The Thermal Conductivity Theory of Non-uniform Granular Flow and the Mechanism Analysis

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1 Coun. Theor. Phys. Beijing, China) 40 00) pp c International Acadeic Publishers Vol. 40, No. 4, October 5, 00 The Theral Conductivity Theory of Non-unifor Granular Flow and the Mechanis Analysis ZHANG Duan-Ming,, LEI Ya-Jie,, YU Bo-Ming, and PAN Gui-Jun Departent of Physics, Huazhong University of Science & Technology, Wuhan 40074, China State Key Laboratory of Laser Technology, Huazhong University of Science & Technology, Wuhan 40074, China Received January 0, 00) Abstract According to the fractal characteristics appearing in non-unifor granular syste, we found the fractal odel to study the effective theral conductivity in the ixed syste. Considering the quasi-equilibriu, we bring forward the fractal velocity probability distribution function. The equipartition of energy is eployed to the nonunifor granular syste, and the granular teperature is derived. We investigate the theral conductivity in granular flow due to the oveent of the particles, naely the heat transfer induced by the streaing ode only. The theral conductivity in the ixed syste changes with the fractal paraeters such as the solid fraction ν, structural character paraeter η, and fractal diension D of size distribution. These paraeters depict the characteristics of the theral conductivity in the actual coplex granular syste. Coparing our conclusion with the correlative experiental data and the theoretical conclusion of binary ixture of granular aterials, the results can qualitatively confir the generality of our prediction on the granular syste. PACS nubers: i, 6.4.Hv Key words: fractal granular, non-unifor granular flow, velocity distribution function, effective theral conductivity Introduction The granular flow can be considered as a special twophase flow. The granular syste is coposed of acroscopic particles e.g. detritus, powders, snowballs, etc.). In the fields of industry, agriculture, ining, and so on, a great deal of interesting phenoena induced by the otion of particles inspire experts with ore and ore enthusias. Further acquainting with the otion of particles in flow has becoe the disquisitive focus in theoretical works, experients, and coputer siulations in the past decades. 5] Although the particles are acroscopic, it is rational that the concepts fro dense-gas kinetic theories are applied to the research of granular systes for the coparability between the otion of particles in granular flow and that of olecules in a gas and the agnitude nuber of the particles in flow. Due to the coplexity of the granular flow, the present study is ainly liited in the unifor or finite ixed coponents syste. In Ref. 5], Hunt and Hsiau used ean-free-path arguents to develop analytical expressions for the effective theral conductivity for low-density granular flows, in which the effective theral conductivity due to the otion of particles is developed fro dense-gas kinetic theory. In Ref. 6], Hsiau further discussed the effective theral conductivities of a single species and a binary ixture of granular aterials. He found that effective theral conductivity changes both with the Biot Fourier nuber and with the species concentration. But all these discussions were liited in the unifor or finite ixed coponents systes, which cannot provide perfect theories for the actual granular systes. This work is focused on extending the investigations of the unifor granular syste to the coplex granular syste. Experients show that for any aterials, in the procession of acroscopic coinution, the geoetrical and atheatical characteristics exhibit self-siilarity. 7,8] In Ref. 9], according to the fractal characteristics of the coplex granular systes, Zhang et al. established the theory of effective theral conductivity of the non-unifor granular flow. But the discriination between the granular teperature and the particle teperature was ignored. By the equipartition of energy, the aelioration was finished in Ref. 0]. In this paper, on the basis of the previous discussions, we elaborately depict the physical echaniss of the fractal odel for the effective theral conductivity of the non-unifor granular syste and discuss the influences to the effective theral conductivity induced by the correlative fractal paraeters. The fractal odel for the theral conductivity is founded according to the fractal behavior of the non-unifor granular flow, based on which the velocity probability distribution function is presented. Because the object in our study is the shear flows with high velocity, high stiffness ratio, The project supported by National Natural Science Foundation of China under Grant No and the State Key Laboratory of Laser Technology Huazhong University of Science and Technology) under Grant No. 97D Corresponding author, e-ail:zhangd@public.wh.hb.cn Eail: yajielei@cn.co

2 49 ZHANG Duan-Ming, LEI Ya-Jie, YU Bo-Ming, and PAN Gui-Jun Vol. 40 and low-density solid fraction, we can obtain the granular teperature by use of the equipartition of energy. Only the heat transfer due to the streaing ode is considered in the heat transfer study. The heat flux of per category of particles with different ass and the enseble unit are derived, fro which the effective theral conductivity is obtained. The changes of theral conductivity with correlative paraeters, the solid fraction ν, the structural character paraeter η, and the fractal diension D of size distribution are discussed, which depicts the characteristics of the effective theral conductivity in the actual coplex granular syste. By coparing our conclusion with the correlative experiental data and the theoretical conclusion of binary ixture of granular aterials, we can qualitatively confir the generality of our prediction to the granular syste. Theoretical Model We first assue that the particles are sooth and elastic spheres and that they have different velocities and properties, siilar to the olecules of gases. If Φ represents a property such as ass, oentu or energy) of the syste, it can be defined as 5] Φ Φf ) C)d N C, ) where n is the nuber density of particles, f ) C) is the velocity distribution function of unifor granular syste, and C is the particle fluctuating velocity vector. The product f ) C)d C represents the probability of particles per unit volue with the fluctuating velocity within the velocity interval of dc centered at C, where dc dc x dc y dc z, C c c, c is the local velocity and c is the enseble average velocity. Under the condition of quasi-equilibriu, the odified Maxwellian velocity distribution function of unifor granular syste is 5] f ) C) n πθ) exp / C υ) Θ ], ) where υ is the diffusion velocity, n is the nuber density of particles, and Θ is the granular teperature which is used to quantify the particle average fluctuating kinetic energy per unit ass, defined as Θ C, ] and it differs fro the particle teperature T, which allows a syste to explore phase space. In a granular aterial, k B T 0 precludes such exploration, which iplies that entropy considerations can easily be outweighed by dynaical effects that now becoe of paraount iportance. For typical sand, the average kinetic energy of a particle, Θ, is at least 0 ties k BT at roo teperature, ] where k B is the Boltzann constant. Fig. The configuration for the theral energy flux. Here we consider one-diensional shear flows, i.e., the diffusion only exits in y direction, υ υ y, as shown in Fig.. Under the condition of quasi-equilibriu, the diffusion velocity is uch less than the fluctuating velocity, υ y C y, then the velocity distribution function in Eq. ) is decoposed into a local Maxwellian velocity distribution function f 0) C) and a correction ter f c) C), 5] f ) C) f 0) C) + f c) C), ) where the Maxwellian ter f 0) C) is f 0) C) n πθ) exp / C Θ ), 4)

3 No. 4 The Theral Conductivity Theory of Non-unifor Granular Flow and the Mechanis Analysis 49 and the correction ter f c) C) is f c) C) n πθ) exp C ) υy C y / Θ Θ. 5) Now consider a granular flow coposed of non-unifor particles, in which the particle ass distribution is continuous ax ). For siplicity, we assue that the surface of the particles is sooth, the aterial of per particle is identical, but the size of particles is different. The ass of a particle is finer particles increases ore quickly, however, when the value of approaches the axiu ax, the values of y with different D reach an asyptotic value. Because in the granular syste, the nuber of particles with the axiu size r ax and the total nuber of particles are both defined to be constant, i.e. n 0, N πr ρ p, 6) where ρ p and r are the ass density and the radius of particles, respectively. The experients 8] showed that granular aterials exhibit soe fractal characteristics. The size distribution of particles in granular flow satisfies the size-frequency characteristics by fractal theory r ) D Y Nr r) N n 0, 7) r ax where Y Nr is the ratio of N r to N, N r is the nuber of particles whose size is saller than r, N is the total nuber of particles, n 0 is the nuber of particles with the axiu size r ax, and D is the fractal diension of size distribution, and < D <. Apparently, Y Nr also denotes the probability of a particle having a size saller than r. Let r r in. Fro Eqs. 6) and 7), we have thus Y Nr r in ) N n 0 rin r ax ) D 0, n 0 ax ) D/ N. Let η n 0 /N, then we obtain ax η /D, 8) where η is an iportant paraeter that describes the structural character of the granular syste, 0 < η <. When η, the granular flow becoes a non-fractal syste. Thus, only when η, is equation 7) applicable. 8] Fro Eqs. 6) and 7), the probability of particles per unit volue within + d can be expressed as y d dy N r d d N n 0 ax ) D/ D D )/ d, 9) y N n 0 ax ) D/ D D )/. 0) Figure shows the probability density y versus ass in granular flow. In Fig., n 0, N 000, the diensionless ass of the particles is defined to vary between and 5, and the values of D are.08,., and., respectively. Fro Fig., it can be seen that y decreases with the increase of, and that when the fractal diension D of size distribution increases, the probability of Fig. The probability density of the particles versus ass. We assued that the ass distribution is independent of the fluctuating velocity and that the particles are isotropically distributed throughout the flow. The nonunifor granular velocity distribution function satisfies Eq. ). The nuber density of particles is the product of the total nuber of particles in the syste N and the probability density y, n Ny. Thus the probability of particles distributed in the intervals of + d and C C + dc can be expressed as f ), C)dd C Ny πθ ) / exp C υ ) ] ddc Θ, ) where, Θ denotes the granular teperature of the particle with ass. Apparently, f ), C) is the correlative probability density. Here, the granular teperature Θ of the particles with ass is defined as Θ C C f ), C)d C. ) And under the condition of quasi-equilibriu, for onediensional shear flows the diffusion only exits in y direction, thus the fractal velocity distribution function of non-unifor granular syste is f ), C) Ny πθ ) exp C υ ) ] / Θ Ny πθ ) / exp C x + C y υ y ) + Cz ], ) Θ

4 494 ZHANG Duan-Ming, LEI Ya-Jie, YU Bo-Ming, and PAN Gui-Jun Vol. 40 where y is deterined by Eq. 0), and υ y is the diffusion velocity, υ y C y. By linearizing, we can derive f ), C) Ny πθ ) / exp C x + Cy + Cz C y υ y + υy ] Θ Ny πθ ) exp C ) exp / Θ C ) yυ y Θ Ny πθ ) exp C ) + C ) yυ y. 4) / Θ Θ Effective Theral Conductivity of Nonunifor Granular Flow In granular flow, there are four universal kinds of heat transfer echanis: i) The heat transfer due to the oveent of the fluid. Since the heat capacity of the fluid is uch saller than that of the particles, the heat flux transferred by the fluid can be neglected; 5] ii) The conduction between particles during collisions. Because of the short duration of the collision tie and the sall contact area, this conduction is negligible; ] iii) The theral radiation. Because in our discussion the teperature of the fluid is the roo teperature, the theral radiation is so sall at low teperature that can be neglected; iv) The heat transfer due to the streaing ode. Copared with the forer echaniss, this echanis acts the ost iportant effect in the low-density granular flow when the heat capacity of fluid is quite sall and the environental teperature is not high. 5] Therefore, in the following discussions, only the effective theral conductivity due to the otion of the particles is taken into account. According to Ref. 5], for Biot nuber B i hl p /k p, which is less than 0., where l p is characteristic length, and k p is theral conductivity, the luped syste analysis is applied and the energy equation becoes c p dt hat f T )dt, 5) where is the particle ass, c p is the specific heat, A is the total surface area, T is the particle teperature, T f is the local teperature of the surrounding fluid, and h is the heat transfer coefficient between the particles and the fluid, h can be obtained by experients. The righthand side of Eq. 5) represents the heat transfer between the particles and surrounding fluid, and the left-hand, the heat absorbed by particles. The teperature gradient only exists in the y direction, as shown in Fig., consider a particle initially at fluid teperature T f that oves a short distance l to a new position with fluid teperature T 0 T f + l y dt/dy), where l y is the y coponent of l. Assuing that C y is constant in this sall local region and using t as a characteristic tie equal to l y /C y, according to the literature, 6] we have the particle teperature T T 0 C y σ dt exp l )] y, 6) dy σc y where σ c p /ha. The diensionless group l y /σc y l/σc is the product of Biot nuber and Fourier nuber, where the Fourier nuber is F 0 k p ta/l p c p. Using l y /σc y l/σc in Eq. 6) and assuing that the characteristic l is the free path λ, when B i F 0, equation 6) reduces to T T 0 l y dt dy. 7) The excess energy carried by the particle to this position relative to the surrounding fluid is dt ) e c p T T 0 ) c p l y, 8) dy where l y λ cos ξ, and ξ is the angle between C and the y axis as shown in Fig. ). Then equation 8) becoes e c p λ cos ξ dt dy. 9) Siilar to the properties of the streaing transport, the heat flux in the y direction is found by integrating the product of C y and the excess energy e carried by particles over the entire velocity space q y n ec y ec y f ), C)d C π n c p λ Θ / dt dy, 0) where n, λ, and Θ denote the nuber density, free path, and granular teperature of the particle with the ass, respectively, and n Ny. Then for non-unifor granular flow with the continuous ass distribution of particles ax ), the enseble heat flux Q y in the y direction can be found by taking average over ass Q y 8 ax ax q y d d 9π c pλ dt dy ec y f ), C)d C ax Ny Θ / d, ) where ax. The granular teperature Θ for per unit ass in the interval + d, is applied to quantify the average fluctuating kinetic energy of the particles. In accordance with the kinetic theory of gases, ] the equipartition of fluctuating kinetic energy for a binary ixture of granular aterials is assued ] to be αθ α βθ β, ) where α and β represent the first and the second kinds of particles in a granular flow, respectively. Eq. ) reveals

5 No. 4 The Theral Conductivity Theory of Non-unifor Granular Flow and the Mechanis Analysis 495 that in each direction the fluctuating kinetic energy of particles is identical in the binary ixture. Thus the average granular teperature of that syste is defined as ] Θ αn α Θ α + β n β Θ β α n α + β n β, ) where n α and n β denote the nuber density of two categories of particles, respectively. In the recent literature, 4] the coputer siulations show that the equipartition of energy does not exist in noral and tangential odes in non-unifor granular syste. Only in shear for high stiffness ratio φ, φ k t /k n where k t and k n are elasticity coefficients of springs in noral and tangential direction, are the energies approxiately equal. Under this condition, we extend Eq. ) to Θ Θ i Θ i k Θ k G, 4) where G is a constant. The equipartition of fluctuating kinetic energy for granular flow in which the ass distribution is continuous can be expressed as Θ G ax ). 5) The granular teperature of the i-th kind of particles in a granular flow is defined as Θ i C i f ) i, C i )dc i. 6) By generalizing Eq. ), the average granular teperature of the non-unifor syste is Θ n Θ + n Θ + + k n k Θ k n + n + + k n k, 7) where n i i,,..., k) denotes the nuber density of the i-th kind of particles. By introducing Eq. 5) into Eq. 7), the average granular teperature of non-unifor granular flows can also be expressed as Θ Gn + n + + n k ) n + n + + k n k Ḡ, 8) where is the average particle ass of the syste. For the non-unifor granular flow with the fractal ass distribution, in Eq. 8) can be expressed by integral, ax y d, 9) where y is presented by Eq. 0). Then equation 8) is expressed as ax ) Θ G y d. 0) The free path λ is connected with the fluctuating velocity of particles. Using the atheatical ethod siilar to Chapan s 6] and in view of the influence of the granular bulk in the flow the solid fraction ν and the radialdistribution function g 0 ν)), the free path is derived as λc) ϖ πnd Eϖ)g 0 ν), ) where ϖ C/ Θ, and Eϖ) denotes the function Eϖ) ϖ e ϖ + ϖ + ) ϖ 0 e τ dτ. ) For low-density shear flows ν < 0.5), the radialdistribution function is proposed as ] g 0 ν) ν ν), ) and for shear flows of high fraction ν > 0.5), g 0 ν) is proposed as ] g 0 ν) ν ).5ν ν, 4) where ν is the axiu shearable solid fraction for spherical particles. Equation ) is evaluated by cobining Eqs. 5), 8), and 9), 8 Q y 9π c dt ax pλ dy Θ) / / Ny d / π c dt Θ) ax pλ / Ny d dy π Nc dt pλ dy Θ / ) / π ρ dt / pνc p λ Θ. 5) dy ) / Introduce a paraeter β /, where ax / y d, and Eq. 5) is rewritten as Q y π ρ dt pνc p βλ dy Θ /. 6) For siplicity, the free path λ of different particles is substituted by that of unifor syste with the average ass, 5] r λ νg 0 ν), 7) where r is the average size of the unifor syste, and g 0 ν) is given by Eq. ). By cobining Eqs. ), 6), and 7), the fractal heat flux of the non-unifor granular flow in y direction is Q y β π ρ r ) pc p dt Θ/ νg 0 ν) dy. 8) Copared with the Fourier law, Q y k eff dt /dy, the effective theral conductivity k eff of the non-unifor granular flow is k eff ρ pc p r Θ / β 9 πg 0 ν). 9) Here for the fractal syste with the continuous ass distribution, by Eq. 9), β is β ax / y d. 40) ax y d

6 496 ZHANG Duan-Ming, LEI Ya-Jie, YU Bo-Ming, and PAN Gui-Jun Vol. 40 Inserting Eqs. 8) and 0) into Eq. 40), β becoes β ηd D) D η /D. 4) η /D For B i F 0, equation 9) indicates that k eff is proportional to the square root of the granular teperature, which is in agreeent with the result in Ref. 6]. Then, for the unifor syste, β, equation 9) becoes k eff ρ / pc p r Θ 9 πg 0 ν), 4) which is exactly the result by Hsian and Hunt. 5] This reveals that our result is a general forula. Substituting Eqs. 6), 0), and ) into Eq. 4), we obtain k eff c p Gρp β ν) 9 ν)π r, 4) ax where r 4πρ p y d) / /4πρ p ) /, and β is the fractal correction factor given by Eq. 4). 4 Coparisons and Discussions Based on Eq. 4), the characters of the effective theral conductivity k eff of the non-unifor granular flow versus correlative paraeters are discussed in the following sections. 4. Coparison of k eff with k eff According to Eq. 4), figure shows the effective theral conductivity k eff plotted against the total solid fraction ν for the radii of the particles r,, and 5 see curves A, B, and C). For siplicity, the group c p Gρp ρ p ) / is noralized to be. Assuing that the axiu diensionless radius in the syste is 5 and the iniu is, the structural character paraeter η 0. and the fractal diension of size distribution D.8, then the corresponding curve of the non-unifor syste is curve D in Fig.. and in the non-unifor syste, and it decreases ore sharply with the saller radius of the particles. The variational trends of the four curves are consistent and the curve D presenting k eff of the non-unifor syste just locates in the range, k eff r in) < k eff < k eff r ax). It should be noted that the echanis of heat transfer is the streaing ode. On the one hand, with the increase of ν, there are ore particles transporting the theral energy in the flow field; on the other hand, the free path also decreases with the increase of the solid fraction, λ /νg 0 ν)] see Eq. 7)), which reduces the heat transfer rate because of the particle collisions. Therefore, the final variational trend is k eff decreases with the increase of the solid fraction ν. The sae variational trend and the hoologous variational range in the two categories of granular systes qualitatively prove taht our condlusion is a general result for the granular syste. 4. Influence of η on k eff Let D.4, η 0., 0.0, 0.00, and 0.000, according to Eq. 4), figure 4 shows the fractal effective theral conductivity k eff versus the total solid fraction ν. The four curves A, B, C, and D in the figure represent η 0.000, 0.00, 0.0, and 0., respectively. Fro Fig. 4, we can see that, for any η value, k eff coherently decreases with the increase of the solid fraction ν when D is constant. Saller η corresponds to larger k eff. It indicates that k eff increases with the greater dispersion of finer particles. Actually, fro Eq. 4), the theral conductivity k eff is inversely proportional to the average radius r, which eans the saller average radius of the fractal syste iplies greater k eff. Fig. 4 The effective theral conductivity k eff versus the solid fraction ν for different η values. Fig. The effective theral conductivity k eff versus the solid fraction ν. It is seen that the theral conductivity decreases with the increase of solid fraction both in the unifor syste 4. The Relation Between k eff and D According to Eq. 4), figure 5 shows the fractal effective theral conductivity k eff versus the fractal diension D of size distribution for ν 0.5. The three curves A, B, and C in the figure represent η 0.00, 0.00, and 0.0, respectively, which is consistent with the result in the forer discussion. Fro Fig. 5, it can be seen, for a given

7 No. 4 The Theral Conductivity Theory of Non-unifor Granular Flow and the Mechanis Analysis 497 ν, k eff increases with the increase of the fractal diension D. The reason is that higher D iplies greater dispersion of the finer particles, which contributes the oveents of particles by the streaing ode and then enhances the theral conductivity in granular flow. experients. Only the connection between the Prandtl nuber P r µc p /k eff and the solid fraction ν was given, as shown in Fig. 6. Thus, by P r and ν, we copare our results with their analyses and experients. Fig. 5 The effective theral conductivity k eff versus the fractal diension of size distribution D for different η values. 4.4 Coparison of Theories with Experients Wang and Capbell ] experientally exained the effective theral conductivity k eff of the unifor granular flow and the apparent shear viscosity µ as the functions of shear rate. Hsiau and Hunt 5] deduced the expression of k eff by kinetic theory see Eq. 4)). But neither the velocities nor the granular teperature were easured in their Fig. 6 Coparisons of the theoretical predictions for the apparent Prandtl nuber with the experiental shear-flow easureents. The shear stress depends on both the kinetic and the collisional contributions and is expressed as 5] / du P xy rρ p g ν, e p )T dy, 44) where g ν, e p ) is a function of solid fraction ν and the coefficient restitution between particles e p and is deterined as 5] g ν, e p ) 5 π 96 δ δ) g 0 ν) δ δ ν + 64 δ 5 δ δ + ) ] ν g 0 ν), 45) π where δ + e p )/ and g 0 ν) is deterined by Eq. ). According to Eqs. 4) and 44), the Prandtl nuber of the fractal ode is P r 9 πg 0 ν)g ν, e p ) β, 46) where β is the fractal correction factor given by Eq. 4). Fro Fig. 6, it is clear that in the range of 0. ν 0.5, our results the coefficient restitution e p equals and 0.6, respectively) are closer to the experiental data than those of Hsiau and Huntin s theory. 5] The range of the solid fraction is deterined by Eq. ), which is suitable for low-density ν < 0.5) shear flows. In literature, 5] the discrepancy between the kinetic theory estiates and the experiental easureents is attributed to the finite-sized annulus, in which the shearing of the flow and the heat transfer near the wall ay vary significantly fro that in the bulk region. The difference between our theoretical results and experiental easureents, besides the above reason, ay be attributed to the following two reasons: ) Our theory is suit for granular ulti-ixtures, but the objects of the experients are unifor granular flow; ) Our odel is for rapid shear flows, and the echanis of heat transfer is the streaing ode, so the velocity of particles significantly influences theral conductivity in flow. However, the experiental equipent restricts the velocity of the particles. Because there have not been suitable experiental data available, the Wang s easureents only confired our results qualitatively. 4.5 Coparison of Fractal Results with Theories of Binary Granular Mixture For the binary granular ixture, r r, ν and ν are the solid fraction of different particles, respectively, and ν is the total solid fraction, ν ν + ν. When B i F 0, the effective theral conductivity is 6] k eff 8 π i, 0 i ρ p c p d i ν i γ i Θ/, 47) where 0 +, γ i 0 α i ω 5 i e ω i dωi.

8 498 ZHANG Duan-Ming, LEI Ya-Jie, YU Bo-Ming, and PAN Gui-Jun Vol. 40 According to Eq. 40), n ν + n ν β n + n ν + ν n + n ν r ν + ν ) r ν r ν ν + r ν ). ν 48) Fig. 7 The diensionless theral conductivity versus the solid fraction for r /r.0, B if 0 0, and for ν /ν 0, 0.6 and 0.8, respectively. According to Eqs. 9) and 48), figure 7 shows the fractal diensionless effective theral conductivity k eff /ρ p c p r Θ / versus solid fraction ν for r /r.0, B i F 0 0, and for ν /ν 0, 0.6 and 0.8. The curves arked with B, C, and D correspond to our odel, and the curve arked with E is for the binary granular ixture. Figure 7 shows that the four curves are in quite agreeent in the range of 0. ν 0.5. The range of the solid fraction is also deterined by Eq. ), which is only suitable for low-density ν < 0.5) shear flows. That is to say our fractal results can be deduced to the binary granular ixture. 5 Conclusions On the basis of the fractal characteristics presented by the non-unifor granular syste, a fractal odel for the ulti-ixture under the quasi-equilibriu has been derived and the fractal velocity distribution function is proposed in this work by this odel, we studied the physical echaniss of the fractal odel for the effective theral conductivity of the non-unifor granular syste and discussed the influences to the effective theral conductivity induced by the correlative paraeters, e.g. the solid fraction ν, structural character paraeter η, and fractal diension D of size distribution, which depict the characteristics of the effective theral conductivity in the actual coplex granular syste. By coparing our conclusion to the correlative experiental data and the theoretical conclusion of binary ixture of granular aterials, we can qualitatively confir the generality of our prediction to the granular syste. References ] M. Farrell, G.K.K. Lun, and S.B. Savage, Acta Mech ) 45. ] S.S. Hsiau and M.L. Hunt, J. Fluid Mech. 5a) 99) 99. ] D.G. Wang, and C.S. Capbell, J. Fluid Mech ) 57. 4] J. Payan and P.J. Willia, Physica A75 000) 47. 5] S.S. Hsiau and M.L. Hunt, J. Heat Transfer 5b) 99) 54. 6] S.S. Hsiau, J. Multiphase Flow 6 000) 8. 7] H. Xie, R. Bhasker, and J. Li, Minerals and Metallurgical Processing, February 99) 6. 8] T. Allen, Particle Size Measureent, rd ed., Chapan and Hall, London, New York 98). 9] D.M. Zhang, Z. Zhang, and B.M. Yu, Coun. Theor. Phys. Beijing, China) 999) 7. 0] D.M. Zhang, Y.J. Lei, and B.M. Yu, Coun. Theor. Phys. Beijing, China) 7 00). ] M. Heinrich and R. Sidney, Rev. Mod. Phys ) 59. ] J. Sun and M.M. Chen, Int. J. Heat Mass Transfer 988) 969. ] S. Chapan and T.G. Cowling, The Matheatical Theory of Non-unifor Gases, Chapter 6, rd ed., Cabridge University Press, Cabridge 970). 4] N.P. Kruyt and L. Rothenburg, J. Solids and Structures 8 00) 4879.