Simple approximate solutions to the Beek—Bakker problems

The Chemical Engineering

Simple Approximate JUN’ICHI

Department (Received

Journal, 28 (1984)

127

127 - 130

Solutions to the Beek-Bakker

Problems

HATANAKA

of Chemical Engineering, August

University of Osaka Prefecture,

17, 1983)

ABSTRACT

Approximate closed-form expressions for mass transfer coefficients in a laminar fluid phase with a tangentially moving interface are proposed making use of Ruckenstein’s method of quantitative order-of-magnitude analysis and the correlation method of Churchill and Usagi. The velocity profile along the interface is approximated to be linear with the distance from the interface. Good agreement is obtained between the present approximate solutions and numerical results.

1. INTRODUCTION

One of the basic mass transfer problems between immiscible fluid-fluid phases is characterized by the fact that the interface of the phases moves tangentially. The laminar velocity profile near the interface in any phase considered is approximately expressed as u(x, Y) = u,(x) + a(x)y

(1)

where u,(x) and a(x) denote the interfacial velocity and interfacial velocity gradient respectively and y is the vertical coordinate from the interface to the phase considered. The linearized approximation of the velocity profile is only justified for the case of large Schmidt number and short contact length. The approximation is useful for most liquid phases. Several solutions ,to the related mass trans‘fer problems have been presented. Beek and Bakker [ 1,2] first derived the mass transfer coefficient in the form of an asymptotic series in two cases: with both u,(x) and a(x) constant as in Couette flow, and for u,(x) constant and a(x) 0: l/~“~ as in laminar bound0300-9467/84/$3.00

Sakai, Osaka 591 (Japan)

ary layer flow. These two problems have also been solved numerically by Byers and King [ 31 and by Ueyama and Hatanaka [ 41 respectively .

All these solutions, however, do not seem to be convenient for practical applications because the solutions have not been expressed in closed form. In this work we intend to obtain a solution of closed form for a more general problem in which both u,(x) and a(x) are arbitrary functions of x. This problem may be called a generalized Beek-Bakker problem. The method of solution used in this work is the “algebraic” method or the “quantitative” order-of-magnitude analysis proposed by Ruckenstein and coworkers [ 5 - 71. This method is essentially of approximate nature, but it has been shown to give reasonably accurate solutions to several heat transfer or mass transfer problems of the boundary layer type [ 71.

2. SOLUTION

OF

THE

FIRST

BEEK-BAKKER

PROBLEM

First, we consider the case where both u,(x) and a(x) are constant, which was originally solved by Beek and Bakker [l]. The problem reduces to solving the following linear differential equation:

ac

(24, + ay) ax

a2c

=D aY2

(2)

where u, and a are constants and are assumed to be known. Since our considerations are restricted to mass transfer in a single phase of semi-infinite extent, the dimensionless concentration C(x, y) should satisfy the following boundary conditions of boundary layer type : 0 Elsevier

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in The Netherlands

128

C(x, 0) = 1 C(0, y) = C(X, -)

(3)

= 0

Following Ruckenstein [5], we assume that all derivatives in eqn. (2) can be replaced by algebraic quotients and that both y and ay can be replaced by 6(X) (the concentration boundary layer thickness) and ay2 by 62. Then Ea6 AU s+-__-=___ X X

D

(4)

h2

where A and E are numerical constants to be determined so that the extreme solutions satisfy eqn. (4). The local mass transfer coefficient k is assumed to be given by k=

D 6(x)

(5)

Following Beek and Bakker [l] the dimensionless mass transfer Y as Y=

we define coefficient

kx1’2

(6)

hPP2

Putting

Y into eqn. (4) we have EBli2

A -+-= Y2

1

(7)

Y3

where B is another dimensionless parameter also used by Beek and Bakker [l] and defined as B=

a2Dx us3

(8)

As seen from eqn. (7) Y is a function only of B. The two limiting solutions of Y for B = 0 and B -+ 00 are well known from penetration theory and the Leveque solution respectively. These solutions for Y at the two extremes of B are sufficient to determine the constants A and E: y=yO=

-

1

whenB=O

*l/2

(9)

Bl/6

Y=Y,=

9”V(4/3)

whenB-+m

(10)

From eqns. (7), (9) and (10) we have A zz Ye2 = _

1

(11)

+/2

E=

YO03

_

B1/2

=

1 9l?(4/3)3

(12)

Putting yields

eqns. (11)

and (12)

into eqn. (7)

B1/2

1

-7fY2 + 9r(4/3)sY3

= l

(13)

or more compactly (14) Equation (13) or eqn. (14) gives an approximate solution of the mass transfer coefficient of the first Beek-Bakker problem. Unfortunately, these equations cannot be rewritten to express Y in a simple explicit form but give values of Y only through interpolation. To obtain an explicit form of eqn. (14), we can use the simple interpolation method developed by Churchill and Usagi [ 81, which has been found to be very useful for correlating both computational and experimental data in various transport phenomena. We assume that Y is approximately expressed by Y” = Y,” + Y,”

(15)

where it is a numerical constant which should be such that the differences between the values of Y numerically determined from eqn. (2) and in eqn. (15) are minimal. Most simply, IZ is determined from the value of Yat YO= Y-as n=

log 2 log( Y/ Yo)

at YO= Y,

(16)

One of the most appropriate values of n was found to be 2.5, although Flores and Gottifredi [9] proposed n = 3. The algebraic solution (eqn. (14)) and the interpolation solution (eqn. (15)) are now compared with a numerical solution. The numerical solution of eqn. (2) was obtained by the use of a Crank-Nicholson scheme (three terms) similar to that used by Byers and King [3] (six terms). The algebraic solution (eqn. (14)) always gives lower values of Y than those obtained by the numerical solution. Its maximum relative deviation is less than 0.2%. The interpolation equation (eqn. (15)) with n = 2.5 gives higher values of Y for small B and lower values for large B than those obtained by the numerical solution. The maximum relative deviation of eqn. (15) from the numerical solution is less than 1.5%.

129 3. SOLUTION FOR GENERAL CASE

VARIABLE

us AND

a:

The basic differential equation for mass transfer with variable u, and/or a must be rewritten as

The boundary conditions are the same as eqn. (3) for which u, and a are constant. Equation (17) seems to be more complicated, but when we apply the assumptions of the algebraic method that du,/d.r = u,/x and da/dx = a/x, as before, we obtain an equation equivalent to eqn. (4). If the coefficients A and E in eqn. (4) are allowed to be functions of X, the first Beek-Bakker problem can be extended to the case with variable u, and/or a. We define Y* in place of Y as

(18) Two limiting solutions with variable u, or a are given by the generalized penetration theory [lo] and the Lighthill solution [ 111. These solutions are written in the form of Y* as

$5

PO=

B*

y*,

=

a=0

(19)

u, + 0

(20)

l/6

9i’ar(4/3)

where B* is defined in place of B as B*

=$([us dx/ug$(u-*x/[u”’ dj3

Equation (22) with eqns. (19) - (21) provides an approximate solution for mass transfer with arbitrary variables u, and/or a. The accuracy of eqn. (22) cannot be checked for all possible cases. Only one special example was examined, the mass transfer problem in one phase of two-phase cocurrent laminar boundary layer flow, which has been analysed by Potter [12]. The inter-facial velocity u, is constant and the interfacial velocity gradient a is inversely proportional to the square root of x. Thus, B* in eqn. (21) is a constant in this case, and a simple numerical integration gives the “exact” solution [ 41. The deviations from the numerical solution of the values of Y* obtained from eqns. (22) and (23) were always found to be negative. The maximum deviations of Y* obtained from eqn. (22) and eqn. (23) are less than 1.2% and 2.3% respectively .

The algebraic method does not give the solution for negative a, because the limiting solution at a < 0 and u, + 0 does not exist. If --a is replaced by 13 (>O), eqn. (4) is rewritten as

Au, + Eb6 ___ X

D =_

X

(24

6*

The solution of eqn. (24) by the algebraic method is formally given by

(g?)? ($zp

(25

where u in B* should be replaced by b. Equation (25) is of no use when B* is large but was found to be useful only when B* is smaller than about 0.02, for the case of constant u, and a.

(21) Using these new parameters, we obtain the following equation which has the same form as eqn. (14):

(22) The parameters Y* and B* reduce to Y and B respectively when both u, and a are constant. Equation (22), of course, also reduces to eqn. (14) when both u, and a are constant. We can derive an explicit approximate equation similar to eqn. (15): y*25

= ~*~2.5

+ yf,2.5

(23)

4. CONCLUDING

REMARKS

Two approximate solutions of closed form have been obtained for a general mass transfer problem through a laminar fluid phase with a tangentially moving interface. Since the velocity profile was assumed to be linear with the distance from the interface, the results are applicable to the liquid phase only. The interfacial velocity and the interfacial velocity gradient may be arbitrarily varied. This problem should be referred to as the generalized Beek-Bakker problem.

130

The accuracy of the solutions by the algebraic method and the interpolation method is not always ensured, although Ruckenstein estimated the error of the algebraic method to be within 10% [7]. When both u, and a are constant the approximate solutions give sufficiently accurate values for the mass transfer coefficient. Although for the general case the errors seem to be greater than those for the case of constant u, and a, we may expect that the approximate solutions (eqns. (22) and (23)) are useful for practical purposes. REFERENCES 1 W. J. Beek and C. A. P. Bakker, Appl. Sci. Res., Sect. A, 10 (1961) 241. 2 W. J. Beek and C. A. P. Bakker, Appl. Sci. Res., Sect. A, 12 (1963) 139. 3 C. H. Byers and C. J. King, AZChE J., 13 (1967)

628. 4 K. Ueyama and J. Hatanaka, J. Chem. Eng. Jpn., 11 (1978) 331. and R. Rajagopalan, Chem. Eng. 5 E. Ruckenstein Commun., 4 (1980) 15. 6 E. Ruckenstein, AIChE J., 26 (1980) 850. 7 E. Ruckenstein, Chem. Eng. Sci., 37 (1982) 1505. 8 S. W. Churchill and R. Usagi, AIChE J., 18 (1972)

1121. 9 A. F. Flores

and J. C. Gottifredi, Lett. Heat Mass 141. W. E. Stewart, J. B. Angelo and E. N. Lightfoot, AIChE J., 16 (1970) 771. M. J. Lighthill, Proc. R. Sot. London, Ser. A, 202 (1950) 359. 0. E. Potter, Chem. Eng. Sci., 6 (1957) 170.

Transfer,

10 11 12

9 (1982)

APPENDIX

a A b B C D E k n U

US X

Y Y

A: NOMENCLATURE

(&lay),, velocity gradient at the interface (s-l) constant or function of x (dimensionless) ---a (s-l) a2Dx/us3, parameter (dimensionless) dimensionless concentration of solute molecular diffusivity (m2 s- ‘) constant or function of x (dimensionless) local mass transfer coefficient (m s-l) numerical constant (dimensionless) tangential velocity along the interface (m s-l) interfacial velocity (m s- ‘) coordinate along the interface (m) coordinate normal to the interface (m) k&2/(u,D)“z, mass transfer coefficient (dimensionless)

Greek symbols complete r 6

gamma function (dimensionless) concentration boundary layer thickness (m)

Subscripts interface s, zero limit of parameter B or B* infinite limit of parameter B or B* 00

Superscripts * generalized

quantity

in eqns. (18) - (21)