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Fundamentals of Mass Transfer

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Fundamentals of Mass Transfer CHAPTER8 Fundamentalsof Mass Transfer Diffusion is the processby which molecules,ions, or other small particlesspon- taneouslymix, moving from regionsof relativelyhigh concentrationinto regionsof lower concentration.This processcan be analyzedin two ways. First, it can be describedwith Fick's law and a diffusion coeffìcient,a fundamentaland scientificdescription used in the fìrst two partsof this book. Second,it can be explainedin terms of a masstransfer coef- ficient, an approximateengineering idea that often gives a simpler description. It is this simpleridea that is emphasizedin this part of this book. Analyzing diffusion with masstransfer coefficients requires assuming that changesin concentrationare limited to that small part of the system'svolume nearits boundaries.For example,in the absorptionof one gasinto a liquid, we assumethat all gasesand liquids are well mixed, exceptnear the gas-liquid interface.In the leachingof metal by pouring acid overore, we assumethat the acid is homogeneous,except in a thin layernext to the solidore particles.In studiesof digestion,we assumethat the contentsof the smallintestine are well mixed, exceptnear the villi at the intestine'swall. Such an analysisis sometimescalled a "lumped-parametermodel" to distinguishit from the "distributed-parametermodel" using diffusion coefficients.Both modelsare much simplerfor dilute solutions. If you are beginning a study of diffusion, you may have troubìe deciding whetherto organizeyour resultsas masstransfer coefficients or as diffusion coefficients.I havethis trouble too. The cliché is that you should use the masstransfer coeffìcient approach if the diffusion occursacross an interface,but this cliché has many exceptions.Instead of dependingon the cliché, I believeyou shouldalways try both approachesto seewhich is betterfor your own needs.In my own work, I havefound that I often switch from one to the other as the work proceedsand my objectivesevolve. This chapterdiscusses mass transfer coefficients for dilute solutions;extensions to con- centratedsolutions are deferredto Section13.5. In Section8.1, we give a basicdefinition for a masstransfer coefficient and show how this coefficientcan be usedexperimentally. In Section8.2, we presentother common definitionsthat representa thicket of prickly al- ternativesrivaled only by standardstates for chemicalpotentials. These various definitions are why masstransfer often has a reputationwith studentsof being a difficult subject. In Section8.3, we list existingcorrelations of masstransfer coefficients; and in Section8.4, we explainhow thesecorrelations can be developedwith dimensionalanalysis. Finally, in Section8.5, we discussprocesses involving diffusion acrossinterfaces, a topic that leadsto overallmass transfer coeffìcients found as averagesof more local processes.This last idea is commonly called masstransfer resistances in series. 8.1 A Definition of Mass Tlansfer Coefficients The defìnitionof masstransfer is basedon empirical argumentslike thoseused in developingFick's law in Chapter2. Imagine we are interestedin the transferof mass 211 lrl 8 / Fundamentalsof Mass Transfer e.1/ A DeÍir irom someinterface into a well-mixed solution. We expectthat the amounttransferred is (a) proportionalto the concentrationdifference and the interfacialarea: / rateol mass\ _ ,. ( inrcrfacial\ / concenrrarion\ (8.1-I) \transferred/-"\ area i \ diflerence) where the proportionality is summarizedby À, called a masstransfer coeffìcient. If we (c divide both sidesof this equationby the area,we can write the equationin more familiar symbols: Nr:k(cri-ct) (8.r-2) where N1 is the ffux at the interfaceand c1; and cl are the concentrationsat the interface Fig and in the bulk solution,respectively. The flux N1 includesboth diffusion and convection: mas it is like the total flux nt exceptthat it is located at the interface.The concentrationc1; is air- at the interface but in the samefluid as the bulk concentrationc1. It is often in equilibrium exc( with the concentrationacross the interfacein a second,adjacent fluid phase;we will defer is tt discussionof transportacross this interfaceuntil sectiong.5. The flux equationin Eq.8.1-2 makespractical sense. It saysthat if the concentration zero. Thus, differenceis doubled,the flux will double. It also suggeststhat if the areais doubled,the total amountof masstransferred will doublebut the flux per areawill not change.In other 4. words, this definition suggestsan easy way of organizingour thinking around a simple constant,the masstransfer coefîcient ft. k Unfbrtunately,this simple schemeconceals a variety of approximationsand ambigui- This valueii ties. Before introducingthese complexities, we shall go over someeasy examples. These examples The time are important. Study them carefully beforeyou go on to the hardermaterial that follows. (r Example d 8.1-l: Humidification Imaginethat water is evaporatinginto initially dry air in ; theclosed vessel shown schematically in Fig. 8.1- I (a). The vesselis isothermalat 25"C,so the water'svapor pressure is 23.8 mm Hg. This vesselhas 0.8 liter of water with 150cm2 of surfacearea in a total volume of 19.2 liters. After 3 minutes,the air is five percent The air is in saturated.What is the mass transfercoeffìcient? How long will it take to reach ninety percentsaturation? r- Solution The flux at 3 minutescan be found directly from the valuessiven: We usethì. vapor ( I / air \ concentration volume I Nr: \ / \ / Reanangins ^ ^- /23.8\ / | mol \ /273 \ U.U)t" tt lt ,60 _)ttg.4tirersl - r / \ 22.4tirers / \ 298/ :4.4.10-E mol/cm2-sec The concentrationdifference is that at the water'ssurface minus that in the bulk solution. It takes trr:: That at the water'ssurfàce is the valueat saturation;that in bulk at shorttimes is essentially 'ler 8.1 / A Defnition of Mass Transfer Coe.fficients 213 :r'J is (a) Humidilication (b) Packod Bed t-E''ut .lt) | I r fiàldìf'l rie i: (c) (d) A '-.rliar LiquidDrops Gas Bubble rg1 o"o.l , | ?"o i t) l' à,arI ::: JÙC Fig. 8.1-1.Foureasy examples. We analyzeeach of thephysical situations shown in termsof -,iolll masstransfer coefficients. In (a),we assumethat the air is at constanthumidity, except near the IS air-waterinterface. In (b),we assumethat water flowing through the packed bed is well mixed, --Ltnl exceptvery closeto the solidspheres. In (c) and(d), we assumethat the liquid solution,which i.'Ier is thecontinuous phase, is at constantcomposition, except near the dropletor bubblesurfaces. l-:iIilll zero. Thus, from Eq. 8.1-2, we have i. !he (2:'8 t t:' . il c'f 4.4. l0-" mol/cm2-sec: k . ?l -o\ 22.4.l0' cm' 298 ''--nle \160 / k :3.4. l0-2cm/sec :::ui- -;-- - This value is lower than that commonly found for transferin gases. - \c The time requiredfor ninety percentsaturation can be found from a massbalance: ,:i.tt / accumulation\ / evaporation\ I l:l I \ In gaspnase / \ rate / a -:.1ltl ;Vc1 : l1Y, dt -'..0 : Ak[ct(sat) - c1] - tÌll --:'llI The air is initially dry, so / :0, cr:0 We usethis conditionto integratethe massbalance: Ll _1_"_(kAlv)t-( - I c1(sat) Rearrangingthe equationand insertingthe valuesgiven, we find '' t:-Lmú- \ kA \ cr (sat)/ 18.4.101 cm3 Inl| - 0.9) (3.4. l0-2 cm/sec). (150cm2) : 8.3. 103sec :2.3hr It takesover 2 hoursto saturatethe air this much I+ 8 / Fundamentalsof Mass Transfer 8.1/ADt Example 8.1-2: Mass transfer in a packed bed Imagine that 0.2 centimeter-diameter This valuc spheresof benzoicacid arepacked into a bed like that shownschematically in Fig. 8.1- 1 (b). definition The sphereshave 23 cm2 surfaceper 1 cm3 of bed. Pure water flowing at a superficial A tang, velocity of 5 cm/secinto the bed is sixty-two percentsaturated with benzoicacid after it dissolvine haspassed through 100centimeters of bed. What is the masstransfer coefficient? in problen Solution The answerto this problem dependson the concentrationdifference that it is rh usedin the definition of the masstransfer coefficient. In every definition,we choosethis so muchl differenceas the value at the sphere'ssurface minus that in the solution. However,we Benzoi can definedifferent mass transfer coefficients by choosingthe concentrationdifference at is relatirel variouspositions in the bed. For example,we can choosethe concentrationdifference at base.b1 L the bed's entranceand so obtain carbonor by one mr Nr:klcr(sat)-01 dissolutitrr 0.62c t sat){5 cm/sec)A Third.and 1 : (ct(sat) (23.rr/.*.x loo .,,-')A intertace. aremuch whereA is the bed's crosssection. Thus intellectu " k:1.3. 10 3cm/sec can be a..: chemical. This definitionfor the masstransfer coeffìcient is infrequentlyused. Alternatively.we can chooseas our concentrationdifference that at a position e in the bed and write a massbalance on a differentialvolume AA,zat this position: Erample \\ater.3ì . amountof (accumurarior,: ( -, / \ -ìminut3. "r,"lr"'fJ;"r, ) \dissolution/ 5 o : A(ctrnl-r,rol ) 11a1.;olrt \ l.- l:,1--l where a is the spheresurface area per bed volume. Substitutingfor N1 from Eq.8.l-2, dividing by AAz, and taking the limit as Az goesto zero,we find dc, ka -c1l ' : ^lct(sat) 47. This is subjectto the initial conditionthat z :0, ct:0 Integrating,we obtainan exponentialof the sameform as in the fìrst example: c1 -_r | -(--\kaluo)z c't(sat) given, Rearrangingthe equationand insertingthe values we find 5. ::l::i l *:('n)',l'-'' I \a:/ \ r'1(sat)/ 5 cm/sec ln(l - 0.62) (23 cmzlcm3)(100cm) : 2.1' 10-3cm/sec 8.I / A Definition rf Mass Transfer Coefficients 215 .:'lC f This value is typical of those found in liquids. This type of mass transfer coefficient . bt. definitionis
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