Remarks on using the film model in physical and chemical mass transfer in the liquid phase

Chemical Engineorrny Science, Vol. 45, No. 5, pp. 1417-1419, Printed in Great Britain.

1990.

C00%2509/90 $3.00 + 0.00 0 1990 Pergamon Press plc

Remarks on using the film model in physical and chemical mass transfer in the liquid phase (Received

PHYSICAL

MULTICOMPONENT OPERATIONS

5 May

MASS

1989;

acceptedfor

TRANSFER

Experimental data on liquid mass transfer in binary systems lead to the conclusion that the binary mass transfer coefficient depends on the square root of the diffusivity: k.. L, - D?:5 ‘, From

the definition

.

(1)

18 September

1989)

lack of an interrelation between the fluxes, the systems of diffusion equations can be solved and proper transfer rates of components can be determined. For concentrated systems with the cross effects only in the case when all the binary diffusion coefficients are the same, the above procedure yields an exact solution, while in the other cases it is only an approximation.

of the binary mass transfer coefficient: MASS

D-.

k,=

+

oij

it can be seen that the thickness of the film, 6,, depends on the diffusivity: &.-D?..5 (3) I, . u This conclusion gives the essential difficulty in calculation of multicomponent physical mass transfer. In the literature [see Krishnamurthy and Taylor (1985) and Zanycki et al. (1987)] it is assumed that the basic method for the determination of mass transfer in multicomponent physical systems is the one using the following equation: J = N - N,y = CkO(Y* - Yb).

(4)

A detailed description of calculation of these vectors and matrices can be found in the above-mentioned papers. Equation (4) is obtained after integration of the Stefan-Maxwell (S-M) equation:

cg = 2 li=

kfi

with the boundary

publication

Yi”“, 1

YKN, (i=

IK

,. .,

1

TRANSFER

WITH

REACTION

1 In the liquid phase the following

Case

A + y,B n)

A CHEMICAL

In the consideration on mass transfer with a chemical reaction it is assumed that the film thickness is the same as for the purely physical mass transfer. The solution of mass balance equations for film under steady-state conditions allows one to determine concentration profiles in the film and, as a consequence, mass transfer rates and enhancement factors. However, also in this case-similarly to the physical multicomponent mass transfer-there are problems connected with the film thickness. These problems occur even in such simple cases as absorption with a chemical reaction in dilute systems when no interrelation betwen particular fluxes occurs. Let us consider first the simplest case-absorption of component A with instantaneous reaction of component B in an inert component C. The other cases are considered further.

(5)

reaction

takes place:

products.

The classical equation describing ponent A is as follows:

the transfer rate of com-

conditions x=0

Yi = Y:

x=b

yi = yf.

N,=kLAC;

It should be stressed that some formal trick was used in the solution of the S-M equation. As been mentioned above, from experimental investigation it follows that the thicknesses of subsequent binary films are different for particular pairs of components in the multicomponent mixture, while in the integration of the S-M equation one film thickness 6 was employed (see the second boundary condition). However, in the further transfdrmations of this equation to each binary diffusion coefficient Dij an individual film thickness 6, is attributed and owing to this the binary mass transfer coefficients kij are used. A question now arises: can binary mass transfer coefficients be used in calculation and description of multicomponent mass transfer? For dilute multicomponent systems, in which there are no cross effects between mass fluxes, this question can be answered in a positive way. Despite the fact that film thicknesses differ for particular components, due to

Equation (6) is obtained taking the film thickness for component A as 6,. As a result of a simple consideration (Westerterp et al., 19X4), with film thickness 6, of component B, the follwing equation was obtained: N,

= kL,C

Naturally, when the diffusion coefficients of components A and B are equal, both relations are equal. One should consider which of these equations is more justified. When D, %- D,, then the reaction plane will be placed at the end of the film (near to the bulk) and the enhancement factor is equal to 1, which results from eq. (6) but not from eq. (7). In the opposite case, when D, + D,, eq. (7) is more valid from the physical point of view. The following question arises: which thickness of the film should be used in calculations? The answer to that question can be two-fold. The first one is the following: from the safety

1417

1418

Shorter Communications

point

of view the greatest thickness of the film should be be stressed that there is a lack of a rational meaning in that solution. The second one is that the relation should be searched which better solves the whole problem. Premises for such a method are provided by the considerations similar to the multicomponent physical mass transfer. A consideration similar to the case of physical mass transfer (Fig. 1) leads to the equations used. It should

(9)

Y&‘, = N,

(10)

a+b=l.

(11)

In eqs (8) and (9) the same assumption as in multicomponent physical diffusion [integration of eq. (5)] is used:

After some consideration, form is obtained:

from eqs (SHl

Fig.

2. Concentration profiles for instantaneous reactions A + C -t products and B + C - products.

1) the following

(15) and, using proportionality

(l), the final form is (16)

N,

= k,,Cj;

[I

+ S(Z>““].

(14)

Equation (14) allows one to calculate the mass transfer rate of component A in the case of instantaneous reaction with component B. In the extreme cases (e.g. D, g D, and D, S DA) it leads to the solutions obtained from eqs (6) and (7). Case 2 In this case, there is absorption of two gases and instantaneous reaction with a third component (Fig. 2):

(17)

(18)

WAN,, + YCBN, = NC

a+b=l. After some consideration, easily obtained:

(11)

the following

equations

can be

A + y,--C -+ products B + yceC In this case the following cess:

products.

equations

describe the whole pro-

(1% and

X

. (20 YC&; D, ‘.’ +ctD, d 1

1+

Case 3 In this case, there is absorption of one gas and its instantaneous reactions with two gases (Fig. 3): A + y,B + products A + yc’cc * products. In this case the whole (15), (17) and

process is described

by eqs (9), (1 l),

jQNg+Nc. Ye From Fig.

1. Concentration

profiles for instantaneous A + B -+ products.

reaction

N,

eq. (21) the following

= k,,Cf[l

+ &(2>“.’

(21)

Yc

equation

is obtained:

+ s(&c)“‘].

(22)

Shorter Communications vector of molar fluxes total molar flux, kmol/(m2s) distance from the interface, m vector of molar fractions molar fraction

N N, X

Y Y Greek 6 Y

r3

%I Ci

“cs

1419

letters film thickness, m stoichiometric coefficient

Subscripts A, B, C i, j, k L Superscripts * b

component component liquid

A, B, C, respectively i, j, k, respectively

interface bulk of the liquid

Fig. 3. Concentration profiles for instantaneous reactions A + B + products and A + C + products. Institute of Chemical and Process Technical University of tbdi ul. Wolczahska 175, 90-924 Lbdi,

CONCLUSIONS A new method of calculation of enhancement factors in the case of instantaneous reactions is proposed. The rates of mass transfer in three cases: absorption of one gas and its reactions with one component, absorption of two gases and their reactions with one component, and absorption of one gas and its reactions with two reactants are presented. The equations are more rational than the previous ones. NOTATION a

b C D J k k”

distance of the reaction plane from the interface, m distance of the reaction plane from the bulk of the liquid, m concentration, kmo1/m3 diffusivity, m2/s vector of molar diffusion fluxes mass transfer coefficient, m/s matrix of multicomponent finite flux mass transfer coefficients

Chemical Engineering Science, Vol. 45, No. 5, pp. 1419m1421, Printed in Great Bntain.

Applicability (First

Poland E. NAGY

Research Institute of Technical Chemistry of the Hungarian Academy of Sciences PO Box 125, H-8201 Veszprem, Hungary REFERENCES

Krishnamurthy, R. and Taylor, R., 1985, Simulation of packed distillation and absorption columns. Ind. Engng Chem. Process Des, Dev. 24, 513. Westerterp, K. P., van Swaaij, W. P. M. and Beenackers, A. A. C. M., 1984, Chemical Reactor Design and Operation, p. 399. John Wiley, New York. Zarzycki, R., Chacuk, A. and Starzak, M., 1987, Absorption and Absorbers. WNT, Warsaw.

‘Author

to whom correspondence

6 March

should be addressed.

OOOs2509/9a $3.00 + 0.00 0 1990 Pergamon press plc

1990

of the external-diffusion

received

R. ZARZYCKI+ A. CHACUK M. STARZAK Engineering

1989; accepted

The simple external-diffusion (or film) model after Glueckauf and Coates (1947) has been employed in the literature by many researchers. Attractive for its simplicity, this linear driving force (LDF) mode1 has often been used without consideration on its range of applicability. Recent comparisons (Rice, 1982) of the exact solution and the LDF (or parabolic profile) mode1 suggests that using the same experi-

model in

in adsorption

revisedform 9 August

studies 1989)

mental data gives rise to different values of fitted diffusivity (Hills, 1986). The purpose of this communication is to address this point-the conditions for the validity of the extemal-diffusion model. Needless to say, engineers prefer to use the simplest model if it has a strong physical basis, since more exact, complicated models often lead to intractable results.