Taylor diffusion in a laminar falling film

Chemical

Engineering Science.

1973. Vol. 28, pp. 1763-1764.

Pcrgamon Press.

Printed in Great Britain

Taylor diffusion in a lamiuar falling film (Received 14 August 1972) and p-th moment in the film: AN INVESTIGATION is made here of the dispersion of a solute in a laminar falling film. This information is fundamental, for example, in the treatment of the residence time distribumP(r) =&=li c,dq. tion of such a film. The effect of the lateral diffusion of a solute in a stream Using Eq. (7), we obtain from (4) similar to Aris: with a developed velocity profile was made originally by Taylor [l-3]. For laminar flow in a straight tube of circular cross-section, the apparent diffusion coefficient was found with some limitations to be aV/48D. Aris[4] showed that this result was correct without restriction on the value of CL, or on the initial distribution of solute. He presented a more with the conditions: detailed treatment of the flow in a tube of arbitrary crosssection and flow profile. The case of a laminar falling film can CP(?. 0) = e&J) be studied similarly. and: Consider the viscous flow of a fluid along an inclined infinite plane (x, z), parallel to the axis Ox. In steady flow the !!5=0 n=o velocity u is a function of y only: an n= 1. U(Y) = Wl +x(Y)].

(8)

(9)

(10)

(11)

(1) Averaging over the film thickness, with the condition (11):

The diffusion equation can be written: !$a=p(p-l)T,_,+p_I.

Taking D constant (+I= 1) and an origin moving with the mean velocity of the film, we define dimensionless variables:

(12)

The bar denotes the average over the film thickness. dition (10) is now: m,(O) = mm.

8 = (x- Ut)/a T = yla

T = Dt/az p = UalD

(3)

Con-

(13)

Solving Eqs. (9) and (12) step by step in a manner similar to that of Aris, we have for the moments:

and Eq. (2) becomes:

ac azc px!K+azC -=-a7

at2

8.5

a+

m, = constant (unity)

with the conditions: C(S, 1). 0) = Cl&?, 7)) $+=O

for?=0 T)= 1.

(5) (6)

1

(15)

(16)

Forr+Oo: 0) m I- = - $.~r-~ C A,mv4 cos mr

1

The p-th moment of the distribution of solute in the filament through n at time t: (7)

(14)

m co(r). r) = 1+ C. (A, cos mmq)e-m’** m=1 m ml = - 3wF4 2 A,m-4 cos mm ( 1- e-“l”*).

(4)

so that the maximum on the distribution ultimately a distance proportional to Ua’/D.

1763

(17) curve moves

Shorter Communications To obtain the second moment, giving information about the variance, we have to solve Eq. (9) for p = 1 and with the velocity profile of a laminar falling film: x(1)) =$(I-3772)

(18)

~-~=f~(l-3ne)c,.

(19)

Solving in terms of the particular integral and complementary function: 1

cz - m,,+;jp

7

(

z-q*

1 +p*

1

dq

(20)

and substituting c1 into (12) with p = 2 and neglecting terms which converge to zero, we obtain: dm, --& = 2+&z

I

‘1 o j(l-3~$j--qp+;n’

+O{exp (-m*&)}

)

(21)

2a2u2 1050.

This relationship for the apparent diffusion coefficient K of a laminar falling Iilm is the same as that for a pipe derived by Taylor[l], except for the constant value. Comparison of the two constants, 21105 relating to the Iilm, and l/48 for the pipe, shows approximately a 9 per cent decrease of dispersion, which reflects the smaller velocity variation across a laminar film. An experimental verification might be dit%cult, owing to this small difference. It is interesting to note that our result coincides exactly with that for the limiting case of a parallel slit extrapolated to single phase flow, in the study of spread of a chromatogram by Aris [5]. Acknowledgement-I am deeply indebted to Prof. Dr. J. R. Boume (Head of Laboratory) for valuable discussions and critical comments. J. PRENOSIL

NOTATION constants thickness of the tilm (L), radius of the tube (L) concentration (mole/l) moment of concentration distribution in position y molecular diffision coefficient (L’/t) K apparent diision coefficient (LW m moment of concentration distribution in the film 0 origin of coordinate system time (t) : average velocity (L/t) velocity (L/t) t: variance rectangular coordinates (9 X.Y,Z A,B

c”

and: m, - 2(1+&++8.

(22)

B is negligible by comparison, because the variance of the initial concentration distribution is small. Coming back now to the variance V of the distribution of solute about the moving origin and using Taylor’s notation, we have: V=

J

“dy

+- (x- Ut)*C(x, y, t) dx I -m

(23)

and: *im!dV=D+2alZIZ=K 1OSD CIO2 dt

(24)

where K is the apparent diision coefficient (it can be also shown that any distribution tends to normality as in the case of flow in a pipe). The molecular diffusion coefficient D is usually very small and K becomes:

0”

Greek symbols 1). 6 coordinates (dimensionless) or. velocity (dimensionless) r time (dimensionless) x velocity distribution function 8 exponential function ‘p function of the variation of the diffusion coefficient Subscripts m P

REFERENCES [l] [2] [3] [4] [5]

_

Techn. them. Laboratorium ETH Zurich, Switzerland

dr)

- 2( 1 +j&)

K

TAYLOR TAYLOR TAYLOR ARKS R., ARIS R.,

G., Proc. R. G., Proc. R. G., Proc. R. Proc. R. Sot. Proc. R. Sot.

Sot. 1953 A219 186. Sot. 1954 A223446. Sot. 1954 A225 473. 1956 A23567. 1959 A252538.

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