A numerical study is performed for the facilitated diffusion of carbon-dioxide in an ultrapure water film flowing downward along a vertical surface. The model adopted for the present work includes the effects of the reaction rate for the hydration of CO(2)in a liquid. Comparison of the total concentration of CO(2) for different film thickness indicates that the effects of the hydration is more remarkable for the thinner liquid film

Inaba, Hideo

Nozu, Shigeru

English

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering,Okayama University,VoI.26, No.1, pp.21-26, November 1991Facili tated Diffusion ofCarbon-Dioxide in aLiving Body Liquid FilmShigeru NOZU* and Hideo INABA*(Received October 11 , 1991)SYNOPSISA numerical study is performed for the facilitated diffusionof carbon-dioxide in an ultrapure water film flowing downward alonga vertical surface. The model adopted for the present work includesthe effects of the reaction rate for 'the hydration of CO2 in aliquid. Comparison of the total concentration of CO2 for differentfilm thickness indicates that the effects of the hydration is moreremarkable for the thinner liquid film.1 INTRODUCTIONTransport of carbon-dioxide in liquids has received considerable attention inconnection with the biological and industrial systems. The transport of CO2 throughsolutions is augmented by the hydration of CO2• The hydrated CO2 , in the form ofbicarbonate ion, diffuses with CO2 dissolved physically in the liquid. Previousresults for the behavior of CO2 in liquids have shown that the dissociation of CO2is far from chemical eqUilibrium, and the kinetics of the reactions have beenrevealed [1~3]. However, little study can be observed for the hydration of CO2 in aflowing liquid film. The present work was undertaken to provide a model forpredicting the behavior of CO2 absorbed in a living body liquid film.2 FORMULATION AND NUMERICAL METHODThe problem is formulated in corresponding to the coordinate system shown inFig. 1. A gravity~controlled liquid film with thickness 0 flows downward along avertical surface. In Fig.l, x and y show the coordinates parallel and normal to the*Department of Mechanical Engineering2122 Shigeru NOZU and Hideo INABAsurface at y = 0, respectively. Ultrapure water isadopted as a solvent. To simplify the treatment thefollowing assumptions are made:(a) Rate of CO2 absorbed by the film is substantiallysmall compared to the film flow rate so that the filmthickness is unchanged along x-direction.(b) The film thickness is so small compared to theplate length that the relation a/ay » a/ax holds.(c) Velocity profile in the film is written as(1)FallingfilmyxFig.1 Physical modeland coordinate systemwhere u is the liquid velocity in x-direction, g thegravitational acceleration, V the kinematic viscosityand n = y/o. The profile given by equation (1) wasderived from the gravity-controlled film flow model withnegligible shear-force at the liquid-gas interface.(d) Undissociated carbonic-acid contributesnegligibly to both the total CO2 in the film and thetransport of gas [1].(e) Concentration of metal ions in the film can be neglected in comparison withthat of hydrogen ion. This assumption is possible for the wall surface polishedusing the electrolytic-abrasive combined method.Diffusion process is characterized by the gradient of the electrochemicalpotential. The potential of species j, Vj , is defined as(2)where VjO is the standard electrochemical potential, R the universal gas constant, Tthe absolute temperature, [c j ] the concentration, Zj the charge number, 1 theFaraday constant and $ the electrical potential. The diffusion flux in y-direction,Nj , is obtained from(3)where Dj is the diffusion or ionic self-diffusion coefficient of species j in aliquid.In order to evaluate .the rate expression showing the depletion of CO2 bydissociation reaction, the equilibrium and kinetic properties of the carbon dioxide-bicarbonate system must be examined. Several reactions taking place in the systemhave been observed, in which the next two steps for the present problem are ratelimiting [2].Facilitated Diffusion of Carban-DilJ%ide in a Living Body Liquid Filmk1CO2 + H2O+ H2C03+-*k1andk2CO2 + OW+ HC03-+-*k223(4)(5)*where k1 ,2 and k1 ,2 denote the forward and reverse rate constants for the abovereactions, respectively. In comparison to these reactions, the dissociation ofwater and also the second dissociation of carbonic-acid are sufficiently rapid sothat the dissociation equilibria can be assumed asand(6)(7)where Kw and KH2C03 denote the equilibrium constants for water and carbonic-acid,respectively.On the basis of the rate law of the reactions given by equations (4) and (5)and also the dissociation equilibria by equations (6) and (7), the conservation ofCO2 dissolved physically in the film can be written asNeglecting the effects of H2C03 , a total carbon balance in the film givesO([C02 ] + [HC03-]) 02[C02 ] O{O[HC03-] _ 1 (d¢)}u - D -- + 11. - - [HCO ] - -ox - C02 oy2 ~HC03- oy oy 3 RT dyElectrical neutrality requires the following conditions:andCombining equations (6) and (11) yieldsandWhen the flux of each ionic species is formulated in terms of equation (3) andsubstituted into equation (10) then introducing equation (11), we get(9)(10)(11)(12)(13)24 Shigeru NOZU and Hideo INABAYl50.5'I'x= 2.0 m 1.0r- 0.5-0.20.1r- 0.05 0.03oo,......No8........,......'...,oUI"-'YIS0.5x= 2.0 m 1.0 0.5oo-,......No8,....,No8Fig.Z Variations of [COZ] and [HC03-] with y, 0 0.1 rumd<t>dyRT (DHC03- - DOH-) a[HC03-]/ay + (DOH_ - DH+) a[H+]/ayt (DHC03- - DOH_)[HC03 ] + (DOH- + DH+)[H+](14)Equations (8) and (9) were solved to provide the concentration distributions ofCOZ and HC03-. Initial and boundary conditions are[COZ] 0, [HC03-]°at x°(15a, b)NCOZ 0, NHC03-°at y°(16a, b)[COZ] [COZ]i' NHC03-°at y 0 (17a,b)Solutions of equations (8) and (9) subject to the initial and boundaryconditions (15) ~ (17) were obtained numerically. A finite difference schememarching in x-direction were employed to solve them simultaneously and iterativelyat each x-step.3 RESULTS AND DISCUSSIONNumerical calculations were conducted under the conditions of [COZ]i = 10- 8 Mand T = 353.15 K for 0 = 0.1 and O.Z rum. Figures Z and 3 show the variations of[COZ] and [HC03-] with y at various x-locations for 0 = 0.1 mm and O.Z rum,respectively. A rapid penetration of CO2 into the film can be observed withincreasing x, whereas the production of HC03- proceeds gradually. In both cases,the [HC03-] value near the liquid-gas interface is larger than that near the wallsurface at smaller x, whereas the [HC03-] is seen to be almost unchanged along y atlarger x. This indicates that the production rate of HC03- in the film isFacilitated Diffusion of Carbon-Dioxide in a Living Body Liquid Film 250.5x= 8.0 m5.0~102.0~1.0~0.5... 0.05,-I 0.2"..Aoo,...,No8.......,.....,' ....ou:cw0.5oor-tNo8-,.....,NoU'-'Fig.3 Variations of [C02] and [HC03-] with y, 0 0.2 mID10°r---------------------------,with dissociationWithout dissociationequilibrium6:: 0.1 mm---- ------------ -=-;:.;-;:=----- --=-----.././././././Nou19.00-0-,.....,.....o~ 10-1......,+,......,oUN:c'-'+~o8x mcontrolled by the fluid velocity along with the values of [C02], [HC03-] and [OH-],and the rate constants of the reactions.Figure 4 presents the axial variations of the total [C02] in the film, [C02]+[H2C03 ]+[HC03-], for 0 = 0.1 and 0.2 mID. The results obtained on the basis of themodel without dissociation are also shown for comparison. The [C02]+[H2C03 ]+[HC03-]26 Shigeru NOZU and Hideo INABAvalue increases monotonically with x, then reaches gradually the prediction obtainedfrom an equilibrium analysis. Comparison of the results with and without dissocia-tion indicates that the effects of the facilitated diffusion is more remarkable forthe thinner film at larger x. This is in accord with the results showry in Figs.2and 3.The present model can be extended to a biological system with slow chemicalreactions. Also, it can be adopted to study the facilitated diffusion of carbon-dioxide in a condensate film by taking the effects of the variation of filmthickness into consideration.REFERENCES[1] Meldon,J.H., Chem. Engng. Commun., 16, (1982), pp.263-300.[2] Kern,D.M., J. Chem. Education, 37, (1960), pp.14-23.[3] Otto,N.C., and Quinn,J.A., Chem. Engng. Sci., 26, (1971), pp.949-961.

Facilitated Diffusion of Carbon-Dioxide in a Living Body Liquid Film

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