedpackings,as shownin Figure4.10, have the advanlagethat lhe flow channels are vertical and there is much less tendencyfor the liquid 1()flow to the walls than with
110Chemi@lEngineedns Prccesses
(e)
(f)
Fieure4.7. (a) Cenmic Raschisdngs; (r) cflmic kssing nn$ (.) Cermic Berl sa.tdle;(d) p,ti riie Qrstic): (d) P'u rins (n€lat)i (.f) M.ral Nurbr nns$ (t) nddc Nrtrer ring
Flowthroughporcusmedia111
Figtte4,1.
continLed
nndom packings. The propertiesof some conmonly usedindustrial packngs are shown in Table4.1. The size of packing used influencesthe height and diameter of a column, the pressure drop and cost of packing. Generally, as the packing size is increased,the cost per unit volume of packing aDdihe Fessure &op per unit height of packing fie reduced,and the mass transfer efficiercy is reduced.Reducedrllass tansfer efficiency resilts in a taller column being needed,so that the overall column cost is not always reducedby increasing the packing size. Nolmalty, in a column in which the packing is nndomly arranged, the packing size should not exceedoDe-eighthof the column diameter.Above this size, liquid distribution, and hencethe masstransferefliciency, deterioratesrapidly. Since cost per unit volune of packing doesnot fall much for sizesabove50 mm whereasefficioncy continuesto fall, ftere is seldomany advantagein usingpackingsmuch Iargerthan50 mm in a rardomly packedcolurnn. For laboralory puryosesa numberof specialpackingshavebeendevelopedwhich are,in general,too expensivefo.large diarnetertowers.Dixon packings,which are Lessingrings made from wire mesh, and KnitMesh, a fine wire mesh packing, are typical examples. Thesepackings give very high interfacial areasand, if they are flooded with liquid before operation, all of the surface is aclive so thal the tmnsfor characteristicsare very good even at low liquid rates. The volume of liquid held up in such a packing is low and the presswedrop is also lo\r. Someof thesehigh efficiency woven wire packings have be€n usedin coltmns up to 500 mm diameter. Crid pnckings,wiich xrc relxtivelJ,casy |(r fabricalc.arc rsurLllyrLscdin col mns of sqoxlescdion. rnd freqrertlJ in coolirg rolverswhich are describerlin Volumel, Chapler l:1. They may be made from wood, plastics, carbon, or ceramic maferials, and, becauseof tbe rclatively large spacesbetwe€nthe individual grids, ihey give low pressure drops. Furlher advantageslie in their easeof assembly,their ability to accept fluids with
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g
5
114Chemical Engineering P.o@sses suspended solids,andrhejreaseof weltingevenat very low liquidrares.The mainproblem is that of obtaininggood liquid disrributionsince,ar high tiquid raies,rhe liquid iendsto cascadeftom one grid to the nex! withour being broker up into tine dropleis which are desirable loi a highinlerfffiat.ufldce.A1 c\dmpteot c cootinprouerpaiking..Cootflo 3'02r is shownin Figure4.8. This is simitarto rhe stmcturedpackingJdescribed tater, anclconsistsof vacuumformedPVC sheersctampedrogctherwithin a meralandplasrics frame to folm a module which can be 0.6 m or 1.2 n i; deprh.Srrucruredpackinis may be broadly classified into either the knined or rhe non-lnitied type, ana bott typis rnay be assembledin a segmentedway or in a spiral form. In the lafter, corugut.d ;[ip, oi dbbonscoil abouta centrea\is to form a flal cakeof the requisitctowerdiameterwhich is usuallylessthan I m_Theseelementsare thenstackedone upon the otherto Drovide lhe nece+ar)beddeplh.In rherigid Llpeot sLrucrureo pacUng.rhcse.orrugnr"j.F..,. of melalor plasticare assemblcd to form intersedingopenchannels.The she€tsmav. in addilion. be perforaled andrhe)provideuniformtiquidflowo\er bolhsiderwhilerapour flou. upwards andprovides inr mareconracr $ilh rt-etrquid.One,urh rlpe of pacUng. Mellapak(33) is shownin Figure4.9, and otherssuchas cempakil4l -" ut.o uuoiUtf!. Low pressuredropsof lypically 50 N/m2per theorericajstageare possiblewirh HETp.s, rangingfrom 0.2 to 0.6 m, voidagesin excessof95 per cent, and high specific surface areas.The resulting higher capaciry and efficiency wiih structured packingsis, howeverachievedat higherinirial capitalcosrthanwith the olher packingsdiscussed in rhis sectionl15].
FiguE 4.8,
Msco Coolno3 extended surface,cootineroper pactring
porousmedia 115 Flowlhrcugh
Figu€ 4.9. structured packinss (4) meral gauze (r) carbon (.) corcsion esisDn! ptdslic
4.8.3.Fluidflow in Dackedcohmns
It is important to be able to predicr rhe drop in pressurefor rhe flow of the two fluid streamsthrough a packedcolumn. Earljer in this chapterthe drop in pressurearising from the flow of a single phasethrough granularbedsis considercdand the samegeneralform of approachis usefully adopted for the flow of two fiuids &rough packed columns. It was noted that the expressionsfor flow through ring,type packings arc less reliable than those for flow through beds of solid particles. For the typical absorptjoncolunn there is no very accurateexpression,but there are severalcorrelationsthat are useful for design purposes.In the majority of casestlre gas flow is turbulenr and the general form of rhe relaiion belween the drop in pressure-AP and the volumetric gas ffowrate per unit area of colurnn c is shown on curve A of Figure4.10. ,AP is rhenproportionalto r!3 appro\imarely. in rgreemenr wirh lhe (urre A of Figure4.lO al hieh Relnoldsnumbars. If, in addition to the gas flow, liquid flows down the tower, (he passageof the gas is not
J 1 6C h e m i c E n rgo c e s s e s . l n g i n e e r iP
e E
v
ligur 4.10. Prcssurcdropsin wet packines(logdilhnricaxet
significandyaffectedat low liquid ratesand the pressuredrop line is similar to line A, althoughfor a given valueof 116the valueof AP is somewhatincreased. When dle gas rale reachesa cetain value. the pressuredrop then dses very much more quickly and is proportiondlto i/U5,as shownby the secdonXY on curvc C. Over this section the liquid flow is interferingwith the gasflow and the hold-upof liquid is progessively inffeasing. The fre€ spacein the packirgs is thercfore being continuously taken up by the liquid, and thus the resistance to flow risesquickly.At gas flows beyondY, AP rises very steeply and the liquid is held up in the column. The point X is known as the loadingpoint,and point Y as the floodingpoint for the givenliquid ffow. If the flowrate of liquid is increased, a similarplot D is obtainedin which the loadingpoint is achieved at a lovr'ergasratethoughat a similarvalueof -AP. Whilst it is advantageous ro havea reasonable hold-upin the columnas lhis promotesinterphase contact,i! is nol praclicable to operateunderffoodingconditions,andcolumnsarebestoperatedover the sectionXY. Sinceihis is a sectionwith a relativelyshof rangein gas flow, the safe practiceis 1() design for operation at the loading point X. Ii is of interest to noie that, if a column is floodedand then allowedto drain. the valueof AP for a given gasflow is increased overthatfor an cntirclydry packingas shownby curveB. RosEandYoung[16]correlated rheir experimenial pressuredrop data for Raschig rings by the following equation:
-o",:-o"r(t*f) where:
4P,,, is the pressure&op affossihe wet drainedcolumn. AP,r is the Fessuredrop acrossthe dry column.and d,, is ihe nominalsizeof ihe Raschigrings in mnl.
This effect will thus be most significant for smau packings.
(4.34)
porous Flowthrough nedia 117 Thercfie severalwaysof calculatirgthe pressuredrop acrossa packcdcolumnwhen gasandliquid are llowing simultaneously andthe colunrnis operatingbelow lhe loading point. One approachis to calculatethe pressure drop for gasllow only andthenmuitiply lhis pressuredrop by a factorwhich accountsfor the effcctof the liquid ffow. Equarion4.19 may be used lbr prcdicting the pressuredrop lor the gas only. and rhen rhe pressurcdrop with gasandliquid Rowingis obtainedby usingthe correctionfaclorsfor the liquid flow rale givenby SHERwooD and PigfordllTl AnotherapFoachis thalof MoRRrs andJacksonllslwho arrangedexperimental darafor a wide rangeof ring andgrid packingsin a graphicalform convenienlfor the calcularion of the numberof velocilyheadslr' lost per unit height.ofpacking.?{ is subsrirured in rhe equation:
-LP::NpcuLl wherer - AP pc rc I
= = : :
(4.3s)
pressuredrop, gasoensrry, gasvelocity,basedon the cmply columncrcss-sectional area,and heightof packing.
Equation4.35 is in consistentunits-For cxample,with pc in kg/m3,,G in mA, I in m. and ly' in m-r. -AP is rhenin N/m'?. Elnpirjcalcorelationsof experimental datafor pressurcdrop havealsobeenpresented by Leva[]9l.rndby ECKERT et al.[20]or Pall rings.Whercthe datar.reavailable,the nosr accuratemethodof obtainingthe pressure dropfor fiow througha bed of packingis from lhe manufacturer's own literature.This is usuallypresented as a logarithmicplot of gas rate againstpressurcdrop,with a parameterof liquid nowrateoD the graphs,as shown in Figure4.16, althoughit shouldbe stressedihat all of thesemcthodsapply only ro conditionsal or below the loadingpoint X on Figure4.i0. ff applied!o conditionsabove the loadingpoinl the culculatedpressure drop wouldbc too low. It is thereforenecessary first lo checkwhctherthe columnis operatingat or belou,thc loadingpoint,andmethods ol predictingloadingpointsare now consjdercd. Loading and floocling points Althoughthe loadingand floodingpointshavebeenshownon Figure4.10, rhereis no compleielygeneralised expressionfor cdlculatingthe onsetof loading,allhoughone of the following semi-empirical co elationswill often be adequate. MoRRrsandJAcKsoN,'' gavetheirresultsin rheform of plotsof rt(,J(r/,Jr)al the loddingratefor va.iouswelling ratesLw (m3/sn). ,,6 and,r. are averagegasand liquid velocitiesbasedon the empty (/kclpi) columnand t: is a gasdensitycofiectionfactor,wherepi is the densiry of air at 293 K. A uselul graphicalcorelation for fiooding rareswas first presentedby SHERwooD et dl.[21]and later developedby Lobo€rdl.l22l for random-dunpedpackings,as shown in Figure4.1I in which:
"s^i" " I,JE) W) f:n(e)"i',r*oa
118Chemical Enqineering Processes
dH2A-N2 DH2O CO'
;6t g 0.01
0.001
b Butydcacid-A r 4cH3oH Ak v Turbin€oil Air >B 100O Air < 10 Coil Alr o O i l N ol. A r r Oil No. 1 CO, E oilNo r H2 a oilNo 2 At ? OllNo.2 CO, + oit No.3 Air
0,01
L' 14
d 1F; Figlre 4.11. oeneralhedcorreiaiionforiooding raresin pacled towc4 l
where:16 is the vclocity of rhe gas,calculatedover rhe wholecrcss-secrion of rhe bcd, .S, is ihe surfaceareaof the packingper unit volumcof bcd, due !o gravity, I is the acceleration I,' is lhe massrateof flow per unit areaof the liqujd, G' is the massrateof fiow per uni! areaof the gas,and p. is the viscosityof waterat 293 K approximalelyI mN s/m'?,and sulfix G refers to the gas and suflix a !o the liquid. possibleconditionsof operation.In theseexpresThe areainsidethe curverepresenls pc/pr pr/pu sions.the ratios and havebcenintroducedso that the relationshipcan be appliedfor a wide rangeof liquids andgases.It may be notedthat,i[ the effecrivevalue of a is increasedby usinga rotaiingbed,then higherflowrurescan be achievedbefore the onsetof flooding.
s e d r a1 1 9 F l o wt h r o l q hp o r o um
There are several ways of calculating the prcssuredrop acrossa packed colunn when gasandliquid are fiowingsimultaneously and the columnis operatingbelow the loading point. One approachis to calculatethe prcssuredrop lbr gasflow only and ihen multiply rhis prcssuredrop by a factor whicb accountsfor rhe effec!of the liquid flow. Equalion4.19 may be usedfor predictingthe prcssuredropfor the gasonly, and rhenthe prcssuredrop with gasard iiquid flowing is obtainedby usingrheconectionfactorsfor thc liquid flow rarcgiren b) SHLRiooD andPiglorJllTl Anotherapproachis thatofMoRRrsandJacksonll8]who arrangedexperimental datafor a wide rangeof ring andgrid packingsin a graphicalform convenicntfor lhe calculalion of the numbcrof velocityheadsN lost pcr unii heightofpacking.lr' is substitured in rhe
^P - +Npcu? where: - AP pc uc I
: : : :
(,r.35)
pressuredrop, gasclenslty, gasvelocity,basedon the emptycolumncross-sectionalea,and heishtof packing.
Equation4.35 is in consisteniunirs.For example,with pc in kg/nr, ,6 in m/s, / in m, and lr' in m r, -AP is rhenin N/m'?. Empiricalcorelationsofcxperimentaldaiafor pressure drop havealsobeenpresentcd by Levalg] andby ECKERT e/ al.[20]or Pall rings.Wherethe daiaarc available,the mosl accuralemelhodof obtainingthe pressure drop ior fiow througha bedofpacking is from the manufhclurer's own literature.This is usuallyprcscnledas a logarirhmicplo! of gas rate againslpressurcdrop, with a parameteroi liquid flowrateon the graphs,as shown in Figure4.16, althoughir shouldbe stressedthat a1lof thesemelhodsapply only to condltlonsat or below the loadingpoint X on Figure4.10.If appliedto conditionsabove the loadingpoint the calculatedpressure drop wouldbe 1oolow. It is thereforcnecessary first to checkwhctherthe columnis operatingat or belowthe loadingpojnl, andmethods of predictingloadingpointsare now corsidered. Loaclingan.l flooding points Althoughthe loadingand lloodingpoinishavebeenshownon Figure4.10, thereis no completelygeneraliscd expressioD for calculatingthc onsetof loading,althoughone of the followirg semi-empirical conclalionswill often be adequalc.MoRRrsandJ.rcrsori'!' gavetheirresultsin lhe folm of plorsof ry'(uc/rr) al the loddingratefor variouswetting ratesI,$, (m3/sm). ,G and ur are averagcgasdndliquid velocitiesbasedon rhe emprt columnand 1/ = (.//kclp/) is a gasdensityconectionfacror,wherepr is rhe densiry of air at 293 K. A useful graphicalcorrelationfor floodlng ratcs was first presenredby SHERwooD e, dl.[21]and later devclopcdby Lobo€rdl.t22l for random-dumped packings,as shown in Figure4.ll in whichi
(s)€l(4r)o',,o'*oo "*'"" #Je)
120Chemical EnsineeirqProcesses
Liquid distribution Provisionof a packingwith a high suface arcaper unit volume may not resultin good conlaclingof gas and liquid unlessthe liquid is disrributeduniionnly over rhe surface of lhe packng. The needfor liquid distributionand redisrriburionand correcrpacking sizehasbeennotedprcviousiy.The effectivewettedarcadecreases as rhe liquid rareis decreasedand, for a given packing, dere is a minimum liquid rate for effective use of the surface area of the packing. A useful measureof lhe eflectivenessof wetting of the availableareais rhe wettingratc /-,, definedas: Volumetricli{tuid rateocr unit cross-sectional areaof column Packingsu aceareaper unit volumeof column L
Apr.sB S,
(4.36)
Thus the welling rate is analogousro the volumctricliquid rate per unir length of cncumference in a weltcd wall colunn in which the liquid flows down the surfaceof a cyljnder.If the liquid rale were too low, a continuousliquid film would not be formed aroundthe circumferencc of the cyiiDderand someof the lrea wolLldbe ineffective. Similareffeclsoccurin a packedcolumn, althoughthe flow parterns andarrangement of the surfacesarether obviouslymuchn1orecomplex.MoRRrs andJackson[l8]haverecommendedminimumwettingratesof 2 x l0-5 mr/s m for rjngs25-75 mm in diamelerand giidi ofpirch le* rhar 50 nm. cnd 1.3 l0-s n'/\ m tor ldger pacungs. The distributionof liquid over packingshas been studiedexperimenlallyby many workers and, for instance,Toun and Lmuex(45.46)showedthat lor a single point feed rhe distribulionis siven bvl Q,:
t exp( a2 x'z)
(4.31)
where 0, is the fraciion of thc liquid collectedat a disrancer liom rhe centreand c and a are constanlsdependingon the packlngarrangemcnl. Nonnant2TlMANNNGand Cannon[28]and othershaveshownthatlhis maldisiribution is onecauscoffalling rransfer coefficientswith tall towers. A nomograph which relates liquid rate, tower diameter and packing size is given in Figure4.13[10]The wefiingrareLw may be obtainedas dn absolutevaluefrom the inner right hand axis or as wetinq fnction from the outerscale.A valueof wetting fraction exceedinguniiy on lhat scdleindicatesthatthe packingis satisfaclorilywer. h shouldbe notedthat many organicliqulds havc favourablewettingpropeniesand vr'cuingmay be effectiveat muchlowcr rates,lhoughmatelialssuchasplaslicsandpolishedsrainless slecl are difficult to wet. Figure4-13does,however,represenithe best availabledataon the subjectof welling.In the exampleshownin Figure4.13,the arrowedlinc conesponds ro the caseof a liquid flow of 0.018m3/sin a colunn of 1.6m diamererand a packingsize of 25 mm, which givesan approximatewettingrateof 5 x 10-5 mr/m s, corresponding to a total wettingof morethan I on rhe oursiderighGhandscaleithis is saiisfacrory.
poro!smedia 121 Fiowihrough
,{;13
o', E
z,it:-q B
-e E
o04 Iv{m3hzs) Fieuo 4,13. Nomographtbr rhe estinaiionot the degEeof weftingin I pmkc.tcojutul29l
In manyindustrialapplications of packedcotumrs,it is deslrablcto know thc volumerric hold-upofihe liquid phasein thecolumn.This infonnationmightbc reeded,for exanple, if the liquid were involvedin a cbcmicalreaclionor if a conrrolsysremfor rhe column werc being designed.For gas,liquid systemsihe hold-upof liqujd Hi,j for conditions belowthe loadingpoint hesbecnfoundlgl to vary approximately as the 0.6 powerof lhe liquid rate.andfo. rings and saddlesthis is given approximateiy by: /./\
06
H",:0.1$l;)
(4.38)
wherci ,' is rhe liquid flowmte(kg/n,s)t d is the equivalentdiamelcrof rhepacking(nxr), an{l ,?,, is the hold-up(mr of liquid/m3of column). Thus with 2-5mm Raschigrings, L' of 1.0kg/m2s and d :20 mm, 4, hasa valueof 0.021mj/mr.
122Chemical Enginee.ing Pfocesses
4.9 FLOWINPACKED BEDS The chemical and energ/ industriesdeal predominantlywith nultiphase and multicomDooentclstemsin whichconsidemble atlenlion is devoted lo bcreacing rheinlerfacidt con;cr betweenihe phasesto enhanceproperrytransfen and chemicalreactionsat theseextended surfaceinterfaces. As a.esult,packedbedsareext€nsiveb/ usedin th6 chemicalprocessindustries.Someexamplesaregasabsorption,cataMc reactors,anddeepbed ltration. 4.9.1 FriclionFaclorCo elations Tfie friction factor for packedbeds,fpb, is de ned by a3 DplLPl
J"p b i . . . - -
' ' oulr
(4.3e)
where € is the po.orr4, (or yoid yolune fraction), Dp is tbe particl€ diamerer,and r, is the superfcial wlocitJ. The s,rpercial velociry is obrainedby dividing rhe volumetric ow rate by the total ooss-sectionalareaofthe b€d.Note that the actual ow areais a fraction ofth€ totalcross-sectional area. For packedbeds,the Reynoldsnurnberis de ned by ^Repr- Dpu"o -liJ+ I
14.401
For laminar oq the relationshipb€tweenthe liiction factor and th€ Reynoldsnumber is givenby
R e ,< a1o
r*=P
(4..41)
whichis Lno\ n asrheKozcn\-Carnanequa on. In the caseofturbulent ow,i.e-,Rej,,> 1000,th€ relationship betweenRerb andt, glvenby theBurke-Plumner equationin the for.m
f tu= 1.'75
Rep,> 1000
js
(4.42)
T\e so-calledEryun equarion(1952) is simply the summationofthe Kozeny-Camanand the Burke-Plummerequations 150
(4.43)
porousmedia 123 Ftowthrough
Exsmple4.l A columnof0.8 m2 cross-section and30m heightis packedwith spherjcal particles ofdiameter 6 mm.A uid withp:1.2kg/m3 andp= 1.8x l0-5 kg/m.s ows thrcugbthebedatamass ow rateof0.65kg/s.Ifthe pressure dropis measured as3200pa, calculate theporosityofthe bed: a) Analytically,
b) Numerically.
Solution Assumption 1. The systemis isodtemal. Analysis The supe.cial velocity throughthe packedbedis u ,=
0.65 ,1.2^,
= 0 . 6 7 rm r "
Substitution of the valuesinto Eqs.(4.35)and (4.36)givesthe friction factorand the Reynolds numberasa functionolporosityin theform 11.)200'l
.'
Dpl^P. , .r'164\r-e/ .. ./ i t = .' I ro" to rpb= r , puI T-.1{Lr"oatt,3o)l=
I _,_.r l,0q1l{L.2'.l. R '"^=D,ut_L _ 1,6_:_.10
u r-.
1
r.r-;ro-Jl
| \
.-"uolr-,J
rll
(2)
Substitution ofEqs.(1) and(2) intoEq.(4.6-5)gives e 3 0 . 4 i 6 €+ 2 2 . 4 5 5 €_ i . 9 7 9 : 0
(3)
a) Equation(3) can be solvedanalyticallyby usingthe prcceduredescribedin Section 4.7.1.2in AppendixA. In orderto calculatethe discriminant, the tems M and ly' mustbecalculated fromEqs.(A.7-5)and(A.7-6),respectively: { 3)(2.455 ) {0.476)2 M =_: +_0.793 9 ir'_
_ r q i ( 0 . 4 7 6 1 2 . 4 5 5i 2r 7 r { t . g 7 q ) + ( 2 r { 0 . 4 1 6 , . _O.rno
Therefore, thediscriminant is n=m3 i-t't2:q0.79y3+ e.7gg\2=1.137 SinceA > 0, Eq.(3) hasonly onerealrootasgivenby Eq.(A.7-7).Theterms,SandZ in this equationarecalcLllatedas
'{ o . z o o + , , 1 3 7 r- " ' 1 1 "[t' I = 1 r u - . 6 1 ' ' . - 1 o . t s ou \ t n l t 3 _ 0 . o 4 4 s = t r uI
124Chemical Enqineenng Proesses
porosityofthe bedis Hencetheaverage ._t.2jt
n 416
0 . 6 4 4t = - : _ 0 . 7 4 0 I
b) Equation(3) is rearangedas F ( 4 : e 3 - 0 . 4 1 6 ?+ 2 . 4 5 5< - 1 . 9 7=9o
(4)
FromEq. (A.7-25)the itemtionschemeis |-
'r ='t
0.02ci-r f (q-r.t F (t i i . , , r ) - F ( 0 9 9 . / ,
(5)
Assuming a startingvalueof€, :0.7, theiterations arcgivenin thetablebelow:
0----dr | 2 3
0.'746 0.'745 0.'t45
4.9"2 HeatTransferCorrelation for heattransf€rin packedbeds: Wf itaker(3l) (1972)proposed thefollowingcorrelation Nup, : (0.4R";f + o.2R#) P.o4
(4-44)
TheNusseltnumberin Eq.(4.6-6)is definedby th\Dp ( Nupr:--rl
(4.45)
€
(4.40)is validwhen Equation 3.7
Pr,z0.7
A11Fopertiesin Eq. (4.40) areevaluatedat the averagefluid tempemturein the bed. Calculationof the heattranskr rute Oncethe averageheattransfercoefcient rs determined,the rateof heattransferis calculatedftom Q = axvlh\^TLM
(4-46)
wherc y is the total volumeof the packedbed and a, is the packingsurfaceareaper rmit volumedened by 6al .) "'=--
(4'4',7)
Flowihroughporousmedia l2s
4-9.3 MassTranslerCorrelation (12X1977) proposed Dwivediand Upadhyay a singleconelationlor bothgasesandliquidsin packed and fluidizedbedsin termsofthej-factoras ..
€Jit ft:
0.765 'n-+ ' " ' b \O8 )
|
0.165
ta_ae\
{Re;'rot36
ThetermsJi,, andRe;D;nEq.(4-48)aredoned whichisvalidfor 0.01< Re;, < 15,000. Dy
-,,=(f)s*'r
(4-4e)
and
(4-50)
Re|6 =
masstransfer Calculation of the masstranskr rate Oncetheaverage coefficient is determined, the late of masstransferofspecies,4, in, is givenby
(4-st)
rhe: a,V \k"l(^ce\mMt
Example4.2 A packedbedof porosity045 ;n a pipe,2.5 cm in ;nternaldiameterby usingnaphthalenespheres 5 mm in diameterPureair at 40'C flowsat a supercial velociryof9 m/s throughthebed.Determine thelengthofthepacked bedrequired for theavemge concentration of naohthalene vaDorin theair to reach25%ofthe saturationwlue. Solution Physicalproperties (,4 )inair( 6) at40"C (313K) is Diffusioncoefficient ofnaphthalene / r r r-\ l / 2 {DAB)ltJ:rrA6rjool- | -r0.b2, ro '){=)
:6.61 ,10 6m2/s
\JUUl
Forairat40'C (313K): r : 16.95x 10-6m2ls TheSchmidtnumberis "-
1 6 . 0 5 : t 0o ^ - _ ---" D e s - 6 6 1' l 0 o
Assumptions 1. Steady-state conditionsprevail. 2. Thesystemis isothermal. 3. Thediameterolthe naphthalene sphercsdoesnot changeappreciably.
126Chemical Engineering Processes
Analysis System:Air in the packedbed Understeadyconditions,the conservation statement for naphthalene, species,4, becomes _ Rateofmoles of,4 in: Rateofmoles of,4 out The terms in Eq. (1) are expressedby
Rateof molesof,4 in = d,V \k,JQ:ci ru Rateof molesof"4 out= QGA)out: (irD2/4) po@A)ab
(l)
(2) (3)
Since the concentrationat the surlace ofthe naphthalenespheresis constant,the expression for (Ac,a).rr',becomes
(4)
Substitution ofEqs.(2H4) intoEq.(1)andnoringrhatv = (1rD2/4)L gjve
r-- '" r"fr- "^''"'l cA \*c)a,
L ) Notethat for a circularpipe,i.e.,atr= 4/ D, the aboveequationreducesto Eq. (5)
rs)
The interfacial areaper unit volume, du, is calculated as
6 ( l , ) 6 r l- 0 . 4 s ) , ^ "-*-oa*=oo" To determinetheavengemasstransfercoefrcientfrorn Eq.(4.6-l0), rst it is necessary to calculate theReynolds number Dpuo f0.005)19) (cpr=-_Gl3\lI=_zo)) Substitution ofthis valueintoEq.(4.6-10)gives
',",,: ffi *ffi =d##+#ffir : oorso given (4.6-l
in which
byEq.
i). Therefore, theaverage masstransfer coeficientis (o ( l . r- o o l 8 6 - - & . ,- - o l 8 6 ) r q )
^s"-, rd.EiIJG'-"'/'
The length ofthe bed is calculatedfrom Eq. (5) as
r: -
t-u666,ln(l
- o 2s)=o 02rn
Colnln€nt The use ofa packedbed increasesthe masstransfer areabetweenair and solid naphthalene.This in tum causesa dmstic decreasein the length ofthe equipment.
porousmedla 127 Flowlhrough
Further rcading Branan CR. Rules ofThurnb for Chenical EngineeB. Honslon, TX: Cxlf Pxb Co., I994. Leva M. R€consider packed lower pressuredrop correlations. Chem EnB Prog 88: 65.-72, t992. couhon JM. JF Rlchardson. JR Blackhusr, JH Harkef. chemical Eneineerins. vol.6. .1Lhed. New York: Perganon Pres, 1991. REFERENCES r. Conlson JM, JF Richa.dson, lR Bl.ckhus!, JH Harkei. Chemical Engineerlng. Vol. L 4th ed. Nea York: Pe.samon Press. 1991. 2. KRAMERS, H.: Pbri.d 12 (1946)61, Helt transiir riom sphcrcsto flownrginedia. from drcps. R.: Cherr.Eie. Ptue.48 (1952)l4l, 1t3. Evaporation 3. R{Nz,w E. and MARSHALL,N. 4. CunA, A. S. and THoms,G.: A.t.Ch8.J1,91.t963)151. Directanalogybetweenmassandhearfiansfer to bedsof spheres. 5. zENe F. A. and OrH\jER,D.F.: Fl,idiltioa a..t FLui.lparidle s)sr?,r (Reinhold,1960). L. E. md YouNc,E. H.1Pn e$ E4tip,ent DeJrg\ v,rs,l D,stgn (Chapmrn& HaI, 1959). 6. BRoVNELL, 7. MoLYMG, F : Cncni.a/ PlanrDerts,, Vol, I (Buftesor1hs,1963). 8. BS 5500: 1978:a!r;,n lvcldedPmrlE v?rreli (BnrishStandadsInstitLion, London). 9. LEVA.M.: ?blecr Pa.trnss an.1Pa.kel rover Deliqi lJ.S Stonewtu€Co., 1953). 10. Nofion ChemicalPoccssProductsDiv., Box 350,Akrcn, Ohio:Hydrontl Ltd,, King St.,Fc.ton.Stokcon lor desieningpacked!owe6. 11, EcGRr, J. S.: Cncn.E,is. Pr,s. 57 No. 9 (1961)54. Designtechniques 12. CooLRo3 is a proddctoivhco LId,, Crotdon Snircy. 13. Mellapakis ! regisrered lradena* of Snlzer(UK) Ltd., llnborcugh, Hants, Cnnbria. 14. cenpak is a Eeisieredtradenrt ol Clitsch(U() Ltd. Knkby Srophcn, 15. CENG. C. K., C/€,,. E s, Alrat), 91 No 5 (March5 198.1)40. Plcted colunn intemals. of packcd 16. RosE,H. E. and YouNc.P.E: Pttc.lnsl. Mech C,is.1B (1952) ll4. Hydrauliccharactcristics toweN opemtingundercount€rcurcntilow condnions. R. L : abriptioh atu| *ta.ria, (Mccrae'Eill, 1952). 17. SsERwooD, T. K. lnd PlcFoRD. 18. MoRRrs,G. A. and J^c$o"_,J:,4rJa?r,", tu,c6 (Butiesodhs, 1953). 19. LEv , M.: Cr,n trg. Pms. Symp.Ser.No. 10,50 (195,1) 51 59. Flow thronghnrigatcddunpedpacting Pressu€drop. loading.Roodnrg 20. EcKERr.J S.. FoorE,E. H.. and HUNlNcror.',R. L.: Cr?n t a, P.,s. 54, No. I (Jan.l95E)?0 5. Pall rings newlype of lower packing. 2l. sHERworD, T. K., SlnpLEy,C, H., and HoLLowa. F. A. L.: htd. Erg. Chen. 30 (193a)165 9. Flooding ' c o , r i e . i r p d c \ e d. o ' | t r . 22. LoBo.W. E.. !RrEND,L, HASHMALL, F., lnd zENz.F.: I/"hs. A,t. 1n!. Ch.h. Lne. 1711945)693-710. Liniting capacnyoidunped lowerpackings. pcrtbrnancc. 23. EckEkr,J. S.: Ch.ft. Eng.P/,s.59 No 5 (i963) ?6. Towerp.ckings comparative corelations. 24. LE\a, M: Chen. EnB.Pas. 88 No, | (1992)65. Rccotrsidcr lackedrowerpressuF-drop F.: Truht.Ah. tat. Chett. ErE. 35 (1939) 709 | 8. An nnprovcddeviceto 25. TouR,R. S. lnd LERNTAN. to liquid now in packedtowers. demo.stratcthe laws of iequencydistribution.With snecialreferencc 26.TouR.R.S.andLERMAN,F.:Trans.A,t.h6lChem.En:1,35(1939)719-42.Theunconnneddislribnlion oi liquid nr towerpacking. 2?. NoNAN, w, S.i zarr, /r st. ChenLEng. 29 11951)226 39. Tbe perfom!.ce of grid p&kcd iowes, 28. MANNT._C, R, E. andcANNoN,M. R,: ,r1t'& cr?,,. 49 ( l95l) 347 9. Distillalioninprcvene.! by control oi lhase cha.nellingin p&ked cohmns. 29. LE\A,M.t TowerPatkingt and Pd.kedIbrer D.sigr (.U.5.SlonewlE Co., | 953). Div.. Boi 350,Akron,Ohio:HydronylLid., Kins St.,ienton, Stoteon' 30. Norror Chenicrl Prccesslroducrs 31. Whitaker.S.. l9T2.Forcedconlectionheattransiirrcorclationsfor o w in pipes,past at plates.s ingle cylinders, snrelespheres.and for o w in packedbedsand tube bundlcs,AIChE Journal I 8, 36132. Dllivedi, PN. and S.N. Upadhyay.1977,Particlc-uid nass transfefin xedand uidized beds. Ind. Eng. Chcm. ProcessDes Dcil i 6. 157
128Chemi€lEngineedng Processes
PROBLEMS Porous Media I . A packed bed is composedor crushedrock with a d€$ity of I 75 lb-/fi3 of such a size and shape tha! the ave.age ralio of surface area to volune for rle Frlicles is 50 in.']/in.3-The bedis 6 ft deep,hasa porosiryof0.3, and is covered by a 2 ft deeplayerof water rhat drainsby gravirythroughihe bed.Calculale the ffow rate of water thiough lhe bed in gpn/frt, assuminSit exitsat I ahn prcsure. 2. An impnrily in a walerslreamat a v€ry smallconcentmiionis to be removeditr a charcoaltrickle bed nlter. The filter is in a cylindricalcolunn tlra! is 2 f! in diameter. and lhe bed is 4 fl deep. The water h tept at a level thal is 2 ft above the rop ot the bed, and it trickles through by Sraviry flow. If lhe charcoal particleshave a geometricsurfacearea 1o volume raiio of48 in. ' and they pack with a porosityof0.45. whal h the flow raie ofwaler throlgh the column, in gpm? 3. A lrickle bed nlter is conposedora packedbed of broken rock. The shapeof the rock is suchthal the averageratio ofthe surfacearea10volumefor ihe rock parliclesis 30 in.-' The bedis 2 ft deep,hasa porosityof0.3, and is coveredby a layer of water that is 2 ft deepand dmins by gravity thmugh lhe bed. (a) Deternrinelhe volune flow rate of the water rhronghthe bed per unit bed aiea (in spm/ft'). (b) If lhe water ls pumped upw-ardlbrough th€ bed (e.s. io flush jt ouri. calcllale th€flow rate(in spm/# of bedarea)!ha! will bercquiredto fllidiz the (c) Calculalethe correspotrding flow ra|e rhar would sweeplhe rock particles away with the water.The rock densityis l20lbn/ftr.
PackedColumns 4. A packedcohmn ihat is 3 fi in diamelerwith a packinghe;Bhtof 25 f! is usedto abso$ an impu ly Fon a methane8asstreamuslngan aminesolnlionabsorben1.The sas flow rare is 2000scfnr,and the ljquid has a densiryof 1.2 g/cm3 a n dd v i c o . r ) o l 2 c P l ' h ec o l u r n no p e , a l ed. t I a r r d n o8 0 l - . d e e r m i n e, h e liquid ltow rate ai which floodingwould occurin the colnmn and the pressur€ drop al 50% ofthe floodingliquid rate for the foilowingpackings: (a) 2 in. ceranricRasch'grinss (b) 2 in. plasticPall rings 5. A packedcolumn is usedto scrubSO, from air by using iraler. The sas flow rale is 500scfm/ft'. and the column operatesat 90"F and I alm. Illhe column containsNo. I plasticIntalox packing,what is the maximum liquid flow rale (per uni! crosssecrio! of colunn) thal could be usedwithou! floodins?
pofousmedia 129 Flowthfough
A st.ippingcolumnpackedwlrh 2ln. meralPall ringsusesair a! 5 psigand 80'C to sirip an impurity lion an absorber oil (SG:0.9, viscosity= 5 cP, ?: 20'C). If the flow rate of lhe oil is 500 lb,i/nin and that of the air is 20 (a) Whal is the minimnn colnmndiameterihat can be usedwithout flooding? (b) If the colnnn diancter is 50'%greaterthan ihe minintrn size.whar is ihe pre$ue drop per ft ofcohtmn height? 7. A f a c \ e dc o l u a n L a \ 0 t s T l a ' a r e e r J n d4 n h r r r J n dL o n . a i n c .m 2 5m Raschigringsistrsedira gasabsorplionprocesstoremole an inpurily fron rhe gasstreamby absorbjngit iD a liqnjd sohcnt.The liqnid, whjch has a vlscosity of 5 cP ard SG: Ll, entersrhe top of tbe colnmnat a rate of 2.i kg/G mr), and the gas,which can be assrmedio havethe sane properiiesasair, eniersthe botrom oflhe colnnn ai a rate of0.6 kg(s mr). Thc column opcmtesat atmo sphericpresstreand 25'C. Deiermine: (a) The pressnredrop throush the colnmn,in inchesofwarer. (b) Howhish the liq id rateco ld be inoeasedbelbrethe collmn wolld flood. 8- A packedcolnmn is usedio absorbSOr fronr nne gasnsing an ethanolanine solution.The collmn is 4 ft i! diameter,has a packedheight of 20 ft, and is packed\rith2 in. tlasticPall rings.Theflnegasisata lenperatnreof I80"F ard hasan aaerage molccnlarwcighrof 31.The amincsohtion hasa specincgravjty of 1.02and a vjscosityat thc opcraiiDgtcmperatlreof 1.5cP. lfrhe gasDrlst leavethe cohtnn al25 psig and a florv rale of 10.000scfn, detemino: (a) Thc naxinum alloivablcflow rate of the liqnid (ir spm) rhat would .es r in a prcsnre drop that is 50% ofrhat at which floodingwotrld occur. (b) The hoBcpowcr that would bc rcquircdfor thc blower ro move the sas r h r r . s h r h ec o L u n rinf r h eb l o * e r i . 8 0 0 or l t . i ( f l . A packedabsorptiontower is nsed !o removeSOr from an air stream by absorplionin a solvent.Thetoweris 5 fl in diameterand60 ft high and contains 1.5 in. plastic Pallrings. The lemperalue ard !.essxrein rhe tower are 90"F atrd l0 psig.The gasstrean flow rate is 6500sclm.The liquidsSG is 1.25.and i c o .r ) ( 2 ' c P . " (a) Whdi is the liqnid flow rale (in 8pn) at which lhe colnmn will flood? (b) lf the cohmn op€ratesat a liquid flow ralc tliat is 75% of lhe flooding !al!e. wha! is lhe rotal pressured.op llroLrghlhe lower in psi? l0_ Apacked absorytioncolnnnremovesan imtnrity from a eassirean byconlacl with a liqnid solvent.The columnis 3 f1in djamelerand contains25 ft ofNo. 2 plasllcSuperInlalox packing.Tl1esas has aDMW of28, enlersthe colxmtrat 120'F,and leavesal l0 psis at a rate of 5000scfn. The liquid hasan SG of 1.15 and a liscosily of0.8 cP. Delermnre: (x) The flow rate of rhe liquid in spm that would be 50% of thc flow ratc at which the colunn would flood. (b) The pressuredrop thrcugh the columD.in psr. (c) The horsepowerofthe blowerreqniredto nove rhe gasthroBghlhe colunrn if il is 60% eincierr.
130Chemi@lEnsineerins Prccesses
NOTATION A
arca,IL'?l
4
particle surfac€ area/per unit volme, [1/L]
, d Dh q
diameter, [L] parficle diueter, |Ll hy&aulic dianeler, [L] €n€rsy .lissipated pe; ;nit mass of flnid, tF LIM : I,: fitl
G
sasmassnu{, IM/r'zrl
r Jiu z ar,ar, tr' n xRe,pM P O r , We ,.t, z
permeability,[L'1] porousmediafriclioDfactor,Eq. (131li. t l lensth.[L], liquid nals flrx tM/L1l massofsolidstMl ioradonrate,rpm.l/tl nunber of ffanes. t l porousmed;aReynolds-nunber, Eq. (13-13), [-] pressure.tF/r = M/Lt'l volumetricflow rate, IL3u conpressibility parameter, Eq. (13-44), [-] time, [i] wettedperimeter,[L] coordina!€ dnections, [L]
o o I p 1//
porosily or void rraction. [l polenlial (: P+ psz). F /t: : M/Lt'l viscosity, [M/Lt] densily, tM/Lrl sphericilyfactor,I l
^()
(), O,
Subscripts 1,2.3 f i s
referencepoinls filrer frame side iftenntial superficial
pofousmedia 131 Flowthrough
NOMENCLATURE in M, L, T,0
B
Ti]l,] cross{€ctionEl ma of bed d colnm c@fficien! in equarion4,49 Pmeabiliry coefficiert (eguation,l.2) Constant in eguarion4,28 (15 for sphericalparricle9
c Specific hea! a! co.stant pre$ure Cefficien! in equarion4,49
D DL DR
Axial dispe4ioncoemcient Radial dGpe6ion oefficieni
di
Dimerer of rube or column Equivalent dianeter of po€ slee = e/ss a usedby Noninal p@ting size (e.g. dianeter for a Raschig ring) Fncrional voidage of bed of parricles or packing
L2 Lt L2
-. J4
L2T-
0-1
^ L2TI LzT-\ L L L L L
i ...
Wall coftcrion faclor (equdion 4.23)
kgs
G
MTI "t-
Cas nds flowra€ undd noodingconditions
Nf- 2T I a2
Accelerdtion due to g vity Liqnid nold-upin b€d,volune ofliquid pe.uni!
r KO
Heat imnsfer coefficient M6s tnnstlr coefficient j f4tor for nas tEnsf€r j-factor for heat trmsrer Constmt in ffop equations4.1 and 4,2 Dimensionle$ @nsunt id equation4,6 Koredy consimi in equatiod 4-9 Shapefactor in equation 4.22 Consistencycoefiiciont for poweFhw nuid
t4r
i' .
30 t
TM-rr;T
MLT J'-' jT' 2 t/f--
132 ChemicalEngjneeringPro@sses I t/ t! a; ,,, I l' l, N z a/ LP -LP,t LP,,, gr
Liquid nass nownte Liquid massvelocity vohneric liquid rate Volun€tncliqnid rat€per unit area weftins rare(,2/s!) Lengtbof bed or heightol collnn packing Le.gth of now pdsaAe $rcush bed Length of cncuhr tubc Nunb€r of velocily headslost thrcuet unit heigbr of bed (equation4.47) Floq behaviour index for nower'law-nnid Expone.rin equation4.17
R Rr
Pressuredrop acrcss bed or colunn P F . . u c d r o pa c . , . b @ o i L l ) p l c u n S Pressuredrop ucro$ bed of wel packing Fractio. of liquid collect€d at dnb.ce r fton centE line of peking in equation4.49 Drag force per unit ma of tube Fall Drag force per unit surfaceaea of panicles
S S, S"
Surface rea per u.it volume of panicle or packing Surface arca per unii volune of bed Gpecific surlace) Surface ea of coniaine. per unit volune of b€d
!.
Avenge fluid velocity basedo. .ross sectionallEa A of Meanvelocityof duid in tube Volumetic Rowrateot gas p€r unit dea of cr$s s4tion Volumedc noqrlte of liquid per unit trm of crcss secrion Metu veloci.y in pore channel Voltrne of duid flowins in tine r Side of cube Coef6cient in equation .r. | 7 Coefficiem i. equation 4 17 ,1.37,4:19 Coenicientof D in equarions
i
MTI ML2Tr L3T t LT-' tlT t L L L L-i
"^t
v'
t"_,t. MLrT-2 ML-IT'2 ML rT 2 Mr--rT 2 Lo-rT-2 L Lr L-' Lr T lrI ltT I Lrr
ff1
T]T I L:) L L ML-rT-r Tr ML-'T- I ML-I ML rT 2
A"', R4MR (Rcr),, s. s/ S, Suf6x
Densiqrconmtionfetor J(p6lp/) C6hof nunbe. (parlicle) (Volume l, Chapter 9) Nusclt nDnberror tube wall (44lr) Nuselt nnnber (panicle)(illt) P*teI ntnber (ucrtleDL) at @ctkDRJ Pfmdtl .nmber (crrlt) Reynordsnunber for flow ths\*, tnbe (u,r/ tL) Modifed Reynoldsnunber basedon pore sizeas usedby Cman (equadon4.13) Modifed Reynoldsnunbe. basedon paniclesize(!.zlplr.) Metznerand ReedReynoldsnunber (equton 4,27) Reynoldsnumhorfor powe.law fluid in a granularb€d (equation4.28) Schnidt nunber (p/pD) shesoo.l nunber (parlicle)(nDzrlD) Sruron number(pznitlet ttt/Crput, A refeA LoJir JL2Sl K G @rarsio gds t rcfe4 ro liquid u refes to water a! 293 K 0 refeB to stmdard condilions
