Internal energy transport in adiabatic systems: Thermal taylor dispersion phenomena

Pergamon

Inr. J. Non-Lineor

Mechanics. Vol. 29, No. 5, pp. 639-664, 1994 Copyright 0 1994 Elsevier Science Ltd Pnnted in Great Britain. All rights reserved W2&7462/94 $7.00 + 0.00

002&7462(94)00013-l

INTERNAL ENERGY TRANSPORT IN ADIABATIC SYSTEMS: THERMAL TAYLOR DISPERSION PHENOMENA R. P. Batycky, D. A. Edwards and H. Brenner Department

of Chemical

(Received

Engineering,

Massachusetts Institute MA 02139, U.S.A.

2 October

1993;

in

revised

form

of Technology,

Cambridge,

9 May 1994)

Abstract-This paper outlines a general theory of thermal convective
NOMENCLATURE B B c ; D 6*, b* ik

I j, J j,, J, k K k, K e

’j

M, n nj il PIII Pe :4 90 aqo

Contributed

vector field defined in equation (2.57) or (3.59) vector constant defined in equation (2.58) or (3.60) specific heat capacity solute concentration vector constant in equation (2.84) molecular diffusivity scalar or dyadic dispersivity unit vector in kth direction dyadic idemfactor local- and global-space flux vectors of T or P polyadic local- and global-space moments defined in equations (2.29) and (3.30) scalar thermal conductivity partition coefficient in equation (1.11) or heat capacity ratio in equation (4.11) local- and global-space thermal conductivity dyadics characteristic local scale length basic lattice vector m-adic total moment of P defined in equation (2.38) or (3.39) distance in the direction normal to a surface integer in equation (3.3) unit normal vector on local-space boundary aqo triplet of integers in equation (3.6) conditional probability density or Green’s function defined in equation (2.17) or (3.15) m-adic local moment of P defined in equation (2.26) or (3.28) Peclet number local-space matrix of bounded configurational variables local-space “volume” element bounded local-space “volumetric” domain local-space “areal” boundary encircling q0

by K. R. Rajagopal. 639

640

R. P. Batycky

et al.

1:::;( );

global-space matrix of unbounded configurational variables global-space “volume” element unbounded global-space “volumetric” domain global-space “areal” boundary encircling Qm, local position vector in a unit cell volume element in a unit cell circular cylindrical coordinate in Table 1 Reynolds number position vector discrete position vector specifying, say, the centroid of the nth unit cell continuous position vector Ieplacing discrete position vector R” in equation (3.74) volume elements in R and R spaces faces of a unit cell directed area1 vector of the jth face of a unit cell directed element of surface area on a unit cell face time temperature initial, uniform temperature volumetric internal energy density defined in equation (4.1) local- and global-space vector velocity fields mean thermal propagation velocity vector fluid velocity vector field volume small volumetric domain domain of entire periodic medium in equation (3.7) mean interstitial fluid velocity vector through porous medium dyadic field defined in equation (3.65) axial coordinate in Table I thermal diffusivity scalar or dyadic effective thermal dispersivity scalar or dyadic m-adic local-space field defined in equation (2.30) Dirac delta function of argument ( ) Kronecker delta Kronecker delta with a “vector”-index argument unit normal vector on a particle surface in a porous medium mass density superficial domain of a unit cell combined faces of a unit cell volume fraction in main text or azimuthal circular cylindrical coordinate angle in Table a q-space field in equation (2.34) continuous phase discontinuous phase momenta1 order normal direction local-space or unit cell domain local space global space thermal unbounded global-space domain parallel perpendicular

(. .)C ( )M (...)* (...) (...)” (...)’

convective contribution molecular (or conductive) contribution refers to a macroscale phenomenological coefficient initial value at time 2 = 0 steady-state value achieved at long times transposition operator

(...) (.Y-Y) U,(. sym(

denotes a mean value or a macroscale value nonconvective contribution scalar convectivediffusive linear operator defined in equation (2.34) symmetrization operator for a dyadic defined in equation (2.35) denotes a norm in equation (2.44) or a jump in the value of a function denotes a union of the local and global spaces.

Q

dQ gr br R Re R R _” R d3R, d3i% s,, 2s t T TO ll u, u tr* v V AV !!o V* W z a, a G(*,OS* LT...) 6 sE, Y P 2, 4 (“.A ( )d (. )ln ( )” 1:::; (A! (.“)T

II( 0

)

1II

)

1

in equation (3.38)

1. INTRODUCTION In a previous analysis [l] (hereafter referred to as Part I), the authors applied generalized Taylor dispersion theory to the problem of internal energy transport in an insulated circular cylindrical tube containing a flowing fluid. The present paper generalizes that specific problem to a much broader class of problems, including both continuous (duct flow) and discontinuous (spatially periodic) systems. Two applications of the generic macrotransport theory to thermal transport processes in spatially periodic media and composite materials are presented to illustrate the theory.

Thermal

Taylor

dispersion

phenomena

641

1.1. Motioation Consider a static packed bed or composite material for which the continuous (c) and discontinuous (d) phases each possess identical, uniform, temperature-independent thermal diffusivities, a, = ad = a, say = constant. As such, the local temperature distribution T at every point of each phase obeys the unsteady-state heat conduction equation aT

(1.1)

-=uV2T, at

where c1is constant throughout. Of particular interest in engineering applications is the effective thermal diffusivity Cr*of the bed, namely the constant appearing in the unsteadystate heat conduction equation

(1.2) governing the mean temperature distribution T at each “point” of the composite bed, regarded as a continuum. Given the present circumstances, involving a thermally homogeneous bed, it would appear natural to suppose that the effective diffusivity Cc*of the bed is merely the constant pointwise scalar diffusivity a common to both phases; that is, &*

2

a

(1.3)

In this same context, were the respective materials composing the continuous and discontinuous phases interchanged to produce a new bed, it would appear that the resulting “inverse” bed should possess exactly the same effective diffusivity (1.3) as the original bed. Remarkably, neither of these conclusions is true. Indeed, depending upon specific circumstances, I%*for one of the above two bed configurations may be less than c1by many orders of magnitude, whereas it may be of the same order as a for the other bed. Nor is it even generally true that the effective diffusivity of either bed will be a scalar, despite the scalar nature of the local diffusivity a; rather, for nonrandomly configured beds, Cr*may prove to be anisotropic. We have purposely posed these questions and facts in a purely abstract c/d setting in a manipulative attempt to inhibit critical thinking that might otherwise be provided by focusing on a concrete physical example; that is, we have deliberately attempted to mislead the reader to encourage his/her interest in what follows. However, had we initially posed the example cited below to illustrate the pertinent issues, it is certain that any knowledgeable reader would have intuitively (and immediately) seen through the generic c/d ruse chosen to motivate the potential reader, for the following extreme example renders the otherwise subtle ruse immediately transparent. Consider the case illustrated in Fig. l(a), where the continuous, interstitial phase is (static) air and the discontinuous phase consists of cast-iron pellets. Inverse to this is the bed depicted in Fig. l(b), where the roles of the cast-iron and air have been interchanged. These cases are deliberately chosen for their dramatic effect, since air and iron are so seemingly disparate in their thermophysical properties. Yet, surprisingly, to a high degree of approximation, cast-iron and air possess the same thermal diffusivities [2], namely about a z 0.2 cm2 s-l. (This equality can be rendered even more exact by changing the air pressure so as to alter the density of the air, as required.) Nevertheless, despite the equality of a for the two phases, it is intuitively clear that bed (a) will possess a significantly smaller effective thermal diffusivity than bed (b), and moreover, that Cr*for the cast-iron continuous bed (b) will be only slightly smaller than a itself. A little thought reveals the source of the above paradoxes. The basic pointwise conduction equation is peg= J=

-V-J, -kVT.

(1.4) (1.5)

642

R. P. Batycky

et al.

cast iron ellets intcfstitia P air

cast iron matrix

Fig. 1. (a) A packed bed consisting of cast iron pellets (shown shaded) with static air (shown white) occupying the interstices; here, air constitutes the continuous phase. (b) A packed bed consisting of a continuous cast iron phase with the voids filled with static air; here, the air constitutes the discontinuous phase. In each of these two modes of arrangement the cast iron and air phases possess essentially identical thermal diffusivities, a z 0.2 cm’s_ I.

These indeed

combine

to yield equation

(1. l), wherein

(1.6) when the thermal conductivity k is a pointwise constant, independent of position within each of the two phases. However, whereas (x is (approximately) the same for air and cast-iron, the individual thermophysical properties p, c and k are quite different for these two materials [2]. Most importantly for our purposes, k is discontinuous across the phase boundary, so that the required pointwise normal-flux interphase boundary condition, J, (air) = J, (cast-iron),

(1.7)

for points lying on the phase boundaries necessitates the existence of discontinuity in the normal component aT/c?n of the temperature gradient at such phase interfaces. As such, equation (l.l)-which implicitly requires continuity of first derivatives of the temperature field-is violated at a phase boundary, and hence is not, in fact, applicable at all points of the system. [Technically, (1.1) applies at all points, with the exception of a set of points of measure zero constituting the phase boundaries.] This deliberately dramatic, effective-medium c( = constant illustration (which is quantitatively resolved in Section 3.6 as a special case of the subsequent theory) shows the need for a comprehensive general theory of thermal macrotransport phenomena. [The reader who still believes that there is no “paradox” to resolve, and that all is indeed intuitively obvious, would then do well to contemplate the following molecular diflision example. Consider the case of a solute diffusing through a quiescent two-phase medium, while possessing the property that its molecular diffusivity D is everywhere constant and is the same for both the continuous and discontinuous phases; that is, D, = Dd = D, say, = constant. As such, the local solute concentration distribution C at every point of each phase obeys the unsteady-state molecular diffusion equation c7c -= & whereas

the effective diffusivity

with fi* = constant

o* obeys the equation

the effective molecular

(1.8)

DV2C,

diffusivity

[3-61

of the solute, and c the mean solute

Thermal

concentration.’

Taylor

dispersion

643

phenomena

Again we raise the question of the equality

o* L

(1.10)

D,

and again the answer is generally in the negative, but this time for a very different reason. Here, the nonequality “paradox” is encountered in circumstances where unequal solute partitioning occurs between the two phases, so that the interfacial boundary condition is Cd = KC,,

(1.11)

rather than that of the continuity of C itself across the phase boundaries! interphase solute distribution coefficient. In such circumstances

Here, K is the

(1.12) with r. a representative volumetric domain (and dV a volume element); rather, when K a constant,

is

(1.13) wherein rd = $ro and t, = (1 - $)ro, with 4 the discontinuous phase volume fraction. The commonality of the thermal and material “paradoxes” is the existence of a “capacity factor”, namely pc in the thermal case and K in the solute species case. In one case, this capacity factor appears in the differential equation, whereas in the other it appears in the interfacial boundary condition. As subsequently discussed in the first portion of Section 4, this distinction is, however, somewhat artificial; rather, in a clearly defined conserved property context, the capacity factor is really interfacial in both cases.] 1.2. Preliminary considerations The general form of the microscale internal energy equation (neglecting viscous dissipation and work terms) may be written as [7]

,cg+ “.VT) =

(1.14)

V.&.VT),

where kT is the thermal conductivity dyadic and v the local fluid velocity vector. We shall henceforth suppose that p and c are independent of time (although not necessarily of position). For our purposes a more useful form of the previous equation may be obtained by using the continuity equation V*(pv) = 0 (1.15) to rewrite (1.14) in the form a(PcT) at + V. JT - (v.Vlnc)pcT

(1.16)

= 0,

in which JT “zf pcvT+ Concomitantly,

whereas

in the case of material

transport

(1.17)

kT.VT. phenomena

one has that [4]

CdV for the relation between the pointwise and mean concentration fields C and C, respectively, appearing in the effective diffusion equation (1.9X an equally simple volume-average counterpart between Fand Tdoes not exist for the comparable thermal problem owing to the intrusion of the thermal capacity factor pc. Thus, the functional generic relation existing between these macro- and micro-temperature fields must be established in any comprehensive thermal macrotransport theory. * As in (1.7) with J ‘zr - DVC, we require boundaries,

namely

.I.,, = J,, d.

continuity

of the normal

flux components

across

the phase

644

R. P. Batycky

et al.

Here, the thermophysical properties p, c, kT, as well as v, are assumed to be specified functions of position. The last term in (1.16) plays a role similar to that of a first-order chemical reaction, or source term, creating (or removing) internal energy pcT from the fluid continuum. This term is nonzero if gradients of c exist along a streamline. Henceforth, we will limit subsequent attention to the class of problems for which v-vc so that this “source” Under the various

= 0,

term disappears from the governing equations. hypotheses made, the internal energy equation pcg+V-J,=O.

2. CONTINUOUS

(1.18)

may be written

as (1.19)

SYSTEMS

2.1. Generic coordinate system Following the generic scheme employed in material macrotransport processes [4], we introduce global- and local-space independent variables (“coordinates”) (Q, q) E R, with R the position vector characterizing a generic microtransport system in physical space R; that is, R=Q@q. (2.1) The global subspace Qw, representing the domain of Q (Q E Q,), will always be unbounded, whereas the local subspace qo, representing the domain of q(q E qo) will generally be bounded. The respective subspaces spanned by Q and q will be supposed mutually orthogonal in the sense that Q-space and q-space operations are commutative; that is, Q and q are regarded as independent variables. Since V = (@R),, we also introduce the global- and local-space vector gradient operators:

a iiQ

()

vQ=

q,l’

(2.2)

(2.3) such that v =

vQ@v,.

(2.4)

The phenomenological coefficients (v, kr, p, c) appearing in (1.17) and (1.19) are henceforth assumed to be functions of only the local-space coordinate q-being independent of Q (and t). Also, we deconstruct v and kT into their respective projections (U, u) and (K, k) onto the Q and q subspaces. Moreover, it will be further assumed that respective q- and Q-space thermal transport processes are uncoupled. Thus, the following decompositions are effected: vrU@u,

(2.5)

k, E K@k,

(2.6)

JT E JOj,

(2.7)

wherein upper- and lower-case letters refer, respectively, to the Q- and q-space projections. In this notation, according to equations (1.19) and (1.17) the temperature field Tis governed

T(Q,q,t)

(2.8)

by the equation ~-‘g+

VQ*J + V,-j = 0,

(2.9)

Thermal

Taylor

dispersion

645

phenomena

characterized by the respective global- and local-space flux densities J = pcUT-

(2.10)

KSVoT,

j = pcuT-

(2.11)

k*V,T.

As a specific example, Table 1 constitutes an equivalence table, relating the present generic notation to the specific notation apropos of the tube-flow thermal dispersion problem addressed previously in Part I (see fig. 1 of that paper). 2.1.1. q-space boundary condition. With aq, the boundaries of the local-space domain q0 and n a unit outward-drawn normal vector on 8q 0, we have for the case of an insulated system that (2.12) n-j = 0 on aq,. It will further be supposed that (2.13)

on aqO,

n-u = 0

whence equation (2.11) requires satisfaction of the temperature boundary condition n-k+V,T=

0

(2.14)

on &I,.

2.1.2. Q-space boundary condition. With 8Qm denoting the (indefinitely large) external boundary of Q,, we require that

lQ”l{(T- To),(J - wUTo)l + {O,Olon 8Qm (i.e. as IQ1 + cc)

(m=O, 1,2,. . .), (2.15)

where To denotes the uniform temperature of the system prior to the introduction heat into the system at time t = 0.

of any

2.1.3. Initial condition. Suppose at time t = 0 that heat is added to the system over some region of the fluid. This is equivalent to specifying an initial temperature distribution, at t = 0,

T = T(Q,q,O)

(2.16)

where this distribution is subject to the requirement (2.15).

Table 1. Equivalence between generic and specific notation for the transport of heat in an insulated circular cylindrical tube; (R,&z) denote circular cylindrical coordinates with z drawn along the tube axis; additionally, (iR,i,,i,)represent unit vectors in this coordinate system

Application: Generic

Q P Qm

internal

Equivalence energy transport

Table in an insulated Specific

1zl {R, 41 {-CC
FQ dq

dz RdRd+ d

VQ

v,

U(q) u(n) K(q) k(q) p(q) 4l)

a

i,dz

ia

i,-- + i,--

i?R RW i,U(R,4) 0 i,ik(R,+) (id,+ i&)k(R,4) p(R4) 4R4)

circular

tube [l]

646

R. P. Batycky

et al.

2.2. Green’s function formulation As in Part I, it proves useful to reformulate the microscale temperature problem in terms of a Green’s function [S]. In this context, define P as the kernel appearing in the following integrand:

VQ> 9,t)- To= Q 9 p(q')c(q')P(Q,q,tlQ',q')CT(Q',q',O) - TolWdQ'. (2.17) ssI ” Following similar arguments to those of Part 1, we can express terms of the closely related function

peg

+ V,.J

(2.9))(2.16)

f’(Q,atls’)

P = (in which we have set Q’ = 0 without

equations

in

(2.18)

loss of generality),

to obtain

+ V,.j = S(Q)S(q

- q’)d(t),

(2.19)

where”

with boundary

J=pcUP-K.VoP,

(2.20)

j = pcuP - k.V,P,

(2.21)

conditions

lQl"{P,J}+10,0) asIQI+= @=0,1,2,...), and the pre-initial

(2.22a) and (2.22b)

n.k*V,P

= 0

on “7q0,

(2.23)

P = 0

V(Q.q;t

< 0).

(2.24)

condition

Integration of (2.19) over the entire domain, conditions, yields

together

ss < Q, 9

p(q)c(s)P(Q,s,tlq’)dsdQ

with use of the preceding

=

boundary

(2.25a) and (2.25b)

0

Accordingly,

the quantity

whose volumetric

density

is pcP is conserved

2.3. Moments of the probability density 2.3.1. Local moments. Define the m-adic local moment P,(q,tlq’) Differentiation

Ef

s Q,

Q”f’(Q,q,tlq’)dQ

for all time.

of P as

(m = 0, 1,2,

. 1.

(2.26)

with respect to t gives

dPm _ at

Direct substitution recurrence equation

s

Q Q’$dQ.

,

of (2.19) into the above produces which P, (q, t I q’) must satisfy: pc$

where drnOis the Kronecker

+ V,.j,

= rm + 6,,6(q

the following

- q’)d(t).

partial

differential-

(2.28)

delta and j, = pap,,,

rm = -

- k.V,P,,

s Qr

5 The use of the same symbols confusion.

(2.27)

/

Q”(V,-JNQ.

(2.29) (2.30)

(J,j) here as in (2.10) and (2.11) for slightly different fluxes should not lead to any

(sp’z) b-z)

(sfvz) PUE (“&Pa uoy?&]u~ uodn u!t?$qoa~

(op’z)

UIOJ~ :o =

Ob

(ZP’ZI

‘W”a

+

‘dm’d)

s

ob

ui

UIICS~ = __3P zEUP

= - 1P

‘bpodncld

(IP’Z)

‘EUP

s

(OP’Z)

= _ JP “MP

.bp%d “de

f6E'Z)

San!8 1 03 IDadsal qIy uo!wy_Ia.IaJJ!a Ob

(8f ‘Z)

*(.

. . “Z’[‘(j

=

tu)

bpLbl~‘b)“d(b)~(b)d s

%I!MO[lOj

GXJJ SB d JO “MI

~UXUOLU

[l?lOJ

XN-U4

aq$

= Ol~)“%u 33P

luauroru 1v301aql Jo a%tmna

:“a

pal@M

‘(0 > 3 tb)A

(LE’Z)

N.IIJaa

‘s~ucXLK.W 1010~

‘Z’E’Z

0 = "d

uo!lfpuo~ py!u!-ad put: Obe UO

(9f’Z)

0 =

“d’A.3.U

uog!puoo krepunoq aq~ 01 XDafqnspah{os aq 03 a.z (ff’z)-(~f’z)

suoyenba

‘(LV + v)f J; VWLS

(SE’Z)

:v ~pelCp due JOJ ‘a~oumqyn~

.A uopmnj acwds-b 6ue JOJ

SE pauyap s! “&=.~o~emdo leaug acmds-b aql 2mqM

‘(l)q(,b - b)q = Odbx f

(IE’Z)

$nd :SMOl[OJ

SE

a.m swawout 1~301Maj ls~y aq$ 6q paysges suoymba

aqi .XOJsuo!ssaldxa l!sqdxa

R. P. Ba~yycky et

648

ai

and C?is some characteristic linear dimension of the local space, Pa may be expressed in the asymptotic form JWh&Ct

= K?(4) + exp,

(2.46)

where “‘exp” denotes terms in qI (;I’and t that are ~xpon~t~a~~~ ~~t~~~~ed in time, Thus, (23 1) asymptoticaliy adopts the following steady-state form: V,*j$ = 0,

(2.47)

where i?(q) ““=” pcuP$ - k-V,Pg>

(2&q

and subject to the boundary condition **k+V,P?

=O

onl?qn_

(2.49)

Moreover, from (2.38), together with f2.43a), (243b) and (2.&S&one obtains the normahzation requirement pcP$dq s 90

“-*1,

(2.50)

whwe we have defined

as the local-space average of PC, and

wli”

I:

Fnrroduce (2.46) into (2.41) to obtain

r~te~a~o~ of(2.54) yiefds MI N ut -t iEi+ exp, where B is a constant of integration, yet to be determinedS Subject to a posteriuri verification+ we assume the following asymptotic fcr J?, :

P,[q,efq’S 2%~~~q~~~~ -5 WqjS + exp,

(2.56)

triaf solution

psj

wherethe function B(q) remairxs to be determincd:cl.Introduction of the latter into (2.38) with rmr: 1, together with use of the normalization condition (2.5OL shows upon comparison with (2.56) that the: constant B is given by the expression

Thermal Taylor dispersion phenomena

649

Trt derive the ~uat~o~s gover~jng the fJ_field, subs~tut~ (2.46) and (2.57) intu f232f to obtain, exactly: (2.59) i$+,B - V,=fP$‘bV,B) = pcP$(U - a),

upon using (247). This is to be solved subject to the boundary condition Pga*k~V~lB = 0

on aq,,

(2.60)

derived from (2.X& ~2-~~, (2.57) and (249) (with sz = ff T%e fatter pair of~~a~~~s serve to uniquefy determine If(g) only to within an arbitrary additive (although p~ysi~~ly irrelevant) constant.

The

constant d can be expressed in the form a = P

+ die,

(2.63)

where BM is a purely ~o~du~t~ve [i-e. ~o~~o~v~~tjve~ ~o~~~~~t~on and ~3~is a purely ~nv~t~ve ~on~~but~on. To a~orn~~~sh this separation of ~on~onveet~ve and ~onv~t~v~ efFects, add and subtract &’ from PO”appearing in the first term in (2.62)+Using (2SI) we may then write the ~ond~~~~vepart as (2.64) where

and the convective part as

The latter c&n be expressed in an afte~~t~v~ form upon multiplying (2.59) by I#, ~~bstitut~~~ the result into (2.66), and simplifying via use of the boun&ry condition (260). Eventually, one obtains

Following dose@ the a~~~~~~ts advanced in Section 4 of %rt i, we wish t0 obtain from the microscale description f2.19)-(2.24) a comparable generic macrotransport energy equation gaverning a macroscale tern~ratu~~ f(Q,t). To this end, we defke a macroscale Green’s function P as

where the eoastant PC* reniains to be d~t~~i~ed. ~iffer~~t~~t~ the above equation with respect to time, substitute from (2.191, anal.use the boundary conditions (Z.X?a),(2.22b) and

650

R. P. Batycky

e/ al.

(2.23). This yields w*g

+ V&i

= s(Q)s(t),

(2.69)

wherein (pcUP s 40

J(Q, t/q’) zf

- and in which {P, J) satisfy the global-space --

lQ”l{P,J}

- K.V,P)dq

boundary

+ (60) as IQI +x

(2.70)

conditions

(m=0,1.2,.

As in Part I, for long times [cf. (2.44)], p will eventually hence eventually attain the asymptotic form

..).

(2.71a) and (2.71b)

become independent

of q’, and

(2.72)

p = p(Q, t).

Moreover, we assume (subject to a posteriori verification via a moment-matching scheme) constitutive form that J z J(Q, t) can be expressed in the macroscale convective-diffusive J = p*u*p-

K*.V,P,

(2.73)

where PC*, U* and K* are respectively, a constant scalar, vector and (symmetric) dyadic to be determined, and VQ = (a/aQ),. Together with (2.69) this yields the so-called model thermal macrotransport equation governing the macroscale Green’s function: _

E+

pc* i

?t

u*.v,p

- ---

= K*:V,V,P

The latter is to be solved for p(Q, t) subject as well as the pre-initial condition

to the boundary

P=o

“zf E* I 0,

conditions

(2.71a) and (2.71 b)

(2.75)

P of the above model equation

(m = 0, 1,2,

Q”&Q, t)dQ

The goal is to asymptotically match these total moments moments M, (2.38) of P, so that ii&,(t)

(2.74)

(t 2 0).

(r -=z0)

derived from (2.24) and (2.68). Define the total moments i%l, of the solution M,(r)

+ s(Q)s(t)

= M,(tlq’)

+ exp

as

).

of p with the corresponding

as t -+ ;c.

(2.76) total

(2.77)

That this matching proves possible to dominant terms in t, independently of q’, ultimately furnishes a posteriori verification of the asymptotic applicability of the macrotransport equation (2.74) in addition to simultaneously furnishing formulas explicitly expressing pC*, U* and K * in terms of the microscale data, namely pc, U, II, K and k-each being prescribed functions of q. Differentiate (2.76) with respect to time and use the model equation (2.74) to obtain the following generating equation: di%I L=l.‘c* dt

(2.78) IQlm{~,vQp}

+ {O,O}

as IQ1 -+ c.

(2.79a) and (2.79b)

m = 0:

Setting

m = 0 in (2.78) and using (2.79a) and (2.79b) gives dn?, ~ dt

= d(t).

(2.80)

651

Thermal Taylor dispersion phenomena

Integration yields (281a) and (281b) Comparison with (2.43a) and (2.43b) shows that n;i,=M,

V(t),

(2.82)

exactly. m= 1: Set m = 1 in (2.78) and use (2.79a) and (2.79b) to obtain (2.83) Integration yields (2.84)

R,=U*t+c,

with the vector C a constant of integration. Comparison with (2.56) in light of the matching condition (2.77) yields i3‘*=0 (2.85) and c = B.

(2.86)

Hence, from (2.55): fi* =

~(q)c(q)~~tq)U(q}dq. I 90

(2.87)

m = 2: Set m = 2 in (2.78) and use (2.79a) and (2.79b) to obtain d&& -= dt

2U*IYJ*t + 2sym(FJ*B) + 2sym$.

(2.88)

Comparison with (2.61) in the light of the matching condition (2.77) yields K* zz 3 pc* *

(2.89)

2.5. Relationship between T and T Analogous to arguments given in Part I, we can relate the coarse-grained, macroscale field F(Q, t) to its microscale precursor r(Q, q, t) by first defining F in terms of P as i’(Q, t) - To Ef L ~(q’)c(q’~ P(Q- Q',tlq'}[~(Q',q',o) - TcJdq'dQ', (2.90) ro ISQh qb to obtain

Tli"(Q,t) = I& q dq)c(q)T(Q,q,t)dq, s0

(2.9 1)

where we have chosen -p7;;*=pc.

(2.92)

Ki*=KM+KC,

(2.93)

It follows that where KM is given by (2.65) and, following use of (2.67), Kc is given by the expression symKdq -t F*

Po”(v,B)t.symk.(V,B)dq,

> where the B-field is to be determined by solving equations (2.59) and (2.60).

(2.94)

652

R. P. Batycky

et al.

It is an immediate consequence of (2.74) and (2.90) that i7;asymptotically macrotransport equation

satisfies the

(2.95) subject to the boundary condition ?:-+ T, = const.

asIQI+co,

(2.96)

at t = 0,

(2.97)

and the initial condition r = T(Q, 0)

in which T(Q,O) is related to the initial microscale temperature t = 0 in (2.91).

3. ~ISC~NTI~U~US

SYSTEMS

field T(Q,q,O) by setting

[S]

3.1. Lattice geometry Consider a spatially periodic porous medium composed of generally curvilinear unit cells. The position vector R of any point in the system can be represented by the decomposition R = R, -t- r,

(3.1)

where r is a local position vector within a unit cell and R, is a discrete lattice vector defining the location of each lattice point: R, = n,l, + nziz + ~313, ni=O,

+l,

i2,

+3 ,...

(3.2)

(i=1,2,3),

(3.3)

with {f,,12, is} denoting a trio of basic lattice vectors. The superficial volume of the cell is zo = Ii x 12x I,.

(3.4)

The unit cell is bounded externally by the closed surface i?r, and consists of the six faces, sij (j = 1,2,3); hence. 3 dT"

E

C

(3.5)

S_j@S+j.

i= I

The location of the nth cell in space can be specified either by the position vector R, or, equivalently, by the triplet of integers (nl,n2,n3).

= In}.

(3.6)

With ro{n} the domain contained within the nth unit cell, the entire spatial domain V0 encompassed by the periodic medium may be represented as

where

The directed area of each of the six faces of the unit cell is defined as si = 12x I,,

s2 = 13x II,

sj = II x 12,

along with S-j = - S+j. It follows from these, together reciprocal to one another in the sense that ili*sj = Gijto

{i;j = 1,293).

(3.9aH3.9c)

with (3.4), that Ii and Sj are (3.10)

Thermal

Taylor

dispersion

653

phenomena

3.2. Microscopic formulation

Similar to (1.19) and (1.17) the energy equation for the temperature field T( R, t) may be written as pcg+V.J=O,

(3.11)

where J = pcvT-

L-VT,

(3.12)

in which V = a/aR is the gradient operator. In the present discontinuous theory, no need exists to separate the phenomenological coefficients into local- and global-space components; hence, from (2.5) the vector velocity field U = u in the continuous theory is here replaced by the fluid velocity v at each point R; likewise, from (2.6), K = k, so that k - kT appearing in (3.12) is simply the thermal conductivity dyadic at a point R. The microscale fields p, c, k and v are each assumed to be spatially periodic, with period R,. Since internal energy may freely permeate the unit cell anywhere (with equal partitioning in the continuous and discontinuous phases), no need arises to distinguish between fluid and particle domains (in contrast with the comparable material transport theory). Finally, if closed regions exist within the unit cell that are insulated against heat flow across their boundaries, one need only set the thermal conductivity in these regions to zero. Equations (3.11) and (3.12) are to be solved subject to the boundary condition T+T,,

as IR(-+cc,

(3.13)

together with the arbitrarily prescribed initial condition T=

T(R,O)

at t=O,

(3.14)

the latter subject to the requirement (3.13). 3.3. Green’s function formulation Once again, we formulate the problem in its comparable defining a function P as T(R, t) - To =

p(R’)c(R’)P(R,tlR’)[T(R’,O)

s VO

where d3R denotes a physical-space reformulated in terms of P ass pc;

Green’s function form by

- T,,]d3R’,

(3.15)

volume element. This enables (3.1 l)-(3.14) to be

+ V-J = S(R - R’)@t),

J = pcvP - k-VP,

(3.16) (3.17)

subject to the more stringent requirements IR-R’I”(P,J)+{O,O}

asIR-R’I-+co

(m = 0,1,2, . . . ),

(3.18a) and (3.18b)

together with the pre-initial condition P=O

V(R; t < 0).

Integration of (3.16) over the infinite volume I’,,, accompanied conditions (3.18a) and (3.18b), yields

s

p(R)c(R)P(R, VO

tlR’)d3R =

0 (t < O), 1 (t 2 O),

(3.19) by use of boundary

(3.20a) and (3.20b)

showing the quantity pcP to be conserved for all time.

n Note the difference between the flux J defined in (3.17) and that defined in (3.12). Our use of the same symbol for these two different fluxes should not cause any confusion. NLM 29:5-B

R. P. Batycky et al.

654

3.3.1. Reduction to intracell density. Use of the decomposition P(R, t[R’) to be expressed in the equivalent form

(3.1) enables

the density

PEE P(R,,r,tlR;,r’).

(3.21)

Moreover, since all the phenomenological coefficients appearing in (3.1 l)-(3.14) have been assumed to be spatially periodic, each is, at most, a function only of the local position vector r-independent of R,; hence, it follows that P is dependent only upon the linear displacement R, - Rb, and not upon R, and Rh separately. Consequently, P may be regarded as being of the functional form P E P(R, With this functionality, in the form

- RL,r,tlr’).

the microtransport

equation

(3.22)

(3.16) may be expressed

alternatively

peg +

V *J = 6,,,,6(r - r’)S(t).

(3.23)

Here

02

v=

(3.24)

ar

is the r-space

gradient

IR, - R;I”{P, Continuity

operator.

Conditions

J} -(O,O}

asn-+{

R.,r

(3.18a) and (3.18b) may now be expressed + ~8, f a,

of P across the unit cell faces requires

f a}.

(3.25a) and (3.25b)

that

P(R,-I,,r+I,,tlr’)=P(R,,r,tlr’)

(k = 1,2,3)

(3.26)

for each point r lying on &O. Similarly, if p, c, k and v are assumed to be continuous the cell faces, the continuity of J [defined in (3.17)] is assured by requiring that VP(R,

- lkrr + Ik,tlr’) = VP(R,,r,tlr’)

V(r EC%,,)

3.4. Moments of the probability density 3.4.1. Local moments. Define the m-adic local moment P,(r,

t/r’) Ef x(R, n

- RA)“P(R,

as

- RH,r,tlr’)

(k = 1,2,3).

across

(3.27)

of P as (m = 0, 1,2,

. . ),

(3.28)

where the summation operation here plays a role comparable to integration over the global space in (2.26). Differentiation of (3.28) with respect to time, followed by substitution of (3.23) into the resulting expression, furnishes the following partial differential equation governing P,(r, t Ir’): pc%

+ V*J,

= dmo6(r - r’)d(t),

(3.29)

where subject

to the pre-initial

J, = pcvP,

- k+VP,.

(3.30)

P, = 0

V(r;t < 0).

(3.31)

condition

As a consequence of (3.26)-(3.28), the local moments conditions on the cell faces at, [S]:

lIPI

satisfy the following jump boundary

IIPOII = 09

(3.32)

IIV~OII = 0

(3.33)

II = -

II VP1 II = -

llrP0/l, II V(rPo) II.

(3.34) (3.35)

655

Thermal Taylor dispersion phenomena llpz/I

(3.36)

= llP~P~/~~ll~

(3.37)

IIVP2 II= IIV(PI PI/PO) //I

where the “jump” in the value of any tensor-valued field F between equivalent points lying on opposite pairs of cell faces is defined as P(&,r /(F I/ zf

+ 4) - P(R”,r), (3.38)

or F(r + Ij) - F(r).

3.4.2. Global foments. Define the m-adic total moments M, of P as the following weighted average of the local moments P,: M,(tjr’) zf s lo

(m = 0, 1,2, . . . )*

dr)c(r)P,(r,tlr’)d3r

(3.39)

Differentiate this with respect to time, substitute (3.29) into the resulting expression, and use Gauss’ divergence theorem to obtain

dMn -=

-

dt

ds- J, + &,,s(t). i 8%

(3.40)

Use of the identity (3.41)

enables the former expression to be rewritten as

dMm _-.---_=

ds- IIJmII+

-

dt

(3.42)

&oW),

Owing to their spatially periodic nature, each of the functions p, c, k and v possess identical values at geometrically equivalent points lying on opposite cell faces; thus, we may derive the jump-function condition (3.43)

/IJ,\/ =~cvIlP,lI -k~IlVP,lI, whence (3.42) adopts the form dM, _= dt

ds - k . IIVP,,, II + ijmOs(t).

_

(3.44)

m = 0:

From (3.44) with m = 0, together with boundary conditions (3.32) and (3.331, it follows upon integration that

0 (t < Oh

=

M

0

i

1

(t 2 0).

(3.45a) and (3.45b)

For times t satisfying the inequality (2.44), in which &’is some characte~stic linear unit cell dimension (e.g. rA13),we have analogous to (2.46) that Po(r, t Ir’) cli P,“(r) + exp, where “exp” denotes terms in r, r’ and t that are exponentially (3.29) asymptotically adopts the steady-state form V*J$ = 0,

(3.46) attenuated in time. Thus, (3.47)

wherein J?(r) “GfpcvP,” -k-VP;,

(3.48)

R. P. Batycky et al.

656

with boundary conditions IIPO”II = 0%

(3.49a) and (3.49b)

11VP? 11= 0.

Moreover, from (3.39) together with (3.45a), (3.45b) and (3.46), one obtains the normakzation r~uir~e~t pcPrd”r s 10 m=

(3.50)

= 1.

1:

As a consequence of (XX?), equation f3.34) may be expressed as

IIPI II = - POIIr II. In addition, differentiation (X33), yields

(3.51)

of the right-hand side of (3.35), together with use of (3.32) and (3.52)

IiVP, 11= - (~~~~~~r~. Substitution OFthe tatter two expressions into {X43) with nt = 1 gives

IIJIII = -JolId.

(3.53)

In conjunction with (3.42) we thereby obtain

It follows from (3.43), (3.32) and (3.33) that /IJo 11= 0, enabling Je to be moved within the jump function operator in the integrand of (3.54). Subsequent application of the identity (3.41) in its inverse sense thereby gives (3.55) Introduction of(3.46) into (3.30) and (3.48) jointly with the divergence theorem, followed by use of (3.47), ultimateIy yields

where we have defined (3.57)

Integration of (3S6) gives (3.58)

M1 TZat + fi -+ exp,

where B is a constant of integration to be determined. Subject to u postsviori verification, assume the following asymptotic triai sofution for Pi: Pi@, t\r’) N P$fr)[Ot

4 Bfrf]

i exp,

(3.59)

where the function B(r) remains to be determined, Substitution of the latter into (3.39) with m = 1 and comparison with (3.58) shows that @ is related to I&by the expression

s

BzL1: p(r)c(r)P~(r)Bfr)d3r. %

(3.60)

The eqnatir.ms governing the B-&Id may be found by su~tit~ti~~ the asymptotic forms (3.46) and (3.59) into equations (3.29) (with M = l), (3.51) and (3.52) to obtain V*(P$k.VB)

- JgnVB = pcP,“B,

(3.61)

which is to be solved subject to the boundary conditions !/Bt! = - ilrk llVBj/ = 0,

(3.62) (3.63)

657

Thermal Taylor dispersion phenomena

As in the continuous case [cf. equations (2.59) and (2.60)], the field B(r) is uniquely defined only to within a physically irrelevant additive constant. m = 2: Set m = 2 in (3.44) and use (3.36) and (3.37) to obtain IIVU’oW) II - PCPOVIIW II1,

ds.(k.

(3.64)

wherein (3.65)

W dGfPIP1/P;. Equation (3.64) may be rewritten in the asymptotic form d$+z;[

ds.(Pzk.()VW”()

j=l

(3.66)

-J$IIWm(l)+exp,

S+j

where, with P p (r, t) representing the leading term on the right-hand side of (3.59),

--

= UUt’ + 2tsym(UB)

+ BB.

(3.67)

From this it is obvious that

IIW”II = - 2tvm(~llrIO + IIBBII

(3.68)

IIVW mII = IIV@B) II.

(3.69)

and Substitute (3.68) and (3.69) into (3.66) and use (3.57) together with the identity (3.41) and the divergence theorem to obtain dMz ‘y 2UOt dt

V*[P$k*V(BB)

+

- J$BB]d3r

+ exp.

(3.70)

s to

Pre- and post-multiply (3.61) by B, add the two results and simplify the resulting expression to obtain the identity V.[P$k.V(BB)

- J,“BB] = 2P$[(VB)tssymks(VB)

+ pcsym(BU)]

(3.71)

Combining the two latter expressions and using (3.60) thereby yields

dM2 dt N 2UUt

+ 2sym(UB)

+ 2ci!+ exp,

(3.72)

where we have defined d d&f P$(VB)‘.symk.(VB)d3r s

(3.73)

[cf. equations (2.61) and (2.63)-(2.;7)]. 3.5. Macroscale formulation As in Section 2, we now wish to obtain the equations governing the macro- or Darcyscale temperature field I@& t), wherein B

d&fR,

(3.74)

is regarded as a continuous variable, representing the macroscale position vector [S]. A macroscale Green’s function p may be defined as p(fi

_ R’, tlr’)

d2f1 w*

s

p(r)c(r)P(B 70(n)

where the constant pC* remains to be determined.

- B’,r,tlr’)d3r,

(3.75)

658

R. P. Batycky

Similar to (2.72) P ultimately totic form

We postulate port equation

(subject

becomes

et al.

independent

of r’, and hence attains

the asymp-

-P 2 P(R - R’,t).

to a posteriori

verification)

(3.76)

that p is the solution

of the macrotrans-

_ @*g

+ V-5 = 6(R - R’)6(t),

(3.77)

where J=pc*U*P-K*.VP, subject

to the boundary

(3.78)

conditions

-Iii-R’I”(P,J)-+{O,O)

asIR-R’I-+cc,

. ).

(m=0,1,2,

(3.79a) and (3.79b)

In the above, (3.80) denotes the macroscale gradient operator. Define the total moments %I,,, of p as l%&,,(t)2’ pc*c(ii n or equivalently

- ii’)mP(R

- ii’,t)

(WI= 0,1,2,

_-

(w - it’)mf’(R - R’,t)d3R V,,

. . )q

(m = 0,1,2,

. . ),

since, when viewed from the macroscale, a unit cell of volume 70 constitutes volume element d3ii. These total moments may be asymptotically matched to their counterparts comparable microscale problem (3.39); that is, R,(t)

‘v M,(tlr’)

+ exp

(3.81)

(3.82)

a “differential” formed in the

as t -+ cc,.

(3.83)

Hence, if we define (3.84) then, as in the continuous transport coefficients n*

theory of Section 2, the matching and a* in terms of the microscale

process furnishes phenomenological

the macrodata as

u* = n,

(3.85)

a* = &

(3.86)

and

where n and ti are given by equations (3.57) and (3.73). The respective fields P,“(r) required therein are defined by equations (3.47)-(3.50) and (3.61)-(3.63).

and B(r)

3.6. Relationship between T and T Similar to arguments employed in Section 2.4 of the continuous theory, the macroscale temperature field F may be expressed in terms of the microscale temperature field T by first defining i; in terms of p as T(R, t) - TO “2’ i

8

-p(r’)c(r’)P(R .i 7”

- R’, tJr’)[ T(R’, r’, 0) - T0]d3r’,

(3.87)

Thermal

Taylor

dispersion

659

phenomena

so as to eventually obtain the desired expression (3.88)

p(r)c(r) T(R, r, r) d3r, wherein pc*

s

dzfL p(r)c(r)d3r. ro To

(3.89)

It follows from (3.87), (3.77) and (3.78) that Tasymptotically satisfies the macrotransport equation - --aT -pC+ at+ U*+VT = K*:VVT. (3.90) ( 1 subject to the boundary condition T+T,=const. and the initial condition

- T= T(R,O)

aslRl-+co

(3.91)

at t = 0,

(3.92)

where T(R,O) is related via (3.88) to the prescribed initial microscale temperature T(R,O) (3.14) [with T(R,r,O) = T(R,O)] together with (3.1) and (3.74).

4. FLOW

THROUGH

POROUS

MEDIA

POSSESSING

THERMOPHYSICAL

CONSTANT,

field

PHASE-SPECIFIC

PROPERTIES

In the special but important case where the thermophysical properties p, c and k of each phase are constant within their respective domains, the generic analysis of Section 3 becomes identical with that for the convective-diffusive transport of a molecular solute through the interstices and particle interiors of a porous medium [3]. Because of this fact, existing solutions [9] of such species transport problems may be adapted to analyze comparable thermal transport problems without the necessity of obtaining solutions anew. The formal equivalence of the two classes of problems may be established most readily for the present case of uniform phase-specific thermophysical properties by defining the volumetric internal energy density” of phase i as u

dLfpcT.

(4.1)

In such circumstances, equation (3.11) becomes

au

dt+V-J=O, in which J=vu-a-Vu,

(4.3)

with a the thermal diffusivity dyadic defined in (2.45). These equations apply separately to each phase, with affices c and d appended to each of the various quantities appearing above to distinguish between the respective continuous (i.e. interstitial fluid) and discontinuous (solid, bed-particle) phases. Explicitly, 2+VJ,=O,

Ji = viui - ai *VUi

(i = c, d).

(4.4)

It For the case of constant properties, this derives from the fundamental thermodynamic relation du = pcd T for specific internal energy changes occurring at constant volume. Note that one could equivalently define u = pc(T - To) to be the internal energy density, since the latter is defined only to within an arbitrary additive constant.

R. P. lhtycky

640

et al.

Assuming no convective flow through the interiors of the bed particles, we have for the respective velocity fields appearing above that Vd= 0,

Y,.= v,

(4.5)

in addition to requiring the kinematic boundary condition V’V = 0

on s,,

(4.6)

where s, denotes the bed-particle surface separating the ~ontinnous and dis~(~lltinuoL~s phases, and I represents a unit normal vector on sP. In terms of the temperature field T, continuity conditions prevailing at the phase boundaries require that r, = TC on s, (4.7) and v.k,-VT,

= v+k,+VT,

cm sps

(4.8)

From (4.1) et seq., these translate into the f&owing boundary conditions imposed upon the respective internal energy densities ui:

and in which

is the ~o~urne~r~cheat capacity ratio. In equations (4.4), (4-9) and (4. IO),the fol~o~~~~~ changes of~o~~tion may be introduced so as to efl’ect the transition from the constant properties t~ze~~~~~ problem tu the comparable (although not necessarily constant properties) solute transporf problem: u+c

a-+D

(solute ~on~entrati~)~)

(4.12)

(solute diffusivity dyadic).

(4. $3)

Moreover, in the concentration problem, K will be termed the interphase solute partition coeffcient. In this notation, equations (4.2) and (4.3), together with the boundary conditions (4S), (4,6), (4.9) and (4.10) are formally identical with those for the comparable solute dispersion problem in porous media treated previoudy [3]. Indeed, in the constant, thermal properties case, Brenner and Adfer f3f (see their section 1tf discuss the problem of heat transfer in packed beds, thus preceding the later analysis by Mei flct] of this constant properties case. Among other things, Brenner and Adler show, in agreement with the subsequent results of Mei, that the mean velocity o* at which the thermal disturbance is propagated is given by the expression (4. t 4) here, jl?i;*= &W,, “+ #W&d*

(4. t 5)

in which & is the volume fraction of phase i, and 3* is the Darcy-scale i~t~rstjt~a~velocity of the carrier fh.tid; equi~a~ent~~~C&T” is the seepage velocity.

The analogy between species and thermal transport phenomena for the case of constant phase-specific thermophysical properties permits one to adapt the species transport numerical calculations of Edwards crt al. [9,I to the present thermal transport case. Their dispersion calculations [93 for the three types of spatially periodic arrays shown in Fig. 2 were performed for the case where the bed particIes are impenetrabIe to sotrite

Thermal Taylor dispersion phenomena

Interstitial Auid _ =6-G=% velocity Wctor, -* V zr,

661

&a

(a) Square array

Fig. 2. Two-dimensional cross-sectionaI representations of the unit cell geometry for: {a) simple square array; (b) staggered array; (c) hexagonal array. The circularcylinders are each of radius a and the mean interstitial velocity v* is parallel to the appropriate symmetry axis af the lattices, as shown.

transport, corresponding formally to circumstances for which Dd = 0, and where the partition coefficient K = 0, For the thermal case, this corresponds to the limiting circumstances for which cr, = 0 and K = 0, where .K is the heat capacity ratifi defined in equation (4.1 I). More precisefy- the caIcuIat~ons of Edwards d uf. appIy when Q/w;c 6 I

(4.X)

K d IL.

(4.17)

k&C < t .

(4.18)

and ~qui~aIe~t~~, these require that

In those circumstances for which (4.17) applies, equation (4.14) shows that fro = $?t7*,

(4-19)

Thermat ~i~~~r~~~ are thus propagated t~~~ngh the ~~~er~t~c~ at the same rnem interstitial velocity as the carrier fluid. The functional dependence of the thermaf dispersivity upon such parameters as bed porosity, Reynolds number, Peclet number and array type is shown in Table 2. Interpretation of these results is parallel with that given in Edwards et al. [9]_

~~er~~~ r~er~~l d~~~s~~~~~~ ofa ~er~~d~e~~~~ kyered ~o~~~s~re Consider the periodic composite medium shown in Fig+ 3, consisting of alternate layers of the solid phases c and d of respective thicknesses I, and l,,in the z-direction, aad extending to infinity in all. three spatial dimensions. Within each of the two phases (i = c, d) the respective thermopbysic~ properties pi, ci and ki (aad hence CQ)are each taken to be constant, ~~dd~tio~ally~v, = fd = 0 in the present n~nconve~~ve case.) ~onsequentIy, the comparabI~ effective molecular di~usiyity cafcuIations of Brenner and Adler [3; see their section Sj may be adapted to the present thermal case to yield . , ** -+ a* = I,l,&: -t- (I - l,i,)aLr (4.20) 4.2.

R. P. Batycky

662

et al.

Table 2. Nondimensional numerical values of the longitudinal (a;) and the lateral (CC?:)thermal dispersivities as a function of bed porosity d,, Reynolds number Re = 2aP*p/p, and Peclet number Pe = ZaP*/a for various arrays of circular cylinders c$‘

0.804 0.804 0.804 0.804 0.804 0.804

Re

Pe

0

Square

0

0 0 0

0.804 0.804

20.9 100 0 20.9 100

0.599 0.599 0.599 0.599 0.599

0 0 0 0 0

0.400 0.400 0.400 0.400 0.400

0 0 0 0 0

10 100 100 100 1000 1000 1000 0 10 100 1000

array

Hexagonal

F;fla

a:/a

0.8357 0.9464 6.0692 301.76 644.82 868.47 17,754.7 56,641.3 67687.7

0.8357 0.8440 0.9746 1.2164 1.1259 0.9906 1.7139 1.2439 1.4892

0.7095 0.8605 7.1223 359.37 26,921.8 0.9368 10.356 524.54 34,567.7

array

Staggered

L?TiX

,-;/m 0.7823 0.8403 4.3465 215.47

ffla

0.7823 0.8328 1.0469 1.375 1

18,955.6

1.8328

0.7095 0.7112 0.8062 1.0100 1.2413

0.7135 0.7753 4.2237 176.53 14.299.9

0.7135 0.9775 1.1930 1.6662 2.0813

0.5728 0.6130 0.7038 0.8188

0.6390 1.5613 57.734 3246.39

0.6371 0.8547 1.0839 1.3828

array a:la

__ ._~

0.7085 0.7216 1.5403 48.8357 4330.56

0.7085 0.8495 1.9607 2.8567 3.6340

0 10 100 1000

in which** (4.2 1)

rf= 4ckc+ h&d

(4.22)

&pcc, + f#JdP&j

corresponding to the expected formulas for series and parallel conductivities, pi E li/(Ic + Id) (i = c,d) the volume fraction of phase i. Consider the special case where the two layers possess the same thermal diffusivity, as originally discussed in Section 3.1. for air and cast iron: xc = ad = a. With this assumption,

the contributions

to a* adopt

the contributions

the respective

forms (4.23)

to &* adopt

the respective

forms

CT = a,

** The significance capacity is

of these equations

can perhaps

(4.24)

be most readily appreciated

PC* = @C&C, + &P&i> whereas

the effective thermal conductivity

dyadic

is

k* = i,i,Ef

+ (I - i,i,)k:,

in which and

corresponding,

respectively,

to thermal

conductances

in parallel

a, say,

(4.23)

a, = ad = 2.

With this assumption,

with

and series.

by noting

that the effective heat

Thermal

Taylor

dispersion

phenomena

663

Fig. 3. A spatially periodic layered medium extending to infinity in all directions and composed of two distinct materials c and d, each possessing uniform thermophysical properties pC. c,, k, and p,,, cd, kd, respectively. Thicknesses of the respective phases are 1, and I,,.

0.0

0.1

0.2

0.3

0.4

0.5

$e of Fig. 4. Functional

0.6

0.7

0.6

0.9

1.0

4b

dependence of cEf/a upon the volume fraction C#Jand thermal conductivity K for the case of a layered medium for which LX,= ad E LX,say.

ratio

(4.25) where [cf. (4.1 l)] K de’ PdCd = ;. PA

c

(4.26)

The functional dependence of tL: upon &Cand K is shown in Fig. 4. For this layered medium, Cr: is symmetric about K = 1 as well as about the values 4C = 4,, = 0.5. Additionally, cc:/a < 1. The respective limiting values of equation (4.25) are (4.27)

R. P. Batycky

et ul.

-

(4.28)

(K B 1).

_1k, ”

&$dPdCd

These two values can be very different. As a physical example, take air and cast iron, possessing respective thermal diffusivities [Z] a, = 0.194 cm2 s- ’ (air) and CX~ = 0.225 cm2 s-l (cast iron).” Whereas on the microscale each material possesses roughly the same isotropic thermal diffusivity (a z 0.2 cm2 s-l), on the macroscale the effective dispersivity a* is highly anisotropic (corresponding to the case K Z+ I), possessing only a very small effective conductivity parallel to the z-axis. The consequent small thermal diffusivity Et perpendicular to the layers arises from the very small thermal conductivity k, of air in conjunction with the rather larger density &$of cast iron. Acknowledgements-This of Energy.

research

was supported

in part by the Office of Basic Energy Sciences of the Department

REFERENCES 1. R. P. Batycky, D. A. Edwards and H. Brenner, Thermal Taylor dispersion phenomena in an insulated circular cylinder. I. Theory. Inc. j. Heat Mass Transfer 36, 4317-4325 (1993); II. Applications. ibid. 36, 4327-4333 (1993). New York (1984). 2. R. A. Perry and D. W. Green, Perry’s Chemical Engineers’ Handbook. McGraw-Hill, 3. H. Brenner and P. M. Adler, Dispersion resulting from flow through spatially periodic porous media. II. Surface and intraparticle transport. Phil. Trans. Roy. Sot. Land. A307, 149-200 (1982). 4. H. Brenner, A general theory of Taylor dispersion phenomena. ~hysicoche~. ~ydrodyn. 1,91-123 (1980). 5. H. Brenner, Dispersion resulting from flow through spatially periodic porous media. Phii. Truns. Roy. Sot. Lmd. A297, 81-133 (1980). 6. H. Brenner, A general theory ofTaylor dispersion. II. An extension. Physicochem. Hydrodyn. 3,139-157 (1982). 7. R. B. Bird, W.-E. Stewart and E. N. Lightfoot, Transport Phenomena. Wiley, New York (1960). R. Haberman, EIementarv Applied Partial Differential Eauations. Prentice-Hall. New Jersev (1987). ;: D. A. Edwards, M. Shapiro and H. Brenner, Dispersion of inert solutes in spatially periodic: two-dimensional model porous media. Trunsport Porous Media 6, 337-3.58 (1991). 10. C. C. Mei, Heat dispersion in periodic porous media by homogenizatjon me&hod, FED (Fluids Engineering Division of the ASME) Ma~ri~huse Transport in Porous Media 186, 11-16 (1991).

+’ The other thermal

properties

p< = 1.29 x lo-’

are 121:

gcme3,

pd = 7.86 gem-‘,

(;=

LOSJg-‘K

c,=0.453Jg-‘K-i,

i,

k, = 2.62x iOmJ Wcm-’ k,=0.802Wcm-‘Km’.

K-t;