Mass Transfer Coefficients (MTC) and Correlations I

Transcription

1 Mass Transfer Mass Transfer Coeffiiens (MTC) and Correlaions I 7-

2 Mass Transfer Coeffiiens and Correlaions I Diffusion an be desribed in wo ways:. Deailed physial desripion based on Fik s laws and he diffusion oeffiien.. Engineering approah based on he mass ransfer oeffiien k Aim of his leure: Inrodue he mass ranfer oeffiien k Give examples how k is used Oulook: mass ransfer orrelaions 7-

3 The 'mass ransfer oeffiien' model assumes ha all subsanes are well mixed exep near he inerfae. Changes in onenraion are limied o he region near he inerfae. Examples: Absorpion: liquid-gas inerfae Leahing (aid reamen o rerieve meals from ores): aid soluion is homogeneous exep near he ore parile surfae Digesion: well mixed ill he inesine s wall Typially he mass ransfer model is appliable when we deal wih mass ransfer ACROSS an inerfae

4 Mass ransfer model: N k i k N : rae of mass ransferred per area (flux) a he inerfae i : onenraion a he inerfae of phase A : onenraion in he bulk of phase A k : Mass Transfer Coeffiien (MTC) [m/s] N inludes boh diffusion and onveion! Comparison wih Fik's firs law: j D D = diffusion oeffiien [m /s] 4 7-4

5 Dilue Soluions z=: = 5 7-5

6 (a) Example "Humidifiaion" A lier vessel onains lier of waer a 5 C. The free surfae of he waer is 5 m.. The iniially dry air above he waer is 5% sauraed afer 3 minues. Wha is he average flux? (Waer vapor pressure a 5 C = 3.8 mmhg).. How long does i ake o reah 9% sauraion? 6 7-6

7 Quesion : Average flux? When no informaion, assume seady sae and se a mass balane: Amoun of waer evaporaed: N area ime Waer in he air: Air volume onenraion Waer evaporaed = waer in he air Area: A = 5 m ; ime: = 3 min = 8 s Air volume: V = ( - )L = 9L (negleing vol. of evaporaed waer) Conenraion:.5,sa.5 n,sa V.5 p,sa R T 3.8 Torr Torr Nm K mol K N m mol.64 3 m 7 7-7

8 Flux from mass balane: mol m V 3 m 4 mol N 4.5 A.5 m 8 s m s How muh waer evaporaed? M ~ V N A m 3. m 4 3 mol.5 m m s 3 8 kg mol 8 s 3 3 kg m Mass ransfer oeffiien (deermined from iniial sae a shor imes): k N mol s,sa.5 4 m mol m s 4 gasphase m () 8 7-8

9 Quesion : How long does i ake o reah 9% sauraion? As ime now is in quesion WE MUST onsider he non-seady sae Aumulaion in gas - phase d d d d waer in air vapor Evaporaion rae A N V A k,sa Boundary ondiions: A = = Inegraion wih respe o B.C. gives: exp A k V (),sa 9 7-9

10 Now solve equaion () for ime ( =9% sauraion): V A k ln,sa wih k from (7.): 9.5 m 8333 s.3 h 3 m m s ln.9,sa,sa I akes almos.5 hours o reah 9% sauraion. 7-

11 (b) Example "Mass ransfer in a paked bed" Consider spherial benzoi aid rysals of. m diameer paked in a bed-like sruure. The spheres have a=3 m surfae area per m 3 of bed. Pure waer flowing a a superfiial (average hrough he ross-seional area of he empy ube) veloiy of v =5 m/s ino he bed is 6% sauraed wih benzoi aid afer L= m of he bed. Wha is he mass ransfer oeffiien? v =5m/s 7-

12 Key poin: Sele he appropriae Δ Always he differene beween he onenraion AT he sphere surfae and ha IN he soluion is seleed. However, his is differen along he bed axis!! For example, a he enrane of he bed: N k, sa (3) The flux N an be alulaed wih a mass balane: Benzoi aid ha lef he spheres = Amoun of aid in he soluion 7-

13 Benzoi aid ha lef he spheres: N exhange area ime exhange area surfae area bed volume m exhange area a L A 3 3 m bed lengh L ross seional m A area A bed lengh L m ime superfiial veloiy v 5 m s s Amoun of aid in he soluion: volume flow oule on. ime volume flow v A oule onenraion.6,sa 3 7-3

14 Combine: N a L A A v N v a L m 5 s m 3 m.6,sa 3.35,sa Comparing wih (7.3): gives k =.35-3 m/s 3 m N k, sa This k is he average MTC, obained from a mass balane for he overall ("blak box") sysem

15 Anoher way is o wrie a mass balane on a differenial volume Az in he bed: v=5 m/s aumulaion flow in minus flow ou amoun of dissoluion v v A z a N A z zz where a is he exhange area per bed volume of 3 m /m

16 Now, N k,sa Divide by Az wih z : d dz k a v,sa Inegrae using boundary ondiions: a z = =,sa exp k a v z Solve for k k v a z ln,sa This is he loal MTC. As z inreases, /,sa inreases and k dereases. A z=m: k 3 m 5 m s ln(.6) 3 m m. 3 m s 6 7-6

17 () Example "Mass ransfer in an emulsion" Mass balane: Bromine is being rapidly dissolved in waer. Half he sauraion onenraion is reahed in abou 3 minues. Bromine is presen in he form of droples wih oal surfae area A in a volume V of waer. Wha is he mass ransfer oeffiien? Mass balane: Bromine dissolving: d d Bromine leaving he droples: V N A A k,sa 7 7-7

18 Combine d d V wih A k or a k,sa a = A / V drople surfae area per uni volume Inegrae his equaion wih = = exp k a,sa Rearranging k a ln ln 3 min,sa,sa We anno separae k a: very imporan wih hemial reaions d d Be areful! a=f(), as he drople size hanges. Here we assume ha k a is onsan min 4 s 8 7-8

19 We have seen ha he MTC is no a physial quaniy like he diffusion oeffiien. Is value as well as is unis depend on he definiion for a speifi siuaion

20 Mass Transfer Coeffiiens Oulook Mass ransfer orrelaion for gas in a paked ower based on experimenal observaions: In he above "srubber" uni NH 3 is separaed from a gas sream by washing he gas wih waer. 7-

21 . The onenraion differene beween bulk and inerfae is differen along he olumn. A loal MTC should be used. Loal onenraion differene Loal mass ransfer oeffiien The loal k does no hange muh ompared o oher variables. In ase here is no suffiien informaion o deermine he loal MTC ("blak box"), use he average MTC, as in Example (b). The inerfaial area beween waer and gas is unknown. Thus, i is diffiul o define he flux per uni area. Combine he area wih k. Remember our las example (). 3. Diffusion indued onveion will also affe k. We will address his laer in he lass. For now we will work wih dilue sysems. 7-

22 Example Averaging he MTC Imagine a porous solid onaining a soluion of onenraion while he ouside onenraion is Diffusion will ake plae (unseady sae) and he flux is (Chaper ) N D,i, As a resul he loal (in erms of a pariular ime raher han a pariular posiion) mass ransfer oeffiien is k D Noe ha a = k = 7-

23 7-3 How is k relaed o? k D d D d N d N k k D N Afer a long ime he average flux an be defined as: k N 3