Radiation heat transport in disordered media

ELSEVIER

Advances in Water Rmources, Vol. 20, Nos 2-3, pp. 171-187, 1997 Copyright 0 1996 Elsevier Science Ltd

Printed in Great Britain. AU rights reserved PII: SO303-1708(96)00029-Z 0309-1708/97/$17.00+0.00

Radiation heat transport in disordered media

William Stieder Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

(Received 15 July 1995; accepted 26 February 1996)

Ra.diation heat transport through the internal void spaces of particle beds, fiber beds, packed beds, reactors and porous media with opaque, diffusely reflecting, gray body surfaces and large solid dimensions (TwI/X* > 100) is considered. A variational principle formulated for the effective radiation conductivity, based on the local particle surface radiosity, differential view factor and solid temperature, permits a rigorous solution of the dependent, long range multiple scattering problem. The conductivity results, applied to a bed of randomly overlapping spheres, agree exactly with pseudohomogeneous results in both the isotropic and anisotropic scattering limits and shed rigorous light on the anisotropic phase function expansion theory. Explicit calculations, performed for several other standard packings, e.g. fiber beds, exhibit a parallel upper and series lower bound ovI:r the various particle shapes and dispersion structures. Results show that an empirical equation first suggested by Vortmeyer (German Chem. Engng, 3, (1980) 124-137), but generalized herein from one P to four PO, PI, Pz, P3 coefficients, which vary substantially with the various industrial packings, will provide a suitable generalization of the emissivity factor of krad for engineering conductivity modeling of radiation heat transport. Copyright 0 1996 Elsevier Science Ltd

Key words: radiation void conductivity, dependent scattering, multiple scattering, an:isotropic scattering, packed bed reactors, particle beds, fiber beds, porous media.

NOTATION

a

a0

A B, B* Bo,BL

c

C d d2r d3r

de,

.

i k

ke

cylinder radius. sphere radius. (= -3(7T4), defined in eqn (lob). radiosny and trial radiosity. radios&y of the end planes x = 0 and

I l.4

average interparticle clearance. (= 4a;r3), defined in eqn (10~). particle diameter. element of surface on Co, C or CL. element of volume. element of angle for the normal to cylinder, perpendicular to the k plane. element of solid angle. integrals defined in eqns (63) and (64). defined in eqn (38). radiation flux incident on a unit surface. unit vector pointing from x = 0 to x= L. total beat flux across the slab. solid conductivity. effective slab conductivity.

k rad K(r, r’)d2r

L

n0

nj

P

pi

_ PO

Pj

r, r’ SO

T, T’

171

void radiation conductivity. differential view factor between two surface points r and r’, given in eqns (3a)-(c). slab thickness. density of sphere centers. (j= 1,2,..., ) density of fiber centers with axial orientation vector Wj and of radius a, per unit area ofj plane. structure coefficient in Vortmeyer’s eqn (1). (i = 0, 1,2,3) structural coefficients in variational &ad eqn (37). probability that no sphere center lies within a volume ZI. probability that no circle centers lies within an area Aj ofj plane. position vectors. exposed sphere surface area per total volume. exposed surface area for cylinders with orientation Wj per unit total volume. local solid temperature and trial solid temperature.

172 W. Strieder

To, TL, r temperatures of planar surface at x = 0, x = L and their average.

V v, % X

total slab volume. solid volume. void volume. slab width coordinate.

Greek

r, 6r, s2r

6B, ST

:

(= rd/X,h) dimensionless particle size. (= C0) radiosity gradient across the slab, defined in eqn (13). variational functional, first-order variation and second-order variation. variations in radiosity and temperature. surface emissivity. average pore diameter.

v(r), 77, I, 17’ unit surface normals pointing into the

e ej

void at r and r’, respectively. [= i(TL - To)/21 defined in eqn (26). polar angle of cylinder in the wj direction.

1 INTRODUCTION

Radiation heat transport within a distributed solid-void system is important in a number of modern technologies and industrial process.28 High temperature radial heat transfer in a packed bed chemical reactor or heat exchanger,3 the fluidized bed regeneration of petroleum cracking catalysts by burning off the carbon deposits (Botteril17) and high temperature heat transfer processes in noncatalytic reactions within porous materials such as the reduction of metal oxides or the roasting of lead sulfides45)53 are cases of high temperature chemical industrial processes, where a knowledge of the effective radiant heat transport coefficient would be useful for improved design and optimization efforts. Chemical vapor deposition of ceramic matrix material within preformed fiber mats for the manufacture of fibre reinforced ceramics46,47 is carried out at 1200 K. Procedures to control the reaction in order to reduce residual void regions depend, in part, on reliable values of both the radiative conductivity and its change with the evolving void structure. Also, in materials processing by self propagating high temperature reaction waves for the synthesis of advanced ceramic materials, the cylindrical compacts used to make the silicon nitride reach 4000K as the reaction wave passes.63 Accurate equations for radiant heat transfer as well as the melting and reaction rates are essential to develop this new technology. Predictive models for radiation heat transport in various packings are needed for advanced thermal insulations’6’31,32 that will conserve energy resources and curb production costs in many industries.

In most engineering applications, it is common that the large particles, present in most packed and

extinction coefficient. average thermal wavelength.

P3 P vector pointing from r to r’ and magnitude. [= (P - wj * pWj)l vector projection of p in j plane. Stefan-Boltzman constant. absorption coefficient. scattering coefficient. total surface area (exposed or overlapped) of spheres per unit total volume.

c CO? CL

total surface area of cylinders with axial vector Wj per unit total area. void-solid interface. planar solid surfaces located at x = 0 and x = L.

(. . .)

void fraction. unit vector that gives direction of the central axis of cylinder. surface integral defined by eqns (30) and (39)

fluidized beds such as chemical reactors and coal combustors, are opaque and diffuse.54 For such cases, one usually assumes a gray body surface model, characterized by a local surface emissivity, E, diffuse reflections and a cosine law distribution of emitted photons with a fourth power average energy. If in addition the average particle dimension d is signifi- cantly larger than the thermal wavelength Xu, of the radiation, i.e. CY_(= nd/&) > 100, geometrical optical theory with the differential view factor will accurately describe the radiation transport process. Kirchoff’s law, that equates the coefficients of absorption and particle surface emissivity, also follows from this mode1.44

Moving from the individual particle surface to the description and modeling of the entire porous medium or random bed, we presume the scale of the micro- structure of the void-solid distribution to be much smaller than the bed dimensions.54 The medium is taken to be sufficiently random and large to be optically thick. Within the solid phase, Fourier thermal conduction with coefficient k occurs. In order to define a void radiation conductivity, we must assume the steady state temperature drop AT across the local average bed dimensions are much smaller than the bed temperature T, i.e. AT/T << 1. Under these conditions the use of a bed effective thermal conductivity k, with an identifiable void thermal radiation conductivity krad is a standard procedure45 in engineering heat transfer calculations. Our objective is to construct rigorous methods and derive expressions for these effective conductivity coefficients useful for the current engineering models of high temperature heat transport in random, disordered void-solid systems.

Radiation heat transport 173

C. L. Tien54 in a recent review of radiation transmission in void-solid beds, has suggested the division of the modeling literature into microstructural theories with an explicit void-solid structure (Vortmeyer 64 in an earlier review called these cell models) and quasihomogeneous models based on either the transfer equation or the more approximate two-flux equation with empirical or model derived scattering and ad sorption coefficients. Radiation contributions predicted 'by structural models date back to Nusselt39 in 1913. The first packed bed models, which included the void volumes as parallel plates13 or spherical cavities,43 incorrectly predicted vanishing void radiation at zero surface emissivity. Improved cell models,12t65166 which used uniform sized solid spherical particles in a regular lattice provided a more realistic void geometry, but still did not account for free radiation passage from one cell to the next. Hence as pointed out by Vortmeyera long range scattering remained badly underestimated. To the wealth of examples of radiation heat transfer calculation based on microstructural models, reviewed by Vortmeyer64 and Tien,54 we add several new Monte Carlo simulations by Kudo,34135 a molecular dynamics simulation of heat transfer by Kotake33 and the volume averaging paper by Whitaker.68 The variational calculations5’ of this paper also fall under the microstructural model class.

For the quasihomogeneous model based calculations several points must be made. Firstly, the appropriate conditions for the application of the transfer equations,44 regarded as the most general, physically exact quasihomogeneous radiant heat transfer equation, has been the topic of a number of papers in the last 10 years. The discussion began” with the observation, ‘the classical theory of radiative heat transfer in particulate media is based on the assumption that particles are separated by very large distances (relative to the particle diameter)‘. The fundamental derivation of the transfer equation requires an infinitely dilute particle bed. However, to develop the discussion further, Brewster stated in9 ‘it has been experimentally shown” that the assumption of negligible multiple scatter and shadowing (i.e. infinite dilution) within an elemental volume containing many particles is merely an artifice of the derivation of the transfer equation . . .’ and again in the same paragraph, ‘further support for this approach may be found in Brewster et al., 1982, where it is demonstrated the two- flux solution of the transfer equation predicts measured transmittance in packed beds’. These comments stem from continuing efforts reviewed by Tien,54 to set particle density limits on the independent and dependent scattering regimes. In other words, can one apply the transfer equation outside of the dilute particle limit, to the practical higher particle density range of packed on fluidized beds on the basis of the existing empirical data?

The focus in this manuscript is on the important case of large ((u > 100) opaque solids with rough diffusely reflecting surfaces, where geometric optics describes the

scattering. For this case, we have not found convincing experimental evidence of the contention of Brewster, Tien and others that on the basis of experiments (see Figs 4 and 5 or Ref. 54), independent scattering, i.e. the transfer equation with calculated or measured coefficients, can be used with accuracy in high density beds (solid volume fractions up to 0.7). For the case of larger (Y, the present status of their research has been summarized by Cartigny et al.,” ‘the Rayleigh-Debye approximation is not applicable in this region, there is presently no established analysis and only limited experimental data are available.’ In fact only three experimental points in Fig. 5 of the companion experimental paper72 lie above cx = 10, none of which apply to diffuse, opaque surfaces. Two of these data points are for nonabsorbing latex spheres. The single case of a large absorbing particle presented in these studies 9-l 1,54,55,72 .

IS polydivinyl benzene particles of mean particle diameter 11.15 microns made absorbing by implanting blue dye in the particle. Diffuse, opaque, industrial particles - compressed ceramic pellets, ore particles, catalyst supports, coal, etc. - have surfaces very different from the plastic particles of Brewster et al. 1o,54

The radiation transfer equation will be reliable within the fundamental conditions of its derivation (independent scatterings and high void fraction), but outside of this range its use is strictly empirical. Even for large particles any empirical justification of either the transfer equation,” or the two-flux model,’ will require a sufficiently large number of separate experiments to cover the relevant engineering parameters. For example, the influence of shadowing on the radiation transport coefficients should be different for absorbing and nonabsorbing particles. Additional particle charac- teristics not yet covered in the experiments9-1’>54155>72 that will also influence scattering from large, opaque, gray particles, include the surface scattering mechanism. The phase function of diffuse surface scattering,” which has strong backscatter, will surely shadow more than the phase function of a specular surface.69 Indeed some surfaces have both specular and diffuse components60-62 of reflection. Of course the various ranges of particle size44 scatter differently and must be separately examined.

Even for the experiments9-11’72 on plastic spheres, there is confusion. Tien and Drolen” stated, ‘in a similar experiment Ishimaru and Kuga3’ also reported dependent effects. For small (u (a = O-529) their results are in good agreement with those of Brewster and Tien. However, for cr > 3.6 they found dependent scattering effects occurring at much lower values of the solid volume fraction. For instance at cx = 3.5 18, they showed dependent scattering effects at a solid volume fraction M 0.02 (c/& < 2.6)‘. (Note that c/X,, is the interparticle clearance to wavelength ratio). These numbers do not agree with those of Tien,54 who claimed dependent

174 W, Strieder

scattering begins at much higher solid fractions (C/&h < 0.5).

The theory to be developed herein is microstructural, exact and rigorous in both form and method. One consequence of the above discussion is that it should be compared with the quasihomogeneous, transfer equation model only in the limit of dilute bed densities, i.e. particle volume fractions of 0.02 or less. Finally, on a separate issue for quasihomogeneous models, Tien and Drolen55 in their recent critique, point out that the two- flux approximation of the transfer equation assumes semi-isotropic scattering and may be inaccurate for the larger anisotropically scattering particles often found in chemical and industrial systems. This also may effect any comparison of the quasihomogeneous two-flux model results with our calculations.

On the other hand, for the microstructural model- based theories, one aspect of the effective conductivity and radiation conductivity calculations is the explicit treatment of multiple scattering. Beek,4 in a design paper for packed bed reactors has pointed out that there was no experimental basis for including the particle surface emissivity in the expression for thermal radiation conductivity. In contrast, Vortmeyer@ has reviewed the various microstructural models for radiation transport in packed solids and in this context stated that the long range effects of scattering from a void region to nearby void regions had not yet been included in any theory of radiation transport. His point was to say long range multiple scattering had not yet been correctly treated for the calculation of krad in microstructural models. Indeed, Beek’s and Vortmeyer’s observations are related in that a rigorous solution to the multiple scattering problem will result in the derivation of the correct bed surface emissivity factor and krad. As it does for the simpler flat plate problems, the radiosity formulation for diffuse scattering4 developed in this paper avoids the difficult sum over successive multiple surface scatterings, but now for arbitrary solid surface geometries. If instead of the radiosity, the equations were cast only in terms of the temperature, the radiant thermal conductivity expression becomes an infinite sum over terms j = 0, 1,2, . . . Each term contains the surface emissivity coefficient e2( 1 - e)j, and a product of view factors, one view factor for each of the j + 2 straight line segments of the path the photons travel between adsorption and emission. Then in addition, this combination is integrated over all possible paths. This type of expanded form of the radiant conductivity was obtained by Whitaker6* as a formal result, though no calculations were performed for explicit model structures. A variational principle can be written that parallels Whitaker’s result. However, because of the difficult necessary integrations over the complex paths and thej summation, it is a formidable task to evaluate. The variational principle for k, formulated in terms of the radiosity avoids these difficulties.

In certain cases, e.g. well mixed fluidized beds,‘8)‘9Y27136 where a finite temperature drop exists between the bed edge and the surrounding vessel wall, a second transport coefficient, the bed edge emissivity Q, is needed. A few parallels exist between the two coefficients, krad and eel, for example, eeff calculation models are based on either microstructural 8,9 or quasihomogeneous scattering models.516 However, kti is present within the bulk energy transport equation, whereas the edge emissivity acts from the boundary condition.‘4YX126141,52,73 Physically, the krad transmission problem for a thick bed is built around corrections to a linear temperature profile, while the E eff photon emission, scattering and bed departure process (by Kirchoff’s law equivalent to bed intrusion, scattering and absorption) is dominated rather by an exponential radiosity behavior.M171 Most of the detailed applications appear in the fluidized bed literature, apart from the k, problem. Efforts to measure E,~ have been far less successful than k,, leading to some troubling inconsistencies in experimental results pointed out in a recent review by Saxena.42 A separate variational principle7’ does indeed exist for the calculation of E,~. There is also an additional means to calculate eeff based on complementary multiple scattering summations, 7o whose truncation generate upper and lower estimates. Since the interest in E,~ appears to be largely in the fluidized heat transfer literature, and since the calculations are somewhat different, eeff will not be treated herein. Significant references have already been included above for the interested reader.

Briefly in outline, fundamental equations are formulated in section 2, and recast in terms of a rigorous variational principle69 for k, and krad in Section 3. Applied in Sections 4 and 5 to a bed of randomly overlapping spheres,69 the variational equations provide exact forms for krad in a gray gas at both the isotropic (E + 1) and anisotropic (6 + 0) scattering limits with an analytical emissivity factor in between. Besides the problem of multiple scattering in sphere beds, the variational method permits an analytical examination of the influence of bed geometry and packing structure74 on k, and krad. Six different models for particle shape and random dispersion structure are considered (see Table 1). Finally, in an empirical effort to correct the well-known46 parallel plate equation15,45 of Damkoehler for void radiation conductivity and plate spacing a, to include radiation path holes across the bed, Vortmeyer@ has suggested the equation

k rad P+(l -P)e -= cu 2(1-P)-6(1-P)’ (la)

or for later use in discussion

k rad -Yzz C.6 (lb)


Table 1. Radiation void thermal conductivities Xlld and associated constants PO, PI, P2, P3 for spheres, various cylinder distributions and Vortmeyer’s eqn (1)

PO eqn (32)

PO Pl p2 p3 Monte Carlo eqn (33) eqn (34) eqn (35)

[58, 591

x rad

eqn (31) [or eqn UN

10s

POC flux 11 to fiber axes

POC flux I to fiber axes

12 2 13 3

2 = 2 . 467 4

2.43 -

2

162; T2 = 0.7630 0.763* 0 -T2

- 16+n2 l

PROC flux 11 to fiber axes - 6 = 1.221 1.16* 0 - -Gl 8 + G, 8 + G,

1

PROC flux I to fiber axes 2Go - = 0.7488 0.749* 0 ~ -Go 1 + Go 1 +Go

1

IOC 12 E

= 0.923 1 0.923* 6 1

13 26 1

Vortmeyer correction to parallel plate & = 0.028t 1t 1t c6 1+9E +

4 --

2 [ 36 - 186 1 *Obtained from Monte Ca.rlo Simulation58’5g and the limit 4 + 1. t Values from empirical equation suggested by Vortmeyer@ to correct for holes in Damkoehler parallel plates.

where S is the average pore diameter (4 x void volume/ total internal surface area). Vortmeyer has suggested a P value@ of about 0.1. Our variational results will support this form, but show that there is not one, but at least three parameters PO, PI and P2. Further, the values of these parameters will depend on the particle shape and dispersion structure.

2 FUNDAMENTAL EQUATIONS

Transport of radiation in the void spaces of a high temperature porous solid will always require some consideration of bed structure. Suppose just outside the ends of a large porous slab (Fig. 1) of total volume V, solid plane surfaces C,-, and CL are positioned at x = 0 and L, respectively, and that i is a unit vector pointing across the slab in the positive x-direction. The total volume V is divided into two subregions, a region of solid V, and a void region V+. The interface C between these two regions makes up the void-solid interface. We assume X0, C and CL are opaque gray surfaces, the radiation from which is emitted and reflected diffusely according to Lamberts’ cosine law.4 The emitted flux depends on the absolute temperature T of the surface, the surface emissivity E and the Stefan-Boltmann constant g, in the combination WT4. Kirchhoff’s law states that the same surface element will absorb only a fraction E of the incident radiation, reflecting the fraction (1 - E). If H represents the radiant flux incident on a unit surface, then for a

diffusely reflecting surface the radiosity B, the radiation diffusely leaving a unit surface, is given by

B=eaT4+(1-E)H (r on C). (2)

The fraction K(r’,r)d2r of radiation diffusely distributed. from a unit surface element located at r’.

x=0 x=L Fig. 1. Randomly overlapping solid spheres (or the j plane showing the cross sections of fibers with axial orientation uj).

176 W. Strieder

that travels a straight line free path, and arrives at a second surface element d*r located at r, can be used to formulate the radiant exchange between surfaces. Since we are assuming diffuse scattering at the surfaces, K is given by the cosine law

K(r’, r) = K(r, r’) (3a)

= - h(r) . PI Mr’) - ~ll(v~> WI (if r’ can see r)

= 0 (otherwise) (3c)

where v(r) and q(r’) are unit normals, respectively, at the points r and r’ on the surfaces Co, C and CL, pointing into the void and p = (r’ - r). Of the diffuse radiation B(r’)d*r’ leaving d*r’ of Co, C and CL, only the fraction B(r’)d*r’K(r’,r) will arrive within a unit area at r on C, then the total incident radiant flux at r on C is

J K(r’, r)B(r’)d*r’ = H(r) (r on C).

c0+c+cL

(4)

When the total incident flux H from eqn (2) is substituted into eqn (4) and the radiosity B is subtracted from both sides, an integral equation in terms of the radiosity and temperature is obtained

s K(r’, r)[B(r’) - B(r)]d*r’

CrJ+C+CL

= & [B(r) - uT4(r)] (r on C). (5)

Note from its definition as a probability and the symmetry property (3a), that the function K(r’,r)d*r’ will sum to unity over the surfaces Co, C and CL.

The steady-state energy balance at a point r within the solid V, and Fourier’s law with solid conductivity k, give

V.(kVZ-) = 0 (r in V,). (6)

The thermal boundary condition equating the net radiative flux from the void-solid surface C at r to the normal flux from the solid

-kq.VT=B-H (r on C) (7)

or from eqn (2) in terms of the radiosity and temperature

krpVT=+ ~ [B(r) - gT4(r)l (r on C). (8)

Equations (5), (6) and (8), together with the steady surface conditions that the radiosity is assigned to be uniform at

B(r) = B. (r on CO)

and

B(r) = BL (r on CL) (9b)

are in principle sufficient to determine B and T. In practice the complex geometry of the surface C and solid volume V, prevent an outright solution.

To calculate the effective conductivity and the thermal radiation conductivity the fundamental equations must be linearized in the temperature. Indeed it is only in this form that the void radiation conductivity krad can be defined in terms of a local absolute temperature cubed. Steady average surface temperatures To and TL, respectively, of the edge surfaces Co and CL and an average slab temperature T = (To + TL)/2 are defined. The linearization presumes the temperature variation across the slab is small compared with the average slab temperature T, i.e. AT/T[= (T - T)/i;] is small. We retain in eqns (5) (6) and (8) only terms zeroth- and first-order in the temperature variation about T. As a consequence in both eqns (5) and (8), aT4 can be replaced by

aT4=A+CT

where

(104

A = -3aT4 (lob)

and

c = 4aT3. (1Oc)

Also, as a result of the linearization, any temperature dependence of k or e can be neglected and the constants in eqns (5), (6) and (8), evaluated at the average slab temperature. Equations (5), (8) and (lO(a-c)) are linear in temperature and are the starting point of the variational analysis. Since the effective and thermal radiation conductivities do not depend explicitly on the temperature drop across the slab, no loss in the generality of the conductivity equations occurs when the problem is linearized.

3 VARIATIONAL FORMULATION

In this section we will derive a variational upper bound on the total heat flux, the net rate heat passes through both the void and solid per unit total cross section of a slab with arbitrary pore geometry. The net heat flux J across the plane at x = 0 is just the difference of the flux in minus the flux out

J = (BoV)-‘L d*r co

X d*r’K(r, r’)Bo[Bo - B(r’)] (11) C-b&

where I’ is the total volume of the slab and BO and BL are the radiosity values on Co and CL, respectively. The


flux into the slab at x = L is derived by a similar procedure. This flux can be obtained from eqn (11) by interchanging the 0 and L subscripts; however, this gives the negative of J. Combining these two equations we find

-J.p= V-’ d2r G+&

X f

d2r’K(r, r’)B(r)[B(r) - B(r’)] co+c+c,:

(12) where

p = (BL - Bs)L-‘1. (13)

The upper bound on the heat flux is based on the variational functional

VI’{B*, T*} = 5 d2r c +C+C II L

X f

d2r’K(r,r’)[B*(r’)-B*(r)12 Co+I:+&

+ &lx d2r[B*(r) - A - CT*12

+ C v ‘d3rk[VT*12 (14) s

where the trial temperature T * must be continuous and at least piecewise continuously differentiable in V, and the trial radiosity B* must satisfy

B*(r) = Bo (r on CO)

BL (r on CL) ’ (15)

The volume element d3r is summed over the solid volume Vs.

To derive a variation principle,51 the trial functions T * and B” are written in terms of the true temperature T and radiosity B and the variations 6T and SB, i.e.

T’=T+ST (164

and

B*=B+SB (16b)

where from eqn (15), 6.B = 0 for r on Co or CL. These trial forms (16a) and (b) are substituted into the quadratic functional I‘ of eqn (14), which is then expanded into three terms,

I’{T*,B’} =I’{T,B].+SI’+621’, (17)

a zeroth I’{T, B}, first-order ST and second-order S21? (= r{6T, 6B)) in th e variations 6T and SB. A complete proof that an extremum variational principle exists requires three steps:

(1) establish the stationary property,

ST=0 (18)

that the 6T and 6B first-order terms in r vanish, (2) show that the second-order terms in I? are positive

(or negative) definite, e.g. for an upper bound

b2r > 0 (19) and

(3) relate the zeroth-order term l?{T, B}, i.e. JY in terms of the true temperature and radiosity, directly to the engineering quantity of interest, e.g. J of eqn (11).

To show that the term SF from eqn (14), first-order in the respective variations SB, ST of the trial functions about B, T, vanishes we write

d2r

X f

d2r’K(r, r’)[B(r’) co+c+cL

- B(r)][SB(r’) - 6B(r)]

E +1--E z

-1 d2r[B(r) -A - CT(r)]

x [6B(r) - CST(r)] + C 1,

v d3rkVT - V(ST).

(20)

Upon interchange of the integration variables r, r’ and application of the symmetry condition (3a) of K in the 6B(r’) term of the first integral of eqn (20), and integration by parts followed by the divergence theorem in the third integral, the first-order expression Sl? becomes

i vsr = d2r co+cL

X s C,+C+CL d2r’GB(r)K(r, r’)[B(r) - B(r’)]

cl)+c+cL d2r’SB(r)K(r, r’)[B(r) - B(r’)]

+ & jE d2rSB(r) [B(r) - A - CT(r)]

& - l-_~

J c d2rC6T(r)[B(r) - A - CT(r)]

+ C s c d2rbT(r)r) - kVT(r)

- C s

v d3rST(r)V - [kVT(r)]. (21) s

As any trial radiosity must satisfy boundary conditions (9a) and (b)

AB(r) = 0 on Co and CL, (22)

the first integral in eqn (21) is zero. The second and third integrals together vanish due to eqn (5) with eqn (lOa),

178 W. Strieder

the fourth and fifth terms combine to cancel from eqn (8) with eqn (lOa), and the last term in eqn (21) is zero by eqn (6). From the interpretation of K as a probability, the integrals of eqn (14) are clearly positive and the second-order term in the variation of l? is also positive, hence

r{B, T} ,<r{B*, T*}. (23)

To relate I’{B, T} to the heat flux J, the extremum form of eqn (14) is rewritten, again using property (3a) of K, integration by parts and the divergence theorem, much in the same manner as in the derivation of eqn (21)

vr{B, T) = J

d2r co+-%

X J

d2r’K(r, r’)B(r)[B(r) - B(r’)] co+c+cL

+ d2r J s

d’r’K(r, r’)B(r)[B(r) - B(r’)] c Co+C+CL

+ e Ix d2rB(r)[B(r) - A - CT(r)]

- &lx d2r,4[B(r) - A - CT(r)]

- &ix d2rCT(r)[B(r) - A - CT(r)]

+ C J

c d2r7’(r)q(r) .kVT(r)

- C y d3rT(r)V - [kVT(r)]. 1,

(24)

The first integral in VI’{B, T} is identical to expression (12) for - VJ . /3, the second and third integrals vanish from eqns (5) and (lOa), and the fifth and sixth integrals combine to zero from eqns (8) and (10a). Upon substitution of eqns (8) and (lOa) into the fourth integral in eqn (24), we note as there are no heat sinks or sources within the solid V, the fourth term sums over the void-solid interface to zero, and the last integral in eqn (24) is zero from eqn (6). The variational upper bound on the heat flux is

(25)

4 TRIAL FUNCTIONS AND VOID RADIATION CONDUCTIVITY

The evaluation of the integrals in the upper bound (14) is considerably simplified if we pass to the limit of a very long slab (let L become large compared to typical bed dimensions, e.g. particle size and average pore diameter). Due to the angular distribution of the radiation diffusely emitted from Co and CL and blocking by the solid, radiation from the edges will penetrate only an

infinitesimal distance across the slab before striking a surface on C. The contributions to the upper bound integrals of eqn (14) from the end surfaces Co and CL go to zero as L-l. In a thick slab the radiation heat conductivity should not depend on the nature of the end surfaces. The end conditions in the thick slab are discussed briefly in the Appendix where it is shown that the radiosity difference (BL - B,,) can be replaced by the black body emission difference o(T; - Ti). Then the linearization about the average slab temperature, 0 from eqns (13) and (10a) becomes

p = C(TL - TO)L-‘i = CB (26)

where C is given by eqn (10~). For a thick slab the effective thermal conductivity k, = IJ/el, from eqns (14), (25) and (26) is bounded above by the variational principle

02k, < -$ J J

d2r c

d2r’K(r,r’)[B*(r’) - B*(r)12 c

+ $&/xd2r[B*(r) - A - CT*(r)12

+ $ s

v d3rk[Vr*(r)12. s

(27)

The only thermodynamic driving force for the steady- state is the temperature difference across the slab. This simple thermal dissipation implies that the dominant contribution to the temperature is a linear profile. A reasonable selection for the trial temperature is

T*(r) = To +8-r. (28)

A direct substitution of the trial temperature (28) along with the linearized form (10) into the interface eqn (8) provides an equivalent trial radiosity

B*(r) = A + CT*+d.q(r) (2%

where w is an adjustable parameter and v(r) is the surface unit normal to r on C pointing into the void. The linear part of eqn (24) allows for a smooth variation across the bed, while the third term B’ fluctuates (due to 0 - 11) with the random local structure of the void-solid interface C.

When the trial functions (28) and (29) are substituted into the variational integrals (27) the integrand of the volume integral is a constant. In terms of the surface normals Q and 77’ at r and r’ along with p = r’ - r, the surface integrals are of the general form

Mp,v, rl’)) = bV’ lx d2r

x J

c d2r’W, r’)qb, v(r), v(r’)l. (30)

The function q in eqn (30) can be [i . p6-‘12, [is pS_']

[i. h- - d)l, [is (17 - d>12 or [i.v12. Also $J is the void fraction, s is the surface area per unit total volume and S(= 4+/s) is the average pore diameter.


Once an optimum value of w is selected for eqn (27), we obtain a general variational form for the void-solid system effective conductivity, valid for an arbitrary dispersion geometry,

The P integrals depend only on the bed geometry, and in terms of the surface average (30)

?= ([i.p6-l]2) _(ii’(7i-_)l[i’pS-11)2, 0 * (4 - v121) (32)

P, = 2([i.p6-‘12)(1 i- P2) -PO,

P2 = ([i.(7j--)]2)-1PJ - 1,

with

(33)

(34)

P3 = 2([i.q12). (35)

Note that a knowledge of the values of the four Ps is sufficient to determine the four bracketed surface averages.

The use of the linear trial temperature (28) in the solid volume of the variational expression (the third term of the right hand side of the eqn (27)) gives a solid conductivity in parallel form.21 The other two integrals include the surface scattering processes. As the trial radiosity term (29) can be derived directly from the substitution of the trial temperature (28) into the linearized (10~) form of (8), it is completely consistent with the linear temperature profile in the solid. When a random two phase suspension consists of two Fourier solids (k and krad), the upper bound variational principle with a linear trial temperature gives the parallel bound form

‘% < h&d + ( 1 - +)k. (36)

It is interesting to note that the inequality (31) is also a parallel bound if we tak.e the square bracketed term in eqn (3 1) to be the void radiation conductivity. From yet another perspective, a plot of k, with increasing k of the solid for a fixed T, E and void-solid geometry will initially be below the 45” diagonal line. The k, curve for fixed geometry, temperature and E will exhibit a single cross over when k, = k = krad, and this is in fact one way to define krad. The application of inequality (31) at this point also affirms that the square bracketed term in eqn (3 1) is a variational upper bound estimate of k rad .

(37)

A comparison of the rigorous expression (37) with Vortmeyer’s suggested eqn (1) and P M 0.1 gives one set of P coefficients. It is of some interest to calculate these

P’s for six appropriate packing geometries. Equation (37) will have six additional different sets of P’s along with their geometrical interpretation.

5 P,,, Z-j AND Pz FOR EXPLICIT RANDOM BEDS

We assume the void-solid bed is statistically homogeneous i.e. the statistical properties do not vary with linear displacement. Beyond this, practical interest suggests a number of symmetry groups for the bed packing, as they occur industrially or give some special insight. An elegant Baysian analysis” has been presented for isotropic beds and applied to spheres, but as our bed structure classifications must be more inclusive, we will use a broader brush here.

A random bed is constructed by cutting a large slab from a random, infinite void-solid suspension. The infinite suspension is generated by placing solid objects without correlation at random, allowing them to overlap freely. The integral (30) can be expressed in a form appropriate for a thick bed by recalling the analytical expression (3b) for K and noting that K is zero (3~) unless r can see r’. We define

h(r,r’) = 1 r’ can see r 0 otherwise (38)

and write

Mp, rl, d)) = - VT’ j-x d2r

x Cd2r’q.Prl’.pp-4h(r,r’)4(P,qlri). J (39)

We now proceed to evaluate the surface averages for six random structures using identical spheres or cylinders:

(9

(ii)

(iii)

(iv)

(v)

(vi)

solid spheres of equal radius placed completely at random and allowed to freely overlap (10s); all cylinders are given the same orientation with Oj parallel to the i direction across the slab thickness (FOCII); all the cylinders are given the same orientation, but perpendicular to the i direction (FOCI); each cylinder is given a random orientation in a plane with all vectors of rotation parallel to i, i.e. the fibers are oriented mutually at random perpendicular to the applied thermal gradient (PROC-L); each cylinder is randomly oriented in a plane with the planes of rotation all parallel both to the vector i and to each other (PROC]]); the cylinders are oriented isotropically, i.e. each cylinder axis is selected independently with any direction in three dimensions equally likely (IOC).

180 W. Strieder

Spheres

(i) IOS One packing model (Fig. 1) is obtained when l’~ identical solid spheres of radius a0 are placed randomly in a volume I’, and allowed to freely overlap one another. This uncorrelated sphere model is well-known and its applications are discussed in Refs 21 and 49-5 1. It suffices to point out the probability PO, that any volume ZJ is free of sphere centers is:21>49-5’

PO = exp(-vno). (40)

That a point be in the void region requires that sphere centers be excluded from a spherical volume of radius ao, and the probability that any randomly selected point be in the void is just the porosity

4 = exp(-4&no/3). (41)

The total area, overlapped or not, per unit total bed volume ao(= 47ra$no) and (as the other spheres are uncorrelated) its exposed (not overlapped) fraction C#J is just the solid area per unit bed volume

s = C&70 = 47r&o~, (42)

hence, a hydraulic radius

6 = 4$/s = (7n&,)-‘. (43)

To evaluate the several integral forms of (30) the exposed surface area in d3r from eqn (42) must be multiplied by the probability that v lies within d2q. As all 47r orientations are equally likely, we have from (30)

(44)

= l/3 (45)

consistent with the condition of isotropy and the unit magnitudes of both i and 7.

For the combination of all surface points r and r’, lying, respectively, within the volume elements d3r and d3p(= d3r’), exposed or overlapped, and with unit normals, respectively, between d2q and d2q-’ we have

2 d2r_i d3rgof$d3pco;l;;.

The event defined by the function h(r,r’) includes; first the points r and r’ must lie on two different spheres in order that h of eqn (38) be non-zero; furthermore the occupied spheres cannot block the free path p(= r’ - r) between the points, which requires

P.17>0

and

p.lj'<O.

Fig. 2. Volume free of sphere centers or looking down on wj to the area in the j plane free of centers of fibers with

orientation wj.

In addition, the event that all sphere centers lie outside of a free volume (f.v.) consisting of a right circular cylinder of radius a0 (Fig. 2) about the free path vector p, capped at both ends by hemispheres of radius ao, completes the specifications for h(r, r’) as defined in eqn (38). Its frequency from eqn (40) is then

Po(f.v.) = exp{no($7& + rru~p)} (46)

and this permits the explicit formulations of integral (39)

MP,rl,rl’)) = - (r.rI’)-‘ci/d3r[d3P

x 4(P7 53 d)P - VP * df4

x exp{ -no(ffu~ + m&)}. (47)

Then for the cases of the forms of q suggested after eqn (30), we have

([i . pS-1]2) = 2/3

([i.pS-‘][i.(q’--)I) = -4/9

and

(48)

(49)

([i . (7’ - q)12) = 26/27 (50)

The resulting forms of PO, Pi, P2, P3 and Xrad, obtained from eqns (31)-(35), are listed in the first row in Table 1 for isotropic overlapping spheres (10s).

The void thermal radiation conductivity in the anisotropic scattering limit (E --) 0) gives the coefficient 12/13. This result, first proposed by Derjaguin2’ has been shown to be the exact solution” for Knudsen diffusion in a dilute bed of spheres. Tien and Drolen55


have pointed out that there is a complete equivalence between Knudsen diffusion of a gas in a porous or packed solid and radiant heat transport with diffusive, totally reflecting walls. Abbasi and Evans’ have used this equivalence in Monte Carlo simulations of radiant heat transport in packed beds and porous solids. In the isotropic scattering case (E + l), the 4/3 coefficient is a well known exact result4 for an isotropic dilute bed of particles.

Overlapping cylinders (fibers). To model a fibrous material, a large slab of volume V is cut from an infinite bed of very long right circular cylinders of radius a placed at random and allowed to freely overlap. The inside structure of the fiber bed depends on the orientation of its fibers. In addition to its random placement, the orientalion of the axial vector wj is assigned to each cylinder. The index j sums over a discrete or continuous set of orientations. As pointed out above, a number of possibilities are of interest. For example, the structure is unidirectional when all fibers are given the same orientation; in a second case fibers may be oriented at lrandom perpendicular to the direction of net flow, and fibers may also be oriented at random in three dimensions with any direction of the axial vector equally likely.

Consider those fibers with axial orientation vector Wj and the j plane perpendicular to this direction (Fig. 2). Those fibers with oriemation Wj appear as randomly overlapping circles of radius a, with centers randomly placed in the j plane. The statistics of this geometry is sufficiently straightforw,ard to allow analytical expressions for the necessary probabilities to evaluate the variational integrals of eqn (39). The probability Tj that no circle center lines within an area Aj can be written6’ in terms of the density ‘9 of circles per unit area within the j plane.

Pj = exp(-njAj). (51)

The void fraction 4 can 'be interpreted as the probability that a randomly chosen point in the fibrous material falls in the void, or the probability that in any given j plane no circle has its center within a distance a of the random point. Since the j planes are statistically independent and from ES of eqn (51),

4 = exp (

- C flj7MZ2') , j I

(54

where the summation is over all assigned fiber orientations of the model.

The surface area aj, overlapped or not, of those fibers with orientation wj within a unit total volume of the slab is

Uj = 27TUlj. (53)

From its product with the void fraction (52), we obtain the exposed portion the void-solid interface due to the Wj fibers,

Sj = Uj$ = 27Rl?lj+, (54)

the total void-solid interface area per unit total slab volume,

S= C Sj = C2TUljQ. (55) i i

From its definition and eqn (16) the average pore diameter,

6=4$/S=2 CTil?lj . ( )

-1

j (56)

We consider the two points r and r’ of the integral form (39), lying on the surface, exposed or overlapped, of different cylinders, which may or may not have the same orientation. The possible choices of q within the volume element d3r will depend upon which cylinder surface the point r is found. The total surface area, overlapped and non-overlapped, within d3r of those cylinders with axial orientation vector wj is given by g_ d3r. Any unit normal ~~ within ajd3r must lie in the j plane perpendicular to wj, and only the fraction dBj/2r of this surface will have a surface normal qj between 0, and ej + dej. We have from eqn (30) then for the sum over all j orientations,

(57)

In a similar fashion, to evaluate eqn (39), we need also the total fiber surface area, overlapped or exposed, within d3r’(= d3p) of those cylinders with axial orientation wi and with unit normal qi between t?” and B)k + de;, given by akdt$d3p/2r.

Only those pairs of surface points, where r and r’ are exposed and the straight line free path between r and r’ is unobstructed, are counted in the integral (30). The probability P( f.v.), that the two points are exposed and can see one another is zero unless p - Qj 20 and p - qi < 0, since otherwise at least one point will be screened from the other by its own cylinder. If these conditions are satisfied, then P (f.v.) is the probability that no third cylinder has its center axis within a distance a of the line joining the two points. To obtain the frequency of this event for a bed of fibers with multiple orientations, each of the statistically independent j planes must be examined to determine if the path is free. The Wj fibers will not block the path p, if in the j plane all circle centers lie outside of an area (Fig. 2)

182 W. Strieder

made up of a rectangle, whose length is the projection pj of the vector p onto the j plane and width is 2a, capped on both of the 2a sides by a semicircle of radius a. As each of the circle centers is placed in the plane at random, independently of each other, the probability that such an area is free of circle centers from Equation (51) is

Fj(f.V.) = eXp[-2apjnj - 7Ta*KZj],

and from the statistical independence of the various orientations22’23

P(f.v.) =

1

exp C[-Zapln, - mz2nl] if p*vjZO and p-q;<0 I

0 (otherwise)

(58)

The integral (39) also requires a sum without restriction over all the possible j, k pairs and we have

x (Vj ’ P)(%4 *PI q(p 71, v;j P4 ’ I’

x exp C[-2aplnl - 7ra2nr]. 1

(59)

The integrations over 0j and 19!! are each performed over all possible angles 0 to 27r, respectively, in the j and k planes, subject to the respective restrictions p - qj > 0 and p. rjk d 0. The p integration is done over the infinite slab volume.22’23

(ii) POCll Consider first cylinders placed at random with axes mutually parallel and an applied gradient also parallel to the cylinder axes. For heat transfer down a very long channel with a constant cross sectional shape, the temperature profile is also one dimensional and linear. In addition, i and the surface normal r] are always perpendicular, so that the bracketed averages in eqns (32)-(35) all vanish except ([i - pS-‘I*). From its overlapping cylinder form (58)

([i.pS-‘I*) = ~*/8, (60)

and the variational principle (31) then gives, not a bounding estimate, but the exact equality

k, = @S7r2/4] + (1 - 4)k, (61)

which is listed along with a PO = 7r2/4 in the second row of Table 1. In the limit E -+ 0, the radiation transport problem becomes mathematically equivalent to Knudsen mass diffusion in the same geometry. Simulations have been performed by Tomadakes and Sortirchos58’59 on

all the cylinder bed structures in Table 1 and their results for PO are given in the second column. Simulation errors in some cases run as high as 2-5%, but the results are in excellent agreement.

(iii) POCl In a second case parallel cylinders are placed at random and allowed to freely overlap, but the flux is perpendicular to the fiber axes. The coefficients PO, PI, P2, P3 listed in the third row of Table 1 can be used to determine the bracket averages (q) from eqns (32)-(35). These values were of course first determined by direct application of eqns (57) and (59). Note from column three that PI is zero and E only appears in the denominator of krad. The PO simulation values listed in column two of Table 1 and marked with an asterisk, have been taken in the dilute cylinder bed limit. From its agreement with the PO simulation values, the anisotropic scattering (e -+ 0) form of krad is exact for a dilute bed. For the opposite limit of no scattering (6 -+ l), since each fiber has a small radius (a << L) compared to the distance for thermal change, it is practically isothermal and emits isotropically, and C&*/8 is the correct two-dimensional isotropic scattering thermal conductivity. Between these limits krad increases monotonically with E, and is at least a very good approximation. As typicals6 fiber bed void fractions are 90% and as at these high porosities overlap is not dominant in our model, the POCl form of krad is a reasonable form for the radiation void conductivity in this particular fiber bed geometry.

(iv) and (v) PROC Each cylinder is randomly oriented in a plane and the cylinder planes of rotation are mutually parallel. The cylinders are allowed to overlap freely. In a fourth case the average flux is in a direction parallel to the planes, and in a fifth the flux is perpendicular across the planes of orientation. Assuming a homogeneous angular distribution of cylinder axes, we replace the nj by bdrj/n in eqns (57)-(59) and sum over j by an integration over rj from 0-n.

nj = pdrj/n O<yj<.ir. (62)

The coefficient PI is zero in both flux directions leaving a monotonic increase of krad with E from the denominator. The radiation void conductivity in column six of Table 1 can be obtained analytically in terms of a constant Go or Gi in each case,

dw = 0.5984. . . , (PROC I)

(63)

and

G, =*I1 (‘-‘*) dv= 1441...,(PROC]]), (64) 4 oE(l-7J2)


where E(m) is the complete elliptic integral of the second kind.2 The agreement of krad for the anisotropic scattering limit (E --f 0), with the simulation results,58,59 is obtained again in equivalent dilute beds. In the limit of no scattering (E + l), the linear trial temperature and krad are exact for a dilute bed with isotropic scattering.

(vi) IOC For isotropic random orientation, each of the freely overlapping, solid right circular cylinders is placed at random into the bed with an orientation independent of those of the other cylinders already present. For a homogeneous angular distribution of cylinder axes, nj is replaced by csin <jd&dyj/(2T) and the sum over j by integration over drj fro:m 0 to 27r and dcj from 0 to 7r/2.

(65) The void radiation thermal conductivity in the anisotropic scattering limit (E + 0) gives the coefficient 12/13.

The 12/l 3 result, known to be the exact solution for a dilute bed of spheres, agrees with the simulation value of PO in Table 1 and is also the appropriate solution for a dilute three-dimensional isotropic bed of cylinders. In the isotropic scattering case (E -+ l), the 4/3 coefficient, a well-known exact result4 for an isotropic dilute bed of particles and also provildes the correct isotropic scattering solution of a dilute IOC cylinder bed. It is interesting to note that Pt for this geometry does not vanish and its 0.5 value is necessary tso obtain the 4/3 limit. Also for any other void fraction, the case of krad in eqns (31)- (35), always at least gives an upper bound.

DISCUSSION OF RESULTS

To demonstrate the effects of the different solid dispersions on the radiation void conductivity, the dimensionless radiation conductivity k&( CS) is plotted vs the particle surface emissivity E in Fig. 3. Vortmeyer’s equation6” labeled 0, provides a small improvement to the Damkoehler-parallel plate equation.3 The Damkoehler equation, which runs from 0 at E + 0, monotonically to O-5 at E = 1, lies slightly below 0 and provides a void-solid series lower bound on the dispersion radiation conductivities, we have derived. The dimensionless void radiation conductivity down along the central axes of a bed of overlapping parallel cylinders, POCll labele’d @ on the plots, in parallel transport with the solid provides a horizontal line (2.467) upper bound for the various dispersions considered. Unlike the two-Fourier phase upper bound,2’ this bound can vary with void shape. Indeed for a non- dispersed solid such as isolated parallel capillaries krad/( C6) can have a much lower value of 1.333.

“-:; 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1

E

Fig. 3. Plots of the dimensionless void radiation conductivity kJC6 vs particle surface emissivity E for various dispersed solid geometries. @ IOC, Isotropic bed of spheres; @ POClI, Flux down fiber axes of parallel fibers; @ POCI, Flux perpendicular to parallel fibers; @ PROCII, Flux parallel to fiber planes of rotation; @ PROCI, Flux perpendicular to fiber planes or rotation; @ IOX, Three dimensional isotropic fiber rotation; @Vortmeyer equation P = 0.1. Equations listed

in Table 1.

The overlapping sphere bed, 10s and labeled 0, is included as the granular example. The variational result in the fifth column of Table 1 can be directly related to the simple linear anisotropic scattering approximation17 of pseudohomogeneous radiation transport theory [note discussion in the Introduction. Pseudohomogeneous theory for our large diffuse opaque particles, only holds in the limit cj + 1.1 For a dilute sphere bed (4 --+ 1) and with (ra&) -’ in place of the average sphere diameter S from eqn (43), the 10s variational equation for krad has the form

krad = 48aT3/{ [9 + 4( 1 - c)](ra&)}, (66)

where C = 4aT3 from eqn (10~) has been included. The absorption coefficient gcu of pseudohomogeneous theory can also be expressed in terms of a0 and no

cr, = 7n&zoQ,. (67)

Tien and Drolen” have pointed out, that the absorption efficiency Q, for a large particle can be set equal to the particle emissivity 6. They further note that the scattering efficiency coefficient 55 Q, from

us = dnoQs,

can be taken as (1 - Q,), so that

(T$ = ra$zo( 1 - Q,).

(68)

(69)

184 W. Strieder

The extinction coefficient K is by definition the sum of the absorption oa and scattering coefficients g’s, and from eqns (67)-(69)

K, = CT, + us = 7ra;ns. (70)

The ‘had variational upper bound expression becomes with eqn (70)

krad<48~~3/{[9 + 4(1 - E)]K}. (71) For an optically thick slab of large spherical particles (nd/& > 100) and using the linear anisotropic scattering approximation, Dayan and Tien (eqn (27) of Ref. [17]) have derived the approximate equality.

krad = 16crT3/{[3 - ~(1 - E)]K}, (L.A.S.). (72) In eqn (72), we have replaced Dayan and Tien’s single scattering albedo w. with its straightforward emissivity dependence” of (1 - 6). For the simple linear anisotropic scattering approximation, the phase function has been approximated by unity, plus a correction term linear in the cosine of the scattering angle. The coefficient z of the cosine term must be assigned a value. For 4 -+ 1 and E + 1, &ad from eqn (72) already coincides with the exact result (71). As Derjaguin’s coefficient is exact in the limit 4 + 1, E + 0, we can require eqn (72) in this limit to coincide with the exact form of Derjaguin*’ and this sets a value of z at

z = -413. (73)

Dayan and Tien17 note that z --+ 1 represents strong forward scattering, while z --+ - 1 implies strong back- ward scattering. Strictly speaking the absolute value of z should be less than or equal to 1 to avoid negative values of the phase function, but Dayan and Tien17 observe that because of the approximate nature of the linear phase function, the absolute value of z in many cases exceeds unity. In addition to very strong anisotropic backscattering, the variational upper bound result (71) establishes the simple linear anisotropic scattering approximation result (72) with z = -4/3 as a rigorous upper bound on the thermal radiation conductivity, that reduces to the exact physical forms in both of the scattering extremes, E = 0 and 1. These features suggest that inequality (71) might be useful as an estimate of the radiant conductivity.

Dispersion structure of long fiber-like cylinders are the object of the curves labeled 0-0. Chemical vapor intrusion25’ *9,37,% 57 to produce advanced fiber reinforced ceramic composites begins with fiber mats to produce cores and nozzles in rockets, heat shields and brake disks for supersonic aircraft and race cars. As the ceramic vapor infiltration requires high temperatures (1200°C) and low pressures, krad in the fiber mats is needed. From the point of view of radiation scattering theory, the strict application of geometrical theory4 to obtain krd requires fiber diameters greater than 9~. As the fiber diameters in the mats are between 10 and 150 h and grow during densification, the kradequations will be applicable.

The curve labeled 0, which provides the k,ad/(C6) values for the flux across the axes of the same parallel cylinders, always lies well below 0. For beds with each cylinder lying in parallel planes, but oriented mutually at random relative angles, the curves labeled @ and @ represent the flux parallel and perpendicular to the cylinder planes of rotation. That cylinders perpendicular to the flux block more effectively implies that @ should lie above 0. From 0-0, effects of fiber rotation in the plane of a dilute fiber bed, where heat transfer is in that plane, is demonstrated. But from @ and 0, the same fiber rotation for flux perpendicular to the fiber plane gives a small effect. The curve @ for cylinder beds with three-dimensional isotropy, i.e. each cylinder oriented independently at random in three dimensions, has the same anisotropic and isotropic scattering limits as the three-dimensional isotropic sphere bed 0. However, as Pi is 0.5 and P2 is small at 0.038, the three-dimensional isotropic cylinder bed curve @ is nearly a straight line, whereas the sphere bed 0, has a positive curvature between the limits. An overview of Fig. 1 in the anisotropic limit (6 + 0) shows a real diversity in the actual radiation conductivities with shape, (i-0, @ and arrangement, 0-8. But even for the isotropic case (e -+ 1) significant effects of dispersed solid geometry remain in krad/(CS) beds. Unfortunately the parallel surface equation or the simple modifications 0, seriously underestimates krad for any E, but particularly at lower E, estimates from the parallel @ plate will be very misleading. The e variation of krad from 0, i.e. the slopes of @ in Fig. 3, are not realistic either.

SUMMARY AND CONCLUSIONS

(i) Long range multiple scattering

The problem of multiple scattering raised by Vortmeyer@ and emissivity dependence of krad by Beek4 have been resolved. Rigorous variational solutions of the complete multiple scattering problem for all E have been presented. For the case of spheres, the results are completely consistent with the other simulations58359 and theories 17, *0,44,55 But the method provides far more physical insight into the nature of the krad coefficients [eqns (31)-(35)]. In other words, Vortmeyer knew some new coefficient was necessary to deal with multiple scattering, but the actual values and influence are larger than he anticipated. In addition, not one, but at least four are necessary.

(ii) Multiple scattering effects of particle shape and dispersion structure

With the development of the method, the significant effect of shape and dispersion structure on krad for cases other than 10s could be addressed (Table 1 and Fig. 3).


The existence of parallel and series type bounds are reasonable in hindsig:ht. The anisotropy of a flat random, fiber bed (X,, for PROCll and PROCI) is clearly exhibited in curves @ and @ of Fig. 3. Most industrial cases will lie between @ and 0, but multiple scatter will give rise to significant structural influences on krad and its variation with E within these limits. Also, an overall picture of the interplay of fiber or granular beds and enclosures emerges from Fig. 3.

(iii) Higher solid densities and void-solid interactions

So far the usefulness of the krad forms have been interpreted at lower solid fractions, but the variational method and bounds are valid at any solid fraction. Hence the variational methods provide a means to approach dense beds’. The interfacial resistances suggested in the arguments of Argo and Smith,3 that cause thermal gradients within the solid particles to be less than those in the voids, must be addressed. Higher solid fraction studies must clarify the role that emissivity plays in interfacial resistance to heat transport, e.g. solid transport in a dispersed bed of particles must vanish as E --+ 0.40 The general equations for high temperature heat transport must be consistent at the various limits of 4 and E with those of the: gray gas4 and packed beds,17 if we are to fully understand the thermal phenomena in these systems. Our research group is investigating these important interfacial ejfects at higher solid fractions. Finally the direct use of truncated multiple scattering summations,38’70 which have been successfully used in other transport problems in disordered media, is being considered.

ACKNOWLEDGEMENT

Acknowledgement is :made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, under whose support most of the research was done.

REFERENCES

1. Abbasi, M. H. and Evans, J. W., Monte Carlo simulation of radiant transport through an adiabatic packed bed or porous solid. AZChE .Z., 28 (1982) 853-854.

2. Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, National Bureau of Standards, 1964.

3. Argo, W. B. and Smith, J. M., Heat transfer in packed beds. Chem. Engng Progress, 49 (1953) 443-451.

4. Beek, J., Design of p,acked bed catalytic reactors. Adv. Chem. Engng., 3 (1962) 203-271.

5. Borodulya, V. A. and Kovensky, V. I., Radiative heat transfer between a fluidiized bed and a surface. Znt. J. Heat Mass Transfer, 26 (1983) 277-287.

12. Chan, C. K. and Tien, C. L., Radiative transfer in packed spheres. ASME J. Heat Transfer, 96 (1974) 56-58.

13. Chen, J. C. and Churchill, S. W., Radiant heat transfer in packed beds. AZChE J., 9 (1963) 35-41.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

Ghen, J. C. and Chen, K. L., Analyses of simultaneous radiative and conductive heat transfer in fluidized beds. Chem. Engng Comm., 9 (1981) 255-271. Chu, H. S., Stretton, A. J. and Tien, C. L., Radiative heat transfer in ultra-fine powder insulations. Znt. J. Heat Mass Transfer, 31 (1988) 1627-1633. Cooper, C. F., Pores and thermal conductivity. J. Brit. Ceram. Sot., 3 (1966) 115-123. Dayan, A. and Tien, C. L., Heat transfer in a gray planar medium with anisotropic scattering. Trans. ASME J. Heat Transfer, 97 (1975) 391-396. Decker, N. and Glicksman, L. R., Conduction heat transfer at the surface of bodies immersed in gas fluidized beds of spherical particles. AZChE Symp. Series, 77(200) (1981) 341-349. Decker, N. and Glicksman, L. R., Heat transfer in large particle fluidized beds. Znt. J. Heat Mass Transfer, 26 (1983) 1307-1320. Derjaguin, B., Measurement of the specific surface of porous and dispersed bodies by their resistance to the flow of rarefied gases. Comput. Rend. Acad. Sci., URSS, 53 (1946) 623-626. DeVera, A. L. and Strieder, W. , Upper and lower bounds on the thermal conductivity of a random two-phase material. J. Phys. Chem., 81 (1977) 1783-1790. Faley, T. and Strieder, W., Knudsen flow through a random bed of unidirectional fibers. J. Appl. Phys., 62 (1987) 4394-4397. Faley, T. and Strieder, W., The effect of fiber orientation on Kundsen permeabilities. J. Chem. Phys., 89 (1988) 6936-6940. Filtres, Y., Flamant, G. and Hatzikonstantinou, P., Wall to fluidized bed radiant heat transfer analysis using the particle model. Chem. Engng Comm., 72 (1988) 187-199. Fitzer, E. and Gandow, R., Fiber reinforced silicon carbide. Ceram. BUN., 65 (1986) 326-335. Flamant, G. and Menegault, T., Combined wall to fluidized bed heat transfer bubbles and emulsion contributions at high temperature. Znt. J. Heat Mass Transfer, 30 (1987) 1803-1811. Grace, J. R., Fluidized bed heat transfer. In: Handbook of Multiphase Systems, ed. G. Hestroni. Hemisphere, 1982, 8-72. Hlavacek, V., Aspects in design of packed bed reactors. Z E. Chem. Fund., 62 (1970) 8-25.

6. Borodulya, V. A., Kovensky, V. I. and Makhorin, K. E., Fluidized bed radiative heat transfer. Proc. 4th Znt. Conf on Fluidization, eds D. Kunis and R. Toel, Engng Foundation, New York, 1983, pp. 379-387.

7. Botterill, J. J., Fluid-Bed Transfer. Academic Press, New York, 1975.

8. Brewster, M. Q., Effective emissivity of a fluidized bed. ASME Winter Ann. Meeting, New Orleans, LA, HTD-40 (1984) 7-13.

9. Brewster, M. Q., Effective absorptivity and emissivity of particulate media with application to a fluidized bed. ASME J. Heat Transfer, 108 (1986) 710-713.

10. Brewster, M. Q. and Tien, C. L., Radiative transfer in packed fluidized beds: dependent versus independent scattering. ASME J. Heat Transfer, 104 (1982) 573-579.

11. Cartigny, J. D., Yamada, Y. and Tien, C. L., Radiative transfer and dependent scattering by particles: part 1, theoretical investigation. ASME J. Heat Transfer, 108 (1986) 608-613.

186 W. Strieder

29. Hopkins, E. R. and Chem. J., Sic matrix/Sic fiber composite: a high heat flux, low activation structural material. .Z. Nucl. Muter., 141-143 (1986) 148-151.

30. Ishimaru, A. and Kuga, Y., Attenuation constant of a coherent field in a dense distribution of particles. .Z. Opt. Sot. Amer., 72 (1982) 1317-1320.

31. Kaganer, M. G., Thermal Insulation in Cryogenic Engineering. IPST Press, Jerusalem, 1969.

32. Kingery, W. D., Klein, J. D. and McQuarrie, M. C., Development of ceramic insulating materials for high temperature use. Trans. ASME, 80 (1958) 705-710.

33. Kotake, S., Compulation in molecular dynamics and heat transfer. In: Computers and Computing in Heat Transfer Science and Engineering, eds W. Nakayama and K. T. Yang, CRC Press, Boca Raton, FL, 1993, pp. 141-151.

34. Kudo, K., Taniguchi, H. and Kim, Y., Side-wall effects on the transmittance of radiative energy through a three dimensional packed bed. Trans. JSME, 55 (1989) 2819- 2823.

35. Kudo, K., Monte Carlo simulation of radiative transfer through packed spheres. In: Computers and Computing in Heat Transfer Science and Engineering, eds W. Nakayama and K. T. Yang. CRC Press, Boca Raton, FL, 1993, pp. 173-185.

36. Marthur, A. and Saxena, S. C., Total and radiative heat transfer to an immersed surface in a gas fluidized bed. AZChE J., 33 (1987) 1124-1135.

37. Melkote, R. R. and Jensen, K. F., Gas diffusion in random fiber substrates. AZChE J., 35 (1989) 1942-1952.

38. Nam, C. W., Physics of inhomogeneous inorganic materials. Prog. Mater. Sci., 32 (1993) l-l 16.

39. Nusselt, W., Z. Bayer Revisions-Ver., 17 (1913) 125. 40. Ochoa, J. A., Stroeve, P. and Whitaker, S., Diffusion and

reaction in cellular media. Chem. Engng Sci., 41 (1986) 2999-3013.

41. Ozkaynak, T. F., Chen, J. F. and Frakenfield, T. R., An experimental investigation of radiation heat transfer in a high temperature bed. Proc. 4th Znt. Conf Fluidization, eds D. Kunii and R. Toef, Engng Foundation, 1984, pp. 371- 378.

42. Saxena, S. C., Srivastava, K. K. and Vadivel, R., Experimental techniques for the measurement of radiative and total heat transfer in gas fluidized beds: a review. Experimental Therm. Fluid Sci., 2 (1989) 350-364.

43. Schotte, W., Thermal conductivity of packed beds. AZChE J., 6 (1960) 63-67.

44. Seigel, R. and Howell, J. R., Thermal Radiation Heat Transfer. McGraw-Hill, New York, 1992.

45. Smith, J. M., Chemical Reactor Kinetics. McGraw-Hill, New York, 1981.

46. Sortirchos, S. V., Dynamic Modeling of Chemical Vapor Infiltration. AZChEJ., 37 (1991) 136551378.

47. Sortirchos, S. V. and Tomadakis, M. M., Modeling transport reactions and pore structure evolution during densification of cellular or fibrous structure. Mater. Res. Sot. Sym. Proc., 108 (1990) 73-89.

48. Stinton, D. P., Besman, T. M. and Lowde, R. A., Advanced ceramics by vapor deposition techniques. Amer. Ceram. Bull., 67 (1988) 350-360.

49. Stoyan, D., Kendall, W. S. and Mecke, J., Stochastic Geometry and Its Applications. Elsevier, Oxford, 1967.

50. Strieder, W. C. and Prager, S., Knudsen flow through a porous medium. Phys. Fluids, 11 (1968) 2544-2548.

51. Strieder, W. and Aris, R., Variation Methods Applied to Problems of Difliron and Reaction, Springer, Berlin, 1973.

52. Szekely, J. and Fisher, R. J., Wall to bed radiation heat transfer in a gas-solid fluidized bed. Chem. Engng Sci., 24 (1969) 833-849.

53. Szekely, J. and Evans, J. W., A structural model for gas-solid reactions with a moving boundary-II The effects of grain size, porosity and temperature on the reaction of porous pellets. Chem. Engng Sci., 26 (1971) 1901-1913.

54. Tien, C. L., Thermal radiation in packed and fluidized beds. ASME J. Heat Transfer, 110 (1988) 1230-1242.

55. Tien, C. L., and Drolen, B. L., Thermal radiation in particle media with dependent and independent scattering. Ann. Rev. Numer. Fluid Mech. Heat Transfer, 1 (1986) l-32.

56. Tien, C. L. and Stretton, A. J., Heat transfer in low temperature insulation. In: Heat and Mass Transfer in Refrigeration and Cryogenics, eds Bouugard, J. and Afgan, N. H., Hemisphere, 1987, pp. 1-13.

57. Tomadakis, M. M. and Sorterchos, S. V., Effects of fiber orientation and overlapping on Knudsen, transition, and ordinary regime diffusion. In: Chemical Vapor Deposition of Refractory Metals and Ceramics, eds Besmann, T. M. and Gallios, B. M., MRS, Pittsburgh, 1982, pp. 221-226.

58. Tomadakis, M. M. and Sortirchos, S. V., Effective Knudsen diffusivities in structures of randomly overlapping fibers. AZChE J., 37 (1991) 74-86.

59. Tomadakis, M. M. and Sortirchos, S. V., Transport properties of random arrays of freely overlapping cylinders with various orientation distributions. J. Chem. Phys., 98 (1993) 616-626.

60. Tsai, D. S. and Strieder, W., Radiation across a spherical cavity having both specular and diffuse reflectance components. Chem. Engng Sci., 40 (1985) 170-173.

61. Tsai, D. S. and Strieder, W., Specular and diffuse reflections in radiant heat transport across and down a cylindrical pore. I. E.Chem. Fund, 25 (1986) 207-218.

62. Tsai, D. S., Ho, F. G. and Strieder, W., Specular Reflection in radiant heat transfer across a spherical void. Chem. Engng. Sci., 39 (1984) 775-779.

63. Varma, A., Cao, G. and Morbidelli, M., Self-propagating solid-solid noncatalytic reactions in finite pellets. AZChE Journal, 36 (1990) 1039-1049.

64. Vortmeyer, D., Radiation in packed solids. German Chem. Engng, 3 (1980) 124-137.

65. Wakao, N. and Kato, K., Effective thermal conductivity of packed beds. J. Chem. Engng Japan, 2 (1969) 24-32.

66. Wakao, N. and Vortmeyer, D., Pressure dependency of effective thermal conductivity of packed beds. Chem. Engng Sci., 26 (1971) 1753-1765.

67. Wax, N., Selected Papers on Noise and Stochastic Processes. Dover, 1954

68. Whitaker, S., Radiant energy transport in porous media. I. E. Chem. Fund., 19 (1980) 210-218.

69. Wolf, J. R., Tseng, J. W. C. and Strieder, W., Radiation conductivity for a random void-solid medium with diffusely reflecting surfaces. Znt. J. Heat Mass Transfer, 33 (1990) 725-734.

70. Xia, Y. and Strieder, W., Complementary upper and lower truncated sum, multiple scattering bounds on the effective emissivity. Znt. J. Heat Mass Transfer, 37 (1974) 443-450.

71. Xia, Y. and Strieder, W., Variational calculation of the effective emissivity of a random bed. Znt. J. Heat Muss Transfer, 37 (1994) 451-460.

72. Yamada, Y., Cartigny, J. D. and Tien, C. L., Radiative transfer with dependent scattering by particles: part 2, experimental investigation. ASME J. Heat Transfer, 108 (1986) 614-618.

73. Yoshida, K., Toru, W. and Kunii, D., Mechanism of bed- wall heat transfer in a fluidized bed at high temperatures. Chem. Engng Sci., 29 (1974) 77-82.


74. Zheng, L. and Strieder, W., Knudsen void gas heat transport in fibrous media. ht. J. Heat Mass Transfer, 37 (1994) 1433-1440.

APPENDIX A

In engineering models for thermal conduction, the thermal energy flux J per unit total cross sectional area of a void-solid system is customarily related to the overall gradient (TL - Tc)/L across a slab with

J = -k,(TL - T,,)/L. (Al)

On a physical basis we presume, because the slab is thick, that the effective conductivity k, does not depend on slab thickness L. If the conductivity in the solid -kq - VT is replaced by the thermal flux J, eqn (8) can

be written for the edge plates Co and CL, respectively,

Ba = CTT; - (1 - Q)J/E,, (‘42) and

BL = CT; + (1 - q)J/cL. (AZ)

Equation (A2) is subtracted from eqn (A3), the resulting equation is linearized in AT/T and (Al) is substituted for the thermal flux, to give

BL - BO = 4aT3(T, - T,,) - [(l - Q)/Q

+ (1 - dIEolke(TL - To>/L. (A4)

The second term on the right-hand side of eqn (A4), of order L-' , can be neglected when L is large, and from eqns (13) and (IOc) for a thick slab

p = C(T, - T,,)i/L. (W

Radiation heat transport in disordered media

Documents

Transcript of Radiation heat transport in disordered media