Effect of heat transfer additives on the instabilities of an absorbing falling film

Pergamon

Chemical En#ineerin# Science, Vol. 50, No. 19, pp. 3077 3097, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(95)00146-8

E F F E C T O F HEAT T R A N S F E R A D D I T I V E S O N THE INSTABILITIES O F A N A B S O R B I N G F A L L I N G F I L M WEI JI* and FREDRIK SETTERWALL Department of Chemical Engineering and Technology, Royal Institute of Technology, 10044 Stockholm, Sweden (Received 2 January 1995; accepted in revised form 21 April 1995)

Abstract--The instabilities of an absorbing falling film in the presence of a surface-activeagent are studied at low or moderate Reynolds numbers. This work is an extension of the previous stability analysis for a falling film in the presence of a surfactant (Ji and Setterwall, 1994, J. Fluid Mech. 278, 297-323). The present model considers the absorption of water vapour which includes both mass transfer and heat transfer. There are three unstable modes for an absorbing falling film (without surface-activeagent): one is the surface wave which is inherent to free surface flow, and the other two are associated with the Marangoni instability. One of these occurs for large positive absorption Marangoni numbers, being induced by the surface tension force resulting from the concentration gradient of water. The other one occurs for large negative absorption Marangoni numbers, and it is induced by the surface tension force resulting from the temperature gradient produced by the releasing heat of absorption. The effects of surface-active solute on these modes of instabilities are determined.

1. INTRODUCTION A study of the instabilities of falling films in the presence of a surface-active solute has been performed by Ji and Setterwall (1994), using the numerical technique of Li and Ji (Ji, 1994), who developed a tridiagonal solver for the Orr-Sommerfeld equation. The processes considered are the adsorption of the surfactant at the liquid-vapour interface and its desorption into the gas phase. As was previously shown by Emmert and Pigford (1954) and Lin (1970), the surfactant stabilizes the surface by suppressing the formation of surface wave. On the other hand, the evaporation of the surfactant may introduce a new mode of instability due to the Marangoni effect. The wave patterns for the falling film may thus become more complicated in the presence of a surfactant with long wavelength surface waves as well as Marangoni waves which have a shorter wavelength than the surface wave. In the present work, we intend to study the linear stability for an absorbing falling film and, more specifically, for a falling film of an aqueous salt solution which absorbs water vapour. The effect of a surface-active solute on the instabilities of such a falling film will be investigated, The purpose of our work is to understand a practice common in absorption refrigeration. It is well-known that surface-active agents, when added in small quantities, increase the capacity of an absorption refrigerating machine (Bourne and Eisberg, 1966). Interfacial or Marangoni convection is generally thought to be the mechanism for the enhancement of mass transfer ob-

*Corresponding author. Present address: Avd. KB, ABB Corporate Research, 721 78 Viister~ts, Sweden.

served in the falling films of the absorber (Burnett and Himmelblau, 1970; Zawacki et al., 1973). Kashiwagi et al. (1985) used interferometry to visualize heat and mass transfer processes during the absorption of water vapour into a quiescent pool of an aqueous lithium bromide solution containing 1-octanol and confirmed the occurrence of interfacial convection. Hihara and Saito (1993) and Kim et al. (1994a, b) observed recently the mass transfer enhancement due to the Marangoni instability in their respective experio mental study for water vapour absorption into a falling film of an aqueous lithium bromide solution with 2-ethyl-l-hexanol as additive. In general, a surface-active solute could have either a stabilising effect or a destabilising effect on a gas-liquid mass/heat system depending on the conditions of the system and on the properties of the solute. It may amplify disturbances when it participates in mass transfer, leading to signifcant increases in transfer rates, as, for example, in the case of ether evaporating from an aqueous solution, see Davies and Rideal (1961). What is taking place is interfaciai turbulence or Marangoni convection. On the other hand, the mass transfer through the interface is at the same time inhibited by the accumulation of the surface-active solute at the interface. It provides a barrier at the interface for the species being transferred (Sada and Himmelblau, 1967), but more important is that by back-spreading against a disturbance insoluble surfactants smoothen ripples, for example, on a falling film of liquid (Emmert and Pigford, 1954; Tailby and Portalski, 1961). The analysis of Brian (1971) and Brian and Ross (1972) revealed that surface convection of solute in the Gibbs layer has a profound stabilising effect on Marangoni convection. The sur-

3077

3078

WEI J] and F. SETTERWALL

face excess of adsorbed solute opposes surface movements and delays or inhibits completely the appearance of the instability. Brian's analysis was verified by Imaishi et al. (1983) for the desorption of several solutes. The process of practical interest, absorption of water vapour in an aqueous salt solution containing a small a m o u n t of surface-active solute, consists of two mass transfer processes in opposite directions as well as the process of heat transfer. While absorption of water vapour provides the main mass flux, the flux in the opposite direction corresponding to evaporation or desorption of surface-active solute is much smaller. It is an open question whether the latter actually occurs or not. Bourne and Eisberg (1966) described the additive octyl alcohol (which was used in their experiment to increase the refrigeration capacity of the absorption refrigeration machine) as "relatively volatile" with respect to the aqueous lithium bromide solution. We treat here the additive as a volatile component. The three transfer mechanisms could all trigger Marangoni convection, and they certainly interact to create an unusual situation. A study (Ji et al., 1993) of the Marangoni instability in an absorbing laminar jet of a salt solution with a surfactant showed that the instability may occur even if there is a strong adsorption of surfactant, provided the absorption Marangoni n u m b e r exceeds a critical value. Both the previous reports (Ji et al., 1993; Ji and Setterwall, 1994) have shown that desorption of surfactant could trigger a Marangoni instability. In addition to this, there may be an inversion in surface tension behaviour of an absorption system when a surfactant is present. As an example, Figs l(a) and (b) show the surface tensions (Yao, 1991; Yao et al., 1991) of an aqueous LiBr solution in the presence of 1-octanol. Slopes of surface tension with the LiBr concentration and with temperature may change both in magnitude and in sign when 1-octanol is being added. The changes of surface tension would change the stability behaviour of the liquid film. A clear understanding, based on a theoretical analysis, should lead ultimately to the prediction of the occurrence of the Marangoni convection and of its effect on the absorption rate. Previous studies of the Marangoni instability in a mass/heat transfer system (Sternling and Scriven, 1959; Brian, 1971; Brian and Ross, 1972; Imaishi et al., 1983; McTaggart, 1983; Castillo and Velarde, 1985; Dijkstra, 1988; Ho and Chang, 1988; P6rez-Garcia and Carneiro, 1991; Ji et al., 1993) mostly treat static films, or films with an initially homogenous velocity, In a falling film flow with an initially inhomogenous velocity, the surface wave produced by the film flow itself and the Marangoni wave which may be induced by concentration/temperature gradients could interact with each other. The surface wave in combination with thermocapillary instability has been studied by Lin (1975) and Kelly et al. (1986) for long wavelength disturbances in the problem of a liquid film flowing down an inclined heated plane. The same problem but for disturbances of finite wavelength has been dis-

100

,

~ ~ ~

80

.~ ~

60

zx

40

[]

20

x I 10

~ ~ ~o

~

~

~

~ o o

o

0

o

o

o ] ~

zx ,x [] ,x x [] x D x t I I 20 30 40

zx 0

×

s~

I 50

t, I 60

70

(a) LiBr eoneentration (wt%) Fig. l(a). Surface tension vs concentration of LiBr at 25°C with 1-octanol concentration as parameter, from Yao (1990): (O) without octanol; ([~) 100 ppm; (A) 40 ppm; ( x ) 500 ppm. I

I

I

I

I

I

I

90 ~

o

~

o

o

o

-

'~

"

~

70

.~ ~ ~ ~

50

A- . . . . . . . .

30

m- - - - - ~ ~ ~" ~ ~ ~ ~ ~

10

15

l 20

I 25

k 30

t 35

I 40

I 45

I 50

-

5/5

(b) T e m p e r a t u r e (°12) Fig. l(b). Surface tension vs temperature for 50% wt LiBr (aq) with 1-octanol concentration as parameter, from Yao et al. (1991): (O) without octanol; (A) 10 ppm; ([3) 40 ppm; ( x ) 80 ppm; ( + ) 400 ppm.

cussed in detail by Goussis and Kelly (1991). In the stability analysis by Ji and Setterwall (1994) for a vertical falling film in the presence of a surfactant, the Marangoni instability induced by the desorption of surfactant is investigated as well as the surface wave. The mathematical model developed here includes the falling film flow, the absorption of water vapour, the desorption and the Gibbs adsorption of surfaceactive agents. Instabilities for such a falling film systern are investigated by surveying the critical conditions for instabilities and the most unstable waves. It is hoped that this work could provide a comprehensive understanding of the mechanisms by which surfactants improve the efficiency of absorption in falling films. 2. FORMULATION OF THE PROBLEM 2.1. M o d e l The physical system modelled is a vertical falling film. It is supported by a wall on one side and the other side is a free surface to a water vapour phase.

Effect of heat transfer additives The liquid, an aqueous salt solution, absorbs water vapour, during which process heat is evolved. The solution also contains a surface-active solute, which is adsorbed at the interface and desorbs into the vapour phase, 2.2. Equations We shall use Cartesian coordinates when writing the equations. Let x and z be the two coordinates parallel to the wall with x in the direction of gravity and let y be the coordinate perpendicular to the wall. The origin of y is placed at the free surface of the primary flow which has a parabolic velocity profile (Yih, 1963):

Vty) = u,, 1 - ~

2.3. Boundary conditions We have the following conditions at the free-surface boundary y = r/(t, x, z): the kinematic condition, the normal stress balance, the tangential stress balance, the mass balances and the heat balance. The first three conditions may be formulated in a straightforward manner, assuming that the surface deformation r/ is infinitesimal and that the drag of the surrounding gas on the surface of the falling film is negligible: 0r/ v = ~-~ + u. Vr/ (7) 2 [-0v

P-Po-

(1)

Um is gd2/(2v) , with 0 the gravitational acceleration, and v the kinematic viscosity. All transport properties (#, k, D~, Da) are assumed to be constants. In the energy equation, the dissipation term due to viscosity

V. u --- 0 the incompressible Navier-Stokes equations: 0t + u. Vu = Ou

I_Vp + vV~u + g

(2)

(3)

p the advection~liffusion equation for water: 0cw O"-t + U" VC w = D w V 2 c w

(4)

the energy equation:

°r+u.Vr=~v~r

Ot

(5)

0ca + u. Vca = DaV2 ca (6) 0t In the above equations, u = (u, v, w), where u, v and w denote the velocity components in the x , y and z directions, respectively, t is the time, p the pressure, p the density, cw the bulk concentration of water, -

-

Dwthediffusivityofwaterinthesaltaqueoussolution, T the temperature, k the thermal conductivity, cp the specific heat of liquid solution, CA the bulk concentration of surface-active solute, and DA the diffusivity of the surfactant in the solution.

0v'~ ~r/

(0w

~y'~0r/]

~y+ ~z]Oz J

+ ~-z:J = 0

(8)

Ox/ Ox +" ~ + ~

2p ~

-"

~x + N ) N

OxOy]

+ ~xx +

= 0.

(O_~y Ow'~Or/

(Ou

(9)

Ow'~Oq

(Oy

O~y)

2# 0z ] 0z

/Oa

/~ ~z + ~ x ] ~ x +/~ ~z +

~3r/Oa\

+ ~ ~z + ~zz~y]/= 0

(10)

where a is the surface tension. The assumptions used in deriving the boundary conditions for the surface-active solute and those for water and heat are somewhat different. In the first case, we assume that (i) the surface excess of surfactant is constant (Yao, 1991; Ji and Setterwall, 1994); (ii) the concentration of the surface-active solute in the vapour phase is zero; and (iii) there is an equilibrium at the interface between the concentration in vapour phase and the concentration in liquid phase. In the latter case, water and heat, we assume local equilibrium at the interface. This implies that, when the pressure in the vapour phase is constant, the equilibrium concentration of water is a function of temperature only. For a "linear absorbent" (Grossman, 1983), the heat of absorption is constant and we have the equilibrium relation at the interface,

pcp

and the advection-diffusion equation for the surfaceactive solute:

(0u

PL~y- ~y+~xJ~x-

- ~,~

aswellasthebeatofdilution(Kashiwagietal.,1983) are neglected. The volume change of the film upon absorption of water vapour is not considered. Since the bulk concentration of the surface-active solute is supposed to be very small in comparison with the bulk concentration of the aqueous salt solution, it is assumed that the presence of the surface-active solute in the film has no effect on the mass diffusion process of water in the solution. The basic equations for the problem are the equation of continuity:

3079

c~ =

-Ce -- - CO T-t Te - To

Ce

To T~ - To

Te - -

Co

(11)

where Co is the initial concentration of the water, To the initial temperature, c~ the equilibrium concentration of water in the solution at temperature To, and T~ the equilibrium temperature of the solution at the concentration Co. Equation (1 I) is assumed to be valid also for a solution containing surfactant. Assuming that the heat of absorption is transferred into the film only, the heat balance at the interface is written as

k(OT \-~y

Or/OT Or/OT'~ Ox Ox Oz Oz/I D f0c~

0r/0c~

\ 0rl 0c~ ]

= q w~ -~y

Ox Ox

-~z ~z f

(12)

3080

WEI JI and F. SETTERWALL

here q is the heat of absorption. It is noted that for vapour absorption with a large concentration level of absorbate in the solution, the lateral convective term at the free interface may need to be accounted for, as indicated by Braunner (1991). However, for simplicity, the lateral convective term is not considered in the present model, i,e. only the diffusive flux is considered in writing eq. (12). The mass balance for the surface-active solute is written as follows, taking into consideration both adsorption at the interface and desorption from the solution into the vapour phase (Ji and Setterwali, 1994):

D l{ScA --

The boundary conditions at the wall (y = d) are the usual ones. The no-slip and non-permeable velocity conditions require that u = 0, v = 0, w = 0 there is no mass flux through the wall: ~cw -- = 0 ~Y 3ca= 0 0--y-

t? [ Oq "~ OrlOu

Ow d ['Orl ~ OqOwl + ~ z + ~zk-~z v) + Oz Oy J

(13)

where k~ is the gas-phase mass transfer coefficient of the surface-active solute, m the solubility coefficient which is defined as the ratio of the concentration in liquid phase to the concentration in the gas phase at equilibrium, and Fo the surface-excess concentration. The stress balances, eqs (8)-(10), contain the surface tension. For the present ternary system, it depends on the concentration of water, on the temperature and on the concentration of surface-active solute,

(x, y, z) = (x*, y*, z*)d u = u* U~, p = p*(p U~), t = t*(d/Um) cw = c*(ce - Co) + Co

(20)

T = T*(Te - To) + To

ca = C*Cao

Assuming a linear relationship and using the equilibrium equation (11), we may write

(O~CA)odCA

(14)

where the variables with asterisks are non-dimensional ones. In the following, the asterisks are dropped for convenience. Each of the non-dimensional variables is then represented by the sum of the unperturbed value and a perturbation:

with

u = U + u' (dd-~) = (O0-~) - Te--Tp(c3~r~

0

P=Ps+P' ew = c~ + c" T=T~+T'

(15)

Ce -- Co \OT]o"

0

Using eq. (14), the boundary equations (9) and (10) become

2 for

Ou'~drl

( -t?v ~ Y+O U )

#k~yy - ~xx)~xx + # ~x

(Ow ~x + ~z)-~z Ou'~q

d~r ~ (t3c. + ~x-~f) OrlOC.'~ + (\d--~-d~/o\-~-x

+ \~cAJo\ Ox + ~xx-~y) = 0

-- ff'zz J -~z +

(19)

2.4. Perturbation analysis We describe here the main steps in the derivation of the linearized perturbation equations for the sake of completeness. The equations in the last two subsections are made dimensionless through the following substitutions:

tr = a(cw, T, cA).

+

(18)

~T O. O--~=

+ Fo ~x + -~xk~x v) + g--£Oy

da = ( d c a ) o dc~

(17)

and the wall is adiabatic:

OqOca a~Oc_A~=kocA xox e z a z / m Ou

(16)

~

(9a)

+ -~x ] ~x + la ( ~z + ~YY)

o \ ~ Z +~z-'~-y)

+ [ #-~a)O~-~Z + ~Z-~-y,] = O.

ca = ca~ + c~

xwhereU=U(y)ex,

direction. The steady unperturbed profiles c,~, Ts and ca, are all assumed to be functions of y only and may be written in dimensionless forms as follows, using the solutions given in Appendix A:

cw,(y) = cw~(O) 1 -- erf -2 ~Y

T" (Y) = ( I - c'~"(O)) [ l - erf ( 2 ~ with

(10a)

(21)

cw,(0) = E v / ~

+ ~

r

(22a)

]_]'~q (22b)

Effect of heat transfer additives and

3081

1 if"

cas(y) = erf(yl) + exp(g 2 - y~)[1 - eft(g)]

(22c)

CAM

/

o.8

with 0,6

Yt

2

~

g=Yx

'

+BiA

0.4

k where Xp is a dimensionless film length to be fixed in the calculations, Pew is the mass Peclet number for the water, Per the heat Peclet number, E the normalized heat of absorption, PeA the mass Peclet number for the additive, and BiA the Blot number. The dimensionless parameters are defined by

U,nd Pew=--ff-~,

Per-

pcpUmd qDw(ce - Co) k , E= k ( T ~ - To)

T

0.2

0

o.2

Figure 2 shows the profiles of the basic state, cal-

1

1

Pe r = 104,

-

t25e)

Et3C'~ /

0--y =

10 6,

O2cw~ 02T~'~

0y + ~E 0y 2

~

)~/

(25f)

--tl

__ = ~3y

/~u'

,3w'\

+ F ~ - x + -| 0 z]

(25g)

and the boundary conditions at y = 1 u' =0,

tgc~ ~y =0,

0c~ ~=0,

0T' ~y =0.

(26)

(24b)

In the perturbation equations, Re is the Reynolds number, Cr the Crispation number, Ma the absorption Marangoni number, MaA the surfactant Marangoni number, and F the adsorption number:

(24c)

Re = - -Umd ,

2 ,

d--t + U. Vu' + u'. VU = - Vp' + ReeV u

ac'~ + dc'~ + v, dCws = 1 2 , dt U--~x Oy pe V c~

o.8

OC'A BiAC'A+

(24a)

au'

o.6

dc~ ~

'=-r' cw

OT'

V'u'=0

o.4

Fig. 2. Basic state (x = 1000, E =0.05, Pe w= BiA = 10, Pe A = 106).

BiA=mDA"

culated using the values of Table 1 and at BiA 10. We choose xp = 1000 in the calculations, which corresponds to a 1 m length of the film if d = 1 mm. Combining eqs (20) and (21) with the equations in Section 2.2 and the boundary conditions in Section 2.3, we obtain the following dimensionless perturbation equations where the terms quadratic in the perturbation quantities have been neglected:

T

Y

kod

PeA=D-fA'

~

0

(23) u.d

I

(27a)

V

c3T' ~3T' + v,dT~ ~ 1 V2 T' a-~ + U-~x ~--}-= Per

(24d)

dc'a U ~cca' v' C3CA~ 1 dt + ~3x + ~3y = p e a V2c'A

(24e)

--

--,

Ma =

(d~r/dcw)o (ce -- Co)

#U~

MaA =

~l

v' = ~ + U~xx

Or'

2 ~3v'

d2U

Re Oy Ou'

---~ + ~ •

1 /02q

q + -~y + Ma ~ x fOC'A 8rlaCA~'~

F

~32r/'~

(25b)

]

+ Ox ~y /I (25c)

gv' ~w' . . f Oc'~ OtlOcw~"~ ~--z + -~y + 'v*a~--~z + ~-~*--~-y ) 0

,

FoUm

(27d)

(27e)

CaoDA

(da/dT)o(T. #Um

Ma =

-- MaA ~"~Z + OZ Oy ,]

(27c)

We are thus defining both Marangoni numbers: Ma and MaA, using concentration differences. An alternative definition of the absorption Marangoni number (Ji et al., 1993) is

\

-- Maa[-'h--5-~\~.,+ ~x-~-y ) = 0

#Urn

(25a)

+ ?Tz ] = o { Oc'~

'

(~a/~cY4)oCao

with the boundary conditions at y = 0,

p,

(27b)

Cr = #Urn

1"o)

with (25d)

d-T o =

~

o

Te - To kOCw/o

(28)

3082

WEt Jl and F. SETTERWALL

in which case one focuses on the heat evolved during absorption. The two definitions are equivalent as one can show that

The equation system is homogeneous in z direction and in x direction as well because of the assumption that the basic profiles depend only on y. The perturbations may then be assumed to be of the form

(v', c', T', c'a, ~l) = [F(Y),f~(Y),fr(y),fA(y), ~/o] x exp[i(kxx + k~z - tot)] (29) where kx and kz are the real wave numbers in the xand z-directions, respectively, and to the complex growth rate. Substituting eq. (29) into eqs (24)-(26) yields the equations _

k 2

)2 F = (kxU -- t o~./-i':T/\uy_~fd2F- k2F)

1 d2fw Pew dy 2

kx ~

1 d2fT

[ ~e~ k 2 + i(k. U - to) ] fw - Oc~ Oy F = 0

[ k2

k2 Ma [ ~ f(O) +--~--[_kxU(O)-tok2Maa r -- iq4 +

q3

1.

I k~,U(O) - to

+

lwT ]

F(0) + I a -- FgA(O)F'(O)]

]

~er + i(kxU - to) fr - OTsFc~y

= 0 (3oc)

PeA dy 2

k~ U"(0) F"(0) -- k2F(0) + k ~ U ( 0 ) - to F ( 0 )

(30a)

f

(30b)

Per dy 2

3. SOLVING PROCESS 3.1. Integral condition The simultaneous equations (30a)--(30d) together with their boundary conditions (31a)-(31f) and (32a)-(32e) constitute an eigenvalue problem. The method used for solving this problem has been described in detail earlier for a falling film with a surface-active solute but without absorption (Ji and Setterwall, 1994). The funcfionsfw, f r andfA couple to the fluid quantities through the boundary conditions (3 lc) at the interface, and do not appear in the internal equation (30a) for the velocity disturbance. Therefore, we may deal with the internal equation alone but with an integral condition. This is for the present case: --

dEU -

k2 = k2 + k2 In eqs (31) and (32) the primes now denote differentiation with respect to y.

_ (dd__~)o(Te _ To)=(~_~f~)o(Ce_Co).

iRe\dyd 2

where

+ i(kxU -- to) fA -- ~

F = 0 (30d)

where=0

(33)

ql = EgT(O)o~(O) + 9~r(O)gw(O) q2 = EOT(0)o'(O)c%(0) -- 0r(0)0w(0) T~(0)

q3 =

0',4(0) - -

BiAga(O)

q4 = c)s(0)O~(0) -- C~(0)#A(0).

the boundary conditions at y = 0: F (0) = i(kx U(0) - to) qo,

(31 a) k4

-F'"(O)+[3k2+iRe(k~U(O)-to)]F'(O)--~rrno=O

The three functions {O~, 9T, OA} are the adjoint functions of {fw, fT, fa }, respectively, and they satisfy eqs (30b)-(30d) in the case of F = 0, i.e.

g,-[k2+iPe~(k~U-~o)]o~=O,

(31b) -- F"(0) -- k2F(0) + ikxU"(O)~lo

(34a)

9~" - [ k2 + iPeT(kxU -- to)]Or = 0, 9~(1) = 0

-- kEMa [fw(0) + t/oC~,(0)]

(34b)

+ k2Maa[fA(O) + t/oC),(0)] = 0

(31c)

fw(O) + fT(O) + [T',(0) + C~(0)] t/o = 0 (31d) --fiT(O) + Ef'~(O) + [Ec~,(0) -- T~'(0)] qo = 0 (31e)

--f'a(O) + BiAfa(O) + [Biac'a~(O)

f~-(t) = 0

9~ - [ k2 + iPeA(kxU -- to)]ga = 0,

9)(1) = 0. (34c)

The two terms lwr and IA in eq. (33) are integrals and defined as follows: t.1

Iwr = Jo [ow(O)Perffr(y) T'~(y)

-- cj~(0)] ~/0 -- F F ' ( 0 ) = 0 and the boundary conditions at y = 1: F'(1) = 0,

9~,(1) = 0

(31f)

F(1) = 0, f~(1) = 0, f ] ( 1 ) = 0, (32a--e)

-- Egr(0)eewg,~(y)c'~(y)] F(y)dy Ia = PeAJo ga(y)c'a~(y)F(y)dy

(35) (36)

which involve all the values of F along y-coordinates.

Effect of heat transfer additives 3.2. Numerical scheme The numerical technique developed by Li and Ji (1994) for solving the Orr-Sommerfeld equation in the case of a falling film flow is utilized here in the solution of the extended eigenvalue problem when absorption is taking place and a surfactant is present. In their solution for the Orr-Sommerfeld equation, one first writes the equation as a set of second-order equations, One then uses a Taylor expansion to discretize the differential equations into an algebraic one with block-tridiagonal form which can be solved by Gaussian elimination. The previous calculations (Li and Ji, 1994; Ji and Setterwall, 1994) have shown that the scheme is stable, very easy to implement, and fast. We shall not describe the details of the numerical scheme here, it is already done in the cited papers, 3.3. Analytical solution in the case of kx = 0 When kx = 0, the basic velocity is absent from the governing equations. An analytical solution of F which satisfies eqs (30a), (31a), (31b), (32a) and (32b) is obtained:

F(y) = CoCq {cosh [k~(1 - y)] - cosh [fl(1 -- y)]} + Co~2 {k~sinh [fl(1 - y ) ] - flsinh [kz (1 - y)] }

(37)

where co is an arbitrary constant, and

fl = x / k ~ - itoRe ~1 = - 2 k 2 f l c osh (fl)+ (2 k2 -itoRe)flcosh (kz) k3 + i - ~ r [k~sinh ( f l ) - flsinh(k~)]

ct2 = - 2flk~sinh(fl) + (2k 2 - itoRe)sinh (k:) k3 + ~ rcosh (fl) - cosh (kz)]. Substituting eq. (37) with Co = 1intoeq.(33) yieldsthe dispersion equation -

-

2k2c2 [kzsinh (fl) - flsinh(k~)] + 2k2cl [cosh(fl) - cosh(k~)] + itoRe [c2 kz sinh (fl) - c~ cosh (fl)] k~ Ma[-[ F(0) q2 J-] + lwr + . q~ I_ lto k~MaA[-| FgA(O)F'(O) -- F(O)q']l 0. (38) + qa L l a - ito d =

The marginal state for this case may be stationary (to = 0). The dispersion equation is written as follows: 2 -- 2sinh(k~)cosh (k~) + M__aa(_ 2Crq2 + k~I~r) k~ qx

+ MaA[2Crq4 - FgA(O)(k 2 - sinh2(k~)) + k~lA] q3 = 0 (39)

3083

and sinh(kz)

F(y) = (1 - y)sinh(k,y)

y sinh [k~(1 - y)] ks which is used when calculating the integrals. The above analytical solutions for the case of kx = 0 are employed as a benchmark to test the numerical program.

4. RESULTSAND DISCUSSION The instabilities of a simpler failing film system, desorption of a slightly soluble surfactant, have been investigated for transverse waves in a previous report (Ji and Setterwall, 1994). In a comparison with a falling film that does not undergo any heat or mass transfer process, the advection-diffusion equation of surfactant adds a series of eigenmodes that we have called "diffusion waves". We found that only the first one of the modes may become unstable when certain conditions on concentration and properties are satisfled for a particular surfactant. Adding a second mass transfer process to the system previously studied involves then another mechanism, "diffusion waves" related this time to the absorption of water vapour. We shall investigate how these diffusion waves interact with each other, for both the transverse waves and the longitudinal waves. For the sake of clarity, we divide all the possible modes for falling film absorbing water vapour and desorbing a surfactant into three groups: (i) Group I includes the modes that are found for pure falling film flows, the word "pure" meaning here a film without any heat or mass transfer, (ii) Group II includes the modes that originate from the simultaneous advection-diffusion equation of water and of heat. Groups I and II together include all modes for a falling film absorbing a vapour (here steam). (iii) Group III includes the modes originating from the advection-diffusion equation of surface-active solute. Groups I and III together include all modes for a falling film in the presence of a surface-active solute. This case has been treated before (Ji and Setterwall, 1994) for transverse waves. In the present analysis, we first study the falling films with absorption but without surfactant. We shall investigate the modes of group II and the effect of the absorption on the surface wave (the first mode of group I). The effect of a surfactant on the stabilities of an absorbing falling film is the target of the present work. For this case, the flow of the film, the absorption of water, the desorption of surfactant and the adsorption of surfactant interact with each other. The unstable modes from the three groups shall be studied in detail. Values of the parameters used in the calculations are given in Table 1.

4.1. Absorbing falling film without surfactant We begin with a study of the modes of group II. When the absorption Marangoni number is zero (Ma = 0), the eigenmodes can be calculated directly

3084

WEI Jl and F. SETTERWALL

from the eigenequations for the simultaneous advection-diffusion of water and of heat with the unperturbed velocity field. The eigenmodes satisfy the equation , qt = Eor(O)g'(O)+ gr(0)9~(0)= 0. The disturbed velocity F is trivial for this case, and the functions {fw,fr} are equal to {gw, gT}- Figure 3 shows eigenfunctions (fw) of the first three modes of the group, being calculated in two cases: kx = 1, kz = 0; and kx = 0, kz = 1. The eigenvalue for each mode is given under the corresponding plot. The numbering of

Table 1. Values of the parameters used in the calculations xp = 1000 Cr = 0.2 E = 0.05 PeA = 106 Per = 104 Pe~, = 10 6

0.4

the modes here is according to the values of their imaginary parts when M a = 0. The function fw becomes increasingly oscillatory with increasing mode number for the both cases. The figure shows that f~ decays rapidly along y-direction for transverse waves (kz = 0), but it is steadily distributed along y-coordinate for lateral waves (kx = 0). This is completely due to the nonuniformity of the basic velocity. If the velocity is a constant, the eigenfunctions of the case of k~ = 0 will be the same as the case of kx = 0, and the eigenvalues of the two cases will differ only in a constant due to the advection term, kxU, see eqs (30b) and (30c). The modes develop when M a ~ O, i.e. when the simultaneous advection-diffusion equation of water and the equation of temperature couple with the fluid equations through the boundary conditions at the interface. Table 2 gives the eigenvalues of the first two modes (of group II) for the two cases: (kx, k,) = (1,0) and (kx, kz) = (0, 1), being calculated at Re = 1, using the parameters in Table 1, and at several absorption Marangoni numbers. Eigenmodes with mode n u m b e r

05'

-

/

C

-0.2

o

. . . . . . . . . . . . . . . .

-0.5

-0.8

t

0

I

-1

~

0.1

0.2

0

I

I

I

I

0.2

0.4

0.6

0.8

Y

Y

1St ( k x = l , 1%=0) (o=(0.998276, -1.72961D-03) 1

1st (k~=0, k , = l )

(o=(0,-3.16686D-06)

1

0.5

\

0.4

o

-0.8

,

0

,

1

0.1

-1

0.2

I

0

0.2

0.4

Y

I

0.8

1

Y

2nd (kx=l,kz=0) (o=(0.995840,-4.17233D-03) 1 0.4

0.6

2nd (kx=0, kz=l) e~=(0,-2.01946D-05) 1

-

\

-~

-0.2

-J

-0.8

, ~" / 0

t 0.1

,

0.5

-

-

-0.5 -1

0.2

Y 3rd (kx=l, kz=0) (o=(0.993505,-6.51318D-03)

0

0.2

0.4

0.6

0.8

Y 3rd (kx=0, kz=l) (o=(0,-5.20322D-05)

Fig. 3. Eigenfunctionsfor thefirst threemodesofgrouplI(Ma = 0): ( - - - - ) real part oftheeigenfuctions; (- - -) imaginary part. The eigenvalue for each case is also shown under its respective graph.

Effect of heat transfer additives

3085

Table 2. Eigenmodes (to) of group II, Re = 1 First mode

Second mode

Ma

k~ = l, k~ = O

kx = O, k, = l

kx = l, k, = O

0

(0.998276, - 1.730 x 10 -3)

(0, - 3.16686 × 10 -6)

(0.995840, - 4.172 x 10 -3)

(0, - 2.01946 × 10 -5)

0.1 1 10

(0.999048, - 2.265 × 10 -4) (1.001552, 5.040x 10 -a) (1.008114, 2.115× 10 -2)

(0, 7.21711 x 10 -a) (0, 4.31392 × 10 -3) (0, 1.84563 × 10 -~)

(0.994296, - 4.19 × 10-3) (0.993390, - 3.616 x 10 -3) (0.993178, -- 3.483 × 10 -3)

(0, - 3.89449 × 10 -6) (0, -- 3.86246 x 10 -6) (0, -- 3.85941 × 10 -6)

- 0.1 - 1 - 10

(0.999532, - 4.465 x 10 -3) (1.006634, - 7.354 x 10 -3) (1.025918, - 1.071 x 10 -2)

(0, - 3.82687 x 10 -6) (0, - 3.85571 × 10 -6) (0, - 3.85873 x 10 -6)

(0.995488, - 1.817 x 10 -3) (0.988665, 9.031x 10 -4) (0.977705, 1.386x 10 -2)

(0, -- 2.65104 x 10 -5) (0, - 2.67424 x 10 -5) (0, - 2.67663 x 10 -5)

higher than 2 are always stable (not shown in the table) in the tested parameters. The stabilities of the system are investigated here for both positive and some negative values of M a . When M a is positive, meaning that surface tension increases during absorption, the first mode of the group II may become unstable, i.e. the imaginary component of co becomes positive, for the both cases: (kx, kz) = (1, 0) and (kx, ks) = (0, 1), see Table 2. The unstable wave may be induced by the Marangoni effect through absorption of water vapour, similar to the Marangoni effect induced by the desorption of surfactant (Ji and Setterwall, 1994). When M a is negative, the second mode of the group II may become unstable in the case of(kx, k~) = (I,0). This may be a thermal instability: a temperature gradient is generated by the heat of absorption released at the interface. A negative absorption Marangoni n u m b e r is found when a higher temperature at the interface results in a higher surface tension, see eq. (28). This is similar to the case treated by Pearson (1958) in which the film is heated from the wall but with surface tension a decreasing function of temperature. An explanation of the physical mechanism for the instability was given by Davis (1987). The eigenmodes of Table 2 demonstrate also that the Marangoni waves, either for positive M a or for negative M a , are almost attached to the surface of the film: the wave velocity in the case of k~ = 0 is zero [real(w) = 0], while the wave velocity in the case of kz = 0 is close to the surface velocity of the falling film [real(co)/k~ is close to 1-1. We shall from now on call the first mode of group II "solute diffusion wave" and the second unstable wave of group II "thermal diffusion wave". The first mode may be unstable for large positive absorption Marangoni numbers and the second one may be unstable for large negative absorption Marangoni numbers. The surface wave and the two Marangoni waves of the absorbing falling film shall be discussed in the following, each separately. 4.1.1. S u r f a c e wave. The unstable surface wave of a falling film occurs only when k~ ~ 0, since the driving force for the disturbance is in the direction of the basic flow (Goussis and Kelly, 1991). We then discuss

2.5

........

k~ = O, k, = l

j ........ ~ . pure falling film

....... A

----0---Ma= 5, MaA=O ~ -~ ~~ "~

"~ "~

2

,it --

~/

Ma=-5,MaA=0 --

~/

-

M a = 5, M a A = I

...... 1.5

M i ~ _ =~ _ : . . : .

1 17"/ / 0.5

./"

.i" ~

~ 10

........

.

.

.

.

/ ~" / ........

0 1

~ 100

.......

1000 Re Fig. 4. Neutral curves of the surface wave in the (kx, Re)plane with M a and M a a as parameters.

the mode in the form of transverse waves, i.e. in the case of kz = 0. Effect of absorption on the surface wave is found to be very small, see Fig. 4 which shows the neutral curves of the mode with M a and M a A as parameters. The results in the case of M a A ~ 0 in Fig. 4 shall be discussed later. When discussing the surface wave here, one should keep in mind that there are two mechanisms to cause the instabilities of long-wave disturbances: the basic flow shear stress and the surface tension force (Goussis and Kelly, 1991), although there is only one unstable region shown in Fig. 4. This has been demonstrated before for a falling film in the presence of a surfactant (Ji and Setterwall, 1994). For the present case of an absorbing falling film without surfactant, it may be illustrated by the results in Table 3. By model 1 in the table, effect of absorption on the flow is only due to the modification of the basic concentration at the free surface by the surface deformation: the term r/oCL~(0) in eq. (31c), the perturbed concentration: the termfw(0) in eq. (31c), being not taken into account. By model 2 of Table 3, the whole effects of absorption

3086

WEl Jl and F. SETTERWALL

including both the terms are considered. Comparing the results of pure film with the model 1, we may conclude that the two mechanisms: the basic flow shear stress and the surface tension force, reinforce each other in causing instabilities of long transverse waves for positive Ma, while the two mechanisms deplete each other for negative Ma. Comparing the results of model 2 with the model 1 and with the pure film, we may conclude that the perturbed concentration [ fw (0) ] suppresses the effects of the basic concentration [r/0c',,(0)]; consequently, the absorption has a very small effect on the instabilities of long transverse waves. 4.1.2.

Solute

diffusion

wave.

We

have

shown

100 10 Ma

1

........ ~ ........ k =0 kz--0,Re=l ks--O. Re=100 /

\.: 0.1 0.01

in

Table 2 that the "solute diffusion wave" may be unstable at large positive absorption Marangoni numbers Ma, for both the cases of the transverse waves (k, = 0) and the longitudinal rolls (k~ = 0). Figure 5(a) shows the neutral curves in the (positive Ma, k) -plane, for the "solute diffusion wave" together with the surface wave. In the case of kz = O, the unstable region of surface wave (in small wave numbers), being at the left of the neutral curve, is larger for a higher Re, as obtained before in Fig. 4. While the unstable region of "solute diffusion wave" (in moderate or large wave numbers), being at the upside of its neutral curve, decreases somewhat when Re increases from 1 to 100. The two unstable regions may merge into a single unstable region when M a is very large. A better understanding of the transverse "solute diffusion wave" may be obtained by Fig. 5(b) which shows the neutral curves of the mode in (k~, Re)-plane. The unstable region is larger for a higher Ma, and the Marangoni instability occurs preferably at low Reynolds numhers. When k~ = 0, there is no unstable surface wave. The neutral state is stationary and is independent of Re, see eq. (39). Figure 5(a) shows that for the "solute diffusion wave" the unstable region of longitudinal waves (k~ = 0) is larger than the transverse waves (k~ = 0), being in accordance with the results of Goussis and Kelly (1991), who indicated that there is stabilising effect from the basic flow on the Marangoni instability of the transverse waves due to the basic shear stress at the free surface and the Reynolds stress in the bulk of the fluid. This can be further demonstrated by Figs 6(a) and (b) which show the wave number of most unstable wave and the growth rate of most unstable wave, respectively, for both the Marangoni instability of longitudinal rolls (presented by real lines) and the transverse waves (presented by dashed lines). The wave number and the growth rate all increase as M a increases in both the cases, but they are larger in the case of kx = 0 than the case kz = 0. The differences between the two cases is smaller for a larger Ma. In Fig. 6(b), the growth rate of the most unstable wave for the surface wave of the pure film is also shown. One can see that the Marangoni instability becomes comparable to the surface wave at large Ma.

...... ~ ........ ~ ........ ~ ¢- \ - \ I-. - - \ :~ V! , • 'i I\'. ~.. i

0.001

...... t

0.1

.......

(a)

,

.... ,

........

1 10 Wave number, k

i

10o

......

1000

Fig. 5(a). Neutral curves of the surface wave and the "solute diffusion wave" in (Ma, k)-plane (Ma A = 0). 1000

........ . . . .

,x ~-

i

........

i

........

M-~=! . . . . . . .

100 Ma=O.03

~ ~

10

"~ ~

1

_ ~ - - - . . . . . . Ma=l

0.1

........ 1

t 10

........

~ ....... 100 1000

(b) Re Fig. 5(b). Neutral curves of the "solute diffusion wave" in (k~, Re)-plane (Ma~ = 0).

The Marangoni instability of long-wave disturbances in the form of transverse waves (k~ = 0) has been demonstrated in the last section by Table 3 when discussing the surface wave. F o r the Marangoni instabilities in the form of longitudinal rolls (kx = 0), Fig. 5 shows that there is only one unstable region for all finite wave numbers. Furthermore, it is found from the calculations that the stationary neutral curve of the case k~ = 0 is independent of the Crispation number Cr when MaA = 0, the term q2 in eq. (39) is zero when ~ = 0. This means that there would be Marangoni instability of long-wave disturbances even if there is no surface deformation (Cr = 0). The two mechanisms revealed by Goussis and Kelly (1991), for the present model one is associated with the modification of the basic concentration at the deformed surface, and the other one is associated with the interaction of the basic concentration with the perturbation velocity field, may all trigger the Marangoni instability of long-wave disturbances in the present case (MaA = O, k~ = 0). When calculating the most unstable waves of the

Effect of heat transfer additives "solute diffusion wave" in the case of kx = 0, we find that there are two peaks of the growth rates with varying wave numbers in the unstable region for a fixed M a and a Re, as shown in Fig. 7(a). Both the growth rates at the two peaks increase as M a in100

........

~

........

~

........

Ma=l, either kx=0 or kz=0

,',

Ma=0.1, kx--0

3087

creases. The first (left) peak may be thought to be a vestige of the Marangoni instability of long-wave disturbances resulting from the modification of the basic concentration at the deformed surface. The second peak corresponds to the results in Fig. 6(b)(when kx = 0) for the "solute diffusion wave". The calculations indicate that the first peak is always lower than the second one under the tested parameters, and may thus be less important. Figure 7(b) is the same plot as Fig. 7(a) but using Cr as parameter. The effect of Cr is significant only in the range of small wave numbers. When Cr = 0 the first peak disappears, there must be no modification of the basic concentration at a rigid surface.

Ma=0.1, kz=0 Ma=0.03, kx=0 10 -

........

J 10

(a) 1

........

~

........ Re ........

t ...... 1(30 ~

1000

........

Ma=l, kx=0 or kz=0 ...... . .(surface wave)

0.1 " .~ o

. I /-

~ ~ ~ Ma=0.03, kz=0

0.01 =

Ma=0. 1, kx=0

form of longitudinal rolls.

- .

Ma=0.1, kz=0 Ma=0.03, kx=0

0.001 Ma=0.03, kz=0

~ ~. \

10 .4

I 1

(b)

10

100

4.1.3. Thermal diffusion wave. It has been shown in Table 2 that the second mode of group II, named here as "thermal diffusion wave", may become unstable at large negative M a for the case of kz = 0 but not for k:, = 0. Figures 8(a) and (b) show the neutral curves of the mode in ( - Ma, k~)-plane with k~ as parameter and in ( - Ma, kz)-plane with ks as parameter, respectively. Figure 8(a) shows that a large kz extends the unstable region in the ( - Ma, kx)-plane. Figure 8(b) shows that the dependence of the unstable region of the mode in the ( - Ma, kz)-plane on kx is not completely monotonic, but anyhow, it becomes smaller as k x decreases, and disappears when k~ = 0. The unstable 'thermal diffusion wave' will never occur in the

1000

Re

Fig. 6. Dependence of the most unstable waves: (a) its wave number and (b) its growth rate on Re with Ma as parameter, for the "solute diffusion wave" (Ma A = 0).

In the present absorption system, the concentration of water and the temperature are not independent with each other, being coupled at the boundary of the interface, see e q s ( l l ) and (12). Any change in the temperature at the interface will always be accompanied with a certain change in the concentration of water. We have therefore a c o m p o u n d Marangoni number, Ma. As revealed by Goussis and Kelly (1991), the mechanism for the Marangoni instability of moderate-wave disturbances is associated with the interaction of the basic concentration/temperature with the perturbed velocity. The "thermal diffusion wave" which is associated with the basic temperature may not be developed to be unstable because of the opposite effect of the basic concentration, cw~, which has

Table 3. Eigenmodes (09) of surface wave, effect of absorption R e = 1, kx=0.1 Ma

- 1 - 0.1 0 (Pure film) 0.1 l

Model 1

Re = lOO, kx = l

Model 2

Model 1

Model 2

(0.203233, - 0.054118) (0.197680, 0.004851) (0.198120, - 0.000924) (0.197600, 0.004849)

(1.34365, - 0.03089) (1.15512, 0.01581)

(l.12847, 0.03284) (1.12821,0.03262)

(0.197591, 0.004849) (0.197591, 0.004849)

(1.12818,

0.03259)

(1.12818,0.03259)

(0.197070, 0.010595) (0.197582, 0.004848) (0.192681, 0.061187) (0.197502, 0.004846)

(1.10311, (1.03145,

0.05473) 0.26304)

(1.12815,0.03257) (1.12788,0.03234)

Model 1: only considering the effect of basic concentration, the term fw(0) in eq. (31c) not being taken into account. Model 2: both the terms r/oC'~(0) andfw(0) in eq. (31c) being taken into account. CE$ 50-19-F

3088

WE1 Jl and F. SETTERWALL ' ''''"'I

' '''""I

' ' ''""I

......

"I

' ''~

0

:

-i

~

.9.

Ma- 1

-~

(0 i

Ma

I '''"'I

........

l

.......

'I

.....

'"I

-

L

-i .3

~

i_

kz,

-4

1 10"6

. . . . . . . . . . . . . . .

0.01

,

.......

1

"

I

,~,

0.01

k$ ..... I

'

'''""1

'

....

1,,,,

..... l

100

(a) ,

-5

O.1

(a) '"1

'

'''""1

'

''J

to i

0

~

,

,

,

.............. 1 kx

, ,,~/

,

t

10

,

,

~ ,

,,,

Ma

10.4

-3

I t 0.02

-4

0.I

I

l 1ff6

,

0.01 (b)

1

-5

100 kz

. . . . .

0.I (b)

'

I

' ' '[

I0

kz

Fig. 7. Growth rates of Marangoni instabilites (MaA= 0, k~ = 0) using (a) absorption Marangoni number as parameter (Cr = 0.2); (b) Crispation number as parameter (Ma = 1).

Fig. 8. Neutral curves of "thermal diffusion wave" (Ma~ = 0, Re = 100): (a) in (Ma, k~)-plane with k~ as parameter, and (b) in (Ma, k=)-plane with k~ as parameter.

a steeper gradient than Ts along the film depth, see Fig. 2. One may predict that if the basic concentration of water is uniform in the film, the basic non-uniform temperature may cause the Marangoni instability of the "thermal diffusion wave" either in the form of transverse waves or in the form of longitudinal rolls, However, the present result is that the unstable "thermal diffusion wave" occurs even under the opposite effect of the steeper basic concentration of water, and it will not occur in the form of longitudinal rolls (k~ = 0). The calculations indicate that the surface deformation (Cr) has a small effect on the "thermal diffusion wave". By an examination of the eqs (30)-(32) in the case of r/o = 0, we find that all the different terms between the two cases of k~ --- 0 and ks = 0 include the basic velocity, U or d 2 U/dy 2. It is found further that the non-uniformity of the basic velocity in the transport equations, eqs (30b) and (30c), plays a determinant part in producing the "thermal diffusion wave". When the basic velocity is taken to be a constant in the velocity equation (30a), the

"thermal diffusion wave" may still occur; but if it is done to eqs (30b) and (30c), the "thermal diffusion wave" will never occur. Since the basic velocity can have an effect o n t h e system only when kx ~ 0, see eqs (30a)-(30d), the unstable "thermal diffusion wave" will not occur in the form of longitudinal rolls (kx = 0). Figure 9 shows the neutral curves of the "thermal diffusion wave" in the ( - Ma, kx)-plane using the normalized heat of absorption, E, as parameter. As may be expected, the unstable region of the "thermal diffusion wave" is smaller for a smaller E: less heat released means a smaller temperature gradient in the film and, consequently, a smaller driving force for the thermal Marangoni instability. In the limiting case E = 0, the mode will never be unstable. Figures 10(a)-(d) show the most unstable waves and the neutral curves for the "thermal diffusion wave" with the absorption Marangoni n u m b e r as parameter. As the magnitude of the negative M a increases, the wave n u m b e r and the growth rate of the most unstable wave all increase, but its wave velocity de-

Effect of heat transfer additives

3089

hexanol (Kim et al., 1994). The data of the LiBr solution in the absence of a surfactant in the two tables are not same, since they are calculated from the -2 surface tensions of the different authors who used different experimental instruments. One should also know that the calculated Ma in the tables may be for -4 1 reference only, since the value depends much on the Ma -t equilibrium relation of water at the interface. The -6 -1 uncertainty of the interface concentration as well as the vapour pressure makes it impossible to make an accurate calculation. In addition to this, the equilib-8 rium surface tensions are used in the calculations, i.e. • 15 .q kinetics of the additive is not considered at all. Any, how, a change in absorption Marangoni number in-10 fluences the Marangoni instabilities of the film since 1 10 both the "solute diffusion wave" and the "thermal kx diffusion wave" depend on this number. It has been Fig. 9. Neutralcurves of the"thermaldiffusion wave"in the shown in Section 4.1.3 that the "thermal diffusion (Ma, kx)-plane using E as parameter (MaA, Re = 100, wave" is weaker than the "solute diffusion wave". If kz = 0). adding a surfactant could give Ma a positive instead of its original negative value, conditions for the stronger of the Marangoni instability are met and the "solute diffusion wave" may become unstable. (ii) Adcreases. The unstable region in the (kx, Re)-plane is sorption of the surfactant at the vapour-liquid interlarger for a more negative Ma, as shown in Fig. 10(d). face and its desorption into the vapour influence the When Figs 10(a) and (b) are compared with Figs surface tension, thus affecting the instabilities of the 6(a) and (b), we find that with the parameter values of system. The desorption of the surfactant may trigger Table 1 the "thermal diffusion wave" is weaker than the Marangoni instability in falling film flows (Ji and the "solute diffusion wave" for a same absolute value Setterwall, 1994) but adsorption of the surfactant inof Ma. Both the wave number and the growth rate of hibits this instability. The effects of a surface-active the "thermal diffusion wave" are much smaller than solute on the three unstable modes of the absorption that of the "solute diffusion wave", system: the surface wave, the "solute diffusion wave" It should be pointed out that this "thermal diffusion and the "thermal diffusion wave", will be discussed wave" may not necessarily degenerate into the second below, each separately. mode of the group II when Ma from a non-zero value tends to zero. It may degenerate into the third, or 4.2.1. Surface wave. A surface-active solute stabilfourth, or even a higher mode of the group, depending ises the surface wave, as shown in Fig. 4. The unstable on the values of the parameters: k~, Re, Ma, etc. The region of the mode in the (kx, Re)-plane decreases modes of group II are actually not totally independent when a surfactant is present, i.e. when MaA # O. This of each other, as adjustment of parameters may con- is valid for both positive and negative values of Ma. vert one mode into another. Assigning the name The calculations indicate that the stabilising effect of "thermal diffusion wave" to one of them is therefore the surfactant on the surface wave is largely due to its done with some arbitrariness, adsorption, i.e. the parameter F, while the effect of Bin on the surface wave is small, as obtained in our previous analysis (Ji and Setterwall, 1994) for a falling 4.2. Absorbin 9 falling film with surfactant film in the presence of a surfactant but without abWe shall now study the instabilities of an absorbing sorption. falling film in the presence of a surfactant. A surface-active solute could affect the stability of an ab4.2.2. Solute diffusion wave. In our previous study sorbing system in two ways: (i) It changes the surface of a falling film with a desorbing surface-active solute tension and consequently, the absorption Marangoni but without absorption (Ji and Setterwall, 1994), we number of the system. This has been mentioned in the found a "diffusion wave" which may become unstable Introduction, as an example, the surface tensions of an for a large surfactant Marangoni number, being due aqueous LiBr solution are changed when the surfac- to the desorption of surfactant. In Section 4.1.2 of the tant of 1-octanol is added (Yao, 1991; Yao et al., 1991), present report, we study the "solute diffusion wave", see Figs l(a)and (b). The absorption Marangoni num- which may become unstable for a large positive abber is negative in the absence of a surfactant in the sorption Marangoni number, being due to the absolution, it may become positive, for example, when sorption of water vapour. We find here that the two 40 ppm (mg/l) of 1-octanol is added, as shown in "diffusion waves", one for water and one for the surTable 4. Another example is given in Table 5 for the factant, are not independent. There are combinations same solution but with the surfactant of 2-ethyl-I- of parameter values for which they can be converted 0

.... ~

........

~

I -] -1 1

3090

W~] JI and F. SETTERWALL lO

'

I

I

'

....

I

. . . . . . . .

I

'

'

' ' ' "

,

~: "~

8

'~

6

,

1,1,,

E~

100

(a) ........

I

........

~:

,

i , i i , ,

10 .4 1

10

I

.

.

.

.

.

.

.

.

.

.

.

.

--.

.~~

Re .

.

.

.

I

........

Ma--2

10

~ ~

099

1000

100

(b)

0.991

"e'

,

-o.s

~.

0.992

' ' ' " 1

0.001

1000

Re 0.993

'

-1

"~'~ ~*~o

10

'

0.01

....... ' ....... ~

3

i

°

-1

1

~

I

~

"~"~ 45~-o.~-/5, ~:

i

I

........

I

.......

-1

_.._._.. -0.5 1

0.989

........

1

I

........

10

(c)

......

I

100

,

.......

1000

I

........

10

100

1000

Re

Re

Fig. 10. Dependence of (a--c) the most unstable waves and (d) the neutral curves in (kx, Re)-plane on the absorption Marangoni number, Ma, for the "thermal diffusion wave" (Ma A = 0).

Table 4. Surface tension gradients and absorption Marangoni numbers for 50 wt% LiBr aqueous solution with 1-octanol at T = 25°C, estimated from the data of Yao (1991) and Yao et al. (1991) and the data of LSwer (1960) with vapour pressure 20 mmHg Concentration of 1-octanol (ppm) 0

40

100

500

0a ~-T~(NmZ/mol)

- 1.42 x 10-3

6.70 x 10 -4

6.24 x 10 -4

4.30 × 10 -4

0tr ~-~(N/(m K)

- 1.50x 10 -4

3.68× 10 -4

4.18x 10 -4

2.76x 10 -~

- 1.65'

0.29

0.14

0.53

Ma

into each other. As a result, they could degenerate into a n o t h e r pair of wave modes, one stable a n d the other unstable. T h e consequence for the system presently investigated is t h a t there is only one unstable "diffusion wave" related to the mass transfers a l t h o u g h there are two m e c h a n i s m s to trigger the M a r a n g o n i instability: a b s o r p t i o n of water v a p o u r a n d desorp-

tion of surface-active solute. W e m a y still call the m o d e a "solute diffusion wave". Figures 1 l(a) a n d (b) show the neutral curves in the (Ma, kx)-plane (with kz = 0) a n d in the (Ma, kz)-plane (with kx = 0), respectively, using M a a as parameter. In the calculations, we choose R e = 100, Bia = 10 a n d F = 2000. F o r the case of k, = 0, Fig. 1 l(a), the critical

Effect of heat transfer additives

3091

Table 5. Surface tension gradients and absorption Marangoni numbers for 50 wt% LiBr aqueous solution with 2-ethyl-1hexanol at T = 25°C, estimated from the data of Kim et al. (1994) and the data of LSwer (1960) with vapour pressure 20 mmHg Concentration of 2-ethyl-l-hexanol (ppm)

~coa c (Nm 2/tool

Oa

~-~(N/(m K)

Ma

10

0

10

30

50

100

200

- 6 . 7 3 x 1 0 -4

5.26x10 -4

3.07x10 -3

3.24x10 -3

2.78x10 -a

2.50x10 -3

- 7.2 x 10- 5

4.52 x 10- ~

2.82 x 10-4

3.06 x 10-4

2.85 x 10-4

3.77 x l0 -a

- 0.78

0.63

3.65

3.84

3.26

2.73

...... ,

........

,

'

." ' " j 2

.....

"

10

1 Ma

.,...,,..... ........,

1

~ M n A , .

0.1 0.01

0 ,

Ma

/MaA.

0.1 ~ ~.!

0.001

. . . . . . . "'

: ....... ,

0.001

. . 1:..., . . . :. ....... . ., . . .

...... I . . . . . .

i

......

0.001

, , ,1..... I

..... I i. . . . . . . ~

....

0.001

' ' !1'""'

..... ' :. . . . . . . i

......

• • - Ma

0

0.01

' 1

o.ol

,,o.,/

!'1

! O.Ol

,

i\

i

• ,XO.1

0.1

-Ma

0.1

~.

'

}!.

". 1

1 10

1 ........ i

O.1

(a)

1

.......

I

10

...... i

100

......

"-.

I0

1000

kx

,,

O.1

(b)

,'

...... , ....... a

1

10

.......

,

100

....

1000

kz

Fig. ll. Neutral curves of the "solute diffusion wave" with the surfactant Marangoni number, Ma4 as parameter (Re = lO0, BiA = 10, F = 2000). (a) in the (Ma, kx)-plane (k~ = 0), (b) in the (Ma, kz)-plane (kx = 0).

a b s o r p t i o n M a r a n g o n i n u m b e r , the m i n i m u m in the neutral curve, increases at first as MaA increases from zero, but t h e n decreases. F o r the case of kx = 0, Fig. 1 l(b), the critical a b s o r p t i o n M a r a n g o n i n u m b e r decreases m o n o t o n i c a l l y as MaA increases. In Section 4.1.2 for a n a b s o r b i n g falling film w i t h o u t surfactant, we o b t a i n t h a t the "solute diffusion wave" m a y be u n s t a b l e only for large positive Ma. O n e effect of the surfactant is f o u n d here, for b o t h the cases of Figs 1 l(a) a n d (b), t h a t the m o d e m a y be unstable even for negative Ma. C o m p a r i n g the two figures, the effect of the surfactant is m o r e r e m a r k a b l e for the M a r a n goni instability in the form of longitudinal rolls [kx = 0, Fig. 1 l(b)-I t h a n t h a t in the form of the transverse waves [ks = 0, Fig. 1 l(a)]. W e shall then discuss the "solute diffusion wave" in the form of longitudinal rolls (kx = 0) in m o r e detail,

Figure l l ( b ) s h o w s t h a t the surfactant m a y e n h a n c e the M a r a n g o n i instability of m o d e r a t e - w a v e disturbances (k~ ~ 1-10), while it m a y at the same time inhibit the instability of large wavelength disturbances (kz ~ 0.1) a n d the s h o r t wavelength disturbances (k= ~ 102). The calculations indicate t h a t the stabilising effect of the surfactant o n the long waves is related not only to the a d s o r p t i o n (F) of the surfactant but also to the surface d e f o r m a t i o n (Cr). T o d e m o n strate the effect of the surface deformation, a plot of the neutral curves in the (MaA, kz)-plane is given in Fig. 12 with Cr as parameter, being calculated at Ma = O, Re = 100, Bi A = 10 a n d F = 0. The real lines in the figure show t h a t the instability of long-wave disturbances is inhibited by the surface d e f o r m a t i o n (Cr :~ 0). This seems to be in conflict with the previous conclusion (Goussis a n d Kelly, 1991) t h a t the instabil-

3092

WEI JI and F. SETTERWALL

ity of long-wave disturbances may be triggered due to 10 r -" - z - \ ~ [ ] ~ i ~ the modification of the basic concentration/temper| 0.1 , ~ 0 |1 / ature at the deformed interface. However, there is ', 0.1 a difference between the present model and the model ..... " of the previous authors, that the basic concentration 1 of surfactant, CA~, is not a linear function of the film depth, y, i.e. the second derivative of the concentra- Ma A tion, c ~ , is not zero. This second derivative multiplied 0.1 by the displacement of the free surface, c)~r/o, represents the modification of the derivative of the basic concentration at the deformed interface, and appears 0.01 in the boundary equation (310 of the mass balance of the surfactant. Assuming now a positive displacement of the free surface at a local point (t/o < 0), i.e. the interface is sunken into the liquid at the point, since 0.001 0.001 0.1 10 100 c)l~ < 0 (see Fig. 2), the term c~t/o will then contribute a positivef~(0), see eq. (31f). Since the basic concenkz tration of surfactant increases along the film depth, by Fig. 12. Neutral curves of the "solute diffusion wave" in conservation of the mass of surfactant, there must be (MaA, kz)-plane using the Crispation number, Cr, as paraa subtraction of surfactant from the interface. The meter (Ma = 0, Re = 100, Bi A = 10, F = 0, kx = 0): ( - - - - ) reduction of the concentration will result in a higher c~s(O) # O; (- - -) c~(O) = O. surface tension at the point (MaA > 0 , i.e. -- da/~Ca > 0), a surface tension force opposing the sunken deformation will then be produced. Another 15 ..,,,~, ........ , ~,,,,,,,~[, ~ ........ ~ . . . . . . . . . term in eq. (31f) involving the surface deformation is / lff5/ 103 ,' BiAC'AstIO which will be stabilising as well by the same 10 mechanism. The results of Fig. 12 may thus be understood as follows. When C~s(O)= 0, presented by the 5 dashed lines in the figure, the Marangoni instability of long-wave disturbances may occur. The unstable r e - M a 0 h ~ o u ~ - -/ gion is larger for a higher Cr, i.e. the surface deformawit tion destabilizes the instability, being in accordance surfaetant ~-"" ~ .i with the previous authors (Goussis and Kelly, 1991). -5 When c~s(0) # 0, the stabilising effect related to the surface deformation, the term c~,t/o of eq. (31f) to-10 gether with the other stabilising effects, Biac'astlo of eq. (31f), viscosity and adsorption if F # 0, may sup-15 ........ ~ ........ ~ ........ ~ ........ ~ ........ J press completely the instability of the long-wave dis0.01 0.1 1 10 100 100 turbances, even though the surface deformation is kz a condition to trigger the instability. Consequently, Fig. 13. Neutral curves of the "solute diffusion wave" in the unstable region for small wave number distur- (Ma, kz)-plane with the adsorption number, F, as parameter bances becomes smaller as Cr increases, as shown in (Re = 100, Ma A = 1, BiA = 10). Fig. 12. For large wave numbers, the effect of Cr becomes smaller, tending to be negligible, because the surface deformation will be suppressed by the surface Figure 13 shows the dependence of the neutral state tension effect which is stronger for a larger wave of the "solute diffusion wave" on the adsorption of the number. In our previous report (Ji and Setterwall, surfactant, the parameter F. If there is no surface 1994), we obtained similar results as the ones present- excess (F = 0), the unstable region in the (Ma, kz)ed by the real lines in Fig. 12 but for the Marangoni plane is extended in comparison with the surfactantinstability of transverse waves (k, = 0). We gave there free case (presented by the dashed line), being due to an explanation by means of the stabilising effect from the desorption of the surfactant. When F # 0, the the basic flow which would be larger for a larger unstable region becomes smaller as F increases. The surface deformation. But here, since the basic flow stabilising effect of F is more remarkable for shorter would have no effect on the instability of longitudinal wavelength disturbances (larger k~). While the rolls (k~ = 0), we find that it is the term c~r/o in stabilising effect by the surfactant on the long-wave eq. (31f), being associated with the non-linearity of the disturbances (small k,) is mainly due to the term c~(0) basic concentration and the surface deformation, t/o in eq. (31f), as discussed before. Our previous rewhich suppresses the Marangoni instability of long- port (Ji and Setterwall, 1994) demonstrated that the wave disturbances. Certainly, this is also valid for the Marangoni instability induced by desorption of the transverse waves, our previous explanation is not surfactant may be completely suppressed by its adcomplete, sorption supposing F to be very large. The Maran/

Effect of heat transfer additives 11) ,

. . . . . . . .

. . . . . . . .

. . . . . .

, =~ BiA

, ' ,' , ,' ,

5 /

~

1~

100

Ma

y 0

_

~ .........

....

-5 0.01

,

, ,,..,.I

0.1

,

.,.H,,J

1

,

,,,

10

goni instability in the present case, an absorbing falling film + a surfactant, may be induced either by the absorption of water or by the desorption of the surfactant when F is small. But it can be only due to the absorption when F is very large, as the case of F = 10 ~ in the figure. Figure 14 shows the dependence of the neutral state of the "solute diffusion wave" on the desorption of the

ii ,i...t..]

surfactant, t h e p a r a m e t e r B iaa .

without

surfactant is nonvolatile, different feature than the cases of BiA ~ 0 is that the absorption Marangoni

surfactan

l l m .

3093

,

100

1000

kz

WhenBia=O,i.e.

ifthe

number, Ma, tends towards a finitevalue as kz tends to zero, there being no "extra" reduction of the unstablc region at small wavc numbers. This may be easily understood from thc discussions for Fig. 12. When Bi~ = 0, the basicconcentration,CAs,is homogeneous, both the terms BiAc'a~rlo and c~sr/o in eq.

Fig. 14. Neutral curves of the "solute diffusion wave" in (Ma, kz)-plane with the desorption number, Bi,t, as parameter {Re = 100, M a A = 1, F = 2000).

(31f) are zero and thus have no stabilising effect on the instability of long-wave disturbances. The only stabilising effect by the surfactant is due to its adsorp-

100

10-1

10 .3

1

10 "5

0.001

0.01

0.1

(a)

1

Ma A ~

i * ' '' ' ' " I

. . . . . . . . . .

,

. . . . . . . .

0.001

0.01

(b)

' ' ''''"I

' ' ''''"

0.1

1

Ma A 10_9. ~ , , ,,,,,, ~I

g~ ,I ,,,,i, .

, , r,,,,

- o.1

10

"1"

-

"" :"

10"3

kz

0.01

"

co i

/

j.

""

~

~

-

.t

/ o

10 . 4

/

/

/ / I 1

0.001

(c)

,

,

, ,,,,,I

,

,

, .....

0.01

i

0.1

MaA

,

.......

10-5

1

,I

0.001

(d)

.......

I

. . . . . . . .

0.01

I

. . . . . . .

0.1

MaA

Fig. 15. The wave numbers and thegrowth ratesofthemost unstablewaves ofthe"solutediffusionwave" as functionsof the surfactantMarangoni number, M a A {Re = I00, BiA = I0),(a,b) usingthe adsorption number (F)as parameter(Ma = 0.I)and (c,d) usingthe absorptionMarangoni number, Ma, as parameter (r = 2000).

3094

WEI J] and F. SETTERWALL

tion, F, which exists even with no surface deformation, Figure 14 shows also that the change of the neutral curves with the BiA may not be monotonic for some wave numbers. This may be explained as follows, A general conclusion in our previous report is that the desorption of surfactant destabilizes the Marangoni instability, being due to its contribution in producing a non-uniform basic concentration of the surfactant. However, there are also some by-effects from the desorption which may inhibit the instability, for example, the terms BiAc'adlo and c~,r/0 in eq. (31f) which may suppress the instability of long-wave disturbances, and the term BiAfa(O ) in eq. (31f) which may reduce the surface tension driving force for the Marangoni instability of moderate-wave disturbances. The dependence of the total effect of the desorption of the surfactant, either destabilising or stabilising, o n BiA m a y

thus not be monotonic.

small Ma. The destabilising effect is due to its adsorption, as demonstrated in Figs 15(a) and (b). When r is very small (close to zero), the wave may be more unstable for a larger MaA: the wave number and the growth rate increases as Man increases. The relation between the most unstable waves and Man depends also on the absorption Marangoni number, Ma, see Figs (15c) and (d). The surfactant inhibits the Morangoni instability when Ma is large. On the other hand, when Ma is very small, tending to the case of a falling film with surfactant but without absorption, the growth rate of the most unstable wave increases always as Man increases, see Fig. 15(d), as obtained before (Ji and Setterwell, 1994).

As

8

........

,

.......

an example, the calculations give that c~s = 7.236, rp

cAs =

-- 106.1

BiA = I0;

when

and

r=o

p

c~s = 17.06,

~

c ~ = - 78.35 when B i A = 100. Comparing the two cases, the first derivative cAs in the latter case (BiA = 100) is larger and thus the surface tension force in producing the Marangoni instability of long-wave disturbances is larger, although the stabilising effect BiAc'adlo is also larger; while the second derivative c~s of the latter case is smaller and thus the stabilising effect of c~dlo is smaller. Consequently, the unstable region in small wave numbers in the case of Bi~, = 100 is larger than BiA = 10. In the previous paper (Ji and Setterwall, 1994), we did not report the non-montonic behaviour, because the range of the parameter Bi~

without surfactant

~ ~

7

~ ~:

6

10o0

3~.i 5

........ 1

0.01

' 10 BiA

........

........ 100

,

,

i',

,

,1,.

I"=0

e x a m i n e d there is too small to find it.

Without surfactant

Figure 15 shows the wave numbers and the growth rates of the most unstable waves, for the "solute diffusion wave", as functions of the surfactant Marangoni number, MaA, using F and Ma as parameters. The surfactant mostly decreases the wave number and the growth rate except in the cases of very small F or very

moo ~

0.001

O

r~

/ 3ooo/ 10.4

........

,

........

i

........

,

10 BiA

........

100

M a A = 0 (without surfactant) ~

~"

0.9925

........

,

........

10 0.01 0.992

r=0

"~

y

without surfactant

io 0.9905

........ , 1

10

........ , 100

....... 1000

_

........ 1

,

.......

10

100

~l~iA

Re Fig. 16. Neutral curves of the "thermal diffusion wave" in (k~, Re)-plane with Ma.4 as parameter (Ma = - 1,

Fig. 17. W a v e number, growth rate and wave velocity of the most unstable waves of the "thermal diffusion wave" as functions of Bia with F as parameter (Re = 100,

BiA = 10, F = 2000).

Ma = - 1, Ma,4 = 0.01).

Effect of heat transfer additives 4.2.3. Thermal diffusion wave. Figure 16 shows the neutral curves of the "thermal diffusion wave" in (kx, Re)-plane using Man as parameter, being calculated at Ma = - 1, BiA = 10 and F = 2000. The unstable region becomes smaller for a larger Man. The stabilising effect of the surfactant is due to its adsorption, as demonstrated by Fig. 17: a plot of the wave number, the growth rate and the wave velocity of the most unstable waves as functions of BiA using F as parameter, fixing Re, Ma and Man. When F ~ 0: all three parameters of the most unstable wave are lower than those of the surfactant-free case. A larger F and a smaller BiA result in a larger stabilising effect on the "thermal diffusion wave",

5. CONCLUSIONS A linear-stability analysis has been performed for an absorbing falling film with or without surface-active agents. There are three unstable modes for a falling film (at low or moderate Reynolds numbers) absorbing vapour without surfactant: one is the surface wave, and the other two are associated with the Marangoni instability, the "solute diffusion wave" and the "thermal diffusion wave", respectively. The absorption has a very small effect on the surface wave. The Marangoni instability of "solute diffusion wave" is due to the absorption of water vapour and may occur for large positive absorption Marangoni numbers. The Marangoni instability of "thermal diffusion wave" is due to the temperature gradient in the film produced by the heat released during the absorption and may occur for large negative absorption Marangoni numbers. The "solute diffusion wave" assumes preferably the form of longitudinal rolls than the form of transverse waves. While the "thermal diffusion wave" may assume the form of transverse waves, but it will never assume the form of longitudinal rolls. When the two Marangoni instabilities are compared in a same magnitude of the absorption Marangoni number, we find that the "thermal diffusion wave" is much weaker than the "solute diffusion wave". A surface-active solute has a stabilising effect on both the surface wave and the "thermal diffusion wave", being largely due to its adsorption. It could have either a stabilising effect or a destabilising effect on the "solute diffusion wave", depending on its properties. When a surfactant having a large Bin and a small F is added into the absorption system, the "solute diffusion wave" may be unstable even at a negative absorption Marangoni number. The higher the evaporation rate and the smaller the surface excess, the larger the enhancing effect is on the "solute diffusion wave", The effect is more remarkable for the Marangoni instability of the "solute diffusion wave" in the form of longitudinal rolls. The present work studies the interfacial convection in the process of practical interest: absorption in a falling film, providing an understanding of the effect of a surface-active solute on the process. Without surface-active solute, Marangoni instability could occur

3095

in theory: the "solute diffusion wave" or the "thermal diffusion wave". But an actual system may not meet the conditions for the Marangoni instabilities, or it may but the instability is very weak. A surface-active solute could change the surface tension behaviour of the system, causing the system to fall into a domain where Marangoni instability is possible. In addition to this, a surface-active solute may extend the condition for the occurrence of the Marangoni instability of the "solute diffusion wave". The results of the present work may be served to understand the observations of the experimental studies (Hihara and Saito, 1993; Kim et al., 1994) for water absorption into a falling film of aqueous lithium bromide solution with 2-ethyl-lhexanol as additive. When there is no surfactant, the absorption Marangoni number is negative, the "thermal diffusion wave" might become unstable but it is very weak. As observed by Hihara and Saito (1993), the falling film in the absence of additives had low absorption rate, and assumed "transverse waves" as the flow rate increases which may be thought of as the long surface wave. When the surfactant is added to the solution, both the experiments obtained a higher absorption rate (about a fourfold increase) and observed the "rivulets" moving over the surface of the film in the lateral direction, which may be thought of as originated from the Marangoni instability of the "solute diffusion wave" in the form of the longitudinal rolls. There are two probable causes by the surfactant in producing the "solute diffusion wave": (i) Ma could become positive and at the same time the surfactant, if it is volatile, may lower the critical absorption Marangoni number, even reaching a negative value, thus the absorption may trigger the instability and (ii) desorption of surfactant may trigger the instability.

Acknowledgement--We are very grateful to Dr H. Bjurstr6m for valuable discussions.

BiA ca ce Co

Cp cw

Cr d Dn Dw E

fw,fr, fA 9 gw, gr, 9A

NOTATION Biot number [ = k~d/(mDA)] bulk concentration of surface-active solute equilibrium concentration of water at temperature To initial concentration of water specific heat of liquid solution bulk concentration of water Crispation n u m b e r ( = pU,,/a) depth of the film diffusivity of the surfactant in the solution diffusivity of water in the salt aqueous solution normalized heat of absorption F = q D w ( ~ - Co)1 [_ k(Te - To) _] functions defined in eq. (29) gravitational acceleration functions defined in eq. (34)

3096

WEI Jl and F. SETTERWALL

IA

integral defined by eq. (36) integral defined by eq. (35) thermal conductivity

IwT k kx kz

real wave numbers in x-direction real wave numbers in z-direction gas-phase mass transfer coefficient of the surface-active solute

ko m

solubility coefficient defined as the ratio of the concentration in liquid phase to the concentration in the gas phase at equilibrium

Ma

absorption Marangoni number I = (d-~w) o(ce

de

MaA

-

-

Co)/(#Um)

1

surfactant Marangoni numbers

I = -- (fl~C~)o(CA°/(ItUm)l p

pressure

Pea

mass Peclet number for the additive

( = Urad/DA) Per Pew

heat Peclet number ( = pcpU,~d/k) mass Peclet number for the water

( = Umd/Dw) q

Re t T Te To u

U

heat of aborption Reynolds number ( = U,~d/v) time temperature equilibrium temperature at the concentration Co initial temperature (u, v, w) with u, v and w the velocity components in the x, y and z directions respectively unperturbed velocity

U,~

[ = #d2/(2v)]

x

coordinate parallel to the wall and in the

y z

direction of gravity coordinate perpendicular to the wall coordinate parallel to the wall

Greek letters

v

adsorption number [ = FoU=/(cAoDA)] surface-excess concentration surface deformation kinematic viscosity

p cr co

density surface tension complex growth rate

F Fo

Superscripts and subscripts ' s

perturbation in eqs (21)-(29), or derivative in eqs (31)-(36) steady unperturbed value

REFERENCES

Bourne, J. R. and Eisberg, K. V., 1966, Maintaining the effectiveness of an additive in absorption refrigeration systems. U.S. Patent 3 276 217. Brauner, N., 1991, Non-isothermal vapour absorption into falling film. Int. d. Heat Mass Transfer 34, 767-784.

Brian, P. L. T., 1971, Effect of Gibbs adsorption on Marangoni instability. A.I.Ch.E.J. 17, 765-772. Brian, P. L. T. and Ross, J. R., 1972, The effect of Gibbs adsorption on Marangoni instability in penetration mass transfer. A.I.Ch.E.J. 18, 582-591. Brian, P. L. T., Vivian, J. E. and Mayr, S. T., 1971, Cellular convection in desorbing surface tension-lowering solutes from water. Ind. En#ng Chem. Fundam. 10, 75-83. Burnett, J. C. and Himmelblau, D. M., 1970, The effect of surface active agents on interphase mass transfer. A.I.Ch.E.J. 16, 185-193. Castillo, J. L. and Velarde, M. G., 1985, Marangoni convection in liquid films with a deformable open surface. J. Colloid Interface Sci. 108, 264--270. Davies, J. T. and Rideal, E. K., 1961, lnterfacial Phenomena. Academic Press, New York and London. Davis, S. H., 1987, ThermocapiUary instabilities. Fluid Mech. 19, 403-435. Dijkstra, H. A., 1988, Mass transfer induced convection near gas-liquid interfaces. Ph.D thesis, Groningen. Emmert, R. E. and Pigford, R. L. 1954, A study of gas absorption in falling liquid films. Chem. Enang Prog..50, 87-93. Goussis, D. A. and Kelly, R. E., 1991, Surface wave and thermocapillary instabilities in a liquid film flow. J. Fluid Mech. 223, 25-45. Grossman, G., 1983, Simultaneous heat and mass transfer in film absorption under laminar flow. Int. J. Heat Mass Transfer 26, 357-371. Hihara, E. and Saito, T., 1993, Effect of surfactant on falling film absorption. Int. J. Refrig. 16, 339-346. Ho, K. L. and Chang, H. C., 1988, On nonlinear doublydiffusive Marangoni instability. A.I.Ch.E J. 34, 705-722. Hozawa, M,, Inoue, M., Sato, J. and Tsukada, T., 1991, Marangoni convection during steam absorption into aqueous LiBr solution with surfactant. J. Chem. Engng Japan 24, 209-214. Imaishi, N., Hozawa, M., Fujinawa, K. and Suzuki, Y., 1983, Theoretical study of interfacial turbulence in gas-liquid mass transfer, Applying Brian's linear-stability analysis and using numerical analysis of unsteady Marangoni convection. Int. Chem. Engng 23, 466--476. Ji, W., Bjurstrom, H. and Setterwall, F., 1993, A study of the mechanism for the effect of heat transfer additives in an absorption system. J. Colloid Interface Sci. 160, 127-140. Ji, W. and Setterwall, F., 1994, On the instabilities of vertical falling liquid films in the presence of surface-active solute. J. Fluid Mech. 278, 297-323. Kashiwagi, T., Kurosaki, Y. and Shishido, H., 1985, Enhancement of vapour absorption into a solution using the Marangoni effect. Nihon Kikai Gakkai Ronbunshu B51, 1002. Kashiwagi, T., Watanabe, H., Omata, K. and Lee, D. H., 1988, Marangoni effect in the process of steam absorption into the falling film of the aqueous solution of LiBr. KSME-JSME Thermal and Fluid En9. Conf., Seoul, Korea. Kelly, R. E., Davies, S. H. and Goussis, D. A., 1986, On the instability of heated film flow with variable surface tension. Heat Transfer 1986 Proceedings of the 9th International Heat Transfer Conference, San Francisco, Vol. 4, p. 1936. Kim, K. J., Berman, N. S. and Wood, B. D., 1994a, Experimental investigation of enhanced heat and mass transfer mechanisms using additives for vertical falling film absorber. Proceedings of the International Absorption Hear Pump Conference, AES-Vol. 31, pp. 41-47. Kim, K. J., Berman, N. S. and Wood, B. D., 1994b, Surface tension of aqueous lithium bromide + 2-ethyl-l-hexanol. J. Chem. Engng Data 39, 122-124. Li, J. and Ji, W., 1994, A tridiagonal solver for the Orr-Sommerfeld equation, in Ph.D thesis by Ji, W., Theoretical study of effects of surface-active solutes on absorption system, KTH, Stockholm, Sweden.

Effect of heat transfer additives Lin, S. P., 1970, Stabilizing effects of surface-active agents on a film flow. A.1.Ch.E.J. 16, 375-379. Lin, S. P., 1975, Stability of liquid flow down a heated inclined plane. Heat Mass Transfer 2, 361. Llorens, J., Marts, C. and Costa, J., 1988, Discrimination of the effects of surfactants in gas absorption. Chem. Engn9 Sci. 43, 443-450. McTaggart, C. L, 1983, Convection driven by concentration- and temperature-dependent surface tension. J. Fluid Mech. 134, 301-310. Pearson, J. R. A, 1958, On convection cells induced by surface tension. J. Fluid Mech. 4, 489-500. Perry, R. H. and Chilton, C. H., 1973, Chemical Enoineers' Handbook, 5th Edition, pp. 3-224. P6rez-Garcia, C. and Carneiro, G., 1991, Linear stability analysis of B6nard-Marangoni convection in fluids with a deformable free surface. Phys. Fluids A 3, 292-298. Sada, E. and Himmelblaa, D. M., 1967, Transport of gases through insoluble monolayers. A.I.Ch.E.J. 13, 860-865. Scriven, L. E. and Sternling, C. V., 1964, On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321-340. Sternling, C. V. and Scriven, L. E., 1959, Interfacial turbulence: hydrodynamic instability and the Marangoni effect, A,I.Ch.E.J. 5, 514-523, Tailby, S. R. and Portalski, S., 1961, The optimum concentration of surface active agents for the suppression of ripples. Trans. Inst. Chem. Engrs 39, 328. Weast, R. C., 1980, CRC Handbook of Chemistry and Physics, 60th Edition. Whitaker, S., 1964, Effect of surface active agents on the stability of falling liquid films. I & EC Fundam. 3, 132-142. Yao, W., 1991, Work notes, Yao, W., Bjurstr6m, H. and Setterwall, F., 1991, Surface tension of lithium bromide solutions with heat-transfer additives. J. Chem. Engn 9 Data 36, 96-98. Yih, C. S., 1963, Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321-334. Zawacki, T., Leipziger, S. and Well, S. A., 1973, Inducement of convective motion in static absorbers. Paper presented at the 4th Joint AIChE-CSChE Chem. Eng. Conf. APPENDIX A: UNPERTURBED SOLUTIONS FOR THE CONCENTRATION OF WATER, THE TEMPERATURE AND THE CONCENTRATION OF SURFACTANT

When the equations given in Sections 2.2 and 2.3 are normalized according to eq. (20), using the one-dimensional velocity of eq. (1), we obtain the dimensionless equations for the unperturbed concentrations and the temperature as follows, for c w and T: uOc~, 1 (02ew O2cw~ ax = ~ e ~ \ - ~ x 2 + - ~ y 2 ]

(A1)

uOT 1 //¢~2T t~2T'~ 0-~ = ~rer [,~'2-x 2 + ~y2)

(A2)

with the boundary conditions at y = 0: aty=

cw = 1 - T, __

1: Ocw=0, ~y

~__T= EOC~ Oy Oy OT

--=0, ~y

(A3, A4) (A5, A6)

3097

and for ca, . Oca I ( O e C A O2ca'~ u '~-x = PeA \ OX2 + ~ye J

(A7)

with the boundary condition tOca . at y = 0: - - = t3y BtAcx ~CA ^ at y = 1: ~ y = o.

(A8) (A9)

The dimensionless parameters appearing in the above equations are explained in Section 2.4. The mass Peclet numbers and the heat Peclet number may be estimated as Pew ~ 106, PeT ~ 10'* and Pea ~ 106 (Perry and Chilton, 1973; Weast, 1980). We may give approximate solutions to the unperturbed equations by assuming U = 1 in the film, neglecting the mass diffusions in the x-direction as well as the heat conduction in the x-direction, and using a semi-infinite layer approximation. The assumptions may be questionable for the energy equation (A2) as the heat Peclet number is not large enough to allow them. This will not be considered here, since the purpose of the present work is to study the stability of system, but not the accurate solution of the transport equations. With the initial conditions at x = 0, cw = Co, T = To,

CA = Cao

or in dimensionless form, Cw = 0, T = 0,

CA = 1

we obtain the following solutions by the Laplace transform technique: lcw(x, y) = cws(0) 1 - erf{-- y ~] (A10) [ \2 x/w/~,]J [T(x, y) = (1 - cws(0)) 1 - e r f f - - --y .'~l (A11)

[_

\2~J_]

with c~(O) =

x~r Ex/~

+ x/~r

and CA(X, y) = erf(yl) + exp(92 -- y~)[1 -- eft(0)] (A12) with Y'

y 2,~/x/Pea'

g=Yl+

BiA ~ x urea

where erf is the error function. The solution (A12) has been given in our previous report (Ji and Setterwall, 1994). The profiles depend on both x and y. However, for large values of Pew, Per and PeA, the dependences of the profiles on x may be quite weak in comparison with the y. In the stability analysis, we may assume that the basic eoncentrations and the basic temperature are functions of y only.