Mass Transfer Processes

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1 Mass Transfer Processes S. Majd Hassanzadeh Department of Earth Scences Faculty of Geoscences Utrecht Unversty

2 Outlne: 1. Measures of Concentraton 2. Volatlzaton and Dssoluton 3. Adsorpton Processes 4. Knetc Mass Transfer

3 Measures of Composton Mass concentraton (of a substance n a flud): Mass fracton (of a substance n a flud): Molar concentraton (or molarty): Mole fracton:

4 Measures of Composton Mass concentraton (of a substance n a flud): mass of a substance per unt volume of the flud phase, C [ML -3 ]. For a N-components flud phase, we have: N ρ C Mass fracton (of a substance n a flud): mass per unt mass of the flud phase, ω [MM -1 ] or [-]. Unts: fractonal number, %, ppm, ppb. Subject to: ω = = = 1 C N ω = 1 / ρ = 1

5 Measures of Composton Molar concentraton (or molarty): No. of moles of the substance per unt volume of the flud phase (the soluton), c [moll -3 ]; g-moles/cm 3, g-moles/l, etc. c = C / M w Mole fracton: No. of moles of the substance per total number of moles n the flud phase, [-]. χ = c / c ω w w = = N M w χ M w = 1 Molalty (or molal concentraton): No. of moles of the substance per unt mass of the solvent (not the soluton), commonly denoted by m [mm -1 ]. χ M χ m M = n M solv w

6 Measures of Composton for gasses Ideal Gas Law: P V = g g g n RT P g n g V g s the gas pressure [MT -2 L -1 ]; (unts: Pascal, mbar, etc.) s the number of moles of the gas, s the gas volume, T s temperature n degrees Kelvn, R s the unversal gas constant (n SI system: Joules per Kelvn per mole)

7 Measures of Composton for gasses P g Partal pressure: [MT -2 L -1 ]; (unts: Pascal, mbar, etc.) s the pressure of a gven gas component as f that component flls the whole gas volume by tself. So, from deal gas law: g g = g P V n RT Dlaton Law: The total gas pressure s equal to the sum of all partal pressures: P g = P g n g = n g g n χ g = = n PV g / RT PV/ RT = Pg P g g

8 Mass transfer processes n sol Aqueous Phase Volatlzaton/ Dssolutoon Henry s Law Gas Phase Adsorpton (KD) Solublty & Dssoluton Raoult s Law Vapor Volatlzaton/ Pressure Dssolutoon & Raoult s Law Sold Phase NAPL

9 Volatlzaton Lqud-Gas Mass Transfer - Equlbrum Pure lqud n contact wth vacuum If a pure lqud s n contact wth vacuum, t evaporates untl the correspondng vapor pressure s reached. Vapor pressure, P vap, s reported n tables n the lterature. It ncreases wth temperature. Vapor pressure Temperature

10 Exercse Vapor pressures of benzene and PCE are 100 mmhg and 18 mmhg, respectvely. It they are each n a separate closed contaner n equlbrum wth headspace ar, what are ther mole fractons n the ar?

11 Volatlzaton Lqud-Gas Mass Transfer - Equlbrum How about f a mult-component lqud s n contact wth a gas? At equlbrum, both components wll be present n the gas phase. Each component has a partal pressure, whch s now smaller than the vapor pressure How small? That s prescrbed by Raoult s Law.

12 Volatlzaton Lqud-Gas Mass Transfer - Equlbrum Raoult s Law: χ lq g = vap lq P P χ where s the mole fracton of component n the lqud. g = vapχlq = gχg P P P Exercse An organc lqud s prepared by mxng 100g of benzene and 300g of PCE. They are then put n a closed contaner. What are the mole fractons of components n the headspace of the contaner? Molecular weght of Benzene and PCE are 78 g/mole and 165 g/mole, respectvely

13

14 Henry s Law The smple form of Raoult s Law s not vald for dlute solutons. The complete form s: P = a P χ / φ g sol v sol

15 Equlbrum of dlute mult-component lqud wth a gas If the component s n small concentratons (solute), then we must employ Henry s Law g = χsolh Henry s coeffcent H depends on temperature and gas pressure Varous forms of Henry s law are possble, relatng dfferent measures of concentraton to each other: P χ g = χsolh χ p g = χsol H χ g = sol C C C H P

16 Lqud-lqud mass transfer; Dssoluton The maxmum concentraton of a snglecomponent NAPL n water s equal to ts solublty, S [M/L 3 ] The maxmum concentraton of a component from a mult-component NAPL n water s gven by a specal form of Raoult s law: χ n max = S χn s mole fracton of the component n NAPL Raoult s law s applcable only f: χ 0 C n

17 Lqud-lqud mass transfer; Dssoluton Maxmum aqueous concentraton of a component present at small amount n a mult-component NAPL s gven by a specal form of Henry s law: C w where s the concentraton of component n water and s ts concentraton n the NAPL. H ds C n C = C H w n ds s Henry s constant for dssoluton.

18 Dstrbuton of NAPL components n sol Aqueous Phase Henry s Law Volatlzaton/ Dssolutoon Gas Phase Adsorpton (KD) Solublty & Raoult s Law Vapor Pressure & Raoult s Law Sold Phase NAPL

19 Exercse on Raoult s Law: Example: A solvent contans the followng components (collectvely called BTEX): The solvent s n equlbrum wth ar at the pressure of one atmosphere (760 mm Hg). Calculate the mole fracton of BTEX components n the ar. If ths solvent s n contact wth water n a contaner, what would be the equlbrum concentraton of ts components n water?

20 Equlbrum adsorpton Often, t s assumed that adsorpton can be modelled as an equlbrum process,.e. the component n the lqud and the component n the sold are n equlbrum. In fact ther chemcal potentals are equal. Chemcal potental s a functon of concentraton. Ths leads to: concentraton of adsorbed solute s an algebrac functon of the solute mass concentraton n the flud: s = f C ( ) whch s also called equlbrum adsorpton sotherm. s s mass fracton of adsorbed solute (adsorbed mass per unt mass of sold phase) and C s aqueous phase concentraton If changes n s (and C ) are not large, a lnear approxmaton may be made, n whch case we ll have: K d [L 3 /M] s called dstrbuton coeffcent. s = KC d

21 Mass transfer processes; Knetc Concentratons change wth tme. Intally, a component s present wthn a phase and starts to partton nto other phase(s), OR there s a change from equlbrum. For example, consder knetc adsorpton

22 Batch experments on vrus adsorpton (Data from Reza Sadegh) C/C L2 L10 S2 S Tme (h)

23 Knetc adsorpton The change of concentraton of a solute n sol s gven by the followng mass balance equatons: dc n = rads + rdes dt b ds ρ =+ rads rdes dt One may assume that adsorpton rate depends on C and desorpton rate depends on s: rads = f C ( ) r = r r = f C f s s ads des rdes = ( ) ( ) f s ( )

24 Knetc adsorpton Commonly, lnear approxmatons are employed such that: = ρ b s a d r nk C k s where k a [T -1 ] and k d [T -1 ] are adsorpton and desorpton rate coeffcents, respectvely. Ths equaton may be rewrtten as: b nk a b rs = ρ kd C s = ρ α KDC s (1 n) ρbkd ( )

25 Mass transfer processes; Lnear Knetc A lnear knetc formula, typcally has the followng form: kn max r, α = λ( C Cα ) where : λ s a knetc rate constant [1/T] C C α max s the maxmum possble concentraton often equal to the equlbrum conc. s the actual concentraon

26 Mass transfer processes; Lnear Knetc Knetc dssoluton from a mult-component NAPL: r ds n ds w, = λ (?? C ) rds, n = λds ( S χn Cw) Knetc volatlzaton from water: r vol w vol g, = λ (?? C ) rvol, w = λvol ( HCCw Cg )

27 Mass transfer processes; Lnear Knetc Knetc volatlzaton from a mult-component NAPL: r vol n vol g, = λ (?? C ) p g χnpvap = = = ( ) =, mol mol max g mol n RT / V n W RT / VW C RT / W max, g = χn vap mol / C p W RT rvol, n = λvol ( χn pvapwmol / RT Cg )

28 What are sources of knetc adsorpton? Upscalng s the man cause of knetc behavor Consder spread of an adsorbng solute n a ppe The velocty profle s gven by: v r = 2(1 v ) R 2 R

29 Transport of an adsorbng solute n a ppe, cylndrcal coordnates 2 2 c r c c 1 c + 2(1 v ) = D + r 2 t R z z r r r c s D = B.C. at the ppe wall r t r= R r= R D = dffuson coeffcent [L 2 /T] R = radus z = longtudnal coordnate s = adsorbed solute concentraton [mass per unt area] We assume lnear equlbrum adsorpton at the wall: s = kc d r = R

30 Example of concentraton feld, c(z*,r*,t*). Reacton rate s scale-dependent, as s clear from gradent of concentraton wthn the tube. The concentraton averaged over the entre radus of the pore, represents the concentraton that would be sampled as effluent.

31 Non-symmetrc behavor of breakthrough curve of average concentraton resultng from smulaton n Sngle Tube model wth an nstantaneous concentraton source.

32 Transport of an adsorbng solute n a ppe, equaton for average concentraton 2 c c c + v = Ddsp α K 2 Dc s t z z s = α ( K ) Dc s t ( ) D dsp = Taylor dsperson coeffcent [L 2 /T] α = 9.0 (1+ 4 k) pe 0.95 K D

33 Relaton between K D and κ K* D s lnearly related to kappa and s ndependent of peclet number. And the curve obtaned by fttng s: K*D

34 Presence of stagnant zones and/or low-permeablty zones n the porous medum; Physcal Non-equlbrum

35 Advecton-Dsperson- Adsorpton It s possble to have a combnaton of equlbrum and knetc stes ρ s s s s = r = ρ α (1 f ) K C s ( ) b k k b sorpton D k x = + = e k e D s fk C C b se F b φ + ρ + = ρ α (1 f ) KDC s t x x ( ) k

36 HOMEWORK A sol sample from a hazardous waste ste s analyzed and found to have a total carbon tetrachlorde (CTC) mass fracton of 15 ppm (mg/kg). The followng sol and NAPL propertes are determned: porosty= 0.35, ρ b =1600 kg/m 3, S w = 0.52, T = 25 C, CTC vap. pressure= 15,100 Pa, Mol. Weght of CTC = g/mole, H C = 1.24, K D = m 3 /kg, and ρ CTC =1584 kg/m 3. Determne whether CTC s present as a separate phase and f so, what s ts saturaton n the sample.

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