Modeling of Absorption of NO 2 with Chemical Reaction in a Falling Raindrop

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1 Korean J. Chem. Eng., 0(), (003) Modelg of Absorption of NO with Chemical Reaction a Fallg Radrop Vishnu Pareek, Vod K. Srivastava* and Adesoi A. Adesa** Department of Chemical Engeerg, Curt University of Technology, GPO Box U987, Perth, WA, 6845, Australia *Department of Chemical Engeerg, Indian Institute of Technology Hauz Khas, New Delhi, India **School of Chemical Engeerg and Industrial Chemistry, University of New Sh Wales, Sydney 05, Australia (Received 8 August 00 accepted 9 August 00) AbstractA model has been proposed for the scavengg of NO a fallg radrop. After absorption, aqueous NO undergoes a second order reaction to form various ions such as NO, NO 3 and H. The model is based on the unsteady state convective diffusion equation, which was solved for given boundary conditions by usg implicit alternate direction (ADI) method. The circulation of fluid side and side the radrop has been taken to account to realistically describe the flow field the numerical doma. The model predictions dicate that the ph of a radrop is a direct function of the drop size and bulk concentration of NO. The model predicted a ph of ab 4.9 for a 00- micron radrop fallg through a 0-ppb ambient concentration of NO. For the same ambient concentration of NO, a 0-micron radrop would have a ph of ab The predictions also suggested that for all practical purposes the gas phase resistance may be taken as the rate-controllg step. The predicted values of gas-side mass transfer coefficient compared well with the estimated values usg standard mass transfer correlations. Key words: Absorption, Acid Ra, Scavengg, Radrop, Fite Difference INTRODUCTION Large quantities of pollutg gases (such as SO x and NO x ) are troduced to the atmosphere through a variety of dustrial and automobile activities. These gases can directly affect the welfare of the ecosystem around the tense source through dry deposition. However, the widespread effect on the ecosystem is caused by the absorption of these gases rawater, which can decrease the ph of rawater appreciably, thus leadg to a phenomenon called acid ra. As rawater is equilibrium with the atmospheric carbon dioxide, which yields a natural acidity with a ph of ab 5.6 [Sefeld, 986], a ph of less than 5.6 may be considered an dication of SO and NO x absorption. A number of attempts have been made to describe the mechanism of trace gas scavengg a fallg radrop for SO absorption and its subsequent transformation to various chemical species [Hales, 97; Baboolal et al., 98]. However, the data on the absorption of NO x a radrop are scarce, even though many areas of the world they have been found to contribute significantly to the acidity the rawater [Ahmed et al., 990]. Furthermore, a maority of the reported studies on NO x the literature focussed on the measurement of chemical composition of rawater [Saxena et al., 996]. Relatively few laboratory studies have been carried to establish the pathways by which the oxides of nitrogen are converted to various species after they are absorbed water [Bambauer et al., 994; Schwartz and White, 98, 983]. Although useful qualitative formation can be obtaed from such measurements, especially with regards to the deposition rate, no quantitative formation is available on the efficiency with which a particular gas the atmosphere is scavenged by ra. To whom correspondence should be addressed. v.pareek@curt.edu.au To overcome this difficulty a number of acid ra models have been reported [Metcalfe et al., 00]. Most of these models are statistical nature and are used by various environmental agencies the formulation of acid ra control policies. The Hull Acid Ra Model (HARM) [Metcalfe et al., 00], extensively used by the UK government for the control of acidification of ra, is a receptor oriented Lagrangian statistical model. These models, however, do not estimate the efficiencies with which drop of a given size scavenges NO. In this paper we present a model for scavengg of NO a fallg radrop. The model is based on the unsteady state convective diffusion equation, which was solved by usg a fite difference approximation scheme called implicit alternate direction (ADI) method [Carnahan et al., 969]. The only model puts are radrop size and prevailg NO concentration. A fallg radrop exhibits vigorous ternal circulation. Therefore, to realistically describe the flow-field side and side the radrop, the velocity relations reported by LeClair et al. [97] were corporated the model. The model is capable of predictg ph levels likely to be achieved by a radrop fallg through a given ambient NO concentration; hence, it can be used to set the ambient limits on the emission of NO for the allowed levels of ra ph. THEORETICAL CONSIDERATIONS. Model Equations and Boundary Conditions For diffusion of NO side a fallg radrop, the general convective diffusion equation [Bird et al., 960] (on neglectg azimuthal coordate) reduces to: A, = D C A, t AL A, C - A, -- A, θ θ r v θ r r v A, r A, k r θ C A, () 38

2 and for side the radrop we have: A, - = D C A, t AG A, - -- C -- A, -- A, - r θ r θ r v A, v r - θ ---- A, - r θ Modelg of Absorption of NO with Chemical Reaction a Fallg Radrop 39 C A, and C A, denote the concentration of NO side and side of the radrop, v r and v θ are components of velocity the radial and angular directions, respectively. Eqs. () and () were solved by usg followg boundary and itial conditions: () () At t=0, C A, =0 (for all a r 0 and π θ 0) () At θ=0 and π, ( A, / θ)=0 (for all a r 0 and t>0) (3) At r=0, ( A, /)=0 (for all t>0 and π θ 0) (4) At r=a, D A, AL A, AG (for all t>0 and π θ 0) (5) At t=0, C A, =C Ao, (for all r>a and π θ 0) (6) At θ=0 and π, ( A, / θ)=0 (for all r>a and t>0) (7) At r= C A, =C Ao, (for all t>0) Fig.. Flow field side a fallg radrop. Introducg dimensionless variables D T = AL t , N Peclet L a C A, C Ao, v r, = av , o N D Peclet G AL the Eqs. () and () become:, = C = , C = , C A, H = C Ao,, A C Ao, C Ao, av , o D AG D V R = -----, V θ = -----, D = AL r k, R = - and k = C Ao, a a V o v θ V o D AG = C C T R θ θ R R R V θ N -- Peclet, L V R R R θ kc D AL (A) Inside the radrop: σ V R = ---- ( R )cosθ ( σ) σ V θ = ---- ( R )sθ ( σ) Outside the radrop: 3 σ V R = ----R ----R 3 ( σ) ( σ) cosθ 3 σ V θ = ----R ----R 3 4 ( σ) 4 ( σ) sθ (3) (4) (5) (6) D = C C T R R θ R θ R R N --- Peclet, G V V R θ R R θ (A) with the new itial and boundary conditions: (A) At T=0, C =0 (for all R 0 and π θ 0) (A) At θ=0 and π, ( / θ)=0 (for all R 0 and T>0) (3A) At r=0, ( / R)=0 (for all T>0 and π θ 0) (4A) At r=a, D = H R A R (for all T>0 and π θ 0) (5A) At t=0, C = (for all R> and π θ 0) (6A) At θ=0 and π, ( / θ)=0 (for all R> and T>0) (7A) At R= C = (for all T>0) As poted earlier a fallg radrop exhibits vigorous ternal circulation as shown Fig.. The flow-field side the radrop is also affected by drag force of atmospheric air. The dimensionless components of velocity profiles both side and side the radrop are given by [LeClair et al., 97]: Fig.. A typical numerical grid. Korean J. Chem. Eng.(Vol. 0, No. )

3 330 V. Pareek et al. σ is the viscosity ratio (water to air). The termal velocities V o of fallg radrops were estimated by usg the method led by Beard [976].. Solution Scheme The partial differential Eqs. () and () were solved by usg a fite difference approximation scheme called the implicit alternate direction (ADI) method [Carnahan et al., 969]. A sample of numerical grids utilized the calculations is shown Fig.. The scheme consists of two steps: () Explicit scheme for θ-direction and implicit scheme for r-direction the first half of the step-time. () Explicit scheme for r-direction and implicit scheme for θ-direction the next half of the step-time. As this scheme is a hybrid of explicit and implicit schemes, it enoys the advantages of both and was found to be very stable, with the convergence typically achieved with 50 iterations. Discretization of the space side and side the radrop was achieved by usg an geniously developed Fortran code called ADIAN (see Appendix A). The application of the scheme to Eqs. () and () gave a set of lear equations, with a tridiagonal coefficient matrix, which were readily solved usg a subre called MATRIX [Srivastava, 98]. 3. Physical and Chemical Data The diffusivities of NO air and water were calculated by usg the Fuller method and Wilke-Chang equation, respectively [Bird et al., 960]. The estimated value of D AL at ambient temperature (0 o C) was m s and that of D AG was m s. The value of Henry s equilibrium constant H A for the absorption of NO water was taken as.0 0 mol L atm [Scheartz and White, 983]. The aqueous oxidation of NO has a second order ketics [Sefeld, 986] with the rate constant k = mol L s. RESULTS AND DISCUSSION The performance of the model described above was evaluated with respect to variation drop size and ambient concentration of Fig. 3. Radial concentration profiles of NO (aq) after a time of 0.5 s (Drop size=00 micron, ambient NO concentration =80 ppm). Fig. 4. Radial concentration profiles of NO (aq) after a time of 0.5 s for different drop sizes (Angular coordate=0, ambient NO concentration =80 ppm). NO. Fig. 3 shows radial concentration profiles side and side a 00-micron radrop fallg through an ambient NO concentration of 80 ppm after a fallg-time of 0.5 sec. Inside the radrop (referred as bulk liquid Fig. 3), the concentration profiles for various angular coordates showed a maximum at a dimensionless radial distance of ab This effect may be attributed to the ternal circulation with the radrop. It is apparent from Fig. 3 that the begng of its fall (t<0.5 s) NO concentration decreased with creased angular coordate. This is because the lower hemisphere (θ π/), the fluid flows the opposite direction (Fig. ) to diffusion, while the upper hemisphere (θ π/) both momentum and diffusional transport are the same direction, thus enhancg mass transfer. As may be expected, this effect was less pronounced the smaller radrops. (cf. Fig. 4). Furthermore, the model predictions dicated that NO concentration was essentially variant with angular coordate side the radrop. Fig. 3 also shows that the gas phase NO concentration dropped rather sharply from 80 ppm the bulk to ab 0.7 ppm at the gas-liquid terface suggestg that gas film resistance was probably the rate-controllg step. For mass transfer to a spherical drop, Frossg s correlation [Bird et al., 960] gives, k g =0.5 m s (K g = mol m 3 s atm ). Assumg that the film theory describes the mechanism for NO transport from the gas to the radrop, the data from Fig. 3 may be used to estimate k g =D AG /z as 0.69 m s (K g = mol m 3 s atm ). The good comparison between predicted data and literature correlation further strengthened the confidence the unsteady state convective diffusion model proposed this work. Moreover, applyg the criterion K g P -- Ao << k C Ao, for the gas-film controllg step, the present data provide the LHS as and hence it may be confirmed that gas-side resistance was the controllg step. Angular concentration profiles associated with the radial profiles Fig. 3 are shown Fig. 5. A snapshot of the angular concentration profiles shows that the aqueous NO concentration decreased with angular coordate (θ) irrespective of the radial distance selected. However, apparent tersection of some of the profiles was March, 003

4 Modelg of Absorption of NO with Chemical Reaction a Fallg Radrop 33 Fig. 5. Angular concentration profiles of NO (aq) after a time of 0.5 s (Drop size=00 micron, ambient NO concentration =80 ppm). Fig. 6. Variation concentration of NO (aq) at different radial coordates (Drop size=00 micron, ambient NO concentration =80 ppm). due to a combation of unsteady state conditions and ternal recirculation of the liquid with the radrop. As may be seen Fig. 6, steady state was obtaed after ab sec. As a result of NO absorption, the concentration of H -ions side the radrop creased. A concentration history of H -ions is Fig. 8. Average fal ph Vs drop size. shown Fig. 7. As may be readily seen the Figure, the smaller drops had a higher concentration of H -ions at a given time. For example a 0-micron drop achieved an H -ion concentration of ab 0 5 mol L after a fallg time of s, for the similar duration the H -ions concentration for a 00-micron drop was only ab mol L. Furthermore, the smaller drops achieved an equilibrium concentration of H -ions earlier than the larger drop. The crease H -ions concentration led to a decrease the water ph. Fig. 8 shows the effect of radrop size and bulk concentration of NO on the fal average ph (allowg a drop time of 5 s) achieved by a fallg radrop. The ph dropped exponentially with the crease ambient NO partial pressure and with decrease radrop size. It is therefore clear that the ph of radrop is very sensitive to the crease NO concentration and drop size. For example, a radrop of average size of 00-micron can achieve a ph as low as 4.4 when fallg through an NO concentration of ab 00 ppb. The effect of liquid side Peclet number N Peclet, L =av o /D AL on NO concentration is shown Fig. 9. By defition, Peclet number this case is a direct measure of drop size and its termal velocity; therefore, it may be taken as a direct measure of the circulation effects with the radrop. As dicated above, the angular gradient concentration of NO was maly due to the circulation effects. Fig. 7. Variation concentration of H ions (Ambient NO concentration=00 ppb). Fig. 9. Effect of Peclet number on concentration of NO (aq) after a time of 0.5 s (Drop size=00 micron, ambient NO concentration=80 ppm). Korean J. Chem. Eng.(Vol. 0, No. )

5 33 V. Pareek et al. At lower values of Peclet number, when the circulation effects were negligible, various constant A curves cocided with each other as shown Fig. 9. For higher values of Peclet number the concentration profiles correspondg to different values of angular vector A diverged away from each other, dicatg the domance of circulation effects. Interestgly, the concentration profile for A=π/ was almost dependent of the Peclet number. This was apparently due to the fact that at A=π/, all the velocity-vectors with the ra drop were parallel to each other and were potg vertically upwards or downwards and there was no component of velocity potg the direction of diffusion. Hence, the concentration profile at A=π/, was free of the circulation effects. CONCLUDING REMARK An acid ra model for the scavengg of NO has been developed. The model is based on the unsteady state convective diffusion equation. The underlyg partial differential equations were solved by usg the implicit alternate direction method to avoid numerical stabilities. The ternal circulation effects played an important role the begng of the absorption of NO the fallg radrop. The predicted results showed that the ph of a fallg radrop was very sensitive towards variation bulk concentration of NO and drop size. Sce the model is based on the unsteady state absorption of species, it adequately describes the mechanism of trace gas scavengg. APPENDIX A The alternate direction implicit (ADI) method utilizes the advantage of both implicit and explicit fite difference schemes. In general, it is stable, convergent and is not computer memory tensive [Carnahan and Wilkes, 969]. In this study, the ADI method constituted the folowg two steps:. Explicit Scheme for θ-direction and Implicit Scheme for r-direction for the First Half of the Time-step -. For Inside the Drop On employg the scheme to Eq. (A), we have, n AC, B C n, i, D C n, i, = E n. A = T ( R) B = F ( R) D = F ( R) E n n n n = G i C, H C,, i I C,, i k( C n, ) F = N --- Pe V ---- R, R i ( R) 4 R G i = T R i ( θ) H = J R i ( θ) I = J R i ( θ) (A) cot θ J = ( ) ( θ) N --- V Pe θ, 4R i θ The superscript n represents the time-step, subscripts i and represent radial and angular discretizations respectively. Superscript (n /) LHS of Eq. (A) represents the value at half time-step. -. For Outside the Radrop Similarly, the application of the ADI method side the radrop (Eq. A) yielded, (A). Implicit Scheme for θ-direction and Explicit Scheme for r-direction for the Second Half of the Time-step -. Inside the Radrop On usg the implicit scheme for θ-direction and explicit scheme for r-direction for the next half of the time-step, we have, -. For Outside the Radrop Similarly, for side the radrop, R i n AC, L B C n, i, D C n, i, = L n. n n n n = G i C, H C,, I C,, n n n M i C, N C,, i O C,, i = P n M i = T R i ( θ) N = J R i ( θ) O = J R i ( θ) n n P = QC, S C n, i, U C n n, i, k( C ) Q= T ( θ) S = F ( R) U = F ( R) n n n M i C, N C,, i O C,, = W n W (A3) (A4) On usg boundary conditions (A-7A), Eqs. (A-A4) reduced to a set of tri-diagonal matrices, which were then solved by usg MATRIX [Srivastava, 98]. The flux condition at the surface of the radrop was corporated by an iterative method. NOMENCLATURE a : radius of radrop [m] A : angular coordate [radian] C A, : concentration of NO side the drop [mol L ] C A, : concentration of NO side the drop [mol L ] C Ao, : itial concentration of NO side the drop [mol L ] C Ao, : itial concentration of NO side the drop [mol L ] C :C A, /C Ao,.. n n = QC, S C n, i, U C n, i, March, 003

6 Modelg of Absorption of NO with Chemical Reaction a Fallg Radrop 333 C :C A, /C Ao, D :D AL /D AG D AL : diffusivity of NO water [m s ] D AG : diffusivity of NO air [m s ] D p : ra drop diameter [m] H A : Henry s constant for absorption of NO water [mol L atm ] k k : dimensionless rate constant, k = C Ao, a k : rate constant [mol L s ] k g : gas side mass transfer coefficient [m s ] K g : gas side mass transfer coefficient based on partial pressure [mol m 3 s atm ] N Peclet, L : liquid side Peclet number (av o /D AL ) N Peclet, G : gas side Peclet number (av o /D AG ) P O : partial pressure of oxygen [atm] r : radial vector measured from the center of radrop [m] R : dimensionless radial coordate (r/a) t : time [s] T : dimensionless time (D AL t/a ) v r : radial component of velocity [m s ] v θ : angular component of velocity [m s ] V o : termal velocity of radrop [m s ] V R : dimensionless radial component of velocity, (v r /V o ) V θ : dimensionless angular component of velocity, (v θ /V o ) z : mass transfer film thickness [m] Greek Letters θ : angular vector σ : ratio of viscosity of water to viscosity of air REFERENCES D AL Ahmed, A. F. M., Sgh, R. P. and Elmubarak, A. H., Chemistry of Atmospheric Precipitation at the Western Arabian Gulf, Atmospheric Environment, 4A, 97 (990). Baboolal, L. B., Pruppacher, H. R. and Topalian, J. H., A Sensitivity Study of a Theoretical Model of SO Scavengg by Water Drops Air, Journal of the Atmospheric Sciences, 38, 856 (98). Bambauer, A., Brantner, B., Paige, M. and Novakov, T., Laboratory Study of NO Reaction with Dispersed and Bulk Liquid Water, Atmospheric Environment, 8, 35 (994). Beard, K. V., Termal Velocity and Shape of Cloud and Ra Drops Aloft, Journal of the Atmospheric Sciences, 33, 85 (976). Bird, R. B., Stewart, W. E. and Lightfoot, E. N., Transport Phenomena, John Wiley & Sons, New York (960). Carnahan, B., Luther, H. A. and Wilkes, J. O., Applied Numerical Methods, John Wiley and Sons, New York (969). Hales, J. M., Fundamentals of the Theory of Gas Scavengg by Ra, Atmospheric Environment, 6, 635 (97). LeClair, B. P., Hamielec, A. E., Pruppacher, H. R. and Hall, W. D., A Theoretical and Experimental Study of the Internal Circulation Water Drops Fallg at Termal Velocity Air, Journal of the Atmospheric Sciences, 9, 78 (97). Metcalfe, S. E., Whyatt, J. D., Broughton, R., Derwent, R. G., Fnegan, D., Hall, J., Meter, M., O Donoghue, M. and Shutton, M. A., Developg A Hull Acid Ra Model: Its Validation and Implications for Policy Makers, Environmental Science and Policy, 4, 5 (00). Saxena, A., Kulshrestha, U. C., Kumar, N., Kumari, K. M. and Srivastava, S. S., Characterization of Precipitation at Agr, Atmospheric Environment, 30, 3405 (996). Schwartz, S. E. and White, W. H., Ketics of Reactive Dissolution of Nitrogen Oxides to Aqueous Solution, Advanced Environmental Science and Technology,, (983). Schwartz, S. E. and White, W. H., Solubility Equilibria of the Nitrogen Oxides and Oxy-Acids Dilute Aqueous Solution, Advanced Environmental Science and Technology, 4, (98). Sefeld, J. H., Atmospheric Chemistry and Physics of Air Pollution, John Wiley and Sons, New York, 95 (986). Srivastava, V. K., The Thermal Crackg of Benzene a Pipe Reactor, Ph.D. Thesis, University of Wales, Swansea (UK) (98). Korean J. Chem. Eng.(Vol. 0, No. )