1038
JOHNG. KIRKWOOD AND BRYCE: CRAWFORD, JR.
case of several species of molecules. Also, formulas are easily derived for the “partial integrals” e12 alone, translatioiial kinetic energy alone, the
Vol. 5G
momentum of a single species of molecule alone, etc. In such cases, some, hut not, all of the identities (103)-( 106) will hold.
THE MACROSCOPIC EQUATIONS OF TKANSPOHT” BY JOHN G. KIRKWOOD AND BRYCE CRAWFORD, ,JR. Slerling Chemislry LaboratorU, Yale Universily, New Haven, Connecticut and School of Chemistry, University of Minnesota, Minneapolis, Minnesota Received Auouat 18, lBdB
The equations of continuity, motion and of energy and entropy transport in multi-component, chemically reacting fluid mixtures are formulated on the basis of the postulate of partial local thermodynamic equilibrium. In this approximation, it is assumed that a thermodynamic temperature can be defined a t each point in the fluid and that the intensive thermodynamic functions, chemical potentials, partial specific enthalpies, partial specific entropies, etc., possess their equilibrium values a8 functions of the local temperature and densities of the several components. It is shown that the pofitulate of partial local equililwium uniquely determines a dissipation function in terms of the diffusion and heat currents, the chemical potential and temperature gradients, the viscous stresses and the local free energy increments of the several chemical reactions. When phenomenological relations between the viscous stresses, heat current, diffusion currents and the rate of strain, temperature gradient and chemical potential gradients are introduced into the equations of trmsport, a system of partial differential e uations is ohtained which determines the velocity field and t,hc local thermodynamic state of the fluid in terms of initial and%oundary conditions. The analyAis leads to a generalization of results previously ohtained by IGckart, Curtiss and Hirschfelder. and others. Chemical reactions itre treated in terms of a relaxation-time spectrum which characterizes the lag behind equilibrium.
The extension of thermodynamics to apply to irreversible processes has received considerable attention since the early work of Eclrart,‘ Tolman and Fine,2 and Onsager.3 We may refer to the recent monographs of Prigogine,* de Groot5 and Denbigh.6 In the present paper we formulate the equations of transport from the purely macroscopic standpoint, using the approximation of local thermodynamic equilibrium, with a somewhat different view. Our object is to obtain the general equations goveriiing the behavior i n time and space of any multicomponent, chemically reacting fluid, to the approximation indicated. The Hydrodynamic Equations We consider an element of fluid, and assume we may adequately describe its state at time t by giving its specific energy E , its density p, its composition in terms of weight fractions x,, and its velocity u. For a system in equilibrium, these quantities will be constant in space and in time; for our flow system, they will vary both in space and in time, with the substantial derivative, d/dt, being connected to the time-derivative a t a fixed point, b/dt, by the usual relation d/dt = d / d t
+u.0
(1)
The flow pattern may be characterized in terms of the heat current, diffusion currents and chemical reactions occurring. The material currents might he described in terms of the local mean velocity * This study was begun at the California Institute of Technology in 1950-51. when B. L. C. was a Guggenheim Fellow on Sabbatical leave from the University of Minnesota. (1) C. Eckart, Phue. Rev., 88, 267, 269, 919 (1940). (2) R. C. Tolman and P. C. Fine, Rev. Modern Phua., SO, 51 (1948). (3) L. Onsaqer. Phys. Rev., 37,405 (1031); 38, 2265 (1931). (4) I. Prigogine, “$tude Thermodynamique des processes irreversibles,“ Desoer, Liege, 1947. (5) 5. R. deGroot. “Thermodynamics of Irreversible Processes,” Interscience Publishers, Ino., New York, N. Y..1951. (6) K. G . Denbigh, “Thermodynamics of the Steady State,” John v i l e y and Sone, Inc., New York, N. Y.,1961,
u, for each component a; and indeed we shall have occasion to consider these total .fluxes p,~,. But we
assume that these velocities u, do not greatly differ, and we shall describe the material currents in terms of the mean velocity u given by p l l = Zapallu; p a = Z a p
(2)
and the diffusion currents j, = pa(ua - u ) ; Zaja = 0
(3)
Thus motlon is divided into the convective motion described by u and the diffusive motions j,. We shall ultimately linearize our equations with respect to the latter. To describe the chemical processes, we may first write the chemical equation for the rth reaction as 0
=i
ZavhAa
(1)
where A , is the chemical symbol for component a and v, is the number of grams of a produced per gram of reaction. (We note that 2,vL = 0.) Next we introduce the progress variable Xr, equal to the grams of reaction occurred per gram of original sample; then in a sample of mass m the change of mass of a,ma, will be dm, = m&vbdAr
(5)
Our first task is to relate the c’langing state of the fluid to the flow pattern. We may begin this with the aid of the basic relations of hydrodynamics, the equation of motion and the equatiow of continuity of matter and of energy. Equations of Continuity of Matter.-The overall equation of continuity of matter, for the fluid element as a whole, is of course
However, we may also apply the continiiity principle separately to each component, a, relating the change of local density of that compownt, pm, to the divergence of the total flux paua and t8hel o c ~ l source q5m.
5
+
THEMACROSCOPIC EQUATIONS OF TRANSPORT
Dec., 1952
The local source 4a is just the mass of CY produced per unit volume per unit time by chemical reaction. In terms of our progress variables, we may apply eq. (5) t o our element of fluid and we see that +a
(7)
pZrv&Xr
So the equation of continuity of
CY
becomes
dpa b1 + V . ( p a ~ a ) p Z r ~ & X r
or, after some rearrangement to express this in our chosen variables dxa
p-
dt
+ V.ja
du
tit
= V.u
($)
dT PlM
+ ( g ) T , m d p + ZaZ'adma
(14)
If the system is closed, dma = 0, and moreover we may write dEt = dpr
- pdV
(15)
whence p~,v:jr,
(9)
So we have the rate of change of composition given in terms of the diffusion currents and the chemical reactions. The Equation of Motion.-The equation of motion for t81ieconvective motion of t,he element, as a whole is p--
where the partial specific energies E, are precisely those defined in equilibrium thermodynamics. We may also re-examine the energy equation by generalizing from relations valid for equilibrium states. For an equilibrium system of mass m and total energy Et, we have dEt
(8)
1049
+px
or
- = V.(u - puu) + PX
Inserting this in (14), we find dEt
dp,
- pdV
+ ZaEEadma
(17)
If we include kinetic energy and external forces, the direct generalization is
at
where u is the stress'tensor and pX describes the force acting on the element due to external fields. Presumably these outside forces might act selectively on the different components; the total effect of selective forces p a x a would then be p X = Zapaxa.
The Energy Equation.-From hydrodynamics alone, we may write down the energy balance for an element of fluid and simplify it with the aid of the equations of continuity and of motion to give
If we now linearize with respect to j,, and utilize the continuity and force equations, we obtain dE
p-
dl
-
- V . q - Z a V . j a E a + Zaja*Xa
= U:VU
(19)
in which a separation has been effected between the energy transport associated with diffusion currents, and that due to pure heat flow. There exists a certain arbitrariness in the defidE p= u:V u + Zaja.Xa - V-jE (12) nition of heat flow for open systems, even in the dt of equilibrium states. Our definition, based where j E is an energy-transport vector including the field on the first law of thermodynamics, seerns to us to effects of both heat flow and diffusion; E is of course follow from a clear-cut operational distinction bethe specific internal energy. (If the external tween heat and work. Certain authors prefer to forces X a are derivable from a potential, this is not included in E.) This is as far as one can go from base the definition of heat flow on the second law, hydrodynamics; after introducing the local-equilib- in the form d S =: d p ' / T + Z a Sa d ma rium assumption, we shall re-examine this equation. The heat flow then becomes identified with q' rather than q (see equations (23) and (24) below). The Assumption of Local Equilibrium We now introduce the assumption that all of the While it is true that q' enters as a fundamental thermodynamic functions of state exist for each quantity in the linear relations between currents element of fluid, with values determined by the and forces (equations (26) below), the difference local state variables (say p and E ) according to the term p2Ja Vu is clearly operationally distinguishequilibrium relations. Thus the local temperature able as the work associated with the volume inT may be defined by the same function T(p,E) crement produced by diffusion of the several comas is given by measurements on equilibrium systems. ponents into the element a t constant T and p . If the heat flow is defined as q', then the energy The validity of this assumption has been dis- equation (18) must be treated rather art.i&ially, cussed elsewhere4; ultimately, of course, it must be validated either by experiment or by microscopic with partial enthalpies Ra replacing the E,, and with a somewhat unrealistic deletion of the work theoretical analysis. The question thus lies beyond associated with the diff usion-produced volume the scope of our discussion here. increment from the work term of which it is propBy this assumption, then, we may write E as a erly an operational part. However, if used confunction of T, p and xu; indeed, we may write sistently, either definition of the heat flow leads to correct results, R = ZaXaEa (13)
JOHN G.KIRKWOOD AND BRYCE CRAWFORD, JR.
1050
The Entropy Equation.-The local-equilibrium assumption also permits us to define the specific entropy of our element of fluid through the relation dE = TdS
- p d(l/p)
+ Z a p u dza
(20)
We may note that the chemical potential pu used here does not include the potential of the external forces X; also that in using the term p d(l/p) we are limiting ourselves to a fluid. Since we have expreasionsfor (dE/dt),(dp/dt) and (dx,/dt) we may easily get an equation for (dS/dt), In order to have this in the most useful form, we shall manipulate the terms to get an equation of the form dS
-
0
Vol. 5G
The Phenomenological Relations Experience shows that in many systems the flows and forces of equations (23) are linearly related. We must treat scalars, vectors and tensors separately, for the entities of different tensorial character cannot interact (Curie’s theorem). We shall also defer discussion of chemical reaction. Thus we may now introduce the phenomenological relations u
-
-[p
+
(51
- P)(V*U)II + 214
d = sym ( V u ) -q’ = QCQV In T ZBoa V T ~ A -ja = nao V In T Z&5 Vw‘p
- V-ja
+
i.e., an “entropy source” term and a term involving
+
(25%) (25b) (26)
Equations (25) express the conventional Newtonian the divergence of an “entropy flow”. The entropy stress tensor; q and (p are the coefficients of shear source term we have written for convenience as a and volume viscosity, respectively. Equations dissipation function divided by T; we shall arrange (26) express the general linear dependence of heat for c9 to be in the form flow and diffusion on the gradients of temperature a EijiaFi (22) and chemical potential. The flows and gradients have been chosen to be where the ji are “currents” and the Fi “conjugate forces,” gradients being very satisfactory as forces. those appearing together in the dissipation funcThis will permit us to apply Onsager’s relations to tions of (23). For this choice, Onsager has shown3 that the matrix 52 of the coefficients in (26) is the ji and F1 so chosen. If we carry out this algebra, we find the form symmetric; i e . , that f l a p = flua. These reciprocity relations are most useful; their application to specific (21) easily achieved, with 9a sum of four terms problems, especially in steady-state flow, has been *I = (u p 1): v u discussed elsewhere.4--8 +, = - - p ~ r ~ ~ r i I The introduction of these relations has also been used to find the requirements for @ to be positive.’ +a = -ZajwVTp That 9 should be positivei.e., that the occurrence $4 = -q‘.V In T of irreversible processes should cause an increase of js = (q‘/T) zajaXa (23) entropy in each element of fluid-may be construed where as a (slightly strengthened) form of the second AFr = E a ~ b p a law. Introducing (25) into (23) yields V T ~ & VpA f s’,V T
+
+
-
Vpa q’ = q
- X a-+ & V T
- ZajapVa
+I
(24)
The four contributions to 9 represent the dissipation due to convection, chemical reaction, diffusion and heat flow, respectively. In the first, both flow and force are (second-order) tensors; in the second, both are scalars; in the two last, both are vectors. The inclusion of the external force X, in the gradient v p ; is made here for convenience. If X, is derivable from a potential, one could express this inclusion by saying that p ; is the chemical potential including the external potential. Inclusion or exclusion is merely a matter of convenience in writing the equations; we have written ours in the fashion which seemed to us most easily adapted to specific problems likely to arise. In particular cases, &s when the external force is a rapidly varying function of time, explicit treatment of the X, may be strongly indicated. We may note that electrical conductance is included in our equations in the diffusion terms. If component CY carries a charge 2, (per gram), then the current i is given by ZaZaja; in the presence of an external electric field of strength E, the force X, = ZaE will apply to this component; and the dissipation term Z&Xa included in gives the electrical dissipation i-E.
+
+
= (P(V.u)* 21 [ G
- f (Ir *)I]: [L
- f (tr &)I]
(27)
which is essentially positive if the viscosity coefficients 7 and p are positive. Introducing (26) into (23) gives for $3 $4 a quadratic form in the this is positive if the gradients vln T and v&; matrix fl is positive-definite. We may note that a different set of (vector) currents and forces may be used. The desirable properties are that the dissipation function be of the form (22), and that the matrix of coefficients in the linear relations similar to (26) be symmetric and positive-definite. If we transform the currents ji to a new set ji’ by any orthogonal homogeneous linear transformation, and the forces Fi to new forces Fi’ by the same transformation, these properties will be retained. The Transport Equations We now wish to recast our equations to obtain a general set of transport equations describing the state of the fluid as a function of space and time. We choose for our independent variables U, T, p and x,; other variables such as p and p a may be written as function8 of these. Actually, we already have equations for the change of x a and u, equations (9) and (10). We need to eliminate p and S from equations (6) and
+
(7) C. (1960).
F. Curtiss and J, 0,Hirschfelder,
J.
Clem. Phys.. 18, 171
Wee.,
(21). To this end, we use the thermodynamic relations ~ K CdVT = K(P1’ dS) BT(dp/p) + &(p d$,)[BT i?, ~ ? 3 a ] (28) PKCV dp = B(pTdS) i- Pcp(dP/P) + Za(p dza)[pCpTa - PZ’Zal (29) Here p and K are th. usual volume-expansion and compressibility coeficients
+
-
from equilibrium, or the %g” of the rth reaction in its attempt to maintain equilibrium. The equation determining A: is easily found from the condition that AFI = 0 at Xr = A::
Agi = Zavag&9; Qu#=
The iiitroductioii of equations (6)) (9) and (21) thus leads to d 2‘ pKCv-&
[Kq&
Q’ = ZaZg v(gv:gap (31)
dt
-
- pCpAVr)
Zrpir(BAHr
Z a [@j,.V HA
+ (V.ja) p C , ~ ~ ’ l
(32)
where Vi?: is defined analogously to v ~ Land AH,, AVr analogously to AFrl equations (24). We may also rewrite equation (10) in the NavierStokes form P
=
PX - v P
+ BV + (; + 7
$9)
v (v .u)
(33)
and we may add a formal expression of chemical kinetics db dt
(apa/aX#)Tmxr
The introduction of (9) leads to the set of equations for the A::
- Sl’(v.u)] - KV.9’
- &PA, ( K A H ~ BTAV,) - Za [ & , V a L + (V.ja)BTval PKCY--dP = [B $1 - pcp(V*u)I - 6V.q’
du
1051
THEMACROSCOPIC EQUATIONS OF TRANSPORT
iY52
fr
(T;
(34)
Pa)
Equations (31) to (34) and (9), together with (26), form a complete set of partial differential equations describing the behavior of the variables T,p , x u arid u in terms of functions of these variables and of the external forces Thus the velocity field and the thermodynamic state of the fluid at every point are determined as a function of initial and boundary conditions. Chemical Reaction The equations we have thus found are reasonably satisfactory save for t,hat governing reaction rate; chemical kinetics does not fit into the scheme of linear relations between flows and forces. Without discussing the microscopic reasons for this, we accept the experimental fact that the approximation of setting ilr proportional to AF, is not a good one. We therefore approach the problem by dividing Xr into two variables
x,.
+
Xr = X h (35) where XO, is the equilibrium value of the progress of the rth reaction, for the temperature, pressure and over-all composition of the fluid element. The new variable Fr will then measure the deviation
which may be combined with (31)to (33) and (9) to obtain the complete description of the fluid assuming no lag in the chemical reactions. Chemical lag may then be treated in terms of relaxation behavior. If we expand the rate function f r of equation (34) about the equilibrium point (A:, Ai, ...), and linearize, we have hr
ir
+k
~ ~ n t i
(38)
where the derivative is evaluated at the equilibrium point. We may note that the linearization used here sets the rates proportional to concentrations; this will be exact for some types of reaction, and should in any case be a better approximation than linearization in AF,. Equation (38) gives us a set of linear equations in the 6,with a forcing function :k which is known from the set (37). Such a set of equations can be solved in terms of “relaxation times.” If we define the determinantal function A(z) Iz 6” - 6r.I (39) with minors Ara(z) and rods K ~ then , we may write the solutions of (38) by the method of Laplace8 transforms
where the P, are the values of is at t = 0, &(z) is the transform of i:and , the relaxation times 7. a -1/Kn (41 1 Thus the effect of the initial conditions will die away according to these relaxation timea; and when the functions AB are inserted, the lag of the chemical reactions behind the equilibrium values will also be expressed in terms of the spectrum of relaxation times. (8) ‘2. Doetach, “Theorie und Anwendung der Laplace-Transformation.” Dover, New York. 1943, pp. a29 ff.
