Chapter -10-Humidification and Cooling Towers-Appendices

8/13/2019 Chapter -10-Humidification and Cooling Towers-Appendices

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378Chemical ngineering.o@sses

Simultaneous heat ard mass transfer also occurs in exoihermic or endo-themic heterogeneousrcacting systems alrd in the absorption or adsorptionirom concentfated asstreams. hese opics aie addressedn separatetlus-trations but we retain ihe air-$rater system as the central theme of this

10.1 The Air-Water System: Humidi{ icat ion andDehumidif i cat ion, Evaporat ive Cool ing'10, '1.1TheWet"Bulb empratureWe stafi oul deliberations y examinint the events har occurwhena flowinggascomes n contactwith a liquid surface.Th readerwill be aware rompersonalexperienc hai this process esulis n a drop in the remperatuieofthe liquid, often feferred io as evaporaiive cooljng. The chill we experiencewhen v\mo blowso\er oJr pFr.p.rhg bod,e. . one rd"r i reslat;o; f rh]:

Let us assume that both the water and the air are inirially at the san1etemperature.During the first sta8eoI evaporatior he eneryy re+lired forthe process,.e., he laient heat AH", wiu come rom the liquid itself, whichconsequentlyxpeiencesa drop in temperatule. Thar decline, once it istriggered,h'ilt causea correspondingamount of heaftransfer to rake placefrom the air to the water At this iniermediate stage, he latent heai ofvaporizaiion s provided bofi by the iquid iiself and by heat transfer romme warmer gas.As the liquid temperaiure continues io drop, the rate of heat rransferacceleratesntil a stage s reachedwhere the entire energy oad is suppliedby the air itseu.A steacly tate s attaind n which the rate of evaporationis exacuybalancedby the rate at which hai is rransfefed from the sas rol} le iq, d. Ihe l iqL.d . r l -"n:drd ro oFar rs wer-oulb Fn-pe-crrurejh,.and the corespondh8 air temperature s referrd o as the "dry-bulb tem-peratur," ?db. he difiercnc ?db ?*b)constituteshe drivins force or rheheatbeLnSrnrslerredrom Inegar o lhe rcu'd.Thrs - indr,"red Lr igure10.1,wl1ichalso shows the associated umidities of the air,y*bGg Hro/kgair), the saturation humidity prevailint at the surface of the liquid, and ydb,the humidity in the bulk ail The wet-bulb temperature and its associatedsaturation humidity play a central role in humidilication and dehumidiJica-tio4 in waier cooling operafions,as wel as n1drying pfocesses. heseartaken rp in subsequent illustraiions.The relation among 7;b, Ywb, nd the systemparameterss established yequalmSha dieor he. l l r ,m>fe-Jr om".. {o w,rrero t}e r. le of evnpordhor,i.e., the mte at which moisture is transferred from the water surface to theair. Thus,


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379 HumidifielionndCoo'ing oweF

EffectiveGasFilm

Humldity

Tcr

krM,

FICURE, IO-1Tcmperaturca.d humidiry dist.iburionarcund a warcr drop dposcd ro a flowing airstrem

Rat of evapomtion = Rate of heat hansferftr4(y*b - ydb)^H. = tu4(Tdb T*b) (10.i)

where fty s the mass bansfer coefficient in units of kg H,O/m, s dy.CanLeling erms dnd real.ranSing e obtain(10.2)

00.3)

where the difference 7,6 - T*b is referred ro as rh wet-bulb depression.Wnote uomEqudfionl0.)hdt hphumidiryof rheair vooian, n prin-cipl, be stablished ftom masured vatues of Tdb,T;b, and y_b" the tarterbeinq obtained from the relation

n = (rJ..)", v"."*",, -(P*)." M.-

where Prio is the vapor pressure of water, available Irom tables, and M =molar maas. ?db s measured by exposinS a dry themomerer to th ftowinsair, while T*u is obtdined n sinr;ldr tdsh;onusinq a LhemometercovereJwith a moisLwick. More recenlde\i(es for medsuringVrely on c}anges nelchical properlies of the sensor elenlent v,/ith the moisture content ;f air.Both wet- and dry-bulb properties appear in the hulnidity charrs that arctaken uD sho.tlv.


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380 ChemicalEngneeihg Proesses

'10.1.2The Adiabatic SaturationTernperature nd the PsychrometricRatioBefore addressing the properties and construction of the hlrmidity charts,we consider a small vadation on the simple contact of water wftlt flowingair, which led to the wet-bulb conditions. In this modiJied arranqement,shown n Figure10.2. stredm f nir rshumidif ied n cont.ctwirh c;nstan(lyrecirculated water. Both the water and the exiting gassh.eamattain adiabaticsaiuration temperature, 4, which is lower thaJ| the dry-bulb temperaturcol lheenlering rr becau5ef evapof.f i \ cool int.ff care is takn to inhoduc the make-up water at the same adiabaticsatumtion tempenture, and the datum temperatu is set at ?.,, a simpleenergy balance wilt yield

Rate of nergy in - Rate of energy out = 0lc,(rdb ?:,)+ ydb^4,1 tq(4, - 4,) + Y"r'H,l= 0

which on rearrangement leads to the expressron(10.4)

4" v,,= qE, 4, ^H,, (10.s)where C" is the specific heat of th ai , also temed humid heaL in units ofkIlkg dry air. Plots of ftis equation appear in the humidity charrs discussedin the nexi i]Iustration-The striking similadty between the adiabatic saturation and wet-bulb rcla-tions, Equation10.s and Equation 10.2, ed to a detailed exaininarion of themtio of the two slopes, l/ft{", also known as &e psychrometic ratio. Thesestudies culminated in ihe finding ftai for the watepaft system, and onlyt'orttdi sysfer,, its value is approrimately uniqa Thus,

Psychromehic Ratio ft/kyq = 1

Flcl lRr l0.2llow sheet howinS he attaimnt ol adiabatic ahration conditio.s.

(10.6)

OuiletGas


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Ihis exprcssion, known as ihe Lewis relaiiorL when used to compareIquation 10.2and Equationl0.s, eads o the conclusion hat the adiabaticsatumtion arld wet-bulb temperatures are essentially identical. The l-ewisrelation has other importarlt implications as rveli, as will become apparentin Ilustration 10.4dealing wiih the desiSnof watr-cooling o 'ers. t is seenthere that $e underlying model equations can be enormously simplified bymaking use of the T-ewis elatlon.10.1.3 The Hunidity Chartthe psydlrometric or humidiiy chadsarc dispiayed n FiguJe10.3 nd Figure10.4 or the low and high temperatur ranges shown. To familiarize ourselveswith the propefiies of these diagrams, we stari by defining and deriving asei of variables, which appear implicitly or expliciily in the two figures.AbsabteH midity YThis quantity was aheady referrd to in connection with the wt-butb tm-peraiureand is redefinedhere or convenience:

381 HumldlncationndCooling oweE

YlksH-o ksdn rr)= lq PH'") o p - n

0.70 t72 0.74 0,76 078 0.80Dry-Brtb mpefatuf"C

FICURE0,3Huniiditychafr:ow remperahreange. F.o CanierCorporarioD.i& temission,)

(10.7)

'6rc


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382ChemilEngineeringrocesses

350 360 370 J30 390 400

50 60 70 30 90 T00 110Dry-BubTempemtureC

:^>'Eo.o5I6

FIGURE 0.4Hrtuidity charr high temperarure range. (Ircn Carir Corporarion. With permission,)

where pH:o is the partial pressure of water vapory appears as the dght-hand side ordinate jn the humidity charrs.ReldtioeHmliditlt RHTo obtain a sense of the relative degree of saturation o{ th air, we defhe


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383 Humidiflion ndCooling owers

E"RH i \oo (10.8.)where P4o is the saturationvapor pressureof water at [he tempemturRtI varies over the mnge 0% (dry air) to 100% (tully saturated air) andappears in the humidity charts as a set of parametric cuNes fiat risesmoothly from left to righi.

Deu Point TdpThis is the temperature at which moist air, cooled at constant Pr and Y,becomessaturated, .e., attains 100%,elative humidity. Its value is estab-lished by moving from the initial delining point of a givm air-water mixtureon the humidity chad along a horizontai line to tle eventual intersctionwith the curve or 100% elative humidiryHunlid Vol ne \'/HThe hmid volme oI moist air is the specific volurr-Ie in m3lkg dry airmeasued at Proi= 101.3 Pa (1 atm) and L\e temperature7 of the mixture.Values of yH appear in the humidity chats as a set of lines of ngative slope.Humid Heat C,This quantity, which has already been encountered in connection with theadiabalicsaturation emperature, s the specifichatof moist air expressedin units of kj/kg dry airHumid hat doesnot usually appear expiicitly in the cha.tsbut is con-taind in the nthalpies shown there. It can be calculated from the folowingequation:

(10.9)EnthalpVHWith the humid heat in hand, we are in a position to Iormulat the enthalpyoI an air water mixture- With ?'lchosen as the datum temperaturc for bothcomponentsand addint sensibleand latent heatswe obtain

q(kjlkg dry air)= 1.00s 1.88Y

H(kllkc dry air) = q(? T1)+ YAiJ,ISensible Latent

heat heatwhere the datum temperature is usually set equal to 01C forwater and dry air

(10.10)

both liquid


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384ChemilEnsineeing roesses

Values of the enthalpies of various air-water mixtures are read from theleft-hand oblique ordinate.AdiabaticSaturatlan empenture $ and Wet-Bulb tmperaturedrPlots of the adiabatic saturation line, Equationlo.2b, appear in the humiditychads as lines xtending Iaom the abscissa o th 100% rclative humiditycurve. The point oI intersection with that curve defines the wet-bulb tem-peratur Twb,which is also the adiabatic saturation temperaturc 4,.Example 4.1This concrete example iltustrates the various uses to which the hurniditychats may be putr We choos moist aii with a relative humidity of 25% anda (dry-bulb) tempemtur of 50]c and proceed to calculate various propertiesof interest using the chait shown in Figurc 10.4.

Absol te HunliditV y. This value is read Irom the right side rcctangularordinate, which yields

DewPoint TdD.We tollow the hodzontal line through the point Y = 0.0195,T = 50lC io its inte$ection with the 100%rlative hurniditv curvand obtain

Y = 0.0195 g H,O/kg dry air (10.4a)

(10.4c)

(10.4b)This corresponds to the temperature at which, on isoba c cooling of themoist air, the firsi condensation f water occurs.WetBulb tetnperature ",r.Here the procedure s to follow the adiabaticsaturation line to its intersection wi& the 100% relative hu]nidity curve. We

Note ihat the wet-bulb temperature is not identical to the dew po;nt.WaterPartialPrcssurcpts.o. his quantity canbe obtaineddirctly fromihe absolut humidity aJ|d Xquation 9.3a. Solving it for p .o yields

' " r ' l 8 +2 929x 0.0195x 01.3

l 8 + 2 9 x 0 . 0 1 9 5= 2.09 kPa (10.4d)


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385 HumidifcaljonndCooling oweE

Humid Valnne VH. The plots for yr are shown as steep ines of negativeslope.The poinr y = 0.1095, = 50 C is locatdbei\aeen he linesfor VH= 0.90and 0.95.Linear nteryolationyields ihe valueVjJ= 0.94sm3lkg dry air (10.4e)

Enthaw H. This value is read from the oblique left-har|d ordinate oIFigure 10.4ar|d comes o

Water Remo?aLS|uppoie the air mixture considered herc is to be cooledand dehumidiJied to T = 15lC and RH = 20%. The water to beremoved can tien be calculatedas follows'

H = 103 Jlkg dry air

())h"r = 0.0021

(10.4f)

(10.49)Y)ntur= 01095Water o be rcmoved:

(1')'.,r"r (4rr",' = 0 0i9s - 0.0021 0.0174 g H,o/kg air (10ah)Altemativelt the result may be expressed in volumetdc units bydividing by the humid volume oI the original mixt'lre.0.0174/0.945= 0.0184 kc H,o/m3 initial miature.

Watet RemooalHeat Load. In addition to ihe arnount of rr'ater to bercmoved, an impofiant parameter in the dsign of a dehumidifica-tion unit is d1eassociated eat oad. That quantity carlbe computedfrom the relevant enihalpies read from the humidity chart. Wehavefor ihe casecited(H)i"ld"l= 103 kj,/kg dry air, (I4[."r = 20 3 k]/kg dry air {10.4i)

ueat removed = (F,);u"l- (H)r,,.r 103 20.3= 82.7kl/kg dty an (10.4j)Altematively, llsing volumetric units

Heat removed = [(I,i",u - $I)t^]/v s = 8z.z 0.9as= 87.5kl / n3 (10.4k)


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386Chemicalnginee.ingrocsses

'10.2CoolingTowersCooling towers alrd spraypondsare devices or the cooling of water by transler ofheat, rom hot recirculated ooling water fiom f'actoryplant facilities.Tle c.w.gains sensjbleheat lrom various heai exchangeG;by exposing the sudace of hotwaler o inconing / sunounding ir whichcools he waler by padly evaporatinghehot water so as to rerrove the latent heat of water and alsoa part of sensibleheat ofhot water is removed. This is due to difference in tenperanre between hot waterand ncoming/suroundingair to the cooling owersystem r spraypond.The otalheat hus removed o the air from the hot recirculatedrater n the cooling ower,orsplay pond is the sum of latent heat of evaporation and sensible heat. The airtemperaturemust be lower than the hot water temperature.The cooling of water incooling ower or spraypond akesplaceuptoand below he wet bulb temp.ofair inwamest summermonths.Cooling owershave wo typesofair flow: crossflow ndcounterflow. In crossflow towers, the air moves hodzontally across he downwardflow of water. n counterflow owers,dre af movesvertically upwardagainst hedownward all ofthe watef.Therearenany typesandsizes fcool;ng towersl(r) Induced raft mechanicalower.(ii) Forceddraftmechanicalower.(ttt)Parabolic himney ypecooling ower.(i, Sprayponds or smaller oolingdul'-Out oftwo rypesof mechanical raft cooling owersonly Induceddrafl tower withfan at the top of tower s widely usedbecausehere s no chance f sho circuit ofmojsthot humid air, as ound n forceddrafl ower,with fan at the bottomoftower.rI.D. cooling owerscanbegroupedn a row ol severalowersdepending n coolingduty and water ate,per unit crosssection,s 2- 3 times more than hat of naturaldraft towers. In I.D. cooling fower lllaxinum approach o WB. temperatureseconomicallypossible.Running cosl of nechanical draft C.T. is h;gher due topowercost offans. Hot watef s pui into he sump at the op of tower and t flows

into C.T. d ough nozzles t the op deck.Splash ar ills inside he owergives heincreased rea of contactof water with the air. Wind velociry below 4 kr r issuitable or LD. tower; coolingapproach, pto 5 "F. of W. B. temperature fair ispossibleduring summer months for designpuryose. .D. cooling towers havedouble entry of air through lowerc ai two sides and ail and water flo\rs countercunently. In mecha.ical draft cooling towers,hot water from plant facilities sdirectly aken o the op and lows through wo flappervalves nto top deckhavingplaslicnozzles or flow ofhot water nto the basin.Coolingwater evel n top deckis kept ai 3 -5"height.The concentradon f compormds ccLming n circulatingwaier systens hai cancause caling r corrosion fequipmentmustbe controlled t a desirableevel.Thisconcentralionevel, developed n eachsystem,s basedon the qualityof makeupwater and the water reatingchemicals sed o conhol corrosionor scaling,Theconcentralions usually epoted asconcenhation ycLes nd efers o thenumberoftimes he compoundsn the nakeup waterareconcentratedn the biowdownwater-For example,flhe concentrationn ihe makeupwaterwere 125ng,&g and the


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387HumidificationndCoolingTowef

concentration of the blowdown were 500, the concentration cycles would be500/125or 4 cycles.The conpoundsareconcentratedy the lossof \rater throughevaporalion nd windage.The evaporationoss n a cooling tower is calculatediiom the ratio of specific heat o d]e heatofvaporation. The specific heatof water is4.186kJ(kg .'C) and he heatof vapomtions 2326 kl/ltg. The ratio4.186/23260.0018/'C indicates hat 0.18olo vaporadon ccurs or every degreeof coolingtaking placeacross he tower.MechanicalDraft TowersFans are used o move the air through the mechanicaldraft tower. The pedomanceof the tower has a grealer stabiliry because t is affected by fewer psychrometricvariables. The fans provide a means of regulating the air flow. Mechanical drafttowen are characterizedas either forced draft or induceddmft.Forceddraft owers(Fig. 10-5) The an is located n theair stream ntering hetower. This tower is characterizedby ligh air entrance velocities a|ld low exitvelocities,herefore,he towe6 aresusceptibleo recirculationhus havinga lowerpedormance tability.The aoscan alsobe subjecto icingunderconditions flo\i,ambient enperature and high humidity.Inducedclraft owers(Fig. 10-6) The fan s located n the air streameaving hetower.This causes ;r exit velocitieswhich are hree o four timeshigher han heirair entrance elociiies.This improves he heatdispersion nd rcduces he potentialfor recirculation.nduceddraft owers equireaboutone kw of input for every 18000 m3A ofair.3Coii shed towers (Fig. 10-8) This applicationexists n many older coolingtowers.The ahnospheric ojls or sections re located n the basin of the coolingtower.The sections recooledby flooding he surface fthe coils with cold water.Reasons or discontinued use were scalingprcblems,poor temperaturecontrol, andconshuction costs.Th;s rype tower can exist both as mechan;calof naturaldmft.

Fig.(10.5)Mechanicalorced draft counterrlow ower

l*l


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388Chemicalngineeringrocesses

Fig.('10.6)echanicalnduced raftCounterflowowe..

Fig.{l0.7)Mechanicalnduceddraft counre.flow ower

Fig. 0- Mechanicatdraftoitshed ower


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389Flumidifkationnd Coo nETower

NaturalDraftCoolingTowersThese are hyperbolic tall, usually concrete, owers with hot water sprayingarangement at the top through wh;ch water falls thrcugh the enpty tower withoutany splash bars. Air enters at lhe bottom ofthe lower al1d lows upward by the &aftproduceddue to difference oftemperatue between op water zone and bottom zoneofcold water. Operation and maintenancecost is low but drift loss ftom iop is moreand cooling approach s lower. Wind velocity, higher than 4 Km,ftr is suitable butprecipitationoss s morc. Fjrst cost s higher. n naturaldmft cooling ower hotwater pump, at the bottom of the C.T., boosts he pressure or sending hot water tothe top sprayen due to higher head requiredto lift the hot water. This increasesheoperalion cost. Cold water pump usually recirculates the cold water to the plant.Thermometersare used for measudng Hot and Cold tvater tempetature and pitottube is used for measurementof cold \i,ater flow. Hot and cold water headerpresstuemeasulementle usuallyprovidedAtmospheric pray owe.s(Tig. 10-9) Cooling owe$ of this typearc dependentuponatuospheric onditions. o mechanical evices reused o move he air.Theyare used when small sizes are required and when low performancecanbe olerated.Hyperbolic natural draft towers (-F;9. 10-10) - These towers are extremelydependable nd predictablen their thermaiperfomance.A chimneyor stack sused o induce air movement through the tower.

t , t { f

Fig. 10-9 Atnosphericspray ower


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390ChemjcaEngineerinerocesses

Fig. 10-10 Hyperbolic aturaldraft ower10.3. SpraypondHerehotwaters sprayedhrough series fspraynozzlesn coldwater ondandtemperature f water comesdown due to sensible eat hansfer and evaporationwhich removeshe atentheat.The heatdury s lessandoperafionalost s the east.Coldwater s recirculatedo theplant facilities tom spraypond. n generalor alltypes of cooling rvater systems mergency ooling water circulationptmps withstand y emergencyower, reusuallyequ;redo run hepunps/criticalquipmentonpowerailure.A pit pumpprovisionor draining f coldwaterbasjns usuallykept aswell as coldwateroutletstrainersor cold wateroutlet o cold waterpump,areprovided.10.4.DesigncriteriaCooling approach: t ;s the difference between cold water and wet bulbtemperaturef water.A minimum f5'F cooling pproachu ng summer onthsis possible.Dew point temperature: enperatureat which air-water mixtue issaturated ith water vapour.Cooling range: t is the differencebetween ot watertemperature nd cold water emperature.Wet bulb depression:t is the differencebetween ry bulb andwet bulb emperature.he followingparametersre requiredfor designingcooling ystem.(DQuantityof water o be cooled,m3/hr.(i,)Wetanddry bulbdesignemps. election.(ij,)Air velocityhroughhe owers.m/sec.(ir) Maximum ndminimum rybulb emperature.(1,)Hot waterandcold water emps equired.(1,i)Height ftower,m and ower ill volume,m'.(yi, psychrometrichart f air atatmosphericressure.


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391ChemilEngineerlrg rocesses

10.2: Operation of a Watet-Cooling TowerAs previously mentioned, warm process water that was used in a plant forcooling purposescan be restored o its original temperatureby contactingit with an airsiream, which causes t to undergo evaporative cooling. Theoperation is generally carded out in cooling towers containing stacked pack-ings of large size and voidage to minimize pressure drop. We prcpose hercto model the operation oI such a tower and, in the course of the modeldevelopment, introduce the reader to some ingenious simplifications basedon the Lwis relation (Equation10.2c).As in all packed-columnoperations, he fundamental model equationsconsist of differential balances taken over each phase, the principal noveltyhere s ttie simultaneoususe of massand energybalar'ces.Th pedinent variables and the differential elments arornd which thebalancesare taken are d;splayed n Figure10.5a.Water Balanceooet Gts Phase kg H,OhnI s)This balaJ|ces no different from similar massbalances sed n packed-gasabsorbers nd distillation columns and iakes he fonn

Rateof water vapor in - Rateof water vapor oui = 0

- [c.v...]=o (10.11)

c. - r. "(vt vr

Ic""1.I1.""-lwhich upon iniroduction of the auxiliary mass transfer mte equation, divi-sion by Az, and letting Az J 0 yields tlte usual form of ODE applicable to

(10.12)

wherc'\^ - Y is the humidiry dnving force.

W er Balallceoxer Water PhaseThis balarlc s omitted since he watr lossesare usually less han 1%.


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392 HumidificationndCooling owers

Gas PhaseEnergy Balance kl/m'1 )Here we must be carcful io include both sensibleheat transfer as well as thelatent heat brought inio the air by the water vapor. We obtain

Rate of enerty in - Rate of energy out = 0

feal.c...1,L+^4N""eI lc ,q l=o (10.13which, after applying t]rc same procedure as before, yields

I_aU]

a.Column ariablesWalerTem eratueTL

b. OperatingDiagram

Here T. and Tc are the water and air tempemtures, rcspectivel, and Hequals the enthalpy of the moist air at a Siven point in the tower.Liquid PhaseEnergy Balance E/111'1)A completelyanalogousderivation to the gas?haseenergy balanceyields

FICURE 10,5 Vdiables and oPelahng diagid for a Packed cooling tower

c"ff-uo1r,-r.1- t,Kya(Y\=a

LCp,# -u,(r,-r.) - ,Ii,Kya(v-v) =o

(10.14)

slope (LcLlG)mn

(10.1s)


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393Chemlcalngineeringrocesses

wherewe havereplaced he iquid enthalpyHl by Cp1 T1- ?r). Themodel is completed by adding the relevant equilibdum relation,which coincides wiih the 100%RH curve in the humidity charts andis also available in analytical form. Thus, for the equrlibrium relationv. =f(r) (10.16)

Equation 10.12, Equation10.14,Equation 10 15, and Equation 1016with the previously given expression for H (Equation 10 10)of five equations in the five state variablesY, y*, Tc, TL' and HAlthoueh a nume{ical solution of these equations is today easily accom-olished,elrlv workers in the field had to castabout for altemativemeansof solving the mode1.To do this, they used the ingenious devhe of introduc-inq the L;wis ielation into the gas-phaseenergybataJrce,which has the effectof-combining Tc and v into a single variable, the air enthalPy H We sketchLheDrocedure elow,Lrsinr,nlert"Lial aluPs r Pla"euf yt d].tdTl lo dc'om-moiaLe the 6lin (oeffi.ienis4 and /r, u,ed in the LFwr- relahon We u'e ihdtrelation to replace ft by ftyc, and obiain in the first instance

(10.17)where, as seenfrom Xquation 10-10, he bracketed terans( ) represeni enthal-pies of air-water mixtures.We thereforecanwrite

L,dH -k o l , . c l ^J l - ) ) Lc . l . AHv) | 0' d z

G . = - k , a l H , - H ) = 0 (10.18)where H, H can be considercd an enthalpy driving force, which rePlacesand combines the temPerature and humidity diiving folces in the orisinal

We now assume hat the two-fiIm theory can be aPPlied o this system,wiih the resuit that Equation10.18can be cast n L\e torm

c.*-x, ,@.-n)=o (10.1e)where Kp is now the ovemll mass hansfei coeflicient ar'd H* the gas enthalPyin eoldibrium wii i rhebul l water emperdiurPTh'is equation is of ihe same fornis gas-phasedifferential balancesencounteied in gas absorPtion and distillation, so ihat the design proceduresusec{ here can 6e replicated, provided an aPPropriate oPemting iine can beconstructed.That line is obtained rom an overall two-phase ntegral heatbalance aid takes the form


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G"@1-A = LCL(ru rL)for paft of the column, aJtd or the efltne towr

Gs@1- H) = LCLQI rr2)The gas-phaseenergy balance (Equation10.19)anintgated to yield the familiar HTU-NTU relation:

394 HumidifietionndCooling owers

(10.20)

(10.2i)in tum be Jomully

c tE" , )H

H. =I{,)

(10.221

The model is completed wift the addition oi the equilib uln rlation:(10.23)

which is constructd from &e 100% RH culve oI the psychrometric charts.The original st of five equations, tlree o{ which are ODEs, have thus beenreduced to the tlue relations(Equation10.21, Equation 10.22 and Eq.10.23). ,^lhat is mor, th set is now cast in the {amiliar form of an HTL}-NTU expressior joined to an opemting line and equitbnum relation. Thegraphical procedure used to solve this much simpler set is oudined in Fituie10.5b and {ollows the usual routine oI drawing an operating line, this one ofslope LCllG., ftrough the point (Hr, ?a), aJld evaluating the NTU integralusing the erthalpy ddving force read from the operating diagram. Note thatit is now GMh, not lMr., which consponds to an irJinitely hith towex


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ChemicalEngineering rocesses95

NotationA mass ansfer areaa interfacial areaCs specifrc eatofairCr- specilrcheatofwaterC* specificheatofwet airGs mass low: rateof vaporwaterH, latentheatofvaporizationHTU heightoftranslerunith heat transfercoefftcientk mass ransfer oefhcientL nass flow rate iquid waterM molecularweightNTU number of transler unitT temperaturep" vaporpressure f waterP1 total pressurepartialpressure f water'relat;vehumidityhumidityheightofcolurrn

Subscriptas adiabatic atumtiondb dry bulbG gasL liquidwb wet bulb -

pRHYZ

-tmt/m'kJ,&gdry air KlJlkg dry air Kkl&g wet ir Kkg/skJ/kgm1(9m- .s YL(9skg&molemKatmatmatmkg wet air /kg dry airnt


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396 Humidifielionndcoolinq owers

ReferencesR.E. Treybal. Mass TttflsferOryations . Mccraw-Hill, New York, 1952.Seealso 3rd edition, 1979.F.M:Whi,te. HeatanAMas, Tn$/r. Addison Wesley, Readil& MA, 1988.S. Middleman. lnffi ctio to Mass ana Hut rnnsftr. lohr.Wiley, NewYotk, 1997.E.L. cussler. Dillltsion: Illars Tn sjbt in Fluid sJsisus. 2nd ed., Cam-bridge University Press,New York, 197.H.D.Betu. Heatand MassTra,ger (translaied fromthe cerman). Spring-er, New York, 1998.A.F. MilIs. Btlsic leat and Mass Transfd. 2nd ed., Prentice Hall, Engle-wood Cliffs, Nj, 1999.J.D. Sader and EJ. Henley. Sepfratio4 Plocess Pnnciples. loh\ Wtley,New York, 2000.


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AppendixAMATHEMATICALRELIMINARIES

A.1 CYLINDBICALNDSPHERICAL OORDINATEYSTEMSFor cylindrical oordinates,he variables/, d.z) ate related o the rectangularoordinates(r, ),,.) as ollows: t*

0 = arcIan(, x)

0


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398Chemical ngineedngrocesses

. . - - . t _ . - 1 . .

ux is in the /-directionux is in the 9-directionux is in thed-direction

FlgureA.1 Thecyl'ndical ndspheical oodinalo ysioms.

IR ' )s ;n0dedOt":,:::"' (A.1-e)

(A.2-1)

A.2 MEANVALUETHEOREMff /(r) is continuous n tho intwal a < r < ,, then he valueofthe integrationof/(r) overm intewal r :4 to ,r : , is

, =l.' rcto': ct l"u ,: 1y11uo1where(/) is the average alueof / in the interval a ( .r ( b.


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(4.2-2)

Flgure .2. Themean alue l lhe unction(J).ln ilgure A.l nore rnarJ, J (.r) d.I $ rne lrea under he culve between } and ,. on theotherhand, /)(D a) is the areaunder he rectangle f height /) andwidth(, - a). Theaverage alue of /, (/), is de ned such hat these wo areas rc equalto eachother.It ispossibleo extend he de nition ofthe meanvalue o two- and hree-dimensional asesas

| | r,,, r> "o,\ f): --- 'r- and (/):JJdxdyI I I f rx 'Y,ztdxaYaz

llla'a*,


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400ChemiqlEnsineedngrccesses

A.3 SLOPESONLOG-LOG NDSEMI-LOG RAPHPAPERA mathematicalransformationhat converts he logarithmof a number o a length in ther-direction sgivenby

where r is the distancen the r-directionandLr is thecycle ength or the r-coordinale.Therefore,fthe cycle engths takenas 10 cm, he distancesn the -direction or vanousvalues fr aregiven n TableA.1.Theslope fa straightine,m, on og-loggraph apers

(A.3-1)

los11,r loelr /(," - ( , , \ L-m: -' '' :| --::--------: l--:-f o g1 2 l o g r \ t , , ( ^ t l L , (4.3-2)

I t t r - t t t \ 1\ x 2 - i c l , / L r

ft1""a'o*:l."ao'

* l,'i,'c,oo,=l"ou'uja'*lwo.,lff t'ro,t1ff

On heother and, heslope fa straightine,m, on semi-log raph aper )-axis s ogarith-mic) slogl'2 logyl (A.3-3)

A.4 LEIBNITZ'S ULEFORDIFFERENTIATIONF NTEGRALSLet (,r, t) becontinuousndhave continuouserivative //at in a domain fthe t plane,which ncludeshe ectangle (.x ( b,4 < I < t2.Then orrl < r < 12

(A.4-1)ln otherwords, differentiationand nte$ation can be nterchangedfthe limits ofthe inlga-tionare x ed,On heother and, fthe limitsofthe integraln Eq. A.4-1)aredependentn ime, hen

(4.4-2)

rr 0.00 3.01 4.77 6.02 7_78 8.45 9.03 9-54TableA.l. Dislancesn he -dictionJora ooartlhmic-axis


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4.5 Nuneical Ditferentiation t ExDeinental DataIf /: /(i) onty, henEq. A.4-2) educeso

f , ta^: lart l f f f larr\ (A.4-3)

(A.s-l)

A.5 NUMEBICAL IFFERENTIATIONFEXPERIIlIENTALATAThedetemination fa mte cquireshe differentiationfthe odginalexperimentalata.Asexplained y De Nevers 1966),givena tableofr ) data, he valueof d)/dr can becalculatedy:

l. Plotting he dataon graphpaper, rawing smooth urve ilrough hepointswith thehelpofa French u e, and hendrawing angento this curve.2. Fitting heentire etofdatawith anenpiricalequation,uch sapolynomial, nd hendifferentiatingheempirical quation.3. Fittingshol1 ections fthe data y using rbit raryunctions.4. Using he differenceablemethod,.e.,plotting he diff'erencesnd smoothinghemgraphically.DeNevers lsopointsout he act hatalthoughhevalueofl),/lr obtained y anyoftheaboveour methodss approxinately qual o each ther.hevalue f12y/,"1.x2s ertremelysensitiveo themethod sed.In thecase fthe graphicalmethod,herearc an n nite number f waysof drawing hecurve hrough he datapoints.As a result, he slopeof the angentwill be affected y thenechanics f dlawiru he curvedine and he angent.Theavailability fcomputer rograms akeshesecond nd hirdmethods eryattractive.Howevetsince he choice fthe functionalormolthe equations highlyarbitrary,he nalresults almost ssubjectivendbiased s hatoblained singa French urve.Two melhods, amely he Douglass-Avakian1933)andWhitakeFPigford1960)meth-ods,arewodhnentioningaspartofthe third apploach. othmethodsequirehe values fthe ndependeDtariable, ,be equall|spacedby namountAi.

A.5.1 Douglass-Avakian ethodIn thismethod,he valueof dy/dx is detemlined y tting a fourth-degreeolynomialoseven onsecutiveatapoints,with hepoint n questioDs hemid-point, y least quares.fthe nid-points designatedy 1., then hevalueofd)/d,v at hisparticularocations givenby

d) J9 t \ .L, t ) r | - 4e\LX'J )dx I512Lr


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A.dgtlEmRlffPjtrE',tu{'EffiBSdIn this case, parabolas tted o ve consecutiveatapoints,with thepoint n question sthemid-point, y least quares.hevalueofd)/dr ati. isgivenbydr_Lxrdx 104.r (A.5-3)whereX is dened by Eq. A.5-2).ExampleA.1 Given he enthalpy f steam t P :0.01 MPa asa functionoftemperatureas ollows, etermineheheatcapacity t constantressuret 500"C.

Tcc) (t e)

THfc) Q/e)100200300400500600

2687.52879.53076.53279.63489.13705.4

'700 3928.7800 4159.0900 4396.41000 4640.0I 100 4891.2

500 700rcc)

SolutionInenealcapacltll consEmtressufe.P. sde ned s i t / d )p. l herelore.etermrna-tion of Cp requiresnumericaldifferentiationof the F1versushe f data.Graphical methodTheplotof l/ versls is given n the gure belowTheslope fthe tangento thecurve tI:500.C gives r = 2.t2 J/e.K.

4200

A 38oo

3000

t00 900 1100


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A.5 NumencalDifte@nliatbn t E\perinenlal DalaDouglass-AvakianmethodThevaluesequiredo useEq. A.5-1)aregiven n the ablebelow:

Xy200300400500600'700800

28',79.530'76.53279.63489.13'105.43928.74 l59 .0

-8638.s-3279.63705.4'7E57.412,47',7

77,746.5-24,612-3279.63705.4

31,429.6112,293

-3-2- l

0I23

I = s968.7 L = 4r,789.9Therefore,he heat apacity t constantressue t 500'C isgivenby

^ 397 r tX r r -49 { IX ry l { J97 r {5c08 .7 )4q ) r41 .789 .9 ,^ , . , . , ,(P - l 5 l 2a r - ( l 5D r \ r co r =z r r J /g^

Whitakr-Pigfo.d methodBy takingX : f and ' : A theparametersn Eq. A.5-3)aregiven n he ollowing able:

Xy300400500600700

3076.53279.63489.13',705.43928.'7

-2- l

l2

-6153-3279.63'.705.47857.4

L:2130.2Therefore,he useofEq. (A.5-3)givesheheat apacity t constantressuresIXt 2130.2I 0A i (10 ) (100 )The difference table methodTheuseofthe dilferenceablemethods explainedn detail y Churchill l974).To smooththe databy using his method,he divideddiJferencesI1/AI shown n the ablebelowareplotted eNus emperahren the gwe.


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404chemical ngineeing rccessesAT AH LH /^r100

200300400500600700800900

10001100

268',7.s2879.53076.53279.63489.13705.43928.74159.04396.44640.04891.2

100100100r00100100100100100100

t92t97203.r209.5216.3

230.3237.4243.6251,2

|.92t.9'12.03L2.0952.t62

2.3032.3742.4362.5t2

loo 3oo t*t""rtt ton tt t

Each ine representshe avera4e alueof r|fr ld.f over he specied tempemtuteange.Thesmoothcurve shouldbe dmwn so as o equalize he arcaunder hegroupofba6. From thegure, heheatcapacity tconstantressue t500 C is 2.15Vg.K.


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4.6 Begrcssionand CoretationAFDedixA 405A.6 BEGRESSIONNDCORRELATION

Toprcdicthemechanismfa process, e oftenneedoknow he elationshipfoneprocessvariabieoanother,.e.,how hereactoryieldepelds npressllle. relationshipetweenhetwovariables and , measuredvera range fvalues, anbeobtained y prcposinginearrclationshipsst, becauseheyare hesimplest. heanalyses e use orthisarecol/eldtor,which ndicates hetherhere s indeed linear elationship,nd cgzss;or,which nds theequation fa straightine hatbest ts heobserved-]) data.A.6.1 SimpleLinearRegressionTheequation esc binga straightire is

(A.6-1)where.rdenotesheslope fthe line and denoteshe)-axis ntercept. ostofthe time hevariablesx and do nothave linear clationship. owever,ransfomation fthe variablesmay esult n a linear elat ionship.ome xamplesftansformation regjven n TableA.2.Thus, inear egressionanbeapplied ven o nonlinear ata.4.6.2 Sumol SouaredDeviationsSuppose ehavea setof obseNations1, rz,13, .., . ,v.The sumof thesquares ftheirdeviationstom somemean alue,.ru,s

s: l{ri ,.)2i = lNowsuppose e wish o minimize t with respecto themean alue ru, .e.,

(4.6-2)

^ s : ^ .:u : ) _z\x i - x^ ) :2 (Nx^

- I \ - " .

-i.,')- / (A.6-3)(A.6-4)

of the deviations s thelerefore, he meanvalue hat minimizeshe sumof the squaresarithmetic ean,.TableA.2.Transiormaln oi nonline equationso inearorms

l I .1 o g ) : a l o g x + l o g a

I rsr h l ineoI ,, I :, rin"*


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406Chemicalngineeing rocessesA.6.3 The Methodot LeastSquaresTheparameters andD n Eq. (A.6 l) areestimatedy the method f leastsquares.hesevalueshave o be chosensuch hat the sumof the squares f the deviations

S : L l y t - (ax i+b \ l '

I -n- , -- - - _ - L t y i _ a r i - b t ^ iA C\",-o - -2Dry, - o,, t,

EquationsA.6-6)and A.6-7)canb simplified s" \ - .? . r \ - , . - \ - " . " / - ' ' / 1 ^ ' - z ) ^ t ) ti i i

- \ - " . . r v r - \ - , , ." L n t | , . v L - r .Simulraneousolution fEqs. A.6-8) nd A.6-9) ives

- N(I;x;y,) - (I ixi)(I;y,)u - -------::-t=---Nr f , r r i . r f ,x fl l iri rLr: | - (f, :r i ){L ri) i )Ntf ,x f - t l ,x ; r2

(A.6-5)is minimum.This is accomplished y differentiating he unction S with rcspect o a and ,,andsetting hesederivatives qual o zero:

(4..66)(A.6-7)

(A.6-8)(A.6-9)

(A.6-10)(A.6-11)

Example A.2 Experimentalmeasurcmentsf the densityof benzene aporat 563 K aresivenas ollows:

(atm)30.6431 .6032.6033.8935.1'l36.6338.39

V(cm"/mor)116410671013956900842771

(atn)40.044 t .7943.5945.4847.0748.07

'7m646591506443386

Vlcm-/morl


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4.6 Regrcssiq an.t ConetatbnAssumehat hedata beyhevirialequationf state,.e-,

PVBC- l * - r - -RTVi2anddeterminehevirial coef cientsB andC.SolutionTheequatior of statecanbe reananged s

Note thatthis equationhas he form

/ P V \ - Cl_ 1 lv:B+_-\ / r / V

l P V \ - l)= l - - r )v and r : -. /v

Y=B+Cx

T:rking : 82.06 rr3.atm/mol ) the cquired alues rccalculateds ollows:,i r i x 103 xt l i I x 106-265.4-288.3-288.9-285.6-283.4-279.9

-273.8-268.5-261.4-254-243.1

0.859o.9370.9871.0461 . l l tL1881.2971 .4141 .5481.692t.9'762.2572.591

0.2280,0.2'702-0.28520.2987-0.3149-0.3324-0.3593-0.3873-0.4151-0.4424-0 .50 r9-0.548'7-0.5984

0.7380.8780.9'751.0941.2351 .4111.6822.0012.3962.8633.9065.0966 .712

I ) i= 3500 .3 I . { r :0 .0189 I1 , } r : -4 .qS3 l Ix ,2 :30 .99^ t0 6Thevalues f E andC arc

. r l , r ; r r l ,x , i r t l , . r1r ( l ; r ;y ;ri ,xl) _ (l;;i),_ (_1500 .3 J0 .qqx0 6 l_ {o .0 l 8q ) r_4 .q831 r 1 r? " - l / - ^ rI ] , , 30 .qs l 0 u ) - (0 .0189 )2


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408Chemicalnqineedlq rocesses- N ( l ; r , y1 ) - ( l r r , ) ( l r y ; )N(Lixi) - (Lixi) '( r 3 ) t -4 .q831) (0 .018s ) ( -3500 .1 )^^ , ^^: i U . l - 2 { C m / m o l , -( l l ) ( l 0 .qe l0 -o ) (0 .018e)

The methodofleast squares analsobe applied o higher orderpolynomials.For example,consider econd-orderol) omialy=u rz+bx+c ( ,4 .6 -12 )

To nd the constants , ,, and c, the sum ofthe squared eviations

s-LIv ' -1"^ l + ^, , t l ' (A6-13)mustbeminimum.Hence,

as as a5a": , ,b : a , :o (A6-14)Partial ifferentiationfEq. (A.6-13)gives

, l , l +af" l +. r i - l j t , (A.6-i5)o f " r . \ - . 2 ' \ - - \ - " ' t 6 -16)L i v 1 , . i , \ 1 , ^ ' _ 1 2 ^ t ) t

" l ' l +t ln+,N:lrt (A.6-r)These quations ay henbesolvedor theconstants, ,, andc.Ifthe equations ofthe form

y:ar(" +b (A.6-18)lhen he ammeters?. . and canbedelermineds ollons:

L Least quaresalues fa andD canbe ound or a se esofchosen alues fr.2. Thesumofthe squares fthe deviationsan henbecalculatedndplotted ersus tond theminimumand,hence,hebest alueofn. Thecorrespondingalues fd and,are eadily oundby plotting hecalculatedalues ersus and nterpolating.Altematively, q. ,4.6-18)might rst bearranged s

log(y- r) = n ogr + logd (A.6-19)and he east quaresalues f r?and ogd aredeterminedor a sedes f chosen alues fr,etc.


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4.6 BegrcsslonndConelalionAppedixA 409

Example A,3 I It is proposedo correlate he data or forcedconvectionheat transfr o aspheren terms fthe eqDationNu= 2 +aRe,

The ollowingvalueswereobtainedrom McAdams 1954) or heat ransfer romairspheresy forced onvection:

SolutionTheequation an be rearranged s

Iog(Nu-2) , ' ogRe+logdNote that this equationhas he fonn

to

y : n x I b

y : loS(Nu 2) j = logRe b:loga'?

0.096910.633471.23045-0.09691

r.266943.69t35I23

I49Lri : 1 ,77Ol t r , ) , =4 .86138 Lrl :u

Thevalues f, and, are (3)(4.86138)(6X1.7670r):0.66368(3)(14) (6)2(14)(r.76701)(6)(4.86138)

(3)(14) (O, : -0.73835 + a =0.1827A.6.4 CorrelationCoetficientIf two variables, and , are elatedn such way hat hepoints fa scattr lot end o fallin a straightine, henwe say hat here s a[ associationetweenhevariables nd hat heyare inearly orrelated.hemostcomrronmeasurefthe strength ftheassociationetweenthevariabless hePearson orrelation oefrcient,. It is de nedby

F * . \ - , , .1 , \ l t - - (A.6-20)(r': $)(r,i-qd)

' r nrs roorems aKenom Lnurcr l ( ry l41.


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410Chemicalngineeringro@ssesThevalueofr can ange rom -l to +1. A valueof -1 means petfectnegatiyeorrela-l iot.Aperfectnegativecorrelationimpliesthat :at+bwherca 0. Whenr : 0, the vadablesare un-corelated. This, however,doesnot imply that the variablesarcunielated. t simply indicatesthat ifa relationshipexists, hen t is not lineat

A.7 THEROOTOF AN EQUATIONIn elgineering roblems, e frequently ncounterquationsfthe form

l(.r):0 (A.7-l)andwant o determinehevalues fr satisfying q. A.7-1). hese alues recalled hen otrof/(r) andmayberealor imaginary. incemaginaryootsappeai scomplex onjugates,the number fimaginary ootsmustalways eeven.fie function (r) maybeapolynomialn or tmaybea ranscendentalquationnvolv-ing rigonometricnd/or ogadthmicerms.A.7.1 Roois ol a PolynomialIf /(-r) is apolynomial,henDescartes'uleof rigndetermineshemaximum umber frealroots:

. Themaximum umber f realpositiveoots s equal o the number f signchangesn/ (x) : o '. Thema-ximumumber f realnegativeoots s equal o thenumber fsign changesn/(-x) = o.In applyinghesign ule,zerocoef cients re egardedspositive.A.7.l.l Q adraticequation The ootsofa quadraticquation

(4.7-2)

(A.7-3)arcgrven s

-b+ ' r ' -4a;2a

If4, ,, andc are ealand f A = 12- 4dc sthediscriminant,hen. A > 0; the rootsarc eal and unequal,. A = 0; the ootsare ealandequal,. A < 0: the ootsarecomplerconjugale.


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4.7 The Raot al an Eguation4.7.1.2 Cubicequatioa Considerhecubicequation

x3+px2+qx+r :oLet us de ne the tems M and N as

, " rq ' p '

, , 9pq 27r 2p ti 4If p,q,andr arereal ndf A=M3+N2 is hediscriminanthen

. A > 0l one oot srcal nd wocompler onjugale.. A = 0; all roots are realand at least wo are equal,

. A < 0; all rootsare ealandunequal.Case i) Solutionsor A > 0In this case, he rootsaregivenby

l\ :S+T- ;p111 -x2 : ) (S+T \ - 1p + l iVSrS- I )l l t -. r r - - (S+ I ) - - a - - i a /3 (S r ) 2 3 ' 2

(4.7-4)

where

(A.7-s)(4.7 6)

(4.7 7)(A.7-8)(4.'7-e)

(4.7-12)(A.7-13)(A.7-14)

(A.7-10)(A .7 -11)

Case ii) Solutionsor A < 0The rootsarcgivenby

/ A \ l_r +2/ Mcos|\rJlP/ A \ ir r :+2/ Mcos( ;+ r20 ') - 1e/ B \ lxr +2V-Mcos(i 240") - i p,=*""o"\m6 is n degrees) (A.7-15)


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412Chemical ngineeringrccessesln Eqs. A.7-12)-(,{.7-14)he upper ignappliesf N is positive, nd he owersignappliesif 1V s negative.ExampleA.4 Cubicequationsfstateare requently sed n thermodynamicso describethePyZ behavior fliquidsandvapors. hese quationsreexpressedn the orm

- RT a(r)V-b Va+fv+ywherehe ermso, P, /, andd(f) fordilTerentypesofequations fstatearegivenby

Eqn.ofState

(A.7-16)

a T)van der Waals 2 0 0 aRed l i ch -Kwong2D0o / "8Peng-Robinson 2 2b -b2 aQ)WhenEq. A.7-16)has hree eal oots, he argest nd hesmallestoots orrespondo themolarvolumes fthe vaporand iquidphases,espectively.he ntermediateoot hasnophysicalmeaning.Predictthe ensity fsaturated ethanolvaport 10.84 tn and140'C using hevanderWaals quation fstate.Thecoefcientsa and, aregivenas

a:9.3424 m6.atm/kmolr and , : 0.0658m3/kmolTheexperimentalalueofthe density fsaturated ethanol apor s 0.01216 lcm3.SolutionFor he vanderWaals quation fstate,Eq. A.7-16)akes he orm

i t _(o*Rr\ i ,*Li _ _o\ P / P PSubstitutionfthe values fa, r, n, andP intoEq. 1) gives

t - r - J . 92 . l 72 0 .86 r87 0 .0567 :oAppiication f the sign ule indicateshat he maximum umberequal o three.Theterms M and N are,, 3q p' ( l)(0.8618) (1.1923)2, , ,__=_ :_0 .84s99

a)of realpositive oots s

(1 )

(3)= 0.775

(4)9pq - 27 - 2p3 _ (9) -3. t923\0.861) (27 (- 0.0s67) (2) 3.1923)3

Thediscriminant, , isa: Mr + N2: (-0.845)3 (0.775)2: 0.003

54

(5)


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A7 TheBaolal an EquahonAppedixA 413Therefore,ll the ootsofEq. (2)are ealandunequal. efore alculatinghe ootsby usingEqs. A.7- 2)-(A.7 l4), d mustbe determined.romEq. ,47- 5)rerr* ^-"arccosV(oJ45p r'6)'

Hence,he cotsarel, : rrr"ro*"".(T) *'.t?2 t no, (7)

/ ? . 8 5 \ i . l 9 2 jv2 : (2 )J0 .84scos{= 1201+- :0 .109 (8 )\ J . / J/ t 8 5 \ J . l 9 2 l

v , - r2 rJ0 .845cos l- : -+2 ,0 ) . , - - -0 .181 re )\ J , / JThemolar olume fsaturatedapor, r, conespondso he argestoot, .e.,2.902m3/kmol.Sincehemolecular eight,M, ofrnethanols32, hedensity fsaturatedapor, s, sgivenby

32 :0 .01103 /cmr (10)(2.902)(l 103)

(6)

Mo':E=A.7.2 NumericalMethodsNumericalmetbods houldbeusedwhen heequationso besolved recomplex nddo nothave irectanalytical olutions. arious uner'icalmethods ave eendevelopedor solvingEq. A.7-l). Sorne fthe mostconvenientechniqueso solve hemical ngineeringroblemsaresummadzedy Serghidesl982),GjumbirandOlujic 1984), ndTao 1988).Oneofthe most mpodant roblemsn theapplication fnumerical echniquess conyl,Sence.Itcan epromotedy nding agood tafting alue nd,/or suitableransformationfthevadable r theequation.Whenusingnumericalnethods,t is alwaysmportanto useengineeringommon eDse.The ollowingadvice ivenby Tao 1989) hould lways e renremberedn theapplicationof numericalechniques:

. Toerr sdigiLal.o atch he rrorsdi ne.

. An ounce ftheory s worth100 b ofcomputer utput.. Numericalmethods re ikepoliticalcandidates:hey'll ell youanything ou wanttoneat4,7.2.1 Newton-Rdphsonmethod heNet,,ton-Rdphsonethods one fthe mostwidelyusedechniqueso solveanequation fthe orm (.r) : 0.Tt s based n heexpansionfthefunction(x) by Taylor eries round nestimater I as

f (x) : f (xk r l+ (* r i* r )91ax l tL t ,( x - xk i 2d2 f I- L \ r , , , , - (4.7-11)


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414ChemilEnqineeringroessesIf we neglecthe derivativesigher han he rst orderand etr :.r* be he valueofr thatmakes(r) = 0, thenEq. A.7-17) ecomesf(xt r)x t :1 r , 1- - i iT;1.,, (A.7-rE)

with ,t > 0.Itemtionsstart with an nitial estimate d and he rcquirednumberof iterations o get.xt isdependent n the following error control methods:. Absolute rror control: Convergences achieved hen

lxk xk-r < ewheree is a smallpositivenumberdetermined y the desiredaccuracy.. Relative error controlr Convergences achievedwhen

(A.7-r9)

l*;.1"*'""where

u,= I toz-'with n being he number f co.rectdigits.The result,digits.A graphical cpresentation f the Newton-Raphsonthatthe slopeof the angentdmwn o the curveat ri l

"rop":,uno : {which s denticalo Eq.(A.7-18).

(4.'1-20)

(4.7-2r)rt, is correct o at least , signi cantmethods shown n FigureA.3. Noteisgivenbyf (rr,-t) (4.7 22)

FigureA.3. TheNewton-Raphsonelhod.


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AoDedix 415The Newton-Raphsonmethodhas wo maindmwbacks: i) the rst dedvativeofthe func-tion is not alwayseasy o evaluate, il) the methodbreaksdgwn f (///d,r),r r : Oatsomepoint. To circumvent hesedisadvantages,he Ist derivativeof the functionat r/. I is ex-pressedy lhecentral ifferencepproximations

df .f (xk 1+ L\ -f xr r - a)

^ = I :l100SubstitutionfEq. '4.7-23)to Eq. A.7-18)eadso

0.0Lxp-1f@p-1)/{1.01-{* ) - /(0.99xr-r)

2L (4.7 23)

(4.725)

t (4.7-24)

with t > 0. The main advantages fEq. (,{.7-25) over the numedcal echniquesprcposedto replaceheNeMon-Raphson ethod,.e., he secantmethod, re: i) it requires nly oninitialguess,.r,,nstead ftwo, (i;) the aleofconvergences faster.


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416Chemi@l ngineeins fosses

A.8 METHODS F NTEGRATIONAnalytical evaluationof a de nite integral

t: J" fE)dx (A.8-1)ispossiblenly or imitedcases. henanalytical valuations mpossible,hen he ollowingtechniquesanbeused o estimatehevalueofthe integral.A.8,1 MeanVaiueTheoremAsstatedn Section .2, f /(r) is continuousn he nte ala


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4.7 The Boat of an EquatianAppedixA 413Therefoie, ll the ootsofEq. (2)are ealandunequal. efore alculatinghe cotsby usingEqs. A.7-12)-(A.7-4), d mustbedetermined.rolnEq. A.7-15)

TN' rc:rtPo-a 'cco 'y 'r , rz t arcco 'y '{0Ja5, .1*a 'Hence,he ootsare

71:rrr",o*"".(T)*"ltt:r.rot (j):ra."t*"""(T*,ro) f :o.'oe (s)

/ tR5 \ r r o )1y ._ {2 ,J0 .845cos |. - -

t -2a0} - : .- -

_ 0 . 8 te )\ J ' / JThemolarvolume fsaturatedapor, ", correspondso he argestoot, .e.,2.902n3/kmol.Sincehemolecular eight,M, ofmethanols 32, hedensity fsatumted apor, s, sgivenby

: 0 .01 03 /cmr

(6)

Vz

32, t :E: (10)(2.902)(l 103)A.7.2 NumericalMethoclsNumericalmethods houldbeusedwhen heequationso besolved recomplex nddo nothave irect nalytical olutions. arious umericaimethods ave een evelopedor solvingEq. A.7-1).Some f the mostcoDvenientechniqueso solve hemical ngineeringroblemsarcsummarizedy Selghides1982), jumbirandOlujic 1984), ndTao 1988).Olteofthe most mpoltant roblemsn theapplicationfnumericalecluriquess cor?yer,gelrce.t canbepromoted y nding agood tarting alue niVor suitable atNformationffhevariable r theequation.Whenusingnumericalnethods,t is alwaysmpoftanto useengiieerjng omnonsense.The bllowingadvice ivenby Tao 1989) hould lways e rememberedn theapplicationof numericalechniques:

. Toen isdigiral.o calchhe rfor sdivine.

. An ounce ftheory s worth 00 Ibofcomputer uFut.. Numericalmethods re ike politicalcandidates:hey'll tell you anything ou want ohear4.7.2.1 Newton-Raphsonethod TheNey)ton-Rdphsonetho.ls one fthemostwidelyusedechniqueso solve nequation fthe orm"/(x) = 0. t is based n heexpansionfthefunction(r) by Taylorseries round nestirnater I as

-f ;r) "f r* r)+ (, - 11-11{iax ln t ,e - xk-t)zd2 I- t dr2-",_,-

(4.7-t1)


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414ChemicalEnsinees PrccessesIfwe neglect hederivativeshigher han the rst order and et r : .Ik be the value of, thatmakes(r) : 0, thenEq.(A.7-17) ecomes

with ,t > 0.Itemtionsstart with an nitial estimate o and he requirednumberof iterationso getri isdependent n the following error contrdl methods:. Absolut rror control: Convergences achieved hen

x k - x k 7 < wherc s is a smallpositivenumberdetemined by the desiredaccuracy.. Relativeerror conholi Convergences achievedwhen

(A.7-1e)

fGr t)x t , = x t _ 1 - - j V ]*1.,,

,o - ro r l r , loo r ") x t lt ^es = - 10'-n

(A.7-18)

(4.7-20)

(4.7 2t)signi antA.3.Note

(4.7-22)

with n being he numberof correct igits.The result, t, is correct o at leastndigits.A graphical epresentation fthe Newton-Raphsonmethod s shown n Figurethat the slopeofthe tangentdmwn o the culveat it-l is givenby

. top . ' r no : { / | - I t x * t taxb r 1 xk 1 -xkwhich s denticalo Eq.(A.7-18).

FigureA.3. TheNewion-Raphsonelhod,


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44/73

418Chemical ngineeringro@sses

FlgureA.4. ValJos t he ulcl'on/( r I al liveqJally pacsd oir s.

The areaofthe trapezoid s then. f/(rr) + /(rz)l(xz rr)Atea: 2-

Ifthis procedures repeated t four equally spacedhtervalsgiven n Figure A.5, the value ofthe ntegral s, t " - l l t a t f ( a L t l l \ t l / t d I A r ) + I ta | 2 | x t l \ xr : J t r t x tax - - 2 t 2

, lf (a+ 2Lx)+ f (a+ 3Ax)lAx, [f (a+ 3Lx)+ f (b)lLxo

l b f f t a t l t b \ 1t : J" t (x \dx :Lx l ;+ Ih- ax\ ) tu12ax\ Ik l r^ . r ) ? l (A.8-8)This esult anbegeneralizeds

(A.8-6)

(A.8-7)

, : |"oro ,=o7lrr,* rf ,, *,o") (r)] (A.8-e)


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A.8 Melhodsot lntegtationwhere

(A.8-10)

(A.8-12)

Simpson's uleThe rapezoidalule ts a straightine Ist-orderpolynomial) etweenhe wopoints.Simp-son's ule,on the otherhand, s a second-orderolynomial etweenhe wo points.n thiscase, hegeneml ormula s

r b t - n 1t= l f$ t . t t :+ / {a)+a )-. t " t . ' - -t 2

I t a + t Ax l+ z L J la+ r Lx ) + J (b \i=2.4,6 (A.8-rl)b -a

Note hat his onnula equireshedivision fthe nterval fintegmtionnto aneven umberofsubdivisions.ExampleA.6 Determine he heat required o increasehe tempemtlre of benzene aporfiom 300K to 1000K at atmosphericressure.heheatcapacity f benzeneapor ariesasa function ftemperatures ollows:

500

SolutionTheamount f heatnecessaryo incrcasehe emperaturefbenzene apor rom300K to1000K under onstant ressures calculatedlom the ormula

.r000d: t f i : Jroo earlhe vaf ial ion l ( p asa lunchon l lernperalures shown n lhe gure below:

Z (K) 300 400 700 800 900 1000ap (callmol.K) | O.eS Ze.ll 32.80 3',7.7441.75 45.06 47.83 50.16

7(X)

cp


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A,A Methads f lntegationTableA,3, Floorsndweightactorsor heGauss-Legendrsuadkiurc(Atamowitz and Srequn,1970)

I2

+0.57735 2691 96260_0000000000000+0.77459 6692 t483

+0.3399804354856:t0.861l33 15 40530.0000000000000+0.5384631015683+0.906178459 8664

1.00000000000000.888888888 88890.555555555 55560.652t451548 25460.347858451 74540.56888888888890.47862 6704 93660.236928850 6189

Example A.7 Evaluater : [ ' 1 d ,J1 x +2

using he ve-point(, = 4) Causs-Legendreuadratureomula andcompare t with theanalyticalolution.SolutionSince :2 and : 1, romEq. A.8-i3)

u+3

(T)', 2u+7The ve-pointquadratures givenby

- 12 1, - 1 ^ar- l . i rs ,1J. f t t , : t rThevalues f uri and (a;) aregiven n the ablebelow:

ui F (ui)u14701234

0.00000000+0.53846931-0.53846931+0.906179850.90617985

0.568888890.4'78628610.478628610.236926890.23692689

0.28571429 0.162539690.26530585 0.126982990.30952418 0.148147150.25296667 0.059934610.32820135 0.0',77',75973L';-[ w F(u;) 0 57536417


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422ChemicalngineeringfocessesTherefore,

1 = (0.5X0.5753641)= 0.287 8209Analytically,

:0.28'768207

4.8.4.2 Gauss-Iaguerrequadrature The Gauss-Laguerreuadmture aobe used o eval-uate nteeralsofthe fonn

wherea is arbitrary and nite.

reduces q. A.8-15)o

I :tn(x + 2)1:=1

I: I e-r f (x)h(Thetransfomation

x : +a

="(i)

r=I"* 'rata':"" lo-

(A.8-15)

(A.8-16)

(A.8-17) " l \ u l d u = e ' > w , f l u ;whereherli andz1aregiven n TableA.4.

Table4.4. Roolsand weighl acloB or the causs-LagueiieuadlaturAbramowiizandSlesun, 970)

Roots "r)0.58578 4376 73.414215623 30.41517556732.29428 3602 96.28994 0829 70.322546896 91 . 7 4 5 7 6l 0 l l 5 84.536622969 19.395079t23 10.26356319781.4r340059173.59642577tAl?.085810058 912.640808442 6

0.853553905 30.14644 6094 70.711090099 90.278s1 7335 90.01038565 020.603151043 20.3574t 6924 80.03888908550.000532947 60.52175610530.3986681r0 30.07594 M96820.003617i86 00.00002 3699,72

Example4.8 Thegammaunction,(r?), s de nedbyl= fo*"'" aaf("


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A.8 Methodsal lntegatbnAppedixA 423wherehevariable in the ntegrandsthedummy ariabte fintegration. stimate(L5)by usingtheGauss-Laguerreuadraturc ith :3.

SolutionS inced :0 , then

p=u and F(u)= ttrThe four-pointquadraturcs givenby

f ( l . t = , / -

ui w1 F (ui \ :1@ w;F(ui\0 0.32254769 0.60315410t 1.745',76110 0.35',7418692 4.53662030 0.038887913 9.39507091 0.00053929

0.s67932821.321272532.t29934343.06s137990.34255I010.47224700.082828690.00165300r(l.5)= ti:; ?,' (,r)= 0.8ee2802

Theexacl alue f f(1.51 s0.88622b9255.4.8,4.3 Gauss-Hermiteuadratwe TheGauss-Hermiteuadratureanbeused o evalu-ate ntegralsofthe form

Theweight factorsand appropriate ootsfor theTableA.5.

(A.8-18)rst few quadmtureormulasaregiven in

TableA.5. Bools ndweightaclorsor heGaussHariile quadiurc AbramowitzndSlegun,1970)

I2

+0.707107811+1.2247487140.000000000+1.650681239t0.5246476233+2_020188705:t0.9585746460.000000000

0.88622 92550.29540 9752l. l8l63590060.08131283540.80491409000.0199i24210.3936132320_945307205

ut ie tap: lu1r6 i1Theval esofui and ( i) aregiven n the ablebelow:


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424ChemicalngineenngroessesA.9 MATRICESA rectangular rray ofelementsor functions s calleda twtrit.lf the arrayhasm rows and ncolumns, t is called an m x ,1matrix and exprcssedn the form

(A.e-r)The numbers r functions ij arecalled he elements f a matrix EquationA.9-1) s alsoexpresseds

(4.92)ill which the subscripts and representhe row and he column ofthe matd)L espectively.A maaix having only one row is calleda row matri-xot row vector), while amatrixhavingonfyonecolumn s caileda columnmatrix or column ector).When henumber frows andthe number of columnsare the same, .e., m : r, the mahix is called a squarematrLxor amatrixof order z.A.9.1 FundamentalAlgebraicOperations

1. Twomatrices : (di7)andB: (bi;) ofthe same rderareequal fand only f dij =bi j .2. If A = (a;;) and B = (rii) have he sameorder, he sum of A and B is de ned as

Example .9 If

A + B: ta,j+ bi j l

A -B= (aij - bij)

(A.e-3)If A, B, andC arematrices fthe same rder, dditions commutativendassociative,

A * B: B *A (A.9-4)A+(B+C)= (A+B)+C (A .9 -5 )

3. If A = (ai;) and B = (Di;) have he sameorder, he difference betweenA and B isde ned as

(A.q-6)

A =

dete rmineA+BandA-8 .t;7t -L] IU5 ilnd B=


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Solution

4. lfA : (drj) and 1.s any number,heprcductofA by .1.s de ned as.lA: At: (taij) (A.9_7)

5. Theproduct ftwo mahicesA andB, AB, is dened only if the number fcolumnsin A is equal o tbe number f rows n B. In this case,he wo matrices resaid o beconformable the otder tated. or example,fA is oforder4 x 2 andB is of order2 x 3, then theproductAB is

In general,fa matdx of order(m,r.) is muttiplied y a matrixof order(/,,?), heproducts a matrixoforder(ra,n). Symbolicallt hismaybexpresseds(n,r) x (r,n) = (n, n) (A.9_9)

ExampleA.10 If

f2+2 -t-4f [4 -5' lA+B=lr+3 0+0 l :14 0 |L3+o s+rl Lt 6lf2-2 r+41 f o i lA-B=lr I 0-0 l : l_2 0lL3 0 5-r l L3 4l

f a" a ' .AB: J zr a2,l l br t bp a3 |I a3t a]1 | D2t b22 tDt Iloo, oo,J

f a1b1 lay2b21 a1b12+ DbD a1 f t3+apbp l_ l o , tb1 a22b21 11b12a22bLt ) tb tJ a22b1tI o l ' b , , - aJ2bz toJ tb .2 aeb22 a1 f t3 a ,2b - | tA Y-o llaa b1. t at2b4 aa b1,7 a42b22 a1b1, ar2b23)

^":Ii ;']fllL-r s lL'rf (1)( l )+()(2) l f -r I=i r2)(r i+(0)r2)l : l 2 IL( 1)(1)+r5){2r lL e l

[ r - ] l Tr " lA: l 2 0 l and B: l , lL- l s l L- . rdetemineAB.Solution


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426ChemielEnqineenngrocesses6. A matrixA canbe multiplied y itselfif andonly f it is a squarematrix.TheproductAA canbe expressedsA2. Ifthe relevant roducts rc dened, themultiplication fmafticess associative,.e.,

A(BC) = (AB)C (A.e-10)anddistdbutive,.e.,

A(B+ C)= AB +AC (A.9-11)(B+C)A=BA+CA (A .9 -12 )

but, n general,not commutative.A,9,2 DelerminantsFor each quarematrixA, it is possibleo associatescalar uantity alled he determinantofA, Al.lfthe mahixA in Eq. A.9-1) s a square atrix, hen hedeterminantfA isgivnby

a t 7 4 1 2 a B . . . a 1 n.l27 422 423 ... A2n (A.e-r3)ant an2 4 , , , ann

Ifthe row andcolumncontaining n element rj in a squarematrixA aredeleted,h de-terminalt ofthe remainingsquarearray s called the minor of aij and denotedby Mil. Thecofactor fdij, denoted y Aij, is thendened by the olationA i j : ( t ) t + r M i i (A.9-14)

Thus, fthe sum of the row and column indicesofan elements even, he cofactor and hsminorofthatelement re dentical; thenviseheydifrer n sign.Thedeterminantfa squarematrixA canbe calculatedy the ollowing ormula:A = ) a i L A , r : > a L , A L ) (A.9-1s)

(A.9-16)

where and may stand or anyrow and column, espectively. hereforc, he determinantsof2 x 2 and3 x 3 matricesare

a l l421.j31

atr at2a2r 422412 Ltr3422 423432 433

= a)p22a + aDaBqt + aBa2ta32- a71a23a32 ar2a2933 aBa22a3r (4.9-17)


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ExampleA.11 DetermineA if

SolutionExpandingn he rst row, .e., : I, gives

f 1 o 1' lA: l 3 2 1 lL r I 0 l

3 l- t 0 +1

at bt cla2 b2 c2a3 b3 ca

32t l

0 b t c t0bzczobca

2110 04.9,2.1 Some ropertiesof determinants

1. Ifall elementsn a rowor colurffrarezero,hedeteminants zerc, .e.,at bt c1a2 b2 c2000 = 0 (A.9-t )

2. Thevalueofa determinants notaltered hen he owsarechangedo columns r thecolumnso rows, .e.,when he owsandcolumns re nterchanged.3. The nterchangefany wocolumns r any wo rotvs fa determinanthangeshesignoffhe determinant.4. If two columns r two rowsofa determinantre dentical,he determinants equaloZEIO.5. lf each lementn anycolumnor row ofa determinants expresseds he sumoftwoquantities,he determinant an be expresseds he sumoftwo deteminantsofthe sameorder.-..

6. Adding he samemultipleof eachofanother row doesnot changehecolumns.element f one ow to the coffespond]nglementvalueofthe determinant.The same s toue or the

a 1 l d y b 1 c 1a2+d2 b2 c2a31d3 b3 ca +

a r b 1 c la2 b2 c2a3 b3 ca

dr bt crd2 b2 c2dt, bt ct (A.e-19)

This esult ollows mmediatelyromPropertiesand5.7. Ifall theelementsn anycolumn r rowaremultiplied y any actor,hedeterminantsmultiplied y that actor,.e.,),ay b1 c1La2 b2 c2).a3 b3 ca (4.9-2r)

a t b t c t ) | ( a t * n b t \ b t c taz b2 c2 l= la2+nb, b2 c2a3 b3 q | ) (at+nb) bt ,t

ar b1 cra2 bz c2a3 b3 ca

(A.9-20)

( l lL)at bt c l(1/X)a2 b2 c2(1/),)a3 u cal,. (A.e-22)


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428Chemicalnsineens PrccessesA.9.3 Typesof MatricesA.9.3.1 The ranspose f a matrix 'lhe matn)^ btained rom A by interchanging ows andcolumns cal ledhe,TdnsposefA and enotedy AT.The ransposefthe productAB is theproduct fthehansposesn the orm

(AB)r= BrAr (A.9-23)L.9.3.2 Unit matrfu The unit matrix I of order ? s the square x n mahix having ones nitsprincipaldiagonal and zercselsewherc,.o.,

(A.9-24)

For anymatrix(4.9-25)

A.9.3.3 Slmmetric and skew'symmetricmatrics A squarematrix A is said to be r)tl-metic if

| | o . . . or=l o I ot. . . . . . . . . .\ 0 0 . . . I

A: -A,r ot a i j = - t :L j i

A: A.T ot aij : ai iA squarematrixA is said o be skew-sJmmetricor antisymmetric)f

(4.9-26)

(4.9-21)Equation4..9-27)mplieshat hediagonal lementsfa skew-symmetricatdxarcallzero-A.9.3.4 Singularmatr,,r A squarcmatrixA for which hedeterminantA ofits elementsiszero s termed rirS&Ialmatrix, f Al+0, thenL is nonsingular.A.,93.5 me in))erse dtr,:' Ifth determinantA ofa squaremahix A does ot vanish,i.e.,a nonsingular atrix, t thenpossessesn rverse or reciprom[)matrixA I suchthat

A A - I : A _ I A = Ifie inverseof a matrix A is de ned by

(A.9-28)

Adj AtAl (A.9-29)where Adj A is called the adjoint of L. It is obtainedrom a squarematrix A by replacingeach lement y its cofactor nd hen nterchangingowsandcolumns.ExampleA,12 Find he nverse fthe matrixA given n Example .l l.

\


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SolutionTheminor of A is givenby

Thecofactormahix is

3- 11- t

a,,: l r' I _.tL -

2I10 3- lI- l

2I01

10[ - r 1 5l=l -1 I 1 ll -2 -2 2)

01l00121

lUI1I3

F - r lIAd jA : | - r l

tilr '-r lo o l- |

The ranspose fthe cofactormatrixgives he adjoint ofA asilsince Al = 4, he useofBq (A.9-29)giveshe nverse f A in the ormr _0.25o.2s _0.51o-' : A91: | -o.rr o..tt o.s IlAt I r . :s _0.2s .5A.9.4 Solutionof Simultaneous lgebraicEquatlonsConsiderhesystem fn non-homogeneouslgebraicquations

a ] : t / ' : ta2x2+ " '+ a lnh :c1aTr r+a22x2+ . . + a2nxn : 2thti ;r+ an2x2+ '+ anflh: cn

in which he coefcientsarj and heconstantsi are ndependentftl't2' ,tn butarcotherwisearbihary.n matrix notation,Eq (A.9-30) s expresseds

(A.9-30)

(A.9-31)

AX=C (A..e-32)


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430ChemicalngimeingProces*sMultiplication ofEq. (A.9-32)by the inverseofthe coef cient matrixA gives

x = A-rc (A.9-33)4,9.4,1 Cramer's rulE Cmmer's ule states lrat, if the determinantofA is not equal ozero, he systemof linear algebmicequations asa solutiongivenby

(A.e-34)whereA and Aj are he determinantsfthe coef cientandsulstitutedmaaices,cspec-fi,rely.Thesubstituted @t/ir, j , is obtained y replacinghe tlr column fA by hecolumnofc's. i.e..

(A.9-35)

lA ; l" ' lA l


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AppendixBFLUXEXPRESSIONSORMASS,MOMENTUM,NDENERGY

Table8,1, Conpo.enlsoi l1e sfess nsor or Newbnian auids n rcctangular oordinales

(c)

(E)

(G)(F)

Table82. Coaponpnls l lhe eress e.sorror Newlonianluids n cylindricaloodinaies

(D

(B)

(D)

' ,.:-rl 'P 1,".",lL d r r _ _ Ll a i ) l. r , : / r ; ; rv. ,1f a L ) lz i ,= l r ; - ; rv .v t

/ iut r i r r ' \. , , : . r ' : l \ t + ; /.",:u,: u(P*P)\ d ? d t )

/ iJo . Ax ' \r . ' : r , ,= / i \tu

+dz /Ju, Au" d ,( v . v ) : -+ r+ -dI nt

, .= ulr ' -1rr."t ]t o r ) ll ^ t t I t u B D r \ 2 - 1' e e : - l t \ ; M + ; ) - 1 ' v ' " t )

' , . : -p lz* lw. , t ll d z r l | | / d \ r , r u , I+a :+ ,= t f ; \ i ) + , n )/ ; t u a . I A r . \r a .= t z a : t r \ t j z r - r )/ h ' t u . \t . r : 1 , . : - p l : + , I\ d r t 7 /

. - l t , 1 1 1 6 . J u .( v . Y t : - - ( r u r l + : + -

(B)

(c)(D)

(G)


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432Cheniel Engineedngrccesses430Chemical ngineeing .o@sses

(A)

(G)

l ^ rq 2 - .1f = - I t t z - - - \ y . 9 |L o r r l. . : -" f r l ]a" * " ) ?,o. l" " L \ ' t e r . / J l- . t | " d u . r F c o r F \ 2 , v ) l, , 0 - / f ' \ . " i " d ad r , r , ) - t , " ,I J /us \ l aa lI , a : 1 6 , : - | I | | u \ ; ) + ; d 0l s i n e a / D b \ ] 1 l r ' lt a o : t , t e : - p l , N \ , : _ e ) t , r ^ e M If I a r r 0 / u d \ ' l' 6 ,= ,6= ul , r , "aaO'a. \ ; / l

,o . r - ' , j ? r " , ) t I ^ ' ^ r , r , i "p l+- l14rz J, rsin4 l0 t sil9 ao

Table 8.3, Componenlsol tho slrsss lensor or Newionian luids in spherical

Table 8.4. Fluxexpressionsor energy ranspod n rectangular oordinales

Flux

aQCPT)rrr: -a ar, aTq t : - * a t

a@epT), r : - "7, nT

a@e T)" A z

Total

vepr) ,

@ePDnt

@ePDxz


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Appedix 433

AppendixB. Flux Expressionsor Mass,Momentun,and EnetwTableB.5.Fluxexpressionsoreneqy Enspod n clindncal oodinatosTotal MolecularFlux Flu Flux Constraint

-aTq ,= k i la@e r)S , = a U

d ab. pr) (pcpT)ta

, 1 . = k ;o@A T)S , : - a U

be prJr,

TableB.6.Fluxexprsssionsoreneryyansporl n sphericloodinatesTotal

. "= -ka fa@e T)S , - - " A ,

be PDq'kar

(pC pT)t6t A e

kaTd a1tpr)

Note(pCpT),tb

Pie = constant


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434Chemical ngineedngocessesAwendix B. FluxExpresstonsor Mass,Momatun, and Enetry

Table 8.7. Flux expressionsor mass ransporl n clngular caordinalesTotal Flux Flux

ie,=-pDeaYJA,: -uAB i;

t t " - - p u e B - i

J A \ :- U A B -

car;t i : co ts :at i ,=-DouY

"erjJi,: .D^BTJA " : - " A B7 l

"etq,:-cDAB+4,= D*a#


61/73

AppedirB 435AppendixB, Flux Exprcssionsor Mass,Monentum,and Enerw

Table 8.8. Flux oxprcssionsor mass nspod in cylindrical ootdinatesToral Flux

WA,

lAz: -P UAB UJ A z : - u a B 0 ,r " = cDaB+J A , : - A R x |

DAB AcA

cAr:r a 1 = c u a B J 1r A . : - u 4 0 7 ; '


62/73

436Chemicalnqineennq@@sses

Tabte 8.9. Flux exprcssionsor mass ransporl n spherical oordinatesTotal MoleculaJFlux Flux Flux

wa.JA,= uaBn;

None

t a 0. Dts opt.tet - - , M

tto- - t siro ad- Des opltaa:- r sne M4, : -cDAs7r A , = u A B i

"nui",ce=- . ae,* DaB |cA

"116" A , : - r i t n o M


63/73

AppendixCPHYSICALPROPERTIES

This appenalix ontainsphysical propertiesof some iequently encounteredmateials in theffanspoftofmomentum,energy,andmass The readershould efer to eitherPery's ChernicalEngineers' andbook1997) r CRCHandbook f ChemisrryndPhysics2001)or a moreextensiveist of physical roperties.TabfeC.1contains iscosities f gases nd iquids,as aken rom Reidet aL. 19'7'7).a'ble C.2 containshermal onductivitiesf gases,iquids,andsolidsWhile gasand iquidthemalconductivitiesrecompiledromReid?t al (1977), olid hermal onductivity al-ues re aken romPerry'sChemical ngineers' andbook1997). hevalues f thediffusioncoeffcients iven rrTableC.3 arecompiledtom F.eld t aL (1977),Pe$ys Chemical ngi-neers'Handbook1997), ndGeankoplis1972).Table C.4 contains he physical propertiesof dry air at standardallosphedc pressure.Thevaluesare taken rom Kaysand Crawford(1980)who obtained he data tom the threevolumes f Touloukian f al. (i 970).Thephysical ropertiesf saturatediquidwatef givenin TabteC.5,are aken rom ncropera ndDewitt (1996)whoadaptedhe data tom Lileya1984).

TableC.1.vscosilres fvarious ubstancesT tr, loaK ks/m.s

Sulfur dioxideLiquids

Ethdol

273373303373.53834233 1 33133 r 3353

0.91.311 . 5 1l . 8 ll _ l t1.231.351.634.923_188.565.348.264.65

3033433 t 3348


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:

438 ChemicalEnqineednqP@cessesAppendixC PhyslcalPropefles

Table C.2. Thormalconduclivilis l varioussubslancsTK w/mK

EthanolSulfurdioideLtquids

EthnnolSolidsBrickGlassFiberSteel

2 7 33',73300473293375273293323293293313

0.u210.03200.01670.02 30.01500.02220.00830_1480.1370.1030.1650_152

0.72398

0.03645

300300300300300

Table C.3. Exoerimenlal aluesot binarydiflusioncoetficients l 101.325kPaTK DAs

Air{O2

Air H2ON2 SO2LiquidsNH: HzOBenzoicacid-H2oCO2 H2OEthdol-H2OSolidsBi PbH2 Nickel

317.23 1 33003 1 3296263288298298283293358298

1.7? l0 51.45 10-50.62x 10 52.88x 10 54.24x 10 51.04 10 5

l;77 x ll-el .2 l x l0 91.92 10-e0.84x lo-e] l x 10-201.16x 0-120.21 l0 Y


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Appendixc. PllysicalPropeiieTable .4. Prope esofairalP:101.325kPaTK kg/.3 epkj/kg K t x l 0 31001502002502632',732752802832852882902932952983003033053083r03 1 33 5320323325330333343350353363373400450500550600650700

3.59852.36131.76901.4t19t.3421r.29301.28361.260',71.24731.23851.2256| .21721.20471.19661.18451.17661.1650i.15731.14601.1386r. t2771.12061.10311.0928r.08611.06961.0600l.o29l1.0085r.00000.9'7240.94630.88250.78440.70600.64180.58830.54310.5043

7.06010.3813.3616.06t6;t0l7.2017.3017.5411.6917.1917.9318.0318.1714.2718.4118.53l8_6418.7418.8818.971 9 . l 119.20t9.4319.5719.6619.8920.o220.4720.8120.9121.342 L 7 122.9424.9326.8228.6430.3031.9333.49

|.9624.385'7.55?1r.3712.4413.3013.4813.921 4 . 1 814.3614.631 4 . 8 115.0815.2715.5415.7516.001 6 . r 916.4',7t6.66r6.95t 7 . t 1t 7 62t1.9r1 8 . 1 018.5918.8919.8920.6320.9121.9523.0126_0031.7837.9944.5651.5058.8066.41

1.028l . 0 l1.006r.003t_003l.004L0041.0041.0041.0041.0041.0041.0041.005r.0051.0051.0051.0051_005r.0051.0051.0061.0061.0061.0061.006L00?1.0081_008r.0081.0091 . 0 1 01.013L020|.0291.0391.0511.063i.075

9.22013.7518.1022.2623.24u.o724.2624.6324.8625.00?5.2225.3',725.6325.7425.9626.1426.3726.4426.7026.852',7.092',7.2227.5427_aO27.9528.3228.5129.2129.7029.8930.583l-2633.0536.3339.5142.6045.6048.405r.30

0.7870.763o.'7430.'1210.7200.11'7o.7t60.7r50.7140.7140.7140. '1140.'720.7130.'720 . 7 | 10.7100.711o . 7 t l0.710o.109o.7w0.7090.7080.7080.'7070.7070.7060.?060.7050.7040.703o.7030.7000.6990.6980.6990.7010.'102

A widely usedvaporpressure orelation over imjted temperature anges s the Antoi er atior exDressedn the fbrm

whereProt s n mmHgand is n degrees elvin.TheAntoine onstants , B, andC, Sivenin TableC.6 or vadous ubstances,re akenrom Reidet dl. (1977).


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440 ChemicalEngineerinsPrcctFrdx c' PrrysicatPropediesTable .5, Prooe ies of saturaiediouidwaler

7 x 1 0 3 Cp l r x 1 0 6 1 x 1 0 32752802852882902932952983003033053083 1 03 1 33 r 5320325330335340345350355360365370373375380385390400

1.0001.000r.0001.0001.0011.0011.00I1.002i .003r.0031.00,11.0051.006r.007r.0081.0091 . 0 1 11 . 0 1 31 . 0 1 61 . 0 i 81.021LOZ41.0271.0301.0341.038L0411.U41.0451.049t.053i.0581.067

2502249'l24'852473246624612454244924472438243024262418241,4240721022391J23782366235423422329231'.72304229122182265225722522239222522122 1 8 3

4.2174.2114.1984 . 1 8 94.1864.1844.1824 . 1 8 14. 804.1794.t784.1784.1784.t7E4.1794.1794.1804.1424 . 1 8 44.1864.1884.1914.1954.1994.2034.2094.2144.2174.2204.2264.2324.2394.256

17501652142212251 1 3 11080100195989285580076972169565463t577528489453420389365343324306289n9274260248237217

5695745825905955986036066 1 06 1 36186206256286326346406+56506566606646686',71674679680681683685686688

12.9912.2210.268-707.95'7.566.946.626.1i5.835.415.204.824.624.324.163.423.152.882.662.452.292.142.021 . 9 1l 80

, t.'761.70t.6l1.531.471.34

0.006110.006970.009900.013870.017030-0191?0.02336o.026t70.031650.035310_04'240o.onp0.05620o_06221o_u3730_081320.10530.13510.1719o.2167o.27l3o.33',720.41630.51000.62W0.'75140.90401.01331 . 0 8 1 51.28691.5233|.7942.455

r - K: p",': bar: = rn3/ksii: ktks; ap : kr/ksK;p = kglm.sr :Wm.K

TableC,6, Anloine quationonslanlsRdge (K) A

ElhaiolMethanol

24t 350280 377405 560260-3702:7V36925',7-364360525

16.65t315.9008l7.163415.973218.911918.587516.1426

2940.462788.514190.702696.793803.983626.553992.01

-35_93-52.36-t25.2-46.16-41.68-34.29-71.29


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Properties f Saturated irl

"c w.-60 0.0000067 0.602?-59 0.0000076 0.605653 0.000003? 0.603457 0.0000100 0611356 0.0000114 0.614155 0.0000129 0.617054 0.000014? 0.619353 00000167 0.6226-52 0.0000rc0 06255,51 0.0000215 0623350 0.00002, 3 06312.19 0.0000275 06340-43 0.0000311 0 6369-4? 0.0000350 0 639?-46 0.0000395 0.6426-45 0.0000445 0.6454-14 0.0000500 0.6433-13 0.0000562 0.51142 0.0000631 0.654041 0.0000?03 0656340 0.0000?93 0.659?-39 0.0000337 0.6625-33 0 0000992 0.665337 0.0001103 0.663236 0.000123? 0.6?1035 0.00013?9 0.673931 0.0001536 0.6767-33 0.0001710 0.ri796-32 0.0001902 0632431 0000213 06353-30 0.000234 06331-29 0.0002602 0 690923 0.0002313 0693327 0.0003193 0.6966-26 0.0003533 0.609525 0.0003905 0.702324 0.0004314 0.7052,3 0.0004?62 0.7030-22 0.0005251 0.?109-21 0.0005?3? 0.7rr720 0 0006373 0.?16519 00007013 0.?194-13 00007711 0.7222r7 0 0003,1?3 0725116 0.0009303 0.?2?9-15 0.0010107 0?30314 00011191 0 .?33613 0.0012262 0?364-12 000r34r5 0?393lr 0.0014690 0.?421-10 0.0016062 0.7.1509 0.00r?55r 0.?4?3+ 0.00r9m6 0.?50?-? 0.0020916 0.7535{ 0.0022311 0 7s635 0.0024362 0.7592-.1 0 002?031 0.76203 0.0029430 0.76.192 0.00320?4 0.7677-1 0.0034374 0.7705

0.0000 0.602? -.60.351 0.017 1i0.3340.0000 0.ri056 -59.344 503260.0000 0.6034 53.333 0.021 3r?0.0000 06113 57.332 0.024 -5?.3030.0000 0.6u1 0.0230.0000 0.6170 5r3r9 0.0310.0000 0.6191 54.313 0.036 54.2140.0000 0.6227 53.307 0.041 -53.26?00000 0.6255 52.301 0.0,1 12.25500000 51.295 0.0520.0000 0$r2 50.230 0.059 502300.0000 0.6341 49233 0.067 -49.2160 0000 0.63119 -43.27? 4.075 ,13.2420.0000 0.6393 -17.2?1 0.035 17.1360.0000 0.6,126 .16.25 0095 -.161700.0000 0.6, 55 15.259 0 103 -45.1510.0001 r).84a3 14.253 A-r2r0.0001 a.Esr2 43.247 0.137 l3.t1r00001 0.6540 ,12.2.11 0.153 .12.03300001 0.6569 1.235 0.1?2 -41.0630.6591 40.225 0.192 -40.03?0.6626 -39224 0.216 1000?0.0001 0.6654 13213 02,11 l?.9760.0001 0.6633 31.212 0.270 16.9420.0001 06?12 36.206 0.302 35,9050.0001 0 6?40 -35,200 0.336 -3.1.3640.0002 06?69 -3.1195 0.375 -33.3200 0002 0 6793 33.13900002 0.6326 -32.133 0.46.00002 0.6355 31.173 0.517 30.66r00003 30.171 0.571 29.5910.0003 29.166 0.636 2:.5290.0003 24.$0 a-107 27 4540.0004 0.69?0 27 r34 a ft2 26.3720.0001 0.6s99 26.149 0.367 25.13200004 0.7023 25.113 0.959 -24.13,10.0003 0.7a51 -24.137 1.059 23.0?300005 0.7036 23132 r.1?1 21.9610 0006 0.7rr5 221260.0007 0.?1.14 21120 11250.000? 0.?1?3 r.570 13.545a.aaaa 0.72a2 r9.r09 1.?290.0009 0.7%r r3.r03 1.9020.0010 0?261 17.093 2.092 -15.0060.001 0.?290 16092 2.299 -13.?930 0012 0.?320 ,r5.aa6 2.3u 72.5620.0013 0.?349 14.030 2?69 113110.001,1 0?379 13.075 3.036 10.03S0.0016 0.?409 12.069 3.327 4.7420.001? 0.?, 39 11.063 3642 -7.42r0.7469 -10.057 3.936 6.0?20.0021 0.7490 -9.052 4.353 -1.6930.0023 0.7530 -a.0,r6 4.?64 J.2330.0025 ?.040 5.20200023 0.7t91 5.6?7 4.3570.0030 0.?622 -5.029 1.1640.0033 0.7653 6.751 2,1240.0036 0.?63t 7.353 ,1.3360.0039 0.7?17 -2011 5.9950.0043 0.771S r.006 3.?12 7.106

nepnnbd by lcrDnsion or tne Anedm socieiy ol H.aii4, Rer.i6ctuiins rd Ai.Condiijonilg Ensinees, Airmh, GA , trcn t\.199t As'lR.tE Eaa.lbnh Fn larearats


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442ChemicalEnqines ProcessesPropertiesot SaturatedAirl

0.1129 1.0226

0.2179 1.1533

0 0.003739 0 7734I 0.0040?6 0.77622 0.00431 0.7?913 0.004?07 0.73195 0.005.124 0 ?3?66 0005313 0.?90,1? 0.00623? 0.?9333 0.006633 0.796rI 0.00?157 0.799010 0.007661 0.301311 0.003197 0.304612 0.003766 0.307513 0.0093?0 0.310314 0.010012 0.313215 0.010692 0.3160

16 0.01413 0.313317 0.0r2r?3 0.32r?13 0012939 0.324519 0.01334 0.32?420 0.014753 0.330221 0.015721 0.333022 0.0167,11 0.335923 0.017321 0.333?24 0.013S63 0.3.11626 0.O24r70 0.4414

29 0.025?35 0.355330 0.02?329 0.353631 0.029014 0.361433 0.03267,1 0,36?r34 0.034660 0.3?0035 0.036?56 0.372336 0.033971 0.8?563? 0.041309 0.3?3533 0.0,137?3 0,331340 0.049141 0.337041 0.052049 0.3393.13 0.053365 0.395514 0.061791 0.393315 0.065411 0901246 0.069239 0.904047 00?3232 0.906943 0.0?7556 0.909?49 00320?? 0.912550 0.036353 0.915451 0.09r9r3 0.913252 0.097272 A.92r153 0.10:9.13 0.92395.1 0. 0r954 0.926?55 0.rr ,l 0 92965E O.122An A9%157 0.1292,13 0.935353 0.136351 0.933159 01, 4942 0.9409

9..173 9.4731.006 10197 11.2032.012 10.970 r2.9A23.013 rr.?931.021 12.6125.029 13.6106.036 14.603 20.644?.041 15.6?1 22.7t33.047 16.305 2A.A529.053 13.010 27 A6410.069 19.293 29.35211.065 20.653 31.121D.A?r 22.rAA 34.17913.477 23.619 36.726t. .034 25.236 39.3?015.090 2?.02316.096 23.36? 4,1.96317.102 34.424 11.92613.103 32900 5r.00319114 35.r0r 54.2163?.43,1 57.5552r.r27 39.903 61.03542.527 64.66045.301 63.,14024.146 43.239 ?t.33525.15326.159 54.633 30?9327.165 53.120 35.23524.{2 61.304 39.97629 19 65.699 94.3?330.135 69320 100.00631.192 74.117 105.36932.193 73.?30 110.97933652 116.35?33.?99 123.01135.219 9.r.$6 129.45536.226 99.983 136.2093?.233 106.053 143.29033.23S rr2.114 r5A.1B39.246 rr9.253 153.50410.253 126.430 166.63313.1.005 175.26542.263 t42.AM j4.27643.215 r50.4t5 193.7,144.242 r59.4r7 203.69945.249 rff 3?4 214.16,r46.296 114aa2 225.17947.304 139.455 236.7594311 200.64,1 243,955,19.319 212.435 261.30350.326 225.019 275.34551.33,1 233.290 239.62452.341 252.344 304.$253.349 261.247 32A.59654.357 233.031 337.33355.365 299.772 366.137511.3?3 3r7.5,1S 373.92257.331 336.41? 393.?9353.339 356.461 41435059.337 377.?33 43?.135

Repnnbd by pemnsion of the mencm Socieiy ofHstidg, Ref.igcrating dd Aircondiiionins E4inoerq Aila.ia, GA , froD ihe 1993 ,saR{, Edrdbtuk-Fu dM alr


69/73

ProDerties t SaiuraledAir'

R4rinted by pernnEion df the Aseri@r Socieq of Heating, Refrige.aiin8 and Air'CondtiodnsEnsnreri Ailan 3.GA. Flnm he 199s,{,gti,-{,


70/73

444Chemicl ngineeringro@sse3

AppendixDCONSTANTSNDCONVERSIONACTORS

PHYSICAL ONSTANTS' :82.05 cm3.atm/mol= 0.08205m3.atm/kmol.K: 1.987 allmol.K

- 8 .1 l 4 /mo luasconslanT/< =g.Jl4\ t0 I l*pa.ml7mol.K:8.314 x 1O-5 ar.m3/mol.K= 8.314 10 2bar'.371'-olK:8.314 x 10-6MPa.m3/mol.K= q.8oo7m/s2Acceferatfonl gmvrty8) = 12.t.740ls2

. - . . : 5 .67051 to -8w /m2 .K4stetan-Ijot man consmnto) - 0 .1713 10 B tu /h t 'R

CONVERSIONACTORSI ke/ml- I0 I gTcml l0 I lgTLDensiw , - ' .- ' ,I kg/m' :0.06243 bld

Diffusivity I m2/s: lOacm2/s(Kinematic, ass,Therm 1) | m2 s: 10.7639N / s= 3.87 x I 04 t2 h1J=1Ws: lN rn=10 -3kJ

EnergtHea work I cal:4.184JI H :2j778 x l0 4kW h:0.94783BtuI kJ/kg.K 0.23q atleKHeat apacrt) t kJikeK - 0.23q ru/tb.'RI N - I kgm/s2 l05gcm/s2{dyne)I N - 0.2248bf= 7.23275b R/s2 poundals)


71/73

AoDendixD. ConstunEand C@vetsionFaclorc

Heat ux

Pressure

Tempemture

Velocity

Viscosity

Volume

lw /m2 :1J l s .m2I w /rr? : 0.317 g Btu lh.#1Wm2.K = 1 J/s m2.KHeatransferoefcient 1\t ' t lm2K:2.39x 10 5cal/scmzK1 W/m2.K:0.1761Btu/h P "Rlm- l 00cm-106 tmten$n I m- lq.l7o in= J.2808i

MassMass o w rateMassux

1kg: 1000I ke 2.2046b1 kgls 2.2046b s 1936.6b hr kg/s.m'/0.2048 b/r.d - z.lu..luTt',

lW=1J ls :10 -3kWI kw :3412.2Btu/h 1.341p1Pa=1N/m2I kPa= 103 a:10 3 MPaI atm= 101.325Pa= 1.01325ar:760 mmHg1 atfr : 14.696bf/in21 K= 1 .8 .Rr (oF) :1 .8 ICC)+32

1 m/s:3.60 km/h1 m s= 3.2808 t s= 2.237mi hIkg /ms : IPas1P(po i se )= lg l cm.s1kg /ms :10P=103cPI P (poise) 241 9 blft.h1m3=1000L1 m3 6.1022x w4 in3=3538 ff =264.t? gal

Mass ransfer oefcient 1 m/s= 3.2808 t/s

| $ /mK- IJ /smK=2 lq l 0 I ca l l s mKlnermal onouclrtty I wim K - 0.5778 tu/hn 'F

I mr /s : 1000 /svo fumer rc\ r ra re lm l / s=35 .313 . / s=1 .27121' l o ' f l 31h


72/73

446ChemielEnsineeiisPrccesses

AppendixEDimensionlessNumbersF.. R. = P, L

sc sc- =NuorSh ,: ,rn=#- LK.C',| '

Db 'J#

CppU Uz,*"

(Lc)* ,rcrD

,. ,. = ""i"'''

,, ,t=+

Ralioof inedal fo.ces oljscous orces;Ratioofmomenun fansilr byconvecrion ndby molecula.actlonof vis@sllyor I Rorio fconlecrileransport" to noleule transpofi(of enerey r nass)

Ralioof nomentumdifirsivityto them.l difrusivityRrdo of momenbmdiffirsivityto na$ difflsivilyRatioof to1al rmsfer onoleculd transfer(ofenersy .nas)Ratio f convectivcime calero reactionime cale;nlio of [email protected] raieof generationftlo chenical tactionRatioof diftusion ime scalelo reactionine scale;ntio ol diffusive nnspor1to raieof generationuero chenrical@actionRatioof inredace ranspodlo bulk tnnsponRatio fpsure ocso nenial orcesRatioof ine'1ial orcestognvnadonrl o.cesRatioof lhemal ditrusivilyto mas diffirsivityRatioof inenialto surfaceorces

Formore hcu$ionon dimensionlessumberselevantor ow process,eeB'rd?,21. 1960) ndBrcdkey d HeNhey1988).


73/73

Roierences 447

REFERENCESAbramowitz. . andLA. Stegun.9?0, Hddbookof Mathenaticalunctions.oler Publicarions ew York.Buckler, .J., 969,Theverticalwindpro e of nonthly meaD indsoler theprairies, anada epa4ment fTransport.ch.Memo.TEC 718.Churchill. .W,1974, he nlerpreiatio d Useof RateData:TheRateConcepl. cripta ublishing o..Wash-ington.D.c.DeNe\e6,N.. 1966.Rate alaard de valives, IChEJoumal 2. l l0Incropera,F.P and D.P Dewitt, 2002, Fundamentah f Heal od MassTransfer,5th Ed , Wilev, Ne{ York.Geankoplis,.J.,1972.MassTmnsport henomena,ol Rinehait ndWinston, ewYorkGjumbirM. andZ. olujic, l9ll4,Effectivc ays o olve insle onlined quations.hem Ens 9l (Julv23).51.Kays,WM.andM.E. Crawfod,1980.Conveclile e.t nd MassTransfer,2ndd, Mccraw'Hill.NewYotkLide,D.R.,Ed.,20Ol.CRCHindbook fChenittryandPhysics,82ndd, CRCPrcss, ocaRaton. lorid.Liley.PE.,1984. team able$n SIUnits. chool f Mechanicalnginenng,urdue n versitv' est nfavetteMcAdanN,W.H.,1954,Heatlransmission,3rd d., Mccriw-Hill, Ne$ YorkMickle).H.S..T.S.Slenvood ndC.E.Reed,1975.AppliedMalhematicsn ChenicalEngineering,nd Ed..D.25.TalaMccraw-Hil1. et Delhi.Pe;x R.H.,D.W.Green nd .O.Maloney,Eds..997. er)\ Chemical ngineN' andbook.thEd.'MccrawHill,NewYork.Reid,R.C.. .M.PruusnitzndT.K.Sherwood,977, hePrcPeniesf Gases ndLnFids.3rdEd Mccraw_Hill'Serghides..K., 1982, teratile oluiionsy di.ect ubsliiutioq hem. ng.89 Sept6), 107Tao,B.Y, 1988, indnrgyouf ooLs. hem. n8.95 Apr 25),85Tao,B.Y. 1989, inear rog.arnnns.Chcm. ng.96 July),146

Touloukian.S.. P.E. lley .nd S.C.Srxena, 970. homophtsical ropediesfMatter,Vol 3: ThermalCon_duclivity.Nonmetlllic Liqui(ts andGases.FL/Plnum, eiv YorkTouloukian,5., PE. Lilcy andS.C.Saxena,970, hcrnophysical ropelLiesf Matter, ol 6:SpecilicH#rNonmetalliciquids ndGases,FVPIenum, ewYorkTouloukian, S., PE. Liley andS.C.Saxena, 970, hermophysicaltoPediesf Matter.Vol. Il: ViscosiryNonmctallic iqDlds d Gasos.Fl/Plenon.NowYork.WhitakerS. andR.L.Pigford,19

Chapter -10-Humidification and Cooling Towers-Appendices

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