- Email: [email protected]
A new version of extended irreversible thermodynamics and dual-phase-lag model in heat transfer
12 March 2001
Physics Letters A 281 (2001) 16–20 www.elsevier.nl/locate/pla
A new version of extended irreversible thermodynamics and dual-phase-lag model in heat transfer S.I. Serdyukov Department of Chemistry, Moscow State University, 119899 Moscow, Russia Received 5 June 2000; received in revised form 22 January 2001; accepted 1 February 2001 Communicated by A.R. Bishop
Abstract For a system involving heat transfer, a new postulate of extended irreversible thermodynamics is suggested, which, in case of a stationary states, reduces to the local equilibrium hypothesis. On the basis of this postulate, a transport equation of the dual-phase-lag model of heat transfer is derived. In the proposed formalism, an extended evolution criterion generalizing the known Glansdorff–Prigogine criterion takes place. 2001 Elsevier Science B.V. All rights reserved. PACS: 05.60.+w; 44.60.+k; 05.70.Ln Keywords: Extended irreversible thermodynamics; Heat transfer; Dual-phase-lag model; Evolution criterion
As ε = 0, Eq. (1) reduces to the Maxwell–Cattaneo equation
1. Introduction The linearised dual-phase-lag model of heat transfer [1] is based on the equation ∂∇T ∂q t + q = −λ∇T − ε , (1) ∂ ∂t where T is the equilibrium temperature, λ the heat conductivity, q the heat flux, τ and ε are relaxation times. Using the balance equation, τ
∂T (2) = −∇ · q, ∂t with ρ the mass density, c the heat conductivity per unit mass, let us transform (1) to the following form:
ρc
∂ 2T ∂T ∂∇ 2 T 2 = κ∇ , + T + ε ∂t ∂t ∂t 2 where κ = λ/(ρc) is the thermal diffusivity. τ
(3)
E-mail address: [email protected] (S.I. Serdyukov).
∂q + q = −λ∇T , (4) ∂t and Eq. (3) reduces to the telegrapher’s equation. Eq. (3) is more realistic than the hyperbolic telegrapher’s equation and, for instance, more precisely predicts distribution of heat on the metal surface subject to short-pulse laser irradiation. As ε = 0 and τ = 0, Eq. (1) reduces to Fourier’s law
τ
q = −λ∇T ,
(5)
and Eq. (3) reduces to the classical equation of heat conduction ∂T (6) = κ∇ 2 T . ∂t Eqs. (1) and (4) may also be obtained from extended irreversible thermodynamics [2,3], which rests on the hypothesis that the entropy density is a function
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 7 4 - 3
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
17
of usual thermodynamic variables and corresponding dissipative fluxes. For a system involving dual-phaselag heat transfer, the entropy density s is a function of the internal energy u, the heat flux q and the additional variable Q (the flux of the heat flux, which is a secondorder tensor) [4,5], i.e.,
where u˙ = ∂u/∂t. This postulate is weaker than s = s(u, q), for it reduces to s = s(u) at the stationary state. In this Letter we assume that entropy density is a function of both u, u˙ and u¨ = ∂ 2 u/∂t 2
s = s(u, q, Q).
This function reduces to s = s(u) in case of a stationary state. We use postulate (11), instead of (7), to obtain a new generalized Gibbs equation and an entropy balance equation and derive Eq. (1). Here we show that, within the framework of the proposed formalism, an extended evolution criterion generalizing the known Glansdorff–Prigogine criterion [9] as well as the theorem of minimum entropy production take place. One should note that the choice between different thermodynamic theories should be made on the basis of experimental data. In [2,10], an experiment was proposed (but not yet carried out), which would enable one to make a final choice between postulates (7) and (11).
(7)
The generalized Gibbs equation, corresponding to (7), assumes the form of T ds = du − va · dq − vA : dQ,
(8)
where T = T (u, q, Q), a = a(u, q, Q), A = A(u, q, Q), v ≡ 1/ρ. However, postulate (7) is not required for thermodynamic derivation of Eqs. (1) and (4). For instance, for the stationary state, Eqs. (1) and (4) reduce to Fourier’s law (5) and Eqs. (3) and (6) degenerate to the Laplace equation, ∇ 2 T = 0. On the contrary, the generalized entropy density (7) depends also on the fluxes in the stationary state. Function (7), therefore, does not reduce to the classical function where s = s(u). For this reason the generalized Gibbs equation (8) does not reduce to the differential form T ds = du.
(9)
In other words, postulate (7) of the extended theory implies something more than necessary for thermodynamic derivation of Eqs. (1) and (4). A fundamentally different approach to formulation of extended thermodynamic theory consist in the fact that not the dissipative fluxes, but the time derivatives of usual thermodynamic variables are considered as additional variables. In [6], as an usual thermodynamic variable, temperature is regarded; and as an additional thermodynamic variable, the time derivative of temperature is taken. In [7], N 1 time derivatives of temperature are chosen as additional variables. In [8], a formalism of extended irreversible thermodynamics was considered, based on the postulate that the entropy density is a function of both the internal energy, the partial densities of the components, and their time derivatives. For a system involving heat transfer only, according to this postulate s = s(u, u), ˙
(10)
s = s(u, u, ˙ u). ¨
(11)
2. Entropy balance equation Based on postulate (11) let us write the generalized Gibbs equation as T ds = du + Ω d u˙ + Λ d u, ¨
(12)
where T = T (u, u, ˙ u) ¨ = (∂s/∂u)−1 is the extended temperature, ∂s , ∂ u˙ ∂s Λ = Λ(u, u, ˙ u) ¨ =T . ∂ u¨ For a set of stationary states (u˙ = u¨ = 0), in particular, the generalized equation (12) reduces to the form of (9). Let us consider now the time derivative of entropy Ω = Ω(u, u, ˙ u) ¨ =T
∂s ∂u ∂ u˙ ∂ u¨ = T −1 + T −1 Ω + T −1 Λ . ∂t ∂t ∂t ∂t Using the balance equation of internal energy ρ
∂u = −∇ · q ∂t
(13)
(14)
18
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
and the equations ∂ u˙ ∂ u¨ ˙ ¨ (15) = −∇ · q, ρ = −∇ · q, ∂t ∂t which are obtained by differentiating Eq. (14) with respect to time, equality (13) is transformed into
ρ
∂s = −T −1 ∇ · q − T −1 Ω∇ · q˙ − T −1 Λ∇ · q¨ ∂t = −∇ · T −1 q + T −1 Ω q˙ + T −1 Λq¨ + q · ∇T −1 + q˙ · ∇ T −1 Ω + q¨ · ∇ T −1 Λ . (16) Comparing this equality with the entropy balance equation
ρ
∂s = −∇ · Js + σ, (17) ∂t we obtain the expressions for an entropy flux Js and entropy production σ :
ρ
¨ Js = T −1 (q + Ω q˙ + Λq), (18) −1 −1 −1 σ = q · ∇T + q˙ · ∇ T Ω + q¨ · ∇ T Λ 0. (19) From (18) and (19) it can be seen that the entropy flux and source of entropy depend on both heat flux q ¨ In the special case of and time derivatives q˙ and q. steady state, Eqs. (18) and (19) reduce to the classical expressions.
3. Linear theory Let us consider the expressions for the entropy flux and the entropy source. Eqs. (18) and (19) imply that q, q˙ and q¨ are independent variables of the functions Js and σ . The thermodynamic force ∇T −1 corresponds to the flux q. Let us regard the quantities ∇(T −1 Ω) and ∇(T −1 Λ) as the thermodynamic ¨ forces corresponding to the flux rates q˙ and q. Let us introduce the components qi of the heat flux, and the components q˙i and q¨i of the flux rates (i = 1, 2, 3), and write the entropy production as ∂T −1 ∂(T −1 Ω) qi + q˙i σ= ∂xi ∂xi i ∂(T −1 Λ) 0, + q¨i (20) ∂xi
where x1 , x2 , and x3 are the Cartesian coordinates. As in classical irreversible thermodynamics, one should expect that the flux components are generally related to all the thermodynamic forces: qi = F1i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), q˙i = F2i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), q¨i = F3i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), where F1i , F2i , and F3i (i = 1, 2, 3) are generally nonlinear functions. At a first-order approximation, qi , q˙i , and q¨i are linearly related to the components of the thermodynamic forces ∂T −1 /∂xi , ∂(T −1 Ω)/∂xi , and ∂(T −1 Λ)/∂xi : ∂T −1 ∂(T −1 Ω) qi = + L12ij L11ij ∂xi ∂xi j ∂(T −1 Λ) (21) + L13ij , ∂xi ∂T −1 ∂(T −1 Ω) q˙i = + L22ij L21ij ∂xi ∂xi j ∂(T −1 Λ) (22) + L23ij , ∂xi ∂T −1 ∂(T −1 Ω) L31ij q¨i = + L32ij ∂xi ∂xi j ∂(T −1 Λ) (23) , + L33ij ∂xi where Lαβij (α, β, i, j = 1, 2, 3) are phenomenological coefficients. In an anisotropic medium, Lαβij = 0 at i = j and Lαβii = Lαβ at i = j , where we introduced the coefficients Lαβ . In this case, from (21)–(23), the following simpler equations are obtained: q = L11 ∇T −1 + L12 ∇ T −1 Ω + L13 ∇ T −1 Λ , (24) q˙ = L21 ∇T −1 + L22 ∇ T −1 Ω + L23 ∇ T −1 Λ , (25) −1 −1 −1 q¨ = L31 ∇T + L32 ∇ T Ω + L33 ∇ T Λ . (26) Further, as independent variables, we use the heat ¨ Let us bring flux q, and the heat flux rates q˙ and q. Eqs. (24)–(26) to the form ¨ ∇T −1 = R11 q + R12 q˙ + R13 q, −1 ¨ ∇ T Ω = R21 q + R22 q˙ + R23 q, −1 ¨ ∇ T Λ = R31 q + R32 q˙ + R33 q,
(27) (28) (29)
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
where Rαβ (α, β = 1, 2, 3) are the phenomenological coefficients. Apparently, the coefficients Rαβ and Lαβ constitute matrices, either being the inverse of the other. Using (27)–(29), we make the substitution of the thermodynamic forces in (18) and obtain the nonnegative bilinear form + (R12 + R21 )q · q˙ + (R13 + R31 )q · q¨ + (R23 + R32 )q˙ · q¨ 0.
(30)
As known from linear algebra, the condition of nonnegativity of form (30) imposes some constraints on the phenomenological coefficients Rαβ [11]. As known from the classical theory [12], the flux q corresponds to an even state variable. Hence, the flux rate q˙ corresponds to an odd state variable, whereas q¨ becomes even again. Therefore, according to the Onsager–Casimir theorem, the following reciprocal relations hold: R13 = R31 ,
R23 = −R32 .
(31) Let us now consider the case when the variables q, ˙ and q¨ are related by the fourth equation q, ¨ ∇T −1 = R41 q + R42 q˙ + R43 q.
(32)
Thus, four Eqs. (27)–(29) and (32) relate three vari˙ and q¨ are ables. In this case, only two variables of q, q, independent. 1 We can now proceed to derivation of Eq. (1). Let us assume that R41 = R43 = 0 and, hence, ˙ Let us rewrite the Eq. (32) reduces to ∇T −1 = R42 q. latter equation as ˙ ∇T = k q,
(33)
where k = −R42 T 2 . Assuming k to be constant, let us differentiate (33) with respect to time, then we have ∂∇T ¨ = k q. ∂t From (27) and (34), we obtain
the internal energy u only. Then phenomenological equation (35) is identical to Eq. (1). We come to the following identification: R11 = 1/(λT 2 ), R12 = τ/(λT 2 ), and R13 /k = ε/(λT 2 ). 4. Extended evolution criterion
σ = R11 q · q + R22 q˙ · q˙ + R33 q¨ · q¨
R12 = −R21 ,
19
(34)
1 R13 ∂∇T R12 q˙ + q = − ∇T − . (35) 2 R11 kR11 ∂t R11 T Let us next consider the case when Eq. (35) contains the equilibrium temperature; i.e., T is a function of 1 The linear dependence of the fluxes in the classical theory is covered comprehensively in [12, Chapter VI].
Let us now consider the elements of the nonlinear extended theory. As in classical irreversible thermodynamics, let us postulate the convexity of the entropy density as a function of the variables u, u, ˙ and u. ¨ Then we have the inequality d 2 s = dT −1 du + d(T −1 Ω) d u˙ + d(T −1 Λ) d u¨ 0, whence we obtain ∂T −1 ∂u ∂(T −1 Ω) ∂ u˙ ∂(T −1 Λ) ∂ u¨ + + 0. ∂t ∂t ∂t ∂t ∂t ∂t (36) Let us consider the entropy production P = σ dV of the system as a whole, where V is the volume of the system. Let us introduce the form ∂ ∂ dX P = q · ∇T −1 + q˙ · ∇ T −1 Ω dt ∂t ∂t ∂ −1 + q¨ · ∇ T Λ dV , (37) ∂t which is a part of the time derivative of P . Using the Gauss theorem, let us transform (37) into −1 ∂T ∂(T −1 Ω) dX P =− ∇ ·q+ ∇ · q˙ dt ∂t ∂t ∂(T −1 Λ) ∇ · q¨ dV + ∂t −1 ∂(T −1 Ω) ∂T q+ q˙ + ∂t ∂t ∂(T −1 Λ) ¨ + (38) q · n dΣ, ∂t where n is a unit vector directed outside along the normal to the surface, and dΣ is a surface element. When T , Ω, and Λ on the surface are given (timeindependent boundary conditions), the surface integral becomes zero. Using Eqs. (14) and (15), and inequality (36), we obtain from (38) that ∂T −1 ∂u ∂(T −1 Ω) ∂ u˙ dX P = + dt ∂t ∂t ∂t ∂t ∂(T −1 Λ) ∂ u¨ dV 0. + (39) ∂t ∂t
20
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
The equality sign in (39) corresponds to a stationary or equilibrium state. Thus, in the proposed theory, an extended evolution criterion is valid, which is a generalization of the Glansdorff–Prigogine criterion [9]. Inequality (39) is obtained without assuming the specific form of the phenomenological laws. In the special case when the phenomenological equations are linear, and the matrix formed by the coefficients of Eqs. (27)–(29) is symmetrical (R12 = −R21 = 0, R23 = −R32 = 0), we have dP /dt = 2dX P /dt 0; i.e., the theorem of minimum entropy production is valid.
sidered, the extended evolution criterion and, in particular, the theorem of minimum entropy production are valid. This have suggested that theory formulated on the basis of the proposed postulate is extended to nonlinear irreversible thermodynamics.
5. Conclusion
References
In this Letter, we have formulated the thermodynamic formalism for heat conduction on the basis of the postulate that the entropy density s is a function of u, u, ˙ and u. ¨ This postulate differs essentially from the generally accepted one, since it reduces to the local equilibrium hypothesis at the stationary states. Within the formalism proposed, the new expressions have been obtained for the entropy flux and the entropy source that are functions of the heat flux and its time derivatives:
[1] D.Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, New York, 1997. [2] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, 2nd ed., Springer, Berlin, 1996. [3] I. Müller, T. Ruggeri, Extended Thermodynamics, Springer, New York, 1993. [4] C. Pérez-García, D. Jou, J. Phys. A: Math. Gen. 19 (1986) 2881. [5] D. Jou, M. Criado-Sancho, Phys. Lett. A 248 (1998) 172. [6] I. Müller, Arch. Rational Mech. Anal. 41 (1971) 319. [7] R.C. Batra, Arch. Rational Mech. Anal. 53 (1974) 359. [8] S.I. Serdyukov, Zh. Fiz. Khim. 71 (1997) 1572, Russ. J. Phys. Chem. 71 (1997) 1412. [9] P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, New York, 1971. [10] J. Casas-Vázquez, D. Jou, Phys. Rev. E 49 (1994) 1040. [11] E.F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1965. [12] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.
˙ q), ¨ Js = Js (q, q,
˙ q). ¨ σ = σ (q, q,
As in the classical theory, the entropy source is a bilinear form of generalized fluxes and forces, whence the general expressions for the thermodynamic forces have been obtained. We have considered the linear phenomenological equations and derived the equation for dual-phase-lag heat transfer. In the formalism con-
Acknowledgement This work was supported in part by the ISF and Russian Government (Grant No. 3100).
Physics Letters A 281 (2001) 16–20 www.elsevier.nl/locate/pla
A new version of extended irreversible thermodynamics and dual-phase-lag model in heat transfer S.I. Serdyukov Department of Chemistry, Moscow State University, 119899 Moscow, Russia Received 5 June 2000; received in revised form 22 January 2001; accepted 1 February 2001 Communicated by A.R. Bishop
Abstract For a system involving heat transfer, a new postulate of extended irreversible thermodynamics is suggested, which, in case of a stationary states, reduces to the local equilibrium hypothesis. On the basis of this postulate, a transport equation of the dual-phase-lag model of heat transfer is derived. In the proposed formalism, an extended evolution criterion generalizing the known Glansdorff–Prigogine criterion takes place. 2001 Elsevier Science B.V. All rights reserved. PACS: 05.60.+w; 44.60.+k; 05.70.Ln Keywords: Extended irreversible thermodynamics; Heat transfer; Dual-phase-lag model; Evolution criterion
As ε = 0, Eq. (1) reduces to the Maxwell–Cattaneo equation
1. Introduction The linearised dual-phase-lag model of heat transfer [1] is based on the equation ∂∇T ∂q t + q = −λ∇T − ε , (1) ∂ ∂t where T is the equilibrium temperature, λ the heat conductivity, q the heat flux, τ and ε are relaxation times. Using the balance equation, τ
∂T (2) = −∇ · q, ∂t with ρ the mass density, c the heat conductivity per unit mass, let us transform (1) to the following form:
ρc
∂ 2T ∂T ∂∇ 2 T 2 = κ∇ , + T + ε ∂t ∂t ∂t 2 where κ = λ/(ρc) is the thermal diffusivity. τ
(3)
E-mail address: [email protected] (S.I. Serdyukov).
∂q + q = −λ∇T , (4) ∂t and Eq. (3) reduces to the telegrapher’s equation. Eq. (3) is more realistic than the hyperbolic telegrapher’s equation and, for instance, more precisely predicts distribution of heat on the metal surface subject to short-pulse laser irradiation. As ε = 0 and τ = 0, Eq. (1) reduces to Fourier’s law
τ
q = −λ∇T ,
(5)
and Eq. (3) reduces to the classical equation of heat conduction ∂T (6) = κ∇ 2 T . ∂t Eqs. (1) and (4) may also be obtained from extended irreversible thermodynamics [2,3], which rests on the hypothesis that the entropy density is a function
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 7 4 - 3
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
17
of usual thermodynamic variables and corresponding dissipative fluxes. For a system involving dual-phaselag heat transfer, the entropy density s is a function of the internal energy u, the heat flux q and the additional variable Q (the flux of the heat flux, which is a secondorder tensor) [4,5], i.e.,
where u˙ = ∂u/∂t. This postulate is weaker than s = s(u, q), for it reduces to s = s(u) at the stationary state. In this Letter we assume that entropy density is a function of both u, u˙ and u¨ = ∂ 2 u/∂t 2
s = s(u, q, Q).
This function reduces to s = s(u) in case of a stationary state. We use postulate (11), instead of (7), to obtain a new generalized Gibbs equation and an entropy balance equation and derive Eq. (1). Here we show that, within the framework of the proposed formalism, an extended evolution criterion generalizing the known Glansdorff–Prigogine criterion [9] as well as the theorem of minimum entropy production take place. One should note that the choice between different thermodynamic theories should be made on the basis of experimental data. In [2,10], an experiment was proposed (but not yet carried out), which would enable one to make a final choice between postulates (7) and (11).
(7)
The generalized Gibbs equation, corresponding to (7), assumes the form of T ds = du − va · dq − vA : dQ,
(8)
where T = T (u, q, Q), a = a(u, q, Q), A = A(u, q, Q), v ≡ 1/ρ. However, postulate (7) is not required for thermodynamic derivation of Eqs. (1) and (4). For instance, for the stationary state, Eqs. (1) and (4) reduce to Fourier’s law (5) and Eqs. (3) and (6) degenerate to the Laplace equation, ∇ 2 T = 0. On the contrary, the generalized entropy density (7) depends also on the fluxes in the stationary state. Function (7), therefore, does not reduce to the classical function where s = s(u). For this reason the generalized Gibbs equation (8) does not reduce to the differential form T ds = du.
(9)
In other words, postulate (7) of the extended theory implies something more than necessary for thermodynamic derivation of Eqs. (1) and (4). A fundamentally different approach to formulation of extended thermodynamic theory consist in the fact that not the dissipative fluxes, but the time derivatives of usual thermodynamic variables are considered as additional variables. In [6], as an usual thermodynamic variable, temperature is regarded; and as an additional thermodynamic variable, the time derivative of temperature is taken. In [7], N 1 time derivatives of temperature are chosen as additional variables. In [8], a formalism of extended irreversible thermodynamics was considered, based on the postulate that the entropy density is a function of both the internal energy, the partial densities of the components, and their time derivatives. For a system involving heat transfer only, according to this postulate s = s(u, u), ˙
(10)
s = s(u, u, ˙ u). ¨
(11)
2. Entropy balance equation Based on postulate (11) let us write the generalized Gibbs equation as T ds = du + Ω d u˙ + Λ d u, ¨
(12)
where T = T (u, u, ˙ u) ¨ = (∂s/∂u)−1 is the extended temperature, ∂s , ∂ u˙ ∂s Λ = Λ(u, u, ˙ u) ¨ =T . ∂ u¨ For a set of stationary states (u˙ = u¨ = 0), in particular, the generalized equation (12) reduces to the form of (9). Let us consider now the time derivative of entropy Ω = Ω(u, u, ˙ u) ¨ =T
∂s ∂u ∂ u˙ ∂ u¨ = T −1 + T −1 Ω + T −1 Λ . ∂t ∂t ∂t ∂t Using the balance equation of internal energy ρ
∂u = −∇ · q ∂t
(13)
(14)
18
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
and the equations ∂ u˙ ∂ u¨ ˙ ¨ (15) = −∇ · q, ρ = −∇ · q, ∂t ∂t which are obtained by differentiating Eq. (14) with respect to time, equality (13) is transformed into
ρ
∂s = −T −1 ∇ · q − T −1 Ω∇ · q˙ − T −1 Λ∇ · q¨ ∂t = −∇ · T −1 q + T −1 Ω q˙ + T −1 Λq¨ + q · ∇T −1 + q˙ · ∇ T −1 Ω + q¨ · ∇ T −1 Λ . (16) Comparing this equality with the entropy balance equation
ρ
∂s = −∇ · Js + σ, (17) ∂t we obtain the expressions for an entropy flux Js and entropy production σ :
ρ
¨ Js = T −1 (q + Ω q˙ + Λq), (18) −1 −1 −1 σ = q · ∇T + q˙ · ∇ T Ω + q¨ · ∇ T Λ 0. (19) From (18) and (19) it can be seen that the entropy flux and source of entropy depend on both heat flux q ¨ In the special case of and time derivatives q˙ and q. steady state, Eqs. (18) and (19) reduce to the classical expressions.
3. Linear theory Let us consider the expressions for the entropy flux and the entropy source. Eqs. (18) and (19) imply that q, q˙ and q¨ are independent variables of the functions Js and σ . The thermodynamic force ∇T −1 corresponds to the flux q. Let us regard the quantities ∇(T −1 Ω) and ∇(T −1 Λ) as the thermodynamic ¨ forces corresponding to the flux rates q˙ and q. Let us introduce the components qi of the heat flux, and the components q˙i and q¨i of the flux rates (i = 1, 2, 3), and write the entropy production as ∂T −1 ∂(T −1 Ω) qi + q˙i σ= ∂xi ∂xi i ∂(T −1 Λ) 0, + q¨i (20) ∂xi
where x1 , x2 , and x3 are the Cartesian coordinates. As in classical irreversible thermodynamics, one should expect that the flux components are generally related to all the thermodynamic forces: qi = F1i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), q˙i = F2i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), q¨i = F3i (∇T −1 , ∇(T −1 Ω), ∇(T −1 Λ)), where F1i , F2i , and F3i (i = 1, 2, 3) are generally nonlinear functions. At a first-order approximation, qi , q˙i , and q¨i are linearly related to the components of the thermodynamic forces ∂T −1 /∂xi , ∂(T −1 Ω)/∂xi , and ∂(T −1 Λ)/∂xi : ∂T −1 ∂(T −1 Ω) qi = + L12ij L11ij ∂xi ∂xi j ∂(T −1 Λ) (21) + L13ij , ∂xi ∂T −1 ∂(T −1 Ω) q˙i = + L22ij L21ij ∂xi ∂xi j ∂(T −1 Λ) (22) + L23ij , ∂xi ∂T −1 ∂(T −1 Ω) L31ij q¨i = + L32ij ∂xi ∂xi j ∂(T −1 Λ) (23) , + L33ij ∂xi where Lαβij (α, β, i, j = 1, 2, 3) are phenomenological coefficients. In an anisotropic medium, Lαβij = 0 at i = j and Lαβii = Lαβ at i = j , where we introduced the coefficients Lαβ . In this case, from (21)–(23), the following simpler equations are obtained: q = L11 ∇T −1 + L12 ∇ T −1 Ω + L13 ∇ T −1 Λ , (24) q˙ = L21 ∇T −1 + L22 ∇ T −1 Ω + L23 ∇ T −1 Λ , (25) −1 −1 −1 q¨ = L31 ∇T + L32 ∇ T Ω + L33 ∇ T Λ . (26) Further, as independent variables, we use the heat ¨ Let us bring flux q, and the heat flux rates q˙ and q. Eqs. (24)–(26) to the form ¨ ∇T −1 = R11 q + R12 q˙ + R13 q, −1 ¨ ∇ T Ω = R21 q + R22 q˙ + R23 q, −1 ¨ ∇ T Λ = R31 q + R32 q˙ + R33 q,
(27) (28) (29)
S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
where Rαβ (α, β = 1, 2, 3) are the phenomenological coefficients. Apparently, the coefficients Rαβ and Lαβ constitute matrices, either being the inverse of the other. Using (27)–(29), we make the substitution of the thermodynamic forces in (18) and obtain the nonnegative bilinear form + (R12 + R21 )q · q˙ + (R13 + R31 )q · q¨ + (R23 + R32 )q˙ · q¨ 0.
(30)
As known from linear algebra, the condition of nonnegativity of form (30) imposes some constraints on the phenomenological coefficients Rαβ [11]. As known from the classical theory [12], the flux q corresponds to an even state variable. Hence, the flux rate q˙ corresponds to an odd state variable, whereas q¨ becomes even again. Therefore, according to the Onsager–Casimir theorem, the following reciprocal relations hold: R13 = R31 ,
R23 = −R32 .
(31) Let us now consider the case when the variables q, ˙ and q¨ are related by the fourth equation q, ¨ ∇T −1 = R41 q + R42 q˙ + R43 q.
(32)
Thus, four Eqs. (27)–(29) and (32) relate three vari˙ and q¨ are ables. In this case, only two variables of q, q, independent. 1 We can now proceed to derivation of Eq. (1). Let us assume that R41 = R43 = 0 and, hence, ˙ Let us rewrite the Eq. (32) reduces to ∇T −1 = R42 q. latter equation as ˙ ∇T = k q,
(33)
where k = −R42 T 2 . Assuming k to be constant, let us differentiate (33) with respect to time, then we have ∂∇T ¨ = k q. ∂t From (27) and (34), we obtain
the internal energy u only. Then phenomenological equation (35) is identical to Eq. (1). We come to the following identification: R11 = 1/(λT 2 ), R12 = τ/(λT 2 ), and R13 /k = ε/(λT 2 ). 4. Extended evolution criterion
σ = R11 q · q + R22 q˙ · q˙ + R33 q¨ · q¨
R12 = −R21 ,
19
(34)
1 R13 ∂∇T R12 q˙ + q = − ∇T − . (35) 2 R11 kR11 ∂t R11 T Let us next consider the case when Eq. (35) contains the equilibrium temperature; i.e., T is a function of 1 The linear dependence of the fluxes in the classical theory is covered comprehensively in [12, Chapter VI].
Let us now consider the elements of the nonlinear extended theory. As in classical irreversible thermodynamics, let us postulate the convexity of the entropy density as a function of the variables u, u, ˙ and u. ¨ Then we have the inequality d 2 s = dT −1 du + d(T −1 Ω) d u˙ + d(T −1 Λ) d u¨ 0, whence we obtain ∂T −1 ∂u ∂(T −1 Ω) ∂ u˙ ∂(T −1 Λ) ∂ u¨ + + 0. ∂t ∂t ∂t ∂t ∂t ∂t (36) Let us consider the entropy production P = σ dV of the system as a whole, where V is the volume of the system. Let us introduce the form ∂ ∂ dX P = q · ∇T −1 + q˙ · ∇ T −1 Ω dt ∂t ∂t ∂ −1 + q¨ · ∇ T Λ dV , (37) ∂t which is a part of the time derivative of P . Using the Gauss theorem, let us transform (37) into −1 ∂T ∂(T −1 Ω) dX P =− ∇ ·q+ ∇ · q˙ dt ∂t ∂t ∂(T −1 Λ) ∇ · q¨ dV + ∂t −1 ∂(T −1 Ω) ∂T q+ q˙ + ∂t ∂t ∂(T −1 Λ) ¨ + (38) q · n dΣ, ∂t where n is a unit vector directed outside along the normal to the surface, and dΣ is a surface element. When T , Ω, and Λ on the surface are given (timeindependent boundary conditions), the surface integral becomes zero. Using Eqs. (14) and (15), and inequality (36), we obtain from (38) that ∂T −1 ∂u ∂(T −1 Ω) ∂ u˙ dX P = + dt ∂t ∂t ∂t ∂t ∂(T −1 Λ) ∂ u¨ dV 0. + (39) ∂t ∂t
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S.I. Serdyukov / Physics Letters A 281 (2001) 16–20
The equality sign in (39) corresponds to a stationary or equilibrium state. Thus, in the proposed theory, an extended evolution criterion is valid, which is a generalization of the Glansdorff–Prigogine criterion [9]. Inequality (39) is obtained without assuming the specific form of the phenomenological laws. In the special case when the phenomenological equations are linear, and the matrix formed by the coefficients of Eqs. (27)–(29) is symmetrical (R12 = −R21 = 0, R23 = −R32 = 0), we have dP /dt = 2dX P /dt 0; i.e., the theorem of minimum entropy production is valid.
sidered, the extended evolution criterion and, in particular, the theorem of minimum entropy production are valid. This have suggested that theory formulated on the basis of the proposed postulate is extended to nonlinear irreversible thermodynamics.
5. Conclusion
References
In this Letter, we have formulated the thermodynamic formalism for heat conduction on the basis of the postulate that the entropy density s is a function of u, u, ˙ and u. ¨ This postulate differs essentially from the generally accepted one, since it reduces to the local equilibrium hypothesis at the stationary states. Within the formalism proposed, the new expressions have been obtained for the entropy flux and the entropy source that are functions of the heat flux and its time derivatives:
[1] D.Y. Tzou, Macro-to-Microscale Heat Transfer: The Lagging Behavior, Taylor and Francis, New York, 1997. [2] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, 2nd ed., Springer, Berlin, 1996. [3] I. Müller, T. Ruggeri, Extended Thermodynamics, Springer, New York, 1993. [4] C. Pérez-García, D. Jou, J. Phys. A: Math. Gen. 19 (1986) 2881. [5] D. Jou, M. Criado-Sancho, Phys. Lett. A 248 (1998) 172. [6] I. Müller, Arch. Rational Mech. Anal. 41 (1971) 319. [7] R.C. Batra, Arch. Rational Mech. Anal. 53 (1974) 359. [8] S.I. Serdyukov, Zh. Fiz. Khim. 71 (1997) 1572, Russ. J. Phys. Chem. 71 (1997) 1412. [9] P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, New York, 1971. [10] J. Casas-Vázquez, D. Jou, Phys. Rev. E 49 (1994) 1040. [11] E.F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1965. [12] S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.
˙ q), ¨ Js = Js (q, q,
˙ q). ¨ σ = σ (q, q,
As in the classical theory, the entropy source is a bilinear form of generalized fluxes and forces, whence the general expressions for the thermodynamic forces have been obtained. We have considered the linear phenomenological equations and derived the equation for dual-phase-lag heat transfer. In the formalism con-
Acknowledgement This work was supported in part by the ISF and Russian Government (Grant No. 3100).