Lecture 11. Transport in Membranes (1)
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1 ecture 11. Transport n embranes (1) ass Transfer n embranes uk Fow qud Dffuson through ores Gas Dffuson through orous embranes Transport through onporous embranes - Souton-dffuson for qud mxtures - Souton-dffuson for gas mxtures
2 Transport n embranes oar transmembrane fux æ ö ç è uk fow through pores ( drvng force) : permeabty Types of membrane: macroporous, mcroporous, dense echansms of transport n membranes o separaton Dffuson through pores ( drvng force), : permeance Restrcted dffuson through pores Sze excuson, sevng Souton dffuson through dense membranes
3 uk Fow (1) agen-oseue aw (for amnar fow) 2 D v ( 0 - ) 32m (D: pore dameter : ength of the pore) ressure dfference 0 orosty (vod fracton) e np D 2 / 4 (n: pores per unt cross sectona area) Superfca fud buk-fow fux (mass veocty) vre 2 4 erd ( 0 ) 32m - npr D m ( ) ( : membrane thckness)
4 uk Fow (2) ores may not be cyndrca and straght n rea porous membrane ydrauc dameter d æ Voume avaabe for fow ö 4ç è Tota pore surface area æ Tota pore voume ö 4ç embrane voume è æ Tota pore surface area ö ç è embrane voume Tota pore surface area per unt voume of just the membrane matera (not ncudng the pores) av a ( 1- e ) Tortuosty factor, τ If pore ength s onger than the membrane thckness, t 4e a 2 re ( 0 - ) 2 2 2( 1- e ) ta m v ( 0 - ) 3 re 2 2 2( 1- e ) ta m v
5 qud Dffuson through ores When dentca tota pressures but dfferent component concentratons exst: no buk fow, but dfferent dffuson rates can acheve separaton odfed form of Fck s aw De ( c - c ) 0 Concentraton drvng force e D Effectve dffusvty De K r t é d ù m Restrctve factor K r ê1 - ú, ( dm / d p) 1 êë d p úû 4 c,o c, effect of pore dameter, d p, n causng nterferng cosons of the dffusng soutes wth the pore wa Seectvty rato for soute moecues not subject to sze excuson: S j D K r D K j rj
6 Gas Dffuson through ores (1) If tota pressure and temperature on ether sde are equa De c ( p - p ) 0 arta-pressure drvng force De ( p - p ) 0 RT c, tota concentraton of the gas mxture (/RT by the dea-gas aw) D e e é 1 ù ê ú t êë ( 1/ D ) + ( 1/ DK ) úû Ordnary dffuson Knudsen dffuson Cosons occur prmary between gas moecues and the pore wa
7 Gas Dffuson through ores (2) Knudsen dffusvty d pv D K From the knetc theory of gases as apped 3 to a straght, cyndrca pore of dameter d p v æ 8RT ö ç è p 1 2 verage moecue veocty of moecuar weght D K æ T ö 4, 850d p ç è 1 2 When Knudsen fow predomnates (as t often does for mcropores) æ ö ç è 1 2
8 onporous embranes echansm - bsorpton of gas or qud components nto the membrane - Dffuson through the sod membrane - Desorpton at the downstream face Dffusvtes of water (cm 2 /s at 1 atm, 25 ) - Water vapor n ar : Water n ethano qud : Water n ceuose acetate sod : Souton-dffuson mode : The concentratons n the membrane are reated to the concentratons or parta pressures n the fud adjacent to the membrane faces thermodynamc equbrum for the soute between the fud and membrane matera at the nterfaces
9 Souton-Dffuson for qud xtures orous membrane onporous membrane Concentraton profe s contnuous K If the mass-transfer resstances n the boundary ayers are neggbe K D s the permeabty,, for the souton-dffuson mode o c o K c o D ( c - c ) 0 K D ( c - c ) 0 K D ( c - c ) F c c
10 Souton-Dffuson for Gas xtures (1) orous membrane onporous membrane Contnuous partapressure profe If the externa mass-transfer resstances are neggbe o c c p D ( p - p ) 0 o D p o D ( p - p ) F ( p - p ) F
11 Souton-Dffuson for Gas xtures (2) Separaton factor ( y x ) ( ) a, y x y : moe fracton n the permeate eavng the membrane x : moe fracton n the retentate on the feed sde of the membrane Unke dstaton, y and x are not n equbrum For the separaton of a bnary gas mxture of and D D ( p - p ) F ( ) x F - y D D ( p - p ) F ( ) x F - y When no sweep gas s used y y D( xf - y ) D ( x - y ) F
12 Souton-Dffuson for Gas xtures (3) If the downstream (permeate) pressure,, s neggbe compared to the upstream pressure, F y x and y x Idea separaton factor a *, D D F F The factor depends on both transport phenomena and thermodynamc equbra When the downstream pressure s not neggbe a é ( x y ) - ra *,, a, ê ú ( x y ) - r ë ù û r : pressure rato r F a a *,, é ê ê ê ê êë x æ y ö ç ra è y æ y ö x ç r è y, ù ú ú ú ú úû a a *,, é ê êë x ( a ) ( a, ) ra,, x r ù ú úû
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