Liquid phase controlled mass transfer in gas-liquid slug flow at low reynolds numbers

Int. Comm. HeatMass Transfer, Vol. 22, No. 5, pp. 741-750, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/95 $ 9 . 5 0 + . 0 0

Pergamon

0735-1933(95)00060-7

LIQUID PHASE CONTROLLED MASS TRANSFER IN GAS-LIQUID SLUG FLOW AT LOW REYNOLDS NUMBERS

T. Elperin and A. Fominykh Ben-Gurion University of the Negev The Pearlstone Center for Aeronautical Engineering Studies Department of Mechanical Engineerng P.O. Box 653 Beer Sheva 84105, Israel

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT A model of mass transfer during isothermal gas absorption from slugs rising in a channel filled with liquid at small Reynolds numbers is suggested. Fluid flow in the region below the bottom of gas slugs is assumed laminar and therefore vortex rings are not formed at the trailing edge of a gas slug. It is assumed also that a flow of dissolved gas can be described by a point source of mass which is located at the bottom of a gas slug. Intensity of this point source of mass at the bottom of the first gas slug emerging into a pure liquid is equal to the total mass flux from the surface of the first slug. The second gas slug emerges into a liquid with concentration distribution formed by a point source of mass at the bottom of the first gas slug. The third gas slug emerges in a liquid with a concentration distribution formed by a point source of mass at the bottom of the second gas slug and so on. Using this model a recurrent relation for mass flux from the n-th gas slug is derived and the total mass flux from n gas slugs in a gas-liquid slug flow is determined. INTRODUCTION The importance of the problem of mass transfer in gas-liquid slug flow stems from a wide use of absorption technologies in chemical engineering, gas mixtures separation, industrial gas cleaning, etc. State-of-the-art in mass transfer in gas liquid slug flow is reviewed in our previous paper (see Elperin and Fominykh, 199411]) where mass transfer in gas-liquid slug flow at large Reynolds numbers was investigated. It was assumed that dissolved gas is perfectly mixed in liquid plugs by a vortex in a liquid plug and that the concentration of dissolved gas in each liquid plug is homogeneous. In this approximation recurrent relations for dissolved gas concentration in the n-th liquid plug and mass flux from the n-th gas slug were derived and total mass flux from n gas slugs was determined. Gas-liquid slug flow at low Reynolds numbers can be realized for a case of gaswater slug flow in capillary tubes (d - lmm) or for a case of gas slugs rising in liquids of high viscosity such as glycerol. 741

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T. Elperin and A. Fominykh

Vol. 22, No. 5

The motion of single gas bubbles in vertical tubes in a viscous regime was investigated by Wallis (1969) [2] and by White and Beardmore (1962) [3]. Viscous regime of gas slug motion in vertical tubes occurs in pipes with diameter less than

1.6.(g~p72g'-I) I/3,

where

g'= gap/9~, Ap = P r -[:)g, which may arise with highly viscous liquids. White and Beardmore (1962) [3] and Wallis (1969) [2] suggested a formula for a single gas slug rising velocity in a stagnant liquid in a viscous regime. The motion of single gas slugs in capillary tubes was investigated by Brethertone (1961) [4]. Brethertone (1961) [41 showed that gas slugs in vertical capillary tubes can be stationary when the gravity force is negligible compared to the surface tension force, that is for Eo < 3.37, where Eo = Peg' d2/~. Vertical upward two-phase slug flow in small diameter tubes of 2, 3 and 6 mm diameter was investigated experimentally by Oya (1971) [5]. Oya (1971) [5] observed such modifications of gas liquid slug flow as piston type slug flow, simple slug flow, single fish-scale type slug flow, double fish-scale type slug flow and long piston type slug flow. State-of-the-art in research of slug flows in small diameter pipes is reviewed by Barnea et al. (1983) [6]. Fluid flow at the trailing edge of a gas slug was visualized by Campos and Guedes de Carvalho (1988) [7]. Results of their experimental investigation showed that vortex fluid flow at the trailing edge of a gas slug is formed when Re = 60 while at Re < 60 vortex free laminar fluid flow at the trailing edge of the gas slug was observed. Ammonia absorption by water from an airammonia mixture was visualized by Nakoryakov et al. (1985) [8] by adding colorless phenolphthalein to water. Ammonia dissolution from a gas slug in water was accomplished by alkaline reaction and phenolphthalein was colored in red. For Re < 60 a wake of dissolved gas at the trailing edge of a gas slug was stable and was not washed out. For Re > 70 a wake of the dissolved gas at the trailing edge of the gas slug was washed out by a vortex at the bottom of a gas slug. MASS TRANSFER IN GAS LIQUID SLUG FLOW

Mass transfer during isothermal absorption of a pure gas from a gas slug by liquid was investigated theoretically by van Heaven and Beek (1963) [9], Petukhov and Fominykh (1986) [10], Fominykh (1987) [11] and by Elperin and Fominykh (1994) [1]. In the approximation of a thin diffusion boundary layer in a liquid phase a simple formula was derived for determination of the mass flux from a single slug entering a pure liquid (see Elperin and Fominykh, 1994 [1]). QI = 4D

1/2 1/2.1/2 rc q0 %'

(1)

where (see Fig. 1) tg ~°=~(~;=

2x/2x/2x/2x/2x/2x/2~0I ( @ ) 2

~~

( - ~ - - - z ) X f dZ" ~ d ~ \ z----]' d )

(2)

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MASS TRANSFER IN GAS-LIQUID SLUG F L O W

743

.1

FIG. 1 Schematic View of a Single Gas Slug Consider mass transfer in a linear cluster of gas slugs at low Reynolds numbers assuming that slug flow is stable and that the lengths of the gas slugs and of the liquid plugs are equal (see

Fig.

2). Assume also that fluid flow at the trailing edge of the gas slug is laminar and that vortex at

the trailing edge of a gas slug is not formed. The experimental observations of Nakoryakov et al. (1985) [8] on visualization of dissolved gas flow at the trailing edge of a gas slug at low Reynolds numbers showed that a flow of dissolved gas can be described by a point source of mass at the bottom of a gas slug. The intensity of the point source of mass at the bottom of the first gas slug is equal to the total mass flux from the surface of a gas slug and is determined by formula (l). Distribution of concentration of the dissolved gas in the first liquid plug below the first gas slug is determined by the concentration distribution from a point source of mass in the fluid flow (see Bird et al., 1960 [12] and Fig. 2):

Ol

C= 4 ~ D \ fry~ + ( z + ( )

/

2

(3)

Distribution of the dissolved gas in a liquid plug is shown by a dotted line in Fig. 2. In a plane z = 0, equation (3) yields:

c(z = 0 ) = 4~D~2r2 + g2

(4)

In a frame moving with rising slugs, the second gas slug enters a liquid with a concentration of dissolved gas determined by formula (4). For small r formula (4) yields:

c(z=0)=Ql/1 4~DF.

Ur2 / 4Dg.

(5)

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T. Elperin and A. Fominykh

Vol. 22, No. 5

t\ / ', / \

I

/\ / / /

\ \ \

FIG. 2

Schematic View of Gas-Liquid Slug Flow Fluid flow around a Taylor bubble is described by the following stream function (see Davies and Taylor, 1950 [13]): r 2

~=U

Rr ( klZ "~ ( k i t "~) k I exp ' ~ - ) l [J ~ - ' - R . jJ/

(6)

where kl = 3.832 is a lowest root of Bessel function JI. Solution (6) is obtained under the following boundary conditions:

where u -

r

u= U

at

z --~ -oo

(7)

v=0

at

y = _+R,

(8)

Or '

v=

r

3z

. At the free surface of a gas slug ~ = 0.

Bessel function Jl(z) can be written in the next form (see, e. g., Abramovitz and Stegun, 1964 [14]): Jl(Z) = ~ Z ~.~

(-z2/4) k

2 k=0 k ! F ( k + 2 )

(9)

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MASS TRANSFER IN GAS-LIQUID SLUG FLOW

745

At z = 0 equation (6) yields: / r2 v ( z = o) = u

2

(lO)

kj

Combining equations (9) and (10) and assumung small r we arrive at the following formula: V(z = 0 ) =

Ukl2r4 16R 2

(ll)

Mass transfer from the second gas slug is is determined by equation of diffusion (see Elperin and Fominykh, 1994 [1]): ~c

02C - D - 0% 0~1/2

(12)

with boundary condition c =Cs

at

~g=0

(13)

The variable ~(z) is determined by the following formulas (see Fig. 1): s %(z) = fr2d~) = ~r2usds ' % 0 Initial condition for equation (12) is derived from formulas (5) and (11):

Q1 c = 47tDg

1

Ub2~l/2 ) 4D•

at

~=0

(14)

2R I/2 U1/41:1/2 "~1 Introducing a new function E = c - c s , the system of equations (12)-(14) can be transformed

where b -

as follows: o~

=D--

o~2~

(15)

746

T. Elperin and A. Fominykh ~=0 ~=

at

QI

4rcDg

Vol. 22, No. 5

gt=0

(16)

Cs _ B,J~-

at

~=0,

(17)

where B=

QIUb2 167tD2 f2 • Solution of equation (15) with initial and boundary conditions (16)-(17) reads (see Carslaw and Jaeger, 1959 [15]): 1 c0//'¢)= 2~n-D~- S ( A - B ~ f ~ - ) e x p 0 where A

QI 4r~Df

(gt-~')2 4D¢

-exp

dgt'

4D¢

(18)

Cs

From equation (18) we find that:

~¢=-0 2r0/2D3/2~3/2 A!gtexp ( - 4~_~/ d ~ - B o SII/3/2exp( - 4~-~ ] d~ / ( 19) In order to determine mass flux from the second gas slug we use the following formulas:

?expI- /d :20 , i~3/2

/1/2 expI-~D-~Id~ = (2D~)5/4F 5 D-5/2(O),

(see, e. g., Gradshtein and Ryshik, 1965 [16]), where D 5 / 2 (0) a parabolic cylindrical function and D_ 5 /2 (0) = 25/4 F(7/4) " Using the above formulas we find that 1/2

1

3/4

Q1 )4rtl/2Dl/2~/2_~ -

;/0v

4nDg

(20) 3D5/4g2klF(@)

Since 4rtl/2DI/2~l/2c 0 s = QI, equation (20) can be written as follows:

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MASS TRANSFER IN GAS-LIQUID SLUG FLOW

~//2 Q2 = Q I - Q 1

rtl/2Di/2(

.,1/2n~( 1-

1 )-1/4

747

I/2

3D3/4gkIF(+)

(21)

Introducing the following notation

~1/2 I U1/2~{ 1 )"1/4 0~4_) I/2 ~1/2D1/2( , 3D3/4 g k , l _ . ( 3 . ]

=K,

(22)

allows to rewrite equation (21) as Q2 = QI(1-K).

(23)

Intensity of a point source of mass at the bottom of the second gas slug is equal to Q1 + Q2. Therefore the third gas slug enters the liquid with a distribution of concentration of dissolved gas which is determined by the following formula (Q1 + 0 2 ) c4rt:D(

1

Ub2/ltl / 2 / 4D£ '

(24)

where QI and Q2 are determined by formulas (1) and (21). Similarly can be calculated mass flux from the third gas slug: Q3 = Q1 - ( Q I +Q2) K = QI( 1 - K) 2

(25)

Mass flux during gas absorption from the fourth gas slug is: Q4 = QI - (QI + Q2 + Q3) K = Q1 (1 - K) 3

(26)

Repeating this procedure we can determine mass flux from the n-th gas slug: n-1 Qn = QI - K ~ Qi = Q1 (1 - K) n-1 i=l Formula (27) allows to determine total mass flux from N gas slugs:

(27)

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T. Elperin and A. Fominykh

N Q I ( 1 - ( 1 - K ) N) Q£N = ~-'~Qn = K n=l

Vol. 22, No. 5

(28)

Fommla (28) is valid for 0 < K < 1. Condition 0 < K < 1 is met for long liquid plugs when ( > .u, l / 2 nK~l(/ T )1g

~1/4 0 n 1/2

3D3/4(kIF(@)

(29)

and ~1/2 ~ > 2~t1~--i~1/2

4- 1 I~ 0 - ~

4U3 / 2 R F ( 1 ) ~ 3/4 3D5/4klF(3 )

(30)

Using equations (27) and (28) we obtain that mass flux for N = o,, is equal to zero (Q~ = 0) and that mass flux from the infinitely long cluster of gas slugs is

Qz~-

Q1 K

(31)

If ~---~~, K --9 0 and using Lopitales rule we derive from (28) that QZN = QI N" CONCLUSIONS A model is developed for the analysis of mass transfer during isothermal absorption in vertical gas liquid slug flow for the case of small Reynolds numbers. Simple expressions are derived for determining mass transfer rate from the n-th gas slug in gas liquid slug flow and from N gas slugs. A developed model is valid for a case of long liquid plugs. ACKNOWLEDGMENTS

This work was partially supported by Israel Ministry of Science and the Arts and by Israel Ministry of Absorption. NOMENCLATURE

r,z coordinates, m u,v velocity components, m.s -1 s length of an arc along the slug measured from its leading edge, m g acceleration of gravity, m.s -2

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MASS T R A N S F E R IN G A S - L I Q U I D S L U G F L O W

S

surface area of a slug, m 2

d ~'g

channel diameter, m length of a gas slug, m

N

number of slugs

D

coefficient of molecular diffusion, m2s - I

c

concentration of a soluble gas, kgm -3

Q

mass flux, kgs-I

U

velocity of a gas slug rising in a stagnant fluid, ms-I

t

length of a liquid plug, m

R

channel radius, m = c - c s, kgm -3

Cs

concentration of saturation, kgm -3

g ' = gAg/gr, ms -2 Re = Ud/v

Reynolds number

Eo = p f g ' d 2/(5

Eotvos number

k t = 3.832

lowest root of Bessel function

J 1(z)

Bessel function

Greek Symbols ~t

dynamic viscosity, kgm-lc -i

~(z) function, m4s -I stream function, m3s -1 velocity potential, m2s -I c5

surface tension, N m -1

v

kinematic viscosity, m2c - I

p

density, kgm -3

Ap = p~ - p g ,

kgm -3

Subscripts ~

liquid

g

gas

s

value at interface

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T. Elperin and A. Fominykh

Vol. 22, No. 5

REFERENCES

1.

T. Elperin, A. Fominykh, Mass transfer during gas absorption from a linear cluster of slugs in the presence of inert gas, Int. Comm. Heat Mass Tran,sfer, 21 (1994) 651-660.

2.

G . B . Wallis, One-Dimensional Two-Phase Flow. McGraw Hill, N.Y., 1969.

3.

E.T. White, R. H. Beardmore, The velocity of rise of single cylindrical air bubbles through liquids contained in vertical tubes, Chem. Eng. Sci., 17 (1962) 351-361.

4.

F.P. Brethertone, The motion of long bubbles in tubes, J. Fluid Mech., 10 (196l) 166-188.

5.

T. Oya, Upward liquid flow in small tube into which air streams (lst report, experimental apparatus and flow patterns), Bulletin of the JSME, 14 ( 1971) 1320-1329.

6.

D. Barnea, Y. Luninski, Y. Taitel, Flow pattern in horizontal and vertical two phase flow in small diameter pipes, Can. Journ. Chem. Eng., 61 (1983) 617-620.

7.

J . B . L . M . Campos and J. F. R. Guedes de Carvalho, An experimental study of the wake of gas slugs rising in liquids, J. Fluid Mech., 196 (1988) 27-37.

8.

V . E . Nakoryakov, B. G. Pokusaev, A. V. Petukhov and A.V. Fominykh, Mass transfer from a single gas slug, J. Eng. Phys., 48 (1985) 533-538.

9.

J.W. van Heuven, W. J. Beek, Gas absorption in narrow gas lifts,

10.

A.V. Petukhov and A. V. Fominykh, Mass transfer from single Taylor bubbles,

Chem. Engng. Sci., 18 (1963) 377-390. Heat Transfer - Soviet Research, 18 (1986) 70-75. 11.

A. Fominykh, Mass and Heat TransJer in Gas-Liquid Slug Flow, Ph.D., Novosibirsk University, Novosibirsk, Russia, 1987.

12.

R.B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley and Sons, N.Y., 1960.

13.

R. M. Davies and G. I. Taylor, The mechanics of a large bubble rising through extended liquids and through liquids in tubes, Proc. Roy. Soc., A200 (1950) 375-390.

14.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, N.Y., 1964.

15.

H.S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, 2nd Edition., Oxford University

16.

I. S. Gradshtein and I. M. Ryshik, Tables of Integrals, Series and Products, Academic

Press, 1959. Press, N.Y., 1965.

Received February 23, 1995