3. Mass Transfer with Chemical Reaction

Transcription

1 8 3. Mass Transfer with Chemical Reactin 3. Mass Transfer with Chemical Reactin In the fllwing, the fundamentals f desrptin with chemical reactin, which are applied t the prblem f CO 2 desrptin in ME distillers, are presented. 3. Diffusin Diffusin is mass transfer f a substance frm ne part f a system t anther as a result f randm mlecular mtin r a cncentratin gradient. Slute will flw frm the regin f high cncentratin t a regin f lw cncentratin. The mlar flux is directly prprtinal t the cncentratin gradient [Bae98, Wel0], C n! = D. (3.) x The prprtinality cnstant is the diffusin cefficient D f the slute in the liquid. This relatin is called Fick s first law. It applies t steady state diffusin nly. In many cases the cncentratin will vary bth with time t and distance x. Fick s secnd law f diffusin can be expressed as C t = D 2 C 2 x. (3.2) Diffusin in liquids is very much slwer than in gases, since the distance between the mlecules is substantially smaller than in gases and the free mbility f the mlecules is strngly reduced by the intermlecular frces. ccrding t the Stkes-Einstein equatin fr large, spherical mlecules diffusing in a dilute slutin the diffusin cefficient D depends n the temperature T, the viscsity µ f the liquid and the radius r f the mlecule, D = k T 6 π r µ, (3.3) where k is the Bltzmann cnstant.

2 3. Mass Transfer with Chemical Reactin 9 Wilke and Chang [Wil55] prpsed a crrelatin fr nn-electrlytes in an infinitely dilute slutin, in essence, it is an empirical crrelatin f the Stkes-Einstein equatin, D = µ ( Φ M ) V 0.6 / 2 T, (3.4) where D is the diffusin cefficient f in the liquid in cm 2 /s; µ is the viscsity f the slutin in centipise; T is the abslute temperature in K; M is the mlecular weight f the liquid in g/ml; V is the mlal vlume f the slute at its nrmal biling pint in cm 3 /(g ml) and Φ is the assciatin factr f the liquid. Data n the diffusin f CO 2 in electrlyte slutins have been reprted by Ratcliff and Hldcraft [Rat63]. They fund that the diffusin cefficient varies with µ instead f µ -. Thus, at cnstant temperature, the diffusin cefficient f CO 2 in seawater D CO 2,SW can be determined with the diffusin cefficient f CO 2 in pure water and the viscsities as fllws [Rat63]: DCO 2,W µ SW lg = 0.87 lg, (3.5) DCO 2,SW µ W where D CO2, W is the diffusin cefficient f CO 2 in water, and µ W and µ SW are the dynamic viscsities f water and seawater, respectively. The diffusin cefficient f CO 2 in pure water was given by Mcachlan and Danckwerts [Mc72]: with lg DCO = (3.6) 2,W 2 T T D CO2, W in cm 2 /s and T in K. Figure 3. shws the diffusin cefficient f CO 2 in pure water and in seawater as a functin f temperature fr different salinities based n equatins (3.5) t (3.6).

3 0 3. Mass Transfer with Chemical Reactin D CO2,SW x 0 9 [m 2 /s] g/kg 35 g/kg S = 0 g/kg Temperature [ C] Figure 3.: The diffusin cefficient f CO 2 in pure water and in seawater as a functin f temperature fr different salinities. 3.2 Mass Transfer Theries at a Gas-iquid Interface Chemical desrptin is a cmplex prcess invlving chemical reactin kinetics, mass transfer prcesses, phase equilibria at the brine/vapur interface as well as fluid dynamics. Several useful predictins have been perfrmed t describe the behaviur f highly cmplicated absrptin and desrptin prcesses with chemical reactins by using simplified mdels which simulate the situatin well fr practical purpses withut intrducing a large number f parameters. These are, amng thers, the film thery, the bundary layer thery, the penetratin and the surface renewal thery. Film Thery The simplest and ldest mdel which has been prpsed fr the descriptin f mass transprt prcesses is the s-called film thery. It was suggested by Whitman [Whi23] and first applied by Hatta [Hat28] t absrptin with chemical reactin. The film thery is based n the assumptin that when tw fluid phases are brught in cntact with each ther, a thin layer f stagnant fluid exists n each side f the

4 3. Mass Transfer with Chemical Reactin phase bundary. Mass transfer by cnvectin within this layer is assumed t be insignificant, and accrdingly the transprt is slely achieved by steady state diffusin. Beynd the thin layers the turbulence is sufficient t eliminate cncentratin gradients. Figure 3.2 shws the film thery cnceptualisatin fr the case f absrptin f a gas in a liquid. The interfacial regin is idealized as a hypthetical unstirred layer. The cnstant partial pressure p implies n resistance t mass transfer in the gas phase. Figure 3.2: Film thery cnceptualisatin. In the film thery, the mass transfer cefficient k is directly prprtinal t the diffusin cefficient D and inversely prprtinal t the film thickness δ: D k =. (3.7) δ The dependence f the mass transfer cefficient n the diffusin cefficient predicted by the film thery is nt cnsistent with experimental results [st67, Dan70]. Nevertheless, a number f theretical prblems in the field f chemical absrptin and desrptin invlve such mathematical difficulties as t allw their slutin nly fr the simple film mdel.

5 2 3. Mass Transfer with Chemical Reactin Bundary ayer Thery Bundary layer thery differs frm the film thery in that the cncentratin and velcity can vary in all crdinate axes [Bae98]. Hwever, as the change in the cncentratin prfile is the largest in the x directin, i.e. the crdinate perpendicular t the phase interface, this simplifies the differential equatins fr the cncentratin significantly. Fr diffusin thrugh a laminar bundary layer, the average mass transfer cefficient can be fund frm an equatin f the frm [Bae98, Wel0]: D n m k = c Re Sc (3.8) uρ ν where is the characteristic length, Re = and Sc = are Reynlds µ D number and Schmidt number, respectively. The cnstants c, n and m depend n the type f flw, laminar r turbulent, and the shape f the surface r the channel ver r thrugh which fluid flws. Herein m is /3, i.e. the mass transfer 2 / 3 cefficient varies as D which is typical f bundary layer calculatins [Wel0]. Penetratin Thery In 935, Higbie [Hig35] prpsed a mdel fr the gas exchange between a liquid and an adjacent gaseus phase. The gas-liquid interface is made up f a variety f small liquid elements, which are cntinuusly brught up t the surface frm the bulk f the liquid by the mtin f the liquid phase itself. The mechanism f this replacement is nt relevant at this pint: it may be due t turbulence r t the flw characteristics in the equipment. Fr example, the liquid may flw in laminar flw but is mixed at certain pints, bringing fresh, unexpsed liquid elements t the surface. Each liquid element, as lng as it stays n the surface, may be cnsidered t be stagnant, and the cncentratin f the disslved gas in the element may be cnsidered t be everywhere equal t the bulk-liquid cncentratin when the element reaches the surface. The residence time at the phase interface is the same fr all elements. Mass transfer takes place by unsteady mlecular diffusin in the varius elements f the liquid surface.

6 3. Mass Transfer with Chemical Reactin 3 The mass transfer cefficient k in the liquid phase is directly prprtinal t the square rt f the diffusin cefficient D and inversely prprtinal t the square rt f the age t f the element as fllws k D = 2. (3.9) π t The penetratin thery represented a first step twards the develpment f a turbulence hypthesis which prpses that the turbulent mvements reach the bundary range near t the phase interface. Since turbulent mvements are stchastic by nature, Higbie s cncept that the liquid elements stay the same time at the phase interface is nt realistic. Surface Renewal Thery In 95, Danckwerts [Dan5] prpsed the surface renewal thery which is an extensin f the penetratin thery. It is based n the cncept that the liquid elements d nt stay the same time at the phase interface surface. He prpsed the fllwing analytical frm fr the age-distributin functin: s t () t = s e ψ, (3.0) where s has the physical meaning f the rate f surface renewal, and /s may be regarded as an average lifetime f surface elements. The mass transfer cefficient resulting frm this mdel is prprtinal t the square rt f D and the rate f surface renewal s as fllws k = Ds. (3.) In the penetratin and surface renewal mdels, in which the surface film is replaced by bulk water after a fixed time interval, althugh between these peridic replacements mlecular diffusin still determines the transfer between the film and the gaseus phase, the verall transfer velcity is a functin f the time interval between film renewal events. Since this is shrter than the timescale f diffusin acrss the full width f the film, the film thickness itself is nt a factr. Tr and Marchell [T58] pinted ut that the surface renewal mdel is valid nly when the surface renewal is relatively rapid.

7 4 3. Mass Transfer with Chemical Reactin In many circumstances the difference between predictins made n the basis f the different mdels will be less than the uncertainties abut the values f the physical quantities used in the calculatin. The mdels can thus be regarded as interchangeable fr many purpses, and it is then merely a questin f cnvenience which f them is used. 3.3 Desrptin with Chemical Reactin disslved gas will be desrbed frm a liquid int an adjacent gaseus phase, if the cncentratin f the gas in the bulk f the liquid is larger than that at the phase interface surface. The desrptin f a gas can be caused by lwering the ttal pressure r the gas partial pressure, by increasing the temperature r the inic strength f the slutin r by chemical reactin in the slutin [Sha76]. The desrptin f disslved gas frm a slutin withut reactin is called physical desrptin. When the disslved gas chemically reacts with ther cmpnents in the slutin, the desrptin is called chemical desrptin [Dan70, Sha76, st80]. There are tw mechanisms f gas desrptin frm aqueus slutin. If the difference between the partial pressure f the gas in equilibrium with the bulk liquid and the partial pressure at the surface, i.e. the degree f supersaturatin, is mdest, the gas will be desrbed by diffusin frm the liquid free surface in a way analgus t gas absrptin (quiescent desrptin) [Ish86]. Hwever, if the degree f supersaturatin is large, bubbles will frm in the interir f the liquid and much f the gas will be released by diffusing frm the surface f the bubbles (bubble desrptin). The grwth f gas bubbles can partially change the hydrdynamic cnditins (increase f turbulence, destructin f the bundary layer, etc.) and in this way intensify the diffusinal mass transfer. The supersaturatin can be a result f either an intentinal actin, i.e. flushing r verheating, r can ccur spntaneusly in a definite regin f the liquid phase in which the partial pressure f the gas exceeds the ttal pressure [Zar93]. Thus, bubble desrptin is a prcess very different frm absrptin prcesses in which the area f surface available fr mass transfer is determined by external factrs [Ish86]. It can be assumed that in ME distillers CO 2 is released by quiescent desrptin, because the partial pressure f CO 2 in equilibrium with the bulk liquid des nt exceed the ttal pressure in the evapratr stages. The phenmenn f desrptin with chemical reactin is made up f a number f elementary steps: (a) Chemical reactin f the disslved gas within the liquid phase.

8 3. Mass Transfer with Chemical Reactin 5 (b) Mass transprt f the disslved gas frm the bulk f the liquid t the phase interface. (c) Transprt f the gas thrugh the phase interface. (d) Mass transprt f the gas frm the phase interface t the bulk f the gas phase. Steps (a) and (b) may take place simultaneusly, and thus mutually interfere. The verall phenmenn resulting frm steps (a) and (b) takes place in series with steps (c) and (d). Effects f Chemical Reactin n Mass Transfer The ccurrence f chemical reactins has tw distinct effects n the desrptin prcess [st67, Dan70, st83a, Car87]:. Chemical reactins affect the cncentratin f the disslved gas in the bulk f the liquid. During desrptin the chemical reactins cntinuusly prduce the cmpnent t be desrbed, thus prviding a certain cncentratin f it in the bulk f the liquid and hence a certain driving frce fr the mass transfer. 2. The secnd effect is mre subtle. t a given level f driving frce, the actual rate f mass transfer may be significantly larger when chemical reactins are taking place than it wuld be in the absence f chemical reactins. The rate enhancement may be s large as t actually reduce the mass transfer resistance in the liquid phase t the pint at which it is negligible as cmpared t the resistance in the gas phase. The cncept f rate enhancement intrduced abve is frmalised as fllws. In the absence f chemical reactins, the desrptin flux f the gas is given by ( C C ) n! = k (3.2),B,Ph where k is the physical mass transfer cefficient in the liquid phase withut chemical reactins, C,B is the cncentratin f the disslved gas in the bulk f the liquid and C,Ph is the cncentratin at the phase interface. The actual desrptin flux in the presence f chemical reactins may be larger than the value given by equatin (3.2). The chemical mass transfer cefficient k can be defined and the desrptin flux can be written as ( C C ) n! = k. (3.3),B,Ph

9 6 3. Mass Transfer with Chemical Reactin The rate enhancement factr E is defined as the rati f the chemical desrptin flux t the physical desrptin flux n! k E = = k ( C C ) k,b,ph. (3.4) The average reactin time, the diffusin time and the residence time are very imprtant quantities in the analysis f mass transfer prcesses with chemical reactins [st67]. The average reactin time t R is a measure f the time required by the chemical reactin t cver a certain fractin f its path tward equilibrium. The average reactin time fr a reversible first-rder reactin can be written as t R = (3.5) k + k where k and k - are the rate cnstants f the frward and backward reactins, respectively. The diffusin time t D is a measure f the time available fr mlecular diffusin phenmena t take place befre mixing f the liquid phase makes the cncentratin unifrm. It can be expressed as D t D = (3.6) k 2 where D is the diffusin cefficient f the disslved gas in the liquid. It shuld be brn in mind that, while the diffusin time depends n hydrdynamic cnditins, inasmuch as it is the time actually available fr the diffusin prcess within the surface elements, the reactin time nly depends n the kinetics f the reactins cnsidered, and is nt the time available fr the reactin, but the time required by it [st67]. Finally a third characteristic time shuld be cnsidered, namely the time which is actually available fr the reactin. The latter is bviusly the residence time t P f the liquid in the apparatus cnsidered. It is evident that, if a chemical desrptin prcess has t be cnsidered at all, the value f t P has t be at least f the same rder f magnitude f t R. In fact, shuld t R be much larger than t P, n reactin wuld take place at all in the liquid, and the prcess cnsidered wuld be a prcess f physical desrptin [st67].

10 3. Mass Transfer with Chemical Reactin 7 Resistances t Desrptin The resistance t desrptin at the phase interface is usually negligible and physical equilibrium may be assumed t prevail [st67]. Prvided that the partial pressure f the gas is small, Henry s law applies: C,Ph = H p (3.7) where H is the Henry s law cefficient f the gas in the slutin and p is the partial pressure f the gas in the gas phase. The desrptin flux f the gas is given by n! ( C,B C,Ph ) = k G C,Ph p, R T H = k E B (3.8) where k G is the mass transfer cefficient in the gas phase, p,b is the partial pressure f the gas in the bulk f the gas phase, T is the temperature f the gas, and R is the universal gas cnstant. ssuming that the ttal cncentratin difference is lcated in the liquid phase, the liquid-phase and the gas-phase mass transfer cefficients can be cmbined t define the verall mass transfer cefficient K by ( C H p ) n! = K. (3.9),B,B The ttal resistance t transprt can be expressed as K H R T = +. (3.20) k E k G The liquid-side mass transfer cefficient k is usually between 0-5 and 0-3 m/s. The gas-side mass transfer cefficient k G usually ranges frm 0-3 t m/s [Bra7, Cha82, Sch84]. ccrding t equatin (3.20), fr gases like SO 2, NH 3, and HCl that are highly sluble (high H) r react rapidly (high E), the gas-phase resistance apparently cntrls the transprt. Fr gases that are less sluble (lw H) and d nt react at all r nly slwly ( E ) the liquid-phase resistance predminates and cntrls the ttal resistance.