Implicit Continuum Mechanics Approach to Heat Conduction in

Dec 1, 2009 - law of heat conduction, and for complex materials such as nonlinear fluids, ... the coefficient of thermal conductivity is generalized b...
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Ind. Eng. Chem. Res. 2010, 49, 5215–5221

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Implicit Continuum Mechanics Approach to Heat Conduction in Granular Materials Mehrdad Massoudi* National Energy Technology Laboratory (NETL), U.S. Department of Energy, P.O. Box 10940, Pittsburgh, PennsylVania 15236

Morteza M. Mehrabadi Department of Mechanical Engineering, San Diego State UniVersity, 5500 Campanile DriVe, San Diego, California 92182-1323

In this paper, we derive a properly frame-invariant implicit constitutive relationship for the heat flux vector for a granular medium (or a density-gradient-type fluid). The heat flux vector is commonly modeled by Fourier’s law of heat conduction, and for complex materials such as nonlinear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematic parameters such as temperature, shear rate, porosity, concentration, etc. In this paper, we extend the approach of Massoudi [Massoudi, M. Math. Methods Appl. Sci. 2006, 29, 1585; Massoudi, M. Math. Methods Appl. Sci. 2006, 29, 1599], who provided explicit constitutive relations for the heat flux vector for flowing granular materials; in order to do so, we use the implicit scheme suggested by Fox [Fox, N. Int. J. Eng. Sci. 1969, 7, 437], who obtained implicit relations in thermoelasticity. 1. Introduction Study of granular materials has become an interdisciplinary field. To design equipment such as bins and silos, combustors, hoppers, chutes, hydrocyclones, etc. in an effective and economical way, we need to understand the various factors governing the flow of granular materials. In the past few decades, with the advancement of computers and various computational schemes, alternative approaches such as numerical simulations and statistical approaches have also been used in addition to the classical continuum approaches. Granular materials can be a part of multicomponent flows which because of their increasing importance in many industries have also become the subject of considerable attention. [In multicomponent theories, it is not necessary to have all the components in motion; for example, in composite materials, the different components undergo elastic (or plastic) deformation without involving any motion.] Examples include fluidization,4,5 gas-solid flows,6 pneumatic conveying,7 and suspensions.8 To model (flowing) granular materials, methods in continuum mechanics can be used. In this approach one assumes that the material properties of the ensemble may be represented by continuous functions. Another approach used is based on the techniques used in the kinetic theory of gases.9,10 Review articles by Savage,11 Hutter and Rajagopal,12 and de Gennes,13 and books by Mehta,14 Duran,15 and Antony et al.,16 to name but a few, address many of the interesting issues in this field. In this paper, we use the continuum approach. The two most important constitutive relations which characterize the behavior of nonlinear materials, such as granular materials, are the stress tensor and the heat flux vector. In a number of applications, these materials are also heated prior to processing, or cooled after processing. Very little fundamental work, from a mathematical point of view, has been devoted to these types of problems. Sullivan and Sabersky17 studied the * To whom correspondence should be addressed. E-mail: massoudi@ netl.doe.gov (M.M); [email protected] (M.M.M.).

heat transfer from a plate to various granular materials. Later, Spelt et al.18 generalized this problem by studying the heat transfer to granular materials flowing along an inclined chute at higher velocities. Based on their experimental results, they speculated that the higher velocities caused a decrease in the density of the material and that decrease in density caused the reduction of the heat transfer. Patton et al.19 continued this research by extending the range of conditions investigated previously; an experimental technique was developed to allow density measurements, since the density changes are significant in the rapid flows of granular materials. In the studies relevant to granular materials, generally Fourier’s law is assumed to hold and through various mechanisms such as statistical averaging20,21 or homogenization techniques, experimental curve-fitting,22,23 and numerical simulations24 a modified coefficient for the thermal conductivity is obtained [see Massoudi1,2 and references therein]. There have been many attempts to generalize Fourier’s heat conduction law. For example, the thermal conductivity is assumed to depend on the volume fraction or the shear rate, etc.;25-27 this is very similar to the attempts which have been made in rheology where power-law models have been proposed, whereby the viscosity is assumed to depend on volume fraction or shear rate, etc. Just as in rheology to capture rate-dependent effects models such as the Oldroyd28 and integral models have been developed, there have been attempts to develop rate-type heat flux vectors for various materials. In this paper, we extend the approach of Massoudi,1,2 who provided explicit constitutive relations for the heat flux vector for flowing granular materials; in order to do so, we use the implicit scheme suggested by Fox.3 More specifically, we derive implicit relations for the heat flux vector for a density-gradientdependent material. It is shown that, under certain conditions, the derived equation reduces to the results of Fox. To proceed, we first give the basic governing equations dealing with the thermomechanical aspects of a single component medium, which will be referred to as a granular medium henceforth.

10.1021/ie9014155  2010 American Chemical Society Published on Web 12/01/2009

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2. Governing Equations Although in general the fluid phase plays an important role in determining the dynamics of dilute suspensions, it does not have much influence on the bulk solids behavior. The behavior of granular materials, in general, is governed by interparticle cohesion, friction, and collisions. For these applications, the balance laws, in the absence of chemical reactions and thermal effects, are the conservation of mass, conservation of linear momentum, and conservation of angular momentum. Let X denote the position of particles in the reference configuration. Then the motion is represented by the mapping x ) χ˜ (X,t) and the conservation of mass in the Lagrangian form is F0 ) F det F

(1)

where F0 is the reference density of the material, F is the current density, and F is the deformation gradient tensor given by ∂χ (2) ∂X The conservation of mass in the Eulerian form is given by F)

∂F + div(Fu) ) 0 (3) ∂t where ∂/∂t is the partial derivative with respect to time, and u is the velocity vector. The balance of linear momentum is du ) div T + Fb (4) dt where d/dt is the material time derivative given by d( · )/dt ) ∂( · )/∂t + [grad( · )]u, b is the body force, and T is the Cauchy stress tensor. The balance of angular momentum (in the absence of couple stresses) yields the result that the Cauchy stress is symmetric. The energy equation in its general form is F

dε ) T · L - div q + Fr (5) dt where ε denotes the specific internal energy, q is the heat flux vector, r is the radiant heating, and L is the velocity gradient. For a complete study of a thermomechanical problem, the second law of thermodynamics has to be considered.29 In other words, in addition to other principles in continuum mechanics such as material symmetry, frame indifference, etc., the second law also imposes certain restrictions on the type of motion and/ or the constitutive parameters. In general, the application of the second law of thermodynamics, i.e., the Clausius-Duhem inequality, is a subject matter which has caused some controversy. As pointed out recently in an important paper by Rajagopal and Srinivasa30 (p 644), the usual approach is “to posit some phenomenological law for the fluxes and see what restrictions are imposed on the phenomenological coefficients by the non-negativity of the rate of entropy production.” This approach has been used successfully in many problems, but as pointed out by Rajagopal and Srinivasa30 (p 645), based on observation “materials are characterized not only by how they store energy (as indicated by the equation of state) but also by how they produce entropy.” In our problem, since there is no general agreement on the functional form of the constitutive relation and since the Helmholtz free energy is not known, a complete thermodynamic treatment is lacking. F

3. A Few Remarks about Granular Materials Granular materials exhibit the properties of both a solid and a fluid as they can take the shape of the vessel containing them,

Figure 1

thereby exhibiting fluidlike characteristics, or they can be heaped, thereby behaving like a solid. Granular materials can also sustain shear stresses in the absence of any deformation. The characteristics of the particles that constitute the bulk solids are of major importance in influencing the characteristics of the bulk solids both at rest and during flow. It is very difficult to characterize bulk solids, which are composed of a variety of materials, mainly because small variations in some of the primary properties of the bulk solids, such as the size, shape, hardness, particle density, and surface roughness can result in very different behaviors. Furthermore, secondary factors such as the presence or absence of moisture, the severity of prior compaction, the ambient temperature, etc., which are not directly associated with the particles, can have a significant effect on the behavior of the bulk solids. A granular material covers the combined range of granular powders and granular solids with components ranging in size from about 10 µm up to 3 mm. A powder is composed of particles up to 100 µm (diameter) with further subdivision into ultrafine (0.1-1.0 µm), superfine (1-10 µm), or granular (10-100 µm) particles. A granular solid consists of materials ranging from about 100 to 3000 µm [see ref 31]. This range includes most of the materials used in laboratory experiments and whenever we use the term “granular material” we shall henceforth refer to this range. We should mention that the volume fraction field ν(x,t) plays a major role in many of the proposed continuum theories of granular materials.32 That is, even though we talk of distinct solid particles with a certain diameter, in our theory, the particles through the introduction of the volume fraction field are homogenized, as shown in Figure 1 [see ref 33 for a thorough discussion of homogenizing the microstructure of granular media]. In other words, it is assumed that the material properties of the ensemble are continuous functions of position. That is, the material may be divided indefinitely without losing any of its defining properties. A distributed volume Vt ) ∫ν dV, and a distributed mass M ) ∫Fsν dV can be defined, where the function ν is an independent kinematical variable called the “volume distribution function” and has the property 0 e ν(x,t) < νmax < 1. The function ν is represented as a continuous function of position and time; in reality, ν in such a system is either 1 or 0 at any position and time, depending upon whether one is pointing to a granule or to the void space. That is, the real volume distribution content has been averaged, in some sense, over the neighborhood of any given position. The classical mass density or bulk density, F, is related to Fs and ν through F ) Fsν. In the continuum approach that we have adopted here, the fluctuations in the velocity field are ignored, and the phenomenon of heat transfer is incorporated in the energy equation. To include in addition to the energy equation, the notion of granular temperature would be inconsistent with our approach. The rapid flow of granular materials is generally maintained by particle collisions causing the particles to have highly irregular paths;

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these irregular motions create fluctuations on all field variables (such as velocity, temperature, etc.). This suggests a similarity between the rapid flow of granular materials and the turbulent flow of a fluid. Blinowski34 appears to have been the first to observe this similarity. Ogawa35 and Ogawa et al.36 derived theories using formal procedures of continuum mechanics, while recognizing the discrete nature of granular materials. They defined two different kinds of temperature: one is the usual temperature associated with the thermal fluctuation of the molecules of each grain, and the other is related to the “random” translational and rotational fluctuations of the individual grain. These are proportional to 〈V2〉1/2 and 〈ω2〉1/2, respectively, which are the root mean squares of the translational and rotational velocity fluctuations arising from interparticle collisions. They recognized that the dissipation processes for energy of flowing granular materials are different from those of a liquid or gas. In the case of gas or liquid, momentum transfer is due to the thermal motion of molecules which is depicted in the theory via the dependence of the coefficient of viscosity on the temperature. In the case of granular materials, however, the momentum transfer occurs by particle collision. One of the basic differences between the continuum approach and the kinetic theory approximation, when studying granular materials, is the need for additional governing equations. As a result of the averaging procedure and the fact that the fluctuations of the individual particles are considered, at least one extra equation, named the “pseudoenergy” equation, is added to the list of the basic equations.37 Thus, even in the absence of (real) thermal effects, when solving a boundary value problem using the kinetic theory approach, there are at least four basic equations. These equations are the conservation of mass, the balance of linear momentum and angular momentum, and the pseudoenergy equation. The concept of granular temperature has its roots in turbulence modeling, where the ideas of Reynolds result in decomposing the velocity vector into an average and j + u′, where u j is a fluctuating component.38,39 That is, u ) u the average velocity and u′ is the fluctuating velocity. While in classical thermodynamics temperature has a clear meaning,40 making its appearance in the energy equation, there is no such role or clear meaning for the granular temperature. One can define a scalar d as a measure of the fluctuation of the flow d ) 1/2(u′ · u′). The granular temperature ϑ is related to the kinetic energy d through ϑ ) (2/3)d. In general, the continuum approach is applicable when the packing of the material is reasonably compact, i.e., high volume fraction, and the fluctuations from the mean are not significant. Finally, we would like to mention that the application of kinetic theory of gases to flows of granular materials is plagued by many assumptions, perhaps beyond what the original theory may have stood for. In the past three decades, researchers in the field of granular materials have exploited the techniques of kinetic theory and statistical mechanics. There are certain flow regimes where the collisions of particles are rare, in the sense that the flow is so slow and the particles are so densely packed that one cannot assume the basic assumptions in the kinetic theory are valid.41 However, certain processes such as fluidized beds present a special challenge:5,6,42 before the onset of fluidization, the flow regime is perhaps more in the slow deformation range and after fluidization the flow regime is in the rapid flow range. It is in the rapid flow regime that the kinetic theory approach may be used. Thus, one can look at the kinetic theory based models as tools which may be appropriate for some cases and irrelevant or not appropriate in other cases. Just as one can build a bridge or design a ship without ever having access to the tools

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in statistical theories, it is also possible to study many engineering problems involving granular materials without ever using the kinetic theory approach. From the above equations, it can be seen that constitutive relations are needed for T, q, and r. Modeling of the stress tensor, T, in both implicit and explicit forms, has received much attention.45-48 [In recent years, Rajagopal43,44 has provided a rigorous implicit methodology where constitutive relations for a class of fluids whose viscosity depends on pressure and shear rate can be obtained.] However, constitutive modeling of heat flux vector, q, and the radiation term, r, are still in the early stages of development. In the next section, we give a brief overview of the many possible generalizations of q used in continuum mechanics, and we discuss very briefly the traditional, i.e., the explicit, approach to the formulation of the heat flux vector. 4. Heat Flux Vector Formulation: Traditional Approach The classical theory of heat conduction, first proposed by Fourier and later generalized by Duhamel [see ref 49], assumes that the constitutive relation for the heating flux h is a linear function of the temperature gradient, i.e., h ) K(θ,F) grad θ, where θ is the temperature, F is the deformation gradient, and K is the thermal conductivity tensor. Fourier was the first person50 to state that heat conduction depends on the temperature gradient and not on the temperature difference between the two adjacent parts of a solid body. For an isotropic material, this reduces to the classical Fourier law of heat conduction q ) -k∇θ

(6)

where k is generally assumed to be constant. This equation can be generalized in a variety of ways. For example q ) q(θ, ∇θ, θ˙ , (∇θ˙ ), ...) (7) An equation of this type is an explicit relationship whereby the dependent variable, q, is explicitly given by a function which can depend on the temperature, its gradient, its time derivative, among other variables such as velocity gradient, etc. For example, for thermoelastic solids one can generalize eq 7 to q ) q(θ, ∇θ, F)

(8)

where for an isotopic solid material, after imposing frame invariance, the above equation can be expressed as29 (p 98) q ) q(θ, g, B)

(9)

where g ) ∇θ;B ) FF . Now, it can be shown that an isotropic representation of eq 9 is29 (p 118) T

q ) k1g + k2Bg + k3B2g

(10)

For thermoviscous fluids, a possible generalization of eq 7 is q ) q(F, θ, g, D)

(11)

where D ) (1/2)(L + LT) ; L ) grad u, which similarly reduces to29 q ) β1g + β2Dg + β3D2g

(12)

Of course, the specification (or measurement) of the response functions k1, k2, and k3 or β1, β2, and β3 in the above equations, even for the simplest thermoelastic solids or thermoviscous fluids, remains a challenge. It can also be seen that with any added complexity due to the material’s structure, such as porosity, the equation for q will be more complicated. Granular materials possess not only porosity (volume fraction), but also inherently are anisotropic, and at times display yield stress and

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normal stress effects, and exhibit viscoelastic or viscoplastic behavior.51,52 Recently, Mehrabadi et al.53 proposed constitutive relations for the stress tensor of a density-gradient-dependent fluid. Massoudi1,2 assumed that the heat flux vector q for a density-gradient-dependent fluid (where flowing granular materials can be thought to be a subclass of such fluids) can be given by q ) q(θ, g, F, m, u, grad u)

(13)

where u is the velocity, θ is the temperature, F is the density, and m ) grad F. Then frame indifference54 implies q ) q(θ, g, F, m, D). This is a generalization of a case presented by Bowen55 (p 161) for a compressible, conducting viscous fluid. In our case, we think that the density gradient plays a role, not only in the distribution of the materials, but also in the way it influences the heat conduction. Massoudi1,2 showed that the general representation for the heat flux vector given by eq 13 is56,57 q ) a1g + a2m + a3Dg + a4Dm + a5D2g + a6D2m(14) where a1-a6 are scalar functions of the appropriate invariants of the tensor and vector terms. In the next section, we will derive an implicit relation for q. In order to obtain such relations for granular materials, we discuss briefly some attempts at implicit formulation of the heat flux vector within the domain of solid mechanics (elasticity). We then discuss the method proposed by Fox,3 who obtained implicit relations in generalized thermoelasticity. We use his approach to obtain implicit relations for q for density-gradient-dependent fluids. 5. Implicit Constitutive Relation for the Heat Flux Vector In recent years, due to manufacturing of new materials such as fiber-reinforced composites, microfabrication technologies, nanoscale thermal transport especially in the semiconductor industry, microtime heat transfer processes such as short-pulse laser applications and high speed electronics,58-61 there has been a growing need for more accurate representation and understanding of conductive and radiative heat transfer processes. Petroski62 has pointed out that for many nonhomogenous and anisotropic materials the simple Fourier’s law63 does not hold. Maxwell64 recognized that if the classical Fourier’s law of heat conduction is used in the energy equation, since one obtains a parabolic-type transport equation (diffusion type), when the material is subjected to a thermal disturbance, the effects of it are felt instantaneously everywhere; i.e., the thermal signal propagates with infinite speed. Maxwell suggested a wave-type heat flow, now called second sound. In fact, Maxwell in his book,65 Theory of Heat (p 260), describes this phenomenon: It follows from this result that, in calculating the temperature of point P, we must take into account the temperature of eVery other point Q, howeVer distant, and howeVer short the time may be during which the propagation of heat has been going on. Hence, in a strict sense, the influence of a heated part of the body extends to the most distant parts of the body in an incalculably short time, so that it is impossible to assign to the propagation of heat a definite Velocity. That is, if eq 6 is substituted into the equation of conservation of energy, eq 5, in the absence of internal heating, for a homogeneous isotropic material, one obtains the parabolic heat transport equation k∇2θ ) Fc(∂θ/∂t). In order to avoid the problem of infinite speed, a generalization of Fourier’s equation,66 often referred to as the Cattaneo-Vernotte (CV) equation, has been used: q + τ(∂q/∂t) ) -k∇θ, where τ is a positive

constant. If this equation is then substituted into the energy equation, one obtains a hyperbolic-type heat transport equation, k∇2θ ) Fc[(∂θ/∂t) + τ(∂2θ/∂t2)], which predicts a finite speed VT ) (k/Fcτ)1/2 for heat propagation. In recent years, many different nonclassical thermoelasticity theories of hyperbolictype heat transport have been developed.67,68 Recently, Massoudi and Mehrabadi,69 in the context of generalized thermoelasticity, based on the approach suggested by Fox,3 obtained an isotropic representation for q˚: q˚ ) q˙ - Wq ) a1g + a2q + a3Bg + a4Bq + a5B2g + a6B2q(15) where “°” denotes a (frame-invariant) convective-type derivative, the superimposed dot “ · ” designates d/dt, which is the material time derivative given by d( · )/dt ) ∂( · )/∂t + [grad( · )]u, a1-a6 are scalar functions of the appropriate principal invariants, and B ) FFT, W ) (1/2)[L - LT]. Thus, Massoudi and Mehrabadi69 showed that eq 15 is a generalization of Fox’s equation. In the remainder of this section, we shall extend their results to density-gradient-dependent fluids. The basic thought experiment motivating the formulation of the constitutive relation is that granular materials are not “solid blocks” and they are not “gas molecules”. The methodologies appropriate for the definition, calculation, or measurements of the thermal conductivity of a slab or a gas are not proper with respect to the thermal conductivity of a granular medium. Heat conduction is through contact, and contact in the case of granular materials depends upon the distribution of particles. Whether the particles are at the maximum packing arrangement or are randomly distributed should lead to different values for the components of the heat flux vector. Therefore, in this visualization experiment, we can see that the heat flux must depend in some form on a measure of particle distribution, as well as other important physical parameters. In fact, as mentioned by Tzou61 (p 29), “The finite time required for heat flow to circulate around the low-conducting aerial closures in porous media is another known cause for lagging behavior... This type of delayed response depends on the detailed configuration of the solid particles and the interstitial gas within the material volume.” Let us assume q˚ ) f(F, θ, g, m, q, D)

(16)

where g ) grad θ, m ) grad F, and q are objective vectors and D is an objective second rank tensor. Clearly we are neglecting the effects of the interstitial fluid and we have assumed that m (the density gradient, related to the gradient of volume fraction which is related to porosity) is a measure of particle distribution. According to the representation theorems,56 the generators are

(17) g, m, q, Dg, Dm, Dq, D2g, D2m, and D2q An isotropic vector valued function based on these generators can be obtained: q˚ ) R1g + R2m + R3q + R4Dg + R5Dm + R6Dq + R7D2g + R8D2m + R9D2q(18) where the R’s are functions of the appropriate principal invariants, depending on the generators listed in eq 17. This is a very general constitutive relation for the heat flux vector. In order to make it more amenable to practical engineering applications, in the next section we discuss some simplifying and limiting approximations. One can see certain similarities between this equation and eq 15, which was derived by Massoudi and Mehrabadi.69 While eq 15 is appropriate for an

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isotopic nonlinear thermoelastic material, eq 18 is for a densitygradient-type thermoviscous fluid. As with all implicit constitutive relations, such as Oldroyd-type fluids, one can be certain that the numerical schemes to solve these equations are much more complicated, as the governing equations and the constitutive equations are to be solved simultaneously. Furthermore, the higher order terms in eq 18 require additional boundary conditions.

thought of as a limiting case of flow between two long cylinders where the inner cylinder is heated and the outer cylinder is rotating with a constant speed. This arrangement could be used to measure the viscosity of the material or to dry the particles, for example. We now make the following assumptions: (i) radiant heating r can be ignored; (ii) the density (or volume fraction), velocity, and temperature fields are of the form F ) F(y, t)

(25a)

u ) u(y, t)i

(25b)

θ ) θ(y, t)

(25c)

q ) q 1i + q 2 j + q 3k

(26)

6. Remarks It can be shown that if R2 ) R4 ) R5 ... ) R9 ) 0, eq 18 reduces to the equation derived by Fox,3 which isq˚ ) R1g + R3 q. If we furthermore make the assumptions that R1 ) -1/τ and R3 ) kR, then we obtain the Maxwell-Cattaneo model. Obviously, in both of these cases, the effects of density variation (density gradient) and velocity gradient are ignored. If the second order effects, i.e., those depending on D2 are ignored, then a reduced form can be obtained such that q˚ ) γ1g + γ2m + γ3q + γ4Dg + γ5Dm + γ6Dq (19) where now the γ’s depend on the (reduced) list of principal invariants. This can be rewritten as q˚ ) q˙ - Wq ) γ1g + γ2m + γ3q + γ4Dg + γ5Dm + γ6Dq(20) If, furthermore, we define, as is usually done in granular theories, a limiting equilibrium condition such that

(21) Df0 we obtain a much simplified implicit form for the constitutive relation for q˚ such that

Now, in general

where i, j, and k are unit vectors in the x, y, and z directions, respectively. With the above assumptions, the conservation of mass is automatically satisfied. It then follows that

( )

0 u′ 0 1 D ) u′ 0 0 , 2 0 0 0

(

)

0 (u′)2 0 1 2 D ) 0 (u′) 0 , 4 0 0 0 0 u′ 0 1 W ) -u′ 0 0 (27) 2 0 0 0 2

(

)

where prime denotes differentiation with respect to y. Also, notice that ∇F · ∇F )

( dFdy ) , 2

tr D ) 0

(28)

q˚ ) λ1g + λ2m + λ3q

(22)

With these the reduced form of q˚, given by eq 20, has three components which are

q˚ ) λ1∇θ + λ2∇F + λ3q

(23)

∂q1 γ4 γ6 ∂q1 u′ dθ +u - q2 - γ3q1 u′ u′q2 ) 0 ∂t ∂x 2 2 dy 2 (29a)

or

where it can be seen that we now have an additional term, namely λ2∇F, compared to that of Fox.3 Furthermore, to use this for granular materials, we can replace the vector m ) grad F by vectorn, where n ) grad ν is related to mthrough m ) grad F ) grad Fsν ) Fs grad ν ) Fsn

(24)

where we have assumed that Fs is constant. As an example, let us look at the heat transfer in a packed bed situated between two long horizontal parallel plates where the lower plate is fixed and heated and the upper plate is set into motion and is at a lower temperature than the lower plate (see Figure 2). As the flow starts, the particles begin to roll over each other and slide, and due to the nonlinear effect (dilatancy) the upper plate will have to move outward (upward) to allow for the motion of the particles. This problem can be

Figure 2

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()

()

∂q2 ∂q2 u′ dθ dF +u + q1 - γ1 - γ2 - γ3q2 ∂t ∂x 2 dy dy γ6 u′q1 ) 0(29b) 2

()

∂q3 - γ3q3 ) 0 (29c) ∂t It seems that in this form it is not possible to obtain an analytical solution to this system of equations for q. It is evident that the equations for the components of the heat flux vector are coupled with each other and, in addition, they would have to be solved with the conservation equations. However, if we assume that (i) F is constant, (ii) θ, q1, and q2 depend on time only and q3 ) 0, and that (iii) γ6 ) 0, then the above equations are significantly simplified and are reduced to ∂q1 1 - ξq2 ) - q1 ∂t τ

(30a)

∂q2 + ξq1 ∂t

(30b)

() 1 ) - ( )q τ

2

where ξ ) u′/2, and γ3 ) -1/τ are assumed to be constants. As shown by Fox,3 these two equations can be solved using the methods of complex analysis [if we writeq ) q1 + iq2, where i ) (-1)1/2], where

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[ ( 1τ + iξ)t]

q ) a exp -

(31)

where a is a (constant) complex number, which is the initial value of q. This equation implies that the magnitude of the heat flux vector q decays exponentially in time, and because of the frame-invariance requirements [see eq 15], its direction is in the same direction as the angular velocity, while depending on the shear rate ξ. It can be seen that for more complicated cases, where the above simplifying assumptions are removed, one must resort to numerical solutions of the equations. Nomenclature b ) body force vector B ) left Cauchy-Green tensor ) FFT D ) symmetric part of the velocity gradient F ) deformation gradient g ) grad θ l ) identity tensor k ) thermal conductivity L ) gradient of velocity vector m ) grad F q ) heat flux vector t ) time T ) Cauchy stress tensor u ) velocity W ) skew symmetric part of velocity gradient x ) spatial position occupied at time t ν ) volume fraction F ) bulk density F0 ) reference density θ ) temperature div ) divergence operator ∇ ) gradient symbol

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ReceiVed for reView September 9, 2009 ReVised manuscript receiVed November 6, 2009 Accepted November 13, 2009 IE9014155