The penetration theory for gas absorption accompanied by a first-order chemical reaction with a diffusing catalyst

ChemrcalEngrneenng

Scrence,

1974

Vol 29, pp 275-279

Pergamon Press

Pnnted m Great Bntam

THE PENETRATION THEORY FOR GAS ABSORPTION ACCOMPANIED BY A FIRST-ORDER CHEMICAL REACTION WITH A DIFFUSING CATALYST A TAVARES

DA SILVA*

Instltuto Superior TBcmco, Llsboa, Portugal (First received 9 February 1973, accepted 11Apnl 1973) Abstract-The penetration theory solution IS presented for gas absorption accompanied by a firstorder chemical reaction catalysed by a species &ffusmg m the hqmd phase A full numerical solutlon was obtained for the enhancement factor, the factor by which the chemical reaction increases the rate of absorption Asymptotic expansions were obtamed by analytical methods, for both small and large values of the contact time The results show that the enhancement factor Increases mdefimtely with the contact time and that It IS substantially iugher than the enhancement factor predicted by the film model For large values of the contact time the prechcted enhancement factors differ by a factor of two

INTRODUCTION Theoretical models of the gas-absorption process have been widely used to predict the effect of a chemical reactlon on the rate of absorption One aspect of these theoretIcal predlctlons has been the

species m solution the concentration uniform according to the equation

remarkable agreement between the predlctlons based on the film model and those based on the penetration theory, of the effects of chemical reactions on the rate of absorption Except m the case where there IS an appreciable difference between the dlffuslvltles of the solute gas and of the dlssolved reactants the effects which have been predicted by the two theories do not differ by more than a few per cent[3,4,9] The purpose of this paper IS to present the penet-

species B reacts mstantaneously and irreversibly with a species also of umform concentration, giving a species C according to the equation

ration theory solution pamed by a first-order

for gas absorption

A +F’roducts,

B + C + Other Products,

of which 1s

(1)

(2)

species C 1s a catalyst for reaction (l), the rate of reaction of which 1s of the general form r=kc[CI[Al+k,[Al

accom-

chemical reactlon catalysed by a species which 1s diffusing m solution, to point out the behavlour of such a system, and to compare the penetration theory solution with the known film model solutlon As it wtil be seen, the predictions of the effect of the chemical reaction based on the two theories for this system show a substantial dlfference, which 1s m contrast to the results obtained for other cases

In most cases of importance k, *kc[C] Such a situation 1s encountered, for instance, when carbon dioxide having a small quantity of chlorine 1s absorbed mto a carbonate-bicarbonate solution hypochlorlte will then be formed, catalysmg the reaction between CO* and water [8] The film model solution of this problem has been presented by Tavares da Sdva and Danckwerts[8] and 1s expressed m terms of Airy functions

THEORY to be considered IS that m which two gaseous species, A and B, dissolve mto the hquid phase, species A reacts irreversibly with a

Penetration theory equations The penetration theory model has been described

The problem

Departamento de Engenharla *Present address Quinuca, Faculdade de CMncias e Tecnologia, Coimbra, Portugal

in vaflous references[2,3,6] In the present problem the partial differential equations descnbmg this model can be wntten m the followmg dlmenslonless form $=$+a(c+p)

275

A

276 a% rcg=z

TAVARES DA SILVA

ac

subject to the lmtlal and boundary

(4) condltlons

At

8 =O,y >O,a =O,c =0

(5)

At

8>O,y=O,a=l,c=l

(6)

At

B>O,y+~,$=0,$=0

for which numerical solutions tend in general to be less accurate, leading to asymptotic expansions of the enhancement factor, were also determined by analytical methods which are described elsewhere [7] In the case of small contact times this involved the evaluation of single and double mtegrals, which had to be carried out numerically RESULTS AND DISCUSSION

(7)

The result which 1s desired 1s the enhancement factor E This 1s the factor by which the rate of absorption of species A IS Increased as compared to its physlcal absorption In this problem the value of E cannot be calculated by a material balance because since the species with which A reacts 1s of umform concentratlon there 1s no possible way of calculating the amount of A m the combined state E has then been calculated by means of the equation

and 1s a function of the dlmenslonless contact time tJ and of the dimensionless parameters rc and p Solution of the equatrons The solution of Eq (4) consistent with the mitral and boundary condltlons (5)-(7) is, for all (y, 0), the well-known form c =erfc

Figure 1 shows the results for p =0 The results are expressed as E vs M at a number of different values of rc M IS related to the dimensionless contact time, 0, by M = d/4

(10)

It is seen that the value of E increases mdefirntely as M Increases approaching the asymptotic value E,= 2m As expected this asymptotic value 1s independent of rc since for very large contact times the dlmenslonless concentration c is everywhere equal to unity, the diffusion of species C not tiectmg the absorption of species A For not so large values of the contact time mcreasmg the dlffuslvlty ratio rc at the same value of M has the effect of mcreasmg E, which can be understood by noting that If the dlffusivlty of C increases its concentration also increases everywhere

‘t

’ ( 2rre m>

representing pure diffusion of C mto the semlinfinite medium The differential equation (3) with c given by Eq (9) was solved numerically by fimtedifference methods. These methods, which are described elsewhere 171,were not straightforward and showed some mterestmg features due to the fact that in the problem considered here the enhancement factor increases indefinitely as 0 increases and the fact that it has to be obtained by calculating the derivative as expressed m Eq. (8) Standard convergence tests and comparisons with known asymptotes led to the conclusion that the results are accurate to within 2 per cent All the computations were carried on IBM 360144 or IBM 370/165 digital computers Asymptotic expansions of the solutions of Eqs (3) and (4) for both small and large contact times,

44.34

Fig

.g

1 Penetration theory solutlon, p =

0

Figure 2 shows the results for p = 1 It IS seen, by comparison with the curves m Fig. 1, that the asymptotic value of E IS approached more rapidly, 1 e for lower values of 8, as 8 mcreases than m the case in which p =0 It is also seen that m this case the variation of E with rc for the same value of the dlmenslonless contact time is less than in the case m which p = 0 These effects are due to the fact that increasing the value of p has the effect of mcreasmg the overall rate of reaction of species A and making its rate of absorption less dependent on the concentration of the diffusing catalyst C

271

Penetration theory for gas absorption

The asymptotic expansion for E for small values of the contact time obtamed by the methods described m Ref [7] IS E = 1-s

Ma;(O)+

M*&(O)-

The coefficients a{(O) and a;(O) m the terms m M and MZ depend on the parameters rc and p and are shown respectively m Tables 1 and 2 Expansion (11) wdl be vahd with sufficient approxlmatlon for values of M such that the third term 1s about 5 per cent of the value of the second term In Fig 4 1s shown the comparison between the numerical solution and the asymptotic expansion valid for small values of the contact time, represented by Eq (11) m which only the first 3 terms were considered, for the case rc= 1, p = 0 It 1s seen that the agreement 1s very good for values of M less than about 0 25 and it can be said that the

Fig 2 Penetration theory solution, p = 1

Figure 3 shows the results for rc = 1 at a number of different values of p As expected, for the same contact time, the enhancement factor E mcreases as p mcreases, because the overall rate of reactlon of species A mcreases

Fig 4 Penetration theory comparison between the numerical solution and the asymptotic expansion for small values of the contact time, rc = 1, p = 0

Fig 3 Penetration theory solutlon, rc = 1

Table 1 Asymptotic expansion of E for small contact times, values of the coefficient a!(O)m expresslon (11) as a function of rc and p P 0

01

1

5

01

- 0 4729

05 1 5 10

-0 7239 -0 8200 - 0 9793 - 10217

-0 5857 -0 8367 -0 9328 - 1 0922 -11346

-16013 - 1 8522 - 1 9485 -2 1077 -2 1501

-6 1148 - 6 3658 - 6 4620 -6 6212 - 6 6636

rc

10

-11757 -12007 - 12 104 - 12 263 - 12 306

Table 2 Asymptotic expansion of E for small contact times, values of the coefficrent a;(O) m expression (11) as a function of rCand p P

rc

01 05 1 5 10

0

01

-2 510x lo-’ -1 906x10-* -5 126x 1O-2 - 3 789 x 10-l -8 238x 10-l

- 1 949 -2605 -2 853 -3064 -3 635

-4 -5 -5 -5 -6

(11)

1

5

719x 10 355x 10 572x 10 960x10 132x 10

-8 519x lo* -8 837x 10’ -8945x102 -9 125x lo2 -9 194x lo*

10 -3 -3 -3 -3 -3

241x10’ 304x 10’ 325x10” 361x10’ 376x 10’

A TAVARES DA SILVA

278

be seen from the solution presented m Ref [8] that the enhancement factor E for the film model 1s only dependent upon the values of M = kc[C]*Da/kr* and M p = kJkc[C]* and IS independent of rc [m Ref [8] E The asymptotic expansion for E which 1s valid for large values of the contact time has been ob- IS given as dependent upon the parameters b = tamed by the methods described elsewhere[7] and (DA/k:)& + kc[Cl*)‘lUdCl*~2 and kdCl*Kk, + kJC]*)=l/(l+p) so that b=M(l+p)3] Figure 6 IS shows a plot of E vs M for various values of p for the film model The asymptotic value of E for very 1 1 5 1 large values of M (or of b m Ref [8]) can be ---____ (12) calculated from the asymptotic expansions of the -4&l +p) 64 fir= (1 +p)“’ Airy functions [l.S] and it 1s easily seen that It 1s Figure 5 shows the comparison between the numerical solution and the asymptotic expansion for large M for the cases rc=l, p=O and p=l It 1s seen that the agreement IS excellent from relatively small values of the contact time, and that the asymptotic expansion approaches the numerical solution from above, as expected asymptotic approxlmatlon 1s likely to be more accurate than the numerical solution for those values of

Fig 6 Film model solution E, = m

Fig 5 Penetration theory comparison between the numerical solution and the asymptotic expansion for large values of the contact time, rC= 1 Companson

with the film model solution

The film model solution of the problem being considered here has been presented by Tavares da Sdva and Danckwerts[S] and 1s given m terms of Airy functions To compare the penetration theory solution with the film model solution it IS convenient to use the mass-transfer coefficient for physlcal absorption for the penetration theory, -

This value 1s 50 per cent lower than the correspondmg value of the penetration theory solution Figure 7 shows a comparison between the film model solution and the penetration theory solution for a large range of values of M It IS seen that there IS a substantial difference between the enhancement factors predicted by the film model and the penetration theory and that, for values of M greater than about 1, the enhancement factor predicted by the film model 1s about half the value of the enhancement factor predicted by the penetration theory This substantial difference between the two predlctlons cannot be compared with the small

(13) to ehmmate the contact time from the definition of M and express it as M = kc[C]*D,,/k,*

(14)

In this form MIS expressed m terms of parameters not characterlstlc of the penetration theory It can

Fig 7 Companson of penetration theory (rc = 1) and film model solutions p = 0

279

Penetration theory for gas absorption dtierences found by previous authors who have compared the solutions of the film and penetration theones for various other cases of gas absorption accompanied by a chemical reaction Indeed, it has always been assumed that there 1s a remarkable agreement between film and penetration theory predlctlons of the effect of a simultaneous chemuzal reactlon upon the rate of gas absorption, specially when the diffusion coefficients of all the mtervenmg species are equal The problem studied m this paper seems to be the first case of gas absorption accompamed by chemical reactlon m which the film and penetration theory predlctlons show a substantml disagreement

E.

NOTATION

[Al a

4(O), a;(O) [Cl C DA DC E

concentration of species A, gmol/l dimensionless concentration of A, [Al/LA]* coefficients m asymptotic expansion for E for small M (see Eq (11)) concentration of catalytic species C, gmol/l dlmenslonless concentration of C, rCl/[cl* dlffuslvlty of A, cm2/sec dlffuslvlty of C, cm’lsec enhancement factor defined by Eq (8)

value of E at large values

erfrl =-:T I One-z2dz erfc q

= 1-erf

7) rate-constant for the reactlon between A and species in solution of umform concentratton, set-’ catalytic rate-constant for the reaction between species A and C hqmd-film mass-transfer coefficient, cmlsec kc[C]*D,JkL2 m general, r&/4 in penetration theory k&[Cl* D&A contact time distance beneath the hquld surface, cm

k, first-order

kc k, M P

Acknowledgements-The author would hke to thank Dr J R A Pearson for suggestmg the treatment of the asymptotic expansions and for various useful dIscussIons and cntlasms, the &rectors of the Computer Laboratory, University of Cambndge, and of Centro de Cticulo da Umversldade TCcmca de Lisboa for provichng computer time Fmancml support through Research Grant TLE-6 from the Institute de Alta Cultura, MimstCno da EducacHo National, Lisbon, and from CormssBo Coordenadora da InvestigaGiio para a OTAN, Lisbon, 1s gratefully acknowledged The work was partly done m the Department of Chenucal Engmeermg of the Umverslty of Cambndge to the Head of which, Professor P V Danckwerts, the author wishes to express his gratitude for the hospltahty rendered

asymptotic of M

rc t X

Y x&[Cl*lD, 0 kdCl*t Superscnpt * Interface REFERENCES

[I] ABRAMOWITZ M and STEGUN I A, Handbook of Marhemaacal Funcflons NatIonal Bureau of Standards, Washmgton, D C 1964 [2] DANCKWERTS P V , A Z Ch E Jl 1955 1 456 [3] DANCKWERTS P V , Gas-Lquld Reactlow, pp 103-104 McGraw-Hill, New York 1970 [4] DANCKWERTS P V and KENNEDY A M , Trans lnstn Chem Engrs 1954 32 549 [5] MILLER J C P , The Awy Integral, Mathemarud Tables, Part-V01 B, Bntish Assoaation for the Advancement of Science Cambndge University Press, Cambndge 1964 [6] SHERWOOD T K and PIGFORD R L , Absorptlon and Extraction, 2nd, pp 317-339 McGrawHill, New York 1952 [7] TAVARES DA SILVA A, Research Report No INV-TLE6, 32/72, January 1973 [S] TAVARES DA SILVA A and DANCKWERTS P V , Chem Engng Set 1973 28 847 [9] TAVARES DA SILVA A and DANCKWERTS P V ,Znst Chem Engrs Symp Ser p 48 The Institution of Chenucal Engmeers 1%8