Gas absorption accompanied by complex chemical reactions-IV

ChemicalEngineering

Science,

Pergamon Press.

1972, Vol. 27. pp. 247-255.

Gas absorption

Printed inGreat

accompanied by complex chemical reactions -IV Unsteady

K. ONDA,

Britain

state

E. SADA, T. KOBAYASHlt and M. FUJINE Nagoya University, Nagoya, Japan (Received 20 April

197

1)

Abstract-For gas absorption accompanied by (m,n)-(p,q)-th order reversible, consecutive and parallel chemical reactions, the approximate solutions based on the penetration and surface renewal theories are derived under the condition of equal diffusivities, and compared with the approximate solution based on the film theory to manifest the generalized solution being suitable for three theories. For the case of reversible chemical reaction, the approximate solution based on the penetration theory is compared with the numerical solution to examine the accuracy of Hikita and Asai’s approximation in the unsteady state. It is found that the deviation of the approximate solution from the numerical solution is only a few per cent. THE

AUTHORS [ l-31 have mathematically discussed gas absorption accompanied by (m,n)(p,q)-th order reversible, consecutive and parallel chemical reactions from the viewpoint of the film theory. In those papers, it has been confirmed that the approximate solutions obtained from the application of Hikita and Asai’s approximation[4] are in good agreement with the numerical and analytical solutions within a few per cent. It is the purpose of this paper to present the approximate solution based on the penetration and surface renewal theories for gas absorption accompanied by the generalized complex chemical reactions similar to those in the previous papers, and to confirm that Hikita and Asai’s approximation is also valid for those in the unsteady state condition as well as the steady state (film theory). Furthermore, these approximate solutions are compared with the film theory results to manifest the generalized solutions being suitable for three theories.

(m,n)-th ORDER

IRREVERSIBLE REACTION

A + vBB 4 Product.

(a)

When one applies the penetration theory to the gas absorption process, the following differential equations can be derived. DA z-z

= k,A”B” = R,

D?%_aB= B &x2 at

vB

R

1’

(1)

(2)

The boundary conditions to be imposed on the simultaneous solution of these differential equations are t=O,xaO;A=A,,B=B,

(3)

t > O,x=O;A=AiyB=Bi,~=O

(4)

t>O,x+yA=A,,,B=Bo

(5)

CHEMICAL

In order to examine whether Hikita and Asai’s approximation is also adequate in the unsteady tpresent

state condition or not, the problem first considered is gas absorption accompanied by a single (m,n)-th order irreversible chemical reaction.

where B = Bi in Eq. (4) is the definition of the

address: Suzuka College of Technology, Suzuka, Mie, Japan.

247

K. ONDA,

E. SADA,

T. KOBAYASHI

concentration of the species B at the gas-liquid interface. In the following discussion, however, this is treated as one of the boundary conditions and assumed to be independent of the gasliquid contact time, although Bi may be considered in fact to be a function of time in the penetration theory. The reaction rate term R, in Eqs. (1) and (2) can be linearized with respect to A by using Hikita and Asai’s approximation as follows; R, = 5

k,Aim-‘BinA E 5A.

(6)

Then, Eq. (1) becomes identical to that for gas absorption accompanied by first order chemical reaction. The concentration profile of A and the reaction factor @ may be obtained by a similar method to that given by Danckwerts [5]. A = 4 [exp (x-~ + exp (-~a) + A0 exp (- &)

) erfc (x/2m+ erfc (x/2met-f (x/2*)

X exp (- 4M,/r)

fi) *)I (7)

(8)

where Ml = r&/4 = MbP.

(9)

On the other hand, if one substitutes Eq. (7) into Eq. (2) in which the reaction rate term is already linearized with respect to A, the following relationship would be yielded

@2=4,LE,

Gl

and M. FUJINE

When one compares Eq. (11) with the analytical solution given by Danckwerts [6] and Sherwood and Pigford[7] for gas absorption with the instantaneous chemical reaction, it may be seen that Eq. (11) is identical to the analytical solution only for the case of r, = 1. When rB = 1 and Ai @ B,,, Eq. (11) becomes nearly equal to that solution as suggested by Brian et al. [8]. The restriction of Eq. (1 l), Ai < B,,, does not occur on the film theory approximate solution even if the value of r, is not equal to unity, i.e. when the reaction rate is considered to be instantaneous, the approximate solution becomes identical to that given by Hatta[9]. Therefore, the assumed condition of B = Bi which may be considered to cause the restriction mentioned above, would be appropriate for the film theory, but for the penetration theory it will come into some question except for r, = 1. However, when one takes into consideration of the derivation of Eq. (10) in which Hikita and Asai’s approximation is used twice, the agreement between the approximate and the analytical solution is remarkable and satisfactory. The accuracy of the approximate solution has been confirmed by Hikita and Asai[4], who showed that, when Eq. (10) is replaced by Eq. (12), the approximate solution for the case of m = IZ= 1 and A,= 0 agrees with the numerical solution given by Brian et a!.[81 within a few per cent, even if the value of r, deviates from unity and the restriction Ai 6 B,, is not satisfied. @= l+(@a-l)(l-bi).

(12)

In Eq. (12), @‘ais the value evaluated from the analytical solution given by Danckwerts and others[6,7]. This would indicate that Hikita and Asai’s approximation is also valid for the penetration theory when the restriction Ai <
5 { &(I-$)}(I--b,)1+

(10) Therefore, when V% + m, Eq. (10) is rewritten by Eq. (11) because the reaction rate may be considered to be instantaneous, i.e. qA + 0, bi+Oandfil+m.

248

(m,n)-f&q)-th

ORDER COMPLEX REACTION

Gas absorption

accompanied

CHEMICAL

by the following

Gas absorption accompanied by complex chemical reactions-

complex chemical reactions similar to those in the previous papers [l-3] is considered in this section. A+v,B

s v,E++F

(b)

A + v~B 2 vcC + Product (c) vatA’ + v;C 2 vBB A+v;B

3 v& (d)

A + v;C 2 Product A+v,B

3 Product (e)

A + vcC 2 Product

IV

t>O,x+yA=A,,A’=A;,B=B,,C=C,, E = E,, F = F,,.

(17)

The condition (16) means that the concentrations at the interface are also assumed to be independent of the gas-liquid contact time in the similar way to Eq. (4), although it is not true. For each reaction system, the approximate solution is obtained by the similar method to those mentioned above and described in the previous papers[l-31. The diffusivity of each species is assumed to be equal to that of gaseous species A, in order to exclude the restriction of the approximate solution such as Ai < B,, and to simplify the problem. 1. Reversible reaction (reaction b)

A + vBB 3 Product (f)

The approximate solutions based on the penetration and surface renewal theories may be obtained by using the Laplace transform. When these solutions are compared with those The material balance equation for each species may be written based on the film theory, it can be seen that each solution is quite alike. The results are summarized in Table 1, where parameters 5 and @,, D &k!&, R +v’R Jax2 at J1 J2 are different in each theory so the parameters (J=A,A’,B,C,E,F) (13) are also listed in Table 2. For gas absorption accompanied by (l,-)(1 ,-)-th order reversible reaction, Sherwood where the sign of reaction rate term RI or R, and Pigford[7] and Huang and Kuo[lO] have must be changed provided that the species “.I” derived the analytical solutions based on the is in the direction of arrow sign. The stoichiopenetration and surface renewal theories, remetric coefficients v, and vi are equal to each spectively (although the former’s solution other except for the species C in reactions contains some mistakes). When one compares (c) and (d). The boundary conditions to solve the approximate solutions for (1 ,-)-( 1,-)-th the above differential equation are order reversible reaction with these analytical solutions, it may be found that the surface t=O,xsO;A=A,,A’=A;,B=B,,C=C,,, renewal theory result is identical to that given E= E,,F= F,, (14) by Huang and Kuo [lo], whereas the penetration theory result deviates from the analytical solution as shown in Fig. 1. This deviation (maximum: about 6 per cent) is considered to be =aF=(-) caused by the assumption of Eq. (16), because (15) ax Sherwood and Pigford’s solution has avoided B = Biy C = Ci, E = Et, F = Fi using this. The approximate solutions obtained (16) by a similar method to that given by them are v~,A’ + v~B 2 Product.

249

K. ONDA,

E. SADA, T. KOBAYASHI

and M. FUJINE

Table 1. Approximate solutions based on the film, penetration and surface renewal theories, (in the last two theories, r,,, r,, r,, r, and r, are taken to be equal to unity) Reaction system

(a)

Approximate solution

Mbi”

@= l--q,+$qdl-bi)

1++-& E

(b) @-(l--q,)

=zqdl-b,)

0 =zqB(ei-eo)

a = Q,, w = a; Cc)

M, and Mj

Mbj”

~-(l-q~)=~o{~‘-(l-qA~)}+~qB(l-br)

and M’ciq

(e)

l+z

(I-bA+zq&,-C,)

@=

l-Y,+:%

@=

a+

@=

rB rc l-e+-q~(l-bb)+v~qB(co-c~) %

(4

(

)

M(b,“+Vc,*)

M ( bin+ Vc,“)

ci =

Mbi”

and M”biq

expressed in Table 3. Although the assumption of Eq. (16) is also used to linearize the reaction rate terms and to obtain the relationships between @ and b,, ei and&, the result for the case of m = p = 1 and n = q = 0 is identical to the exact solution (see Fig. 2). Figure 2 shows the comparison between the approximate solutions in Tables 1 and 3 and the numerical or exact solutions, whose deviations 250

are 2 per cent for (2,-)-(2,-)-th order and 5 per cent for (l,-)-( I,-)-th order. Although the result in Table 3 is considered to be more accurate than that in Table 1, the former becomes more complex to evaluate the reaction factor @ for the case that 4M,/7r(K,21) is very large. Furthermore, the reaction factor Q, for K, < 1 can not be evaluated from the result in Table 3. Therefore, for the case of (m,n)-

Gas absorption accompanied by complex chemical reactions-

IV

Table 2. Parameters to be used in each theory Penetration theory

Film theory L

Surface renewal theory

1

erf (2*)

5

sech a

%

VR tanh a

(

@I

@0(1-q.&

Q(l -q”O

%(l -q,45)

@:

@;(I -q.&‘)

@A(1-q,v5’)

@;tl-q.X)

N.4

dA -DA ( dx ) ==o

f J-1 @t3,J

4

DA/XI.

2v5Jz

l+M,

mm ~+--&)erf(2~)+fexp(-3)

G,

,,

dt

[-D~Jl(r)(%),=j

dr

v&

Table 3. Approximate solution based on the penetration theory for reaction (b), which may be obtained by a similar method to that proposed by Sherwood and Pigford [7] (rB = r, = r, = I )

and whenK,

whenK,=

> 1;

I;

~~i~-q~~[~[~-exp(-~)]+2[1-~erf(2~~]]

where M, = Mb,“( I + vET)

(p,q)-th order reversible reaction, the accuracy of the result in Table 1 is examined. Figures 3-5 show the comparison between the result in Table 1 based on the penetration theory and the numerical solution given by Secor and Beutler[l 11, in which the agreement of the two solutions is quite remarkable, i.e. the deviation is less than 5 per cent. This

would suggest that the result in Table 1 is also accurate and furthermore the assumption of Eq. (16) does not cause any significant error when the diffusivity of each species is assumed to be equal to each other. 2. Consecutive reaction {reactions c and d) The approximate solutions based on the 251

K. ONDA,

E. SADA, T. KOBAYASHI

and M. FUJINE

4

Fig. 1. A * E (penetration theory) m = p = 1, r, = v, = 1 and qA=e,=O. Approximate solution (Table 1); -----exact solution (cf. Table 3).

Fig. 2. A * vEE (penetration theory) r, = 1, vE = p/m = 1 and qA = e, = 0. Approximate solution (Table 1); -.-.-.approximate solution (Table 3); ----numerical solution.

penetration and surface renewal theories are summarized in Table 1 by comparing with the film theory results[2]. In this case, the approximate solutions may be obtained by a similar method to that for gas absorption accompanied by (m,n)-th order irreversible reaction, because, in each reaction step, gaseous species reacts with the reactant being present in the liquid phase. The relationship between bl and ci in reaction (d) is different from that in the previous paper[2]. The result in Table 1 may be obtained by the analogous manner to the derivation of Eq. (lo), i.e. it is not necessary to assume the concentration profiles of the species B and C and the same relationship between bi and C~ is derived from the viewpoints of the film, penetration and surface renewal theories. Furthermore, it was found on the film theory result that the accuracy of the approximate solution in Table 1

is a little better than that of the previous paper to evaluate the reaction factor @. Figure 6 shows the comparison between the penetration theory approximate and the numerical solution for reaction (d). The deviation of the approximate solution from the numerical is only a few per cent, which is however about three times of that for the film theory. This would suggest that Hikita and Asai’s approximation is also valid for the penetration theory but the accuracy of the approximate solution is considered to be less than that for the film theory. 3. Parallel reaction (reactions e andf) The approximate solutions based on the penetration and surface renewal theories can be derived by the similar method to those mentioned above and then summarized in Table 1. Although the accuracy of the approximate

252

Gas absorption accompanied by complex chemical reactions-

IV

4-

5 3-

Q Q

3

2-

I

5

h

I *

Fig. 5.A+B*2E+F(penetrationtheory)m=n=q= p=2.rB=r,=r,=l, v~=Y~=I,v~=~.~~=~,=~,=O Approximate solution (Table and Kq,=l. -- - - - numerical solution (Ref. [ 111).

I

1. 1);

Fig.3.A+BGE+F(penetrationtheory)m=n=p=q= 1, r,=r,=r,=l, vB=vE=vF=l, qA=e,=f,=O and Approximate solution (Table 1); - - - - numK=S.erical solution (Ref. [l 11).

r

Fig. 4. 24 + B S E + F (penetration theory, m = 2, n = p = 1, r,=r,=r,= 1, v~=va=r+= l/2, q*=eo=fo=O andK=l. Approximate solution (Table 1); - - - - numerical solution (Ref. [ 111). q=

Fig. 6. Reaction (d) (penetration theory) m = n = p = q = 1, ~,Y~=Y~=Y;= l,q,=4andc,=O. APProximate solution;----numerical solution.

r,=r,=

253

K. ONDA,

E. SADA,

T. KOBAYASHI

solutions is not confirmed in this case, it will be expected from those mentioned above and described in the previous paper[3] that the deviation of the approximate solution from the numerical solution is only a few per cent. In reaction (e), the relationship between bi and ci is also obtained by a similar method to that in reaction (d), which was confirmed on the film theory to give a more accurate approximate solution than that of the previous paper [3].

and M. FUJINE

gas absorption accompanied by the more complex chemical reactions. The approximate solutions for the case of r, # 1(J = A’, B, C, E and F) can be obtained by similar methods to those mentioned above, although they are not shown in this paper. These solutions are different from the results shown in Table 1. However, it may be considered that they involve some restriction such as Ai G BO which must be removed to evaluate the accurate approximate solution.

CONCLUSION

For gas absorption accompanied by (m,n)(p,q)-th order reversible, consecutive and parallel chemical reactions, the approximate solutions based on the penetration and surface renewal theories are derived under the condition of equal diffusivities, and compared with the film theory approximate solution to manifest the generalized solution being suitable for three theories. For two reaction systems, the approximate solution based on the penetration theory is compared with the numerical solution to confirm the accuracy of the approximation. The following conclusions are drawn from this paper; (1) The form of the approximate solutions is found to be insensitive to the theory as shown in Table 1. (2) The deviation of the approximate solution based on the penetration theory from the numerical solution is only a few per cent which is about the same order as those in the previous papers [ 1,2]. Therefore, although the accuracy of the approximate solution based on the surface renewal theory is not confirmed, this will suggest that Hikita and Asai’s approximation is valid not only for the steady state condition but also for the unsteady state condition. (3) The assumption that the concentrations, at the gas-liquid interface, of the reactant present in liquid phase are independent of the gas-liquid contact time, is confirmed not to affect significantly the evaluation of the reaction factor. This would simplify the problem of obtaining the approximate solutions based on the penetration and surface renewal theories, for 254

NOTATION

A.A’

concentrations of gaseous species, moles/cm3 B concentration of reactant being in liquid phase, moles/cm3 b B/B,, of reactant or interC concentration mediate species being in liquid phase, moles/cm3 Cl& diffusion coefficient, subscript indicating the species, cm2/sec E concentration of a product in reaction (b), moles/cm3 El& concentration of a product in reaction (b), moles/cm3 f F/&

D”

K

kl, k2 k,*, k,*’

A4 M’

MU

kz

_-AiP+9-m-n

reaction rate constants of the first and the second reaction step physical mass transfer coefficients for gaseous species A and A’, see Table 2, cmlsec 5

k,D~Ai”-‘Bo”I(k,*)2

&

v,&A;p-lCoq/ (k,*‘) 2

-

2

P+l

~,ut-.J;~-‘B,~/(k;‘)~

M1, A4: dimensionless Table 2

parameters,

see

Gas absorption accompanied by complex chemical reactions - IV

m, n kinetic orders for the first reaction step NA, NAP absorption rate of gaseous species A and A’, see Table 2, moles/ m3. set kinetic orders for the second reacP7 4 tion step defined by Ao/Ai, AJA: and Bo/Ai %, qA’, % reaction rates of the first and, the RI, & second reaction step, moles/ cm3. set r defined by D/DA, subscript indicating the species surface renewal rate, see-’ ; ,“a4sR;;;;;;o;;;-~!; set 9

:

VcIq--l/bin--l

V

!$

X

distance beneath surface of liquid, cm thickness of liquid film, cm

XL

Greeksymbols see Table 2. stoichiometric coefficients, subscript indicating the species defined by Eq. (6) reaction factors, defined by NA/k,*Ai and N,,/k,*‘A; see Table 2 asymptotic value of Q,for A&G + CCI surface-age distribution function AI/A,

Subscripts A, A’ B, C, E, F

gaseous species reactant or product being in liquid phase i gas-liquid interface 0 liquid bulk , indicating the value for gaseous species A’

KqB*-”

REFERENCES [l] ONDA K., SADA E., KOBAYASHI T. and FUJINE M., Chem. Engng Sci. 1970 25 753. [2] ONDA K., SADA E., KOBAYASHI T. and FUJINE M., Chem. Engng Sci. 1970 25 761. [3] ONDA K., SADA E., KOBAYASHI T. and FUJINE M., Chem. Engng Sci. 197025 1023. [4] HIKITA H. and ASAI S., Kagaku Kogaku 1963 27 823. [.5] DANCKWERTS P. V., Trans. Faraday Sot. 1960 46 300. [6] DANCKWERTS P. V., Trans. Faraday Sot. 1950 46 701. [7] SHERWOOD T. K. and PIGFORD R. L.,Absorption and Extraction, 2nd Edn. McGraw-Hill, [S] BRIAN P. L. T., HURLEY J. F. and HASSELTINE E. H.,A.I.Ch.E.JI 1961 7226. [9] HATTA S., Tech. Repts. Tohoku Imp. Univ. 1928 8 1. [lo] HUANG C. J. and KU0 C. H.,A.I.Ch.E. Jll965 11901. [ll] SECORR.M.andBEUTLERJ.A.,A.f.Ch.E.J/196713365.

New York 1952.

Resume- Les auteurs d&vent des solutions approximatives pour l’adsorption de gaz accompagnee de reactions chimiques paralleles, consecutives ou reversibles de I’ordre (m,m)-(p,q); ces solutions sont basees sur les theories de penetration et de renouvellement de surface dans une condition de diffisivites &ales. Les auteurs comparent ces solutions a la solution approximative basee sur la theorie de film pour demontrer que Ia solution generalisee est valable pour trois theories. Dans le cas dune reaction chimique reversible, on compare la solution approximative basee sur la theorie de la pen&ation a la solution numerique; on examine ainsi l’exactitude de I’approximation de Hitika et Asai dans le cas de l’ttat instable. Les auteurs trouvent que la difference entre la solution approximative et la solution numerique ne represente que quelques pour cent. Zusammenfassung - Fur die Gasabsorption begleitet durch reversible, aufeinanderfolgende und parallele chemische Reaktionen (m,m)-@,q)-ter Ordnung werden auf Grund der Durchdringungsund OberlI%chenemeuerungstheorien N~erungsIGsungen abgeleitet unter der Bedingung gleicher Ditfusionsvermogen, und mit der Nlherungsliisung auf Grund der FiImtheorie verglichen urn die fur drei Theorien geeignete verallgemeinerte Losung zu erweisen. Fur den Fall reversibler chemischer Reaktion wird die Naherungslosung auf Grund der Durchdringungstheorie mit der numerischen Losung verglichen urn die Richtigkeit der Hikita und Asaischen Ngherung im unstabilen Zustand .zu priifen. Es wurde festgestellt, dass die Abweichung der NtierungsIGsung von der numerischen L&sung nur wenige Prozente betr%gt.

255