THE EOUlIIIIlENCE OF THE PENETRIITION THEORIES SAMUEL SIDEMAN
n this paper we shall examine the apparent equiva-
I lence of the penetration and potential flow theories for interfacial mass transfer. The equivalence of the mass transfer coefficients obtained from these theories has been shown for gas absorpton. However, the nature of the penetration theory is such that the equivalence exists only for high Peclet (or low Fourier) moduli. Generally, the equivalence will hold where the ratio of viscosities of the two fluids is small enough to allow neglect of the tangential shear stress at the interface. Since the penetration concept is independent of viscosity, the penetration model may be used in cases where potential flow theory is not applicable. I n such instances there is no longer equivalence. The equivalence of the coefficients for mass transfer between a bubble or drop and a continuous liquid medium, as obtained from Higbie’s penetration theory (5) and the potential flow theory, has been known for some time. Specifically, Garner (4) has observed that the coefficients are identical for “fully circulating” drops when the internal fluid is moving at the same velocity as the external fluid a t the interface. Under these conditions, there is no boundary layer, and potential flow exists. I t has also been noted (6, 9) that, although the potential flow theory takes interfacial acceleration into account, unlike the penetration theory, the physical models are identical. Both models are based on diffusion into an element of fluid sliding over the interface of the two phases (9). This general statement, however, is oversimplified and may be misleading. Basic Concepts
The assumption of an ideal fluid-i.e., one which is incompressible and exhibits frictionless flow, permits simplified solutions of Laplace’s equation, vP = 0. In 54
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
particular, the two solutions are a pair of conjugate, orthogonal functions-viz., the stream and potential functions. The s t r e a m function is familiar through the use ofcertain parametric values of the function to describe the “streamlines” of fluid dynamics. The potential function refers to a hypothetical velocity potential which is analogous to the electrical potential or gravitational potential in more easily accepted examples. These orthogonal functions describe a flow field which closely resembles the actual flow field of a real fluid, provided wall effects are negligible. Since no momentum is transferred a c r w a stream tube c o n i p w d of streamlines, every streamline may represent the physical boundary of the flow field. Whatever the actual physical boundary, the assumption of ideal fluids implies some finite velocity a t that boundary corresponding to the assumed zero shear rate. A gas-liquid interphase in motion most closely resembles the model described above, particularly if the mass transfer occurs under normal flow conditions. Under normal conditions the Navier-Stokes equations are satisfied in both phases. However, even if the liquid has a finite viscosity, there would be no effect on the liquid in contact with the gas phase and Euler’s equation would apply satisfactorily. It is interesting to note that the ideal flow thus obtained in an actual liquid is theoretically restricted, but no restriction can be imposed on the tangential velocity components in an ideal liquid. As pointed out by Levich (7), the viscous forces existing in real liquids establish that the tangential velocity a t the surface of a bubble will be constant but do not assume that it is zero. The constancy of the tangential velocity affects the derivative of the velocity but not the velocity itself. This allows us to assume that the velocity distribution, in the case
The inadequacy of the two-film theory for interfacial mass transfer, especially for the absorption of a gas, resulted in
a number of improved theories, each having certain advantages. The penetration theory, as improved ly Danckwerts, and the potential Jlow theory are the most prominent and have often been assumed equivalent.
The author shows that equivalence
does exist but only under restricted conditions
of flow around a gas bubble, is little different from the velocity distribution in an ideal liquid. It is the physical reality of zero tangential shear stress, rather than zero viscosity, that permits us closely to approach an ideal flow configuration. Reclonpulor Coodinotos
The mechanics of predicting the heat or mam transfer a c r w a gas-liquid interface, in the case of a two dimensional flow, are quite simple and will not be elaborated here. I t is sufficientto note that, for a free falling planar sheet or a cylindrical jet of an ideal liquid with constant velocity under steady-state conditions, the solution is identical with one obtained for unsteady-state m a s or thermal diffusion from the respective physical configurations. This is easily demonstrated for the case of the two dimensional flow of a free film of constant thickness x = 2 s as shown in Figure 1. For the sake of generality, concentration of mass or heat, rather than temperature, will be used in this presentation. Note that with C = pC,T and a denoting diffusivity (thermal or mass), the transfer coefficient is K = h/pC,, where his the familiar heat transfer coefficient. When we use rectangular coordinates (x, z), the film flows in the z-direction at a constant velocity, U, and the equation describing the m a s transfer of component A from the gas phase into the liquid in steady state is given by
since z / U = t. The boundary conditions normally associated with such a problem would be specified as
C = Ca at x = 0, 2 s C=Oatx>O
bC
-=oaatx=s
bx
> 0)
(2)
z = O
(ort=O)
(3)
r>O
(ort>O)
(4)
z
>0
(or t
and solutions of Equations 1 to 4 can be found in many textbooks. For the particular case of short contact times, the boundary condition given by Equation 4 may be modified to yield an immediate solution. As it happens, this modification is the essence of the penetration theory.
where conduction in the z-direction is assumed negligible compared with convection in the same direction. Since U is constant, rearranging the left side of Equation 1 yields VOL. 5 8
NO. 2 F E B R U A R Y 1 9 6 6
55
The concept of the penetration theory was apparently first introduced formally by Higbie (5) and later revived and extended by Danckwerts ( 3 ) and others ( 7 7). As implied by its name, the basis of this theory lies in the fact that, in unsteady-state heat or mass transfer, the depth of penetration into the exposed boundary is dependent on the time of contact. The longer the contact time the deeper the penetration and vice versa. Furthermore, at short contact times, if the penetrating component does not penetrate very deeply, say up to x = 6, where S/s O (4 ’) which obviously implies that mathematically s > m . I t is thus clear that Equations 1, 2, 3, 4’derived from potential flow considerations for short contact times, are identical with the set of Equations 1’, 2, 3, 4’that can be derived directly from the penetration theory for the same contact times. Similar considerations would apply when treating the more practical problem of transfer from a gaseous phase to a liquid film of thickness s flowing over a solid plane. Obviously, the ideal fluid is not affected by the presence of the solid plane. Similarly, if 6 O
(14)
$020
(1 5)
m 2 4 > O
C=O
cp>O
(1 6 )
b2C f f -
C
X
co
d4X
(17)
(1‘) = (23)
dr2
where t = R&/U and the boundary conditions are as given by Equations 2, 3, and 4’. The solution is evidently identical with Equation 5 and the transfer coefficient would be identical with Equation 6 if, as stated earlier, L = 2 R. If, however, texp is taken as
t,,,
=
Lr
(R/U)d 0 = aR/U,
-
Equations 13 to 16 are of a form similar to Equations 1’ to 4’, yielding the familiar solution - = erfc ___
(21)
Note that the solution is identical with the one obtained earlier (Equation 8) for transfer to a flowing film, where now L = 2 R. The solution of this problem, based on the penetration theory, is identical with Equations 8 and 21. This is obvious considering the basic premise of the penetration theory, namely 6/R 1. Since this is the very condition which allows the formal solution of the potential flow problem, the identity of the solutions is not surprising. Moreover, since by statement of the problem y / R > (bzC/dd). This is the very condition which allows the previous (and commonly assumed) neglect of the “axial” molecular diffusion as compared with the convective diffusion in the same direction. Thus, the solution is clearly restricted to high Peclet numbers where the above assumption, as well as that of the thin concentration or thermal layer, holds. 58
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
(1) Bird, R. B., Stewart, 1)’. E.: Lighrfnor, E. K,,“Transporr Phenomena,” p. 537 \$?ley, New Yurk, 1960. (2) Boussinesq, hI., J . M a f h . Pure A))>/. 1, 285 (1905). 43, 1460 (1951). (3) Danckwerrs, P. V., IND. ENG.CHEM. (4) Garner, F . H., Foord, A,, T s y e b a n , hi., J . Appl. Chem. 9 , 315 (1959). (5) Higbie, R., Trans. Am. Znst. Chern. Eng. 91, 365 (1935). (6) Leonard, J. H . , Houghton, G., Chem. En!. Sci. 18, 133 (1963). (7) Levich, V. G., ‘‘Ph~sicncheinicalHydrodynamics,” Prcntice Hall, New York, 1962. (8) Ruckenstein, E., Chem. Eng. Sci. 10, 22 (1959). (9) Sideman, S., Shabtai, H., Can. J . Chem. .!&I 42 (6), 107 (1964). (IO) Sideman, S., Taitel, Y., Inlern. J . Heat M a n Transfer 7, 273 (1 964). (11) Toor, H. L., Illarchello, J. M., Am. Znnrt. Chem. Eng. J.4, 97 (1958). (12) West, F. B.. Robinson, P. A,, Morgenthates, A. C., IND.ENC.CHEX.43, 234 (1951).
