Solvent CounterRDiffusion in Gas Absorption

PIERRE GRENIER

Universid Laval, Qukbec, QUB.

The usual design equations for absorption towers are based on the assumption that only one component is diffusing in the gas phase: the counter-diffusion of the solvent is ignored. A correction factor has been developed, based on a solution of the Stefan-Maxwell diffusion equa- tions for the case of two gases diffusing through a stagnant third one. The speeial case of equimolal counter-diffusion is also considered. The correction factor is obtained by simultaneously solving two non-linear equations. The system ammonia-water-air is used to illustrate the order of magnitude of the error the factor is intended to correct.

he usual design equations for continuous mass exchangers T such as packed towers and similar apparatus are still based o n Whitman’s two-film theory‘’) in spite of current interest in the penetration theory(2). In particular, equations for the esti- mation of the required height of ar! absorption tower are derived considering either the gaseous or liquid film, or b ~ t h ( ~ * ~ ) . In the specially frequent case where the gaseous film is used, the derivation of a workable equation requires the assumption that only one component, labeled A, is diffusing in the gas film adjacent to the interface. A material balance on a differential element of tower volume involves the instantaneous di ffusion rate NA. This in turn is replaced by the following expression applicable to the diffusion of a single gas A through a stagnant

NAa = ___ lN-=- . . . . . .(1)

gas C, DAC Pa Yci CDAC a ( Y A ~ - Y A ~ ) R’TL ycE L Ye”‘

and one finally obtains an equation of the form

kcaSPdh V

. . . . . . . . (2) (1 - YA) ( Y A s - Y A i )

y.4 2 0

in which k,x = (DAL Pa) / (R’TLyC,,)

Unless the gaseous phase is saturated with the solvent, the latter will normally be subjected to a certain amount of evapor- ation and will counter-diffuse into the gaseous phase. This will in turn reduce the diffusion rate of component A and Equation ( I ) strictly will not be applicable. Instead one should use an expression for the diffusion rate of A in a three-component system in which a second component B is also diffusing, though in an opposite direction, through stagnant gas C.

In many cases the use of the binary equation causes no major inconvenience but for some design situations it becomes necessary to estimate the error with some accuracy and this is the specific purpose of this study.

Diffusion in a Ternary System The usual starting point for studying digusion in a n-com-

ponent gaseous system is the set of Stefan-Ma~well(~.‘j) eq- uations

-+ +

Le ealcul des tours d’absorption comporte gBnBralement l’hypothese qu’un seul composant diffuse dans la phase gazeuse ct la contre-diffusion du solvant est n6gligBe. On propose un facteur de correction, Btabli B partir d’une solution des Bquations de Stefan-Maxwell dans le cas de la diffusion de deux gaz B travers un troisieme gaz immobile. Le cas spdcial de contre-diffusion Bquimolaire est aussi Btudi6. Le facteur provient de la solution de deux &qua- tions non linhaires. On indique, par l’exemple du systeme ammoniac-eau-air, l’ordre de grandeur de l’erreur qu’il sert B corriger.

of which only n - I arc significant at constant total pressure. More rigorous equations developed by Curtiss and Hirsch- felder‘7) from the Chapman-Enskog(a) treatment for binary mix- tures, reduce to lquations ( 3 ) for multicomponent gas mixtures which can be considered as ideal. This assumption has already been made in the derivation of Equation ( 2 ) and does not con- stitute a further restriction to its applicability.

A number of authors have presented rigorous or approxiniatc solutions, for the general or for special cases, to the problem of three-component diffusion., Their work has been summarized by Duncan and T o ~ r ( ~ ~ ’ ~ ) . ~ ~ i l l i l a n d ’ s solution(ll) to the Stefan- Maxwell equations for a ternary gas system corresponds to the present conditions, one component being stagnant and N, = 0. He obtained a set of two non-linear equations, each containicg the two unknown rates of diffusion N,, and N,. Calculation of either of these rates thus requires an iterative procedure. For a similar situation, Hsu and Bird(12) gave a slightly different solution with somewhat simpler equations. This paper proposes the use of a modification of these to obtain the desired expression for the transfer rate through the effective gaseous film of the component being absorbed.

It is generally agreed(’3) that binary diffusivities can be used in Equations ( 3 ) so that Djk = nk,. In order to facilitate the handling of equations, the following definitions are introduced:

NA/NB = - m,.. . . . . . . . . . . . . . . . . (4)

c = Z / L , L dz = d Z . . . . . . . . . . . . . ( 5 )

= DHC/DAS. . . . . . . . . . . . . . . . . ( 6 )

s = DBC/DAC ( 7 )

N = NAL/c DSC = a constacit. . . . . . . . . . . . (8)

It should bc noted that Z is one of the Cartesian axes and that z is dimensionless. With the restriction that N, = 0 and that diffusion takes place in one direction only, the Stefan-Maxwell equations reduce to

The solution to those equations must also satisfy the following

The Canadian Journal of Chemical Engineering, August, 1966 21 3

boundary conditions, with the subscript g referring to the bulk of the gas phase, and i to the interface.

z =J 0, yA = YAg, yc = y c ~ . . . . . . . . ..(11)

yC = YCi. . . . . . . . . . ( 12) z = 1, Y A = YA;,

Integration of (10) yields an expression for the distribution of the concentration of component C over the film thickness:

YclYcr = exp [N (s - ;) 21

Substituting into (9) and integrating gives likewise the dis- tribution of component A :

I. - r 1. I Ycr

Ycr exp [(s - :) Nz] m +- -

1 1-Y m-1' 1 --(-) m s-Y

Application of the second boundary condition (12) to (1 3) and (14) gives two transcendental equations:

m _ - = YCG

YAi + 1- - ( - ) 1 1-Y m - 1

m s-Y

YcdYcr = exp [(s - :) N ]

Rearranging yields expressions for N and thus for NA, according to (8):

1

In considering the possibility of introducing these transfer rates at any phase of the derivation of the absorption tower Equation (2), a number of points must be discussed. First, Equations (18) and particularly (17) are rather unwieldy, as compared with the single gas transfer rate expression (1). Furthermore both contain L. the equivalent film thickness, and in this particular instance, the coefficient k, cannot be easily introduced to eliminate it. Finally, these expressions are not straightforward as both contain on the right-hand side, the factor m which is the negative of the ratio NA/NB.

It is however possible to add to Equation (2) a correction factor so designed that it would substitute the transfer rate in a 3-component system for the transfer rate in a binary system:

R = (NA)ABc/(NA)Ac. . . . . . . . . . . . . . . (19)

T w o different and independent formulas for this factor are obtained by dividing Equations (1 7) and (1 8) by

. . . . . . . . . . .

They are:

fi, = ~- m-1 DAC

m DAB 1

In (Y-)

r

yc'ca AC

1

rn -~ Ycr YAc +

1--(-) 1 1-7 m-1 m s--r

and

YCi In -

Although (21) and (22) are independent they must satisfy the condition R = R, = R2. Digital computer solution of this system of equations is easily done, for a given system and nu- merical values of the boundary conditions.

Ammonia-Water-Air System The choice of the system NH3-HzO-air is quite arbitrary.

A solution with a single set of conditions for this system has been published, using Gilliland's Fquations("), but the chosen total pressure and temperature were rather unusual: 155 mm Hg and 40°C. As a large number of absorption operations are carried out under atmospheric pressure, application of Equations (21) and (22) at this pressure will give useful information on the magnitude of the error involved when using the ordinary absorption tower design Equation (2).

First, the various binary diffusivities must be obtained. For the pairs NHrai r and HzO-air the values of these properties are readily available(14). On the other hand DNH3-mz0 is not reported in usual reference works. Consequently, it was decided, for the sake of uniformity, to calculate the three diffusivities by means of the Hirschelder Equation(13).

The values of the Lennard-Jones or Stockmayer parameters q'K and a were taken from Hirschfelder, Curtiss and Bird(16) and those of the collision integral fiA8 which is a function of q'K, were read from tables prepared by Hirschfelder, Bird and Spotz(l3). Calculated values (cm2/sec) at 25°C and 1.0 atmos- phere are as follows: HNVair: 0.264; HzO-air: 0.248 and NH3- H20: 0.301. In the case of ammonia-air, agreement with the experimental values('4) is mediocre (12%) and this was to be expected as the Hirschfelder equation is recommended only in the case of a pair of non-polar substances. This will of course affect the accuracy of the absolute values of the diffusion rates but will not alter the dependency of these rates on the con- centrations of the two diffusing species. For the same reason, recent discussions(16) on so-called experimental values of diffus- ivities were not considered.

Results calculated below are for a temperature of 25OC but apply to other temperatures since diffusivities always appear in equations as ratios of diffusivities r and 5. For this system, these ratios have respective values of 0.8239 and 0.9394.

Results All results are presented as ratios of the diffusion rate of

ammonia for the ternary system to the diffusion rate of ammonia

The Canadian Journal of Chemical Engineering, August, 1966 21 4

1.00

0.98

0.96

0.94

0.92

0.90

0.88 0.88

0.86

yAi =O.Ol

Yeg = O -

- I 0.86

I a , ’Bi , I

0.02 0.04 0.06 0.08

Figure 1-Effeet of water counterdiffusion.

through stagnant air neglecting the counter-diffusion of water vapor. The concentration of the latter was selected as the main variable, and values up to 0.10 mole fraction in the gas phase at the interface were used, corresponding to the vapor pressure of pure water at 46OC. Thus a range of liquid temperatures extending above usual values is obtained, the equivalent con- centration of 20OC being 0.02 mole fraction.

Figure 1 shows the effect of the counter-diffusion of water vapor at various diffusion potentials for ammonia. The solid part of the curves represents conditions such that the diffusion rate of ammonia is greater than the diffusion rate of water. Equimolal counter-diffusion is a special case which will be dealt with in a subsequent section. The effect of water counter- diffusion appears to be very small at high ammonia concen- trations but increases as the latter decrease. These curves correspond to an ammonia mole fraction of 0.01 in the gas phase at the interface, except for the uppermost one (unlabeled), which is for yAi = 0. In this case, there is only one curve for all values of the ammonia diffusion potential.

In Figure 2, the ammonia driving force is again the parameter, but this time, the lower curves represent weak values of the driving force at high ammonia concentration (v.g. { y A I - y A i } = IO.25 - 0.20}). The effect of water vapor diffusion is more important and this situation would correspond to the case of a slightly soluble gas : low driving force in the gas phase and high driving force in the liquid phase.

For all the cases considered above, the bulk concentration of water in the gas phase has been set at y B p = 0. If the gas phase is not dry, the effect of water counter-diffusion is still less as demonstrated by Figure 3 .

1.00

0.98

0.96

yAg = 0.25

Yeg* O

0.02 0.04 0.06 0.08

Figure 2-Effect of ammonia Concentration.

Equimolal Counter-Diffusion

A number of investigat~rs(~J’) have pointed out that Gilli- land’s solution is unsatisfactory for the case of equimolal counter-diffusion, the equations being always satisfied by (NA + NB) = 0. The same criticism applies to Equation (17) which becomes indeterminate under these conditions (m = 1 ) . Furthermore, if the driving forces for both diffusing species are equal, Equation (1 8) gives zero for NA. In order to cope with these special cases, an additional restriction must be applied to the initial differential equations before integration. Substi- tution of m = 1 in Equations (9) and (10) yields

Upon integration with the same boundary conditions, (1 1) and (12), two separate expressions are obtained for the diffusion rate

Eliminating NA between (25) and (26), one obtains the re- lationship that must exist between concentrations and diffu- sivities, in order to have equimolal counter-diffusion:

The Canadian Journal of Chemical Engineering, August, 1966 21 5

1.00

0.98

0.9f

0.94

0.92

0.90

0.88

0.86

yAg= 0.100

yAi = 0.00

’Bi 0.02 ~ 0.04 0.06 0.08 0.10

Figure 3-Effect of water concentration in bulk of gas phase.

Toodg) has already demonstrated the necessity of satisfying a similar equation.

Discussion

Equations ( 1 7) and ( I 8) giving the fluxes of the two diffusing components in a ternary gaseous system have been obtained under a form which makes them particularly suitable for the application of a correction factor in design equations. In other respects they correspond to previous solutions by T 0 0 r ( ~ ) and

by Gilliland(ll) but Ikpation (1 7 ) gives a more direct formulation of N A and has the advantage of being of the same general form as the corresponding Equation ( I ) for binary diffusion.

Calculations for the ammonia-water-air system indicate that under temperature and pressure conditions normally prevailing in an absorption tower, solvent counter-diffusion has no im- portant effect, reducing the ammonia diffusion rate by a maxi- mum of about 5%. This value applies when the gas does not contain any trace of solvent vapor at the tower inlet. As it ascends, the effect is still less noticeable and an average cor- rection factor of the order of 1.02 can be applied directly to the tower height.

The number of paramaters involved does not allow gen- eralization in graphical or tabular form for other systems. In the case studied, the three binary diffusivities were nearly equal. For the absorption of a gas with a more complex molecule, the water-air diffusivity may very well be much greater than the other two and thus give sensibly lower values of R. This can be ascertained readily for any system with known physical pro- perties.

Acknowledgments The author wishes to express his gratitude to Professor Le Goff of

the University of Nancy in whose department this work was initiated during a sahhatical leave. The help of A. Miville-Deschenes in pro- gramming is gratefully acknowledged.

Nomenclature Interfacial area, cm2 per ~n13 of packing volume ’Total gas concentration, gram-moles/cm3 Diffusivity for the gas pair A - B , cm2/sec Packed tower height, cm Boltzmann’s constant, ergs/(molecule - OK) Thickness of equivalent gas film through which dif- fusion takes place, cm Molecular weight Total pressure, atm Ratios of diffusion rate in a ternary system to dif- fusion rate in a binary system, defined by equations 19, 21, 22 Universal gas constant, (~1113 - atm.)/(gram-mole

Absolute temperature, “K Mole fraction in gas phase Axis in the direction of diffusion Dimensionless distance in the direction of diffusion Gradient Characteristic energy of interaction between mole- cules, ergs/molecule Characteristic diameter of molecule, A Collision Integral for diffusion, dimensionless

- “K)

Indices

A , H, C K i : refers to interface j , k , n m

: Components of a ternary system : refers to bulk of gas phase

: dummy indices : refers to logarithmic mean average between positions

g and i 1, 2 : refer to top and bottom of absorption tower

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T. G., “Mathematical Theory of Non- Uniform Gases”, 2nd ed

(10) Duncan, J. B. and Tmr, H. L., A.1.Ch.E. Journal, 8, 38 (1962). ( 11) Sherwood, T. K., “Absorption and Extraction”, McGraw-Hill Com-

pany, New York, 1937. ( 1 2 ) Hsu, H. and Bird, R. B., A.1.Ch.E. Journal, 6, 516 (1960). (13) Bird, R. B., Stewart, W. E. and Lightfoot, E N., “Transport Phe-

nomena”, John Wiley and Sons, New York, 1960. ( 14 ) Perry’s Chemical Engineers’ Handbook, McGraw-Hill Book Com-

pany, New York, 3rd ed. 1950, 4th ed. 1963. (15) Hirsfelder, J. O., Curtiss C. F. and Bird, R. B., “Molecular

Theory of Gases and Liq;ids”, John Wiley and Sons, New York, 1954, pp. 214, 1111.

( 1 6 ) Scott, D. S.; Chen, N. H. and Othmer, D. F.; Sherwood, T. K.: Ind. Eng. Chem Fundamentals 3, 278-89 (1964).

( 1 7 ) Wilke, C. R., Chem. Eng. Pro&., 46, 95 (1950).

( 8 ) Chapman, S. and Cowling

( 9 ) Toor, H. L., A.1.Ch.E. Jolrnal, 3, 198 (1957). ’Cambridge University Press, 1951.

Manuscript received November 29, 1965; accepted March 17, 1966. Based on a paper presented to the 15th C.I.C. Chemical Engineering Conference, Quebec, One., October 24-27, 1965.

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21 6 The Canadiarr Journal of Chemical Engineering, August, 1966