Ind. Eng. Chem. Res. 1991, 30, 873-881 Fitzer, E.; Schaefer, W. The Effect of Crosslinking on the Formation of Glasslike Carbons from Thermosetting Resins. Carbon 1970, 8, 353-364. Foley, H. C. Carbon Molecular Sieves: Properties and Application in Perspective; Perspectives in Molecular Sieve Science; Flank, W. H., Whyte, T. E., Jr., Eds.; American Chemical Society: Washington, DC, 1988; pp 335-360. Franklin, R. E. Study of the Fine Structure of Carbonaceous Solids by Measurements of True and Apparent Densities. I. Coals. Trans. Faraday SOC.1949a, 45, 274. Franklin, R. E. Fine Structure of Carbonaceous Solids by Measurements of True and Apparent Densities. 11. Carbonized Coals. Trans. Faraday SOC.1949b,45,668. Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982; pp 68, 154-160. Horvath, G.; Kawazoe, K. Method for the Calculation of Effective Pore Size Distribution in Molecular Sieve Carbon. J . Chem. Eng. Jpn. 1983, 16, 470-475. Lafyatis, D. S.; Foley, H. C. Molecular Modelling of the Shape Selectivity for the Fischer-Tropsch Reaction Using a Tri-Functional Catalyst. Chem. Eng. Sci. 1990, 45, 2567-2574. Lamond, T. G.;Marsh, H. The Surface Properties of Carbon41 The Effect of Capillary Condensation at Low Relative Pressures Upon the Determination of Surface Area. Carbon 1963a, 1, 281-292. Lamond, T. G.; Marsh, H. The Surface Properties of Carbon-111: The Process of Activation of Carbons. Carbon 1963b, 1,293-307.
873
Lamond, T. G.; Metcalfe, J. E., III; Walker, P. L., Jr. 6A Molecular Sieve Properties of Saran-Type Carbons. Carbon 1965,3,59-63. Moreno-Castilla, C.; Mahajan, 0. P.; Walker, P. L., Jr.; Jung, H. J.; Vannice, M. A. Carbon as a Support for Catalysta-111. Carbon 1980, 18, 271-276. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984; p 189. Schmitt, J. L., Jr.; Walker, P. L., Jr. Carbon Molecular Sieve Supports for Metal Catalysta-I. Preparation of the System Platinum Supported on Polyfurfuryl Alcohol Carbon. Carbon 1971, 9, 791-796. Schmitt, J. L., Jr.; Walker, P. L., Jr. Carbon Molecular Sieve Supports for Metal Catalysts-11. Selective Hydrogenation of Hydrocarbons over Platinum Supported on Polyfurfuryl Alcohol Carbon. Carbon 1972, 10,87-92. Trimm, D. L.; Cooper, B. J. The Preparation of Selective Carbon Molecular Sieve Catalysts. Chem. Commun. 1970,477-478. Trimm, D. L.; Cooper, B. J. Propylene Hydrogenation over Platinum/Carbon Molecular Sieve Catalysts. J . Catal. 1973, 31, 287-292. Yang, R. T. Gas Separations by Adsorption Processes; Butterworths: Boston, 1987; pp 14-17.
Received for review June 5, 1990 Revised manuscript received August 30, 1990 Accepted December 11, 1990
Unified View of Transport Phenomena Based on the Generalized Bracket Formulation+ Brian J. Edwards and Antony N. Beris* Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, Delaware 19716
The Hamiltonian formulation of equations in continuum mechanics through generalized brackets is presented here in order to demonstrate the inherent structure and similarity between a variety of transport phenomena. The bracket formulation presented in this paper is based upon the Poisson bracket description of continuous systems and the entropy dissipation postulated in irreversible thermodynamics. This general formulation is presented for both single-component and multicomponent systems, as well as for systems with internal structure, for example, viscoelastic fluids. Thus, in addition to providing an alternative formulation for transport phenomena (of value for possible new numerical schemes), this paper represents the initial stages of a generalization of nonequilibrium thermodynamics to complex media (i.e,, materials with internal structure), which has never been accomplished t o date in a fully satisfactory manner using traditional approaches. 1. Introduction
For many years, intuition and experience have told us that the physics of transport phenomena is governed by an underlying structure or symmetry that is the same regardless of the type of transport involved (Bird et al., 1960; Sherwocd et al., 1975). Only recently, however, have the methods become apparent through which this underlying physical symmetry can be demonstrated. The major device in this task is the generalized bracket formulation, which has been only recently proposed (in a number of different variations) for dissipative continua (Kaufman, 1984; Grmela, 1985,1986,1989; Beris and Edwards, 1990a,b). The generalized bracket formulation defines the time evolution of an arbitrary functional (integral function) in terms of the Hamiltonian (totalenergy) and the dissipation present in the system. Dedicated to the memory of the late R. L. Pigford.
088~-5885/91/2630-0873~02.50/0
The generalized bracket formulation reduces to the more familiar Poisson bracket description when dissipation is absent. The Poisson bracket formulation of nondissipative continua has been developed only relatively recently (in the past 30 years) despite the fact that ita counterpart for discrete systems, developed at the beginning of the last century by Poisson (1809), had astonishing successes in the development of quantum mechanics at the beginning of this century (Lanczos, 1972). The Poisson bracket formulation has recently stirred considerable interest (Dzyaloshinskii and Volovick, 1980; Morrison and Greene, 1980; Holm et al., 1985; Grmela, 1986,1988; Salmon, 19881, mainly due to the advantages gained through its application to nondissipative systems with respect to nonlinear stability analyses (Holm et al., 1985). The objective of this paper is to reveal the underlying Hamiltonian structure of a variety of transport processes through the generalized bracket formulation. This can serve many purposes, such as the development of better 1991 American Chemical Society
874 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
numerical methods, the establishment of the connections between macroscopic and microscopic theories, as well as enabling the description of transport processes within media with internal microstructure. In particular, we want to close the gap existing between the standard Newtonian fluid transport equations and the derivation of the equations of transport for complex (non-Newtonian) fluids. In so doing, we can begin the development of a general theory that we believe is capable of eventually encompassing all realms of hydrodynamic transport, including thermal diffusion (Soret effect), Dufour effect, heat transfer and diffusion in viscoelastic media, viscoplastic flows, compressible viscoelastic flow, etc. Most of the applications in nondissipative continua are in plasma physics (Holm et al., 1985), magnetohydrodynamics (Morrison and Greene, 1980), and other areas of interest to high-energy physics. Recent work, though, has shown how dissipative systems can be incorporated into a much broader bracket formulation (Grmela, 1985,1986, 1988; Beris and Edwards, 1990a; Edwards et al., 1990) which uses a dissipative bracket in addition to the Poisson bracket. So far, this formulation has been used to describe the fluid mechanics of incompressible, isothermal Newtonian and non-Newtonian fluids (Grmela, 1988, 1989; Beris and Edwards, 1990a,b). There is no work to date where the generalized bracket formulation has been applied toward the more complex problem of coupled hydrodynamic transport equations (Le., mass, heat, and momentum transfer), except for the very recent work by Grmela (1989), who has independently approached this problem using a different bracket formulation. His approach, however, leads to very complex bracket expressions, even for a simple Newtonian fluid, with little insight into how they can be generalized to complex systems. As a consequence, they are limited to heat and momentum transfer. In this paper, we develop a consistent generalized bracket formulation that leads to simple bracket expressions and can apply to all transport equations. 2. Generalized Transport Bracket In this section, we illustrate the form of the generalized bracket that we are using to describe transport phenomena, as well as the underlying mathematical principles which govern its use. The main idea of this approach is to express the dynamics of some arbitrary functional, F, in the form w / d t = l(F,H)l (2.1) where is the generalized bracket and H i s the Hamiltonian (total energy) of the system in consideration. This generalized bracket is split into two subbrackets (2.2) l(F,G)l = IF,GI + [F,Gl ((.,e))
for arbitrary G as well as F , which describe the conservative and dissipative fluid processes, respectively. Let us discuss each of these subbrackets in turn. The bracket describing conservative effects, is the traditional Poisson bracket, although written for a continuous medium in a fixed Cartesian coordinate frame, x. The form of this bracket is now well-known and need only be taken as a given for this work. The reader may refer to a number of papers to see its derivation from the most fundamental principles of classical mechanics (Morrison and Greene, 1980; Holm and Kuperschmidt, 1983; Simo et al., 1989; Edwards and Beris, 1991). In order to interpret the form of this bracket for a given system, let us consider a particular system which can be completely described through a number of primary variables, a, b, c , ... E P, P being the operating space for the {e,.),
system. Thus one can express the total system energy as a functional of these variables H[a,b,c,...I = l hR( a , b , c ,...) dV
(2.3)
where R is the domain of interest with volume element dV. Thus, by setting the dependence of the arbitrary F upon these same variables, we can express the dynamics of F via (2.1) (considering only the Poisson bracket for now) as
where M is the momentum density defined as M E pu; p is the mass density and u the velocity vector field. This expression holds providing that the primary variables are all scalars. (If this is not the case, one must also consider the material objectivity of the primary variable. One may see Edwards and Beris (1991) for inclusion of nonscalar primary variables, as well as section 5.) The functional derivatives of the form 6Hlba are defined (provided that there is no dependence on the gradients of the primary variables, as is the case throughout this paper) as
so that by comparing the two forms of the dynamical equations for F , (2.4), we find the evolution equations for the primary variables, as well as the momentum density. Now we come to the main contribution of this paper, which is to introduce and discuss the appropriate dissipation bracket for describing transport phenomena. As alluded to earlier, the Poisson bracket can only describe the conservative system effects, and the bracket expression of (2.4) must therefore be generalized if one wishes t o describe dissipative processes through the same type of formalism. Let us consider the properties that the dissipation bracket must possess. First, if we let F = H, then we have that W / d t = (H,H}+ [ H A = 0
(2.6)
since the total energy of the system must be conserved. Since { H a= 0 by definition, we must require that [ H A = 0 as well. Furthermore, if we let F = S, where S = Js dV is the entropy functional, we also have that dS/dt = (S,q
+ [Sm 2 0
(2.7)
since the rate of entropy production within the system must be nonnegative. Therefore, since ( S a = 0, we have that [ S , q 2 0. Similarly, defining the total system mass as P J p dV, p being the mass density, we must have that dPldt = 0, which implies that [ P a= 0. It is also obvious from (2.4) that the dissipation bracket, [ F , W , must be linear in F , although it need not be linear in H. With these properties in mind, we can write the most general expression possible for the dissipation bracket from
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 875 a continuum perspective as IF.Hl =
(e.g., Callen (1960), section 16.5) for a detailed discussion of the Onsager relations.
where the L[.]denotes that E is linear with respect to ., and w = (a,b,c,...,M,s) and u = (a,b,c,...,M); Le., w = w s, s being the entropy density. Thus (2.8) is the most general expression possible where we have pulled out the linear dependence on 6F/6s and written it separately. (Note that the minus sign in front of E appears merely for later convenience.) The quantities 6H/6w and V(6H/6w) represent the system affinities or thermodynamic forces, the former being associated with relaxational phenomena and the latter with the fluxes of the primary variables, e.g., Vu, the velocity gradient tensor field. Now if we use the properties described above, we can learn a little more about this expression. For instance, using the property that [S,H] 1 0 (which indicates why we broke the expression up the way we did), we immediately see that \k must be a convex, nonnegative function. Furthermore, via the conservation of energy, we see that
is a convex function, with an absolute minimum equal to zero at equilibrium where 6H/6u = 0 and V(6H/6w) = 0 ; i.e.
.(E6 0
= 0, v-6H = 6W
=o
(2.10)
3. General Hydrodynamic Equations of Transport for a Single-Component System As a first example of the applicability and generality of the above approach, we shall now present a brief discussion of the generalized bracket formulation for the hydrodynamic equations of transport for a single-componentfluid. We wish to describe the transport processes within a compressible Newtonian fluid with viscosity p = p(p,T) and (Fourier) thermal conductivity k ( p , T ) , which are considered to be, in general, functions of the fluid density, p, and the fluid temperature, T. The hydrodynamic equations for an incompressible Newtonian and other fluids are discussed in Beris and Edwards (1990a,b). A single-component fluid is considered in this section. The equations for multicomponent systems are derived in the section 5. Let us first develop the Poisson bracket formulation for an ideal, nondissipative fluid. The primary system variables, in this case, are the mass density, p, the momentum density, M = pu, and the entropy density, s. The operating function space, P, is then defined as P r
M(x,~): p(x,t): s(x,t):
a3; M = o on an, M(X,O)= M~(x) a+; p(x,o)= P O W i n R s E a; s(x,O)= so(x)in R (3.1)
ME pE
where R+ is the space of real positive numbers. The Poisson bracket for this system, describing nondissipative effects, is then written analogously to (2.4):
with
and nonnegative definite second derivatives. Using the thermodynamic definition of the absolute temperature, T = 6H/ds, we can rewrite the dissipation bracket of (2.8), in terms of arbitrary G as well, as
[F,Gl = -E( L [ z,vg]; 6F 6F g 6G, v6G g)+
1;
6F 6G 6G 6G 6G T1 6s E( L [ - vv-) 6w' 6w 6w' 6w
(2.12)
In this paper, we restrict ourselves to systems that are close to equilibrium, i.e., when E is linear with respect to G (or H)as well as F. Hence we can express Z in terms of the functional derivatives as
Note that this bracket, integrated by parts, gives the well-known Poisson bracket for nondissipative simple fluid flow introduced by Morrison and Greene (1980). The Hamiltonian functional represents the total energy of the system. In this context, it is written as a volume integral of the kinetic, potential, and internal energy densities, respectively:
H(p,M,s) =
n
+ ep + v) dV
(3.3)
where
where A , B , C,and D are phenomenological coefficient matrices which, in general, may depend on 6G/6s and the primary variables of the system. One can impose constraints upon the phenomenological coefficients by considering the conservation of mass and material objectivity, as well as the microscopic time reversibility of the medium. In case of the latter, one can arrive at the Onsager reciprocal relations which equate the phenomenological coefficients of opposing fluxes, i.e., Aij = Aji. One can refer to any standard reference on irreversible thermodynamics
ep = -pg,x,
(3.4b)
and u=po
(3.44
0being the thermal energy per unit mass, g the (constant) gravitational acceleration vector, and x the position vector. The potential energy density is here taken as the gravitational potential; however, other potentials can be included as well (for example, electric and magnetic fields).
876 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
Using this Hamiltonian in the equation dF/dt = (F,H)
ffg = afp = f f l f (3.5)
with the Poisson bracket provided by (3.2) yields the following nondissipative constitutive equations for the primary system variables: (3.6a)
and Qlrpyt
= -V,(u,s) at
(3.6b)
where
is the local equilibrium relation (Prausnitz et al., 1986) and denotes specific properties. the circumflex In order to derive the time evolution equation for the internal energy density, U , differentiation by parts is used: ( I )
au aU(P,s) au a p au as - - --+-at
ap
at
as at
= -V,(u,U)
- pv,u,
(3.9)
Of course, taking the dot product of u with (3.6~)yields the nondissipative kinetic energy equation
a
P [ #2u?
+ u,Va(Xu9]= PV,U,
=
- V,(PU,)
+ pug, (3.10)
and addition of this equation to (3.9) gives the nondissipative total energy equation ae -= - V,(pu,) + P g a U , (3.11)
at
where e = ek + U. Now we shall focus on a dissipative viscous fluid, where the no-penetration boundary condition is replaced with the no-slip condition. The dissipative contributions can be picked up through a dissipative bracket of the form described in section 2:
In this expression, Q and a (the subscript i = k = 0 corresponds to an entropy variable, i.e., po = s and p 1 = p ) are phenomenological coefficient matrices, subject to the symmetry conditions:
Qpayt
=
Qaaty
(3.13b)
aCQ67p
(3.14a)
and = &6pf
+ ~6,,6p, + K’6,8ayf
(3.14b)
where K’ = K - 2p/3 ( K is the bulk viscosity). The latter expression can easily be obtained by consideration of the two types of motion that can occur in the system: shearing and normal (expansion) motions. (See, for example, Hirschfelder et al. (1954).) The functional derivatives (for an unconstrained system) are again identical with the partial derivatives of the integrands, and are elements of the P space. Applying the Hamiltonian of (3.3) to the generalized bracket equation of (2.1) yields the following (dissipative) hydrodynamic transport equations in terms of the primary variables: dP
(3.8)
Then, substitution of (3.6) into (3.8) then yields directly the internal energy equation for a nondissipative system at
Qytap
In this one-component system, since p is conserved, Le., d P l d t = 0, a$ = a1,f = a1.f = 0; however, this will be generalized in the next section (see (4.8)). Since we are here dealing with an isotropic medium
Q,pTf
at
=
a&!? =
as
(3.13a)
= -V,(pu,) at P[% L
+ upvpu,
1
(3.15a)
=
-v$ + Pg, + v y [ P ( v y u + a v,u,) + K’6,7VtUt] (3.1%)
and as 1 - = -V,(U&s) + .?;V,(kV,T) + at 1 T [ P ( V ~ U+, V a u y )+ K’~,,V~U,IV,U~ (3.15~) where the thermal conductivity, k, is defined as k = amT. (Note that cyyoo= &,,/T2, where kyoo is the coefficient of Hirschfelder et al. (1954).) Thus we see that the viscosity acts to decrease the kinetic energy and increase the entropy while the thermal conductivity acts solely to increase the entropy. Similarly to the nondissipative procedure, we can now obtain the dissipative form of the energy equation de + V,(u,e) = -V,(PU,) + PUB, + V,(kV,n + at v,{[P(v,U,+v , U y ) + K’6,yVtUcIUa](3.16) which is the same as eq 10.1-1 of Bird et al. (1960). Thus, we have arrived at the general hydrodynamic transport equations through Hamiltonian principles and demonstrated the applicability of the Poisson bracket formulation. Gmrela (1989) has recently communicated to us an alternative development of the hydrodynamic transport equations based on Hamilton’s principle. His development, although being mathematically correct, tends to obscure the exact nature of the dissipation bracket since he never specifies the dissipative potential in the linear limit close to equilibrium. 4. General Transport Bracket for
Multicomponent Diffusive Systems Now, we shall consider a n-component system for which the primary system variables are M, s, and pi, i = 1, 2, ...,
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 877
n, where pi is the density of component i. Consequently, the system Hamiltonian is now defined as
where n
P =
CPi i=l
(4.2)
The primary system variables are now elements of the operating space
S E R
(4.3)
where po s and cy is a nonnegative, ( n + 1) X ( n + 1) symmetric matrix of the phenomenological transport coefficients (Hirschfelder et al., 1954). However, these coefficients are not all independent. Indeed, because of the requirement that the total mass is conserved, i.e., dP/dt = 0, the following n + 1 constraint equations need to be satisfied: n
(with the initial conditions as well) and consequently, the Poisson bracket of the previous section generalizes to where, for an isotropic medium, a&@ = aik6,p Thus there are n2 - n - 1/2n(n- 1)= 1 / 2 n ( n- 1)independent diffusion coefficients a i k , i # 0 and/or k # 0. Also, special care must be exercised when evaluating the functional derivatives of (4.7). In general, the dissipation modes corresponding to different primary variables are highly coupled to each other; Le., p i = pi(s), i # 0. In order to examine the nonnegative character of the dissipation, therefore, we must make the modes independent which involves using the gradients of 6F/6pi, i # 0 , at constant temperature:
IF.GI =
The usual procedure then yields the nondissipative evolution equations for the primary system variables: api = -Va(u,pJ; i = 1, 2, at
as at
- = -V,(U,S)
9.')
n
(4.5a)
(4.9)
(4.5b)
si,
where ( d p o / a p i ) i = the partial specific entropy. A more detailed reasoning may be found in Forland et al. (1988, pp 16-22). Applying the Hamiltonian of (4.1) to the generalized bracket equation, (2.4), using the dissipation bracket provided by (4.7) and (4.9), yields the following (dissipative) hydrodynamic transport equations in terms of the primary variables:
and
where
Thus, we recognize the chemical potential (with respect to mass) as pi = aU/api. Of course, adding all n pi-component evolution equations yields the total mass conservation equation (eq 3.6a). The nondissipative internal energy equation may be obtained from
au = -au n au a ~ . au as (p1,p2 ,...,pn,s) = c- 2 + - at at i=lapi at as at
i = 1, 2, ..., n (4.10a)
(4.6)
from which the evolution equation for U can be shown to be equivalent to (3.9). Similarly,the nondissipative kinetic energy and total energy evolution equations are the same as (3.10) and (3.11), respectively. Now we focus on the dissipative (diffusive) equations. The dissipative contributions can be picked up through a dissipative bracket similar to the one defined by (3.12):
(4.10~)
878 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
The operating space for this problem must be redefined from (3.1) to include C as well. Thus, P is defined as P r
where pi* is the generalized potential defined as
and the affinity, A;, is defined as Aky vypk* + Skv,T
(4.1lb)
Note that aio= Sio/ T and (Yak = &Oh/ T, where &io, are the coefficients of Hirschfelder et al. (1954). Also, note that because of the constraining equations, (4.8), the contribution of the velocity and gravitational potential as they appear in (4.11) to (4.10) is zero and therefore can be neglected. By summing all the species balance equations, (4.10a),the continuity equation for the total mass density is easily obtained as (3.6a), provided (4.8) is satisfied. As in the preceding section, the internal energy equation is obtained from (4.6) as
dU = -V,(u,v) at
- pV,u,
M(x,~): M E a3; M = o on aR, M(X,O) = M ~ ( x ) p(x,t): p E a+; p(x,O) = PO(X> in s-2 s(x,t): sE s(x,O)= SO(X) in R c E a3x R ~ C(X,O) ~ ; = cO(X) in R
a;
(5.2)
where R3 X R3Trepresents the space of real, symmetric, 3 X 3, positive-definite tensors. Since we have introduced C as a new primary system variable, we must rewrite the Poisson bracket of (3.2) as
+ V,(kV,T) +
The above bracket formulation thus generalizes the single-component hydrodynamic transport equations to multicomponent systems (and provides the nth derivation of the simple fluid hydrodynamic transport equations). It is not difficult to incorporate other effects in the above equations as, for example, chemical reactions in the spirit of traditional chemical kinetics (Bataille et al., 1978), by introducing an additional dissipation term to the above equation, (4.7). However, in order to take into account large deviations from equilibrium, the above dissipation term should be nonlinear with respect to G. More details will appear in a forthcoming publication (Gustafson and Beris, 1991). In the next section, we treat the case of a single-component material with internal microstructure.
where the last three integrals (corresponding to a generalization of the bracket introduced by Grmela (1988) to compressible media) guarantee that the principle of material objectivity of the conformation tensor Cis satisfied. This bracket is derived directly from the Lagrangian bracket in Edwards and Beris (1991). Thus, equating
dF
- = (F,HJ dt
AF ap
5. General Transport Bracket for Compressible Viscoelastic Fluids In Beris and Edwards (1990a), a generalized bracket formulation was introduced for incompressible viscoelastic fluid models, extending and elucidating recent work by Grmela (1988). This formulation was then shown to reproduce a variety of continuum viscoelastic fluid behaviors (Beris and Edwards, 1990b). Here, we present a generalization of the approach presented in Beris and Edwards (1990a) to compressible, nonisothermal viscoelastic systems, based on the ideas presented in the previous sections. First, we shall consider an ideal fluid, without dissipation. We shall concern ourselves with models that contain an additional conformation parameter C : H(p,M,s,C) = $[edp,M) R
+ e&) + o(~,s,C)l dV (5.1)
where the internal energy now also depends on the conformation tensor, which is a parameter characterizing the internal structure of the viscoelastic medium. C thus represents the structure density of the complex fluid and can be written as C = p c , where c is the isothermal, incompressible structural parameter of recent work (Grmela, 1988; Beris and Edwards, 1990a,b).
A~ at
AF aM, +--AM, at
+ -AF - +as- AS
at
AF
aca,
AC,, at
(5.4) yields the nondissipative, compressible, elastic medium equations: dP = -V,(U,P) at
(5.5a)
as = -V,(U,S)
(5.5b)
at
ac,,
= -V,(U,C,,) at
+ Cy,Vru, + CarVyUo
and
.[2
+ u,vpu,]
= -vg
+ Pg, + 2v,( c,,
(5.5c)
E) (5.5d)
where
Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991 879 Note the influence of the fluid structure on the pressure as indicated in (5.5e). Also, note that (5.5~)represents the equivalent of the upper-convected time derivative of C for a compressible medium. As such, the first term on the right-hand side is -Vr(urCaB),contrary to the usual term -UyV,C,p The nondissipative thermal energy equation can now be obtained similarly to (3.8), where now U depends on Cas well:
relaxation (fading memory) effects. As such, AaOytis inversely proportional to a characteristic relaxation time of the viscoelastic medium: the higher the value of the relaxation time, the less important this term becomes, going to zero for a perfectly elastic medium. The usual comparison of dynamical expressions for F now yields (5.10a)
Substitution of (5.5) into (5.6) yields
air a t = -V&,O
- pv,u,
a0 + 2 -Cr,V,U,
aca,
(5.7)
Then, by dotting (5.5d) with u , and adding to (5.6), the total energy equation is obtained as r
.
.[2 +
u@v&J= -V$ + pg, + 20, (5.104
and Equations 5.5, 5.7, and 5.8 thus represent the nondissipative transport equations for elastic media. Note that in the above equations the conformation tensor was assumed to be symmetric (for convenience) as it is almost always. In systems for which this assumption is violated, the terms that are multiplied by the factor 2 must be split into two terms by interchanging the indices (Y and 0. We now turn to dissipative systems in order to derive the general transport equations for real viscoelastic systems. As such, we must rely on our experience at specifying the most general form of the dissipation gained from the previous sections and from Beris and Edwards (1990a). In the following, we shall neglect the solvent viscous dissipation and heat conduction dissipation (for simplicity) which can always be added as indicated by (3.12). Hence, we shall only consider the additional contribution to the dissipation bracket due to the internal structural parameter, C. The dissipation bracket for compressible viscoelastic systems is postulated to be of the form
(5.9) Although this is not an entirely new idea (Grmela, 1988; Beris and Edwards, 1990a,b), it is an issue that still requires further investigation to ascertain the exact nature of the dissipative forces in viscoelastic media. The third and fourth terms on the right-hand side of this expression represent the gradient (non-homogeneous) effects of the internal conformational in straight analogy to the previous dissipation terms, and their incorporation implies the need for the specification of an equation involving the conformation tensor at the boundaries. The first two terms represent a new form of the dissipation (spatially homogeneous) that accounts for the nonequilibrium character of C (at equilibrium, C should be some multiple of 6). The first term originates from Grmela (1988) and accounts for
Thus we arrive at the general dissipative evolution equations for a viscoelastic medium. Note, however, that these equations arise from a postulate that the dissipation bracket of (5.9) should be of the form introduced in section 2 and in Beris and Edwards (1990a). Applying (5.10) to (5.6) gives the internal energy equation for a general viscoelastic medium:
a0 at = -V,(U,O - p v , u , + 2 a6 acmR Cr,VrU,
+
Note that A does not appear in the internal energy equation in the present formulation. Now we shall show how these results apply to a specific viscoelastic fluid model, the upper-convected Maxwell model, following Beris and Edwards (1990a,b). -In order to do this, we define the internal free energy, A, as vkBT vpkBT AItr C’- -In det (C’/p) + A(p,s) (5.12) 2 2 where C’= (K/kBT)C, K is the Hookean (elastic) spring constant (which can, in general, be considered as a function of the temperature), Y is the number of springs per mass, and kB is Boltzmann’s constant. The first term on the right-hand side of this equation thus represents the elastic extension of the polymer molecules, and the second term represents the Boltzmann entropy expressed in terms of the conformation tensor C‘. Note that because the free energy is specified instead of the internal energy-which is a common practice with polymer media-special care needs to be exercised in evaluating the functional derivatives involved in (5.10). More specifically, since O=A+Ts (5.13) we have
880 Ind. Eng. Chem. Res., Vol. 30, No. 5, 1991
(5.15) which, if it is substituted back in (5.14), gives
The dissipation tensors are now specified for the Maxwell model as B = 0 and 2 (5.17) Aapyt = E C a y l S o t where A is a relaxation time. Substitution of these two quantities into the above system of equations yields the compressible upper-convected Maxwell equations: aP _ at - -V,(u,p)
... = general primary system variables B = dissipation matrix C = structural density (conformation) tensor C' = dimensionless conformation tensor dV = volume element in x F, G = arbitrary functionals e = kinetic and internal energy density ek = kinetic energy density e,, = potential energy density g = gravitational acceleration vector H = Hamiltonian functional J = linear flux vector K = Hookean spring constant k = thermal conductivity kB = Boltzmann's constant M = momentum density n = number of species in a multicomponent system P = operating space of the problem of interest p = thermodynamic pressure I) = total system mass Q = phenomenological coefficient matrix 8 = space of real numbers Si = partial specific entropy of ith component S = entropy functional s = entropy density s^ = specific entropy T = absolute temperature t = time Q = internal energy density = internal energy per unit mass U = generalized internal energy density u = velocity vector field x = fixed Cartesian coordinate system a, b,
However, using differentiation by parts
(5.18a)
v
( k , T / W a , p (5.18~)
a0 = -Va(uaO) - ~ V , U ,+ v[KC,, - pk,TG,,]V,u, at (5.18d) Of course, these equations reduce to the usual upperconvected Maxwell equations in incompressible situations. Similarly to the Maxwell model demonstrated above, any of the viscoelastic fluid models discussed in Beris and Edwards (1990b) can be recast into compressible form. 6. Conclusions
The major achievement of this work was to show that the general transport equations for both single-component and multicomponent systems can be derived using a generalized bracket formulation and very simple constitutive assumptions for the dissipative phenomena. Our approach is geared toward the extension of the hydrodynamic transport equations to complex systems. In so doing, we have demonstrated the applicability of this approach, as well as gained new insight into various aspects of the hydrodynamic transport equations for complex materials (for example, the pressure dependence on the material structure). This technique can be applied to any system after specification of the system energy and dissipation. As a specific example, the equations for a compressible viscoelastic medium were derived.
Acknowledgment We acknowledge financial support provided by the Center for Composite Materials.
Notation
A
= generalized free energy
Greek Letters a, p, ... = dummy indices a = phenomenological coefficient matrix 8, t$ = general dissipative parameters CP = dissipative potential K = bulk viscosity A = dissipation matrix X = relaxation time p = viscosity pi = chemical potential of ith component pi* = generalized potential of ith component u = elastic connectors per unit mass p = mass density pi = mass density of ith component fl = domain of interest dfl = boundary of fl Other V = gradient operator
Literature Cited Bataille, J.; Edelen, D. G. B.; Kestin, J. Non-Equilibrium Thermodynamics of the Nonlinear Equations of Chemical Kinetics. J . Non-Equilib. Thermodyn. 1978,3,153-168. Beris, A. N.; Edwards, B. J. Poisson Bracket Formulation of Incompressible Flow Equations in Continuum Mechanics. J. Rheol. 1990a,34,55-78. Beris, A. N.; Edwards, B. J. Poisson Bracket Formulation of Viscoelastic Flow Equations of Differential Type: A Unified Approach. J. Rheol. 1990b,34,503-538. Bird. R. B.: Stewart. W. E.:. Liahtfoot, E. N. Tramsport Phenomena; Wiley: New York, 1960. Callen. H. B. Thermodvnamics: Wilev: New York, 1960. Dzyaloshinskii, I. E.; Voiovick, d. E. Poiason BrackeL in Condensed Matter Physics. Ann. Phys. 1980,125, 67-97. Edwards, B. J.; Beris, A. N. Noncanonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity. J. Phys. A: Math. Gem 1991,in press. Edwards, B. J.; Beris, A. N.; Grmela, M. Generalized Constitutive
Ind. Eng. Chem. Res. 1991,30,881-889 Equation for Polymeric Liquid Crystals, Part 1. J. Non-Newtonian Fluid Mech. 1990, 35, 51-72. Fsrland, K. S.; Fsrland, T.; Ratkje, S. K. Irreversible Thermodynamics: Theory and Applications; Wiley: New York, 1988. Grmela, M. Bracket Formulation of Dissipative Time Evolution Equations. Phys. Lett. 1985, l l l A , 36-40. Grmela, M. Bracket Formulation of Diffusion-Convection Equations. Physica 1986, 210, 179-212. Grmela, M. Hamiltonian Dynamics of IncompressibleElastic Fluids. Phys. Lett. A 1988,130,81-86. Grmela, M. Hamiltonian Mechanics of Complex Fluids. J. Phys. A: Math. Cen. 1989,22, 4375-4394. Gustafson, J. B.; Beris, A. N. Theoretical Description of Reaction and Transport Phenomena in Multi-Fluid Systems with Application to Plasma Glow Discharges. Submitted for publication in Chem. Eng. Sci. 1991. Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. Holm, D. D.; Kupershmidt, B. A. Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity. Physica 1983, 6 0 , 347-363. Holm, D. D.; Marsden, J. E.; Ratiu, T.; Weinstein, A. Nonlinear Stability of Fluid and Plasma Equilibria. Phys. Rep. 1985,123, 1-116. Kaufman, A. N. Dissipative Hamiltonian Systems: A Unifying
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Principle. Phys. Lett. 1984, IOOA, 419-422. Lanczos, C. The Poisson Bracket. In Aspects of Quantum Theory; Salam, A,, Wigner, E. P., Eds.; Cambridge University Press: Cambridge, 1972; pp 169-178. Morrison, P. J.; Greene, J. M. Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Phys. Rev. Lett. 1980, 45, 790-794; errata 1982,48, 569. Poisson, S. D. Sur la variation des constantes arbitraires dans les questions de mgcanique. J . 1’Ecole Polytech. 1809,8, 266-344. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; PrenticeHall, Englewood Cliffs, NJ, 1986. Salmon, R. Hamiltonian Fluid Mechanics. Annu. Rev. Fluid Mech. 1988,20, 225-256. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill, New York, 1975. Simo, J. C.; Marsden, J. E.; Krishnaprasad, P. S. The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates. Arch. Ration. Mech. Anal. 1989, 104, 125-183. Received for review May 29, 1990 Revised manuscript received November 1, 1990 Accepted November 13, 1990
Use of ab Initio Quantum Mechanics Calculations in Group Contribution Methods. 1. Theory and the Basis for Group Identifications Huey S. Wu and Stanley I. Sandler* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Group contribution methods are powerful tools for estimating the thermophysical and thermodynamic properties of pure components and mixtures from experimental data on related substances. Among the problems with current group contribution methods is that there is no theoretical basis for defining functional groups and no a priori way of determining when a group contribution prediction may be in error. Here we show that, on the basis of quantum mechanical ab initio molecular orbital calculations on single molecules, we can develop a simple theoretical basis for defining functional groups. Further, by “supermoleculen calculations involving two or more molecules, we can identify situations in which group contribution methods may fail.
Introduction The number of organic compounds and their possible mixtures is essentially uncountable so that determining all their thermodynamic and thermophysical properties from experiment is impossible. Many theoretical methods have been developed for estimating the properties of compounds or mixtures from experimental data on related substances. One such technique, which is the one of interest to us here, is group contribution methods. In this procedure a molecule is considered to be a collection of functional groups, and the properties of a pure fluid, such as its critical properties or the excess Gibbs free energy for mixtures, are estimated by adding the contributions of all the functional groups involved. In this procedure the contribution that each functional group makes is obtained from a study of the properties of other substances or mixtures containing the same functional group. There are a number of levels of group contribution methods. The lowest, and least accurate, is based on atomic stoichiometry. At this level ethanol, for example, is considered to be a combination of two carbon, six hydrogen, and one oxygen atoms, i.e., C2&0. This is a zero-level group contribution method. Though such a group contribution description is used by biochemists in the generalized degree of reduction method (Minkevich and Er-
oshin, 1973), the obvious problem with it is that no information about the molecular structure is involved; thus one cannot distinguish between ethanol and dimethyl ether, which have very different properties but the same stoichiometry. The next or first-order level of description is based on bonds. This is commonly used (Benson, 1965) in estimating the properties of single molecules, such as critical properties, heat capacity, etc. At this level, ethanol consists of five C-H bonds, and one bond each of C-C, C-0, and 0-H;dimethyl ether is composed of a different collection of bonds. The next level of approximation is to consider a molecule to be a collection of functional groups. This is the method used in estimating mixture properties in, for example, the UNIFAC (Fredenslund et al., 1975) and ASOG (Derr and Deal, 1969) group contribution models. However, at this level ambiguities arise. For example, ethanol could be considered to consist of one of each of CH,, CH,, and OH groups, or of a C2H5and an OH group, or a CH, and a CH20H group, or perhaps even the whole molecule should be considered to be a single functional group, as is the case with methanol, dimethylformamide, and furfural in UNIFAC. The important point here is that while groups are uniquely defined at the stoichiometric level and the bond level of group contribution methods, this is not the
0888-5885/91/2630-0881~02.50~00 1991 American Chemical Society
