Fundamentals of Mass Transfer

Transcription

1 Chapter Fudametals of Mass Trasfer Whe a sgle phase system cotas two or more speces whose cocetratos are ot uform, mass s trasferred to mmze the cocetrato dffereces wth the system. I a mult-phase system mass s trasferred due to the chemcal potetal dffereces betwee the speces. I a sgle phase system where temperature ad pressure are uform, the dfferece chemcal potetal s due to the varato cocetrato of each speces. Mass trasfer s the bass for may chemcal ad bologcal processes such as the removal of sulfur doxde from the flue gas, a chemcal process, or the desg of a artfcal kdey, a bologcal process.. Molecular Mass Trasfer For a bary mxture of ad B, the molar flux, N,z, of speces relatve to the z axs s N,z cd B dy + y (N,z + N B,z ) (.) I ths equato, c s the total molar cocetrato, D B s the dffusvty of B, y s the mole fracto of, ad N B,z s the molar flux of B. For a bary mxture D B D B. The dy term cd B s the molar flux, J, resultg from the cocetrato gradet ad the term y (N,z + N B,z ) s the molar flux resultg from the bulk flow of the flud. I a mult-compoet mxture, the cocetrato of a partcular speces ca be expressed mass or molar cocetrato. For speces, the mass cocetrato, ρ, s defed as the mass of per ut volume of the mxture. The total mass cocetrato s the desty of the mxture whch s the total mass of the mxture per ut volume. The total mass cocetrato s related to the speces mass cocetrato by ρ ρ (.2) I ths equato, s the umber of speces the mxture. The mass fracto, ω, s the mass cocetrato of speces dvded by the total mass desty, ω ρ ρ (.3) The mass cocetrato ad the molar cocetrato are related by c ρ M (.4) -

2 For a bary mxture of ad B, the mass flux,,z, of speces relatve to the z axs s dω,z ρd B + ω (,z + B,z ) (.5) The molar flux of speces ca be expressed as N c v (.6) I ths equato, s the absolute velocty of speces relatve to the statoary coordate axs. Smlarly, the mass flux of speces s gve by ρ v (.7) The total molar flux s the sum of the speces molar flux N cv (.8) I ths equato V s the molar average velocty defed by V c v c x v (.9) Smlarly, the total mass flux s the sum of the speces mass flux N cv (.) I ths equato v s the mass average velocty defed by v ρ v ρ ωv (.) Example mxture of oxyge ad troge gas s cotaed a ppe at 298 K ad atm total pressure whch s costat throughout. t oe ed of the ppe the partal pressure p of oxyge s.7 atm ad at the other ed.8 m, p 2.2 atm. Calculate the molar ad mass flux of oxyge at steady state f D B of the mxture s.26 cm 2 /s. Soluto

3 Sce the temperature ad pressure s costat throughout the ppe, the total cocetrato s a costat. c P RT mol/cm Note: R cm 3 atm/mol K The total cocetrato s a costat therefore N,z N B,z. Ths codto s kow as equmolar couterdffuso. The molar flux of s N,z cd B dy dy + y (N,z + N B,z ) cd B For steady state ad costat area of mass trasfer N,z costat. Separatg the varable ad tegratg N,z 2 cd B dy y y cdb( y y2) N,z 5 (4.9 )(.26)(.7.2) 8 The mass flux of O 2 s N,z mol/cm 2 s,z M N,z (32)( ).69-6 g/cm 2 s Example Water the bottom of a arrow metal tube s held at a costat temperature of 298 K. The dry ambet ar outsde the tube s at atm (.3 kpa) ad 298 K. Water evaporates ad dffuses through the ar the tube, ad the dffuso path z 2 z s 5 cm log. Calculate the rate of evaporato at steady state mol/s cm 2. The dffusvty of water vapor () ar (B) at atm ad 298 K s.25 cm 2 /s. ssume that ar s soluble water. 2 N r (B) Water () Soluto The molar flux of (water vapor) s N,z cd B dy + y (N,z + N B,z ) -3

4 Sce ar s soluble water, t s stagat (or odffusg) ad N B,z. Solvg for N,z gve N,z ( y ) cd B dy cd N,z y B dy N,z cd B dy y y2 2 N,z cd B l y y y The ambet ar s dry so y 2. Vapor pressure of water at 298 K s 3.7 kpa, therefore y 3.7/ N,z cdb l y2 y l mol/cm 2 s Example Nckel carboyl () s produced by passg carbo mooxde (B) at 323 K ad atm over a ckel slab. The followg reacto takes place at the sold surface: N(s) + 4CO(g) N(CO) 4 (g) z z N B N Gas flm Nckel slab The reacto s very rapd, so that the partal pressure of CO at the metal surface s essetally zero. The gases dffuse through a flm wth a thckess.625 mm. Estmate the rate of producto of ckel carboyl, mole/s cm 2 of sold surface at steady state. The composto of the bulk gas phase (z ) s 5 mol% CO. The bary dffusvty s D B 2. mm 2 /s. Soluto The molar flux of (ckel carboyl) s N,z cd B dy + y (N,z + N B,z ) From the stochomerty of the reacto N B,z 4N,z. Therefore N,z + N B,z 3N,z N,z ( + 3y ) cd B dy cdb N,z + 3 y dy N,z cd B y, dy cd + 3y B N,z l, y, + 3y 3 + 3y, Sce y,.5 ad y,., we have Betez, J. Prcple ad Moder pplcatos of Mass Trasfer Operatos, Wley, 29, p. 44-4

5 N,z B cd 3 l l mol/cm 2 s Example crystal of chalcathte (CuSO 4 5H 2 O) dssolves a large tak of pure water accordg to the followg equato: () (2) CuSO 4 5H 2 O(s) CuSO 4 (aq) + 5H 2 O(l) Estmate the rate at whch the crystal dssolves by calculatg the flux of CuSO 4 from the crystal surface to the bulk soluto. ssume that molecular dffuso occurs through a lqud flm uformly. mm thck surroudg the crystal. t the er sde of the flm adjacet to the crystal surface the soluto s saturated wth CuSO 4 whle at the outer sde of the flm the soluto s vrtually pure water. qud flm The solublty of chalcathte water at 275 K s 24.3 g of crystal/ g of water, ad the desty of the correspodg saturated soluto s 4 kg/m 3. The dffusvty of CuSO 4 dlute aqueous soluto at 275 K ca be estmated as m 2 /s. Soluto For each mole of chalcathte (molecular weght 249.7) that dssolves, mole of CuSO 4 (molecular weght 59.63) ad 5 mole of hydrato water dffuse through the lqud flm from the surface of the crystal to the bulk of the lqud phase. et CuSO 4 ad B H 2 O the, N B 5N The molar flux of s N,z cd B dx dx + x (N,z + N B,z ) cd B + 6x N,z N,z cd B dx 6x Usg a average value for the total cocetrato c a, the above equato ca be tegrated over the thckess of the lqud flm,. N,z c a D B dx 6x x2 x 2 Betez, J. Prcple ad Moder pplcatos of Mass Trasfer Operatos, Wley, 29, p. 54-5

6 ca DB N,z 6 l 6x2 6x We eed the mole fracto of compoet at the er sde of the flm (x ), a saturated soluto of chalcathte water at 275 K. The solublty of the salt uder these codtos s 24.3 g/ g H 2 O. For a bass of g of H 2 O, we have 24.3 g of CuSO 4 5H 2 O. The mass of CuSO of of the crystal s / g The mass of hydrato water the crystal s The total mass of water s the g. Therefore, x The other ed of the flm s vrtually pure water; therefore x 2. Next, we calculate the flm average molar cocetrato c a. t locato, the average molecular weght s M (.58)(59.63) + (.58)(8) 2.24 kg/kmol. The correspodg molar cocetrato s c 4/ kmol/m 3. t locato 2, the molar cocetrato s c 2 / kmol/m 3. The c a.5(c + c 2 ).5( ) kmol/m 3 ca DB N,z 6 l 6x2 6x N,z 5 6 l kmol/s m

7 .2 Gas dffusvtes Oe of the most commo method to estmate the bary gas dffusvty D B low pressure system was proposed by Wlke ad ee 3 : D B ( ) T / 2 M B PM σ Ω / 2 2 B B D 3 3/ 2 (.2-) I ths equato: M B 2 + M M B D B dffuso coeffcet, cm 2 /s M, M B molecular weghts of ad B, respectvely T temperature, K P pressure, bar σ B collso dameter, a eard-joes parameter, agstrom Ω D dffuso collso tegral, dmesoless The collso tegral, Ω D, s a fucto of temperature ad termolecular potetal feld for oe molecule of ad oe molecule of B. It ca be approxmated by the followg expresso 4 : a Ω D ( *) b T c + exp( dt *) e + exp( ft *) g + exp( ht *) (.2-2) The parameters ths equato are lsted the followg table T* κt/ε B a.636 b.56 c.93 d e.3587 f g h For a bary system of opolar molecular par, the eard-joes parameters ca be obtaed from the pure compoets by the followg expresso: σ B.5(σ + σ B ); ε B (ε ε ) /2 (.2-3) eard-joes parameters for pure compoets may be estmated from the followg correlatos: σ.8(v b ) /3 ε /κ.5t b (.2-3) V b.285(v c ).48 3 Wlke, C. R., ad C. Y. ee, Id. Eg. Chem., 47, 253 (955) 4 Neufeld, P. D.,. R. Jase, ad R.. zz, J. Chem. Phys., 57, (972) -7

8 Example Estmate the dffusvty of carbo dsulfde vapor ar at 273 K ad bar usg the Wlke- ee equato D B ( ) T / 2 M B PM σ Ω / 2 2 B B D 3 3/ 2 Data: σ, o ε/κ, K Mw CS r Soluto σ B.5(σ + σ B ).5( ) 4.52 o ε B κ ε ε B κ κ / 2 (467 97) / K κt T* ε B a Ω D ( *) b T c + exp( dt *) e + exp( ft *) g + exp( ht *) The parameters ths equato are lsted the followg table T* κt/ε B a.636 b.56 c.93 d e.3587 f g h Ω D.282 M B 2 + M M B Substtutg these values to the Wlke-ee equato yelds D B ( )(273) / / 2 / 2 2 ()(4.98) (4.52) (.282).952 cm 2 /s 5 Betez, J. Prcple ad Moder pplcatos of Mass Trasfer Operatos, Wley, 29, p. 2-8