Evaluation of effective thermal conductivity in random packed bed: Heat transfer through fluid voids and effect of packing structure

PTEC-14559; No of Pages 11 Powder Technology xxx (2019) xxx

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Evaluation of effective thermal conductivity in random packed bed: Heat transfer through fluid voids and effect of packing structure Guojian Cheng a, Jieqing Gan b,⁎, Delong Xu c, Aibing Yu b,d a

Institute for Process Modeling and Optimization, Jiangsu Industrial Technology Research Institute, Suzhou Industrial Park 210008, China ARC Research Hub for Computational Particle Technology, Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia Institute of Powder Engineering, College of Materials and Mineral Resources, Xi'an University of Architecture and Technology, Xi'an 710055, China d Monash University - Southeast University Joint Research Institute, Suzhou Industrial Park 210008, China b c

a r t i c l e

i n f o

Article history: Received 3 May 2019 Received in revised form 27 July 2019 Accepted 30 July 2019 Available online xxxx Keywords: Effective thermal conductivity Discrete element method Heat transfer mechanism Porosity Packing structure

a b s t r a c t Effective thermal conductivity (ETC) is an important parameter in packed-bed systems and is affected significantly by packing structure. Heat flow through a packed bed can be divided into three parallel paths: the solidfluid-solid path, the solid-solid path and the fluid path. The models to evaluate ETC in packed beds for the first two paths were previously developed. In the present work, a comprehensive model to consider the heat transfer through the voids of a randomly-packed bed is further developed using the Delaunay tessellation. The results show that heat conduction through the fluid-filled voids becomes significant when the solid-to-fluid conductivity ratio decreases to a certain level (b5). The effect of packing structure on the ETC of packed beds is then investigated based on the packing generated by the discrete element method (DEM). Four cases with different heat transfer mechanisms or paths are considered. The results indicate that the probability density distribution of heat flow is generally dependent on the packing structure, represented by porosity. With an increase in porosity, the heat conduction and radiation via the solid path decrease, while the heat conduction via the fluid-filled voids increases. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Heat flow through a packed bed can be divided into three parallel paths: the solid-fluid-solid path, the solid-solid path and the fluid path [1]. The first two paths include conduction through the solid phase, conduction through the contact surface between contacting particles and conduction and radiation between neighboring particles. These two paths have been discussed previously [2–7]. The heat flows through the two paths are dominant when the ratio of solid to fluid conductivity is large. However, with a decrease of the ratio, the heat flow by the third path becomes more important. Many researches focused on determining the conductivity by simulating the heat transfer process with the diffusion of some binary gas mixture through packed beds [8–11] or by measuring the electrical conductivity of particles made of a good insulator which was dispersed in an electrolyte solution [12–14]. Tsotsas and Martin [15] proposed a relationship relating the effective thermal conductivity (ETC) of a packing of non-conducting spheres to three factors: the conductivity of the fluid, porosity of the packing, and a parameter that distinguishes between packings with the same porosity but different structures. The experimental verification of the above relationship is, however, very difficult because changes in the thermal ⁎ Corresponding author. E-mail address: [email protected] (J. Gan).

conductivity would be hard to distinguish. Zehner [16] postulated a simplified formula to calculate the thermal conductivity from the porosity of a packed bed. However, the results overestimate the effect of nonconducting particles on the conductivity of the bed [17]. Packed beds with different structures usually have different ETCs. For example, changing the gaps between neighboring particles will either enhance or inhibit heat transfer through the gaps. Flattening of particles at the contact points affects the solid-solid heat transfer as well. In fact, all of the factors, such as particle size (or distribution), particle shape, packing methods, and mechanical and surface properties of particles [18–22], that alter the packing structure, can affect the heat transfer of packed beds. Thus, it is desirable to relate ETC of a packed bed to the material properties and packing structure, so that changes in these factors will reflect changes in the effective bed properties. In the past decades, different correlations between ETC and packing structure have been obtain, as summarized by Antwerpen et al. [23]. Various approaches have also been developed. A widely-used approach for evaluating ETC of a heterogeneous system, is the so-called effective-medium approximation [24,25], in which a basic element such as particle is considered as being embedded in an equivalent homogeneous medium whose properties are to be determined. Although this approach produces good results when the local property fluctuations are small, it has been criticized for being based on unjustifiable assumptions [26,27]. Thus such theories cannot be expected to be useful in predicting conductivities of generalized

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packed beds having random packing structures [28]. Another important method (the periodic-structure model) involves idealizing packing structure with the assumption that the impenetrable inclusions (spheres or cylinders) are centered in the cells of a periodic lattice. However, it is not so useful for cases in which the conductivity ratio is very large because the effective property is highly sensitive to the packing structures [29]. Statistical approaches are also widely used to predict the ETC from statistical characterization of the packing structure of a heterogeneous medium. Brown [30] was the first to show the precise dependence of ETC on its microstructure by obtaining a perturbation expansion. Torquato [31] generalized this method for a medium of arbitrary dimensionality. However, perturbation expansion cannot be employed when the phase properties significantly differ in practical packed beds or porous media. Also the infinite set of statistical functions that characterize the microstructure is not clear [32]. To overcome the problems, a new approach has been developed and verified in our previous work for evaluating the ETC of a packed bed based on the Varonoi-Delaunay tessellation [2,3,33]. This approach involves the use of an exact microstructure of the random packing of equal spheres. Four modes of heat transfer, i.e. heat transfer by conduction between particles in a packed bed through three different paths, and radiation heat transfer between particles, have been considered. It has been used to study the effect of microstructure on the ETC of packed beds. However, some problems still need to be addressed. For example, some other heat transfer mechanisms, such as the heat conduction through the fluid voids, need to be considered according to Yagi and Kunii [34]. In addition, it is not clear how the packing structure affects the ETC under different heat transfer mechanisms and how different heat transfer paths contribute to the total heat flow under different packing structures (i.e. bed porosities). In the present work, we extend this approach to take into account the heat conduction through the fluid between voids. This structure-based network development includes both a complete description of the packed bed and a fundamental description of the pore-scale heat conduction mechanisms. Combined with the previously developed heat transfer models [2], the effect of packing structure (represented by bed porosity) on ETC in packings of mono-size spheres is then analyzed. 2. Computational methods 2.1. Delaunay - Voronoi tessellation It has been well established that a packing can be tessellated into the so-called Voronoi polyhedra, each with the following features [2,3]: • A boundary plane of a polyhedron is a perpendicular bisector of the line segment which joins the element point to a neighboring element point.

• A side of a polyhedron is the line segment which is equidistant from the element point and two neighboring element points. • A vertex of a polyhedron is the point which is equidistant from the element point and three neighboring element points. • When a new point is arbitrarily given in the space divided into the voronoi-polyhedra, the closest element point to this point is that of the voronoi-polyhedron which contains this point. • When the element points are located arbitrarily, the polyhedra obtained are convex and their shapes vary according to the arrangement of the element points. Fig. 1a schematically illustrates these features under twodimensional conditions. A three-dimensional packed bed can also be divided into an array of Voronoi-polyhedra of various shapes and sizes that fill the space and are non-overlapping, with each containing a particle. Fig. 1b shows a typical three-dimensional Voronoi polyhedron and its connection with others (each face representing one connection). If a packing is represented by the Voronoi tessellation, its heat transfer should be quantified from modeling the heat transfer within a Voronoi polyhedron and between neighboring Voronoi polyhedra. This approach has been attempted by Cheng et al. [2] in their study of the ETC due to conduction between particles, i.e. the particle-to-particle connection, of a packed bed. When the pore-to-pore connection is concerned, a similar approach, the so-called Delaunay or Voronoi tessellations, is used to map packed beds for pore space analysis [35–42]. The inverse array of the Voronoi tessellation is known as the Delaunay tessellation, which also gives a unique space partitioning for mono-size packing structure of spheres. The Delaunay tessellation consists of a system of space-filling cells, which are either triangles in 2-D with each triangle having its vertices at the disk centers, or tetrahedra in 3-D, with each tetrahedron having its four vertices at sphere centers. These cells are formed by joining the centers of two neighboring particles in the packing, where neighbors can be obtained by the Voronoi tessellation. Each Delaunay cell closely resembles the general description of a pore and the four vertices of a Delaunay tetrahedron are centers of the spheres in the packing. Fig. 2a shows four spheres arranged to form a tetrahedron. It is clear that a void space exists in the center of this group of four spheres. Access to this space is through any of the four constrictions that are created by each of the four sets of three spheres. So a pore space is defined as a central void with a number of distinct constrictions connecting the void to the other four neighboring pore spaces. The narrow constrictions in the pore space correspond exactly to the void areas on the faces of the cells. These faces control the access to the void volumes within cells, so they can be identified as pore throats in the network analog. It should be

k

A

B

F

G O

l

j

C

E D

i

(a)

(b)

Fig. 1. Schematic illustrations of: (a) a two-dimensional packing and its Voronoi elements, with dotted lines highlighting particle O and its surrounding particles A-F, and different connections between particles O and A, and between particles O and G; (b) a single Voronoi element i together with its neighbors j, k, l and so on.

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(a)

(b) Fig. 2. (a) 3D Delaunay Tetrahedron composed of four spheres and a Delaunay tetrahedron cell, and (b) typical tetrahedron-face geometries in a random packing of spheres.

noted that despite a group of four spheres satisfying the nearest-neighbor tessellation criterion, they do not necessarily touch each other. This condition introduces significant variability to the shape of a pore throat. Fig. 2b shows four typical tetrahedron-face geometries in a random packing of spheres. 2.2. Heat transfer mechanisms The Voronoi-Delaunay tessellation offers a solid basis for the present particle or pore scale evaluation of the transport properties of a packed bed. Thus, the heat transfer in a porous medium can be conducted through the following mechanisms or heat transfer modes, as summarized by Yagi and Kunii [34]: (1) Conduction through the stagnant fluid between two point- or non-contacted particles (2) Conduction through the stagnant fluid between two areacontacted particles (3) Conduction through the contact area between two areacontacted particles (4) Radiation between the surfaces of two particles (5) Conduction through the fluid in void space (6) Radiation between adjacent voids (7) Convective heat transfer between fluid and solid particles

To develop a comprehensive model for evaluating the ETC of a packed bed based on the Varonoi-Delaunay tessellation, we first considered four modes of heat transfer, i.e. heat transfer by conduction between particles in a packed bed through three different paths (Mechanism 1–3), and radiation heat transfer between particles (Mechanism 4). These mechanisms (mechanisms 1–4) have been considered in our previous work [2,3,33]. In the present work, we extend this approach to take into account the heat conduction through the fluid in void space, that is, Mechanism 5. It should be noted that for the packed bed with stagnant fluid and at low temperature, the contribution from heat transfer mechanisms 6 and 7 can be ignored. In the following Section, therefore, we will focus on the new and relevant treatments to mechanism 5. For convenience, Table 1 summarizes the calculation of the heat flow for the five heat transfer mechanisms which will be used in this work.

2.3. ETC calculation The Delaunay network model had been used to calculate the relative permeability at pore level in porous media [40,41], it is extended in this work to calculate the heat conduction through the Delaunay network. As shown in Fig. 2b, there is significant variability in the shapes of the cross-section of the pore throats. For simplicity, the heat flow paths

Table 1 Heat transfer models for different heat transfer mechanisms/modes. Heat transfer mechanisms/modes Mode (1): conduction through stagnant fluid between two point- or non-contacted particles

Mode (2): conduction through stagnant fluid between two area-contacted particles Mode (3): conduction through the contact area between two contacted particles Mode (4): radiation heat transfer

Mode (5): conduction through fluid voids

Heat transfer models Z rsf 2πrdr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ij ¼ ðT i −T j Þ 0 ½ R2 −r 2 −rðR þ hÞ=Rij ð1=ksi þ 1=ksj Þ þ 2½ðR þ hÞ− R2 −r 2 =k f where, h = (dij − 2R)/2, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rij ¼ 3V ij =ðπdij Þ, Rij r sf ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. 2 R2ij þ ðR þ hÞ Z rsf 2πrdr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ij ¼ ðT i −T j Þ r si; j ½ R2 −r 2 −rðR þ hÞ=R ð1=k þ 1=k Þ þ 2½ðR þ hÞ− R2 −r 2 =k ij si sj f 4r sij ðT i −T j Þ Q ij ¼ ð1=ksi þ 1=ksj Þ σðT 4i −T 4j Þ ð1−εr;i Þ Ai ð1− F ij Þ þ 2 εr;i Ai 2 ðT i −T j Þ Q ij ¼ πr 2ij;eff k f Lij Q ij ¼

Equation No. Eq. (T1) Eq. (T1a) Eq. (T1b) Eq. (T1c)

Eq. (T2) Eq. (T3) Eq. (T4)

Eq. (T5)

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rij,eff L

(a)

rij, eff

(b)

Fig. 3. A cylinder model for the heat flow path between two adjacent Delaunay cells: (a) defined cylinder (b) pore throat. Fig. 4. Effect of the number of pores on ETC (kf = 1 W/m K).

between adjacent cells are modeled by cylinders, which are defined in Fig. 3. Here, Lij is defined as the effective length of the heat flow path, which equals the distance between the centers of the adjoining Delaunay cells. rij, eff is called as the effective radius of the heat flow path, which is calculated by Eq. (1): r ij;eff ¼

rffiffiffiffiffiffiffiffiffi Scr;ij π

ð1Þ

cubes of different sizes. The calculated results are shown in Fig. 4. Obviously, ETC is not affected by the cube size, as long as the number of pore in the cube is N4000, i.e., the length of a cube is N8 times of particle diameter. This result is consistent with the result of heat conduction through solid, which was discussed previously [2]. Therefore, to obtain useful results, calculations are conducted using a cube involving around 4000 pores. 3. Simulation conditions

where Scr,ij is the area of the cross section of the pore throat. The heat flow in a cylinder of radius rij,eff and length Lij can be calculated by Eq. (2):  Q ij ¼ πr 2ij;eff k f

T i −T j Lij

 ð2Þ

where Ti and Tj are the temperatures of the pore centers of the adjacent Delaunay cells i and j; kf is the thermal conductivity of the fluid. For each heat flow path (pore throat) in the Delaunay Tessellation network, the effective radius and length are derived directly from Finney's original measurements of spheres center coordinates in the packing. None of them is adjusted to fit the experimental data. For convenience, in this work, the central portion of the Finney packing [43], which is used to construct the Delaunay tessellation network, is a cube of length H, obtained by removing the outside particles. The temperature difference between the bottom (Tb) and top plates (Tt) is fixed and the heat flow passes through the cube from the bottom plane to the top plane. The side faces are assumed to be adiabatic to produce a unidirectional heat flux. Under the condition of steady state heat transfer, thermal equilibrium requires that for each pore i n X

Q ij ¼ 0

ð3Þ

j¼1

where the sum runs over the throats j (j = 1 to n) connected to pore i. Applying this equation to all the pores in the cube will yield a set of linear equations where the temperatures of the particles, Ti, are unknown. The solution of these equations will give Ti and the heat flux, q. By definition, the effective thermal conductivity of the considered cube is q ke ¼ ðT b −T t Þ=H

ð4Þ

It is recognized that if the selected cube size is too small, it may not fully represent the packing structure, which will result in different effective thermal conductivities. To find the minimum cube length giving stable ke values, numerical calculations have been conducted using

For the new structure-based method to be implemented, a complete description of a packed bed is required (consisting of an array containing the x, y and z coordinates and the radius of every sphere in a packing). Finney's measurement of the spatial coordinates of the centers of about 8000 spheres was an arduous task and it is unlikely that a physical experiment of this scale will be repeated [40,41]. At present, it is extremely difficult, if not impossible, to obtain the packing structure experimentally. Alternative, the discrete element method (DEM) is used to form different packing structures [44]. In fact, DEM or CFD-DEM combining with the heat transfer models has been used to study the heat transfer and ETC for spheres [2,3,6,7,21,22,45,46] and non-spherical particles [4,5] in packed and fluidized beds. Seven packings generated by Yang et al. [47] by the DEM simulation were used in this study. The packing process is as follows: a packing simulation began with the random generation of mono-size spherical particles with no overlap in a rectangular box with width equal to 15 particle diameters. Then, the particles were allowed to settle under gravity, and during this densification process, they collided with neighboring particles and bounced back and forth. The dynamic packing process proceeded until all particles reached their stable positions with an essentially zero velocity as a result of the damping effect for energy dissipation. To ensure consistent results, the porosity (defined as the volume fraction of voids in the packing) at the initial state was constant, set to 0.94 for all packings considered. This relative high porosity is selected to make sure that at the initial state, there are no particle-particle direct contact and also no cohesive force involved (e.g. the surface separation distance is beyond the maximum cut off distance). Here, the only long-range, noncontact cohesive force considered in this paper is the van der Waals force calculated by [48,49].   64R3i R3j hs þ Ri þ R j Ha ^ ij F ij;v ¼ −  2  2 n 6 2 2 hs þ 2Ri hs þ 2R j hs þ hs þ 2Ri hs þ 2R j hs þ 4Ri R j ð5Þ where Ha is the Hamaker constant, and hs is the separation of surfaces along the line of the centers of particles i and j. While it is recognized

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that an overlap between two colliding particles is possible in DEM, a minimum separation of 1 nm is assumed in the present paper. Periodical boundary conditions were applied along two horizontal directions to avoid the lateral wall effect. The bottom wall is assumed to have the same physical properties as particles. Because “ragged surface” could be formed for the cohesive powders [50], and particles at the top of the bed are loosely packed [51], the porosity is calculated from the rectangular packing volume inside to eliminate the effect of the ragged top. In fact, studies show that the near-wall region has different ETC than the inside of the packed bed, and the ETC correlations are modified to consider this effect [23,52,53]. The simulations were performed for mono-size particles, with their diameters ranging from 1 to 1000 μm. The porosity ranged from 0.395 to 0.812. Since different sized particles have different inter-particle forces, the packing structures are generated by DEM simulation with different particle sizes and a series of porosities are obtained. However, the present work only concerns the effect of packing structure on heat transfer. Therefore, the DEM-generated packing structures have been scaled so that all particles have a consistent diameter of 1000 μm. The effective thermal conductivity ke was calculated using the particles in a cube of length H. The top and bottom sides are kept at high and low temperatures respectively, while the other sides are insulated. Table 2 lists the porosity, the length of cube and number of particles for each packing. As discussed previously [2], the ETC is not affected by the cube size, as long as the number of particles in the cube is N500. To study the effect of packing structures on ETC, four cases are calculated separately to consider different mechanisms by setting different conditions and parameters of packed bed for each packing structure. The four cases are detailed below: Case 1. Solid path conduction is considered, including the three heat conduction mechanisms by particles [2], and the conduction through the voids filled with fluid and radiation contribution can be ignored. Case 2. Fluid path conduction (calculated by the model developed in this work) is the only heat transfer mechanism, and solid path conduction and radiation contribution can be ignored. Case 3. Radiation between particles [3] is the only heat transfer mechanism. The packed bed is in vacuum (no fluid) and particles are assumed to be point-contact only. The thermal conductivity of the particles is infinitely large, the heat resistance of the particles is negligible. Case 4. Combined heat transfer. All the above five heat transfer mechanisms are considered. Solid path conduction and radiation are calculated by the model described in reference [2,3] and fluid path conduction are obtained in this work. Then the contributions from the two paths are added together. Parameters and conditions used in the calculations for each case are listed on Table 3. For all these cases, ETC is calculated according to the same equation given in Eq. (4).

Table 2 The porosity, the length of cube and number of particles for the seven packings. Packing No.

Porosity

Cube Length (Number in particle diameter)

Number of Particles

Finney 1 2 3 4 5 6 7

0.365 0.395 0.427 0.481 0.531 0.573 0.659 0.812

8 8 8 9 9 9 10 12

621 594 565 718 637 592 611 578

5

Table 3 Parameters and conditions used in calculations for each case. Parameter

Case 1

Case 2

Case 3

Case 4

Particle diameter, μm Particle thermal conductivity, W/m K Particle surface emissivity Fluid thermal conductivity, W/m K Top Temperature, K Bottom Temperature, K

1000 1.08564 0 0.02879 400 300

1000 0 0 0.02879 400 300

1000 Infinity 0.8 0 1100 1000

1000 1.08564 0.8 0.02879 1100 1000

4. Results and discussion 4.1. Heat conduction through voids in a packing of non-conducting spheres As mentioned before, the ETC of a packing of non-conducting spheres is a function of three factors: the conductivity of the fluid kf, porosity ε of the packing and a parameter β, which serves to distinguish between systems having the same porosity but different packing structure. For a given packing (i.e. Finney packing), the porosity ε and the parameter β are fixed constants. Therefore, ETC only varies with kf. Fig. 5a shows the relationship between the ETC and the conductivity of the fluid. Obviously, there is a linear relationship between ETC ke and kf and the ratio of ETC and the fluid conductivity equals to 0.172 for the Finney packing (ε = 0.366). Probability density distributions of heat flow of conduction through void fluid under three different fluid conductivities are shown in Fig. 5b. Note here the heat flows Qij between particles i and j is classified into different groups according to their magnitudes. The scaled probability density distribution f(Q/Qm) is used [3], which is defined by f ðQ=Q m Þ ¼ nk =½nðQ kþ1 −Q k Þ=Q m 

ð6Þ

where nk is the number of Q ij having a value of heat flow between Q k and Q k+1 with Q k+1 - Q k representing the bin width of the kth bin when evaluating the distribution and n is the total number of heat flows. Q m is the heat flow averaged over all the heat transfer paths. Note that this treatment can be applied to heat flows corresponding to different heat transfer mechanisms. It is evident that the probability density distributions under different fluid conductivities are the same for the given packing structure. This means that the distribution of the heat flow due to conduction through fluid-filled spaces is determined by the packing structure, while the amount of the heat flow is determined by the fluid conductivity. As mentioned before, it is very difficult to experimentally obtain the ETC of packing of non-conducting spheres, because all solid materials have a measurable thermal conductivity. To examine the validity of the new structured-based approach for conduction through a pore, it is essential to compare the predictions with the experimental results, which combines the heat conduction through the fluid-filled spaces with other heat transfer mechanisms. Crane and Vachon [54] presented a summary of the experimental data in the literature at low ratio of the conductivities of the solid to fluid, which included measured results by Shumann and Voss [55], Wilhelm et al. [56], Preston [57], and Krupiczka [58]. Some experimental data for higher ratio of solid to fluid conductivities are also available in literature [59,60]. There are also a number of existing analytical models, as summarized by Wang and Li [61]. Fig. 6 shows a comparison between the measured and calculated ETC. It is clear that the measured ETCs agree well with the calculated ETCs, which include the conduction through the fluid-filled spaces as well. ETCs without conduction through the voids are slightly less than the measured ETCs when the solid-fluid conductivity ratio is smaller than 5. This indicates that the effect of heat conduction through the voids on ETC increases with a decrease of the conductivity ratio and this heat transfer mechanism plays an important role when the ratio is b5.

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Fig. 6. Comparison between the measured and calculated ETCs. Measured ETC data are obtained from refs [55–58].

Fig. 5. (a) Relationship between ETC and the conductivity of fluid, and (b) effect of the fluid conductivity on the probability density distribution of heat conduction through fluid-filled voids.

Fig. 7 shows ETCs contributed by conduction through void and ETCs contributed by other heat transfer mechanisms. When the conductivity ratio is large enough (N5), the effect of heat conduction through the voids on ETC can be ignored. This justifies the assumptions that the heat conduction through void can be neglected when the conductivity ratio is larger than 10 [2]. However, when the conductivity ratio is small enough (b0.5), the heat conduction through voids becomes the dominant heat transfer mechanism (here, radiation does not exist for a liquid). Further experimental studies are needed to confirm the predictions when the conductivity ratio is less than unity.

[17,32]. The simplest model for calculation of ETC is to assume that the porous medium consists of a fluid layer and a solid layer, either in parallel or in series with respect to the temperature gradient. As pointed out by Deisser and Boegli [62], these two arrangements represent the maximum and the minimum values of the effective thermal conductivity of a packed bed. The two broken lines represent the maximum (layers in parallel) and the minimum (layers in series) values of ETC under the simulation conditions. Obviously, with the increase of porosity, the calculated ETC is close to the minimum value and ETC equals to the thermal conductivity of fluid phase if porosity equals to 1. Fig. 10 shows the probability density distribution for the three different paths for the DEM packing structures. For the solid-fluid-solid conduction between non-contact particles (Fig. 10a), the probability density distributions are similar for the three selected porosities. The probability value increases to a maximum and then decreases with increasing Q/Q m, giving a typical modal distribution. As porosity increases, the distribution shifts to the left a bit. For the solid-fluid-solid conduction between contacting particles (Fig. 10b.) The two probability density distributions for porosity 0.395 and 0.812 are similar, but there is big difference between these two packings and packing with porosity 0.531. For solid-solid conduction between contacting particles (Fig. 10c.), similar to the solid-fluid-solid conduction between contact particles, the probability distributions are different for the three selected porosities. For packings with porosity 0.395 and 0.812, the probability decreases with an increase of Q/Q m value, while for packing with porosity 0.531, the probability increases to the maximum value, then decreases with the increase of Q/Q m. From Fig. 10b and c, different bed porosities show quite different probability distributions, indicating

4.2. Effect of packing structure on ETC Fig. 8 shows the comparison of the probability density distributions of the heat transfer mechanisms for Finney's measured packing (ε = 0.365) [43] and DEM packing (ε = 0.395). It is obvious that both structures have a similar probability density distribution for each heat transfer mechanism. The small difference observed is understandable because of the difference in porosity between the two packings. The good agreement confirms that DEM generated packing structures are acceptable for simulation study of heat transfer in packed beds. 4.2.1. Solid path conduction (Case 1) The effect of packing porosity on the ETC by solid path conduction is shown in Fig. 9. It is clear that the ETC by solid path conduction decreases with an increase of the porosity, since the heat conduction area of solid path decreases with the decrease of the volume ratio of solid and fluid. The result is consistent with the previous experiments or empirical relations between the ETC and the packing porosity

Fig. 7. Comparison of contributions by conduction through the voids and by other heat transfer mechanisms.

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Fig. 8. Probability density distributions of heat flow corresponding to (a) the heat conduction mechanism by the solid path under Case 1 conditions: ε = 0.365 for Finney packing and ε = 0.395 for DEM; (b) heat conduction through fluid voids under Case 2 conditions; and (c) the heat radiation between particles under Case 3 condition.

Fig. 10. Probability density distribution of (a) solid-fluid-solid conduction between noncontacting particles for different packings, and (b) solid-fluid-solid conduction flows between non-contacting particles, and (c) solid-solid conduction between contacting particles for different packings.

Fig. 9. Effect of packing porosity on ETC by solid path conduction, maximum and minimum theoretical values are represented by the broken lines.

different particle contact conditions and void structures. For the case of porosity of 0.531, there is a peak at Q/Q m around 1.0, indicating the solid-fluid-solid contact lens and solid-solid contacts are more uniform than the other two bed porosities. Fig. 11 shows the effect of porosity on the mean heat flow by the three different heat conduction paths. It is clear that the mean heat flow decreases with an increase of porosity for three different heat conduction paths. This is because with an increase in porosity, the average number of contact particles decreases [63]. This reduces the heat transfer paths by solid phase and causes the decrease of the heat flow. For the solid-fluid-solid conduction between contact particles and solid-solid conduction between contacting particles, the decrease of contact

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Fig. 11. Effect of porosity on the mean heat flow by solid-fluid-solid conduction between non-contacting particles for different packings, solid-fluid-solid conduction flows between non-contacting particles, and solid-solid conduction between contacting particles.

number of particles will greatly reduce the heat flow because of the decrease of the connection path between particles. 4.2.2. Fluid path conduction (Case 2) The effect of packing porosity on the ETC by fluid path conduction is shown on Fig. 12. ETC by fluid path conduction increases with an increase in porosity, because the heat conduction paths increase with an increase of the fluid volume. The experimental determination of the effect of the porosity on the heat conduction by void fluid is very difficult, because all solid materials have some measurable thermal conductivity [17]. Considering the similarity of conduction and diffusion, Zehner and Schlunder [16] postulated an empirical formula based on Currie's experimental data [8] for gas diffusion. pffiffiffiffiffiffiffiffiffiffi ke ¼ 1− 1−ε kf

ð7Þ

where ε is porosity of packing. Zehner's empirical formula is shown in Fig. 12 as a dash line. It is clear that the calculated values are comparable with Zehner's results, even though the calculated values are a little lower than the Zehner's results. Fig. 13 shows the probability density distribution of heat conduction through the fluid void for three DEM packing structures and the effect of porosity on the mean heat flow by heat conduction through the fluid voids. The probability distributions are similar for ε = 0.531 and ε = 0.821, but there is some difference between ε = 0.395 and ε = 0.531.

Fig. 12. Effect of packing porosity on the effective thermal conductivity (ETC) by fluid path conduction (Solid line: calculated; dash line: Zehner's formula).

Fig. 13. (a) Probability density distribution of heat flows for different packings: heat conduction through fluid voids, and (b) effect of porosity on the mean heat flow by void fluid conduction.

The mean heat flow by void conduction increases with the increase of porosity, since the heat conduction path increases with porosity. 4.2.3. Radiation between particles (Case 3) The effect of porosity on the effective thermal conductivity by heat radiation between particles is shown in Fig. 14. The effective thermal conductivity by radiation decreases with an increase in the porosity, because the heat radiation between particles decreases with a decrease of the solid volume. Most experimental data or empirical formulas are obtained for a particular packing structure (ε = 0.4), which has been compared with the calculated values previously [33]. Fig. 15 shows the probability density distribution of radiation between particles for the three DEM packing structures and the effect of porosity on heat flow by radiation between the particles. The three probability distributions are similar, and with an increase of Q/Q m, the

Fig. 14. Effect of porosity on effective thermal conductivity by radiation.

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Fig. 17. Effect of packing porosity on the relative contribution of the heat transfer mechanisms to the overall heat transfer.

Obviously, the radiation heat transfer is the dominant heat transfer mechanism for all porosities under Case 4 conditions. With the increase of porosity the relative percentages of radiation and solid-fluid-solid conduction between non-contacting particles increases, while the relative percentages of solid-fluid-solid conduction and solid-solid conduction between contacting particles decrease. The reason for it is that solid-fluid-solid conduction and solid-solid conduction between contacting particles are more sensitive to the particle contact number, which decreases with an increase of porosity. From above analyses, change of the heat transfer conditions will change the dependence of ETC on porosity and this effect should also be observed in the assessment of the relative contribution of the heat transfer mechanisms. As shown from the above examples, the proposed approach can be used to quantity the case-dependent heat transfer in a packed bed. Fig. 15. (a) Probability density distribution of heat flows for different packings: heat radiation between particles, and (b) effect of porosity on heat flow by solid-solid radiation.

probability increases to a maximum value and then decreases. The mean heat flow by radiation decreases with an increase in porosity, since the solid volume decreases. 4.2.4. Combination of five heat transfer mechanisms (Case 4) Fig. 16 shows the effect of porosity on the effective thermal conductivity for Case 4 conditions. In this case, five heat transfer mechanisms listed above, are all included. The effective thermal conductivity decreases with the increase of porosity, because conduction and radiation heat transfer decreases with the decreasing volume of particles. This relationship is confirmed by the recent experimental results measured by Akiyoshi et al. [64]. Also it is clear that under case 4 conditions, the contribution of conduction by fluid voids is very small and can be ignored. Fig. 17 shows the effect of packing porosity on the relative contribution of the heat transfer mechanisms to the overall heat transfer.

Fig. 16. Effect of packing porosity on effective thermal conductivity (ETC).

5. Conclusions The structure-based approach has been extended to calculate heat conduction through the fluid voids in a packed bed, which is modeled by a representative network consisting of an array of Delaunay tessellation cells. Its validity is confirmed by good agreement between the calculated and measured ETC. For a packing of non-conducting spheres, the ETC has a linear relationship with the conductivity of the fluid. However, the heat flow distribution of conduction through the voids is not affected by the conductivity of the fluid and is only determined by the packing structure. Heat conduction through voids increases with a decrease of the conductivity ratio. When the conductivity ratio is small enough, the heat conduction through void plays an important role in heat transfer in packed beds. Then a microscopic structure-based approach for predicting the effective thermal conductivity of packed beds has been applied to packed beds of different structures generated by DEM. A comparison of the calculated results between Finney-measured and DEM packing confirmed that the DEM packing is applicable for simulation of heat transfer in packed beds. Four cases with five heat transfer mechanisms have been studied by setting different conditions and parameters. Generally, the probability density distribution of heat flow is dependent on the packing structure, although the difference is not very significant for some types of heat transfer mechanisms. With an increase in porosity, heat conduction and radiation by the solid path decreases, while heat conduction through void fluid increases. The results confirmed that the packing structure plays an important role in heat transfer in packed beds. Nomenclature Ai the radiation exchange area, [m2] dij parameter in Eq. (T1) to be determined from a known packing structure, [m] Fij a view factor in Eq. (T4) h defined by Eq. (T1a), [m]

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H Ha hs ke kf ks Lij q Q ij Q k, Q k+1 rij,eff rsf rsij R Rij Scr,ij Tb Ti, Tj Tt Vij

Greek ε εr,i σ

G. Cheng et al. / Powder Technology xxx (2019) xxx

edge length of a cube of packed particles, [m] Hamaker constant the separation of surfaces along the line of the centers of particles i and j, [m] effective thermal conductivity of a packed bed, [W·m−1 K−1] thermal conductivity of fluid, [W·m−1 K−1] thermal conductivity of particle, [W·m−1 K−1] the effective length of the heat flow path, [m] heat flux imposed at the bottom plane of a packed cube, [W·m−2] heat flow between particles i and j, [W] defined by Eq. (6), [W] the effective radius of the heat flow path, [m] defined by Eq. (T1c), [m] radius of the contact circle between particles i and j, [m] radius of a particle, [m] defined by Eq. (T1b), [m] the area of the cross section of the pore throat, [m2] average temperature of particles at the bottom plane of a cube, [K] temperature of particle i and j, [K] average temperature of particles at the top plane of a cube, [K] parameter in Eq. (T1) to be determined from a known packing structure, [m3]

porosity of packed bed emissivity of particles i Stefan-Boltzmann constant 5.67 × 108, [W/(m2K4)]

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