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On the admissible values of the heat flux in hyperbolic heat transport
Physics Letters A 177 (1993) 323-326 North-Holland
PHYSICS
LETTERS
A
On the admissible values of the heat flux in hyperbolic heat transport M. Criado-Sancho ’ and J.E. Llebot Departament de Fisica Estadistica, Universitat Authnoma de Barcelona, 08193 Bellaterra. Catalonia, Spain
Received 4 December 1992; accepted for publication 14 April 1993 Communicated by A.R. Bishop
The thermodynamic stability restrictions on the maximum value of the heat flux in nonequilibrium steady states are compared with the restrictions which must be satisfied by the initial values of the heat flux in order to guarantee the positive character of the absolute temperature along the evolution towards equilibrium in hyperbolic heat transport.
1. Introduction
Hyperbolic heat transport has been receiving increasing attention both for theoretical motivations (analysis of thermal waves and second sound in dielectric solids, finite speed of heat transport, etc. ) as for the analysis of some practical problems involving a fast supply of thermal energy (for instance, by a laser pulse or a chemical explosion, etc.). The usual theory of thermal conduction, based on the Fourier law, implies an immediate response to a temperature gradient and leads to a parabolic differential equation for the evolution of the temperature. In contrast, when relaxational effects are taken into account in the constitutive equation describing the heat flux, as for instance in the Maxwell-Catteneo equation, one has a hyperbolic equation which implies a finite speed for heat transport. The literature in this field is rather large. We refer the reader to several reviews and papers on this subject [ l-3 1. The Fourier equation does not imply any limit on the value of the heat flux: it may be indefinitely increased by enhancing the temperature gradient. In contrast, when relaxational effects are taken into account, the maximum value of the heat flux at a given internal ’ On leave from Departamento de Quimica Fisica, Universidad National de Education a Distancia, Senda de1 Rey, sn, 28040 Madrid, Spain. 0375-9601/93/$
energy is limited. In this note we study several limits on the maximum value of the heat flux, arising from several criteria. The classical Fourier law has the well-known form q= -AVT,
(1)
where q is the heat flux and ;1 the thermal conductivity. The simplest generalization of ( 1) including relaxational effects of the heat flux is the MaxwellCattaneo equation [ l-3 1, nj+q=
-IVT,
(2)
with 7 the relaxation time of q. In opposition to the Fourier law, which predicts an infinite speed for the propagation of thermal pulses (i.e. for the high-frequency limit of the phase speed of thermal waves), the Maxwell-Cattaneo equation yields the finite speed u= (A/pcr)“2,
(3)
where c is the heat capacity per unit mass. Note that in (2) the initial value of q must be given in addition to the initial temperature distribution, so that q becomes an independent variable in the description of the system. Though the prediction of a finite speed for heat transport is satisfactory from an experimental point of view [ I-31, eq. (2) presents important problems which do not arise with eq. (1): (a) The classical
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
323
Volume
177, number
PHYSICS
4,5
entropy production, which has the form a=q.VT-‘, may become negative, because q is independent of VT for times smaller than T. (b) An initial positive temperature profile will not always remain positive at later times during the approach of the system to equilibrium. Here, we will see how both features imply a maximum value of the heat flux.
LETTERS
A
21 June 1993
ical meaning of (6 ) is clear: since U is the maximum speed of thermal waves, puU is the energy density times the maximum speed of heat transport. In radiative heat transfer, such a maximum value for the heat flux is well known, with U equal to the speed of light [9].
3. Positive character of absolute temperature 2. Thermodynamic
stability requirements
The use of (2 ) instead of ( 1) implies the need of a new thermodynamics, which is not based on the usual local-equilibrium hypothesis [ 2,4-7 1. Such a formulation, known as extended irreversible thermodynamics (EIT), uses a generalized entropy of the form ps=pses - (r/2;1T=)q-q .
(4)
It is possible to see [ 2 ] that whereas the classical entropy production (i.e., the production of s,~) may become negative, the production of the generalized entropy s is always positive. The negative definite character of the second differential of’the classical entropy @.s,~ is a necessary condition for stability of thermodynamic equilibrium states. Thus, it is of interest to explore the requirements following from 6’s in nonequilibrium steady states characterized by distributions of u and q. The second differential of this entropy s2s in a steady state under a heat flux q,, has the form
After having obtained the maximum admissible value of puU for the heat flux from thermodynamic stability arguments, we will look for another interpretation of this value stemming in a more direct way from the Maxwell-Cattaneo equation. We will study the conditions to guarantee that the absolute temperature will remain positive along the evolution of the system towards equilibrium. We do so through the analysis of a simple example: a unidimensional system with an initial temperature distribution given by T(x,0)=To+6T,,cos(kx),
(7)
with To a constant, uniform value. We assume also that the internal energy per unit volume pu is given by pu=pcT, with c the specific heat per unit mass. Since the integral of (7) from x=0 to x=L, the length of the system, is zero (we assume that k= 2n/l=2mlL, with n= 1, 2, 3, . ..). a perturbation of the form (7 ) does not change the total internal energy of the system. Introducing (2) into the energy balance equation pli= -v-q
--a6q.6q-2ql)
$6u.6q )
one is led, for constant
(5)
where a stands for a= m/AT’. Then, if T, v and 1 are constant one has da/au= -2a(cT)-’ and d2a/au2 = 6a( CT) -’ and 6’s is definite negative only for values of q less than a critical value qcritgiven in general terms by &it =PUU 3
(6)
where we have assumed that u = CT. Other values of the critical heat flux have been found in different situations with c and L dependent on T [ 8 1, but they are not of interest in the present context. The phys324
(8) coefficients
a=T ar rat2+ar=~V2T,
r, 2 and c to (9)
with x the thermal diffusivity defined as x=A/pc. According to (8) we find for the evolution of the temperature perturbation the equation d26T d6T ?dt2 + dt +xk=GT=O, whose solutions
are
6T(t)=a’exp(ta+)+b’exp(ta_), with
(11)
Volume 177, number 4,5
a& = -+7
f
(1 -4,yk27)“2.
+7
(12)
For 4xk2r< 1 there is an exponential decay towards the equilibrium whereas for 4k2xr> 1 the solution decays in an oscillatory way as T(x, t) = To + ST, cos(kx)
exp( -t/27) ,
X [A sin(wt)+cos(wt)]
(13)
where A is a constant which should be determined from the initial conditions and the frequency w is given by co= +7 ( 4k2x7- 1 )1’2 . Here we will only consider the latter situation, because it is in this oscillatory behaviour that problems with the sign of T can arise. The heat flux related with (9 ) may be obtained from (2 ) and it turns out to be q(x, t) =nksT,
sin(&) exp( -t/27),
X [A’sin(ot)+B’cos(ot)]
(14)
with A,=
2A+4or 1+ 4627*
B,= ’
2-4Awr 1+ 40%2 .
(15)
We have assumed that q vanishes on the boundaries of the system (x=0, x=L). Now we ask which restrictions must satisfy q(0) to be compatible with the positive character of T. Note, indeed, that in spite that T in (7) is positive definite at the initial time, it could become negative at some latter times if A is too high, i.e. higher than To/ST,- 1. The expression for the initial q corresponding to this value of A for small initial perturbations, i.e. neglecting terms of order 6T,,/T,, in comparison with 1, is q(x, 0) = -IkT,
21 June 1993
PHYSICS LETTERS A
which is again pu times the speed of thermal pulses, u. It follows that for an initial value of the heat flux higher than puU, any short-wavelength temperature perturbation will lead to temperature profiles with some negative values at some times during the evolution. Then, for steady states with a heat flux q higher than pu U, the superposition of any temperature pulse, even of small amplitude, will lead to negative values of T. Finally, it may be of interest to consider the problem of the maximum heat flux from the point of view of the kinetic theory of gases. Assume a small volume with internal energy per unit mass u=tkT/m at equilibrium. The maximum energy flux attainable in such a system, at constant energy, by organizing the molecular motion in a single direction would be 4,_ =pu( 3kT/m)“’
.
(18)
This value is not attainable in principle, because it would imply the possibility to organize completely the chaotic molecular motion, in contradiction with the second law. But, in a monatomic ideal gas, where the thermal diffusivity is x= 3 (kT/m)7, one has for the speed of thermal pulses U= ($kT/m) ‘/’ (the same as the speed of sound; when viscous effects are included in the calculation, U becomes 1.64 times the speed of sound [ 5 ] ). One has, therefore, that the maximum attainable value of q that we have found is puU
4w7
1 +40272
sin(kx> .
(16)
Then, in the high-frequency limit 07~ 1, a sufficient condition for T being always positive is that the maximum value of q (0 ) should be less than 4max=ilkT,/wr=pcT,(~/pcr)“==puU,
(17)
Acknowledgement
This work has been partially supported by the Direccion General de Investigation Cientifica y Ttcnica of the Spanish Ministry of Education y Ciencia under grant PB 90-676. 325
Volume
177, number
4,5
PHYSICS
References [ 1] D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 ( 1989) 41; 62 (1990) 375. [2] D. Jou, J. Casas-Vazquez and G. Lebon, Rep. Prog. Phys. 5 1 (1988) 1104; G. Lebon, D. Jou and J. Casas-Vazquez, Contemp. Phys. 33 (1992) 41. [3] D.Y. Tzou, Int. J. Heat Mass Trans. 32 (1989); 33 (1990) 877; J.I. Frankel, B. Vick and M.N. Ozisik, J. Appl. Phys. 58 (1985) 3340; R.J. Swenson, J. Non-Equilib. Thermodyn. 3 (1978) 39. [4] L.S. Garcia-Cohn, Rev. Mex. Fis. 34 ( 1988) 344;
326
LETTERS
A
21 June 1993
L.S. Garcia-Cohn and F.J. Uribe, J. Non-Equilib. Thermodyn. 16 (1991) 89. [ 51 I. Mtlller, Thermodynamics (Pitman, London, 1986). [6] P. Salamon and S. Sieniutycz, Extended thermodynamic systems (Taylor and Francis, London, 1992). [ 7 ] R.E. Nettleton, Phys. Fluids 3 ( 1960) 2 16; 4 ( 196 1) 74; B.C. Eu, J. Chem. Phys. 73 (1980) 2958; A.F. Ghaleb, J. Non-Equilib. Thermodyn. 14 ( 1989) 23 1. [ 81 D. Jou, J.E. Llebot and J. Casas-Vazquez, Phys. Rev. A 25 (1982) 508,3277. [9] G.N. Minerbo, J. Quant. Spectrosc. Radiat. Transfer 20 (1978) 541; CD. Levermore, J. Quant. Spectrosc. Radiat. Transfer 31 (1984) 149.
PHYSICS
LETTERS
A
On the admissible values of the heat flux in hyperbolic heat transport M. Criado-Sancho ’ and J.E. Llebot Departament de Fisica Estadistica, Universitat Authnoma de Barcelona, 08193 Bellaterra. Catalonia, Spain
Received 4 December 1992; accepted for publication 14 April 1993 Communicated by A.R. Bishop
The thermodynamic stability restrictions on the maximum value of the heat flux in nonequilibrium steady states are compared with the restrictions which must be satisfied by the initial values of the heat flux in order to guarantee the positive character of the absolute temperature along the evolution towards equilibrium in hyperbolic heat transport.
1. Introduction
Hyperbolic heat transport has been receiving increasing attention both for theoretical motivations (analysis of thermal waves and second sound in dielectric solids, finite speed of heat transport, etc. ) as for the analysis of some practical problems involving a fast supply of thermal energy (for instance, by a laser pulse or a chemical explosion, etc.). The usual theory of thermal conduction, based on the Fourier law, implies an immediate response to a temperature gradient and leads to a parabolic differential equation for the evolution of the temperature. In contrast, when relaxational effects are taken into account in the constitutive equation describing the heat flux, as for instance in the Maxwell-Catteneo equation, one has a hyperbolic equation which implies a finite speed for heat transport. The literature in this field is rather large. We refer the reader to several reviews and papers on this subject [ l-3 1. The Fourier equation does not imply any limit on the value of the heat flux: it may be indefinitely increased by enhancing the temperature gradient. In contrast, when relaxational effects are taken into account, the maximum value of the heat flux at a given internal ’ On leave from Departamento de Quimica Fisica, Universidad National de Education a Distancia, Senda de1 Rey, sn, 28040 Madrid, Spain. 0375-9601/93/$
energy is limited. In this note we study several limits on the maximum value of the heat flux, arising from several criteria. The classical Fourier law has the well-known form q= -AVT,
(1)
where q is the heat flux and ;1 the thermal conductivity. The simplest generalization of ( 1) including relaxational effects of the heat flux is the MaxwellCattaneo equation [ l-3 1, nj+q=
-IVT,
(2)
with 7 the relaxation time of q. In opposition to the Fourier law, which predicts an infinite speed for the propagation of thermal pulses (i.e. for the high-frequency limit of the phase speed of thermal waves), the Maxwell-Cattaneo equation yields the finite speed u= (A/pcr)“2,
(3)
where c is the heat capacity per unit mass. Note that in (2) the initial value of q must be given in addition to the initial temperature distribution, so that q becomes an independent variable in the description of the system. Though the prediction of a finite speed for heat transport is satisfactory from an experimental point of view [ I-31, eq. (2) presents important problems which do not arise with eq. (1): (a) The classical
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
323
Volume
177, number
PHYSICS
4,5
entropy production, which has the form a=q.VT-‘, may become negative, because q is independent of VT for times smaller than T. (b) An initial positive temperature profile will not always remain positive at later times during the approach of the system to equilibrium. Here, we will see how both features imply a maximum value of the heat flux.
LETTERS
A
21 June 1993
ical meaning of (6 ) is clear: since U is the maximum speed of thermal waves, puU is the energy density times the maximum speed of heat transport. In radiative heat transfer, such a maximum value for the heat flux is well known, with U equal to the speed of light [9].
3. Positive character of absolute temperature 2. Thermodynamic
stability requirements
The use of (2 ) instead of ( 1) implies the need of a new thermodynamics, which is not based on the usual local-equilibrium hypothesis [ 2,4-7 1. Such a formulation, known as extended irreversible thermodynamics (EIT), uses a generalized entropy of the form ps=pses - (r/2;1T=)q-q .
(4)
It is possible to see [ 2 ] that whereas the classical entropy production (i.e., the production of s,~) may become negative, the production of the generalized entropy s is always positive. The negative definite character of the second differential of’the classical entropy @.s,~ is a necessary condition for stability of thermodynamic equilibrium states. Thus, it is of interest to explore the requirements following from 6’s in nonequilibrium steady states characterized by distributions of u and q. The second differential of this entropy s2s in a steady state under a heat flux q,, has the form
After having obtained the maximum admissible value of puU for the heat flux from thermodynamic stability arguments, we will look for another interpretation of this value stemming in a more direct way from the Maxwell-Cattaneo equation. We will study the conditions to guarantee that the absolute temperature will remain positive along the evolution of the system towards equilibrium. We do so through the analysis of a simple example: a unidimensional system with an initial temperature distribution given by T(x,0)=To+6T,,cos(kx),
(7)
with To a constant, uniform value. We assume also that the internal energy per unit volume pu is given by pu=pcT, with c the specific heat per unit mass. Since the integral of (7) from x=0 to x=L, the length of the system, is zero (we assume that k= 2n/l=2mlL, with n= 1, 2, 3, . ..). a perturbation of the form (7 ) does not change the total internal energy of the system. Introducing (2) into the energy balance equation pli= -v-q
--a6q.6q-2ql)
$6u.6q )
one is led, for constant
(5)
where a stands for a= m/AT’. Then, if T, v and 1 are constant one has da/au= -2a(cT)-’ and d2a/au2 = 6a( CT) -’ and 6’s is definite negative only for values of q less than a critical value qcritgiven in general terms by &it =PUU 3
(6)
where we have assumed that u = CT. Other values of the critical heat flux have been found in different situations with c and L dependent on T [ 8 1, but they are not of interest in the present context. The phys324
(8) coefficients
a=T ar rat2+ar=~V2T,
r, 2 and c to (9)
with x the thermal diffusivity defined as x=A/pc. According to (8) we find for the evolution of the temperature perturbation the equation d26T d6T ?dt2 + dt +xk=GT=O, whose solutions
are
6T(t)=a’exp(ta+)+b’exp(ta_), with
(11)
Volume 177, number 4,5
a& = -+7
f
(1 -4,yk27)“2.
+7
(12)
For 4xk2r< 1 there is an exponential decay towards the equilibrium whereas for 4k2xr> 1 the solution decays in an oscillatory way as T(x, t) = To + ST, cos(kx)
exp( -t/27) ,
X [A sin(wt)+cos(wt)]
(13)
where A is a constant which should be determined from the initial conditions and the frequency w is given by co= +7 ( 4k2x7- 1 )1’2 . Here we will only consider the latter situation, because it is in this oscillatory behaviour that problems with the sign of T can arise. The heat flux related with (9 ) may be obtained from (2 ) and it turns out to be q(x, t) =nksT,
sin(&) exp( -t/27),
X [A’sin(ot)+B’cos(ot)]
(14)
with A,=
2A+4or 1+ 4627*
B,= ’
2-4Awr 1+ 40%2 .
(15)
We have assumed that q vanishes on the boundaries of the system (x=0, x=L). Now we ask which restrictions must satisfy q(0) to be compatible with the positive character of T. Note, indeed, that in spite that T in (7) is positive definite at the initial time, it could become negative at some latter times if A is too high, i.e. higher than To/ST,- 1. The expression for the initial q corresponding to this value of A for small initial perturbations, i.e. neglecting terms of order 6T,,/T,, in comparison with 1, is q(x, 0) = -IkT,
21 June 1993
PHYSICS LETTERS A
which is again pu times the speed of thermal pulses, u. It follows that for an initial value of the heat flux higher than puU, any short-wavelength temperature perturbation will lead to temperature profiles with some negative values at some times during the evolution. Then, for steady states with a heat flux q higher than pu U, the superposition of any temperature pulse, even of small amplitude, will lead to negative values of T. Finally, it may be of interest to consider the problem of the maximum heat flux from the point of view of the kinetic theory of gases. Assume a small volume with internal energy per unit mass u=tkT/m at equilibrium. The maximum energy flux attainable in such a system, at constant energy, by organizing the molecular motion in a single direction would be 4,_ =pu( 3kT/m)“’
.
(18)
This value is not attainable in principle, because it would imply the possibility to organize completely the chaotic molecular motion, in contradiction with the second law. But, in a monatomic ideal gas, where the thermal diffusivity is x= 3 (kT/m)7, one has for the speed of thermal pulses U= ($kT/m) ‘/’ (the same as the speed of sound; when viscous effects are included in the calculation, U becomes 1.64 times the speed of sound [ 5 ] ). One has, therefore, that the maximum attainable value of q that we have found is puU
4w7
1 +40272
sin(kx> .
(16)
Then, in the high-frequency limit 07~ 1, a sufficient condition for T being always positive is that the maximum value of q (0 ) should be less than 4max=ilkT,/wr=pcT,(~/pcr)“==puU,
(17)
Acknowledgement
This work has been partially supported by the Direccion General de Investigation Cientifica y Ttcnica of the Spanish Ministry of Education y Ciencia under grant PB 90-676. 325
Volume
177, number
4,5
PHYSICS
References [ 1] D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 ( 1989) 41; 62 (1990) 375. [2] D. Jou, J. Casas-Vazquez and G. Lebon, Rep. Prog. Phys. 5 1 (1988) 1104; G. Lebon, D. Jou and J. Casas-Vazquez, Contemp. Phys. 33 (1992) 41. [3] D.Y. Tzou, Int. J. Heat Mass Trans. 32 (1989); 33 (1990) 877; J.I. Frankel, B. Vick and M.N. Ozisik, J. Appl. Phys. 58 (1985) 3340; R.J. Swenson, J. Non-Equilib. Thermodyn. 3 (1978) 39. [4] L.S. Garcia-Cohn, Rev. Mex. Fis. 34 ( 1988) 344;
326
LETTERS
A
21 June 1993
L.S. Garcia-Cohn and F.J. Uribe, J. Non-Equilib. Thermodyn. 16 (1991) 89. [ 51 I. Mtlller, Thermodynamics (Pitman, London, 1986). [6] P. Salamon and S. Sieniutycz, Extended thermodynamic systems (Taylor and Francis, London, 1992). [ 7 ] R.E. Nettleton, Phys. Fluids 3 ( 1960) 2 16; 4 ( 196 1) 74; B.C. Eu, J. Chem. Phys. 73 (1980) 2958; A.F. Ghaleb, J. Non-Equilib. Thermodyn. 14 ( 1989) 23 1. [ 81 D. Jou, J.E. Llebot and J. Casas-Vazquez, Phys. Rev. A 25 (1982) 508,3277. [9] G.N. Minerbo, J. Quant. Spectrosc. Radiat. Transfer 20 (1978) 541; CD. Levermore, J. Quant. Spectrosc. Radiat. Transfer 31 (1984) 149.