J. Phys. Chem. 1992,96, 1431-1437
1431
Gravltatlonal Instabilities In Multicomponent Free-Dlffuslon Boundarles P. L. Vitagliano,+ L. Ambrosone,t and V. Vitagliano* Dipartimento di Chimica, Universitd di Napoli, Federico 11, Via Mezzocannone 4, 801 34 Napoli, Italy (Received: January 14, 1991)
The gravitational stability of multicomponentdiffusion boundaries is discussed. Explicit expressions are given for ternary and four-component systems, using matrix algebra.
Introduction The evolution of convective motions inside an initially stable diffusion boundary (bottom solution with higher density than top solution) has been called double diffusive convection. Such a process is of wide interest in fluid dynamics.' Besides some trivial cases,convection inside a diffusive boundary in general may arise when matter transport in a gravitational field is promoted by at least two independent driving forces. In the simplest cases, such forces can be a temperature gradient and a concentration gradient in a twwxmponent solution or the gradient of concentration of two components in an isothermal three-component system. The first case has been the subject of much experimental and theoretical study in the field of oceanography, with the aim of interpreting (a) the formation of stable boundaries separating sea levels at different temperature and Composition, and (b) the growth of convectivefingers in sea areas where a hot concentrated solution of lower density is stratified over a cold dilute solution of higher Other examples include unwanted convection in lakes and solar ponds, rollover in liquid natural gas tanks, geology (crystallization and magma chambers processes), geophysics (mantle convection and vulcanism), astrophysia (inside stars at least four components may be involved in the process: angular momentum, heat, magnetic field, helium/hydrogen composition), metallurgy (morphology and crystallization). Papers dealing with the gravitational stability of isothermal diffusion boundaries in three-component systems have also been published,&*and recently there has been an increasing interest in this In particular the fluid-dynamics theory providing the conditions for the growth of convection has been extended to a case of generalized transport equations that include cross t e r m ~ ; ~ - ~ J ' namely, diffusion in a three-component system. Diffusion in this system is described by the generalized Fick's equations: Ji = -EDij grad Cj (i, j = 1, 2) (1) where JI and Ci are the flow and concentration of component i. The flows and diffusion coefficients defined by eq 1 are expressed in a volume-fixed reference frame because it corresponds to the experimental laboratory frame fixed with respect to the diffusion cell. Balance equations are expressed in terms of mass-fixed reference frame: Ji Ci(~i- U) (la) and the diffusion coefficients appearing in the single-component mass balance equation (6) are expressed in this second reference frame. Dealing with differential diffusion, namely, constant Dij and small concentration gradients, Fick's equations (1) can be expressed in terms of mass-fmed diffusion coefficients, D,, as well. Expressions for the density in terms of vertical distance and composition are given in the Appendix in terms of the solution Present address: N.L.R., A. Fokkerweg 2, 1059CM. Amsterdam. Holland. FacoltH di Agraria Universiti del Molise, Campobasso, Italy.
0022-3654/92/2096-143 1$03.00/0
of eqs 1 for the freediffusion case and constant D ;the transform equation to obtain the mass-fixed diffusion coeficients from the volume-fixed ones is also given in the Appendix. Presently we have quite sophisticated optical techniques that allow us to measure the four diffusion coefficients in a ternary system to a reasonable degree of accuracy (in binary systems the accuracy on D can be of the order of a few parts in ten thousand).14J5 The use of a laser light source and a photodiode mounted on a scanning motor connected with a computer system16 has further improved the collection of experimental data and allows one to obtain the final Dij data almost in real time. For these reasons it is about the time to approach diffusion measurements in four-component systems, where nine diffusion coefficients must be measured. A preliminary and useful knowledge to approach these systems is a theory on the gravitational stability behavior of four-component diffusion boundaries as a function of the concentration gradients of the various components. This theory is presented in the following pages; matrix algebra has been used for the mathematical expressions in order to compact them. The final expressions can be easily extended to any (n 1)-component system (n > 3). For n = 2 our expressions reduce to those previously found.*v9J1
+
Discussion (A) Assumptions are as follows: (a) In all of the following, flows are defined in terms of a mass-fixed reference frame. (b) We are dealing with a Newtonian noncompressible isothermal fluid; its density is a linear function of the n independent component concentrations and is independent of pressure and temperature: p
= po(l
+ H-C) (state equation)
(2)
(1) Chen, C. F.; Johnson, D. H. J . Fluid Mech. 1984, 138, 405. (2) Veronis, G. J. Marine Res. 1965, 23, 1. (3) Baines, P. G.; Gill, A. E. J . Fluid Mech. 1984, 37, 405. (4) Huppert, H. E.; Manins, P. C. Deep Sea Res. 1973, 20,315. (5) Turner, J. S . Buoyancy Eflecrs in Fluids; Cambridge University Press: New York, 1973. (6) Sartory, W. K. Biopolymers 1969, 7, 251. (7) Vitagliano, V.; Sartorio, R.; Spaduui, D.; Laurentino, R. J . Solution Chem. 1977,6, 671. (8) McDoughall, T. J. J. Fluid Mech. 1983, 126, 379. (9) Vitagliano, P. L.; DellaVolpe, C.; Vitagliano, V. J. Solution Chem. 1984, 13, 549. (10) Comper, W. D.; Preston, B. N. Adu. Polym. Sci. 1984, 55, 105. (11) Miller, D. G.; Vitagliano, V. J . Phys. Chem. 1986, 90, 1706. (12) Vitagliano, V.; Borriello, G.; DellaVolpe, C.; Ortona, 0. J . Solurion Chem. 1986, 15, 811. (13) Vitagliano, V.;DellaVolpe, C.; Ambrosone, L.; Costantino, L. PHC, Phys. Chem. Hydrodyn. 1988, 10, 239. (14) Tyrrell, H. J. V.; Harris, K. R. Dijfusion in Liquids; Butterworths: London, 1984. (15) Kirhldy, J. S.; Young, D. J. Dvfusion in the CondensedSrare; The Institute of Metals: London, 1987. (16) DellaVolpe, C.; Vitagliano, V., to be published.
0 1992 American Chemical Society
1432 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 p o being a constant (see the final table for the definition of symbols). (c) We assume a bidimensional diffusion system where y is the vertical axis (positive in the upper direction) and x the horizontal axis. This assumption agrees with the shape of diffusion cells commonly used in interferometric apparatuses such as Gouy or Rayleigh diffusiometers (Tiselius cells). These assumptionsqllow us to write the total mass, momentum, and individual component mass balance equations: (pu), (PUh
+ (pu,)
= -pf
(total mass balance)
(3)
+ U ( P 4 , + U(PU), + Px = PV2U (balance of momentum along x axis) (4)
( P d f + U(PU)X + U(P,), + Py = PV2U + gP (balance of momentum along y axis) ( 5 )
(C), + u(C), + u(C), = DV2C (single-component mass balance, n equations) (6) Initial and boundary conditions: time t = 0: u = 0; u = 0 (for all x and y values)
Vitagliano et al. two expressions into each other, the pressure term disappears:
+
V 2 q t \k,V2qx - QXV2qy= qV2(V2\k) - g*H.(C’), and eq 11 becomes
(c’),= D
Set (14)-( 15) represents a cellular motion, namely, a set of small vortices of height L and base L / a that grow up for p > 0 and regress for p < 0. Although the knowledge of the actual length of height L is not critical, one can assume that L is of the order of magnitude of the boundary width (z length 3-4). Let us write
=
y
-
-Q,
x
-C
fy/a
=
Note that the velocity components belong only to the convective terms, UD and U D being zero. Furthermore, since we are interested only in the conditions for the onset of convection, all convective terms in eqs 2-7 are assumed as perturbations of the main diffusion terms. On substituting eqs 7 into eqs 2-6,all terms including products of two perturbation terms (second-order terms) are assumed to be negligible. (f) We assume that diffusion proceeds without perturbations up to an initial time, and then we study the evolution of perturbations from this starting time assuming all diffusion terms are constant. Namely, a diffusion boundary develops with orders of magnitude lower than the perturbations,once a perturbation has appeared (quasi-stationary state assumption). With these assumptions eqs 2-7 become :u u’,, = div v = 0 (8)
(nequations)
(18)
P + t l = -g*kL4H*A
(19)
(PI + D)A = (CD),,
(20)
‘12:
k,,, = 4/27u4 N 1/657
-
(C’),+ u’(CD), = DV’C’
(17)
where I is the identity matrix. Note that the maximum value of k (eq 18) corresponds to a2
u = 0, , u = 0, C =
U’U~
A = [p(a + l)/(aL)]C
and substituting (14) and (15) into (12) and (13), we obtain
where Q >> rA,- for any t value. This condition corresponds, in the free-diffusion case, to y f-. (d) Concentration differences through the boundary (CtopCbtm) are small, so that diffusion coefficients and viscosity can be assumed as independent of composition (differential diffusion). (e) All quantities in eqs 2-6 can be expressed as a Taylor series truncated at the second term; the first term is due only to diffusion and the second one includes convection: p’pD+p’ c=cD+c’ P=PD+P’ u=u’ (7)
U:
(16)
+
u = 0, u = 0, C =Ctop
+ u: + u‘u: + u‘u’,, + PrX/pD= l p u ‘ + + U’U; + P>/pD = qV2u’+ g*H.C’
+ l)]
P = pL2/[*2(a2
k = a 2 / [ d ( a 2 1)3]
> 0: x-&y/a
(13)
(14) C’ = C exput) c o s ( ~ a x / Lcos ) ( u y / L ) (n equations) (15)
> 0 and any x) C = Cbttom (for y < 0 and any x) y-+Q,
v e + \k,*(c~),
(B) A particular solution of eqs 12 and 13 is q = exp(pr) sin ( u a x / L ) sin ( u y / L )
C = Ctop (for y
for t
(12)
(9) (10)
(11)
Where the state eq 2 has been substituted in eq 10,and the term g* = gpo/pD is assumed to be constant. Finally, the flow function, q,is substituted in eqs 8-1 1 (where qy= u and 9,= -u; qYx - qV = 0). Equation 9 is differentiated with respect to y and eq 10 with respect to x. Subtracting the
(21)
The system of equations (19)and (20)can be solved for P by inverting eq 20 and substituting into eq 19:
P + tl = - g * k L ‘ H ( ~ 1 + D)-’(cD),
(22)
This notation allows us to generalize the treatment to any number of components, n, searching for the conditions that give all solutions of eq 22 with the real part of P < 0. Case I: Two Components. In this case eq 22 reduces to
(P+ q ) ( P + D) = -g*kL4H(C,), and a square equation in P is obtained: P 2 + P(D + q ) + g*kL4(pD)y/po+ VD = O
(23) (24)
The first and second coefficients of eq 24 are positive. If the constant term (s+kL4(pD)/ p ” + qD)is also positive, both P roots are negative and the diffhon boundary is dynamically stable:
-g*kL4(PD)y/PoqD < 1
(26)
where g* is a negative term. Equation 26 shows that the boundary is always stable if the ~ top solution has a lower density than the bottom one [ ( P ~O
(33)
+ tl Tr (D) > -g*kL4(P~)y/P0 (34) For [-g*L3/9 Tr (D)] >> 1, as is the case of common liquid > bD)y < 0
[2fi(pD)zz/z
P4 + P3[7 + T r (D)] + P2[oT r (D) + D,
> O1
[ ( p D ) ~ / ~< f iO1
(33a) (34a)
Equations 33a and 34a correspond to the conditions for convection growing at the center of the diffusion boundary and for its growing at the borders, as found in previous p a p e r ~ ; ~ .see ~Jl Figure 2. In fact, by substituting eq 32 into eq 33a, one obtains H.D-I(CD), < 0, namely
+
g*kL4bD)y/p0l + P[qoD2+ ID1 + g*kL4(pD)y Tr (D)/po - (g*kL4/p0)x*(pD),] lDlR = 0 (40)
+
where vector X is defined by the expression
X = H.[D-H-']
(see Appendix)
(41)
We have now three inequalities to be satisfied for the roots P being all negative:
R>O 9 Tr (D)
(42)
+ D2 > - g*kL4(PD)y/po
tlD2 + ID1 > -g*kL4(pDly T r (D)/po
systems, expressions 33 and 34 simplify to (pD)zz/Y
Equation 22 becomes
(43)
+ (g*kL4/po)X*(p&
(44) In common liquids the term qD, ID1 is of the order of magnitude of 10-l2, so for [-g*L3/tl Tr (D)] >> 1, eqs 42-44 reduce to
+
(HID22 - H2DZl)(aCI/aY) + (H2Dll - HP12N3C2/dY)
