Turbulent absorption with instantaneous irreversible chemical reaction

The Chemical Engineering Journal, 10 (197% 49-M @ Elsevier Sequoia %A., Lausanne. Printed in the Netherlands

Turbulent Absorption with Instantaneous Irreversible Chemical Reaction J. B. STEPANEK

and S. K. ACHWAL

Department of Chemical Engineering, (Received

University of Salford, Salford M> 4WT (Ct. Britain)

13 April 1975; in final form 14 May 1975)

Abstract The model of damped turbulence near a gas-liquid interface has been employed in an analysis of mass transfer in absorption with an instantaneous chemical reaction in the liquid. It has also been established that simplified formulas can be substituted for the exact solution without any significant error as long as the reaction plane does not lie too close to the edge of the diffusional subbyer.

INTRODUCTION

Enhancement of mass transfer in absorption, caused by the consumption of the absorbed component in an instantaneous irreversible chemical reaction, has been subjected to analysis by means of several models of the flow conditions near to the gas-liquid interface. In a recent paper, Stepanek and Shilimkanl reviewed the film model of Hatta2, the penetration model of Danckwerts3 and the boundary layer model of Friedlander and Litt4. However, there is increasing evidence” lo that the damped turbulence model of I_evich5 is in very good agreement with the real situation near gas-liquid interfaces in liquids in turbulent motion. The model of damped turbulence has been applied by Levich5 in his analysis of absorption into turbulent liquids. It seems, however, that no work has been published on the application of this model to absorption with an instantaneous irreversible chemical reaction in the liquid phase.

If the diffusivities of momentum assumed to be equal, we obtain EM =E,

=Ay2

The model of damped turbulence suggests that the turbulent diffusivities of momentum and mass vary with the square of the distance from the interface6.

(1)

where A can be related to the friction velocity of the flow and the equivalent surface tension; it remains constant for fixed flow conditions. A pseudo-laminar layer at the interface has been defined in such a way that the turbulent viscosity in such a layer is less than its molecular counterpart. At the edge of the pseudolaminar layer the two viscosities are equal, thus giving for the thickness of this layer s,=&

(2)

In a similar way, a diffusional and its thickness is given by 6, = m

sublayer can be defined,

= &SC-+

(3)

The two thicknesses depend on the physical properties of the liquid (viscosity, diffusivity and surface tension) and the flow conditions as represented by friction velocity. In liquids, Schmidt numbers are usually fairly large, thus indicating that turbulent transport of mass is dominant over the major part of the pseudo-laminar layer and that the concentration profile in the outer region of this layer is rather flat. Hence, although the slope of the velocity profile at the edge of the pseudolaminar layer is still appreciable since D,ff = 2v, it can safely be assumed that the concentration at this point is equal to the concentration in the bulk of the liquid. In view of the preceding discussion, the transport of the component A towards the reaction plane can be described by the following equation: NA =-(DA

ANALYSIS

and mass are

The dimensionless

+Ay2)c2 form of eqn. (4) is

er!h=_ (1 + sc*&!$ CDA

(4)

50

J. B. STEPANEK,

where

The enhancement

(6)

=V=A”: A

DA

DA

The boundary

conditions

at y=Oorn=O

x,&=0

at y=y,orn=n,

The corresponding

!!!!a=_ (1

equation

(7) for component

at

n=rlr

xn=xi

at

n=l

(8)

E(arctg Sc$ - arctg Scinr)

Equations (17) and (18) can be applied only when the rate of absorption is relatively low. At high absorption rates, the convective effect accompanying mass transfer has to be taken into account. In such cases, eqn. (18) has to be replaced by E(arctg Sci - arctg Scinr)

equation

’

A + bB + products Nu = - bNA and eqn. (6) can then be written as +

SCB$)

(10)

of eqn. (4) within the indicated limits yields

NA&I

+ CDASCA

t arctgSCAT& =X2

Similarly, eqn. (10) can be integrated I

(11) to give

(arctg Sch - arctg Scbnr) = xi

(12)

CI?B~C$

The case of purely physical absorption by the equation

fib - _ --

(1 +

scA,j’)

CDA

The boundary XA

conditions

=x;

xA=O After integrating

ND1

is described

(13)

will be taken as

at

77’0

at

q=l

(14)

eqn. (13) we obtain

t = xl arctg SC), cDASC~,

-r

$f

arctg Sci

(19)

For a reaction A+bB=pPtqQ

!T$

CDB

Integration

(18)

(9)

For a reaction described by the stoichiometric

E!d! =(1

(17)

arctg Sci

conditions

xn = 0

= arctg Sch

B is

CDB

with the boundary

of eqn. (15) with eqns. (11)

and

2

SCB$)

+

(16)

Hence, by comparison and (12) we obtain E arctg Stir),

are

xA=xz

bNA8 7

factor is defined as

E=NAJhPA

7) =J+1 g

S. K. ACHWAL

(1%

(20)

n is defined as n=p+q-b

(21)

DISCUSSION

Enhancement factors for absorption with instantaneous chemical reaction can be obtained by solving the simultaneous equations (17) and (18). The results are shown in Figs. l-3 for various values of the Schmidt numbers SCA and SC~. In Fig. 4, the results of this study are compared with those calculated from the models of Hatta2 and Danckwerts3 for the case of widely different Schmidt numbers of the two components. The agreement between the results calculated by Danckwerts and those obtained here is very close, the difference being about 56%. This difference decreases as the two Schmidt numbers approach each other since for DA = DB all solutions reduce to the solution obtained by Hatt$. The solution of the simultaneous equations (17) and (18) is a tedious procedure requiring iterations. A substantial simplification can be achieved, without significantly affecting the accuracy in most cases, by

TURBULENT

ABSORPTION

51

WITH REACTION

Fig. 1. Enhancement factor for SCB = 400 and ScA = 400,160O

dividing the pseudo-laminar surface layer into two sublayers and by neglecting the turbulent diffusivity inside and the molecular diffusivity outside the diffusional sublayer. In this case, three different situations may arise: that the reaction plane lies inside or outside the diffusional sublayers of both components; or, eventually, inside one and outside the other diffusional sublayer if their thicknesses are different. Cizse1: reaction plane inside the diffusional sublayers of both components The component A is being transferred to the reaction plane solely by molecular diffusion and the mass flux is given by the equation NA& -_-

_ Xi

DAc

%

(22)

The transport of component B between the edge of the diffusion sublayer and the reaction plane is also by molecular diffusion only. The corresponding equation is NB61 _-_--p

D~c

_ bNAh

D~c

_

XBd

G&r-‘l)r

(23)

and 3600.

In this equation, 77~ is the dimensionless thickness of the diffusional sublayer of component B and xud iS the corresponding concentration. Assuming turbulent diffusion alone to be responsible for the transport of B in the outer part of the pseudolaminar layer, the process can be described by = sc&

- Xnd)

(24)

After eliminating xnd and nr from eqns. (22) (23) and (24), we obtain

4- 1 bNA6, 2 SC~ DAC

SCA

=x;

f’s% +-x %A

*

A

(25)

By a similar procedure, the rate of physical absorption for the same concentration difference is given by

(26) A comparison of eqns. (25) and (26) gives for the enhancement factor

52

J. B. STEPANEK,

0.1

0 01

IO

10 0

S. K. ACHWAL

100 0

1: b31 A Fig. 2. Enhancement

factor

for Scn = 1600 and ScA = 400, 1600 and 3600.

(27)

Equation (27) can be employed if 6, < &d. Elimination of x&j and NA from eqns. (22) (23) and (24) gives for Qr

The condition

of validity is (31)

for DA > Dg, and (32)

When this result is combined with eqn. (3), we can establish that the condition of validity is S&(2Sci

- l)(ScA(s+z))’

<1

for DA CDs. Self-induced convection has been neglected in these equations. It can be taken into account if the concentration term xt/bxt is replaced by

(29) -

Here, SCA,~ is the larger of the two Schmidt numbers SCA and Scn. For large Schmidt numbers, eqn. (27) can be simplified to give

+yB)i~

(30)

In { 1 + (n/b)x$} n In (1 -xX)

The approximate solution is shown in Figs. l-3 (the broken lines). The limit of applicability is given by x; -bx;

_ &(2Scg

- 1) - Scn scA

(33)

53

TURBULENT ABSORPTION WITH REACTION

Fig. 3. Enhancement

factor for SCB = 3600 and

SCA

THIS

=

400, 1600 and 3600.

PAPEP

E lo-

Fig. 4. Comparison of the enhancement factors calculated from eqns. (17) and (18) with values obtained by Hatta2 and Danckwerts3 for ScA = 400 and ScB = 3600.

54

J. B. STEPANEK,

For this condition the approximate and accurate solutions match exactly at SC* = Scn. For SC* > ScB the difference between the solutions remains fairly low and does not exceed 5% for Schmidt numbers ranging from 100 to 3600. A larger difference of up to just over 10% is obtained for the opposite case of SC_&
S. K. ACHWAL

solving eqns. (36) and (37) for q. The result obtained is bxi/x; 77r= bxz/x; The condition

+1

I f 2ScA

(40)

then becomes

Sch,B(bxfq/x; Case 2: reaction plane outside the diffusional sublayers of the two components In this case, the absorbed component traverses the whole diffusional sublayer by molecular diffusion. This process is described by the equation NA61 z

_ ’ - scft (xz

+ I)> 1 I + 2Scft

bxilx:

Here SCA,B is the smaller of the two Schmidt numbers ScA and ScB. For large values of ScA, eqn. (41) can be simplified to give bxzjxo, > 1

- XAd)

(34)

A

where XAd is the concentration of A at the edge of the diffusional sublayer. From here, the component A is carried by turbulent diffusion in accordance with the equation

(41)

(42)

for ScA > ScB, and bx%/xi > 2 DB/DA - 1

(43)

for SUB > SC,. The approximate solution is represented by the broken lines in the lower left part of Figs. l-3. The limit of applicability is given by the formula

Xi

f SC&B - 1 (44)

(35) 2sc$ - 1 After eliminating obtain

XAd from eqns. (34) and (3 S), we

(36) The transport of component B from the bulk of the liquid phase to the reaction plane is by turbulent diffusion. When the stoichiometry of the reaction is taken into account, the process can be described by (37) The elimination of the dimensionless distance 77, of the reaction plane results in the fomula 1

N&

2Sci - I zx; DAC SCA

A comparison

0

+xB b

(38)

of eqns. (38) and (26) gives

.=I+$A

The match between the approximate and accurate solutions is about the same as in the previous case. The two solutions coincide at SC, = SCB; the difference is less than 5% for SCA > SCB and up to just over 10% for SCA < SCB in the investigated range of Schmidt numbers from 100 to 3600. Case 3: reaction plane outside one and inside the other diffusional sublayer The case of the reaction plane outside one and inside the other diffusional sublayer can be approached in exactly the same way as the previous two cases. However, as seen from Figs. 1-3 and in more detail from Fig. 5, the results of the simplified treatment deviate quite substantially from the exact solution. This is probably because the reaction plane lies in the region where the turbulent and molecular diffusivities are of the same order of magnitude for both components. Neglecting one or the other in the critical region evidently leads to serious errors.

(39)

which is exactly the same result as that obtained by Hattaz. The condition for the reaction plane to be outside the diffusional sublayers can be obtained by

CONCLUSIONS

Formulas have been obtained for the enhancement factor in absorption with instantaneous chemical

TURBULENT

ABSORPTION

WITH

REACTION

Fig. 5. Detailed graph of the enhancement factor SCB = 1600 and ScA = 400, 1600 and 3600.

for the reaction

reaction into a turbulently flowing liquid. These formulas take into account the damping of velocity fluctuation near to the interface. It has been demonstrated that, over a wide range of concentration ratios xi/bxz, approximate formulas can be substituted for the exact solution with very little error. In the case of the reaction plane being outside the diffusion sublayers of both components, the simplified formula is identical to the result obtained by Hatta2.

plane close to the edge of the laminar

n

P P

Q 4 SC

&A SCB xA

NOMENCLATURE

A A B b ; DA

DB

E

NA

NOA NB

coefficient in the formula for turbulent diffusivity, T-l absorbed component reactant in the liquid phase stoichiometric coefficient number of moles per unit volume, NLV3 molecular diffusivity, L2T-r molecular diffusivity of A, L2T-’ molecular diffusivity of B, L2T-r enhancement factor, dimensionless molar flux of A, NLe2T-i molar flux of A in physical absorption, NL-2T-’ molar flux of B, NLw2T-r

XX XAd

XBd

Y

sublayer

for

combination of stoichiometric coefficients (=P+q-b) product of reaction stoichiometric coefficient product of reaction stoichiometric coefficient Schmidt number, dimensionless Schmidt number of A, dimensionless Schmidt number of B, dimensionless mole fraction of A, dimensionless mole fraction of A at the interface, dimensionless mole fraction of A at the edge of the diffusional sublayer, dimensionless mole fraction of B, dimensionless mole fraction of B in the bulk of the liquid, dimensionless mole fraction of B at the edge of the diffusional sublayer, dimensionless distance measured from the interface into the liquid, L

Greek symbols. thickness of the pseudo-laminar layer, L thickness of the diffusional sublayer, L 6d turbulent diffusivity of momentum, L2T-r EM

61

56 Em

77 VBd

.I. B. STEPANEK.

turbulent diffusivity of mass, L2T-’ dimensionless coordinate (= _y/S,) dimensionless coordinate of the edge of the diffusional sublayer of B dimensionless coordinate of the reaction plane kinematic viscosity, L*T-’

REFERENCES 1 J. B. Stepanek Eng., 52 (1974)

and R. V. Shilimkan, Trans. Inst. Chem. 313.

S. K. ACHWAL

2 S. Hatta, Tech. Rep., Tohoku Imp. Univ., 8 (1928-29) 1. 3 P. V. Danckwerts, Trans. Faraday Sot. 46 (1950) 300. 4 S. K. Friedlander and M. Litt, Chem. Eng. Sci., 7 (1958) 229. 5 V. G. Levich, Physicochemical Hydrodynamics (English Transl.), Prentice-Hall, Englewood Cliffs, N.J., 1962. 6 J. T. Davies, Turbulence Phenomena, Academic Press, New York, London, 1972,lpp. 184-187. 7 A. A.KozinskiandC. J. King, A.I.Ch.E.J., I2(1966) 109. 8 J. T. Daviesand S. T. Ting, Chem. Eng. Sci., 22 (1967) 1539. 9 A. P. Lamourelle and 0. C. Sandall, Chem. Eng. Sci., 27 (1972) 1035. 10 R. V. Shtiimkan, Ph.D. Thesis, University of Salford, 1975.