Theorem for the local volume average of a gradient revised

THEOREM FOR THE LOCAL VOLUME AVERAGE OF A GRADIENT REVISED VLADIMiR VEVERKA Research Institute of Inorganic Chemistry, 400 60 fist{ nad Labem, Czechoslovakia (Received 26 November 1979;accepted 26 August

1980)

Abstract--The theorem for the local volume average of a gmdient formulated by SIattery[l] is analyzed from the mathematical point of view. It is shown that the expression for the average of a gradient as sum of the gradient of an average and of an interior wall term (“tortuosity”) for a porous material is mathematically uncertain. An exact integral formula is derived and an idealization of the problem is suggested. In reasonably selected cases, Slattery’s formula can still hold as a DlausibIe aaoroximation. Generally, the resulting differential equations in average quantities have to be applied with caution.

INTRODUCTION

THE

The theorem for the local volume average of a gradient was formulated by Slattery ([1], 4.3.3) as Q’B=Q%++

BndS

where B is an arbitrary scalar, vector or second-order tensor field defined on a part of the geometric space, for example inside the pores of a porous material. With each point x of the porous medium, one associates a closed surface S, obtained by translation without rotation of a fixed surface S; the (constant) volume of the region enclosed by S, is V. S, is the surface of the pore walls inside a given S,, D the outside unit normal vector at a point on S,. If Z is scalar, Q’Z is the usual gradient VZ, if 2 = Z,,. ik ei,. . . . eik (ei are orthonormal unit vectors), (1, are Cartesian coorV’Z = ( 8Zi,, i/ d.Xj)ei,. . .e,ei dinates). I have modified this notation because I believe that the definition

TRANSPORT

alaz . . ask. ai E [lo]; we put &JE = R by products definition. A mapping F of class c’ is r-times continuously differentiable; DF is its derivative considered as a linear mapping ([PI: 8.1), D2F= D(DF). With this notation, we have the following Theorem. Let E be a vector space endowed with structure [ 111. UOE E Euclidean open relatively compact[8], ]a,b[ C R an open interval. Let rp: UOX ]a,b[+ E be a mapping of class e such that q, = q(*, t): U,,+~,(&,) be a C’diffeomorphism[lO] preserving the orientation; let D’s be bounded on Uox ]a, b[. Let W be the (open) set of (x, t) E E x R such that t E ]a,b[, x E rp,(U,). Let F: W-&E

is mqre common in engineering literature ([2], see the footnote 1 in [I], p. 637). The local volume average is defined as

I

ZdV

“Ul

where Vu, is the interior of the pores inside a given S,. The formula (1) has become a basis for a standard averaging technique in the theory of transport phenomena in porous materials: see Refs [3-71. It is my aim to show that the formula (1) is not an exact mathematical theorem. Rather, it can be considered as an “approximation” in the sense I will precise. The precision will require some proofs of pure mathematical nature. The mathematical proofs are presented to be checked by mathematicians; I hope the results can be useful for physicists and engineers in applying the formula with due caution.

THEOREM

The formula for the derivative of an integral over a variable domain, usually called the transport theorem, is well known ([l], 1.3.2). I shall start with a rigorous mathematical formulation of this theorem, because it is its application where I find the formula (1) to be inexact. R is the space of real numbers. If E is a real vector t space, @‘E is the tensor product of k identical spaces whose elements are linear combinations of tensor

be bounded

be a CL-mapping such that DF

on W. Then

d W, t) dA (x1 ;iT I 4 UO) = p,(uo)s k 0 dA(x) + j(v.(G’Mx, I 4 Uo)

0 Wx) (4)

where +(x9 0 = 4&P

09 x = ot(D

(5)

(4 is partial derivative with respect to the second variable). A is the Lebesque measure on E ([9]: (14.2), (14.3)). The divergence operator is defined by contraction in (2) v*z= with respect 833

$

I

Z,

to an arbitrary

Ceh. . . . cc orthonormal

(6) basis (e,) of E.

V. VEVERKA

834

Corollary.Let us denote U(r) a,( t&J. If, for a t, E ]a, b[, dU(t,) = W(tl) where U(r,) (the closure of U(t,)) is a compact

with boundary (“compact g bard”, [12] 1,4.10)? and if F(x, t,) and 0=(x, t,) admit a Cl-continuation over an open U, > U(t,) ([S]: (16.4.3)), then for t= t,: d FdA= ;is I U(r)

n. u#‘dp.

(7)

Here, au(t) is the oriented boundary (a compact manifold), n is unit normal vector directed outwardly from U(t). p is the Lebesquean measure[lO]: 16.22.2 on NJ(t) associated canonically with the euclidean structure of E [ 121, I,4.12). Proof. Because the tensor field F, = F(-, 1) is defined with respect to a fixed frame, it is sufficient to consider scalar F. h,(l) is an endomorphism of E; put J,(l) = det (Opt(E)) 10. From the definition of the determinant of an endomorphism [ 111: XVI 47, Theorem 11) it follows that D(det)(u).u = det(u)Tr(u-‘0) for II, u EEnd(E) (space of endomorphisms of E). Hence[lO]: (16.22.1) and 191: (13.8.6))

F(rp(S, t), 0 det U%(5))

=

I

uo

dA(Z)

WzJWS, 0, MQ

these “singular” cases and to consider the formula (1) as a plausible idealization of the problem. But how was the formula obtained? More precisely: What is the transformation velocity vT = dp/ds in (3-3) Ibid.? If we follow the preceding text in [l], the averaging surface S is translated without rotation. So one would expect that the isotopy rp were itself a translation and UT were constant in space; but this is in obvious contradiction with the assumption (3-S) Z’bbid. that vT .II= 0 on the pore walls S,. So it must be assumed (which is not clearly stated in the text) that the isotopy cp is defined in a more complicated manner to satisfy the condition vT *II= 0 on SW. Intuitively, this appears as possible in the interior of the region enclosed by S; but it is generally impossible on S itself at the points where S intersects S,. v,- has to be continuous (even differentiable) at S,, see the Corollary. Perhaps a more sophisticated mathematical analysis would show that the formula (I) holded to within a set of measure zero. But on admitting this formally, we obtain a difficult theory involving partial differential equations with discontinuities, where the divergence theorem cannot be applied etc. On the other hand, the formula (1) appears as intuitively plausible; it only suffers on lack of mathematical precision. I shall now show in what sense it can be rehabilitated. Because (2) VZ = V. (62) where 6 is unit Euclidean tensor, it is sufficient to consider the divergences. Time plays no role in the local volume averaging, so Z will be a given tensor field. Let us first assume that Z is defined on the whole E. With each x E E, let us associate a relatively compact open neighbourhood U, such that aU, = aox where ax is a compact with boundary: if Z is measurable, define

From the definition (5) we find ~l~T(~,

4 = Q&3-’

. &(P(6

I).

The inverse transformation in the latter integral yields the formula (4), considering that for scalar F, V . (vTF) = vT 1VF + FV . vT where uT . VF = (D,F) - UT. Putting F= fi, __ikwe arrive at the general case. The corollary is an immediate consequence of the divergence theorem ([12]; I, 4.10). I call the vector vT defined by (5) the “transformation velocity” even though the parameter t is not necessarily time and the “isotopy” ([lo]: 16.26) rp is not necessarily a motion. dA is what engineers usually denote dV and dp is the usual dS[l]. THE LOCAL

VOLUME

AVERAGE

OF A GRADIENT

I am now able to analyze the derivation of the formula (1) given in [I]: 4.3.3. The first-question arises immediately: Is the volume average B differentiable? If, in the Fig. 4.3.3-l ibid., the averaging surface S is moved across a pore wall, a break may occur in the value of d Perhaps it could be possible to neglect, in some way,

tHtre, by a “compact with boundary” I understand a compact with boundary of class CL (loc.cil.).

where A( U,) is the volume of U,. Let 2 be of class C’. To obtain a reasonable expression for the derivative of (Z), V = A(U,)should be held constant. So let us assume that in a neighbourhood of an x0 E E, all U, are obtained by a translation of U, Because the derivative in a direction depends only on the tangent vector to the trajectory at x0, it is sufficient to consider the isotopies qi(g, 7) = 5 + 7ei. & E U,,

7

E R.

(9)

Hence (5) uT = ei and (7) U(T) = U, for x = x0+ TC,. Because Z is defined on E independently of the translation of U,, the formula (7) with (9) yields

and after contraction

(6)

(11)

835

The local volume average of a gradient revised Let now R be a (topologically) open region of E (imagine the pores of a porous material). If the field 2 is defined on R, designate 1 (z)*(x) -_ - A(UX> oEZdA I

(12)

The hypothesis (H,) adopted, (V . Z)*(x) is given almost everywhere in E by (18) (being of course equal to zero if diJF is empty) and

I

a.u:

n.Zdfi=

I au*

n. x,+Zdp

almost everywhere

(20)

where U: = rr, n R.

(13)

Let R be such that dR = al? where R is a compact with boundary. If xn is the characteristic function of R (x&x) = 1 for x E R, ,yR(x) = 0 for XE R), ,Q&’ can be defined everywhere on E and

With the same assumption on the cl, as above, further assume that in the decomposition = a,U?: + a,u:

let us

(IS)

both

a,u:=auTnu,=aRnu,

(U, is

a,uZ = auf n au,

(au, = au;)

open)

(16)

and (17)

are parts of manifolds oriented outwardly from Uf. So the integral over aU$ can be defined as a sum of two integrals according to (15). We shall further assume that Z admits a C-continuation on E and that the divergence theorem can be applied in the form

(18) I do not discuss the latter point in detail because it leads to formal complications; the formulation is exact, but to decide when the conditions are satisfied is left to physical intuition. If we imagine again a porous structure, U”: is the interior of the pores inside U,., a,U$ represents pore walls inside U, and a,Ur is the exterior boundary of Ur on aUX, It may happen that a nonnegligible part of aR (of the walls, say) lies exactly on a&; this is the “singular” case mentioned above (see the first paragraph of this section). If we slightly move U, in an appropriate direction, the singularity disappears. So it is plausible to assume that the points x E E for which the area of aR n aU, (measured by p) is not zero form a set of measure zero in E (see Sard’s theorem[lO], 1623.1). A little boolean calculus with elementary topological considerations show that a& rl R c a$: C (a& n R) U (cW, fl dR). (One shows as well that &U, = 8R n Lr,.) Our hypothesis will thus read: (HI) The set N of points x E E such that the assumption (18) is not valid or that g(aU, n R) # p(a,U:) is a negligible set in E, i.e. A(N)=O.

because dlJ, n R C &U~, Z is bounded and p is a positive measure. Hence in the integral of {V . Z)* over U,, (V +Z)* can be replaced by the values (18) where (20) is applied. The local average then equals

(14)

(Z)* = (XRZ).

au:

in E

(19)

Now R is compact by hypothesis. Z, originally defined in R only, has been extended over E; this can be done in the manner tha_t Z equals zero outside a compact neighbourhood of R ([lo]: 16.4.3). So 2 is bounded in E. Moreover (loc$.), if W is an arbitrary open neighbourhood of R there exists a C”-function & such that 0 5 $‘w 5 1, tiw is zero outside W and ~Jw(x) = 1 for x E I?. We thus have, for an appropriate choice of norms, n.xRZdp-

s Ilzll,~Lcr w - R) n au,).

(22)

Let US now strengthen the hypotheses on R and U,: (Hz). Th_ere exists a sequence of open neighbourhoods W. of R such that limpI(W.-R)nau,)=o, “-rm

limn(W,-R)=O “.CC (23)

for almost all x E E. (The second limit is independent of x.) Compare with the hypothesis (HI). In a porous material, we may imagine that W,, is contained inside a surface traced outside the pores (i.e. in the solid) at an arbitrarily small distance from the interface aR. Observe that in the “singular” case discussed above (a portion of aR lies on au.), the first condition of {23) cannot be satisfied. The hypothesis (Hz) adopted we have, with W= W, in (22) n . Ilw,Z dfi

(24)

for almost all 5 E E. Because the right-hand integrals are uniformly bounded by (I.Z/F(aUd (consider the translations), the Lebesque convergence theorem ([9], 13.8.4)

V. VEVERKA

836 applies and we have

But &Z satisfies the conditions of validity of (11); moreover, (qjwJ) is defined and of class C’ everywhere in E (see the proof of (4) and consider the translations of U,), hence the divergence theorem ([12]. I, 4.10) applies and we have

On applying the second condition of (23) to the r.h.s. of (26), we may procede in an analogous way as above (see (22), (24), (25)) to obtain lim

“-

I au,

dcc.(&)n* 9 +wrJ dA

I

= From (21), (25X27),

I au,

WEIn.

I

ucXRZ dh.

(27)

((divPXx)

(14) and (12) we finally have

(P ’ W)(x) = _: j-, n * (z)* dp + (dizxx)

(very thin) three-dimensional layer (see a mention in [l]. 2.3.5 or [13]), a physical quantity Z defined in the phase R will satisfy this condition ([lo]: 16.4.3). If the U, are given, the remaining conditions concern R (and Z); see the paragraphs sub (18) and sub (23). The assumption (18) will seem to be trivial to those who are prone to accept the divergence theorem as a mechanical rule; for those who are familiar with the delicate mathematical nature of the general Stokes’ theorem (whose the former one is a special case), the assumption is no more trivial, but still plausible: on the one hand, observe that aU2 is not necessarily a differentiable manifold (imagine the sphere HJ, intersecting the pore walls); on the other hand, recall that the divergence theorem still holds for a cube, a cylinder etc. In simple terms, the remaining conditions require that aR (e.g. pore walls) be a “surface” in the good intuitive sense of the word. Observe that this hypothesis becomes disputable if the diameter of the pores approaches the thickness of the interface layer; in such a microscopic scale, the idea of a “surface” becomes vague. In the latter case however, we probably would prefer to regard the porous material as a continuum (or take recourse to molecular theories). Let us now designate

(28)

where

1 = h(~,)

I

au, n . F dp

for an arbitrary tensor field F (order_ z 1) defined everywhere in a neighbourhood of U,, universally measurable and bounded. Observe that if F is of class C’ we have (8) (div)F = @ * F).

for all x E E, see (16). We have assumed that all U, have been obtained by-translation from a given U, (a&,= al&, U, ope_n, U, compact with boundary), that R is open, t?R = aR where R is a compact with boundary, that Z, defined in R. admits a C’-continuation on the whole E, and that the hypotheses (HJ with (19) and (&) with (23) are satisfied. Under these conditions, the relation (28) is an exact equality and the whole assertion an exact mathematical lemma. For an amelioration of the lemma, see the Appendix. For those who like generalizations, let us mention that no assumption on the (finite) dimension of E has been made, neither any other assumption on the “nature” of R or Z was necessary. DISCUSSION

The result (28) is, mathematically, considerably weaker than would be (1) if valid. I shall now show that nevertheless, it contains an analogous physical information. Let us first review the hypotheses. The conditions concerning the “averaging volumes” U.. are trivially satisfied if U, is an open ball (which appears to be the most natural choice; however, see later). The condition concerning the continuation of Z could be weakened by a more elaborate mathematical formulation, but this condition is plausible: if one imagines the interface as a

(30)

(31)

The formula (28) can now be rewritten in the suggestive form ((V * Z)*) = (div)(Z)* + (di . Z) (32) which is “almost” the formula (l), apart from the double averaging. (Observe again the remark before (8), VZ= V *(SZ).) See Appendix in addition. I propose [14] to call the volume-averaged quantities such as (8) or (12) “empirical values” (of Z) and the operator (div) (30) the “empirical divergence” (in an analogous way, one can define the “empirical gradient” and “empirical curl”). In effect, if measuring a quantity one always takes a sample from a finite volume (say, U,) and the divergence is obtained from the balance of such a UX according to (30) and (31). Recall that many engineers (at least the elder generation) really imagine the divergence as a “limit” of an expression such as (30). Because (V - Z)* and diZ are themselves already integrated quantities, it is temptating to replace their volume averages by local values. From well-known integral inequalities it follows immediately that such an approximation will be plausible if (F = (V - z)* or d,Z) IIF

- F(x)ll e F,

where F, is a “characteristic

for t E Ux

value” of l/Fll considered

(33) as

The local volume average of a gradient revised

relevant and where the symbol 4 (“negligible”) is interpreted according to the knowledge of the problem and physical intuition. (For the norm llFll recall that on a finite-dimensional vector space, all norms are equivalent[l5]: I, 1.6.2.) We then can write (V +Z)* = (div)(Zr

+ diZ

(34)

with ail due caution as regards the symbol = (“approximately equal”). Looking upon the volume averages as on empirical quantities, another natural condition is that of “reproducitiliby”. In other words, if changing topologically the averaging domain UX associated with a fixed point x (e.g. taking a cube of the same or slightly different volume instead of a ball), the difference between the two volume averages has to be negligible if compared with a characteristic value; see (33). Obviously, this may became impossible in a heterogeneous medium if V, is too small. (Observe that this condition is also an argument against excluding the “singular case” discussed above; see Appendix.) So F = (V 9Zj* or F = diZ must not change considerably in the range of the “averaging volume” IJ,. But because F is an integrated quantity, Lr, shall not be too small if a heterogeneous material is considered; otherwise, F would be again a fluctuating quantity. Now what is the goal of introducing the “empirical” (local average) quantities? Finally, we wish to obtain a differential equation; we hope that in a “larger scale of observation” the course of the new (“empirical”) variables will be sufficiently smooth to allow us simplifying the boundary conditions, introducing some (more or less empirical) transport coefficients etc. to “close” the set of equations obtained. This smoothness is the hypothesis which also leads to the approximations such as (34). So the “averaging step” is, in fact. an idealization of the problem. Physically (not exactly mathematically), the situation is analogous to that in passing from molecular models to continuum theory. I believe it is an illusion to hope that a mathematical model of anything real can be obtained by pure mathematical deduction. The starting step (idealization) and the final step (interpretation) depend on good knowledge of the problem, experience and what may be called physical intuition. This philosophy accepted, let us revert to the approximate relation (34). The terms such as V. Z or VZ occur in balance and transport equations. The equation is subjected to averaging and the resuh is an equation in average (“empirical”) quantities. We now idealize the problem on considering the empirical quantities as “exact” local quantities and accordingly, we replace the empirical divergence in (34) by the differential divergence (or gradient) operator. We then expect that also the solulion of the set of the new differential equations will represent a plausible idealization of the (more complex) reality. This philosophy also rehabilitates the formula (I) as regards its application to reasonably selected problems. To point out that a thoughtless application of the (apparently exact) formula (1) may lead to errors was the

837

goal of the critical considerations

above. I believe that the authors cited at the beginning did not commit any of such errors. Porous materials such as sand, fibrous filters and the like containing a fluid characterized by not-toosteep overall gradients will probably well satisfy all the necessary hypotheses. Nevertheless, one need not be surprised if a theory based on averaged differential equations fails in some cases. One should not forget that usually, the resulting differential equation is a secondorder one, so the hypothesis (div) = div (or (grad) = grad) is applied twice; it is a delicate matter to differentiate an approximate functional equality. In the report[l4], I have applied an analogous technique (developed independently several years ago) to transport phenomena in industrial columns such as absorbers with packing and the like. The packing is also a “porous material”, but the scale of observation necessary for averaging is considerably coarser. One also arrives at a second-order differential equation where the second-order terms represent mixing (e.g. axial mixing). My personal knowledge is that the “diffusion-type” models of axial mixing in columns are subject to criticism based on experimental evidence. In effect, some types of packing obviously cannot satisfy the conditions for averaging differential equations. CONCLUSION

It appears that no rigorous mathematical theorem will allow one to derive an exact differential equation in averaged quantities from an equation valid in one phase of a heterogeneous material. A useful approximation (34) of an analogous physical content as the “theorem for the local average of a gradient” is still at hand. In reasonably selected problems, the resuhing differential equation in averaged quantities can represent a plausible idealization of the problem. NOTATION E

n R U, U: z+ V Z

euclidean vector space (fixed) unit exterior normal vector to surface region in E, for example pores in a solid neighbourhood of point x E E (“averaging volume”) U,nR transformation velocity (5) A( U,) (Volume of 4) arbitrary tensor field on E

Greek symbols A Lebesque measure in E (usual integration) p Lebesquean measure on surface (usual surface integration) cp isotopy delined before (4) xR characteristic function of R, see (14) Other VZ V.Z (div) (Z) {Z)*

gradient of Z, see (2) divergence of Z, see empirical divergence, local volume average local volume average

(6) see (31) (8) relative to R (12)

V.

838 diz

VEVERKA APPENDIX

see(29)

K

(topological) closure of set K c E dK boundary of region K, oriented outwardly see (16), for example pore walls inside U, Wl: &U$ see (17), exterior boundary of U: on 8C

The

functional equality (28) can be expressed in more general form. In the second averaging in (21), the domain U. can be reolaced by an arbitrary open neigbbourhood &(” of x (dU2” = a&(“, UXo’compact with boundary). In the same way, we obtain

RJCFERENCES

Ul SIattery J. C.. Momentum, Energy and Mass Transfer in Continua. McGraw-Hill, New York 1972. PI Bird R. B., Stewart W. E. and Lightfoot E. N.. Trclnspori Phenomena. Wiley, New York 1960. Whitaker S., Chem. Engng Sci. 1973UI 139. ;:; Gray W. G., Chem. Engng Sci. 1975Xl 229. PI Friedman F. and Ramirez W. F., Chem. Engng Sci. 1977 32 687.

WI Whitaket S.. Ind. Ennnn Chem. Fundls 1977 16 408. 171Lehner F. i.. Ind. &gig Chem. Fundls 1978 18 41. [81 Dieudonnk I.. El6ments d’Analyse I. Gauthier-Villars, Paris 1969. [91 Dieudonne J., Efiments d’dnafyse II, Gauthier-Villars, Paris 1968. I101 DieudonnC J., EMments d’ilnafyse IIZ. Gauthier-Villars, Paris 1970. [Ill Mac Lane S. and Birhkoff G., Algebra. Macmillan, New York 1969. [I21 Cartan H., Formes di~~rentieltes. Hermann, Paris 1967. [I31 Slattery J. C., Ind. Engng. Chem. Fundls 1967 6 108. [I41 Veverka V., Transport phenomena in industrial columns. Research Institute of Inorganic Chemistry, Usti nad Labem, Czechoslovakia 1977. Ml Cartan H., Calcul difft+entiel. Hermann, Paris 1967.

for all x E E and for arbitrary such ,$I’. Dividing by A(LI:‘)), a generalization of (28) is obtained. Because (see the hypothesis on z)

where the measure of the set of t such that ,y,,,(t)f~~,(t) approaches zero if y approaches x (consider the translations), {V+Z)* is continuous. If C!>) is for example, a ball of center x and radius r, -I 0, V. = A( Ux(“))we have (V .

Z)*(x) = lim $ “- (

I

arr.“, II. (29’ dp +$-jup,

(da dl). (A2)

If dJ is continuous at point x, both right-hand limits exist separately, the second one being (did)(x). But d,Z is not continuous at all Y E E; imagine again the “singular case” in (29). The condition of “reproducibility” (see the paragraph following after formula (34)) leads me to base the discussion on the formula (28).