AN
APPROXIMATE
MASS
PHYSICAL
MODEL
OF
TRANSFER L. L.
V. M.
Novosel'skaya, Gukhman, and
A. I A. V.
Ershov, Novosel'skii
UDC 532.73-3
On the b a s i s of an approximate physical model of m a s s t r a n s f e r , the distribution of liquid m a s s in the effective diffusion boundary l a y e r of a whirled unidirectional s t r e a m is d e t e r mined, with the r e s i s t a n c e of the liquid phase to m a s s t r a n s f e r taken into account. Several physical models have been proposed for d e s c r i b i n g the m a s s t r a n s f e r in g a s - l i q u i d s y s t e m s [1-8]. Without f u r t h e r analyzing them - they have been analyzed in [9, 10] - we will o n l y s h o w here that each of them m o r e or less reliably d e s c r i b e s a certain c l a s s of p r o c e s s e s c h a r a c t e r i z e d by some basic p h y s i c o c h e m i c a l and hydrodynamic p a r a m e t e r s . There is an indisputable interest alive today in r e s e a r c h and development concerning high-efficiency and high-output m a s s exchangers which consist of contact stages with the phases interacting in a parallel flow [11-13]. In o r d e r to e n s u r e a sufficiently thorough phase separation between stages, for producing a c o u n t e r flow of the p h a s e s throughout the device, and for boosting the m a s s t r a n s f e r rate, it is worthwhile to cons i d e r a whirled two-phase s t r e a m [14-23]. On the b a s i s of certain assumptions, we will analyze here a model of the m a s s t r a n s f e r p r o c e s s in a whirled unidirectional s t r e a m , which should be useful for an engineering design of a p a r a l l e l - f l o w contact stage of a m a s s exchanger. A two-phase (gas-liquid) s t r e a m in a field of centrifugal f o r c e s is c h a r a c t e r i z e d by an annular flow mode. Unlike in a free discharge of liquid o r in a counterflow of phases with a negligibly low gas velocity, h e r e the m a x i m u m tangential s t r e s s o c c u r s at the interphase boundary (Fig. la). The flow of liquid and gas in a whirled s t r e a m is r a t h e r helical and, therefore, c h a r a c t e r i z e d by r e g u l a r v o r t i c e s [24, 25] in each phase layer. Besides, the p r e s e n c e of an interphase boundary is also a cause of i r r e g u l a r v o r t i c e s at that surface [10, 25]. In the case of m a s s t r a n s f e r where the liquid phase r e s i s t s it, one may a s s u m e {considering a high gas velocity and an intensive s t i r r i n g of the gas by the turbulent fluctuations due to vortices) that the concentration of the gas o v e r an entire c r o s s section of the gas s t r e a m is equal to its mean concentration o v e r this section, while the high-velocity g a s s t r e a m should be viewed as the s o u r c e of tangential s t r e s s e s which ensure the forced helical motion of the liquid. The existence of appreciable tangential s t r e s s e d and forced v o r t i c e s in the liquid l a y e r causes turbulence to develop in the liquid l a y e r and, consequently, tends to equalize both velocities and c o n c e n t r a tions within the total m a s s of liquid. The liquid l a y e r is not turbulized uniformly throughout its entire thickness. A section of the liquid annulus contains regions where the hydrodynamic flow m o d e can be a s s u m e d invariable but different than in the remaining s t r e a m (Fig. lb). S. M. Kirov B e l o r u s s i a n Institute of Technology, Minsk. Institute of Nuclear Energy, A c a d e m y of Sciences of the B e l o r u s s i a n SSR, Minsk. T r a n s l a t e d from I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 23, No. 4, pp. 713-719, October, 1972. Original a r t i c l e submitted F e b r u a r y 21, 1972. 9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g/est I7th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher, it copy of this article is available from the publisher for $15.00.
1305
---4 a
D
Tcm g
5
c i
C~
Zone i at the wall of cylindrical channel is the viscous sublayer region (thickness 5'). Here at y < 5" the velocity v a r i e s fast f r o m z e r o directly at the solid wall to the mean value ~ r e f e r r e d to the s t r e a m . The thickness of this zone can be determined from the condition given in [8] with the Reynolds number n e c e s s a r i l y of a m a g n i tude Re ~ 1. Zone 2 is the fully turbulent m a i n s t r e a m Both the turbulent fluctuations and the v o r t i c e s velocity field uniform. Without a large e r r o r , in this zone may be a s s u m e d equal to the mean
in the liquid l a y e r . tend to make the the velocity of liquid s t r e a m velocity ft.
Zone 3 at the interphase boundary is the turbulent boundary l a y e r of liquid (thickness 5 " ) . This region is c h a r a c t e r i z e d by i r r e gular vortices. A strong interphase surface tension, however, tends to stabilize this surface somewhat and it d e t e r m i n e s the velocity profile here. Considering that the thickness of this zone is small, we can a c c u r a t e l y enough r e p r e s e n t its velocity profile u = u(5") as u : = u ' - - - - u ' - - u g, with u' denoting the velocity of the phases at their boundary.
Fig. 1. P r o f i l e s : a) tangential s t r e s s e s 1- = ~-(y); b) mean velocity u = u(y); c) concentration in the liquid phase C = C (y).
The c h a r a c t e r i s t i c s of the concentration profile a c r o s s the liquid l a y e r a r e also shown here (Fig. lc).
In the diffusion boundary l a y e r , whose thickness is 5~, directly at the wall within the viscous sublayer (y < 5'0 < 5') the turbulent fluctuations are so small that m o l e c u l a r diffusion b e c o m e s the governing mode of liquid m a s s t r a n s f e r here. The concentration in this l a y e r v a r i e s from a minimum value at a given c r o s s section to the mean value C r e f e r r e d to the entire 9l a y e r . Turbulent fluctuations and intensive v o r t i c e s within the turbulent m a i n s t r e a m equalize the c o n c e n t r a tion over a c r o s s section of that region, and the differential equation of convective diffusion here b e c o m e s C = C = coast. The basic variation in the absorbate concentration o c c u r s within the effective liquid-diffusion bounda r y layer, whose thickness is 5~'. A qualitative analysis of the interaction between phases, based on the theory of the diffusion boundary ~layer [8] and taking into account the flow c h a r a c t e r i s t i c s of a whirled two-phase s t r e a m [18], has shown that the thickness of this diffusion boundary l a y e r here is a function of its hydrodynamic and p h y s i c o c h e m i cal p a r a m e t e r s . At the interphase boundary the concentration of liquid C x is equal to its mean c o n c e n t r a tion in the gaseous phase at a given c r o s s section. At the boundary with the turbulent m a i n s t r e a m , however, the concentration of liquid is equal to its m e a n concentration in the liquid phase. Such a model describing the m e c h a n i s m of interaction between the phases of a g a s - l i q u i d s y s t e m in a whirled unidirectional flow can s e r v e as the b a s i s for an engineering design of a p a r a l l e l - f l o w contact stage with a p r e s c r i b e d velocity profile the same as in the model. It is n e c e s s a r y to determine the length of the p a r a l l e l - f l o w contact stage, where the mean concentration of the liquid phase C v a r i e s from C e to Cf and the stipulated proximity to equilibrium Cf--Ce is equal to ~lf. The differential equation of m a s s t r a n s f e r can be written as b__C_C-I- div (Cu) - - div (D grad C) -----O. Ot
1306
(1)
T A B L E 1. E x p e r i m e n t a l Data P e r t a i n i n g to the Mass T r a n s f e r in a W h i r l e d T w o - P h a s e S t r e a m (CO 2 a b s o r p t i o n in water)
f Mean ax -Mean abHelix 9 . . lsoluteve- Ce" 108' i Cf* 107, C * 9 10 7 lafveloc~ llocity of pitch, 9 ttyof gaStgas' m/. g/ram3 I g/ram3 g/mm 3 mm Im sec |see J 28 65
17,80 18,75
0,60 0,66
53,3 29,4
0,381 0,368
d L, mIi1
446,0 234,5
0,5606 0,585
446,8 0,642 232,9 0,582
In o r d e r to solve this equation, one m u s t c o n s i d e r the following s i m p l i f i c a t i o n s , which a r e e n t i r e l y justified in t e r m s of o u r m o d e l : that the m e d i u m is i n c o m p r e s s i b l e , the flow is steady, the diffusivity is constant, and the longitudinal c o n c e n t r a t i o n g r a d i e n t is negligibly s m a l l e r than the t r a n s v e r s e c o n c e n t r a tion g r a d i e n t . T h e n OC O~C u -- = D , Ox oy,
(2)
w h e r e u is the v e c t o r of a b s o l u t e v e l o c i t y : /g' - - / 2
u = u'
6"
Y-
(3)
The equation of s t e a d y - s t a t e m a s s t r a n s f e r with the given v e l o c i t y p r o f i l e is OC = D O~C (a - - by) ~ 09---U,
(4)
w h e r e a = u ' , b = ( u ' - f i ) / 6 " , and the b o u n d a r y conditions a r e C --=-Ce. 'for x ==0, C~-.~-C* for y = 6 ,
(5)
X>0,
(6)
aC/Oy = 0 for y = 0.-
(7)
It m u s t be e m p h a s i z e d h e r e that the y - a x i s is p e r p e n d i c u l a r to the v e c t o r of a b s o l u t e v e l o c i t y . F o r a n u m e r i c a l solution of Eq. (4) we u s e the f i n i t e - d i f f e r e n c e s s c h e m e [28]: c~C ~-" ~ C(yf+i, x) - - 2C(yf, x) -6 C(!]f-h x) Oy~ h~ ,
(8)
with h = 6 " / ( u + 1) = 6 " / 4 when n = 3. T h e n Eq. (4) b e c o m e s (a - - byf) aC.,(gf, x) = D. C (yf+l, x) - - 2C (yf, x) -6 C (yf-l, x) Ox h~ for X = 0 9for y - ~ 6 w h e r e Yk = 6 " { k - 1 ) / ( n + 1).
C=Ce;
.for. y = 0
C=C*,
(9)
. Cn-H--Cn ...
h
-=0,
(10)
F i n a l l y , s y s t e m (9) b e c o m e s dC__~l = 1 -[D dx a --:- b ~ 4 dC~ dx dCsdx
' 6" a -- b -2 1 a - - b 36
[
1___66 ( C 2 _ 2C 1-6 Co)] ' ( 6")~
( - - ~ (Ca - - 2C.,-6 C1) ,
I
(11)
16 ] D --(6,,) 2 (CA - - 2Ca + C2) .
4 S y s t e m (11) with the b o u n d a r y conditions (10) is solved n u m e r i c a l l y by the E u l e r method, a c c o r d i n g to which the c o m p u t a t i o n step H is e s t a b l i s h e d on the b a s i s of the equation
1307
C flos. ! f ~A
d
i1
7,5 m o 7,5 ~' o 4~ Fig. 2. Concentration distribution C (x, y) i: :he effective liquiddiffusion boundary l a y e r along the height of a simulated contact stage. Simulation I (A): a) L = 20.99 mm; b ) 104.99 mm; c) 446.99 mm. Simulation II (B): a) L = 20.99 mm; b) 83.99 mm; c) 232.99 mm. Concentration Cf 9108 (g/mm2), l a y e r thickness 6 (~). o
425
m,5 o
Ci, f (xo -t- H) = C,, f (x, O) + dC~, f H,
(12)
where H is the interval. The algorithm for a n u m e r i c a l solution yields the concentration profile along the apparatus height, with a p r e s c r i b e d proximity to the equilibrium concentration at the interphase boundary Cx. F u r t h e r m o r e , one can determine with it the length of a p a r a l l e l ' f l o w contact stage and the optimum thickness of the effective liquid diffusion l a y e r 5~' at the interphase boundary. On the basis of this algorithm, a p r o g r a m has been set up for the "Minsk-22" digital computer and n u m e r i c a l r e s u l t s a r e available. It is to be noted that questions concerning the evaluation of simulated e x p e r i m e n t s a r e of i n t e r e s t in their own right and are the subject of a separate article. Available calculations and r e s u l t s of computer e x p e r i m e n t s pertaining to the m a s s t r a n s f e r in a two-phase s t r e a m whirled throughout the entire channel length yield an estimate of many important p a r a m e t e r s c h a r a c t e r i z i n g the hydrodynamics and the m a s s t r a n s f e r in such a s t r e a m . Using the data of a computer e x p e r i m e n t ("Minsk-22" digital computer) shown in Table 1 as an example, the authors have calculated the mean flow velocity in the liquid l a y e r and thus its mean thickness, also the thickness 6~' of the effective diffusion l a y e r . Such a calculation yielded an estimate of the basic p a r a m e t e r s where a l a b o r a t o r y determination would have been e x t r e m e l y difficult. We obtained: a mean thickness of the liquid l a y e r 5 = 572 ~ and a M 11 thickness of the liquid diffusion l a y e r 60 = 16.15 ~ for the f i r s t simulated experiment, 6 = 408.5 ~ and 50 = 15 ~ for the second simulated experiment. In addition, this method has made it possible to plot c u r v e s r e p r e s e n t i n g the concentration of absorbed gas along the height of a contact stage. NOTATION 6 II 50 u u' _Cf(x, y) Ce(x, y)
cX(x, y) D
1308
is the is the is the is the is the is the is the mean is the is the
thickness of the liquid l a y e r , ~; thickness of the effective liquid-diffusion boundary l a y e r , /a; mean velocity, m / s e c ; mean velocity at the interphase boundary, m / s e c ; mean (finite) concentration in the liquid phase, g/mm3; mean concentration in the liquid phase at the channel entrance, g/m3; mean concentration of liquid at the interphase boundary, g / m m 3, in equilibrium with the concentration in the gaseous phase at a given c r o s s section; p r o x i m i t y to equilibrium; diffusivity in the liquid phase, m m 2 / s e c .
LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
CITED
W . G . Whitman, Chem. Met. Eng., 28, 147 (1923). R. Higbie, T r a n s . AIChE, 31, 365 (1935}. V . D . Stabnikov, A b s t r a c t of Doctoral Dissertation [in Russian], Moscow (1939). I . M . Marchell and H. L. T o o t , Ind. Eng. Chem. Fund., 2, No. 1 (1963). M. Kh. Kishenevskii, Zh. Prikl. Khim., 27, No. 4 (1954}. P . V . Damkwerts, AIChE J., 1, 456 (1955). V . V . Kafarov, Zh. Prikl. Khim., 29, No. 1 (1956}. V . G . Levich, Physicochemical Hydrodynamics [in Russian], Fizmatgiz (1959}. V . M . Ramm, Absorption of Gases [in Russian], Khimiya (1966}. V . V . Kafarov, Fundamentals of Mass T r a n s f e r [in Russian], Vysshaya Shkola (1962}. P . A . Semenov, Zh. Tekh. Fiz., 14, No. 7-8, 425 (1944). P . G . Boyarchuk and A. N. Planovskii, Khim. P r o m y s h l . , No. 3 (1962). N . G . Kuz'min and V. A. Malyusov, Dokh Akad. Nauk SSSR, 117, No. 4 (1957). R . Z . Alimov, Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk Euergetika i Avtomatika, No. 1 (1962). N . A . Nikolaev, A b s t r a c t of Candidate's Dissertation [in Russian], Kazan (1965). A . I . E r s h o v and L. M. Gukhman, Inzh.-Fiz. Zh., 10, No. 4 {1966}. V . N . Kiselev and A. A. Noskov, Authors' Disclos. No. 230077, Byull. Izobret., No. 34 (1968). L . M . Gukhman, A b s t r a c t of Candidate's Dissertation [in Russian], Minsk (1969). A . I . E r s h o v and I. M. Plekhov, Authors' Disclos. No. 182108, Byulh Izobret., No. 11 (1966). A . I . Ershov, I. M. Plekhov, L. M. Gukhman, N. P. Ermakovich, B. N. Isaev, and G. A. Lysakov, Authors' Disclos. No. 257439, Byull. Izobret., No. 36 {1969). L . M . Gukhman, A. I. Ershov, and I. M. Plekhov, in: Heat and Mass T r a n s f e r [in Russian], Vol. 4, Nauka i Tekhnika, Minsk (1968), p. 235. A. I. Ershov, L. M. Gukhman, and I. M. Plekhov, Izv. VUZ, Energetika, No. 5 (1968). A . I . Ershov, L. M. Gukhman, and I. M. Plekhov, ibid., No. 6 (1969). L . G . Loitsyanskii, Liquid and Gas Mechanics [in Russian], Fizmatgiz (1959). N . E . Koehin, I. A. Kibel', and N. V. Roze, T h e o r e t i c a l Hydromechanics [in Russian], P a r t 1, GITL, Moscow (1955). B . P . Demidovich, I. A. Maron, and E. V. Shuvalova, Numerical Methods of Analysis [in Russian], Nauka, Moscow (1967).
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