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The effect of the gas side resistance on absorption with chemical reaction from binary mixtures
Chemical Engineering Science, 1975, Vol. 30, pp. 1477-1482.
Pergamon Press.
Printed in Great Britain
THE EFFECT OF THE GAS SIDE RESISTANCE ON ABSORPTION WITH CHEMICAL REACTION FROM BINARY MIXTURES A. TAMIRt Department of Chemical Engineering, Pembroke Street, Cambridge, England
Y. TAITEL Department of Chemical Engineering, University of Houston, Houston, TX 77004,U.S.A. (Received 16December 1974;accepted 5 June 1975) Abstract-A predictive model based solely on the conservation equations of both phases is presented for the study of the effect of the gas-side resistance in multicomponent absorption with chemical reaction in a laminar liquid film. The model is applicable for short exposure times to a variety of flow patterns, e.g. a cylindrical jet, a wetted-wall column, or an inclined plane. The model may be applied to the wetted-wall column and the inclined plane for any contact time. The numerical scheme for the downstream solution has the advantage of utihsing a similarity solution as its starting point. The model is applied to the system CS,-N, where CS, is absorbed in an aqueous solution of amine and undergoes pseudo first order reaction. It was found that the a priori neglect of the gas-side resistance is not justified. The error introduced in evaluating the absorption flux by neglecting the gas-side resistance is dependent on the reaction rate constant, bulk concentration of the species, length of the absorber and the total pressure. Errors aenerallv increased with increasing values of the above narameters. The gas-side resistance is reduced when the gaseous-atmosphere is under forced convection.
INTRODUCTION
When a pure gas is absorbed into a solution, the main
resistance which controls this process is offered by the liquid phase. This means that the behaviour of the gas phase need not be considered. However, in absorption of one or more species from a multi-component gaseous mixture, the above a priori approach is not justified. This is due to the diffusional processes of the species in the gas phase which cause their concentration at the gas-liquid interface to deviate from their concentration in the bulk of the gas phase. Depending on the operating conditions, this deviation might be significant; it is maximal, for example, in the extreme case of instantaneous surface reaction where the concentration of all reactive species vanish. As a result of the above deviation, absorption fluxes of the individual species might be different as compared to those which would have been obtained had one assumed that the bulk conditions prevail also at the interface. Hence, in the case of multicomponent absorption, the knowledge of the interfacial concentrations distribution becomes the key for calculating absorption fluxes. To achieve this goal, a simultaneous solution of the conservation equations of the gas and the liquid phases is required. It should be noted that the phenomena occurring in absorption from binary mixtures are very similar to those in condensation of a binary vapour mixture. In the latter case, the concentration of the less volatile species at the interface is lower than that in the bulk of the condensing vapour. Similarly, in absorption, the concentration of the more soluble species, namely the one which has a low tDepartment of Chemical Engineering, Ben-Gurion University of the Negev, Beer-sheva, Israel. 1477
value of Henry’s constant, would be less at the interface than in the bulk. The phenomena discussed above become more appreciable if the absorption is accompanied by a chemical reaction in the liquid. Theoretical investigations which consider simultaneous absorption of binary mixtures in the presence of chemical reaction are few [l, p. 46; 2, p. 116; 3-8,131. The solutions obtained were either for the penetration or the film model. The common approach to all investigations was to consider only the mass transfer in the liquid phase and to neglect the effect of the diffusional processes in the gas phase. This greatly simplifies the analytical solutions but the error introduced in such calculations is unknown. To the best of the present author’s knowledge, a complete analysis is not available for absorption from a binary gaseous atmosphere in the presence of chemical reaction which takes into account the simultaneous processes occurring in both phases. In the present work, a complete formulation of this problem is made which is based on a boundary layer model in the gas phase [1, p. 1461 through which the various transport processes occur. The boundary layer is formed due to the interfacial drag of the gaseous atmosphere with a laminar absorbing-reacting film. Solution of the equations is demonstrated for the case where the solute gas undergoes a chemical reaction with a dissolved reactant which is irreversible and first order with respect to concentration of the dissolved gas. It should be noted, however, that many reactions which are second order, under some circumstances may also be considered as pseudo-first order. The condition for first order reaction is provided by Danckwerts [l, pp. 452101. It enables us to make an estimate of the exposure time below which a reaction may be treated as pseudo-first
A. TAMIRandY. TAITEL
1478
order. It should be noted that also other kinds of reactions in the liquid may be treated conveniently using the same approach. A forward step-by-step marching scheme is used in the numerical calculations. It has the advantage of utilizing a similarity solution which exists at the inlet region. The above-mentioned similarity solution corresponds also to the so-called “penetration model” regime which is valid for a semi-infinite body. FORMULATION OF THE PHYSICAL MODEL AND METHOD OF SOLUTION
numerical scheme adopted here. If both components are absorbed, the liquid phase becomes a quarternary system. However, due to the relatively low solubilities encountered between gases in liquids, the interaction between the diffusive fluxes may be neglected and each species diffuses in the solvent as if in a binary mixture with an effective diffusivity 6, i = 1, 2. In the gas phase boundary layer, the distribution in concentration of the more soluble species, designated as 1, may be obtained from the following equation:
uaw,,vaw,=Da’w, The physical model that is demonstrated in Fig. 1 a$ ’ ax ay consists of a laminar liquid film with a fully developed velocity profile which enters a wetted wall or a channel type absorption chamber. The liquid film absorbs one or The other equations for total mass and momentum conboth species from a binary gaseous atmosphere. The servation are absorbed species react with a material which has been aU,aV=, dissolved in the film prior to its entrance into the system. ax ay The chemical reaction occurs isothermally. The gaseous au au a34 atmosphere, which is assumed to behave as an ideal u-&+v-=v~. mixture, enters the absorber with a uniform velocity u,. ay ay Upon its entrance, the mixture attains the surface velocity u, of the liquid film, due to the interfacial drag. It is Note that buoyancy forces are neglected in eqn (4). This is assumed that the transport processes in the gas phase are a reasonable approach due to the imposed velocities u, carried out through boundary layers of momentum and and u, which are relatively large. Equations (1) and (2) to (4) are solved under the mass. A constant properties analysis is carried out, but in order to account for differences in concentrations be- following boundary conditions where i = 1, 2: tween the bulk of the gas and the interface, the properties x =o: u = UI, w, = WI,, IVi = I& are evaluated at an average concentration between the (5) bulk and the interfaceb, lo]. y = m: u = Urn,w, = WI, (6) The mass conservation in the liquid of each absorbed species where i = 1,2 is described by y =-L: aR/ay =o. (7) At the interface, namely y = 0, the gas-liquid relationships are: To simplify the solution for sparingly soluble gases, transverse convection effects may be ignored as compared to the axial ones as done in deriving the above equation. The transverse effects were taken into account in the mass flux conservation equation through the interface as given by eqn (9). Note that the soluble component with a local mass concentration % is consumed by an irreversible first-order chemical reaction according to a reaction rate constant ki. However, other kinds of chemical reactions may also be considered by using the same
I_”
u = uv
(8)
-p-$-$+4s~i= _p~~+pvw, (9) which expresses the continuity of the mass flux of each component. The application of the above boundary condition deserves some attention. Equation (1) implies that the transverse velocity in the liquid, 5, is zero. However, practically 5 has a finite value and hence at the interface pfi = pv. As previously mentioned, the assumption made in deriving eqn (1) was that for sparingly soluble gases WiWy is negligible as compared to the axial convection term. For highly soluble gases this assumption is not justified and a complete formulation is given elsewhere [ IS]. Finally, the Henry’s Law
(10)
Fig. 1. Physical model and co-ordinates.
which indicates an equilibrium at the gas-liquid interface is applicable. In our analysis we do not make any distinction between soluble and insoluble species. Indeed, if one wishes to consider an insoluble gas, all that is needed is to equate its Henry’s constant, Hi, to infinity.
1479
Gas side resistance on absorption with chemical reaction
The solution of the above equations is performed in a transformed co-ordinate system. In the liquid we use for small x the (5, i) co-ordinates defined by
xfj,
5=L2u,
The corresponding boundary conditions for i = 1,2 are 5
=o:
f’= UJU,,
w,,=
WI,
R. = iv8
n =cQ: f’ = ucOIuw, w, - WI,
i=&. q=o:
This enables us to obtain accurate concentration gradients at the gas-liquid interface, namely, the absorption fluxes. By small x(or 4) is meant the region where the penetration of a change in concentration applied at the liquid surface has not yet reached the wall (y = -I,). Hence the mass conservation equations for i = 1,2 reads
f’=l,
fw=f+2$$
(19) (20)
w,=w,w.
(21)
Note that W,,:the concentration of species 1 at the interface, is unknown and is obtained in course of the solution as later explained. To solve eqn (12) for small 5 we use for i = 1, 2 i +m:
Wi = Wi”
(22)
On the other hand, for large 6, eqn (12-l) is solved with 8=-l:
aR/ag=o.
(23)
Ki is a dimensionless reaction rate constant defined as
(13)
The continuity in the mass flux for an individual component through the interface is now expressed by eqn (24) noting that pii = pv
It is interesting to note that eqn (12) shows that chemical reaction may be ignored for very small 5 and that diffusion is dominant. Equation (12) is applicable for a wetted-wall absorber or an inclined plane. The cylindrical jet absorber with a uniform velocity profile is characterized by short time of exposure where the term 45rs’<<1. Hence, eqn (12) satisfies also this flow pattern for small 6. For large 5, the numerical solution was conveniently continued in the co-ordinates (t, 8) where 5 is given in (11) and
F.= I
d a% ’ fw ( 2pmdil-Sc-‘T
w,__2. !j
>
(24)
where inter-facial equilibrium requires eqn (10) to hold. The numerical solution which takes advantage of a similarity solution at 5 = 0, is a step-by-step advancing technique. The differential eqns (12, 12-1)and (17, 18) are put in a finite difference form using implicit scheme. The Crank-Nicolson method is used for eqns (12, 12-l) which yielded a three-diagonal matrix. Also the diffusion eqn (17) was transformed into a similar matrix where for the +. (14) momentum eqn (18) a five-diagonal matrix was obtained. Assuming profiles are known at distance 5, the advance Here the mass conservation equation for i = 1, 2 is to 5 t At is carried out as follows: a guess is made for the described by interfacial concentration W,w and hence WZw= 1 - Wlw. By applying Henry’s Law, the equilibrium solubilities W, WZ,Vat the liquid surface are calculated. They are designated for i = 1, 2 as Wiw.equit. At this stage the physical properties are evaluated at a reference concentThe boundary layer equations for the gas phase are ration 0.5 ( Wi,t W,,) [lo]. It should be noted that aldescribed in (5, 7) co-ordinates where .$ is given in (11) though our analysis assumes constant properties, they and might be corrected at each step for their slight variations (15) with x. Knowing the interfacial concentrations makes it possible to calculate the concentration distribution at the gas and liquid phases and the gradients aWi/aq, a%/&ij If u = at,b/ay, u = -ag/ax and a dimensionless stream at the interface. For small 5, aW,/&j is calculated from function f is defined by eqn (12) with (22) where eqn (12-1) with (23) are applied (I = V(vu,x )f(& 771 (16) for large 5. A smooth transition from the (5, ii) grid into the (&-jr) grid was carefully applied at a suitable 5 location. we obtain for the diffusion equation a W,/Sj were calculated at ?j = 0 using a three point formula. Note again that since an accurate derivative is required it is essential to use the (6, q) grid rather than the (5, 7) grid for small 5, namely, for the “penetration model” regime. fwis calculated from eqn (21) and hence from eqn and for the momentum equation (24) new values may be ontained for the W’s. Designating the difference ( ~lw.cquil. - F,w) by K it is observed that F, is a function of the initial guess W, The iteration procedure is terminated if FI approaches zero, and to W,
W.
1480
A. TAMRand Y. TAITEZL
achieve fast convergence the method of False Position [ll, pp. 2-561 is used. As previously mentioned, the solution for 6 = 0 corresponds to a similarity solution in the absence of chemical reaction (see eqn 12). The distribution for this case are calculated similarly as described above.
more soluble species-C& whereas in the upper part they correspond to the insoluble species-N*. The parameter of the curves is the dimensionless first-order reaction rate constant K, which was varied in the range from 0 to 10’. It is clearly observed that the interfacial concentration of CS2 is always lower than its bulk concentration either in the presence or in the absence of chemical reaction APPLICATIONOF THE MODEL (K, = 0). For the insoluble species, namely Nz, the oppoAND DISCUSSION site behaviour is obtained. This phenomenon becomes The absorption into Dimethylamine solutions of CS2 more appreciable if the reaction is faster, namely if KI is from its binary mixtures with N2 was studied. This system increased, and if the absorber is longer. Note also that for was investigated by Kothari and Sharma[l4] and the each case all curves in Fig. 2 issue at 5 = 0 from the same reaction between CS2 in aequous amine solutions was point. This is due to the fact that the chemical reaction is found to follow pseudo-first order behaviour. From the ineffective at the entrance (see eqn (12)), compared to the kinetic data provided by Sharma, as well as CS2 solubility absorption process. It is also of interest to point out the and liquid diffusivity, Danckwert’s condition[l, pp. 45, difference in the behaviour of the curves for CS2and NZin 2101indicated that the reaction obeys lirst order behaviour Fig. 2 corresponding to K, = 0 (no chemical reaction) and, over a wide range of exposure times (up to 2OtlOsec.). The say, K, > 1. It is observed that the ratio W~V/ Wm in the Refs. (11,12) were consulted for the selection of additional latter case is continuously decreasing while the curve for physical and transport properties of the individual species K, = 0 should approach a value of unity, namely the bulk as well as for their behaviour in the mixture. The reference concentration. This is due to the fact that in the absence properties were calculated as outlined before. It was of chemical reaction, the absorbing stream must eventudecided also to perform the calculations for 30°C (14), zero ally become saturated with CSI and therefore its interfainlet concentration of CSI in the liquid solution and total cial concentration will be equal to its bulk concentration. pressures of one and ten atmospheres. At the lower The upper curves in Fig. 2 indicate that the insoluble pressure it was possible to investigate the effect of bulk species, N2, tends to accumulate at the interface. This concentration WI..of CSt for the following values 0+5,0*1, accumulation enables it to diffuse backwards to the bulk 0.05 and 0.01, whereas at ten atmospheres only the last of the gaseous atmosphere and to balance its convective three concentrations could be considered. NZwas taken as stream caused by bulk flow of the soluble species, C$. It insoluble noting that its solubility at 30°C is about 35 times may also be noted that the accumulation of NZ at one lower than that of CS2. atmosphere corresponding to W,, = 0.5 is much higher In Fig. 2 concentration on the interface along the than for Wlm= O-05at same pressure where the mixture is absorber length 5 are shown corresponding to CS, bulk almost N2. For example, at 5 = 0.15 the ratio Ww/W= = 1.3 in comparison to 1.03 in Fig. 2. Other general trends concentration of 0.05. They are conveniently expressed by the ratio W,/ W, where Ww is the weight fraction at which are drawn from Fig. 2 and 3 are: at the same total the interface and W, is the weight fraction in the bulk. pressure, the ratio WW/W, for CS2(at constant 5 and KJ The curves in the lower part ( WW/W, < 1) describe the is significantly reduced as WI.. is decreased. In addition, for similar values of W,=the ratio WW/W= becomes lower as the total pressure is increased. Both trends affect the actual absorption flux of CS2as compared to its maximum value under absorption at bulk conditions. 5 x 10-s ______-_---_-___$ ,’
Fig. 2. Interfacial concentration W, relative to bulk concentration W, for CS, and N, in absorption from a mixture of CS,N, with W,,= 0.05 at 1 and 10atmospheres.
8
Fig. 3. Dimensionless local flux of CS, in absorption from a mixture of C&-N2 with WI,= 0.05 at 10atmospheres.
Gas side resistance on absorption with chemical reaction
The deviation between the interfacial and bulk concentrations of the species corresponds to some magnitude of the so-called gas-side resistance. The existance of such resistence was emphasized by Astarita[8] who found also experimentally that its value was indeed significant. In Fig. 3 we report on the nondimensional local flux mL~(.$)/fi for CS2 in absorption from mixtures of CSZ-Nz. This value was calculated according to the lefthand side of eqn (9). Note that at 6 = 0 all curves emerge from the same point because at the entrance the chemical reaction is unimportant. The magnitude of this point depends on the total pressure and the bulk concentration of CS. As expected chemical reaction enhances absorption mass-transfer and the phenomenon is manifested towards increasing values of the reaction rate constant K,. Unlike the curves which were obtained in the presence of chemical reaction, the curve for KI = 0 in Fig. 3 is continuously decreasing. This corresponds to the fact that in the absence of chemical reaction the absorption will ultimately cease when the absorbing stream becomes saturated. In addition, it was observed that absorption rates of CS2 become higher as its bulk concentration and/or total pressure is increased. As recalled, forced convection in the gaseous atmosphere was also included in the analysis. This effect is demonstrated in Fig. 3 by the dotted lines for the case where u, is tenfold higher than the stream surface velocity u,. As expected, forced convection tends to alter the concentration profile and to increase absorption fluxes. The improvement depends on the bulk concentration, the reaction rate constant and the forced velocity. The practical conclusion which may be drawn is that forced convection is more efficient for longer ducts rather than for very short ones. In Figs. 4 and 5 the ratio mJm10 is plotted vs the absorbing length, where ml is the actual local absorption flux of CS2 and rn,’ is the flux evaluated at bulk conditions. The above ratio reflects upon the error which is introduced in the calculations should one neglect the effect of the gas-side resistance. Note that at a concentration level lower than 0.1 all solid curves coincide. In general it may be concluded that the local absorption flux of CS2 is lower than the one evaluated at the bulk conditions. This effect becomes stronger if K, and the total pressure are increased. By selecting typical values
___-$g -
0
x;
0.01.0~05.04
IO
02
03
0.4
Fig. 4. Mass flux ratio for 123, in absorption from mixtures of C&-N, with WI,= O.Ol,O.OS, 0.1 and 0.5 at 1atmosphere.
1481
Fig. 5. Mass flux ratio for CS2 in absorption from mixtures of C&-N, with W,,= 0.01,0.05 and 0.1 at 10atmospheres.
for the parameters appearing in 5 (eqn 11) and K, (eqn 13) while for k, in the values reported by Sharma[l4] were applied, it was found that K, may vary in the range of lo”-lo4 where 5 is in the order of 0.1. Hence the a priori neglect of the gas-side resistance is not justified. Finally it would be instructive to comment on the application of the above model for the interpretation of experimental data. In absorption studies with chemical reaction one usually measures the total absorption flux rather than the local values. The total absorption flux can then be compared with the theoretical value, if desired. The theoretical value of the total flux may be computed from our model by numerical integration of the local flux as calculated from eqn (9). Note, however, that the latter is equal to infinity at ,$= 0. Hence to compute the total flux one should start the integration from a very small value of .$(which has to be checked) ignoring the quantity which was absorbed up to this point, which is usually negligible. CONCLUDINGREMARKS
Previous analyses considering absorption of binary mixtures, with or without chemical reaction, tend to ignore the gas-side resistance. In other words, calculations were made by assuming that the bulk conditions of the absorbed gases prevail also at the interface. ~4complete analysis of binary mixtures absorption in a laminar film has been made to check this assumption. It incorporates a simultaneous solution of the gas-side boundary layer equations and the liquid stream in which chemical reaction also takes place. The solution was demonstrated over a wide range of operating conditions for the mixture C&-N, where CS, is absorbed in an aqueous solution of amine undergoing pseudo-first order reaction. The major conclusion which may be drawn is that the a priori neglect of the gas-side resistance is not justified and might lead to errors in evaluating the absorption fluxes as shown in Fig. 4 and 5. The error, in general, depends on the bulk concentration of the species, the reaction rate
1482
A.
TAMIR and
v 5 p I(,
constant, the total pressure and the length of the absorber. For absorption in the absence of chemical reaction the error is usually negligible. It was also found that forced convection of the gaseous phase remarkably improves the absorption rate and is most useful for long ducts rather than for very short ones. Acknowledgement-This work was performed while the first author spent his sabbatical year in the Department of Chemical Engineering at the University of Cambridge. The sabbatical year was supported by the Lord Marks Fund which is greatly acknowledged. Thanks are also due to Professor P. V. Danckwerts for the interest in this work and his collaboration, and also to Mr. P. D. Virkar for his very kind help and useful remarks. NOTATION
D binary diffusivity in the gas B binary diffusivity in the liquid f dimensionless stream function H Henry’s constant k pseudo-first-order reaction rate constant K dimensionless reaction rate constant L thickness of the absorbing stream m actual absorption flux (gr./cm* set) m” mass flux evaluated at bulk conditions total pressure S: Schmidt number, v/D U velocity in the x direction value of u at the absorbent surface UW UCC free stream velocity velocity in the y direction ; mass fraction in the gas phase w mass fraction in the liquid phase longitudinal co-ordinate i mole fraction in the liquid Y normal co-ordinate dimensionless co-ordinate 2: mole fraction in the gas phase Greek symbols q dimensionless co-ordinate in the gas phase
+j dimensionless co-ordinate in the liquid phase
TAITEL
Y.
kinematic viscosity Dimensionless axial co-ordinate density stream function
Subscripts and superscripts 0
inlet (at x = 0) bulk of the gas atmosphere indices designating the species liquid-gas interface t in the liquid differentiation sign 1 the more soluble species -CS2 2 the less soluble species or the insoluble species-NZ la of species 1 at the bulk 1w of species 1 at the gas-liquid interface cc
I
REFERENCES [l]
Danckwerts P. V., Gas LiquidReactions. McGraw-Hill,New York, 1970. [2] Astarita G., Mass Transfer with Chemical Reaction. Elsevier, Amsterdam, 1967. [3] GoettlerL.A.andPigfordR.L.,A.I.Ch.E.J. 197117793. [4] Ramachandran P. A., Chem. Engng Sci. 197126 349. [5] Ramachandran P. A. and Sharma M. M., Chem. Engng Sci. 1970 25 1743. [6] Ramachandrafl P. A. and Sharma M. M., Chem. Engng Sci 197126 l%7. ]7] Roper G. H., Hatch T. F. and Pigford R. L., Ind. Engng Chem. Fundls 19622 144. [El Astarita G. and Gioia F., Ind. Engng Chem. Fundls 19654 317. [9] Tamir A. and Taitel Y., Chem. Engng Sci. 197429 699. [lo] Taitel Y. and Tamir A., Int. J. Heat Mass Transfer 1975 18, 123. [ 111Perry J. H., Chemical Engineer’s Handbook, 4th Edn, 1%3. 1121Reid R. C. and Sherwood T. K.. The Properties of Gases and . . Liquids. Wiley, New York, 1964. [l3] Ramachandran P. A. and Sharma M. M., Trans. Instn. Chem. Engrs. 197149, 253. ]l4] Kothari P. J. and Sharma M. M. Chem. Engng Sci. 196621, 391. [15] Tamir A., Danckwerts P. V. and Virkar P. D., Chem. Engng Sci. 1975Xl 1243.
Pergamon Press.
Printed in Great Britain
THE EFFECT OF THE GAS SIDE RESISTANCE ON ABSORPTION WITH CHEMICAL REACTION FROM BINARY MIXTURES A. TAMIRt Department of Chemical Engineering, Pembroke Street, Cambridge, England
Y. TAITEL Department of Chemical Engineering, University of Houston, Houston, TX 77004,U.S.A. (Received 16December 1974;accepted 5 June 1975) Abstract-A predictive model based solely on the conservation equations of both phases is presented for the study of the effect of the gas-side resistance in multicomponent absorption with chemical reaction in a laminar liquid film. The model is applicable for short exposure times to a variety of flow patterns, e.g. a cylindrical jet, a wetted-wall column, or an inclined plane. The model may be applied to the wetted-wall column and the inclined plane for any contact time. The numerical scheme for the downstream solution has the advantage of utihsing a similarity solution as its starting point. The model is applied to the system CS,-N, where CS, is absorbed in an aqueous solution of amine and undergoes pseudo first order reaction. It was found that the a priori neglect of the gas-side resistance is not justified. The error introduced in evaluating the absorption flux by neglecting the gas-side resistance is dependent on the reaction rate constant, bulk concentration of the species, length of the absorber and the total pressure. Errors aenerallv increased with increasing values of the above narameters. The gas-side resistance is reduced when the gaseous-atmosphere is under forced convection.
INTRODUCTION
When a pure gas is absorbed into a solution, the main
resistance which controls this process is offered by the liquid phase. This means that the behaviour of the gas phase need not be considered. However, in absorption of one or more species from a multi-component gaseous mixture, the above a priori approach is not justified. This is due to the diffusional processes of the species in the gas phase which cause their concentration at the gas-liquid interface to deviate from their concentration in the bulk of the gas phase. Depending on the operating conditions, this deviation might be significant; it is maximal, for example, in the extreme case of instantaneous surface reaction where the concentration of all reactive species vanish. As a result of the above deviation, absorption fluxes of the individual species might be different as compared to those which would have been obtained had one assumed that the bulk conditions prevail also at the interface. Hence, in the case of multicomponent absorption, the knowledge of the interfacial concentrations distribution becomes the key for calculating absorption fluxes. To achieve this goal, a simultaneous solution of the conservation equations of the gas and the liquid phases is required. It should be noted that the phenomena occurring in absorption from binary mixtures are very similar to those in condensation of a binary vapour mixture. In the latter case, the concentration of the less volatile species at the interface is lower than that in the bulk of the condensing vapour. Similarly, in absorption, the concentration of the more soluble species, namely the one which has a low tDepartment of Chemical Engineering, Ben-Gurion University of the Negev, Beer-sheva, Israel. 1477
value of Henry’s constant, would be less at the interface than in the bulk. The phenomena discussed above become more appreciable if the absorption is accompanied by a chemical reaction in the liquid. Theoretical investigations which consider simultaneous absorption of binary mixtures in the presence of chemical reaction are few [l, p. 46; 2, p. 116; 3-8,131. The solutions obtained were either for the penetration or the film model. The common approach to all investigations was to consider only the mass transfer in the liquid phase and to neglect the effect of the diffusional processes in the gas phase. This greatly simplifies the analytical solutions but the error introduced in such calculations is unknown. To the best of the present author’s knowledge, a complete analysis is not available for absorption from a binary gaseous atmosphere in the presence of chemical reaction which takes into account the simultaneous processes occurring in both phases. In the present work, a complete formulation of this problem is made which is based on a boundary layer model in the gas phase [1, p. 1461 through which the various transport processes occur. The boundary layer is formed due to the interfacial drag of the gaseous atmosphere with a laminar absorbing-reacting film. Solution of the equations is demonstrated for the case where the solute gas undergoes a chemical reaction with a dissolved reactant which is irreversible and first order with respect to concentration of the dissolved gas. It should be noted, however, that many reactions which are second order, under some circumstances may also be considered as pseudo-first order. The condition for first order reaction is provided by Danckwerts [l, pp. 452101. It enables us to make an estimate of the exposure time below which a reaction may be treated as pseudo-first
A. TAMIRandY. TAITEL
1478
order. It should be noted that also other kinds of reactions in the liquid may be treated conveniently using the same approach. A forward step-by-step marching scheme is used in the numerical calculations. It has the advantage of utilizing a similarity solution which exists at the inlet region. The above-mentioned similarity solution corresponds also to the so-called “penetration model” regime which is valid for a semi-infinite body. FORMULATION OF THE PHYSICAL MODEL AND METHOD OF SOLUTION
numerical scheme adopted here. If both components are absorbed, the liquid phase becomes a quarternary system. However, due to the relatively low solubilities encountered between gases in liquids, the interaction between the diffusive fluxes may be neglected and each species diffuses in the solvent as if in a binary mixture with an effective diffusivity 6, i = 1, 2. In the gas phase boundary layer, the distribution in concentration of the more soluble species, designated as 1, may be obtained from the following equation:
uaw,,vaw,=Da’w, The physical model that is demonstrated in Fig. 1 a$ ’ ax ay consists of a laminar liquid film with a fully developed velocity profile which enters a wetted wall or a channel type absorption chamber. The liquid film absorbs one or The other equations for total mass and momentum conboth species from a binary gaseous atmosphere. The servation are absorbed species react with a material which has been aU,aV=, dissolved in the film prior to its entrance into the system. ax ay The chemical reaction occurs isothermally. The gaseous au au a34 atmosphere, which is assumed to behave as an ideal u-&+v-=v~. mixture, enters the absorber with a uniform velocity u,. ay ay Upon its entrance, the mixture attains the surface velocity u, of the liquid film, due to the interfacial drag. It is Note that buoyancy forces are neglected in eqn (4). This is assumed that the transport processes in the gas phase are a reasonable approach due to the imposed velocities u, carried out through boundary layers of momentum and and u, which are relatively large. Equations (1) and (2) to (4) are solved under the mass. A constant properties analysis is carried out, but in order to account for differences in concentrations be- following boundary conditions where i = 1, 2: tween the bulk of the gas and the interface, the properties x =o: u = UI, w, = WI,, IVi = I& are evaluated at an average concentration between the (5) bulk and the interfaceb, lo]. y = m: u = Urn,w, = WI, (6) The mass conservation in the liquid of each absorbed species where i = 1,2 is described by y =-L: aR/ay =o. (7) At the interface, namely y = 0, the gas-liquid relationships are: To simplify the solution for sparingly soluble gases, transverse convection effects may be ignored as compared to the axial ones as done in deriving the above equation. The transverse effects were taken into account in the mass flux conservation equation through the interface as given by eqn (9). Note that the soluble component with a local mass concentration % is consumed by an irreversible first-order chemical reaction according to a reaction rate constant ki. However, other kinds of chemical reactions may also be considered by using the same
I_”
u = uv
(8)
-p-$-$+4s~i= _p~~+pvw, (9) which expresses the continuity of the mass flux of each component. The application of the above boundary condition deserves some attention. Equation (1) implies that the transverse velocity in the liquid, 5, is zero. However, practically 5 has a finite value and hence at the interface pfi = pv. As previously mentioned, the assumption made in deriving eqn (1) was that for sparingly soluble gases WiWy is negligible as compared to the axial convection term. For highly soluble gases this assumption is not justified and a complete formulation is given elsewhere [ IS]. Finally, the Henry’s Law
(10)
Fig. 1. Physical model and co-ordinates.
which indicates an equilibrium at the gas-liquid interface is applicable. In our analysis we do not make any distinction between soluble and insoluble species. Indeed, if one wishes to consider an insoluble gas, all that is needed is to equate its Henry’s constant, Hi, to infinity.
1479
Gas side resistance on absorption with chemical reaction
The solution of the above equations is performed in a transformed co-ordinate system. In the liquid we use for small x the (5, i) co-ordinates defined by
xfj,
5=L2u,
The corresponding boundary conditions for i = 1,2 are 5
=o:
f’= UJU,,
w,,=
WI,
R. = iv8
n =cQ: f’ = ucOIuw, w, - WI,
i=&. q=o:
This enables us to obtain accurate concentration gradients at the gas-liquid interface, namely, the absorption fluxes. By small x(or 4) is meant the region where the penetration of a change in concentration applied at the liquid surface has not yet reached the wall (y = -I,). Hence the mass conservation equations for i = 1,2 reads
f’=l,
fw=f+2$$
(19) (20)
w,=w,w.
(21)
Note that W,,:the concentration of species 1 at the interface, is unknown and is obtained in course of the solution as later explained. To solve eqn (12) for small 5 we use for i = 1, 2 i +m:
Wi = Wi”
(22)
On the other hand, for large 6, eqn (12-l) is solved with 8=-l:
aR/ag=o.
(23)
Ki is a dimensionless reaction rate constant defined as
(13)
The continuity in the mass flux for an individual component through the interface is now expressed by eqn (24) noting that pii = pv
It is interesting to note that eqn (12) shows that chemical reaction may be ignored for very small 5 and that diffusion is dominant. Equation (12) is applicable for a wetted-wall absorber or an inclined plane. The cylindrical jet absorber with a uniform velocity profile is characterized by short time of exposure where the term 45rs’<<1. Hence, eqn (12) satisfies also this flow pattern for small 6. For large 5, the numerical solution was conveniently continued in the co-ordinates (t, 8) where 5 is given in (11) and
F.= I
d a% ’ fw ( 2pmdil-Sc-‘T
w,__2. !j
>
(24)
where inter-facial equilibrium requires eqn (10) to hold. The numerical solution which takes advantage of a similarity solution at 5 = 0, is a step-by-step advancing technique. The differential eqns (12, 12-1)and (17, 18) are put in a finite difference form using implicit scheme. The Crank-Nicolson method is used for eqns (12, 12-l) which yielded a three-diagonal matrix. Also the diffusion eqn (17) was transformed into a similar matrix where for the +. (14) momentum eqn (18) a five-diagonal matrix was obtained. Assuming profiles are known at distance 5, the advance Here the mass conservation equation for i = 1, 2 is to 5 t At is carried out as follows: a guess is made for the described by interfacial concentration W,w and hence WZw= 1 - Wlw. By applying Henry’s Law, the equilibrium solubilities W, WZ,Vat the liquid surface are calculated. They are designated for i = 1, 2 as Wiw.equit. At this stage the physical properties are evaluated at a reference concentThe boundary layer equations for the gas phase are ration 0.5 ( Wi,t W,,) [lo]. It should be noted that aldescribed in (5, 7) co-ordinates where .$ is given in (11) though our analysis assumes constant properties, they and might be corrected at each step for their slight variations (15) with x. Knowing the interfacial concentrations makes it possible to calculate the concentration distribution at the gas and liquid phases and the gradients aWi/aq, a%/&ij If u = at,b/ay, u = -ag/ax and a dimensionless stream at the interface. For small 5, aW,/&j is calculated from function f is defined by eqn (12) with (22) where eqn (12-1) with (23) are applied (I = V(vu,x )f(& 771 (16) for large 5. A smooth transition from the (5, ii) grid into the (&-jr) grid was carefully applied at a suitable 5 location. we obtain for the diffusion equation a W,/Sj were calculated at ?j = 0 using a three point formula. Note again that since an accurate derivative is required it is essential to use the (6, q) grid rather than the (5, 7) grid for small 5, namely, for the “penetration model” regime. fwis calculated from eqn (21) and hence from eqn and for the momentum equation (24) new values may be ontained for the W’s. Designating the difference ( ~lw.cquil. - F,w) by K it is observed that F, is a function of the initial guess W, The iteration procedure is terminated if FI approaches zero, and to W,
W.
1480
A. TAMRand Y. TAITEZL
achieve fast convergence the method of False Position [ll, pp. 2-561 is used. As previously mentioned, the solution for 6 = 0 corresponds to a similarity solution in the absence of chemical reaction (see eqn 12). The distribution for this case are calculated similarly as described above.
more soluble species-C& whereas in the upper part they correspond to the insoluble species-N*. The parameter of the curves is the dimensionless first-order reaction rate constant K, which was varied in the range from 0 to 10’. It is clearly observed that the interfacial concentration of CS2 is always lower than its bulk concentration either in the presence or in the absence of chemical reaction APPLICATIONOF THE MODEL (K, = 0). For the insoluble species, namely Nz, the oppoAND DISCUSSION site behaviour is obtained. This phenomenon becomes The absorption into Dimethylamine solutions of CS2 more appreciable if the reaction is faster, namely if KI is from its binary mixtures with N2 was studied. This system increased, and if the absorber is longer. Note also that for was investigated by Kothari and Sharma[l4] and the each case all curves in Fig. 2 issue at 5 = 0 from the same reaction between CS2 in aequous amine solutions was point. This is due to the fact that the chemical reaction is found to follow pseudo-first order behaviour. From the ineffective at the entrance (see eqn (12)), compared to the kinetic data provided by Sharma, as well as CS2 solubility absorption process. It is also of interest to point out the and liquid diffusivity, Danckwert’s condition[l, pp. 45, difference in the behaviour of the curves for CS2and NZin 2101indicated that the reaction obeys lirst order behaviour Fig. 2 corresponding to K, = 0 (no chemical reaction) and, over a wide range of exposure times (up to 2OtlOsec.). The say, K, > 1. It is observed that the ratio W~V/ Wm in the Refs. (11,12) were consulted for the selection of additional latter case is continuously decreasing while the curve for physical and transport properties of the individual species K, = 0 should approach a value of unity, namely the bulk as well as for their behaviour in the mixture. The reference concentration. This is due to the fact that in the absence properties were calculated as outlined before. It was of chemical reaction, the absorbing stream must eventudecided also to perform the calculations for 30°C (14), zero ally become saturated with CSI and therefore its interfainlet concentration of CSI in the liquid solution and total cial concentration will be equal to its bulk concentration. pressures of one and ten atmospheres. At the lower The upper curves in Fig. 2 indicate that the insoluble pressure it was possible to investigate the effect of bulk species, N2, tends to accumulate at the interface. This concentration WI..of CSt for the following values 0+5,0*1, accumulation enables it to diffuse backwards to the bulk 0.05 and 0.01, whereas at ten atmospheres only the last of the gaseous atmosphere and to balance its convective three concentrations could be considered. NZwas taken as stream caused by bulk flow of the soluble species, C$. It insoluble noting that its solubility at 30°C is about 35 times may also be noted that the accumulation of NZ at one lower than that of CS2. atmosphere corresponding to W,, = 0.5 is much higher In Fig. 2 concentration on the interface along the than for Wlm= O-05at same pressure where the mixture is absorber length 5 are shown corresponding to CS, bulk almost N2. For example, at 5 = 0.15 the ratio Ww/W= = 1.3 in comparison to 1.03 in Fig. 2. Other general trends concentration of 0.05. They are conveniently expressed by the ratio W,/ W, where Ww is the weight fraction at which are drawn from Fig. 2 and 3 are: at the same total the interface and W, is the weight fraction in the bulk. pressure, the ratio WW/W, for CS2(at constant 5 and KJ The curves in the lower part ( WW/W, < 1) describe the is significantly reduced as WI.. is decreased. In addition, for similar values of W,=the ratio WW/W= becomes lower as the total pressure is increased. Both trends affect the actual absorption flux of CS2as compared to its maximum value under absorption at bulk conditions. 5 x 10-s ______-_---_-___$ ,’
Fig. 2. Interfacial concentration W, relative to bulk concentration W, for CS, and N, in absorption from a mixture of CS,N, with W,,= 0.05 at 1 and 10atmospheres.
8
Fig. 3. Dimensionless local flux of CS, in absorption from a mixture of C&-N2 with WI,= 0.05 at 10atmospheres.
Gas side resistance on absorption with chemical reaction
The deviation between the interfacial and bulk concentrations of the species corresponds to some magnitude of the so-called gas-side resistance. The existance of such resistence was emphasized by Astarita[8] who found also experimentally that its value was indeed significant. In Fig. 3 we report on the nondimensional local flux mL~(.$)/fi for CS2 in absorption from mixtures of CSZ-Nz. This value was calculated according to the lefthand side of eqn (9). Note that at 6 = 0 all curves emerge from the same point because at the entrance the chemical reaction is unimportant. The magnitude of this point depends on the total pressure and the bulk concentration of CS. As expected chemical reaction enhances absorption mass-transfer and the phenomenon is manifested towards increasing values of the reaction rate constant K,. Unlike the curves which were obtained in the presence of chemical reaction, the curve for KI = 0 in Fig. 3 is continuously decreasing. This corresponds to the fact that in the absence of chemical reaction the absorption will ultimately cease when the absorbing stream becomes saturated. In addition, it was observed that absorption rates of CS2 become higher as its bulk concentration and/or total pressure is increased. As recalled, forced convection in the gaseous atmosphere was also included in the analysis. This effect is demonstrated in Fig. 3 by the dotted lines for the case where u, is tenfold higher than the stream surface velocity u,. As expected, forced convection tends to alter the concentration profile and to increase absorption fluxes. The improvement depends on the bulk concentration, the reaction rate constant and the forced velocity. The practical conclusion which may be drawn is that forced convection is more efficient for longer ducts rather than for very short ones. In Figs. 4 and 5 the ratio mJm10 is plotted vs the absorbing length, where ml is the actual local absorption flux of CS2 and rn,’ is the flux evaluated at bulk conditions. The above ratio reflects upon the error which is introduced in the calculations should one neglect the effect of the gas-side resistance. Note that at a concentration level lower than 0.1 all solid curves coincide. In general it may be concluded that the local absorption flux of CS2 is lower than the one evaluated at the bulk conditions. This effect becomes stronger if K, and the total pressure are increased. By selecting typical values
___-$g -
0
x;
0.01.0~05.04
IO
02
03
0.4
Fig. 4. Mass flux ratio for 123, in absorption from mixtures of C&-N, with WI,= O.Ol,O.OS, 0.1 and 0.5 at 1atmosphere.
1481
Fig. 5. Mass flux ratio for CS2 in absorption from mixtures of C&-N, with W,,= 0.01,0.05 and 0.1 at 10atmospheres.
for the parameters appearing in 5 (eqn 11) and K, (eqn 13) while for k, in the values reported by Sharma[l4] were applied, it was found that K, may vary in the range of lo”-lo4 where 5 is in the order of 0.1. Hence the a priori neglect of the gas-side resistance is not justified. Finally it would be instructive to comment on the application of the above model for the interpretation of experimental data. In absorption studies with chemical reaction one usually measures the total absorption flux rather than the local values. The total absorption flux can then be compared with the theoretical value, if desired. The theoretical value of the total flux may be computed from our model by numerical integration of the local flux as calculated from eqn (9). Note, however, that the latter is equal to infinity at ,$= 0. Hence to compute the total flux one should start the integration from a very small value of .$(which has to be checked) ignoring the quantity which was absorbed up to this point, which is usually negligible. CONCLUDINGREMARKS
Previous analyses considering absorption of binary mixtures, with or without chemical reaction, tend to ignore the gas-side resistance. In other words, calculations were made by assuming that the bulk conditions of the absorbed gases prevail also at the interface. ~4complete analysis of binary mixtures absorption in a laminar film has been made to check this assumption. It incorporates a simultaneous solution of the gas-side boundary layer equations and the liquid stream in which chemical reaction also takes place. The solution was demonstrated over a wide range of operating conditions for the mixture C&-N, where CS, is absorbed in an aqueous solution of amine undergoing pseudo-first order reaction. The major conclusion which may be drawn is that the a priori neglect of the gas-side resistance is not justified and might lead to errors in evaluating the absorption fluxes as shown in Fig. 4 and 5. The error, in general, depends on the bulk concentration of the species, the reaction rate
1482
A.
TAMIR and
v 5 p I(,
constant, the total pressure and the length of the absorber. For absorption in the absence of chemical reaction the error is usually negligible. It was also found that forced convection of the gaseous phase remarkably improves the absorption rate and is most useful for long ducts rather than for very short ones. Acknowledgement-This work was performed while the first author spent his sabbatical year in the Department of Chemical Engineering at the University of Cambridge. The sabbatical year was supported by the Lord Marks Fund which is greatly acknowledged. Thanks are also due to Professor P. V. Danckwerts for the interest in this work and his collaboration, and also to Mr. P. D. Virkar for his very kind help and useful remarks. NOTATION
D binary diffusivity in the gas B binary diffusivity in the liquid f dimensionless stream function H Henry’s constant k pseudo-first-order reaction rate constant K dimensionless reaction rate constant L thickness of the absorbing stream m actual absorption flux (gr./cm* set) m” mass flux evaluated at bulk conditions total pressure S: Schmidt number, v/D U velocity in the x direction value of u at the absorbent surface UW UCC free stream velocity velocity in the y direction ; mass fraction in the gas phase w mass fraction in the liquid phase longitudinal co-ordinate i mole fraction in the liquid Y normal co-ordinate dimensionless co-ordinate 2: mole fraction in the gas phase Greek symbols q dimensionless co-ordinate in the gas phase
+j dimensionless co-ordinate in the liquid phase
TAITEL
Y.
kinematic viscosity Dimensionless axial co-ordinate density stream function
Subscripts and superscripts 0
inlet (at x = 0) bulk of the gas atmosphere indices designating the species liquid-gas interface t in the liquid differentiation sign 1 the more soluble species -CS2 2 the less soluble species or the insoluble species-NZ la of species 1 at the bulk 1w of species 1 at the gas-liquid interface cc
I
REFERENCES [l]
Danckwerts P. V., Gas LiquidReactions. McGraw-Hill,New York, 1970. [2] Astarita G., Mass Transfer with Chemical Reaction. Elsevier, Amsterdam, 1967. [3] GoettlerL.A.andPigfordR.L.,A.I.Ch.E.J. 197117793. [4] Ramachandran P. A., Chem. Engng Sci. 197126 349. [5] Ramachandran P. A. and Sharma M. M., Chem. Engng Sci. 1970 25 1743. [6] Ramachandrafl P. A. and Sharma M. M., Chem. Engng Sci 197126 l%7. ]7] Roper G. H., Hatch T. F. and Pigford R. L., Ind. Engng Chem. Fundls 19622 144. [El Astarita G. and Gioia F., Ind. Engng Chem. Fundls 19654 317. [9] Tamir A. and Taitel Y., Chem. Engng Sci. 197429 699. [lo] Taitel Y. and Tamir A., Int. J. Heat Mass Transfer 1975 18, 123. [ 111Perry J. H., Chemical Engineer’s Handbook, 4th Edn, 1%3. 1121Reid R. C. and Sherwood T. K.. The Properties of Gases and . . Liquids. Wiley, New York, 1964. [l3] Ramachandran P. A. and Sharma M. M., Trans. Instn. Chem. Engrs. 197149, 253. ]l4] Kothari P. J. and Sharma M. M. Chem. Engng Sci. 196621, 391. [15] Tamir A., Danckwerts P. V. and Virkar P. D., Chem. Engng Sci. 1975Xl 1243.