Interlude: Interphase Mass Transfer The transport of mass wthn a sngle phase depends drectly on the concentraton gradent of the transportng speces n that phase. Mass may also transport from one phase to another, and ths process s called nterphase mass transfer. Many physcal stuatons occur n nature where two phases are n contact, and the phases are separated by an nterface. ke sngle-phase transport, the concentraton gradent of the transportng speces (n ths case n both phases) nfluences the overall rate of mass transport. More precsely, transport between two phases requres a departure from equlbrum, and the equlbrum of the transportng speces at the nterface s of prncpal concern. When a multphase system s at equlbrum, no mass transfer wll occur. When a system s not at equlbrum, mass transfer wll occur n such a manner as to move the system toward equlbrum. Famlar examples of two phases n contact are two mmscble lquds, a gas and a lqud, or a lqud and a sold. From the standpont of mathematcally descrbng these processes, each of these stuatons s physcally equvalent. Snce oxygen s requred for all aerobc lfe and water s the prncpal medum of lfe, the transport of oxygen between ar (Phase I) n contact wth water (Phase II) s of paramount concern, and ths physcal stuaton wll be hghlghted n the dscusson. et us magne the physcal and chemcal stuaton where ar s frst brought n contact wth oxygenfree water. There wll be a tendency for oxygen to move nto the water to acheve equlbrum between the phases. Ths equlbrum wll quckly be acheved "locally" at the nterface, but wll requre more tme to become establshed n the bulk water. In order to predct how quckly oxygen transports, we wll need to know a relatonshp between the gas phase and lqud phase equlbrum oxygen concentratons. Many equlbrum laws are known. At a fxed temperature and pressure, the equlbrum relatonshp between the concentratons of a speces n two phases can be gven by a dstrbuton curve such as that shown n Fgure I.. The smplest "dstrbuton law" s one whch states that the equlbrum concentratons of speces between the phases are n proporton. Ths lnear law s usually an approprate assumpton when the equlbratng speces s at a low concentraton: c = K I. j c j In Equaton I., c j s the concentraton of speces j n phase, and c j s the equlbrum concentraton of speces j n phase. K s referred to as the dstrbuton or partton coeffcent. For gases and lquds, the dstrbuton law often used to descrbe equlbrum at low solute concentraton s Henry's aw. For the equlbrum of oxygen between gas and water Henry's aw s: y P = p = x H I. In ths case Henry's aw relates the concentraton of oxygen n one phase (mole fracton of oxygen or partal pressure of oxygen n the gas phase) wth concentraton n the second phase (mole fracton of oxygen n the water). The value of mole fracton n the lqud phase n Equaton I. may also be
converted nto more famlar concentraton unts such as mass concentraton (e.g., g/) or molar concentraton (mol/). At 5 C, the value of the Henry's aw constant for oxygen, H, s 4,800 atmospheres. At 0 C, the Henry's aw constant s,700 atmospheres, whle at 40 C, the Henry's aw constant s 5,500 atmospheres. Fgure I. An equlbrum relatonshp. Concentraton of j n phase I Concentraton of j n phase II Queston : What s the equlbrum concentraton (n mg/) of oxygen n water at 0 C, 5 C and 40 C? Soluton: The mole fracton of oxygen n ar s 0.. The pressure of ar wll be assumed to be.0 atm. Thus, the partal pressure of oxygen n the ar s 0. atm. At 0 C: At 5 C: x = 0. atm/,700 atm = 6.4 0-6 mol /mol H (6.4 0-6 mol /mol H ) (000 mg /mol ) (mol H /8.0 g H ) (997 g H /) =.4 mg/ x = 0. atm/4,800 atm = 4.79 0-6 mol /mol H (4.79 0-6 mol /mol H ) (000 mg /mol ) (mol H /8.0 g H ) (997 g H /) = 8.5 mg/
At 40 C: x = 0. atm/5,500 atm =.9 0-6 mol /mol H (.9 0-6 mol /mol H ) (000 mg /mol ) (mol H /8.0 g H ) (997 g H /) = 7.0 mg/ xygen, lke the vast majorty of gases, s more soluble n water at low temperature than at elevated temperature. A. Two-Resstance Theory Interphase mass transfer nvolves three transfer steps. For the specfc example of oxygen transportng from ar to water, these three steps are:. The transfer of oxygen from the bulk ar to the surface of the water.. The transfer of oxygen across the nterface.. The transfer of oxygen from the surface of the ar to the bulk water. Whtman (9) frst proposed a "two-resstance theory" whch has been shown to be approprate for many nterphase mass transfer problems. The general theory has two prncpal assumptons, whch for the case of oxygen transportng from ar to water are:. The rate of oxygen transfer between the phases s controlled by the rates of dffuson through the phases on each sde of the nterface.. The rate of dffuson of oxygen across the nterface s nstantaneous, and therefore equlbrum at the nterface s mantaned at all tmes. In other words, two "resstances" to transport exst, and they are the dffuson of oxygen from the bulk ar to the nterface, and the dffuson of oxygen from the nterface to the bulk water. Ths physcal stuaton may be depcted graphcally by the dagram n Fgure I.. In Fgure I., the gas phase concentraton of oxygen s represented by ts bulk partal pressure, p g. The lqud phase concentraton of oxygen s represented by ts bulk molar concentraton, c l. The nterface gas and lqud phase concentratons are denoted by p and c, respectvely. Snce transport s occurrng from the gas to the lqud phase, the value of p g must be greater than the value of p. Smlarly, the value of c must be greater than the value of c l. Because the nterface mparts no resstance to transport, the two nterface concentratons wll reman n equlbrum, and ther values can be related by Henry's aw (Equaton I.) or some other equlbrum relatonshp. Note that n general these two nterface concentratons are not dentcal, and ndeed p can be less than or greater than c.
Fgure I. Two resstance theory Gas Phase Interface qud Phase p g Concentraton of oxygen p c δ G Dstance δ c l The dstance between the nterface and the locaton n the gas phase at whch the oxygen concentraton equals the bulk oxygen concentraton s the gas phase "flm" thckness (δ G ), whle the smlar dstance for the oxygen n the lqud phase s referred to as the lqud phase "flm" thckness (δ ). Snce the shape of the oxygen concentraton profle s unknown n both the gas and the lqud, the precse concentraton of oxygen at the nterface s very dffcult to determne. Therefore, the flm thcknesses are not known. Remember, f the transport of oxygen s occurrng from the lqud to the gas nstead, then the value of p wll be greater than the value of p g, and the value c l wll be greater than the value of c. The molar flux of speces j (Φ j, a vector quantty, havng unts of moles of materal per area per tme) s known to be proportonal to the concentraton gradent of speces j. Indeed, ths concentraton dfference s sad to be the "drvng force" for mass transport. If we consder the onedmensonal flux of oxygen across an ar-water nterface, then we may defne a mass transfer coeffcent for the gas phase (k G ) as: g Φ = k G ( p - p ) I. And, for the lqud phase (k ): Φ l = k ( c - c ) I.4 For the physcal stuaton depcted n Fgure I., Equatons I. and I.4 represent the flux of oxygen through the gas and lqud phase, respectvely. At steady-state, the oxygen flux n one phase must equal the oxygen flux n the other phase (otherwse, there would be accumulaton of oxygen somewhere n the system), and therefore these two fluxes equate. These coeffcents are often called ndvdual mass transfer coeffcents because they refer to transport n ndvdual phases. 4
The equlbrum dstrbuton curve shown n Fgure I. may be redrawn wth the partcular concentratons found n the physcal stuaton. Fgure I. shows ths representaton, wth the pont ndcatng the two bulk concentratons. Fgure I. qud phase drvng force for mass transfer Partal pressure of oxygen n ar p g p Gas phase drvng force for mass transfer c l c Concentraton of oxygen n water Many other mass transfer coeffcents can be defned dependng on the type of concentraton gradent beng used to descrbe the mass drvng force for mass transfer. Queston : A bubble of pure oxygen orgnally 0. cm n dameter s njected nto a strred vessel of oxygen-free water at 5 C. After 7.0 mnutes, the bubble has shrunk to a dameter of 0.054 cm. Assumng no resstance to mass transfer n the gas phase (why would ths be reasonable?), what s the lqud phase mass transfer coeffcent? Soluton: The system s the bubble, and a mass balance must be made around the system. The mass balance s: ACC = IN - UT + GEN g dc V dt = 0 - Φ A+0 5
where c g s the concentraton of oxygen n the system (.e., the bubble), V s the volume of the system, Φ s the flux of oxygen (out of the system) and A s the cross-sectonal area of the system. The flux may be represented by Equaton I.4, beng careful to note that the concentraton of oxygen (c l ) refers to the lqud phase oxygen concentraton: Φ l = k ( c - c Notng that the concentraton of oxygen does not change wth tme, and that the system volume s (4/)πr, whle the system surface area s 4πr, the materal balance becomes: Dfferentatng and cancelng terms, Integratng from r 0 to r, 4 ( c - c l g dr π c = - 4π r k dt dr k = - dt c ) l ( c - c ) g k ( c r - r0 = - c - c l ) g Now values must be determned for each of the parameters n ths equaton. r = 0.054 cm/ = 0.07 cm r 0 = 0. cm/ = 0.05 cm t = 7.0 mn 60 s/mn = 40 s c l = 0 (The water ntally was "oxygen-free") c g = p /RT (Ideal Gas aw) Snce the gas bubble s pure oxygen, we may assume that p = atm. Then at 5 C, c g = ( atm)/(0.0806 atm/molk)(98k) = 0.04 mol/ c s found by Henry's aw. At 5 C for pure oxygen, = atm/4,800 atm =.8 0 mol /mol H x -5 = (.8 0 mol /mol H ) (mol H /8.0 g H ) (997 g H /) =.6 0 mol / c -5 - - Thus, k = (0.05-0.07)(0.04)/(0.006)(40) =.78 0 cm/s t ) B. verall Mass Transfer Coeffcents As prevously noted, t s usually not possble to measure the partal pressure and concentraton at 6
the nterface. It s therefore convenent to defne overall mass transfer coeffcents based on an overall drvng force between the bulk compostons. For the physcal stuaton of oxygen transportng from ar to water, two overall mass transfer coeffcents may be defned. An overall mass transfer coeffcent may be defned n terms of a partal pressure drvng force or t may be defned n terms of a lqud phase concentraton drvng force. In ether case, the coeffcent must account for the entre dffusonal resstance n both phases. For the gas phase, an overall mass transfer coeffcent may be defned by: Φ = K g ( p - p * G ) I.5 K G s the overall mass transfer coeffcent, p g s the bulk gas oxygen partal pressure (as before) and p * s the theoretcal partal pressure of oxygen n equlbrum wth the bulk lqud phase oxygen concentraton. In other words, p * s the partal pressure n equlbrum wth c l. Note that ths partal pressure does not physcally appear anywhere n the system. For the lqud phase, an overall mass transfer coeffcent may be defned by: Φ * l = K ( c - c ) I.6 K s the overall mass transfer coeffcent, c l s the bulk lqud oxygen concentraton (as before) and c * s the theoretcal oxygen concentraton n equlbrum wth the bulk gas phase oxygen partal pressure. In other words, c * s the concentraton of oxygen n the lqud n equlbrum wth p g. Fgure I.4 shows the equlbrum dstrbuton curve wth these new drvng forces. Fgure I.4 verall drvng force for mass transfer (consderng lqud phase) Partal pressure of oxygen n ar p g p p * verall drvng force for mass transfer (consderng gas phase) c l c c * Concentraton of oxygen n water 7
The overall mass transfer coeffcents may be related to the ndvdual mass transfer coeffcents. et us frst make the smplfcaton that the equlbrum relatonshp between the gas phase partal pressure and the lqud phase concentraton of the transportng speces at the nterface s proportonal. Ths assumpton s Henry's aw, and t s vald n the case where the transportng speces s present n dlute concentraton. For mass transport between other types of phases, a smlar assumpton may often be made. p j = m j c j I.7 If the speces s oxygen, then Equaton I.7 essentally becomes Equaton I., wth a converson beng necessary to relate mole fracton and concentraton. The theoretcal partal pressure of oxygen (p * ) n equlbrum wth the bulk lqud phase oxygen concentraton may now be related to the bulk lqud phase oxygen concentraton (c l ) by: p * = m c l I.8 Also, the theoretcal lqud phase oxygen concentraton (c * ) n equlbrum wth the bulk gas phase oxygen concentraton may now be related to the bulk lqud phase oxygen concentraton (p g ) by: p g = m * c I.9 f course, the lqud phase oxygen concentraton at the nterface s n equlbrum wth the gas phase oxygen concentraton at the nterface: p = m c I.0 We would lke to fnd the relatonshp between K G, k G and k. To accomplsh ths goal, Equaton I.5 may be rewrtten as: * p - p p - p = + I. K Φ Φ G Substtutng Equatons I.8 and I.0 nto Equaton I. yelds: K g l * p - p p - p = + I. Φ Φ G Substtutng Equatons I. and I.4 nto Equaton I. yelds: K G = k G m + k I. 8
To relate K wth k G and k, an equaton for K may smlarly be derved: = K mk G + k I.4 Equatons I. and I.4 show that the relatve mportance of the two ndvdual phase resstances depends on the solublty of the gas n the lqud. If the gas s very soluble n the lqud, such as ammona n water, then the value of m s very small. From Equaton I., the overall mass transfer coeffcent K G s equal to the gas phase mass transfer coeffcent, k G. In other words, the prncpal resstance to mass transfer les n the gas phase, and such a system s sad to be gas phase controlled. If the gas s rather nsoluble n the lqud, such as oxygen or ntrogen n water, then m s large. In ths case, from Equaton I.4, the overall mass transfer coeffcent K s equal to the lqud phase mass transfer coeffcent, k. Snce the prncpal resstance to mass transfer les n the lqud phase, such a system s sad to be lqud phase controlled. Even when the ndvdual coeffcents are essentally ndependent of concentraton, the overall coeffcents may vary wth the concentraton unless the equlbrum relatonshp s lnear. Accordngly, the overall coeffcents should be employed only at condtons smlar to those under whch they were measured and should not be employed for other concentraton ranges unless the equlbrum relatonshp for the system s lnear over the entre range of nterest. C. Correlatons of Mass Transfer Coeffcents If we consder the transfer of oxygen from ar nto water, we mght magne that the dssoluton process can be hastened by agtatng the two phases. Ths agtaton would essentally result n a decrease n the flm thcknesses, and hence an ncrease n the concentraton gradent. Indeed, the mass transfer coeffcents are dependent on, among other thngs, the flow condtons on both sdes of the nterface. Mass transfer coeffcents are therefore correlated for varous geometres to several dmensonless groups. The prncpal dmensonless groups used for the correlaton of mass transfer coeffcents are: Reynolds Number: Schmdt Number: Sherwood Number: vρ Re = μ μ Sc = ρ D AB Sh = k D AB For these relatonshps, s the characterstc length dmenson of mportance (dameter of a sphere, dameter of a cylnder, or length of a flat surface), v s the mass average velocty, ρ s the flud densty, μ s the flud vscosty, D AB s the dffuson coeffcent of "A" n the flud, k s the mass transfer coeffcent n the lqud phase. These dmensonless numbers each descrbe the rato of the mportance of two transport phenomena. The Reynolds Number physcally descrbes the rato of 9
B nertal forces to vscous forces, the Schmdt Number descrbes the rato of momentum dffusvty to mass dffusvty, and the Sherwood Number descrbes the rato of mass transfer rate to dffuson rate. As noted earler, there are numerous defntons for mass transfer coeffcents. Although all mass transfer coeffcents are related, they do have dfferent values. Therefore, one must exercse care when correlatons are used that the correct mass transfer coeffcent s beng correlated. If the physcal stuaton s the dsperson of bubbles n a lqud-contanng vessel, then a more approprate Reynolds Number mght the characterstc dmenson of the mpeller dameter, D, and a reference velocty of N D, where N s the strrer speed (revolutons/mnute). Thus, Reynolds number becomes: Reynolds Number: ρ Re N D = μ ne way to correlate mass transfer coeffcent s by the followng type of equaton: α β Sh = KRe Sc I.5 where α and β are exponents and K s a constant. f course, there are many other types of correlatons. Just as an llustraton, there are a couple correlatons avalable for the lqud phase mass transfer coeffcent of a sphere suspended n that lqud. The sphere may be an oxygen bubble from whch oxygen s flowng nto the lqud, or the sphere may be a mcroorgansm to whch nutrents are flowng from the flud. For the case of small spheres (less than 0.6 mm dameter) under mld agtaton (where the prncpal contrbutor to mass transfer s gravtatonal forces), the followng correlaton may be used: ( ρ - ρg ) ρg d B Sh + 0. Sc μ = I.6 where d B s the sphere (e.g., bubble) dameter. The densty dfference between the phases s an mportant and unque term n Equaton I.6. For the case of spheres above.5 mm n dameter, the bubbles rse exclusvely under gravtatonal forces, and the densty dfference between the lqud and gas phase becomes mportant: ( ρ ) - ρ G ρ g d B Sh= 0.4 Sc I.7 μ In Equatons I.6 and I.7, the term n brackets s referred to as the Grasshof Number, and t s the rato of buoyancy forces to vscous forces n systems of free convecton. Many stuatons arse n whch the total surface area across whch transport occurs s not calculable. 0
For example, n a strred soluton contanng numerous dssolvng ar bubbles, the total surface area for the transport of oxygen s not known. In such stuatons the total volume of the system s often ncalculable as well. In these cases t s convenent to defne a surface area to volume rato, a = A/V. Thus, some correlatons are used to estmate the product of the mass transfer coeffcent and the surface area to volume rato: k a. For example, a smple correlaton for non-coalescng dspersons s: k 0.7 P 0. a = 0.000 vs V I.8 where (P/V) s n unts of watts/m and v S s the superfcal gas velocty (m/s). The unts of k a n Equaton I.8 are s -. The range of power nput for whch Equaton I.8 s vald les between 500 and 0000 watts/m. It s mportant to remember that dozens of correlatons for k and for k a are avalable dependng on the system. It s often qute dffcult to dstngush between the correlatons, and t s not uncommon for a correlaton to yeld a value n error of realty by more than 50%. Queston : Ar bubbles havng a dameters of.0 mm are strred n a vessel contanng water at 7 C. The dffuson coeffcent of oxygen n water s. 0-5 cm /s. a) Estmate the mass transfer coeffcent. -7 b) If the solublty of oxygen at 7 C s.6 0 mol/cm, calculate the maxmum flux of oxygen nto the fermentor. c) If the bubbles are 0. mm n dameter, calculate the mass transfer coeffcent. Soluton: a) For the case of large bubbles, an estmate of the mass transfer coeffcent may be found from Equaton 5.7: The values necessary for ths equaton are: d B = 0.0 cm ρ = 0.994 g/cm ρ G = 0.00 g/cm g = 98 cm/s ( ρ ) - ρ G ρ g d B Sh = 0.4 Sc μ μ = 6.95 0 - g/cms - Sc = (6.95 0 g/cms)/(0.994 g/cm -5 )(. 0 cm /s) = ⅓ ⅓ ⅓ - Sh = 0.4 (0.994-0.00) (0.994) (98) (0.0)(6.95 0 ) -⅔ () ½ = 498-5 So, k = Sh D AB /d B = (498)(. 0 cm /s)/(0.0 cm) = 0.055 cm/s
b) The flux may be found by Equaton I.4: N = k ( c - c l ) The value of k has been found n part a). The maxmum flux occurs when the concentraton dfference s as great as possble. Ths condton occurs when the nterface concentraton c s equal to the saturaton concentraton, and c l s equal to zero. Thus the maxmum flux s: -7 N = (0.055 cm/s)(.6 0 mol/cm - 0) =.4 0-8 mol /cm s c) For the case of small bubbles, Equaton I.6 may be used: ( ρ ) - ρ G ρg d B Sh = + 0. Sc μ All the necessary values have been calculated n part a). We fnd: ⅓ ⅓ ⅓ - Sh = +0.(0.994-0.00) (0.994) (98) (0.0)(6.95 0 ) -⅔ -5 k = Sh D AB /d B = (7.)(. 0 cm /s)/(0.0 cm) = 0.09 cm/s ½ () = 7. Problems:. qud bromne s poured nto a large vessel of water. The water s strred suffcently to produce a mass transfer coeffcent k a of.8 0 - s -. How long wll be requred for bromne to reach a concentraton n the water equal to one-half the saturaton concentraton?. Jasmone s a valuable chemcal n the perfume ndustry used n soaps and cosmetcs. The compound s recovered by suspendng jasmne flowers n water (the contnuous phase) and extractng wth benzene. The mass transfer coeffcent n benzene drops s.0 0-4 cm/s, whle the mass transfer coeffcent n the water phase s.4 0 - cm/s. Jasmone s 70 tmes more soluble n benzene than t s n water. Is the mass transfer controlled by the water phase or the benzene phase?