Article pubs.acs.org/IECR
Particle Scale Evaluation of the Effective Thermal Conductivity from the Structure of a Packed Bed: Radiation Heat Transfer G. J. Cheng and A. B. Yu* Laboratory for Simulation and Modeling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia ABSTRACT: Radiation represents an important contribution to the heat transfer through a packed bed. The packing structure is important in the determination of the radiation heat transfer, but this aspect is not explicit in most of the mathematical models proposed in the literature. Here, a new numerical approach is proposed to calculate the radiation heat transfer in such a bed from its structure using the Voronoi network model. On this basis, the effective thermal conductivity (ETC) is evaluated by taking into account the effects of the radiation heat transfer and conduction through neighbor particles and stagnant fluid. The validity of this approach is verified by comparing the calculated and measured ETCs under different conditions. This approach is used to investigate the effects of variables such as particle thermal conductivity, emissivity and size, and bed temperature as well. The relative contributions of different heat transfer mechanisms are analyzed. It is shown that there is a similarity in the probability density distributions of dimensionless heat flows among particles under different bed temperatures, indicating that packing structure is a dominant factor in controlling the distribution whereas the mean heat flow between particles mainly depends on bed temperature and related material properties. simulations.8,11 Although the results from the continuous model after the correction were in good agreement with those simulated,8,11 Brewster10 pointed out that the agreement holds only when particle size is sufficiently large in comparison with the wavelength of radiation and interparticle clearance. Jones et al.5 modeled the radiation emitted from a bed of spherical particles having a nonuniform temperature distribution using a one-dimensional radiative transfer equation based on the continuum model and found that measured and computed intensities of the radiation agree well only in near normal directions. They concluded that the nonisothermal packed bed consisting of large spheres is not appropriately modeled by the continuous radiative transfer equation, and a noncontinuous radiation model accounting for structural details may be necessary. Theoretical attempts have also been made to obtain the radiative properties such as absorption coefficient a and scattering coefficient b by applying the so-called Mie solution of single particle (or independent) scattering.12 However, when the particles are tightly packed, the Mie solution is no longer valid because of the dependent scattering.13 To solve this problem, Chan and Tien14 used a ray tracing method to calculate the scattering and absorption coefficients for the socalled cubic packing. However, the direct radiation channeling in the cubic packing results in smaller values for the absorption coefficient but larger values for the scattering coefficient than the corresponding experimental values. Yang et al.15 used a random packing of spheres and ray optics in their Monte Carlo simulation. They compared the results for radiative trans-
1. INTRODUCTION Radiation represents an important contribution to the heat transfer through a porous medium or packed bed. Its accurate prediction is particularly needed in the design and control of fluid bed reactors such as high-performance cryogenic insulation,1 ceramic filters for use in diesel engine exhausts,2 and designing efficient solar radiation absorbers in solar thermal and thermo-chemical reactors.2−4 In the analysis of radiation heat transfer through packed beds, two types of models are commonly used in the literature. The first type is a continuous model in which a packed bed is considered to be a pseudohomogeneous medium, where the radiation in the packed bed is described by a set of differential or integrodifferential equations closed by boundary conditions applicable to an absorbing, emitting, and scattering medium.5,6 In particular, Chen and Churchill6 solved the integro-differential equations using a two-flux model that assumes the radiation intensity can be split into forward and backward scattering components as done by Singh and Kaviany.7 Such an approach can generate a relationship between the exchange factor F and the two parameters (i.e., absorption coefficient a and scattering coefficient b, Table 1). Singh and Kaviany8 extended the independent scattering continuum model7,9 to include the dependent scattering. This was done by introducing a correction factor to the original model, and the factor was treated as a function of porosity. Brewster10 showed that this correction factor is inversely proportional to porosity, and it alters the extinction coefficient of the independent scattering model; note that the extinction coefficient is equal to the sum of absorption and scattering coefficients.5 Brewster10 and Coquard and Baillis11 stressed that the effect of nonvanishing particle volume was felt primarily in the extinction coefficient, while other radiative properties such as albedo (or reflection coefficient) and phase function were relatively unaffected by the volume effect as observed in the direct Monte Carlo © 2013 American Chemical Society
Received: Revised: Accepted: Published: 12202
March 19, 2012 July 16, 2013 August 6, 2013 August 6, 2013 dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research
Article
Table 1. Comparison between the Predicted and Measured Radiant Exchange Factors for Ordered and Disordered Packings exchange factor F (result) orthorhombic (ε = 0.400)
a
cubic (ε = 0.476)
random (ε = 0.365)
references
exchange factor F (model)
εr = 0.35
εr = 0.85
εr = 0.35
εr = 0.85
εr = 0.35
εr = 0.85
Kasparek and Vortmeyer18 Chen and Churchill6 Wakao and Kato16 Argo and Smith17 Vortmeyer19 Kasparek and Vortmeyer18 Kaviany and Singh52 present work
experimental 2/d(a + 2b) 2/(2/εr − 0.264) 1/(2/εr − 1) (2B + εr(1 − B))/(2(1 − B) − εr(1 − B)) (εr + B)/(1 − B) Monte Carlo (diffusion) particle scale approach
0.54 1.11 0.37 0.21 0.33 0.48 0.45 0.36
1.02 1.06 0.96 0.74 0.85 0.97 0.94 0.95
0.6 N/A 0.37 0.21 0.35 0.50 N/A 0.37
1.06 N/A 0.96 0.74 0.91 1.03 N/A 1.0
N/Aa N/A 0.37 0.21 N/A N/A N/A 0.32
N/A N/A 0.96 0.74 N/A N/A N/A 0.83
Not available.
Knowledge of the packing structure of a randomly packed bed is essential for any rigorous analysis of the heat transfer within the bed, as recently demonstrated in the analysis of thermal conductive heat transfer in packed or fluidized beds.20−30 These studies, however, are limited to conduction heat transfer between particles. Radiative heat transfer, if considered, was modeled under simplified conditions without going into structural details (for example, ref.27). When a packet of radiant energy is emitted from a sphere, its transfer will be determined, to a large extent, by the immediate surroundings of the sphere; a part of the packet will be absorbed and another part will be reflected by the surrounding spheres. The transmittance of a laser beam through a bimodal randomly packed bed of spheres has been evaluated for the first time using a ray tracing procedure together with a Monte Carlo sampling technique, which determines the random scattering directions of the incident rays on spheres.31 Although several research efforts have been made to obtain the effective thermal conductivity (ETC) of a packed bed,2,3,32 the key feature of all such studies is the evaluation of radiative heat transfer by using either the continuous model3 or a one-dimensional model.32 However, in recent years, how to take into account the threedimensional structure of a packed bed in the evaluation of ETC due to radiation has become an issue of interest. Generally, there are two approaches in this connection. One is referred to here as the microscopic approach, which is based on the basic principles of radiation and considers the geometrical details (particles and their surfaces; pores and their shape) and material properties8,11,33−39 to be beyond the continuous models described above. The other is the macroscopic approach, which is based on simplified assumptions either in radiation or in packing geometry, or both. The microscopic approach is scientifically more attractive. However, it is extremely difficult to implement, and computationally, it is very demanding. This can be seen from the recent studies by various investigators.36−39 Consequently, while useful for better fundamental understanding, to date the microscopic approach has limited success and application in evaluating the ETC of packed beds. Therefore, in engineering application, the macroscopic approach is more attractive, as reviewed by van Antwerpen et al.30 In fact, most of the cell models described above can be categorized into this approach. However, as discussed, they have various problems resulting from unreasonable assumptions. For a given geometric element, the radiation heat transfer can be described in detail. However, it is extremely difficult, probably not possible to our knowledge, to find such a simple geometric element to represent the
mittance obtained by the simulations with the experimental results of Chen and Churchill6 and found that the transmittance varies inversely with the bed thickness as in experiments but is underpredicted. More recently, further efforts have been made to apply the Mie theory to determine the radiative properties, absorption, scattering and extinction coefficients, and scattering phase function in a packed bed composed of mixtures of polydispersed particles of SiO2, ZnO, and C by Jäger et al.3 These investigators then used these Mie-based values in a raytracing Monte Carlo simulation to obtain the radiation transmittance through the packing considered and found that the transmittance is underpredicted for samples containing SiO2 and ZnO (which are highly radiation scattering materials) and overpredicted for samples containing C (which is a highly absorbing material) in comparison with experimental results. Thus, they established the phase functions for SiO2 and ZnO materials to arbitrarily increase the forward scattering and match the overall transmittance with the experimental results. Then, in the case of samples containing C, they reduced the extinction coefficient from the Mie-based value, again to match the overall transmittance with the experimental results. However, the radiative properties thus obtained have not been validated against other methods. The second type of model used in the analysis of radiation heat transfer is called the cell-model method, in which radiation is treated as a local effect that takes place between boundary surfaces of a unit cell and adjacent particle surfaces. In such a model, radiation heat transfer is usually evaluated by considering the radiation exchange between surfaces that is considered to represent the radiation transfer in the unit cell, and then the exchange factor F is determined. The model of Wakao and Kato16 is based on the consideration of the radiant transfer between touching spheres. The model of Argo and Smith17 uses an assumed geometrical configuration, which consists of two parallel gray planes separated by a particle diameter. Kasparek and Vortmeyer18 modified the plane-layer model of Argo and Smith to incorporate transmission of energy through void spaces in a layer, and they confirmed the validity of this modification by experimental measurement for two ordered packings: cubic and orthorhombic. However, in such a cell model, only radiation heat transfer in the simplified unit cell is considered, and the long-range effects in the whole packed bed are usually neglected.19 Therefore, it is still a challenge to establish a general relationship between the radiation exchange factor and the packing structure by the unit cell model. 12203
dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research
Article
complicated structure of a packed bed. Hence, the previous macroscopic approaches largely fail as a general model. To overcome this problem, van Antwerpen et al.40 recentyl proposed a so-called multisphere unit cell model, where some local mean structural properties of a packed bed, such as coordination number and contact angle in addition to porosity, are used in the calculation of ETC. In this work, we propose a new macroscopic approach to evaluate the steady-state radiation heat transfer in a packed bed of uniform spheres. Different from the previous macroscopic studies, we directly use the packing structure, facilitated by the Voronoi tessellation,20 in the calculation of ETC. This gives the so-called particle or pore scale approach for the calculation of ETC. As mentioned above, it has been successfully used by various investigators in the study of conduction heat transfer in packed and fluidized beds.20−30 We here extend this approach to radiation heat transfer, and demonstrate that as long as the network of particles or pores in a packing is considered properly the ETC due to radiation, which may or may not be coupled with conduction, can be determined reliably even though there are some simplifications in microscopic details.
2. THEORETICAL TREATMENTS It has been well established that a packing can be tessellated into the so-called Voronoi polyhedra, each with the following features:41,42 (1) A boundary plane of a polyhedron is a perpendicular bisector of the line segment that joins the element point to a neighboring element point. (2) A side of a polyhedron is the line segment that is equidistant from the element point and two neighboring element points. (3) A vertex of a polyhedron is the point that is equidistant from the element point and three neighboring element points. (4) 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 that contains this point. (5) When the element points are located arbitrarily, the polyhedra obtained are convex and their shapes vary according to the arrangement of the element points. When these features are applied to particle packing, the element points should be the centers of particles. Figure 1a schematically illustrates these features under two-dimensional 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 nonoverlapping, with each containing a particle. Figure 1b shows a typical three-dimensional Voronoi polyhedron and its connection with others (each face representing one connection). Obviously, 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.20 in their study of the effective thermal conductivity due to conduction between particles (i.e., the particle-to-particle connection) of a packed bed. For the pore-to-pore connection, a similar approach has also been proposed in the study of the permeability of a packed bed, where a pore is defined as a Delaunay unit.43−45 This so-called Voronoi−Delaunay tessellation offers a solid basis for the present particle or pore scale evaluation of the
Figure 1. Schematic illustrations: (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; (c) the connection between two neighboring Voronoi polyhedra as a double pyramid model; (d) the simplified connection as a double taper cone model.
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: (1) conduction through the stagnant fluid between two point-contacted or noncontacted 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. These mechanisms may be considered to correspond to those identified by Yagi and Kunii.46 However, they are more explicit, because the structure of a packing is uniquely defined before the mechanisms are considered. To develop a comprehensive model for evaluating the ETC of a packed bed based on the Varonoi−Delaunay tessellation, we first considered three modes of heat transfer (i.e., heat transfer by conduction between particles in a packed bed through three different paths). That work has been reported in the literature.20 In the present work, we extend this approach to take into account the radiation heat transfer between particles (i.e., mechanism 4). In the following discussion, therefore, we will focus on the new and relevant treatments. 2.1. Radiation Heat Transfer between Neighboring Voronoi Polyhedra. As schematically shown in Figure 1, a 12204
dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research
Article
the cube to the top plane. Under steady-state heat transfer, thermal equilibrium requires that for each polyhedron i
packed bed can be represented by a Voronoi element network to establish its particle-to-particle connections (Figure 1a). Consequently, its heat transfer can be examined by considering the heat transfer between particles or Voronoi elements. For example, particle O may have heat transfer with its surrounding neighbors (i.e., particles A−F) determined from the Voronoi tessellation. The possible heat transfer between particles O and G is ignored in the present study. In fact, this heat transfer happens between the two pores respectively constructed by particles OBC and GBC, corresponding to mechanisms 5 and 6, which will be studied in the next step of our study. Threedimensionally, a Voronoi element may have a relatively complicated geometry (Figures 1b and c). For simplicity, as done in the case for heat transfer by conduction,20 in the present work, a double taper cone model is applied to calculate the radiation heat transfer between particles i and j (Figure 1d). The following assumptions are further made for the calculation of the radiation heat transfer between the two neighboring Voronoi elements: (1) The sphere diameter is much larger than the wavelength of radiation. (2) The sphere surface is gray emitting. (3) All spheres are opaque solids. (4) ΔT/T is much less than unity across a sphere layer. (5) The imaginary surface R (ACBB′C′A′) is perfectly insulated and diffusely reflective. Under the above assumptions, the radiant exchange Qij, rad between particles i and j can be solved by the network method:47 Q ij ,rad =
ni
∑ Q ij = 0 where Qij = Qij,con + Qij,rad. Here the heat transfer between particles i and j by conduction Qij,con is calculated by the numerical model A as reported previously.20 Heat flow rates between particles i and j, Qij, are limited to the particles within the packed cube considered. That is, adiabatic conditions are assumed for the lateral, side walls. While the wall or boundary effect is not a concern in the present work, it is recognized that if a cube is too small, it may not fully represent the packing structure resulting in varying effective thermal conductivities. Numerical calculations had been conducted using cubes of different sizes. It was found that ke is not affected by the cube size, as long as the number of particles in the cube is more than 500.20 For the random packing of spheres, this gives the length of a cube as approximately 8 particle diameters, involving 621 spheres. That is, eq 3 will produce 621 linear equations whose solution can give 621 Ti, the temperatures of 621 particles. To be consistent with their study, H is set to 8 particle diameters for all the computations in this work. Note that in some case studies to be discussed below, radiation and conduction heat transfers will be considered simultaneously. Applying eq 3 to all the particles in the cube will yield a set of linear equations where the temperatures of particles, Ti, are unknown. The solution of these equations will give Ti. As shown by Cheng et al.,20 particle temperatures vary linearly with the bed height, with some scatters corresponding to the nonuniform bed structure. Such linear relationships can also be observed in the present work concerning both conduction and radiation heat transfers. Therefore, the ETC of the cube considered is obtained using the definition
σ(Ti 4 − Tj 4) 1 − εr, i εr, iA i
+
1 A i Fij + [(1 / A i Fi R ) + (1 / A jFjR )]−1
+
1 − εr,j εr, jA j
(1)
where Ai and Aj are the areas of surfaces AA′ and BB′ on the spheres. εr,i and εr,j are the emissivities of particles i and j. Fij, FiR, and FjR are the view factors between the surfaces AA′ and BB′, AA′ and ACBB′C′A′, and BB′ and ACBB′C′A′, respectively. For uniform spheres (i.e., spheres of the same geometrical and physical properties) we have Ai = Aj, εr,i = εr,j, Fij + FiR = 1, Fji + FjR = 1, and Fij = Fji. Equation 1 can hence be simplified to Q ij ,rad =
ke =
(1 − εr, i) εr, iA i
+
A i (1 − Fij) 2
q Tb − Tt H
(4)
where Tb and Tt are the averaged temperatures of particles at the bottom and top planes. If Qij,con = 0, which can be readily set in the present numerical experiments, then Qij = Qij,rad, q = qrad, and
σ(Ti 4 − Tj 4) 2
(3)
j=1
(2)
kr =
The radiation exchange area Ai depends on the geometric configuration for spheres i and j, which can be directly derived from the packing structure; the view factor Fij between surfaces AA′ and BB′ can be obtained by extending the numerical method provided by Jones.48 As illustrated in Figure 1, Ai and Fij are not constant, varying with the packing structures in a complicated manner. 2.2. Establishment of Thermal Equilibrium Equations. The packing structure measured by Finney49 is employed here, in connection with our previous study,20 and the Voronoi polyhedra are determined from the coordinates of 4200 spherical particles by means of the method developed by Joe.50 For convenience, here ETC ke is actually calculated using the particles in a cube of length H, obtained by removing the outside particles. A uniform heat flux q is then imposed on all particles intersecting with the bottom plane and passes through
q
rad Tb − Tt H
= ke (5)
Chen and Churchill6 showed that the radiant heat transfer in a packed bed can be treated as a diffusion process and the radiant heat flux qrad can be represented by eq 6:
qrad = k r
dT dH
(6)
where the radiant conductivity kr is defined as k r = 4FdσTm 3
(7)
where F is called the radiation exchange factor and Tm = ∑ni=1Ti/n is the mean temperature of particles in the considered packed bed. Rearranging eq 7 gives the following equation to calculate this factor: 12205
dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research F=
4dσTm3 kr
Article
3.1.2. Radiation in Disordered Packing. Figure 2 shows that the effect of the emissivity of particle surface on the calculated F
(8)
3. RESULTS AND DISCUSSION 3.1. Model Validation. For a packed bed in a vacuum, if conduction heat transfer through a contact area is ignored, then the radiation heat transfer is the only effective heat transfer mechanism. Various models are available for predicting radiation exchange factor F. In this subsection, the validity of the present structure-based approach will be tested by comparing the calculated results with the available results from experiments and other models. 3.1.1. Radiation in Ordered Packing. Kasparek and Vortmeyer18 reported a set of measured radative heat transfer exchange factors in a particle bed with negligible heat conduction. In their experiment, measurements of radiation heat transfer through a number of planar series of welded steel spheres were made. Conduction and convection were eliminated by placing the layers a small distance apart and performing the experiment in a vacuum. The high thermal conductivity of the material ensured that its heat resistance was negligible and the spheres were isothermal. Measurements were made using polished steel spheres (εr = 0.35) as well as chromium oxide-coated spheres (εr = 0.85). Two ordered packings, namely orthorhombic and cubic of porosity ε = 0.4 and 0.5, respectively, were used. Note that their porosities were slightly larger than the normal regular packing, because the layers were spaced a small distance apart to avoid conduction heat transfer through solid−solid contacts. Table 1 shows the comparison between the calculated and the measured results by various models. The results confirm the validity of the present structure-based approach. For the two regular packings, the predictions by the new method reasonably match the experimental results. This is particularly so for the beds of high emissivity (εr = 0.85). For the beds of low emissivity (εr = 0.35), however, the difference between the calculated and measured exchange factors is relatively large. There are various factors responsible for the difference. For example, the bed structure is not exactly the same as described above. Another one is the uncertainty in the determination of emissivity by the experimental measurements.18 On the other hand, the change of the radiation exchange factor F with porosity qualitatively matches the measured results, although the difference is small for the two packings. From orthorhombic to cubic packing, F increases from 0.36 to 0.37 for εr = 0.35 and from 0.95 to 1.0 for εr = 0.85. The trend is also comparable to those determined by different investigators using different approaches. But the previous models are all empirical, with one or two fitting parameters as seen from Table 1. Moreover, comparison with the results of other models shows that the present approach and Monte Carlo method give similar results. Kasparek and Vortmeyer’s model also matches the experimental results achieved by use of an empirical parameter B = f(ε,εr). However, Kasuparek and Vortmeyer18 only gives a figure to determine B for cubic and orthorhombic packing, and the general function B = f(ε,εr) is not available in the literature. It is not clear how their model can be used for disordered, random packing. This consideration also applies to some models listed in Table 1. In fact, the present calculated F for the random packing is smaller than those of the two regular packings. This indicates that packing structure plays a role in radiation heat transfer.
Figure 2. Effect of emissivity of particle surface on radiation exchange factor of Finney’s random close packing (ε = 0.36): ○, present work; ■, Wakao and Kato;16 ▲, Argo and Smith.17
of the random packing (ε = 0.36) by the new model and the unit cell models of Wakao and Kato16 and Argo and Smith.17 It is obvious that the radiation exchange F increases with increasing emissivity εr, but the present calculated F is larger than the value calculated according to the Argo and Smith model and is smaller than that of the Waoka and Kato model. It should be pointed out that neither of the models has been confirmed to be accurate. Therefore, Figure 2 simply demonstrates that the present approach can equally describe the effect. That is, like these two models, Figure 2 can also be used to estimate the radiation exchange factor F for a randomly packed bed of uniform spheres. The combined conduction and radiation heat transfer through a packed bed of spheres is more often studied because of its practical importance. Yagi and Kunii46 measured the ETCs of various packings consisting of iron spheres when the temperature was up to 1273 K. Their measurements were performed in a cylindrical vessel with a heating rod in the axial position. Wakao and Kato16 measured the ETCs of packed beds of uniform glass beads when the temperature was up to 730 K. The measured and calculated ke/kf values are plotted against the mean temperature of a packed bed. As shown in Figures 3 and 4, the calculated results are in good agreement with the experimental results, further confirming the validity of the proposed structure-based approach. 3.2. Model Application. As an example to demonstrate the usefulness of this approach, in this subsection, the effects of variables such as particle thermal conductivity, emissivity and size, and the bed temperature, are investigated. These variables are basic and are directly related to material properties and operational conditions. To be useful, all the results below are obtained using the structural data of Finney.49 The data correspond to the so-called random close packing and are dimensionless, scaled by particle diameter. 3.2.1. Effect of Material Properties. The quasi-homogeneous theories predict that ETC ke is the sum of the effective conductivity ke,c without radiation contributions and the radiative conductivity kr without conduction contributions:19
ke = ke,c + k r 12206
(9)
dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research
Article
Figure 3. Comparison between the calculated and measured ETC (using a packed bed of iron spheres of d = 11 mm and porosity ε = 0.4): ◆, measurements of Yagi and Kunii;45 lines, present calculations.
Figure 5. Comparison of values ke obtained by the present work and eq 9 based on quasi-homogeneous theory for d = 11 mm, εr = 0.8, Tm = 1173 K, and kf = 0.0635 W/(m K): ◆, present work; Δ, eq 9.
Figure 4. Comparison between the calculated and measured ETC using packings of uniform glass beads of size d: line 1, d = 1.52 mm; line 2, d = 3.14 mm; line 3, d = 4.72 mm; points, measurements of Wakao and Kato;16 lines, present calculations.
Figure 6. Effect of solid thermal conductivity ks on the radiative conductivity kr,c.
radiation with a finite-difference solution for conduction. The quantitative agreement further verifies the present structurebased approach. The present work considers four mechanisms or modes of heat transfer: heat conduction between two noncontacting particles through the stagnant fluid Qnsfs, heat conduction between two contacting particles through the stagnant fluid Qcsfs, heat conduction between two contacting particles through the contact area Qcss, and heat radiation between two particle surfaces. Figure 7 shows the relative contributions of these mechanisms to the overall heat transfer as a function of ks/kf. The relative contribution of heat conduction increases with an increase of the solid conductivity ks (note that kf here is fixed), while the relative contribution of radiation decreases. The reason is that the absolute contribution of conduction increases more than the absolute contribution of radiation as the solid conductivity ks increases. Also, it is clear that under the given conditions (Tm = 1173 K), radiation heat transfer is the dominant heat transfer mechanism (>75%). The effect of particle emissivity on ETC is shown in Figure 8. Obviously, ETC increases with particle emissivity, because the radiation heat transfer is enhanced at high emissivities. Figure 9 shows the effect of particle emissivity on the relative contributions of the heat transfer mechanisms to the overall heat transfer. It suggests that under the conditions considered,
where the radiative conductivity kr can be calculated according to eq 7. In this study, the conduction and radiation heat transfer flows are combined to give one thermal equilibrium equation for each Voronoi element, so that the possible interaction of both heat transfer modes can be examined. The calculated results by the present model and eq 9 are shown in Figure 5. Obviously, the two sets of the calculated results match each other well when the ratio of solid and fluid conductivity ks/kf is less than 100. In this case, the interaction of the conduction and radiation heat transfers can be ignored. However, when ks/kf is greater than 100, the difference between the two methods becomes significant. In fact, the difference increases with the increase of ks/kf. This indicates that both the conduction and radiation fluxes are affected by ks/kf, and quantitatively in a different way. Figure 6 shows the effect of ks/kf on the radiative conductivity kr,c quantified according to eq 10: k r,c = ke − ke,c
(10)
The results are consistent with Figure 5. When ks/kf is less than 100, the effect of solid conductivity ks on kr,c can be ignored. When ks/kf is more than 100, kr,c increases with the increase of ks because of the transportation of the energy through particles by conduction. Singh and Kaviany8,51 obtained similar results by using a method that combines the Monte Carlo method for 12207
dx.doi.org/10.1021/ie3033137 | Ind. Eng. Chem. Res. 2013, 52, 12202−12211
Industrial & Engineering Chemistry Research
Article
Figure 10. Effect of particle diameter on ETC: line 1, ETC for coupled conduction and radiation; line 2, ETC for conduction only.
Figure 7. Relative contributions of the heat transfer modes considered to the overall heat transfer as a function of ks/kf: line 1, heat conduction Qnsfs; line 2, heat conduction Qcsfs; line 3, heat conduction Qcss; line 4, the solid−solid radiation between particle surfaces; dashedline, the percentage of total conduction.
Figure 11. Effect of particle diameter on the relative contributions of the heat transfer mechanisms to the overall heat transfer: line 1, heat conduction Qnsfs; line 2, heat conduction Qcsfs; line 3, heat conduction Qcss; line 4, the solid−solid radiation between particle surfaces; dashedline, the percentage of total conduction.
Figure 8. Effect of particle emissivity on ETC.
heat transfer mechanisms to the overall heat transfer. Under the given conditions, when the particle diameter is small (
