Dynamical temperature and generalized heat-conduction equation

International Journal of Non-Linear Mechanics 79 (2016) 76–82

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Dynamical temperature and generalized heat-conduction equation I. Carlomagno, A. Sellitto, V.A. Cimmelli n Department of Mathematics Computer Science and Economics, University of Basilicata, Campus Macchia Romana, Viale dell'Ateneo Lucano, 10, 85100 Potenza, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 29 July 2015 Received in revised form 9 November 2015 Accepted 10 November 2015 Available online 21 November 2015

By means of a dynamical non-equilibrium temperature we derive a generalized heat-conduction equation which accounts for non-local, non-linear, and relaxation effects. The dynamical temperature is also capable to reproduce several enhanced heat equations recently proposed in literature. The heat flux is supposed to be proportional to the gradient of the dynamical temperature, and the material functions are allowed to depend on temperature. It is also pointed out that the heat flux cannot assume arbitrary values, but it is limited from above by a maximum value which ensures that the thermal conductivity remains positive. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear heat transport Thermomass theory Dynamical temperature Flux limiters

1. Introduction In Classical Irreversible Thermodynamics (CIT) [1] the Fourier law qi ¼  κθ;i ;

ð1Þ

relates the heat flux qi both to the gradient of the thermodynamic non-equilibrium temperature θ and to the temperaturedependent thermal conductivity κ. Second law of Thermodynamics forces the thermal conductivity to be a positive-definite function. For rigid bodies, the combination of Eq. (1) with the local balance of the energy in the absence of heat supply

ρu_ þ qi;i ¼ 0;

ð2Þ

with ρ as the mass density and u as the specific internal energy, provides the evolution equation for the temperature, once u is expressed as a function of θ by a constitutive equation. The use of Eq. (1) allows a satisfactory description of heat conduction in several situations at the macroscopic length scale, but it is unable to describe accurately the same phenomenon at nanoscale. In fact, whenever crystalline solids are confined to the nanometer range, heat transport within them is strongly altered by several phenomena as, for example, the phonon quantization, changes in phonon dispersion relation, increased boundary scattering, etc. The behavior of nanodevices is also influenced by nonlinear effects, which lack in the Fourier law. Furthermore, in micro/ n

Corresponding author. E-mail addresses: [email protected] (I. Carlomagno), [email protected] (A. Sellitto), [email protected] (V.A. Cimmelli). http://dx.doi.org/10.1016/j.ijnonlinmec.2015.11.004 0020-7462/& 2015 Elsevier Ltd. All rights reserved.

nano-devices working at high frequencies, the heat flux has no enough time to accommodate to the value given by the Fourier law, so that one has also to take into account the relaxation time of the heat flux [2–4]. As a consequence, several new heat-transport theories appeared in literature in the last three decades. Many of them have been analyzed in Ref. [5], wherein their connection with different approaches to non-equilibrium thermodynamics has been pointed out. Here we pursue the previous analysis by considering the thermomass (TM) theory [6–11], i.e., a recent heatconduction theory which in rigid bodies regards the heat as transported by a gas-like collection flowing through the crystal lattice. In the present paper we derive by proper thermodynamical methods a generalized heat-transport equation, primarily focusing our attention on the concept of non-equilibrium temperature. The correct definition of temperature at nanoscale is, in fact, a further interesting and not yet explored problem [12–14]. The usual definition of temperature, related to the average energy of a system of particles, is valid for systems in (or very close to) equilibrium, when the time derivative of the average energy is negligible. Once the interest lays in the transport of heat through a nanosystem, instead, one has to keep in mind that the system is in very strong non-equilibrium situations. In Refs. [15–19] both classical and enhanced heat-transport equations have been obtained by means of a dynamical temperature β which principally differs from θ by a frictional term which is responsible of the finite speed of propagation of thermal disturbances. The main differences between β, θ and the localequilibrium temperature T are carried out in Ref. [18]. Note further, that according to the terminology of CIT [1,20,21], β can be also regarded as an internal state variable [22–24].

I. Carlomagno et al. / International Journal of Non-Linear Mechanics 79 (2016) 76–82

The dynamical temperature has its own evolution equation which, in the very general case, reads

β_ ¼ f ðΣ Þ;

ð3Þ

where f is a regular function defined on the state space Σ . For instance, if θ and β enter the state space, Eq. (3) can be written as
 F θ; β β_ ¼ ; ð4Þ

τR

with F a suitable smooth function, and τR a temperaturedependent relaxation time related to resistive processes of interaction among the heat carriers, and thei order of magnitude of h which is in the interval 10  12 s; 10  9 s in the case of secondsound propagation at low temperature [5]. The conditions ∂F Z0; ∂θ

∂F r0; ∂β

1. in thermodynamic equilibrium, with θ equal to T [13,25], β is a regular function of T and the order relation r between different temperatures is preserved; 2. the solutions of Eq. (4) are stable. In the simplest case the previous conditions are realized by the following linear evolution equation:

β θ ; τR

ð6Þ

which will be generalized in Section 2 in order to take into account non-local effects. In Refs. [15,16,18] it is postulated that the heat flux is proportional to the gradient of β by means of the following Fourier-type heat-conduction law: qi ¼  λβ;i ;

ð7Þ

where λ denotes the thermal conductivity expressed as a function of the specific internal energy. The one-to-one relation between θ and u allows to set a strict relation between the thermal conductivity κ in Eq. (1) and the thermal conductivity λ in Eq. (7),

 namely, we have λðuÞ  κ θðuÞ . From the positivity of κ θ , it follows that the function λðuÞ is positive definite, too. Once the evolution equation for β has been derived, different heat-transport regimes can be described. For instance, whenever the material functions τR and λ can be considered constant (for example, when they only show slight changes with the temperature), the combination of Eqs. (7) and (6) leads to the well-known Maxwell–Cattaneo equation [18,19]

τR q_ i þ qi ¼  λθ;i ;

In Section 4, we derive a generalized heat-conduction equation accounting for non-linear and relaxation effects. In Section 5 we show that our generalized heat-conduction equation encompasses the heat-conduction equation of the thermomass theory [6–11]. Finally, in Section 6, we discuss the main results through the prism of the concept of non-equilibrium temperature. After observing that the heat flux cannot assume arbitrary values, but it is limited from above by a maximum value which ensures that the thermal conductivity remains positive, we show that such a limit value is determined by the thermodynamic absolute temperature as well as by the dynamical one.

2. Nonlinear gradient-dependent transport law ð5Þ

ensure, respectively, that:

β_ ¼ 

77

ð8Þ

which provides a satisfactory description of high-frequency thermal waves. Although Eq. (8) is capable to describe relaxation effects, it does not account for non-linear effects, which may enter the heat-transport equation both as a temperature dependence in the material functions, and as the presence of non-linear products of the temperature gradient (or of the heat flux). These effects, instead, will be described by our generalized heat equation. The paper runs as follows. In Section 2, we develop the aforementioned model by establishing the state space and the evolution equations characterizing the system at hand. Moreover, we exploit the entropy principle in order to obtain necessary and sufficient conditions ensuring the compatibility of the model with second law of thermodynamics [1]. In Section 3, we provide a solution of the system of thermodynamic restrictions found in Section 2, proving so that the model is physically admissible.

In this section we postulate a suitable evolution equation for β and analyze its compatibility with second law of thermodynamics [1]. Before to proceed in this analysis, let us observe that the considerations of previous section suggest that in an equilibrium system, where the specific internal energy u is a function of the thermodynamic local-equilibrium absolute temperature T, the use of both u and β would be redundant. However, out of equilibrium, where u does not display the same distribution as in equilibrium, β is a truly independent quantity, not redundant with u. Thus, we are allowed to consider a rigid heat conductor at rest, characterized by the following state space: n o Σ ¼ u; β; β;i ; where, in view of Eq. (7), the further state variable β;i means that the heat flux enters the state space. This is in accordance with the basic tenets of Extended Irreversible Thermodynamics (EIT), a recent thermodynamic theory in which includes the dissipative fluxes in the set of the independent thermodynamic variables [1,12,26]. Then, an evolution equation for β of the form (3) can be postulated. Here we suppose that u β A β_ ¼  þ β;i β;i ; ð9Þ σ τ 2 wherein σ, τ and A are regular scalar functions of u. Eq. (9), besides accounting for the possible variations of the different thermophysical quantities with respect to the temperature, enhances Eq. (6) by extending the evolution of β in the realm of weakly nonlocal thermodynamics. Hence, if for example, one assumes
 τðuÞ ¼ τR θ and σ  cv τR with cv being the specific heat at constant volume, then Eq. (9) becomes ðβ  θ Þ A β_ ¼  þ β;i β;i ; τR 2

ð10Þ

which is indeed physically superior to Eq. (6) since, according to Eq. (6) β reduces to θ in non-equilibrium steady states, whereas if Eq. (10) holds, in steady states θ and β are still related by a partial differential equation. Thus, Eq. (9) (or, equivalently, Eq. (10)) describes not only the usual aspect of β dependent explicitly on time, but also a new aspect dependent on its gradient. This feature is related to the fact that in the presence of a gradient the system is locally receiving an energy input and it is yielding an energy output to the nearby regions. This implies a dynamical aspect even in the steady states, where energy input rate is equal to the energy output rate. By means of the constitutive equation (7), Eq. (2) can be rewritten as

ρu_  β;k u;k

∂λ  λβ;ik δik ¼ 0; ∂u

ð11Þ

78

I. Carlomagno et al. / International Journal of Non-Linear Mechanics 79 (2016) 76–82

wherein δik ¼ δki is the Kronecker symbol. The evolution of the state variable β ;i , instead, follows from the gradient extension of Eq. (9), namely, ! β; 1 u ∂σ β ∂τ β β ∂A β_ ;i ¼  2 þ 2 þ ;k ;k ð12Þ u;i  i þ Aβ;k β;ki : σ σ ∂u τ ∂u 2 ∂u τ Second law of Thermodynamics dictates that locally in space and time the entropy inequality

ρs_ þ J k;k Z 0;

ð13Þ

holds, with s being the specific entropy, and Jk being the components of the entropy flux [1,12,27]. These quantities have to be assigned by constitutive equations, whose most general form is     s ¼ s u; β; β ;i ; J k ¼ J k u; β ; β;i : Then, on the state space the entropy inequality (13) becomes ∂s

∂s

∂s

∂J

∂J

∂J

ρ u_ þ ρ β_ þ ρ β_ ;k þ k u;k þ k β;k þ k β;ik Z 0: ∂u ∂u ∂β ∂ β ;i ∂β ∂ β ;k

ð14Þ

The consequences of the inequality (14) can be investigated by applying the generalized exploitation procedures developed in Refs. [28–30]. According to the aforementioned procedures, beside the evolution equations of the unknown quantities, also their gradient extensions, up to the order of the spatial derivatives entering the state space, must be substituted into the entropy inequality. Thus, as a first step, we substitute the governing equations (2), (9) and (12) into the inequality (14). It is easy to verify that the generalized entropy inequality so obtained is linear in the highest spatial derivatives u;k and β ;ik . Those terms are not included in the state space. Since they can take arbitrary values [1,30], their coefficients have to vanish, otherwise the inequality (14) could be easily violated. If we put equal to zero those coefficients, we get the thermodynamic restrictions "   # ! ∂s ∂λ β ∂τ u ∂σ β β ∂A ∂s ∂J ð15aÞ ρ β;k þ 2  2 þ ;i ;i þ k ¼ 0; ∂u ∂u 2 ∂u ∂β ;k ∂u τ ∂u σ ∂u *

ρ λδki

∂s ∂s þ A β ;i ∂u ∂ β ;k

! þ

∂J k ∂ β ;i

+ ¼ 0;

ð15bÞ

 where the symbol T ki denotes the symmetric part of the secondorder tensor Tik. Once the two relations above have been satisfied, the inequality (14) results in the reduced (or residual) entropy inequality " # ! u β Aβ;i β;i ∂s β ;k ∂s ∂J ρ  þ þ ð16Þ þ k Z 0β ;k ; σ τ 2 ∂β ∂ β τ ∂ β ;k which is a unilateral differential constraint defined on the state space and represents the local rate of entropy production along the process. It is worth observing that the left-hand side of inequality (16) includes only those terms of the entropy inequality (14) which do not contain any highest derivative, namely, the second term and the fifth term. Moreover, the second term therein has been rewritten by taking into account the constitutive equation for β_ given in Eq. (10). The thermodynamic restrictions (15) and the residual entropy inequality (16) constitute the set of necessary and sufficient conditions to ensure that second law of Thermodynamics is satisfied along arbitrary processes. In the next section we exploit them in order to obtain useful information on the model.

3. Solution of the system of thermodynamic restrictions In this section we look for a solution of the system (15) and (16) which is simple, is in accordance with the mathematical theory of representation of isotropic functions, and ensures that the principle of maximum entropy at the equilibrium is satisfied in any process [1,23]. We observe that although we have already given a constitutive equation for the heat flux in terms of state variables, so far no any hypothesis on the specific entropy and on the entropy flux has been made. Then, we postulate for these quantities the following constitutive equations:

 s1 u; β s ¼ s0 u; β  β;i β;i ; ð17aÞ 2 h

i ð17bÞ J k ¼ J 0 u; β β ;i β ;i þ K 0 u; β β;k ;



 wherein s0 u; β , s1 u; β , J 0 u; β and K 0 u; β are regular scalar functions whose form will be determined by thermodynamic
 considerations. Furthermore, s1 u; β is positive definite, so that at the local thermodynamic equilibrium (i.e., when the state variables are homogeneous and β reduces to the local-equilibrium temperature T) the total entropy of the system Z SðC Þ ¼ ρseq dc; ð18Þ C

where C is the material volume occupied by the system and
 seq ¼ s0 ueq ; T , reaches its maximum value [1]. It is worth noting that if in Eq. (17b) we set
 J 0 u; β ¼ 0;


 λ K 0 u; β ¼  ;

θ

then the use of the constitutive assumption (7) turns out the classical constitutive equation for the entropy flux of Rational
 Thermodynamics (RT) J k ¼ qk =θ [1]. If, instead, J 0 u; β a 0, and
 still K 0 u; β ¼  λ=θ, then the constitutive equation of the entropy flux proposed by Müller [31], namely J k ¼ qk =θ þ K k , where Kk is known in the literature as entropy extraflux [1], is also recovered. Due to the constitutive equations above, the model is completely determined by the eight material functions λðuÞ, τðuÞ, σ ðuÞ,



 AðuÞ, s0 u; β , s1 u; β , J 0 u; β , and K 0 u; β . It is easily verified by direct substitution that the two relations

    1 ∂s0 ∂λ u ∂s1 ∂A ∂J  0 ¼ 0; þ 2 þ s1 ð19aÞ 2 ∂u ∂u ∂u ∂u σ ∂u      ∂s0 ∂λ u ∂s0 β ∂τ ∂K 0 þs1 2  2 þ ¼ 0; ∂u ∂u ∂u σ ∂u τ ∂u

ð19bÞ

are sufficient to satisfy Eq. (15a). On the other hand, by the same procedure one may verify that the following conditions ∂s0 ∂K 0 ¼ ; ∂u ∂u

λ

∂s1 ¼ As1 ¼ 2J 0 ; ∂u

λ

imply the other one ! ∂s ∂s ∂J ρ λδik þ Aβ;i þ k ¼ 0; ∂u ∂β ;i ∂β ;k

ð20Þ

ð21Þ

so that, a fortiori, the restriction (15b) is also satisfied. Finally, the further inequalities   u β ∂s0  Z0; ð22aÞ σ τ ∂β   ∂s0 u β ∂s1 2s1   Z ; A σ τ ∂β τ ∂β

ð22bÞ

∂J 0 A ∂s1 Z ; ∂β 4 ∂ β

ð22cÞ

I. Carlomagno et al. / International Journal of Non-Linear Mechanics 79 (2016) 76–82

allow to satisfy the residual entropy inequality (16). It is worth observing that, according to the classical postulate of Colemann and Noll [27], the constraints (19)–(22) restrict the material functions and not the thermodynamic processes [1,25,32]. On the other hand, the model presented here can fulfill the conditions in Eqs. (19)–(22), since it has just eight free material parameters. This allows us to conclude that it is physically realizable, since it does not contradict second law of thermodynamics.

4. Generalized heat-conduction equation In order to obtain a generalized partial differential equation which governs the evolution of the heat flux, let us differentiate the constitutive relation (7) with respect to time, and then substitute in it Eq. (12). Under the additional hypothesis u ¼ cv θ, with the specific heat cv positive and constant, after some straightforward manipulations we get ! c τ θ_ ∂λ Aτ q þ qq τq_ k þ qk  v λ ∂u k λ i i;k
  

 β  θ ∂τ cv τ ∂A qi qi θ;k ; þ 2 ¼  λ 1 þ cv ð23Þ τ ∂u 2λ ∂u

79

equilibrium thermal conductivity, defined as


 β  θ ∂τ R κ 0neq ¼ κ 1 þ ; τ R ∂θ



ð28Þ

which is affected by the local term on the right-hand side of Eq. (10), but not on the non-local one. Its effects on the heatconduction phenomenon will be discussed in Section 6. If either τR is constant, or β C θ, then Eq. (24) results in ! c τ θ_ ∂κ Bτ R τR q_ k þqk  v R q þ q q ¼  κ ″neq θ;k ; ð29Þ κ κ i i;k ∂θ k wherein the non-equilibrium thermal conductivity is now dependent on the thermodynamic absolute temperature as well as on the heat flux, and is given by



  τ ∂B τ ∂B R ð30Þ κ ″neq ¼ κ 1 þ R2 qi qi ¼ κ 1 þ β;i β;i : 2 ∂θ 2κ ∂ θ We see that κ ″neq is affected by the non-local term on the righthand side of Eq. (10), but not on the local one. The properties of Eq. (29) will be also explored in Section 6.

5. Comparison with the formalism of thermomass theory

if we set σ ðuÞ ¼ cv τ ðuÞ. Since the one-to-one correspondence between u and θ allows to express all the thermo-physical quantities as functions of θ instead of u, then Eq. (23) can be finally written as ! c τ θ_ ∂κ Bτ R τR q_ k þqk  v R q þ qq κ i i;k κ ∂θ k


   β  θ ∂τR τR ∂B ¼ κ 1þ þ 2 ð24Þ qi qi θ;k ; τR ∂θ 2κ ∂θ

The thermomass (TM) theory [6–11] regards the heat as transported by a gas-like collection flowing through the medium, and characterized by an effective mass density ρh . The local balances of mass and linear momentum for the heat carriers which constitute that gas collection, named thermons, take the same form of those of classical fluid mechanics, i.e.,

once we defined the following functions of the temperature:


 τR θ  τðuÞ; κ θ  λðuÞ; B θ  AðuÞ: ð25Þ

respectively, wherein vi is drift velocity of the thermons, p is the gas pressure induced by the thermal vibration of the lattice, and f i is the frictional force per unit of volume. The coupling of Eqs. (31) with the constitutive equations

Eq. (24) represents the general heat-conduction equation we are looking for. As a consequence of the evolution equation of β, Eq. (24) contains the non-equilibrium thermal conductivity


   β  θ ∂τR τR ∂B qi qi κ neq ¼ κ 1 þ þ 2 τ R ∂ θ 2κ ∂ θ


   β  θ ∂τR τR ∂B þ β;i β;i ; ð26Þ ¼ κ 1þ τ R ∂θ 2 ∂θ from which it is easy to see the influence of the different terms on the right-hand side of Eq. (10). Some considerations are in order.

 For vanishing τR , Eq. (24) reduces to the classical Fourier law 



with temperature-dependent thermal conductivity, namely, Eq. (1). If the relaxation time τR can be considered constant (i.e., if its variations with temperature can be neglected), and if the nonlinear terms are negligible in Eq. (24), then it reduces to the classical Maxwell–Cattaneo equation with temperaturedependent thermal conductivity. If τR is no longer constant, but the non-linear terms are still negligible, then Eq. (24) yields

τR q_ k þ qk ¼  κ 0neq θ;k ;

ð27Þ

which is a semi-linear Maxwell–Cattaneo equation, with nonlinearities only due to the temperature dependence of the thermo-physical quantities κ and τR . In it, κ 0neq is a non-

ρ_h þ ρh;i vi þ ρh vi;i ¼ 0; 

ð31aÞ



ρh v_ i þ vi;j vj ¼  p;i  f i ;

u ¼ ρc v θ ;

vi ¼

qi

ρc v θ

;

p¼

ð31bÞ

γρh c2 ; ρ

f i ¼ ωv i ;

being c the modulus of the light speed, γ the Grüneisen constant, and ω a suitable coefficient describing the resistance of the crystal to the flux of thermons, leads to the following heat-conduction equation:

τtm q_ i  ρcv θ_ li þ qi;j lj þ κ ð1  bÞθ;i þ qi ¼ 0;

ð32Þ

wherein τtm ¼

κ ; 2γρc2v θ

ð33Þ

is the relaxation time in the TM theory, and li ¼

qi κ 2γcv ðρcv θÞ2

≡τtm vi ;

ð34Þ

denotes a length vector which characterizes the strength of the non-Fourier effects introduced by Eq. (32). Those effects become very important when the time scale is comparable to τtm , or the spatial scale is comparable to the modulus of the length vector above [11], from which we infer that the modulus of li plays the role of a heat-flux dependent mean-free path, which is analogous to the role played by the phonon mean-free path in phonon hydrodynamics. However, such a modulus does not coincide with the mean-free path of the thermons, since in Eq. (34) τtm is not able to account for the different mechanisms of collisions between them. Moreover, experimental evidences show that the mean-free

80

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path of the heat carriers depends on the temperature, but not on the its gradient (i.e., on the heat flux) as li in Eq. (34). Finally in Eq. (32) b¼

q2
3 ; 2γρ2 cv θ

ð35Þ

wherein q is the modulus of qi, stands for a dimensionless number which is also called thermal Mach number of the drift velocity relative to the thermal-wave speed in the phonon gas. Beside to describe relaxation effects, Eq. (32) incorporates information on the characteristic length of the system (i.e., nonlocal effects) and accounts for non-linear effects [9–11]. Note that although the magnitude of li in Eq. (32)increases with increasing values of q, it remains bounded owing to the presence of the socalled flux limiters [12,33–35]. In Section 6 we deserve deeper comments to their physical meaning. At this point, it is important to observe that in the present paper we do not aim to analyze either the microscopic framework leading to Eq. (32), or the physical consistency of the constitutive equations (33)–(35). Here we regard Eq. (32) simply as a phenomenological model which accounts for non-linear effects arising from the physical dimension of the system, which are frequent at nanoscale. Bearing in mind this point, we investigate the compatibility of Eq. (32) with the general tenets of continuum physics [36], as already made in Ref. [5] by one of us with other heatconduction equations. Such investigation generalizes that performed in Ref. [37], where a continuum model of Eq. (32) has been developed under the hypothesis of heat propagation along a single spatial direction. The TM heat-conduction equation (32), which by means of Eqs. (34) and (35) can be also written as [6–11] ! ! _ κ θ κ τtm q_ k þ qk  qk þ qi qi;k 2 2 2γρc2v θ 2γρ2 c3v θ ! q2 ¼ κ 1 θ ;k ; ð36Þ 3 2γρ2 c3v θ shows the same mathematical structure of Eq. (24). In particular, whenever the relaxation time undergoes negligible variations with the temperature (or when β C θ), by direct comparison of Eqs. (29) and (36), one may infer that they coincide if, and only if, the following relations hold:

τtm ¼ τR ; τ  ∂ κ κ R ¼ ; κ ∂θ 2γρc3 θ2

κ2 ; 2γρ θ τ

∂B κ2 ¼ : 2 ∂θ γρ c3v θ3 τtm

B0

θ

2

;

κ ¼ κ0 

B0 ρ

θ

τtm ¼

κ2 ; 2B0 γρ2 c3v

If a continuum system which is isolated from its surroundings is left undisturbed for an indefinitely long time, it reaches a state in which neither changes, nor flows occur within it. This condition is named thermodynamic state of equilibrium. The experiments show that when a body in non-equilibrium is isolated, it spontaneously evolves towards an equilibrium state, in which its entropy takes a maximum. Sometimes, it is useful to distinguish between global and local thermodynamic equilibrium. A thermodynamic system is said to be in a state of global equilibrium, if all its points are in the same equilibrium state. Instead, it is said to be in a state of local equilibrium, if for any of its points there exists an equilibrium state in some neighborhood of that point. Temperature is a subtle concept in non-equilibrium states, since in fast phenomena it necessarily differs from the local-equilibrium one [12,26], which is defined experimentally through the Zeroth Law, and, theoretically, as the integrating factor of the differential form dQ, with Q as the heat exchanged in a thermodynamic process by a continuous system C with an exterior heat reservoir [38]. Such an integrating factor exists if, and only if, any cyclic process can be realized in a quasistatic way, so that the system passes through equilibrium states only. In such a case the differential form dS¼dQ/T is exact, and the entropy S can be defined as a state function depending only on the internal energy [38]. The same property holds for the local-equilibrium specific entropy seq , which is such that the total entropy of C is given by Eq. (18). In this way, the local-equilibrium temperature can be defined as dseq : du

On the other hand, in fast phenomena the cyclic processes cannot be approximated by quasi-static transformations, and so the entropy as state function is no longer a well-defined quantity. In these situations, a valuable help is given by the principle of local equilibrium [20,21], which assumes that outside the equilibrium, locally in space and time, the thermodynamic systems can be described by the same state functions which are well-defined at the thermodynamic equilibrium. Thus, the definition of temperature can be extended to non-equilibrium situations, as a chart of local mappings [25]. In heat conduction beyond the Fourier law, the specific entropy is no longer depending on the internal energy only, but it depends as well on the heat flux. For instance, in EIT such a generalized entropy is proved to be [1,12,18,19,26,39]

ð37dÞ

;

6.1. Nonequilibrium temperature

ð37bÞ

The relations above can be solved by straightforward calculations, and lead to the following expressions of B, κ, and τtm : B¼

As final remarks, this section includes more comments about the concepts of non-equilibrium temperature, flux limiters, and rescaled flux variable.

T 1 ¼

ð37cÞ

2 c3 2 tm v

6. Conclusive remarks

ð37aÞ

v

B¼


 positive. In this range, κ 0 ¼ limθ-1 κ θ represents the maximum value of the thermalconductivity. Then, if θ0 is known, B0 can be determined by measuring the heat conductivity as a function of the temperature.

ð38Þ

where B0 and κ 0 are two positive constants. In the second of the
 1 equations above nothing prevents to write κ 0 ¼ B0 ρ θ0 , with θ0 as a suitable constant temperature, in such a way that   κ ¼ B0 ρ θ0 1  θ  1 : This notation makes evident that the model can be applied in the range of temperatures θ 4 θ0 , in which the heat conductivity is

s ¼ seq ðuÞ 

τ

2λT 2

q2 ;

ð39Þ

so that, the non-equilibrium temperature reads

θ¼

  1 dseq 1 d τ  : q2 2 du 2 du λT

ð40Þ

I. Carlomagno et al. / International Journal of Non-Linear Mechanics 79 (2016) 76–82

A similar expression has been obtained in TM theory [10], unless for the numerical factor 1=2 in front of the non-equilibrium term. Such a difference arises since in TM theory the internal energy is defined by a static temperature θ, where the drift kinetic energy which is present in T has been removed [10]. On the other hand, the specific entropy (17a) is a little bit more general with respect to that in Eq. (39), since it can be put in the form

 s1 u; β 2 s ¼ s0 u; β  q : ð41Þ 2 2λ It is easy to verify that the thermodynamic restrictions ((19)–(22) do not allow s0 and s1 to depend on the internal energy only. However, nothing prevents s0 and s1 to take the form

 s0 ¼ s00 ðuÞ þ s01 β and s1 ¼ s10 ðuÞ þ s11 β . If the previous relations hold, s can be rewritten as s ¼ sEIT þ sD , where sEIT ¼ s00 ðuÞ 

s10 ðuÞ 2λ

2

q2 ;

is the EIT part of the specific entropy, while

 s11 β 2 sD ¼ s01 β  q ; 2 2λ

ð42Þ

ð43Þ

represents the additional entropy due to the presence of the dynamical temperature in the state space. Moreover, if both the dependence of s00 on its independent variable u is mathematically the same of s01 on β, and similar conclusions hold for s10 and s11, then, whenever β reduces to T, Eq. (40) results in

1 ds ds ðT Þ θ ¼ eq  1 q2 ; ð44Þ du du with seq ðT Þ ¼ 2s00 ðT Þ, and s1 ðT Þ ¼ s10 ðT Þ. Eq. (44) yields the temperature of TM theory  non-equilibrium  [10], once s1 ðT Þ ¼ τ= λT 2 . Thus, we are led to the conclusion that our model allows to recover the non-equilibrium temperature of TM theory [10] when β reduces to θ. However, in the general case, the non-equilibrium temperature arising in the present model is far more general, and does not coincide either with the localequilibrium temperature T, or with the dynamical temperature β, but depends on both these quantities, as well as on the heat flux, and is given by     1



∂s0 u; β ∂ s1 u; β T¼  q2i : ð45Þ 2 ∂u ∂u λ 6.2. Flux limiters Another interesting consequence of Eq. (24) is the presence of internal constraints that do not allow the heat flux to exceed a given limit value, beyond which thethermalconductivity would become negative [33]. Let us observe that, due to the identifications (37), Eq. (24) can be put in the form ! ! _ κ θ κ τtm q_ k þ qk  ð46Þ qk þ qi qi;k ¼  κ neq θ;k ; 2 2 2γρc2v θ 2γρ2 c3v θ wherein κ neq denotes now the following non-equilibrium generalized thermal conductivity " # !  2
q2 2β q2 κ neq ¼  κ 1 þ β  θ  1  κ ; 3 3 θ θ 2γρ2 c3v θ 2γρ2 c3v θ ð47Þ which requires an upper bound for the heat flux, since second law of Thermodynamics dictates κ neq Z 0. The experimental results in silicon nanowires, for a difference of temperature ΔT ¼ 100 K, confirm the existence of this upper bound [40]. Such a phenomenology, well-known in non-linear heat conduction, is referred to

81

as the presence of “flux limiters” [12], which are due to the finite speed of the thermal perturbations. Indeed, for a given energy density, the heat flux cannot reach arbitrarily high values, but it has to be bounded by a maximum saturation value of the order of the energy density times the maximum speed. Typical situations of flux limiters arise, for instance, in radiative heat transfer, or in plasma physics [35,34]. This phenomenon is especially evident when the phonon mean-free path ℓp is such that ℓp o d, with d characteristic dimension of the system. In such a case the spatial derivatives of the heat flux are negligible with respect to the flux itself so that, in steady states, Eq. (24) reduces to qk ¼  κ neq θ;k :

ð48Þ

On the other hand, second law of thermodynamics dictates that the local rate of entropy production 

1

θ2

qk θ ;k ;

ð49Þ

is always non-negative. Thus, the coupling of Eqs. (48) and (49) leads to the conclusion that κ is always positive. Owing to the positiveness of κ neq , from Eq. (47) it follows that the local heat flux should not exceed the following limit value:   3 2β q2lim ¼ 2γρ2 c3v θ 1 ; ð50Þ

θ

which shows that the maximum admissible value for the heat flux is affected by the difference between the absolute temperature and the dynamical one. For β  θ, the limit value of the heat flux in the TM theory is recovered [33]. The discussion above shows that the TM heat-conduction equation is a useful phenomenological tool in the description of those non-linear phenomena in which a maximum value of the heat flux is expected as, for instance, the radiative heat transfer. These phenomena show that an upper limit for the heat flux is indeed conceivable from the physical point of view, so that a nonequilibrium thermal conductivity of the form (47) is physically admissible. On the other hand, the general heat-conduction equation (24) does not necessarily impose any “a priori” limitation on the heat flux, and this proves that its validity involves a thermodynamic framework which is more general with respect to that of TM theory. Indeed, a limitation on the maximum value of the heat flux also arises from Eq. (39), or Eq. (40), in order to ensure that specific entropy and the absolute non-equilibrium temperature, respectively, does not become negative. These restrictions are theoretically different from that arising from the statement κ neq Z 0, but, since for a given thermodynamic process it is conceivable only one limit in the heat flux, those restrictions can be used to infer interesting information. For example, in the situation described above, from Eq. (39), or Eq. (40), one may check the value of qlim for the process at hand and then, Eq. (50) can be used to evaluate the difference between θ and β. Of course, these limits are basically conceptual, because all formalisms have their own degree of approximation, which could be superseded by more general, higher-order expressions. 6.3. Rescaled flux variable As last remark, we point out the relation between the present model and a previous model developed in Ref. [37] by supposing the heat flux proportional to a vectorial internal variable ci. In that model the state space was

Σ ¼ θ ; ci ;
 while the relation between qi and ci was qi ¼ κ =τtm ci . Thus, a

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comparison with Eq. (7) yields β ;i ¼ ci =τtm , and allows to regard β ;i as a vectorial internal variable too. It is worth observing that in Ref. [37] the evolution of ci is ruled by a balance law, so that it depends also on the higher-order space derivatives of the elements of the state space. Hence, according to the way paved in Refs. [26,41], the internal variable is reinterpreted, in a more modern sense, as a rescaled flux within the frame of EIT. Analogously, although Eq. (12) is not in the classical balance form, it contains the highest-order spatial derivatives u;k and β ;ik , so that it can be also regarded as a rescaled flux in a general sense. This proves that the model presented here does not contradict that developed in Ref. [37], but simply extends it by introducing in the state space a dynamical temperature, beside the absolute temperature. This gives a deeper insight on the concept of nonequilibrium temperature, and also affects the maximum admissible value of the heat flux.

Acknowledgments The authors acknowledge the financial support from the University of Basilicata, Potenza, from the University of Salento, Lecce (I.C.), and from the Italian Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM). In memory of Prof. Witold Kosiński, unforgettable teacher, the first who introduced the dynamical temperature β, [15].

References [1] V.A. Cimmelli, D. Jou, T. Ruggeri, P. Ván, Entropy principle and recent results in non-equilibrium theories, Entropy 16 (2014) 1756–1807. [2] F.X. Alvarez, D. Jou, Memory and nonlocal effects in heat transports, Appl. Phys. Lett. 90 (2007) 083109 (3 pp.). [3] D. Jou, J. Casas-Vázquez, G. Lebon, M. Grmela, A phenomenological scaling approach for heat transport in nano-systems, Appl. Math. Lett. 18 (2005) 963–967. [4] F.X. Alvarez, D. Jou, Size and frequency dependence of effective thermal conductivity in nanosystems, J. Appl. Phys. 103 (2008) 094321 (8 pp.). [5] V.A. Cimmelli, Different thermodynamic theories and different heat conduction laws, J. Non-Equilib. Thermodyn. 34 (2009) 299–333. [6] B.-Y. Cao, Z.-Y. Guo, Equation of motion of a phonon gas and non-Fourier heat conduction, J. Appl. Phys. 102 (2007) 053503 (6 pp.). [7] D.Y. Tzou, Z.-Y. Guo, Nonlocal behavior in thermal lagging, Int. J. Therm. Sci. 49 (2010) 1133–1137. [8] Y. Dong, B.-Y. Cao, Z.-Y. Guo, Generalized heat conduction laws based on thermomass theory and phonon hydrodynamics, J. Appl. Phys. 110 (2011) 063504 (6 pp.). [9] Y. Dong, B.-Y. Cao, Z.-Y. Guo, General expression for entropy production in transport processes based on the thermomass model, Phys. Rev. E 85 (2012) 061107 (8 pp.). [10] Y. Dong, B.-Y. Cao, Z.-Y. Guo, Temperature in nonequilibrium states and nonFourier heat conduction, Phys. Rev. E 87 (2013) 032150 (8 pp.). [11] M. Wang, N. Yang, Z.-Y. Guo, Non-Fourier heat conductions in nanomaterials, J. Appl. Phys. 110 (2011) 064310 (7 pp.). [12] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, fourth revised ed., Springer, Berlin, 2010.

[13] J. Casas-Vázquez, D. Jou, Nonequilibrium temperature versus local-equilibrium temperature, Phys. Rev. E 49 (1994) 1040–1048. [14] J. Casas-Vázquez, D. Jou, Temperature in nonequilibrium states: a review of open problems and current proposals, Rep. Progr. Phys. 66 (2003) 1937–2023. [15] W. Kosiński, Elastic waves in the presence of a new temperature scale, in: M.F. McCarthy, M. Hayes (Eds.), Elastic Wave Propagation, Elsevier Science (North Holland), Amsterdam, 1989, pp. 629–634. [16] V.A. Cimmelli, W. Kosiński, Nonequilibrium semi-empirical temperature in materials with thermal relaxation, Arch. Mech. 43 (1991) 753–767. [17] V.A. Cimmelli, K. Frischmuth, Determination of material functions through second sound measurements in a hyperbolic heat conduction theory, Math. Comput. Modell. 24 (1996) 19–28. [18] V.A. Cimmelli, A. Sellitto, D. Jou, Nonlocal effects and second sound in a nonequilibrium steady state, Phys. Rev. B 79 (2009) 014303 (13 pp.). [19] V.A. Cimmelli, A. Sellitto, D. Jou, Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations, Phys. Rev. B 81 (2010) 054301 (9 pp.). [20] S.R. De Groot, P. Mazur, Nonequilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. [21] I. Gyarmati, Nonequilibrium Thermodynamics, Springer, Berlin, 1970. [22] B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47 (1967) 597–613. [23] J. Verhás, Thermodynamics and Rheology, Kluwer Academic Publisher, Dordrecht, 1997. [24] A. Morro, T. Ruggeri, Second sound and internal energy in solids, Int. J. NonLinear Mech. 22 (1987) 27–36. [25] W. Muschik, C. Papenfuss, H. Ehrentraut, A sketch of continuum thermodynamics, J. Non-Newtonian Fluid Mech. 96 (2001) 255–290. [26] G. Lebon, D. Jou, J. Casas-Vázquez, Understanding Non-equilibrium Thermodynamics, Springer, Berlin, 2008. [27] B.D. Coleman, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal. 13 (1963) 167–178. [28] V.A. Cimmelli, A. Sellitto, V. Triani, A new thermodynamic framework for second-grade Korteweg-type viscous fluids, J. Math. Phys. 50 (2009) 053101 (16 pp.). [29] V.A. Cimmelli, A. Sellitto, V. Triani, A new perspective on the form of the first and second laws in rational thermodynamics: Korteweg fluids as an example, J. Non-Equilib. Thermodyn. 35 (2010) 251–265. [30] V.A. Cimmelli, F. Oliveri, V. Triani, Exploitation of the entropy principle: proof of Liu theorem if the gradients of the governing equations are considered as constraints, J. Math. Phys. 52 (2011) 023511 (16 pp.). [31] I. Müller, On the entropy inequality, Arch. Ration. Mech. Anal. 26 (1967) 118–141. [32] W. Muschik, H. Ehrentraut, An amendment to the second law, J. Non-Equilib. Thermodyn. 21 (1996) 175–192. [33] A. Sellitto, V.A. Cimmelli, Flux limiters in radial heat transport in silicon nanolayers, ASME J. Heat Transf. 136 (2014) 071301 (6 pp.). [34] A.M. Anile, S. Pennisi, M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Phys. 32 (1991) 544–550. [35] C.D. Levermore, Relating Eddington factors to flux limiters, J. Quant. Spectrosc. Radiat. Transf. 31 (1984) 149–160. [36] Y. Dong, Z.-Y. Guo, Entropy analyses for hyperbolic heat conduction based on the thermomass model, Int. J. Heat Mass Transf. 54 (2011) 1924–1929. [37] A. Sellitto, V.A. Cimmelli, A continuum approach to thermomass theory, ASME J. Heat Transf. 134 (2012) 112402 (8 pp.). [38] M.W. Zemanski, R.H. Dittman, Heat and Thermodynamics, seventh revised ed., McGraw-Hill, New York, 1997. [39] G. Lebon, A thermodynamic analysis of rigid heat conductors, Int. J. Eng. Sci. 18 (1980) 727–739. [40] M. Wang, B.-Y. Cao, Z.-Y. Guo, General heat conduction equations based on the thermomass theory, Front. Heat Mass Transf. 1 (2010) 013004 (8 pp.). [41] G. Lebon, M. Ruggieri, A. Valenti, Extended thermodynamics revisited: renormalized flux variables and second sound in rigid solids, J. Phys.: Condens. Matter 20 (2008) 025223 (11 pp.).