Analytical solution to mass transfer in laminar flow in hollow fiber with heterogeneous chemical reaction

Pergamon

Chemical Enoineerincl Science, Vol. 51, No. 16, pp. 3995 3999, 1996 PII:

S0009-2509(96)00241-2

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/96 $1500 + 0.00

ANALYTICAL SOLUTION TO MASS TRANSFER IN LAMINAR FLOW IN HOLLOW FIBER WITH HETEROGENEOUS CHEMICAL REACTION NORMAN W. LONEY Department of Chemical Engineering, Chemistry and Environmental Science,New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.

(First received 8 September 1995; accepted 29 January 1996) Abstract--Usinga regular perturbation technique, an analytical solution is reached for laminar flow mass transfer in hollow fibers containing a chemical reaction. Specifically,the nonlinear boundary condition resulting from the heterogeneous chemical reaction is expanded in an infinite series. This motivates the perturbation technique used to recast the nonlinear system into a series of linear systems. Analytical solutions to some of these linear systems have been published. This result is useful in evaluating the enhancement in separation that is obtainable in facilitated membrane processes. Copyright @ 1996Elsevier Science Ltd

Keywords: membrane, mathematical model, heterogeneous reaction.

I. INTRODUCTION Membrane separation is a naturally occurring process in biological systems. However, separation technology in engineering applications has traditionally been extraction, distillation and absorption. Applications of membrane separation to nonbiological systems have expanded rapidly to include fields such as Chemical Engineering, Environmental Science, Food Science, Electronic Engineering and Water treatment (Sirkar and Ho, 1992). Some mathematical analyses of the various applications are already available in the literature (Davis, 1973; Koojman, 1973; Davis et al., 1974; Cooney et al., 1974; Noble, 1983; Huang et al., 1984; Kim and Stroeve, 1987; Rudisill and Levan, 1990; Urtiaga et al., 1992; Urtiaga and lrabien, 1993). These analyses are divided into two groups based on the complexity of the boundary conditions. Applications with linear boundary conditions yield analytical solutions, while those with nonlinear boundary conditions do not. Membrane separation processes that include chemical reactions (facilitated) are particularly resistant to the application of the usual elementary analytical approaches. As a result of this, a combination of analytical and numerical techniques are sometimes used to carry out the mathematical analysis (Rudisill and Levan, 1990). In this work, a technique is used that deals effectively with the nonlinear boundary conditions of the type resulting from heterogeneous chemical reactions such as in equation (2.1) below.

fluid flows. One can examine the concentration profile in a single fiber and then predict the performance of the separation device. Further, consider a fluid (Newtonian) from which the solute is to be extracted as entering the reactive section of the hollow fiber in fully-developed, one-dimensional, laminar flow. As shown in the schematic, Fig. 1, at z = 0, the fluid contacts the reactive membrane. At such a location, the solute concentration CA is uniform and has the value CAO. AS the fluid flows further into the reactive section, the solute diffuses through the membrane by carrier-facilitated transport and emerges into the second fluid (shell side) which surrounds the hollow fiber. CA is negligible axially on the shell side since the incoming shell side fluid is devoid of the solute, and is at a much higher flow rate. An alternative to this condition is a constant solute concentration on the shell side. Following Way and Noble (1992) in (Sirkar and Ho, 1992), a reversible equilibrium reaction of the form:

A+B

kl

= AB

(2.1)

k2

occurs inside the membrane, where A is the solute, B is the carrier and AB is the solute-carrier complex. The quantities kl and k2 are the forward and reverse rate coefficients. Following Bird et al. (1960), the equation of continuity for the solute (species A) in this system is:

2. T H E O R Y

Following Kim and Stroeve (1987), consider a bundle of parallel hollow fibers through which the 3995

(

2Vavg 1

R2/ Oz = DA r-~r \

Or ]

(2.2)

3996

N.W. LONEY

shell side reactive semi-permeable membrane

r

I:

impermeable wall

z I

fluid side +

A B I I I I I I I

=

'~

, ..... AB

i?i:

shell side

inlet plane

Fig. 1. Cross-sectional diagram of the hollow fiber with reactive walls. The liquid membrane containing the carrier is supported in the porous walls of the hollow fiber.

subject to C A = CAO

~C A

__ DA ~ ~C rA

-0

at z = 0 atr=0

= k,,,Sf(CA)

(2.3) (2.4)

at r = R

membrane configurations for the same experimental conditions. Equation (2.5) is the boundary condition that incorporates the relevant information given in eq. (2.1) as to the rate of disappearance of species A. Usually, f (Ca) is a quotient of two polynomials, and is typically of the form (Kim and Stroeve, 1988):

(2.5)

U(C A) where DA is the solute diffusivity in the fluid, kw the membrane mass transfer coefficient for the solute, and S the shape factor based on the inside radius, R, of the hollow fiber. This shape factor (Kim and Stroeve, 1988; Noble, 1983) behaves as a correction factor for geometry. It permits the translation of experimental data derived in flat membranes to non-flat

D'nCTKeq D~t(1 + KeqCAH) CA

(2.6)

where the prime indicates diffusivity of that substance in the membrane, Keqis kl/k2. The equilibrium (thermodynamic) distribution coefficient, H, is defined as a proportionality factor relating the concentration of species A in the two phases (membrane and fluid). The

3997

Mass transfer in laminar flow quantity Cr accounts for the concentration of species B and AB. Introducing the dimensionless variables: C A .

r

OFo OFx 2 OF2 ____8-~-+e-~-+e - ~ - +

~

(OFo

02C

1 OC

~O~ q ~O~

(2.2A)

C = 1 at ~" = 0

8F1

(2.3A)

+ ....

= 0

o~

at ~ = 0.

-

(2.4A)

~2) OFo

- 0

0{ OFo

-w(1

(2.7) ,:: 2 ( 1 - ~ 2 )

where w is defined as

Oa

0F1

0~ 8F1

g2~

The system consisting of equations (2.2A)-(2.4A) and (2.7) can be recasted by using the following form of the dimensionless concentration: C = Fo + eF1 + g2F2 q-- ""

(2.13)

(2.14)

at ~ = 0

(2.15)

at~=l.

02F1 1 0F1 . = 8{7 -~ ~ ~

0•I

(2.16)

(2.17)

at f = 0

(2.18)

ate=0

(2.19)

+ c 0 F I - c ~ F o 2]

2(1 - ~2) OF2 02F2 8f = ~ 5 - + ~

8F2

0F2

0~

20F2

=w[(1

-0

F2 = 0

(2.8)

such that: OF1

~ o~

at~=l.

(2.20)

/~> 1).

8Fo

1 OFo

+ - --

(2.7A)

a dimensionless group called the wall Sherwood number. The dimensionless quantity, c~, is defined in Kim and Stroeve (1988). In this work e is less than one, and can be associated with the quatity, fl (Kim and Stroeve, 1988) in two cases. In one case one can let e be fl, while in another let e be the reciprocal of fl (for

OC

02Fo

+~)Fo

F1 = 0

Rk.,S

(2.12)

o~ - o~2

8Fo

o~

at ~ = 1

w{(1 + ~)Fo

F o = 1 at f = 0

OC - - - = w[(1 + ~x)C - ~aC 2 --~ ,~20~C3 -- g30~C4 -~- -.. ]

)

Equating coefficients of like powers of e gives:

Then recognizing thatf(CA) can be recast as an infinite series in dimensionless form to be (1 + a)C + aY,%l ( _ 1).e2C,+ 1, eq. (2.5) becomes:

w=

ate=0

+ ~((1 + c~)F, - ~Fo2) + O(e2)}.

~o: 2(1

8C

--

0

(2.11) - \ 8g + e ~ -

OC

....

(2.10)

VavoR2

into eqs (2.2)-(2.4), we get: 2(1-~2) 8f-

1 at f = 0

zDA

_

C = C A O, ~=a'~

Fo + eF~ + ~2F 2 q'- . . . .

- 0

1 0F2 -.

(2.21)

at ~ = 0

(2.22)

at { = 0

(2.23)

-w[(1 +~)F2--2aFoFI+~F~]

at{=l.

..

(2.24}

OC ~Fo OF1 20F2 8-2 = 8~- + eV ( + ~ 7-Z + ...

The system of equations (2.13)-(2.16) is linear, and can be solved using separation of variables to give:

o7 = W

02C O~ / 2

+

~2F o

02El -O~ - +2~ - + ~ - ~ . 2

202F2

Fo = .~B.__ exp ] -

+'"

2

1 (2.25)

which then results in: where the 2. are defined by: 2(1-~

2" [-0Fo

0F1

,[~-+ ~-+

02Fo

~ 2 ~ + ...]

02F,

-- 0~ 2 -'}- 8 - - ~ -

_1(0Fo -[- "'" q-

kO~

OVa

)

q'-SW-'}-"'"

_ 22, ( ~ _ 2 , \ (2.9)

[-3-~;2;2.]

= 0

(2.26)

N. W. LONEY

3998 and B, by:

where i ~ / ~ is replaced by L Equation (2.31) can be inverted with the use of the residue theorem Mickley et al., 1957). That is

B n :

~(l-~2)exp{-2~2}lF,

--

f~ ((1

~2) exp.

I~-@;1;2~21d

4

p(s.) L - l{Fl(s, 4)} = , " o ~ exp {s,~}

~ -2 2 1 {--/ng~ }IFII2---~-;/~n 1),;~nq~-2]jd~ n/> 0

(2.27)

where 1F1[½ - (2,/4); 1; 2,~ 2] is a solution of the confluent hypergeometric equation (Kummers equation). Values of both 2, and B, can be derived as described by Brown (1960), or by employing standard software packages such as Mathematica (Wolfram, 1991). The system described by eqs (2.17)-(2.20) is also linear, but separation of variables is not the appropriate solution technique. Here Laplace transform is more suitable. That is, by transforming the ~ variable while treating ~ as a parameter, one gets the Kummers equation as shown below. Let

L{F~(~, O} -

where p(s,) and q(s,) are the numerator and denominator, respectively, of eq. (2.31).

3. CONCLUDING REMARKS

e '~ FI(~, 4) d~ = f~(s, 4) (2.28)

The method presented here provides an analytical approach to solving the nonlinear system that occurs in carrier-facilitated hollow fibers. This is a mode of transport that can significantly improve fully developed, laminar flow mass transfer in a separation device. It is therefore important to evaluate new systems expeditiously, without high economic penalties, that can occur when a large number of experiments are needed to define design variables. The previously tedious task of finding the eigenvalues [eq. (2.26)] and evaluating the so-called Fourier coefficients [eq. (2.27)] can now be expedited with a few lines inserted in software packages such as

Mathematica.

then eqs (2.17) and (2.18) become d [ \ dF1 ~ ~g-@-)-

(2.32)

2 s ~ ( l - ~2)F~ = 0

(2.29)

where s is a complex number. Upon solving eq. (2.29) and applying the transformed eq. (2.19), results in:

F,(s, 4) = q(s)exp { - i ~ 2 } ~

and one can see that replacing i x ~ with 2 results in ~Fi [½ - (2./4); 1; 2.~ 2] the identical solution of Kummers equation previously obtained, where r/(s) is defined by:

REFERENCES Bird, R. B., Stewart, W. E. and Lightfoot, E. E., 1960, Transport Phenomena, pp. 519-553. Wiley, New York. Brown, G. M., 1960, Heat or mass transfer in a fluid in laminar flow in a circular or fiat conduit. A.I.Ch.E.J. 6, 179 183. Cooney, D. O., Kim, S. and Davis, J. E., 1974, Analyses of mass transfer in hemodialyzers for laminar blood flow and homogeneous dialysate. Chem. Engn9 Sci. 29, 1731 1738. Davis, J. E., 1973,Exact solutions for a class of heat and mass transfer problems. Can. J. Chem. Engng 51, 562 572. Davis, J. E., Cooney, D. O. and Chang, R., 1974, Mass transfer between capillary and tissues. Chem. Enyn9 J. 7, 213 225. Huang, C. R., Maltosz, M., Pan, W. D. and Snyder, W., 1984, Heat transfer to a laminar flow fluid in a circular tube. A.I.Ch.E.J. 30, 833 835.

- ~ff~ exp ti X ~ t ~(s) =

[ix/~-

w(1 + c¢)]1F1[~

3ix/~" ,1;

ix/~ ] - 2i,~(~

following the Laplace transformation of eq. (2.20). Here the quantity F 2 represents the Laplace transform of F 2, where Fo is the solution of eqs (2.13) (2.16). Equation (2.30) can now be expressed in terms of 2 as:

-

Fds, 0 =

exp

{;/2tl

-

2t},

1

-

1ix/~'~ 2x/.~) FIL2[-3- 2x/2' '~'2"ix/2s/ix/s ~--Tj

Kim, J. 1. and Stroeve, P., 1987, Mass transfer in separation devices with reactive hollow fibers. Chem. Engng Sci. 43, 247-257. Kooijman, J. M., 1973, Laminar heat or mass transfer in rectangular channels and in cylindrical tubes for fully developed flow:comparison of solutions obtained for

2 1; 2~ 2] (2.31)

Mass transfer in laminar flow various boundary conditions. Chem. Engng Sci. 28, 1149 1160. Mickley, H. S., Sherwood, T. K. and Reed, C. E., 1957, Applied Mathematics in Chemical Engineering, Mc GrawHill, New York. Noble, R. D., 1983, Shape factors in facilitated transport through membranes. Ind. Chem. Fundam. 22, 139-144. Noble, D. R. and Way, D. J., Facilitated transport. In Membrane Handbook (Edited by K. K. Sirkar and W. S. W. Ho). Van Nostrand Reinhold, New York. Rudisill, E. N. and Levan, M. D., 1990, Analytical approach to mass transfer in laminar flow in reactive hollow fibers

3999

and membrane devices with nonlinear kinetics. Chem. Engng Sci. 45, 2991-2994. Sirkar, K. K. and Ho, W. S. W., 1992, Membrane Handbook, pp. 3 14. Van Nostrand Reinhold, New York. Urtiaga, A. M. and Irabien, A. J., 1993, Internal mass transfer in hollow fiber supported liquid membranes. A.I.Ch.E.J. 39, 521-525. Urtiaga, A. M., Ortiz, M. I., Salazar, E. and Irabien, J. A., 1992, Supported liquid membranes for the separation-concentration of phenol. 2. Mass-transfer evaluation according to fundamental equations. Ind. Eng. Chem. Res. 31, 1745-1753. Wolfram, S., 1991, Mathematica, pp. 6 7. Addison- Wesley, Reading, MA.