Ind. Eng. Chem. Res. 1995,34,3326-3335
3326
On Convection Induced by Molecular Diffusion Juan Camachot and Howard Brenner**$ Departament de Fisica (Fisica Estadistica), Universitat Aut6noma de Barcelona, 08193 Bellaterra, Catalonia, Spain, and Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Irreversible thermodynamic interpretations of experimental data involving molecular diffusion are usually based upon the assumption that in closed containers the local velocity vO of the so-called center of volume (relative to the fixed container walls) vanishes at every point of the system at every instant of time. This assumption greatly simplifies the interpretation of diffusion data, since by referring the convective-diffusive species flux vector to a local reference frame in which the convective component of the flux is VO and hence vanishes, the transport process occurs by molecular diffusion alone. In turn, this furnishes a straightforward, classical linear scheme for determining diffusion coefficients from experimental measurements of transient species concentrations in the closed d f i s i o n cell. Were this not the case, one would have to determine the transient hydrodynamic velocity field v induced by the diffusional process, simultaneous with the solution of the transient species concentration field-a highly nonlinear analysis owing to the coupling between these fields, similar to that occurring in natural convection problems. In this paper, we first give a physical argument proving that VO does indeed describe the volume flux in a mixture. Subsequently, we derive a simple expression-valid for isothermal incompressible binary mixtures-connecting the barycentric or mass-average velocity field v to the volume-average velocity field vO, i.e., relating the mass and volume flows. Later on, after showing that the generic kinematic argument found in the literature ‘proving‘that VO vanishes in closed containers is incompatible with hydrodynamics and even internally inconsistent, we expose a n alternative, more general development incorporating hydrodynamic effects, one that supplies a (necessary but insufficient) compatibility condition based upon the Navier-Stokes equation. This criterion permits one to identify a priori those classes of systems for which the possibility exists that vO = 0. These circumstances are shown to include all laterally unbounded one-dimensional transport processes as well as all unbounded three-dimensional Navier-Stokes flows for which inertial effects are small compared with viscous effects. Such physicochemically ‘low-Reynolds-number’ flows arise in the latter case in circumstances wherein the Schmidt number v/D (v = kinematic viscosity, D = molecular diffusivity) is large compared with unity, a situation that arises for most liquid-phase diffusion experiments but not for most gases.
1. Introduction Molecular diffusion in fluids is one of the fundamental transport processes addressed in the classic textbook Transport Phenomena by Bird, Stewart, and Lightfoot (1960) (hereaffer referred t o affectionately as BSL). Of signal importance is the choice of reference frame in defining the species fluxes, a fundamental issue which arises in the context of separating molecular diffusion from convection. Though BSL devote a good deal of attention to a number of such choices-including stationary, mass-average, and molar-average velocity reference frames-the use of volume-average velocities1 is mentioned only once, namely, in their Question 16.G appearing on page 518 of Transport Phenomena. Yet, in experimentally determining molecular diffisivities in closed diffision cells, the vanishing of the so-called volume-average velocity vector (at each point of the fluid) greatly simplifies the interpretation of experimental diffusion data gleaned in such cells. This simplification arises from the fact that with pi the local mass concentration of species i, the basic transport equation applying at each point of the fluid continuum within the cell is simply [cf. eq 281
* Author to whom correspondence should be addressed. Universitat Aut6noma de Barcelona. Massachusetts Institute of Technology.
aPi - = V*(DVp,) at at least in binary systems (i = 1,2) obeying both Fick’s law [cf. eq 121 and the law of additive volumes. This equation applies even in the presence of mass-average convection (v f O), the latter arising in circumstances where mass density differences exist between the pure solute and solvent. While the concept of a volumeaverage velocity has received much attention in the classic literature of irreversible thermodynamics (deGroot and Mazur, 1962; Fitts, 1962; Katchalsky and Curran, 1965)-these being major textbooks which appeared more or less contemporaneously with the publication of BSL-the emphasis therein and in more contemporary publications in the field is largely on the (continuum-)mechanical issues involved rather than on the kinematics of the subject. Given the large-scale neglect of fundamental discussions of volume-average velocities in the chemical-engineeringliterature, particularly as embodied in BSL, we use the present opportunity to bring several controversial aspects of the subject to the attention of those readers who, together with us, today celebrate the occasion of 35 years of Transport Phenomena. 1.1. Motivation. By way of entree into the class of problems addressed in this paper, consider a cylindrical container possessing a removable partition initially
0888-5885/95/2634-3326$09.00/00 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3327 separating the container into two generally unequal portions (see the Appendix). The right-hand side contains a pure liquid solvent (assumed incompressible) and the left-hand side contains a dispersion of rigid colloidal solute particles uniformly distributed throughout the same solvent, forming a homogeneous colloidal dispersion. Gravity effects, if present, are assumed negligible. If the mass density of the individual colloidal particles is assumed greater than the mass density of the solvent, the center of mass of this system will clearly be initially situated somewhere other than at the geometric center of the system. After the partition is removed and the colloidal particles and solvent molecules are allowed t o interdiffuse, the system will ultimately become homogeneous, with the particles uniformly distributed throughout. The center of mass of this equilibrium system will now be situated a t the geometric center of the system. This movement of the center of mass of the system induced by the diffusion process represents a hydrodynamic phenomenon that can only have been accomplished by the existence, during the process, of a mass-average, hydrodynamic, barycentric velocity field v. This hydrodynamic motion may be regarded as caused by the original solute concentration gradient. As such, it occurs concurrently with the molecular diffision flux, neither occurring without the other. (Of course, had the mass densities of the two species been equal and the law of additive volumes applicable over the pertinent solute concentration range, only molecular diffusion would have occurred.) Thus, the total solute transport process consists in part of convection and in part of molecular diffusion. By definition, the latter is measured relative t o the mass-average velocity. 1.2. Background of the Problem. As the preceding example clearly shows, molecular diffision in closed, isothermal, incompressible, binary systems composed of species possessing unequal mass densities is necessarily accompanied by a comparable mass-average (barycentric) velocity v. Explicitly, any gradients in the total local mass density p couple with the continuity equation
3at2 + Wpv) = 0
(1)
to produce a local mass flow v. Transport problems of this genre, involving molecular diffision in closed systems accompanied by convective flow resulting from the same concentration gradient, are classic. They date back at least to Maxwell's (1860a,b) observation that "the motions of a system ... are of two kinds: one, a general motion of translation of the whole system, which may be called the motion in mass; and the other a motion of agitation, or molecular motion ...". As a result of its importance in physical applications, in particular to the experimental determination of molecular diffusion coefficients, the subject of diffusion-induced convection has received much attention over the years. Thus, despite much early confusion (Onsager, 1945), especially with regard to multicomponent diffusion phenomena, the macroscopic theory of diffusion is usually regarded as having been fully resolved by the contributions of irreversible thermodynamics (deGroot and Mazur, 1962; Fitts, 1962; Katchalsky and Curran, 19651, stemming primarily from Onsager's (1945) contributions. However, as will be seen below, there still exist some obscure points in the classical theory of diffusion. In
particular, the classical argument used to demonstrate that the volume-average velocity vanishes in closed systems for approximately incompressible mixtures contains some inconsistencies. Let us remark, once again, that the vanishing of v" is essential in the theory of diffision in closed cells since, as mentioned in the Abstract and first paragraph of the Introduction, it reduces the mass-transport process to a simple molecular diffision process, without any apparent convective component. The argument relies on two assumptions: (i) that the system exists, at least approximately, in a state of local mechanical equilibrium, i.e., sans acceleration; (ii) that the traceless symmetric portion of the velocity-gradient tensor vanishes identically: 1/2[Vv (Vv)+]- U3IV.v = 0 (deGroot and Mazur, 1962, p 253). The first assumption is in disagreement with the fact, exemplified in Section 1.1, that for unequal species densities an acceleration must accompany the diffisional process. The second might be justified (although no justification is usually cited in the literature) as being a consequence of the first, jointly with the NavierStokes equations, in order that the acceleration vanish; however, the latter is not fully justified since the Navier-Stokes equations also provides an acceleration term arising from the volumetric viscosity contribution (since V*vf 0). In any event, one could ultimately take both assumptions to be working hypotheses subject to a posteriori verification. The next step in the classical development (deGroot and Mazur, 1962, p 255) is to combine conditions (i) and (ii)so as to demonstrate that V x v = 0, a result which is subsequently used to prove that some reference velocity (such as the mean molar velocity or the mean volume velocity v") vanishes identically. The problem arises when it is realized that those assumptions not only lead to the existence of an irrotational laminar flow but that they also determine the analytic form of the barycentric field-namely, v = a bx, with a(t)and b(t) being some arbitrary functions of time, and x being the position vector (deGroot and Mazur, 1962, p 254), an aspect of the analysis which is commonly glossed over. This flow field is obviously extremely restrictive and will rarely satisfy physically imposed boundary conditions, especially the no-slip boundary condition on container walls; furthermore, since this instantaneous velocity field is purely radial, it will generally prove unable to satisfy prescribed initial conditions! Consequently, the classical theory cannot be regarded as satisfactory from the viewpoint of hydrodynamics. In fact, it is not even self-consistent since, as will be seen in Section 2.2, knowledge of the velocity field v = a bx furnishes the total mass density field p itself (aside from a function of time)! Given these facts, the central issue of the present paper is to address the problem of molecular diffusion self-consistently, without using the classical assumptions and without ignoring the underlying hydrodynamics. While it is indeed true that convection induced by concentration gradients is often small, thereby justifying neglect of hydrodynamic considerations, this is not universally so. Witness, for example, Marangoni phenomena (Slattery, 1990; Edwards et al., 1991) occurring in surfactant-containing systems possessing interfaces, where surfactant concentration gradients along the interface can give rise to relatively enormous interfacial velocities. Here, the underlying hydrodynamics cannot be ignored.2
+
+
+
3328 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 Further motivation for a better understanding of the hydrodynamic aspects of molecular diffusion arises in connection with possible microgravity experiments. On Earth, in the presence of gravity fields, density inhomogeneities arising from concentration gradients give rise to a natural-convection flow, one whose magnitude would overwhelm the much smaller hydrodynamic velocity fields arising from molecular diffusion in the absence of gravity. In the effective absence of gravity, such as is approximately simulated in orbiting satelliteswhere centrifugal forces nullify the effects of gravity-the possibility exists of actually observing this diffusiondriven hydrodynamic flow experimentally. Owing t o its relative smallness, no real possibility existed of actually observing this purely ‘diffusional’velocity field until the advent of orbiting satellites. In this connection, it is obviously of some importance t o acquire a more complete understanding of the nature of such hydrodynamic fields, free of the simplifying assumptions of irreversible thermodynamics. 1.3. Preview. The present paper is organized as follows. Section 2 begins by discussing the physical meaning of the volume-average velocity VO. Although the latter is usually interpreted as describing the volume flow a t a point of the mixture, no proof of this statement appears t o exist in the literature. On the contrary, the implicit argument widely used in the literature justifying this interpretation will be shown to be incorrect, and an alternative and rigorous proof given. Once the physical meaning of +‘ is thereby M y established, we derive a relationship between v and VO, valid for general isothermal incompressible binary mixtures. Finally, Section 2 ends with the evaluation of the respective divergences of the fields v and VO. This calculation shows that (i) the barycentric field in an inhomogeneous binary solution is not solenoidal even i f the mixture is incompressible;(ii) V*VO does not generally vanish unless the mixture is ideal, i.e., obeys the ‘law’ of additive volumes. The latter supplies the first of several conditions that must be met in order that the field VO vanishes in closed containers; for if V-VO f 0, then VO obviously cannot vanish. Additional criteria required for the condition VO = 0 to occur in closed containers are broadly analyzed in Section 3. After observing in Section 1.2 that the usual arguments underlying the vanishing of the volume-average velocity in closed containers are not satisfactory, Section 3 offers an explicit criterion for this to occur, furnishing a compatibility condition that does not lead to the inconsistencies of previous developments, while explicitly taking hydrodynamics into account. This is done in two steps. First, by taking advantage of a relation between the respective mass- and volume-average velocities v and VO derived in Section 2, we obtain a Navier-Stokes-like equation governingthe velocity field VO and a related ‘fictitious’ pressure field P O . Subsequently, this hydrodynamic equation is used to obtain a necessary (but insufficient) condition for +‘ to vanish, whereupon several interesting cases wherein this compatibility condition holds are studied. Section 4 is devoted to conclusions. Finally, an appendix provides an example illustrating the use of the hydrodynamic equations obtained in Section 3 for the evaluation of the barycentric velocity and the pressure fields generated by diffusion. This example deals with a one-dimensional diffusion problem, for which configuration the compatibility condition is automatically fulfilled. Related situations for which satisfaction of this
compatibility condition does not guarantee the vanishing of +‘ are also discussed. 2. Volume-Average Velocity vO 2.1. Physical Meaning of vO. Irreversible thermodynamics introduce a volume-average velocity +‘ defined as (deGroot and Mazur, 1962)
v”
cpivivi
(2)
i
pi being the mass density of species i (i = 1, 2, ..., N), namely, the mass of component i per unit volume of the mixture, and vi is the partial specific volume of species i; vi is the velocity vector of species i relative to the fixed container walls, defined as the velocity appearing in the expression
ni= pivi
(3)
with ni being the mass flux vector of i. In the above, we have by definition that CipiYi = 1. The velocity field VO defined in eq 2 is commonly interpreted in the literature as a volume-average velocity, somehow the velocity at which a center of volume moves, and consequently it is regarded as describing the volume flow. As far as we are aware, however, formal justification for this widely-accepted interpretation is not available in the literature; rather, it is implicitly assumed that the quantity (4)
with i
physically represents the volume fraction of species i existing a t the point x a t time t , i.e., c$i is the volume occupied by species i per unit volume of mixture. Therefore, eq 2 may be written as
v” = &Vi i
so that VO is interpreted as the volume-average velocity in the same sense that the barycentric velocity, v = Ciwivi, is a mass-average velocity, wherein wi pi/p (with Cjwi = 1)is the mass fraction of species i, in which p = Cipi is the total mass density. The weak point in this physical argument is that the quantity c$i defined in eq 4 is not the volume fraction of species i, a fact which may be seen as follows: Regarding & as the volume fraction of species i is equivalent to regarding the partial specific volume vi as the volume occupied by a unit mass of species i in the solution; that this is not the case can be seen from the same definition of vi, namely (j=1,2
,..., i - l , i + l , ...,N)
with V as the volume of the system and mk as the mass of species k in solution. For, despite the fact that by Euler’s theorem the specific volume Y of the solution is given by the expression
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3329
v =ewivi
(7)
use eqs 2 and 3 to obtain
1
one is nevertheless not permitted to interpret V i as the specific volume of the individual species i. In effect, there exists no objective way of ascribing a portion of the total volume to a particular species in a non-ideal3 solution. This apparently subtle fact becomes patently obvious when one recognizes that negatives values of the partial specific volume vi are observed to occur in experiments (Home, 1971;Shakhashiri, 1989). At this point, it is natural to ask whether vO, as defined by eq 2, bears any physical interpretation. Despite the wrong argument reviewed above, in the following we now prove that vO does indeed describe the volume flow in the mixture. To do so, consider first the volume change undergone by a multicomponent solution when one adds or withdraws differential masses dmi of the several species existing in solution under isothermal isobaric conditions. From eq 6,this volume variation is simply
dV = c v i dm,
4 =v
+CVji
(11)
i
The constitutive equation governing j i for binary systems under isothermal conditions is (Bird et al., 1960; Cussler, 1976,1984)
ji = -DpVwi
(i = 1 , 2 )
(12)
D being the molecular diffusion coefficient,and wi being the mass fraction of species i defined above. Consequently V" = v
-DpcviVwi i
Now, by identity
vjvwi = V(ViWi)- wivvi Upon summing the latter and using the relation
1
Now, let us analyze what happens in a flowing mixture. For a given surface element dS fixed in space, the mass flow of species i is obviously given by dm, = d S * q
dS*pivi
(9)
1 'Cwivi = i P derived from eqs 4 and 5, one obtains
As regards the changes of volume incurred on both sides
1 c v i v w i = v- - c w i v v i
of the surface per unit time, one concludes after introducing eq 9 into eq 8 that-on, say, the left-hand side of dS-the volume has increased (or decreased) by an amount
Upon applying Denbigh's (1955)generic partial-specific formula t o the case where the extensive quantity is chosen to be the volume, we obtain
i
P
(14)
i
dV = d S * ( c p i v i v i= ) dS*vo i
whereas the opposite happens on the right-hand side of dS. Therefore, the quantity vO can actually be interpreted as describing the volume flow in the mixture, as is usually assumed to be the case. However, its interpretation as a center of volume velocity does not generally hold, as we have shown. This notion would require that we assign a volume to each and every species i in the solution. But this is not possible as eqs 6 and 7 put into evidence. It is only for ideal solutions, where the partial specific volume vi of species i coincides with the specific volume vp, say, of pure i, that such an assignation may possibly be accepted. As such, use of the name uolume-averagevelocity is physically misleading. 2.2. Relation between v and vO. For simplicity in what follows, we here abandon the full multicomponent approach of the preceding sections to focus exclusively on binary systems (i = 1,2). In particular, the present section furnishes a relation between the velocity fields respectively describing the mass and volume fluxes in isothermal incompressible binary mixtures (solutions). To accomplish this, recall that the mass flux vector ni in eq 3 can be written as a sum of a convective flux p i v plus a molecular diffusion flux j i = pi(Vi - v),SO that the mass flux (eq 3) can be expressed as (Bird et al., 1960)
ni= p i v
+ ji
(10)
in which ZJi = 0. Multiply eq 10 by vi, sum over i, and
As v = l l p , and since df = dx-vf for any function fix) and differential displacement vector dx, it immediately follows for the isothermal conditions t o which eq 12 applies that
'("")
c w i v v i = - - - VP i p 2 ap T , ~ ,
Following introduction of the latter into eq 14, eq 13 thereby becomes
vo = v
+ D[v In p - $$]T,wVp]
(16)
For a system wherein either the mechanical equilibrium assumption of irreversible thermodynamics (dp = 0 ) prevails, or else the system is incompressible, so that (aplap)T,,i= 0, one finds that
Consequently, in such circumstances eq 16 reduces to
v = v"- D V l n p
(17)
The preceding equation relating the mass and the volume flows is thus valid for general (ideal and nonideal) binary mixtures under the restrictions of
3330 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
isothermal, incompressible (or isobaric) conditions. Let us remark, however, that had we not proved physically that vO does indeed describe the volume flux in the mixture, the above equation would have to be regarded as purely formal, being merely a consequence of the definition of VO and of the Fick's law constitutive equation (eq 12)for binary diffision under isothermal conditions. Equation 17 shows when the total density p is locally nonuniform, the respective velocities at which mass and volume are transported differ by a term proportional to the molecular diffisivity. On the other hand, in situations where vO = 0 [as is usually claimed to be the case in closed containers for (approximately) ideal solutions (deGroot and Mazur, 1962)1,one would have a barycentric fluid motion
v = -DV In p
(18)
arising solely from the molecular diffision. In such circumstances, one corroborates, as expected, that mass travels in the direction of diminishing density, as it must in order for the closed system to ultimately achieve equilibrium. Observe, as pointed out in Section 1.2, that the classical argument 'proving' that vO = 0 also furnishes the barycentric field, v = a bx. Its introduction into the latter equation requires that the density field be of the form
+
(where dldt = afat v*V denotes the substantial time derivative), and the mass conservation equation for the solute
d In p = - p c ( v i dwi
+ wi dv,) =
(21)
one obtains
Vv = (v2 - vl)Vj, = (vl
- v,)V*(pDVw,)
(22)
Accordingly, the barycentric field for inhomogeneous binary mixtures is not generally solenoidal even if the mixture is incompressible. It is only solenoidal when the mixture is ideal [i.e., obeys the law of additive volumes-see eq 291 and both species possess the same mass densities, so that VI = v2 for all w1. In order to similarly calculate VVO, substitute eqs 19 and 12 into eq 17 to obtain
v" = v
+ DV In p = v + (v, - v2)j,
(23)
Form the divergence of the latter equation and use eq 22 to obtain Vv" = jl.V(vl - v2)
+
with g(t) as a function of time. Accordingly, if the classical argument were correct, the problem of binary diffision in closed containers would trivially reduce t o determining the values of the time-dependent functions a@), b(t), and g(t)! 2.3. Divergences of v and vO. The present subsection serves to evaluate the respective divergences of the barycentric and volume-average velocity fields for isothermal incompressible binary mixtures. In particular, v is shown to be nonsolenoidal, in contrast to what is normally assumed for incompressible systems. Moreover, @ is proved to be solenoidal for ideal solutions but not generally for nonideal ones. This fact establishes the first of several conditions that must be satisfied in order that vO vanishes in closed containers. In particular, whenever V.vO f 0, the field vO obviously cannot vanish at all points. In order to evaluate VT, observe from eqs 7 and 15 that
+ V-j, = 0
dWl dt
p--
(24)
That the right-hand-side of the preceding equation does not vanish in general can be seen from eq 15,which for the isothermal incompressible systems under study yields w,vvl
+
W2VV2
=0
Consequently, eq 24 can be rewritten as
+
V*V" = jl*(l w1/w2)Vv1
(25)
Since vi is, in general, a function of WI, p , and T , V-vO does not generally vanish. For ideal mixtures, however, the partial specific volumes vi are constant and equal to the specific volumes of pure i , namely, vio. In this case, eq 25 shows VO t o be solenoidal, i.e. VT" = 0
(26)
[Inasmuch as vO cannot vanish unless V*VO does, a necessary condition for VO = 0 is that V*VO = 0. And the latter can occur if, and only if, the binary solution obeys the law of additive volumes over the entire range of compositions encountered in the diffusion experiment. That the law of additive volumes is necessary may be seen as follows. From eqs 25 and 12,we have that
i
V-V" = -Dp(l
+w l l ~ 2 ) V ~ l ~ V ~ ,
Consequently, for incompressible isothermal systems
However, for the binary, isothermal, incompressible systems to which eq 25 applies, we have that V I YI(WI), so that
d In p = -p(vl - v2) dw,
Vv, = (dv,/dw,)Vw,
(19)
where use has been made of the binary identity dwl = -dw2. The species denoted by the index i = 1 will be identified as the solute. By combining the latter with the continuity equation (eq l),written in the form
dlnp + v.v = 0 dt
whence VT" = -Dp(l
+ w ~ / w ~ ) ( V Wdvlldwl ~)~
Clearly, the necessary and sufficient condition that
(20)
V-VO vanishes over the composition range of interest
encountered in the experiments is that dvlfdw1= 0 over
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3331
this range; that is, v1 must be independent of composition. Consequently, ~1 = constant = CI, say, where C1 is a constant, independent of composition. By interchanging the arbitrary indices 1 and 2 distinguishing the two species, it is clear that the condition V*v" = 0 equally requires that dvddwz = 0; in turn, this necessitates that v2 = constant = CZ,say, over the indicated composition range, where C2 is independent of composition. The circumstances for which VI = C1 and v2 = C2, where C1 and C2 are constants, independent of composition, is equivalent t o the law of additive volumes. As a special case of the latter, in circumstances where the composition range encountered during the diffusion experiments includes pure "1",it is required that C1 = v l o , where v1O is the specific volume of the pure species "1". This is the situation that prevails in the case of a solution that is ideal with respect to species "1". (Likewise, if the experimental composition range also includes pure species "2", then C2 = v2O, where v2O is the specific volume of the pure species "2";otherwise, one must distinguish between a solution that is ideal with respect to species "1"but not species "2", as in the classical distinction between Raoult's and Henry's laws for binary solutions.) In any event, additivity of volumes, though necessary, is a clearly insufficient condition for v" to vanish, since a vector field may be solenoidal without being zero everywhere.] For subsequent use in discussing the compatibility condition governing the vanishing of v", the continuity equation (eq 1) may be expressed in terms of v" rather than v. This is simply effected by use of eqs 17 and 26 to obtain
?e + v"*Vp= VQVp) at
api
at
+
Y = p-l = Y,o(1 - awl)
(29) where vi = vio = constant (i = 1,2), in which vio is the specific volume of pure i and a = 1 - vl0/vzois an algebraically-signed scalar constant parameterizing the species mass density difference between the pure solute and solvent. The no-slip hydrodynamic boundary condition on the container boundaries requires that v = 0 on aV, where aV denotes the external boundaries of the system. Hence, from eq 17, the condition vO = 0 requires that Vp = 0 on aV. In turn, from eq 29 this requires that Vwl = 0 on aV. Together with w 1 + w2 = 1, this in turn requires that Vwi = 0 on aV (i = 1, 2). This condition necessitates that Wi = constant on aV. But in the experiment described in Section 1.1, and certainly along the side walls, this condition is clearly inconsistent with the initial conditions, in combination with the fact that the system ultimately becomes homogeneous, so that eventually wi = constant for all x. As such, v" cannot vanish along the side walls and hence cannot vanish everywhere, as originally claimed! 3.2. Navier-Stokes-like Equation Governing (v", pol. The Navier-Stokes equations are
_dv + vp = VV2V + (F 1 + VU)V(v.V) dt
(27)
valid only for isothermal, incompressible, ideal binary solutions. One can also show that a similar expression holds for each of the individual species densities pi; explicitly, pi (i = 1, 2) obeys the equation
- v0*Vpi= V*(DVpi)
counterexample that v" cannot generally vanish in a closed container, even for an ideal solution. Consider the hypothetical experiment described in Section 1.1, in which the mixture is assumed to be ideal and hence to obey the law of additive volumes over the entire composition range; explicitly, from eq 7
(28)
It is in connection with this latter species transport equation or-more precisely-its precursor
that v" plays a fundamental role in diffusion theory. Here, jiO= pi(v - v") -DVpi is the diffision-flux vector relative to the volume-average velocity vO in the expression (deGroot and Mazur, 1962; Fitts, 1962) ni = PivO + jiO.For circumstances where vO = 0, eq 28 reduces to a pure molecular diffusion equation, which can be solved for pi without knowledge of v. Therein lies the importance of the vanishing of the volume-average velocity in interpreting experimental diffusion phenomena. The question of the vanishing of v" in closed systems is a central issue in this paper. It is addressed in the next section, where a criterion is developed in the form of a compatibility condition governing the possible vanishing of v" for ideal mixtures. 3. Compatibility Condition Governing the Vanishing of VO 3.1. Counterexample to the Vanishing of VO in Closed Containers. This section demonstrates by
(30)
in which, for simplicity, the shear and dilatational viscosity coefficients and v,,, respectively, have been taken to be constants, independent of position. Note that we have retained the term V*vto allow for the possibility that the fluid may be 'compressible'. With use of eq 17, together with the assumption that D too is independent of position (as well as of time), and the decomposition
P =Po + Pv +PI + PI0 (31) of the thermodynamic pressure p, we find that (vO, p") satisfies the following Navier-Stokes-like and 'incompressible' continuity equations for our ideal, binary solution: = -Vp"
+ l;lV2v0+ F
(32)
and
v-v"= 0 (33) In eq 31, we have defined the individual pressure contributions: pv = -D(?4 pI = D
+ v,)V2
~ In~p v
p: = -DvO*Vp Additionally, in eq 32
+
F = F, F," where, with I the dyadic idemfactor
(34)
P
~
(35) (36)
(37)
3332 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
F, = D2(IV2p - VVp)*VIn p F," = D[pva*VVIn p
(38)
+ Vp-Vv" + (va*Vp)VIn pl (39)
Those terms in p and F bearing the subscripts I and V derive from the respective inertial (pdvldt) and viscous (11 and vu)terms in the origin4 Navier-Stokes equation (eq 30). The further subdivision of these terms into those with and without the superscript 0 , respectively, represent those contributions that vanish identically when vO = 0 and those that do not. Equation 32 has been written in the form of a Navier-Stokes-like equation governing the fields (V, pa), with the vector field F regarded as a fictitious external volumetric force density. As indicated by the appearance of the subscript I in eq 37, this fictitious force is entirely inertial in origin. Of course, if any portion of either FI or Fp ultimately shows itself to be expressible as the gradient of a scalar function, that portion can then be assigned to the corresponding 'pressure' field,pI orpf, respectively, in eq 31. We shall return to this point subsequently. For specified initial and boundary conditions, one can, in principle, solve the fictitious Navier-Stokes and continuity eqs 32 and 33,together with eq 27 (with D = constant):
e!!at + va*Vp= DV2p
(40)
for the five scalar unknowns vO, p a , and p. Following this, the respective binary mass fractions w1 and wp can be obtained trivially from eq 29. Finally, the barycentric velocity v and thermodynamic pressure field p can be obtained from eqs 17 and 31, respectively, thereby completing the solution of the original system of eqs 20, 21, 30, and 12 plus 29 together with the prescribed boundary and initial conditions imposed upon v, p , p, and wi ( i = 1, 2). 3.3. Compatibility Condition for the Vanishing of vO. As discussed in Section 1.2, diffusion theory based on irreversible thermodynamics claims in present circumstances that vO = 0 in closed containers. This assumption will be examined in this section. More generally, in order that vO vanish in the present, nonequilibrium, hydrodynamic case, eq 32 requires that -Vp"
+F =0
(41)
Equivalently, upon forming the curl of the latter equation and using eq 37,we find that the vanishing of vO requires that VxFI=O
coordinate, one finds from eq 38 that with i, a constant unit vector in the x direction
so that the compatibility condition (eq 42) is trivially fulfilled. In such circumstances, one also finds that the pressure field p(x,t)is p(x,t>= D2[p - (5 32 0
+D
h 2
+At)
(44)
with At) as a function to be determined from the initial conditions imposed upon the density and pressure fields. The generally spatially nonuniform nature of this pressure field contrasts with the irreversible thermodynamic case, where the mechanical equilibrium assumption requires that p = constant, independently of x . A detailed example of a one-dimensional problem, including prescribed boundary and initial conditions, for which vO = 0 is presented in the Appendix. (ii) The case of negligible inertial effects. In situations for which inertial effects are insensible compared with viscous effects, all terms in eqs 31 and 37 bearing the 'inertial' subscript I may be suppressed compared with those terms bearing the 'viscous' subscript V. In such circumstances, not only is the compatibility condition (eq 42) trivially satisfied, since FI = 0, but more generally we have that
F=O
(45)
in eq 32 owing to the absence of sensible inertial effects. Equation 41 thus requires that
for the case where vO = 0, when the pressure field p becomes
This nonuniform pressure field should be contrasted with the uniform pressure required in irreversible thermodynamics, where viscous effects are eliminated a priori (and external forces are absent). From eqs 34-36, the criterion that inertial effects be small compared with viscous effects in eq 32 requires that IFIIIIVPVI e 1. Upon using eqs 34 and 38, this criterion requires satisfaction of the inequality
(42)
with FI given by eq 38. The latter constitutes a compatibility condition imposed upon the density field p. This compatibility condition constitutes a necessary but insufficient condition for vanishing of VO. One must also consider the prescribed boundary and initial conditions imposed upon v,p , p, and w1. An example of the necessity for explicitly considering the boundary conditions appears in the penultimate paragraph of the accompanying Appendix. There exist at least two obvious circumstances for which this compatibility condition is satisfied a priori: (i) One-dimensional transport processes. For unidirectional, one-dimensional, convective-diffusion processes, with p = p(x,t), say, where x is a Cartesian
Dp
m=(v,v,)
o
P10 - P2O -
for all x and t; that is, the density is always positive and of order unity, an obviously correct physical inference. From eq A6, the velocity field is found t o be u(x,t)= P
(-414) This field is readily seen to satisfy the velocity boundary conditions (eqs A8 and A10) as well as the initial conditions (eqs A7 and A12), modulo an initial singularity at the original barrier position, x = h. However, this mathematical singularity is without physical consequence owing to the fact that the small fluid region h ( l - E ) .C x < h(1 E) (0 .c E > h enabled us t o ignore any boundary conditions imposed at the cylinder walls R = R,. In contrast, had one been required t o explicitly account for the no-slip velocity boundary conditions v = 0 on the tube wall, eq 17 would have necessitated that, with u0 = lVOl the magnitude of VO
The solution of eqs A2-A5 for the density field p(x,t) is easily found to be (Carslaw and Jaeger, 1959)
P-PZ
h 2
(-49)
Finally, the implicit requirement that the fluid be a t rest at infinity necessitates that
0
&I)."
[ (43 x0 +D-
p(x,t)=po+D2p-
For a typical liquid diffusivity of the order of D = cm2 s-l, and for a fractional density difference (p1O p2O)/p of order unity, one finds from eq A14 that at x = h (the original barrier position) u zz cm s-l, 1s after removing the barrier. As expected, such molecular diffusion-generated fluid motions are inherently small.4 Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960;Chapters 16 and 18. Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Oxford: London, 1959;p 54. Crank, J. The Mathematics of Diffusion,2nd ed.; Oxford: London, 1975;pp 219-256. Cussler, E. L. Multicomponent Diffusion; Elsevier: Amsterdam, The Netherlands, 1976. Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge: London, 1984. De Groot, S. R.; Mazur, P. Non-Equilibrium Thermoydnamics; North-Holland: Amsterdam, The Netherlands, 1962; pp 4344, 239-241, 250-256. Denbigh, K. The Principles of Chemical Equilibrium; Cambridge: London, 1955;p 99. Edwards, D. A.; Brenner, H.; Wasan, D. T. Interfacial Transport Processes and Rheology; Butterworth-Heineman: Boston, MA, 1991. Fitts, D. D. Nonequilibrium Thermodynamics; McGraw-Hill: New York, 1962; pp 5-6, 21-26, 35, 43, 78-82, 84,90-91, 121122. Horne, R.A., Ed. Water and Aqueous Solutions; Wiley: New York, 1971.
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3336 Jost, W. Diffusion in Solids, Liquids & Gases, 3rd ed.; Academic: New York, 1960; p 427. Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard: Cambridge, MA, 1965. Mavrovouniotis, G. M.; Brenner, H. A micromechanical inveatigation of interfacial transport processes: 1. Interfacial conservation equations. Philos. Trans. R. Soc. London 1993,A346,165207. Mavrovouniotis, G. M.; Brenner, H.; Edwards, D. A.; Ting, L. A Micromechanical Investigation of Interfacial Transport Processes: 2. Interfacial Constitutive Equations. Philos. Trans. R.SOC. London 1993, A345,209-228. Maxwell, J. C. Illustrations of the Dynamical Theory of Gases. Part 1. On the Motions and Collisions of Perfectly Elastic Spheres. Philos. Mag. 1860a, 19, 19. [Reprinted in The Scientific Papers of James Clerk Maxwell, Vol. 1; Niven, W. D., Ed.; Cambridge: London, 1890; pp 377-391.1 Maxwell, J. C. Illustrations of the Dynamical Theory of Gases. Part 2. On the Process of Diffusion of Two or More Kinds of Moving Particles Among One Another. Philos. Mag. l w b , 20, 21. [Reprinted in The Scientifi Papers of James Clerk Maxwell, Vol. 1; Niven, W. D., Ed.; Cambridge: London, 1890; pp 392409.1 Modell, M.; Reid, R. C. Thermodynamics and its Applications; Prentice-Hall: Englewood Cliffs, NJ, 1983; p 183. Onsager, L. Theories and Problems of Liquid Diffision. Ann. Trans. N.Y. Acad. Sci. 1946,46, 241-265. Serrin, J. Mathematical Principles of Classical Fluid Mechanics. In Handbuch der Physik, Vol. 811 Fluid Dynamics I; Fliigge, S., Ed.; Springer-Verlag: Berlin; 1959; pp 164-165. Shakhashiri, B. Z. Chemical Demonstrations; University of Wisconsin Press: Madison, WI, 1989; Vol. 3, pp 276-279. Slattery, J. C. Momentum, Energy, and Mass Transfer in Continua; McGraw-Hill: New York, 1972; pp 459 and 461. Slattery, J. C. Interfacial Transport Phenomena; Springer-Verlag: Berlin, 1990. Wall, F. T. Chemical Thermodynamics, 2nd ed.; Freeman: San Francisco, 1965; p 186.
Footnotes (1) Denoted by V. in BSL and herein by V.
(2) To bring the analysis of interfacial phenomena within the purview of continuum hydrodynamics without the added complication of singular surfaces, we may envision the interfacial domain as a diffise, continuous region (Mavrovouniotis and Brenner, 1993; Mavrovouniotis et al., 19931, to which classical continuum arguments should apply. Clearly, the steep gradients encountered in such systems make the quasi-equilibrium arguments of irreversible thermodynamics suspect. (3) In an ideal solution obeying the law of additive volumes (cf. eq 291, it would be natural to define the volume ascribed to a particular component i as being that volume which the mass of i currently present in solution would occupy were it in a pure state at the same total pressure and temperature as that of the solution. However, even here-in this ideal solution-an ambiguity will exist in the proposed definition for circumstances wherein the physical state of the pure species i a t these conditions is such that it is a solid (whereas the hypothetical ‘state’ of the dissolved species, being mobile, may more closely approximate that of a liquid). And if the pure liquid and solid possess different mass densities, which state should be chosen for the definition? Indeed, one can consider the amusing situation where the solution temperature corresponds precisely to the melting point of the solid, so that both physical states can simultaneously coexist (and the species is relatively incompressible in both states, so that the question of pressure is irrelevant). Indeed, this “change of phase” may be the source of negative values of the partial specific volume vi, sometimes observed (Home, 1971; Shakhashiri, 1989). Thus, the lack of objectivity in making the assignation of volume may extend to ideal solutions as well. (4) Except possibly in the case of Marangoni-driven interfacial flows (Mavrouniotis et al., 1993; Edwards et al., 1991) as discussed in footnote 2.
Received for review January 3, 1995 Accepted August 18, 1995@ IE950009N
* Abstract published in Advance ACS Abstracts, September
15, 1995.
