- Email: [email protected]
Expression for effective thermal conductivity of randomly packed granular material
International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Expression for effective thermal conductivity of randomly packed granular material Yuanbo Liang ⇑ College of Ocean and Civil Engineering, Dalian Ocean University, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 19 April 2015 Received in revised form 14 July 2015 Accepted 14 July 2015 Available online 1 August 2015 Keywords: Effective thermal conductivity Parallel-column model Pipe-network model Randomly packed granular material
a b s t r a c t This paper proposes an effective analytical expression for the accurate evaluation of the effective thermal conductivity (ETC) of randomly packed granular material with high solid-to-fluid conductivity ratio. The procedure is a novel combination of two key techniques: the parallel-column model and the pipe-network model. The ETC can be predicted using the proposed expression without microscopic analysis of the granular sample. Several practical issues associated with the accuracy and efficiency of the analytical expression are discussed, and test cases are provided to illustrate the applicability of the expression. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer in the particle system can be encountered in many engineering and scientific applications, for instance, in porous media, soil and geo-mechanics. For particle systems with high solid-to-fluid conductivity ratio, other heat transfer mechanisms are supposed to be largely dominated by inter-particle contact conduction [1,2]. Here, we concentrate on the mechanism of inter-particle contact conduction for particle systems. As pointed out in the literature [3,4], the finite contact area between particles plays an important role for ks =kf > 103 in the particle system. In the past, some attempts have been made in which the particles are modeled as the thermal network [5], in which each particle is modeled as an isothermal disk/sphere. The feature of the thermal network is that the determination of the thermal properties/parameters in the models is based more on an ad hoc manner than on a rigorous theoretical foundation, which may introduce modeling errors that are difficult to quantify and control. Feng et al. [6,7] solved this problem by proposing a novel numerical methodology termed the pipe-network model for the modeling of heat conduction in systems involving a large number of circular particles in 2D cases. In the present paper, we combine two key techniques: the parallel-column model and the pipe-network model for the heat transfer through the contact network of randomly packed granular material. The expression for the effective thermal conductivity of ⇑ Tel./fax: +86 411 84763228. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.059 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
the granular material is derived. The validity of the proposed expression is quantitatively tested by comparing the theoretical prediction of the effective thermal conductivity given by the proposed expression with the reference solution obtained by the finite element method. 2. Review of the pipe-network model The basic formulations of the pipe-network model are summarized in the following. The notation used here is a little different from the notation used by Feng et al. [7]. 2.1. Heat conduction in a simple particle system Consider the lth particle which is a 2D circular particle of radius rl in a particle assembly, this particle is in contact with n neighboring particles, as shown in Fig. 1(a), in which heat is conducted only through the n contact zones on the boundary of the particle, and the remainder of the particle boundary is fully insulated. A polar coordinate system ðR; hÞ is established with the origin set at the center of the particle. Each contact zone between the lth particle and the ith particle, which is assumed to be an arc, can be described by the position of its middle point in terms of an angle hl;i and a contact angle al;i that determines the contact arc length. If the heat flux along the contact zone between the lth particle and ith particle is described by a continuous function ql;i ðhÞ, then the heat flux on the whole boundary of the particle is denoted as ql ðhÞ. ql ðhÞ can be represented as
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Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
shown in Fig. 1(b). For each individual pipe i of the lth particle, the corresponding thermal resistance Rl;i and conductivity kl;i are respectively given by
kl;i ¼
Rl;i ¼
1 Rl;i 1
pj
ð4Þ ! a4l;i a6l;i 3 a2l;i ln al;i þ þ þ þ 2 36 2700 79380
ð5Þ
As pointed out by Feng [6,7], the higher order terms of al;i in Eq. (5) may not be necessary for small contact angles. In this paper, only the first two terms (up to 3=2) is used. The heat flow, Q i;j , between the ith particle and the jth particle in contact is formulated as
Q i;j ¼ kij ðT i T j Þ
ð6Þ
where kij is the coefficient of thermal contact conductance between the ith particle and the jth particle, T i and T j are respectively the average temperature of the ith particle and the jth particle. kij is formulated as:
kij ¼
1 Ri;j þ Rj;i
ð7Þ
3. Expression for effective thermal conductivity of random packed granular material
Fig. 1. Heat conduction in a simple particle system.
ql ðhÞ ¼
ql:i ðh hl:i Þhl:i al:i 6 h 6 hl:i þ al:i
ði ¼ 1:::nÞ
0
otherwise
ð1Þ
The heat flux equilibrium in the particle requires
Z 2p 0
ql ðhÞdh ¼ 0
ð2Þ
The temperature distribution TðR; hÞ within the particle domain X ¼ fðR; hÞ : 0 6 R 6 r l ; 0 6 h 6 2pg is governed by the Laplace equation as:
8 < jDT ¼ 0 in X @T :j ¼ qðhÞ on @ X @n
The expression for effective thermal conductivity of random packed granular material is derived within the framework of the parallel-column model [8]. Generally, the contact network of the granular material is disordered and the heat transfers through the disordered contact network. In the parallel-column model as shown in Fig. 3, the randomly packed granular material is assumed to be composed of particle columns. It is assumed that each column does not transfer heat with its neighboring columns. The thermal conductance of the parallel-column model is the sum of thermal conductance of all particle columns. As illustrated in Fig. 2, for one particle column, the height of the particle column and the external compressive displacement applied to it are L and n, respectively. The particles are numbered 1, 2, . . ., N 1 from bottom to top. In this context, the particles in the column can only move vertically. The thermal conductance K 1 of one particle column is formulated as
ξ
N1
L ð3Þ
3
where j is the thermal conductivity; @ X denotes the boundary of is the temperature gradient along the normal the particle; and @T @n direction to the boundary.
2
2.2. The pipe-network model In the framework of the pipe-network model, one particle can be conceptually represented by a simple star-shaped pipe network model in which the center is connected to each contact zone [7], as
ξ
1
Fig. 2. Schematic of one particle column in the parallel-column model.
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Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
1 1 1 1 ¼ þ þ þ K 1 k12 k23 kðN1 1ÞN
D
ð8Þ Insulated Wall
Employing Eq. (5), Eq. (7) can be reformulated as
kij ¼
pj
θ
u = 0 , u y = −ξ y
ð9Þ
ln ai;j ln aj;i þ 3
As long as the normal relative contact displacement dnij is much smaller than the grain size (dnij ri and dnij rj ), the contact radius aij can be expressed as [8]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r i r j n aij ¼ d ri þ rj ij
uθ = 0
uθ = 0
ux = ξx
u x = −ξ x
L
ð10Þ
Tt
Tb
The contact angle ai;j can be expressed as the arctangent of aij =r i .
ai;j ¼ arctan
y
aij ri
ð11Þ Insulated Wall
Because dnij ri and dnij rj , we have aij ri . Therefore, with the help of Eq. (10), Eq. (11) can be reformulated as
ai;j ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r i r j n d ri þ rj ij
ð12Þ
ri
Employing Eq. (12), Eq. (9) can be reformulated as
kij ¼
pj ln 2 ln dnij þ lnðr i þ r j Þ þ 3
ð13Þ
The linear or Hooke model [9] is the most commonly used model for the discrete particle simulation, if the normal contact stiffness coefficient is assumed to be a constant for all contact points, then
dn12 ¼ dn23 ¼ ¼ dnðN1 1ÞN1 ¼
2n N1
ð14Þ
Employing Eqs. (13) and (14), Eq. (8) can be reformulated as
( ) N X 1 1 2n N1 ð3 ln 2Þ þ ðln di Þ N1 ln K 1 pj N1 i¼1
ð15Þ
where di is the diameter of the ith particle. When the number of particles is large enough, Eq. (15) can be rewritten in integral form over diameter d:
( ) Z dM 1 1 2n ¼ N1 ð3 ln 2Þ þ N1 ½pðdÞ ln ddd N1 ln K 1 pj N1 dm
ð16Þ
where dM and dm denote the largest and smallest particle diameters, respectively. pðdÞ denotes the probability distribution function of the particle size distribution. For simplicity, we define hf ðdÞi as the statistical average of the function f ðdÞ:
hf ðdÞi ¼
Z
dM
f ðdÞpðdÞdd
ð17Þ
dm
We denote by np the number of particle columns in the granular sample. np is expressed as
np ¼
D hdi
ð18Þ
where D is the width of the rectangular granular sample as shown in Fig. 3. The height of the particle column L and the number of particles N 1 are related as
L N1 ¼ hdi
uθ = 0 , u y = ξ y
x Fig. 3. The schematic of temperature boundary conditions imposed on the granular sample.
Combining Eq. (15) with Eqs. (18) and (19), the thermal conductance K e of the rectangular sample of randomly packed granular material can be expressed as
K e ¼ np K 1 ¼
Dpj 1 ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi L
ð20Þ
The theoretical model is based on an ideal arrangement of particles, i.e., the rectangular sample of heterogeneous granular material is composed of a series of parallel particle columns. In reality, the particulate assembly possesses a heterogeneous and irregular meso-structure. The above idealized theoretical model is incapable of dealing with this situation from a theoretical point of view. Therefore, we modified the above model by adding one dimensionless fitting parameter C. C is named as the modified coefficient. With the introduction of the modified coefficient C, the real granular structure is taken into account empirically and the thermal conductance K e can be expressed as
Ke ¼
CDpj 1 ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi L
ð21Þ
The relationship between the effective thermal conductivity ke and the thermal conductance He of the rectangular granular sample is
Ke ¼
Dke L
ð22Þ
Combining Eq. (21) with Eq. (22), the effective thermal conductivity ke of the rectangular granular sample is formulated as
ke ¼
C pj ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi
ð23Þ
The authors show that the very good agreement between the theoretical prediction of the effective thermal conductivity given by Eq. (23) and the thermal reference solution is achieved by setting C to 0.9. Eq. (23) can be reformulated as
ke ¼
C pj ð3 ln 2Þ þ hln di ln eyy lnhdi
ð19Þ where
eyy denotes the compressive strain along the y axis.
ð24Þ
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Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
Table 1 Summary of the parameters used in the numerical example. Parameter
Value
Number of particles Normal contact stiffness coefficient kn Tangential contact stiffness coefficient kt Thermal conductivity j Width of the granular sample D Height of the granular sample L Largest grain size dmax Smallest grain size dmin Temperature T t Temperature T b
1102 5.0 * 106 (N/m) 2.0 * 106 (N/m) 1.0 (W/m K) 0.2 (m) 0.2 (m) 0.008 (m) 0.004 (m) 1.0 (°C) 0.0 (°C)
Prescribed displacement nx (m)
Prescribed displacement ny (m)
Result of finite element analysis (W/m K)
Theoretical prediction (W/m K)
Relative error (%)
0.0015 0.0010 0.0005
0.0015 0.0010 0.0005
0.425 0.40577 0.37825
0.4358 0.4102 0.3727
2.48 1.08 1.49
5. Conclusions This paper has proposed an accurate expression for effective thermal conductivity of circular disks in a randomly packed assembly. The expression is derived in the framework of the parallel-column model. The solution accuracy of the expression is assessed against the finite element method.
In this paper, exx denotes the compressive strain along the x axis. A fair guess is that the effect of exx on the effective thermal conductivity ke is the same as the effect of eyy . Therefore, Eq. (24) can be reformulated as
C pj ð3 ln 2Þ þ hln di lnðexx þ eyy Þ lnhdi
4.2. Numerical result Summary of the parameters of the numerical tests are listed in Table 1. The results of the numerical tests are listed in Table 2.
Table 2 Computed effective conductivities and relative errors.
ke ¼
ny and ny , respectively. The horizontal displacements of the particles on the left and right boundary are the uniformly prescribed values nx and nx , respectively. Different value of nx and ny are tested in the numerical sample. The expression for ETC in this paper is not suitable for unsteady-state problem. Discrete element method [9] is employed to compute the new position of the particles in the assembly under compression. After the new position of the particles is computed, the finite element method is employed to solve the boundary value problem of heat transfer in particle assembly.
Conflict of interest None declared. References
ð25Þ
4. Numerical validation and illustration The theoretical prediction of the proposed expression is compared with the result of the finite element analysis in the numerical example in this section. 4.1. Numerical example The initial arrangement of granular sample used in the numerical test is generated by the inwards packing method [10]. The sizes of particles are evenly distributed in the range ½dmin ; dmax . The granular sample is confined within a rectangular domain with the width D and the height L, as shown in Fig. 4. The thermal boundary condition is that the uniform temperatures T t and T b are respectively imposed on all particles intersecting the left and right walls; Both the top and bottom walls are insulated to produce a uni-directional heat flux. Only the steady-state situation is taken into consideration. The rotations of the particles on the boundary are specified as null. The biaxial compression is applied on the granular sample. The vertical displacements of the particles on the top and bottom boundary are the uniformly prescribed values
[1] C. Argento, D. Bouvard, Modeling the effective thermal conductivity of random packing of spheres through densification, Int. J. Heat Mass Transfer 39 (7) (1996) 1343–1350. [2] T.S. Yun, T.M. Evans, Three-dimensional random network model for thermal conductivity in particulate materials, Comput. Geotech. 37 (7–8) (2010) 991– 998. [3] P. Cheng, C.-T. Hsu, The effective stagnant thermal conductivity of porous media with periodic structures, J. Porous Media 2 (1) (1999). [4] C.T. Hsu, P. Cheng, K.W. Wong, Modified Zehner–Schlunder models for stagnant thermal conductivity of porous media, Int. J. Heat Mass Transfer 37 (17) (1994) 2751–2759. [5] W.W.M. Siu, S.H.K. Lee, Transient temperature computation of spheres in three-dimensional random packings, Int. J. Heat Mass Transfer 47 (5) (2004) 887–898. [6] Y.T. Feng, K. Han, C.F. Li, D.R.J. Owen, Discrete thermal element modelling of heat conduction in particle systems: basic formulations, J. Comput. Phys. 227 (10) (2008) 5072–5089. [7] Y.T. Feng, K. Han, D.R.J. Owen, Discrete thermal element modelling of heat conduction in particle systems: pipe-network model and transient analysis, Powder Technol. 193 (3) (2009) 248–256. [8] Y. Liang, X. Li, A new model for heat transfer through the contact network of randomly packed granular material, Appl. Therm. Energy 73 (1) (2014) 982– 990. [9] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [10] K. Bagi, An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies, Granular Matter 7 (1) (2005) 31– 43.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Technical Note
Expression for effective thermal conductivity of randomly packed granular material Yuanbo Liang ⇑ College of Ocean and Civil Engineering, Dalian Ocean University, Dalian 116023, China
a r t i c l e
i n f o
Article history: Received 19 April 2015 Received in revised form 14 July 2015 Accepted 14 July 2015 Available online 1 August 2015 Keywords: Effective thermal conductivity Parallel-column model Pipe-network model Randomly packed granular material
a b s t r a c t This paper proposes an effective analytical expression for the accurate evaluation of the effective thermal conductivity (ETC) of randomly packed granular material with high solid-to-fluid conductivity ratio. The procedure is a novel combination of two key techniques: the parallel-column model and the pipe-network model. The ETC can be predicted using the proposed expression without microscopic analysis of the granular sample. Several practical issues associated with the accuracy and efficiency of the analytical expression are discussed, and test cases are provided to illustrate the applicability of the expression. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Heat transfer in the particle system can be encountered in many engineering and scientific applications, for instance, in porous media, soil and geo-mechanics. For particle systems with high solid-to-fluid conductivity ratio, other heat transfer mechanisms are supposed to be largely dominated by inter-particle contact conduction [1,2]. Here, we concentrate on the mechanism of inter-particle contact conduction for particle systems. As pointed out in the literature [3,4], the finite contact area between particles plays an important role for ks =kf > 103 in the particle system. In the past, some attempts have been made in which the particles are modeled as the thermal network [5], in which each particle is modeled as an isothermal disk/sphere. The feature of the thermal network is that the determination of the thermal properties/parameters in the models is based more on an ad hoc manner than on a rigorous theoretical foundation, which may introduce modeling errors that are difficult to quantify and control. Feng et al. [6,7] solved this problem by proposing a novel numerical methodology termed the pipe-network model for the modeling of heat conduction in systems involving a large number of circular particles in 2D cases. In the present paper, we combine two key techniques: the parallel-column model and the pipe-network model for the heat transfer through the contact network of randomly packed granular material. The expression for the effective thermal conductivity of ⇑ Tel./fax: +86 411 84763228. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.059 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
the granular material is derived. The validity of the proposed expression is quantitatively tested by comparing the theoretical prediction of the effective thermal conductivity given by the proposed expression with the reference solution obtained by the finite element method. 2. Review of the pipe-network model The basic formulations of the pipe-network model are summarized in the following. The notation used here is a little different from the notation used by Feng et al. [7]. 2.1. Heat conduction in a simple particle system Consider the lth particle which is a 2D circular particle of radius rl in a particle assembly, this particle is in contact with n neighboring particles, as shown in Fig. 1(a), in which heat is conducted only through the n contact zones on the boundary of the particle, and the remainder of the particle boundary is fully insulated. A polar coordinate system ðR; hÞ is established with the origin set at the center of the particle. Each contact zone between the lth particle and the ith particle, which is assumed to be an arc, can be described by the position of its middle point in terms of an angle hl;i and a contact angle al;i that determines the contact arc length. If the heat flux along the contact zone between the lth particle and ith particle is described by a continuous function ql;i ðhÞ, then the heat flux on the whole boundary of the particle is denoted as ql ðhÞ. ql ðhÞ can be represented as
1106
Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
shown in Fig. 1(b). For each individual pipe i of the lth particle, the corresponding thermal resistance Rl;i and conductivity kl;i are respectively given by
kl;i ¼
Rl;i ¼
1 Rl;i 1
pj
ð4Þ ! a4l;i a6l;i 3 a2l;i ln al;i þ þ þ þ 2 36 2700 79380
ð5Þ
As pointed out by Feng [6,7], the higher order terms of al;i in Eq. (5) may not be necessary for small contact angles. In this paper, only the first two terms (up to 3=2) is used. The heat flow, Q i;j , between the ith particle and the jth particle in contact is formulated as
Q i;j ¼ kij ðT i T j Þ
ð6Þ
where kij is the coefficient of thermal contact conductance between the ith particle and the jth particle, T i and T j are respectively the average temperature of the ith particle and the jth particle. kij is formulated as:
kij ¼
1 Ri;j þ Rj;i
ð7Þ
3. Expression for effective thermal conductivity of random packed granular material
Fig. 1. Heat conduction in a simple particle system.
ql ðhÞ ¼
ql:i ðh hl:i Þhl:i al:i 6 h 6 hl:i þ al:i
ði ¼ 1:::nÞ
0
otherwise
ð1Þ
The heat flux equilibrium in the particle requires
Z 2p 0
ql ðhÞdh ¼ 0
ð2Þ
The temperature distribution TðR; hÞ within the particle domain X ¼ fðR; hÞ : 0 6 R 6 r l ; 0 6 h 6 2pg is governed by the Laplace equation as:
8 < jDT ¼ 0 in X @T :j ¼ qðhÞ on @ X @n
The expression for effective thermal conductivity of random packed granular material is derived within the framework of the parallel-column model [8]. Generally, the contact network of the granular material is disordered and the heat transfers through the disordered contact network. In the parallel-column model as shown in Fig. 3, the randomly packed granular material is assumed to be composed of particle columns. It is assumed that each column does not transfer heat with its neighboring columns. The thermal conductance of the parallel-column model is the sum of thermal conductance of all particle columns. As illustrated in Fig. 2, for one particle column, the height of the particle column and the external compressive displacement applied to it are L and n, respectively. The particles are numbered 1, 2, . . ., N 1 from bottom to top. In this context, the particles in the column can only move vertically. The thermal conductance K 1 of one particle column is formulated as
ξ
N1
L ð3Þ
3
where j is the thermal conductivity; @ X denotes the boundary of is the temperature gradient along the normal the particle; and @T @n direction to the boundary.
2
2.2. The pipe-network model In the framework of the pipe-network model, one particle can be conceptually represented by a simple star-shaped pipe network model in which the center is connected to each contact zone [7], as
ξ
1
Fig. 2. Schematic of one particle column in the parallel-column model.
1107
Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
1 1 1 1 ¼ þ þ þ K 1 k12 k23 kðN1 1ÞN
D
ð8Þ Insulated Wall
Employing Eq. (5), Eq. (7) can be reformulated as
kij ¼
pj
θ
u = 0 , u y = −ξ y
ð9Þ
ln ai;j ln aj;i þ 3
As long as the normal relative contact displacement dnij is much smaller than the grain size (dnij ri and dnij rj ), the contact radius aij can be expressed as [8]
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r i r j n aij ¼ d ri þ rj ij
uθ = 0
uθ = 0
ux = ξx
u x = −ξ x
L
ð10Þ
Tt
Tb
The contact angle ai;j can be expressed as the arctangent of aij =r i .
ai;j ¼ arctan
y
aij ri
ð11Þ Insulated Wall
Because dnij ri and dnij rj , we have aij ri . Therefore, with the help of Eq. (10), Eq. (11) can be reformulated as
ai;j ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r i r j n d ri þ rj ij
ð12Þ
ri
Employing Eq. (12), Eq. (9) can be reformulated as
kij ¼
pj ln 2 ln dnij þ lnðr i þ r j Þ þ 3
ð13Þ
The linear or Hooke model [9] is the most commonly used model for the discrete particle simulation, if the normal contact stiffness coefficient is assumed to be a constant for all contact points, then
dn12 ¼ dn23 ¼ ¼ dnðN1 1ÞN1 ¼
2n N1
ð14Þ
Employing Eqs. (13) and (14), Eq. (8) can be reformulated as
( ) N X 1 1 2n N1 ð3 ln 2Þ þ ðln di Þ N1 ln K 1 pj N1 i¼1
ð15Þ
where di is the diameter of the ith particle. When the number of particles is large enough, Eq. (15) can be rewritten in integral form over diameter d:
( ) Z dM 1 1 2n ¼ N1 ð3 ln 2Þ þ N1 ½pðdÞ ln ddd N1 ln K 1 pj N1 dm
ð16Þ
where dM and dm denote the largest and smallest particle diameters, respectively. pðdÞ denotes the probability distribution function of the particle size distribution. For simplicity, we define hf ðdÞi as the statistical average of the function f ðdÞ:
hf ðdÞi ¼
Z
dM
f ðdÞpðdÞdd
ð17Þ
dm
We denote by np the number of particle columns in the granular sample. np is expressed as
np ¼
D hdi
ð18Þ
where D is the width of the rectangular granular sample as shown in Fig. 3. The height of the particle column L and the number of particles N 1 are related as
L N1 ¼ hdi
uθ = 0 , u y = ξ y
x Fig. 3. The schematic of temperature boundary conditions imposed on the granular sample.
Combining Eq. (15) with Eqs. (18) and (19), the thermal conductance K e of the rectangular sample of randomly packed granular material can be expressed as
K e ¼ np K 1 ¼
Dpj 1 ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi L
ð20Þ
The theoretical model is based on an ideal arrangement of particles, i.e., the rectangular sample of heterogeneous granular material is composed of a series of parallel particle columns. In reality, the particulate assembly possesses a heterogeneous and irregular meso-structure. The above idealized theoretical model is incapable of dealing with this situation from a theoretical point of view. Therefore, we modified the above model by adding one dimensionless fitting parameter C. C is named as the modified coefficient. With the introduction of the modified coefficient C, the real granular structure is taken into account empirically and the thermal conductance K e can be expressed as
Ke ¼
CDpj 1 ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi L
ð21Þ
The relationship between the effective thermal conductivity ke and the thermal conductance He of the rectangular granular sample is
Ke ¼
Dke L
ð22Þ
Combining Eq. (21) with Eq. (22), the effective thermal conductivity ke of the rectangular granular sample is formulated as
ke ¼
C pj ð3 ln 2Þ þ hln di ln 2n þ ln L lnhdi
ð23Þ
The authors show that the very good agreement between the theoretical prediction of the effective thermal conductivity given by Eq. (23) and the thermal reference solution is achieved by setting C to 0.9. Eq. (23) can be reformulated as
ke ¼
C pj ð3 ln 2Þ þ hln di ln eyy lnhdi
ð19Þ where
eyy denotes the compressive strain along the y axis.
ð24Þ
1108
Y. Liang / International Journal of Heat and Mass Transfer 90 (2015) 1105–1108
Table 1 Summary of the parameters used in the numerical example. Parameter
Value
Number of particles Normal contact stiffness coefficient kn Tangential contact stiffness coefficient kt Thermal conductivity j Width of the granular sample D Height of the granular sample L Largest grain size dmax Smallest grain size dmin Temperature T t Temperature T b
1102 5.0 * 106 (N/m) 2.0 * 106 (N/m) 1.0 (W/m K) 0.2 (m) 0.2 (m) 0.008 (m) 0.004 (m) 1.0 (°C) 0.0 (°C)
Prescribed displacement nx (m)
Prescribed displacement ny (m)
Result of finite element analysis (W/m K)
Theoretical prediction (W/m K)
Relative error (%)
0.0015 0.0010 0.0005
0.0015 0.0010 0.0005
0.425 0.40577 0.37825
0.4358 0.4102 0.3727
2.48 1.08 1.49
5. Conclusions This paper has proposed an accurate expression for effective thermal conductivity of circular disks in a randomly packed assembly. The expression is derived in the framework of the parallel-column model. The solution accuracy of the expression is assessed against the finite element method.
In this paper, exx denotes the compressive strain along the x axis. A fair guess is that the effect of exx on the effective thermal conductivity ke is the same as the effect of eyy . Therefore, Eq. (24) can be reformulated as
C pj ð3 ln 2Þ þ hln di lnðexx þ eyy Þ lnhdi
4.2. Numerical result Summary of the parameters of the numerical tests are listed in Table 1. The results of the numerical tests are listed in Table 2.
Table 2 Computed effective conductivities and relative errors.
ke ¼
ny and ny , respectively. The horizontal displacements of the particles on the left and right boundary are the uniformly prescribed values nx and nx , respectively. Different value of nx and ny are tested in the numerical sample. The expression for ETC in this paper is not suitable for unsteady-state problem. Discrete element method [9] is employed to compute the new position of the particles in the assembly under compression. After the new position of the particles is computed, the finite element method is employed to solve the boundary value problem of heat transfer in particle assembly.
Conflict of interest None declared. References
ð25Þ
4. Numerical validation and illustration The theoretical prediction of the proposed expression is compared with the result of the finite element analysis in the numerical example in this section. 4.1. Numerical example The initial arrangement of granular sample used in the numerical test is generated by the inwards packing method [10]. The sizes of particles are evenly distributed in the range ½dmin ; dmax . The granular sample is confined within a rectangular domain with the width D and the height L, as shown in Fig. 4. The thermal boundary condition is that the uniform temperatures T t and T b are respectively imposed on all particles intersecting the left and right walls; Both the top and bottom walls are insulated to produce a uni-directional heat flux. Only the steady-state situation is taken into consideration. The rotations of the particles on the boundary are specified as null. The biaxial compression is applied on the granular sample. The vertical displacements of the particles on the top and bottom boundary are the uniformly prescribed values
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