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Determination of material functions through second sound measurements in a hyperbolic heat conduction theory
Ma&l. Pergamon
Cornput. Modellang Vol. 24, No. 12, pp 19-28, 1996 Copyright@1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/96 $15.00 + 0 00
PII: SO895-7177(96)00175-6
Determination of Material Functions through Second Sound Measurements in a Hyperbolic Heat Conduction Theory V.
A.
CIMELLI
Dipartimento
di Matematica.
Universita
della Basilicata
I-85100
Potenza,
Italy
CimelliQpzvx85.cineca.it
K. Fachbereich
FRISCHMUTH
Mathematik,
D-18051
Restock,
Universitlt
Restock
Germany
kurtQsun2.math.uni.rostock.de
(Received February 1996; accepted March
1996)
Abstract-In a previous paper [I], numerical solutions to mitral-boundary value problems sema-empzrzcal model of heat conduction were compared with available experimental results.
for a
In the present paper, we modify the model by introducing more realistic approximations of constltutive functions, based on measured heat conductivities and second sound speeds for NaF at low temperatures (10. 20° K). We achieve good accordance between measured second sound pulses and numerical solutions in the temperature range covered by experiments, and reasonable behaviour even beyond this interval. Especially, a passage to the diffusive regime of the classical Fourier law is possible. Keywords-Second time, Hyperbolic
sound waves, Critical temperature, heat conduction, Diffusive regime.
Semi-empirical
temperature scale, Arrival
1. INTRODUCTION The aim of this paper is to calculate numerical solutions to a hyperbolic system of equations modelling heat conduction in a range of temperatures characterized by the occurrence of heat pulses (second sound waves), and to compare these solutions with experimental data. Such heat pulses occur, e.g., in very pure crystalline solids (Bi, NaF) around the critzcal temperature of the material at hand. IJsually, there is a maximum of heat conductivity, and near this point second sound appears. Due to the increasing heat capacity, the pulse speed decreases with increasing temperature. At a certain distance from the maximum, to both sides, the pulses disappear, and the diffusive regime takes over. In recent years, a new theory of heat conduction was developed by Kosiliski et al. [2] which was designed to be simple, consistent with rational thermodynamics, and yet capable of describing heat pulses. Hence, obviously the new theory had to be hyperbolic, but it must also allow a passage to the classical (parabolic) case given by Fourier’s law. Based on elements of approaches given by [3-51, a theory with an additional scalar state variable p and a kinetic equation for it was introduced. Typeset 19
by AM-w
20
V. A. CIMELLIAND K. FRISCHMUTH
By design, at equilibrium, j3 coincides with the classical absolute temperature B, otherwise fl follows after 0 with a certain delay, controlled by a parameter T (called relaxation time).l This delay introduces hyperbolicity and, if chosen small, controls the passage to the classical case. The equations for the rigid heat conductor are B = ~(~,~),
(I)
c,,8 + div q = r.
(2)
By r we denoted heat sources, and by q the heat flux vector. Moreover, c, = t’(e), where Eis the volumic internal energy, means the specific heat. Besides (1) and (2), the constitutive equations q = -tcvp, Ic =
r;(8),
e=
E(B),
(3)
7ii= IN@, VP>, with the Helmholtz free energy $ and the heat conductivity Classical ~sumptions for these functions E= % Ic. =
K have been postulated.
(Debye’s law),
/Qe3
(8)
have been introduced. Finally, the function
b = f (8,P>,
(9)
where
af -l
(>
7-= 33
00)
can be treated as a variable relaxation time. Well-posedness, characteristic wave speeds, and blow-up have been discussed in [2,6]. Recently, classical analytical solutions have been constructed [7]. Numerical solutions to initial-boundary value problems for bismuth and NaF have been shown to reproduce heat pulses at the corresponding critical temperatures [I]. It was observed in [6] that-under the above restrictions (3)-(5)-the Helmholtz free energy $ must be a homogeneous quadratic function in {V/3], with a coefficient being itself a linear homogeneous function in 8. Further, one infers that the kinetic equation is determined up to an integration constant $sO. Hence, the semi-empirical model with (3)-(6) is d et ermined by the constitutive functions E(Q), ~(6’)) and one additional constant $2,. On the other hand, the functions E and K can be identified from experimental data, e.g., in the case of NaF (cf. [8,9]). But moreover, there are also data available for the speed UE of second sound pulses in dependence of the absolute temperature 6 [8,10]. A comparison yields that by choice of $+,, it is possible to obtain the right speed at a certain temperature (e.g., the critical temperature Bc, 8, = 16.5 for NaF)2. In the present paper, we show that heat pulses with the right speed are obtained in a satisfactory range of temperatures, moreover, for increasing temperatures, the solutions tend more and more to the classical behaviour. ‘Observe that the same was true in Maxwell-Cattaneo-Vernotte models, but for the heat flux vector q itself, not for a scalar (temperature-like) parameter. 2Throughout this paper, we wilt use only dimensionless variables. The values of constants and the parameters of functions correspond to the following units: temperature (0 and @) rn OK, length m cm, time in ,US, energy in J.
Determination
2. APPROXIMATION AND
of Material Functions
21
OF HEAT CONDUCTIVITY WAVE SPEED
At very low temperatures for both the specific heat c+,(S) and the heat conductivity usually a third order monomial is assumed, cf. (7),(S) [1,8]. It, is easy to see that in that case finally only the ratio c = Q/Q
K(O),
enters the model equations.
However, an analysis of the experimental data [S] shows that ~(0) reaches a maximum. The coincidence of that maximum with the most clearly visible second sound appearance leads to the conjecture that it could be essential to substitute (8) with a function that fits better to the measured values of K. A family of functions suitable for modelling the behaviour of K is given by
~(0) = e
p+b
In .5+r
I”2
B
(11)
REMARK 1. We found this form thanks to the observation that in a double logarithmic scale, the data for K.are arranged nearly on a parabola.
In the sense of least squares method, we obtained the best fit of the data from [S] with a = -7.150703,
b = 6.530065,
c = -1.204074.
For the minimization of the error sum, we used a quasi-Newton algorithm (cf. [ll]). In Figure 1, a comparison of experimental data and the theoretical curve is shown, while Figure 2 dismantles that the range of validity of (8) IS rather short and all on the left-hand side of the critical temperature.
Figure 1. Approximation
of n
Of course, if we introduce the above form (11) of the heat conductivity, the ratio of K. and d(e) is no longer constant. There is actually no evidence that Debye’s law (7) was not correct.3 Besides the data for heat conductivity and heat capacity, we need some information from which to deduce the most crucial constitutive element of the semi-empirical model, i.e., the right-hand side f of the evolution equation, or equivalently Q-= l/fe. To this end, we use the measured arrival times of the heat pulses in order to identify 7, and hence, f experimentally. C v=
3A negligible dependence
of rc on ~9was reported in [Q]
V. A. CIMELLIAND
22
400
K. FRISCHMUTH
-
300 -
200 -
100 -
Figure 2. Comparisonof 6s. We denote by U,, the speed of a wave running into material at an equilibrium state, i.e., q G 0. As it was observed in [lo], the factor l/U; enters the expression for the relaxation time in their model, and the same is true for ours. A fit of that factor for the range 10.. .20 (data taken again from [S]) with A + BP leads to the parameters
A = 9.09,
B = 0.00222,
n = 3.1.
However, the wave speed UE itself (cakulated from the data or from A + BP)
is almost ideally
linear, we found the best fit with U, = Q:+ 713for ct = 0.41943,
y = -0.0127398.
Figure 3 shows that the difference between the two fitted curves is smaller than the variation of the data. Thus, we use the simpler linear approximation.
Figure 3. Approximation
of UE.
Determination of Material Functions
3. THERMODYNAMICAL
RESULTS
The present section is devoted to investigating the compatibility second law of thermodynamics,
in the form of the Clausius-Duhem 7j +r$J
The following two statements THEOREM 1. Constitutive
23
of equations (2)-(6)
with the
inequality
+;ve
(12)
are easily proved.
assumptions (3),(4).
and (6) are compatible
with (12) if and onlv if (13) (14) (15)
THEOREM 2.
Constitutive
assumptions (3) and (4) are compatible 1c,=
VA(e)
+
with (14) if and only if
+Nvo)2.
(W
Moreover,
1L2(e) := qejT(e)e-l
(17)
and
T(e,p)
(v>‘.
:=
(18)
On the other hand, due to the constitutive assumption (5), T does not depend on ,J, and from (18). it follows a2.f (19) aeap =O. Consequently, 0,
p) = f,(e)
+ f2(0).
(26)
As for the internal energy E, we assumed (4), in particular, Debye’s law (7) is used. By comparison of that equation with W
E=++eq=+eBdB,
(21)
we infer
e(e) =
$04= q - ~~ _ +@&7p)2,
(22)
If we insert the general form of $J, i.e., $ = $1 + $~(0/3)~/2, and observe that the left-hand side of (22) is independent of VP, we finally obtain the conditions
ql - e$$ e2 -
e$$
= E(e), =
(23)
0.
Hence, @Z obviously must be a homogeneous function of degree one, $9 = $&,e, and consequently, by the definition of $2,
K(e)T(e)
= ti2,e2.
(25)
24
V. A. CIMELLI AND K. FRISCHMUTH
So, finally, we obtain kinetic equation4
a thermodynamic
compatibility
T(0) =
From this, fi can be determined
g ( > up to a constant &?, g
We assume
further
f2 = -fi,
hence,
Let us notice that the above expression that the Principle
of Maximum
Entropy
=
-l
condition
for the right-hand
$‘20~2 =m’
(26)
via integration
of
@!r2K(f9).
the integration
constant
(27) finally
of $2 leads to a specific entropy at the equilibrium
and Gurtin [14], we conclude that the equilibrium from the complete set of assumptions
states
is fulfilled.
does not enter the model.5 function
which guarantees
Hence, following Colemann
are asymptotically
stable.
constitutive
-
approximation of K from data (as a function of 8, only), Debye’s law for E with ~0 from [8], (i.e., E is independent of p and VP), dependence of the free energy on 6 and VP (and not on p), f2 = -fi in the additive splitting of the right-hand side of the kinetic equation + fi(P)
Altogether,
law for heat flux: q = -n(@Vp,
-
fl(Q)
side of the
= fl(@
-
f(B, p) =
flU9,
the model is determined up to one scalar parameter ~$2~. Finally, the model contains two constitutive parameters, EO and /~a~, and the constitutive function K : ES++ Ii%++. One should notice that, in particular, due to the compatibility condition (26), the relaxation time T can no longer be assumed to be constant (cf. [15]). Functions ~(0) and fi(0) are plotted in Figure 4.
15
10
5
5
10
15
20
25
30
Figure 4. T fl
Obviously, r is convex in the from (26), it is clear that this for each function K. possessing Now, we observe that until we shall verify the assumptions
range we are interested in, regardless what choice of $9,, . Moreover, result is independent of our approximation of K: the same is true a local maximum. now, we did not use the approximation of UE. In the next section, leading to (26) by comparison of wave speeds.
4Compare this with [12,13]. Note that the relation used there was derived only for K.= constant. 5This assumption is equivalent to the condition that B and p coincide at equilibrium
Determination
4. VERIFICATION
of Material Functions
OF MODEL
One can easily check that the model equations
25
ASSUMPTIONS
can be written
as a hyperbolic
system
in balance
form with the wave speed
for a material at equilibrium, i.e., VP E V0 E 0. From (28), we obtain for the relaxation time T, the following
expression:
We present here a comparison of both functions. Although the qualitative behaviour of T calcula.ted by using the above formula is different from the theoretical one, we observe that in the range of applicability
of our theory,
the curves almost
coincide
(see Figure
5).
3-
2.5
-
2-
1.5
-
l-
o.5p t t........................,.........,
10
11
12
13
14
Figure 5. Comparison
15 of 7s
8.3 Figure 6. Solution under Dilkhlet
conditions.
16
17
V. A. CIMELLI AND K. FRISCHMUTH
26
15.6
15.2
15
8.3 Figure
7. Solution
under
Neumann
conditions.
8.
Figure
5. HEAT
8. Transition
PULSE
to diffusive
regime.
EXPERIMENT
FOR NaF
The experimental setup to be reproduced is the same as described in [l]. The data for the considered IBVP have been chosen to be in accordance with the heat pulse experiment for NaF reported in [8]. At the left-hand side of a specimen of pure NaF, the temperature is raised from the uniform value 80 to the value 8c+ imph (impulse height) for a short time impd (impulse duration). The length of the specimen is 0.83, the impulse height 1.0, and the impulse duration 0.25. The constant eu for NaF is assumed to be equal to 2.3 (cf. [9]). The obtained numerical results are visualized in Figure 6 and Figure 7 both for Dirichlet and Neumann type conditions. It is evident that in the range of temperature observe well-determined second sound signals .
[14,16], i.e., near the critical temperature,
we
Determination
However, dampe?ied We conclude
when
the temperature
is raised
and do not lead to a significant that we observe
Finally, second sound peaks ure 9, this different behaviour the same phenomenon should behaviour of the solution, for
the transition
of Material Functions
to the value
20, then
effect at the right-hand to the diffusive
27
thermal
waves are strongly
side of the specimen
(Figure
8).
regime.
are much more evident in the case of Dirichlet conditions. In Figis shown. Let us notice that, due to the form of heat conductivity, be observed in the range of temperatures below 5, and indeed, the the model considered, is the expected one.
2.
4.
6.
8.
Figure 9. Dirichlet vs. Neumann, right end
6. CONCLUSIONS A phenomenological model of heat conduction, based on the concept of a semi-empirical temperature, has been used to determine the properties of heat pulses at low temperatures. The material functions determining the model have been obtained by fitting experimental data. For the present model, at temperatures near the critical one, second sound waves are made evident, with wave speeds in accordance with the measured arrival times. When we approach to the extremes of the interval of temperatures covered by experiments, second sound signals are dampened and the classical Fourier heat conduction is recovered. Furthermore, there is a different behaviour of heat pulses depending on the boundary conditions. In the case of Nuemann type boundary conditions, i.e., when the heat flux is assigned at the end of the specimen, we observe a much more evident damping of solutions. On the other hand, in the case of Dirichlet type conditions, i.e., when the temperature is assigned on the boundary, then it is possible to observe well-designed heat pulse profiles.
REFERENCES K Frischmuth and V.A. Cimmelli, Numerrcal reconstructron of heat pulse experiments, Int J Eng SC 33 (2). 209-215 (1994) V.A. Cimmelli and W. Kosinski, Nonequilibrium semi-empirical temperature in materrals wrth thermal relaxation, Arch. Mech. 43 (6), 753-767 (1991). C. Cattaneo, Sulla conduzione de1 calore, Attz de1 Semzn. Matem. e Fzs. dell’ Unw. dz Modem 3. 83-101 (1948). M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal. 31, 113-126 (1968). W. Kosidski and P. Perzyna, Analysis of acceleration waves in materials with internal parameters, Arch. Mech. 24 (4), 629-643 (1972).
28 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
V. A. CIMELLI AND K. FIUSCHMUTH V.A. Cimmelli, W. Kosiliski and K. Saxton, A new approach to the theory of heat conduction with finite wave speeds, In Proceedings of VI International Conference on Waves and Stability in Continous Media (Edited by S. Rionero, G. Mulone and F. Salemi), Acireale, Italy, (1991). L. Berg, Private communication, Restock (June 1994). H.E. Jackson and CT. Walker, Thermal conductivity, second sound, and phonon-phonon interactions in NaF, Phys. Rev. Letters 3 (4), 1428-1439, (1971). R. J. Hardy and S.S. Jaswal, Phys. Rev. B 3, 4385-4387 (1971). B.D. Coleman and D.C. Neumann, Implications of a nonlinearity in the theory of second sound in solids, Phys. Rev. B 32, 1492-1498 (1988). J.E. Dennis, Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, (1983). W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn. 5, 3-50 (1993). M. Chester, Second sound in solids, Phys. Rev. 131, 2013-2015 (1963). B.D. Coleman and M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47, 597-613 (1967). K. Frischmuth and W. Kosiriski, Application of the semi-empirical heat conduction theory, In Proceedings of the 30. Polish Conference on Solid Mechanics, Zakopane, Poland, (1994). M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford Clarendon Press, (1954). W. Kosiriski, Elastic waves in the presence of a new temperature scale, In Proceedings of ICJTAM Symp. on Elastzc Wave Propagation (Edited by M.F. McCarthy and M. Hayes), Galway, Ireland, North Holland, (1989).
Cornput. Modellang Vol. 24, No. 12, pp 19-28, 1996 Copyright@1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/96 $15.00 + 0 00
PII: SO895-7177(96)00175-6
Determination of Material Functions through Second Sound Measurements in a Hyperbolic Heat Conduction Theory V.
A.
CIMELLI
Dipartimento
di Matematica.
Universita
della Basilicata
I-85100
Potenza,
Italy
CimelliQpzvx85.cineca.it
K. Fachbereich
FRISCHMUTH
Mathematik,
D-18051
Restock,
Universitlt
Restock
Germany
kurtQsun2.math.uni.rostock.de
(Received February 1996; accepted March
1996)
Abstract-In a previous paper [I], numerical solutions to mitral-boundary value problems sema-empzrzcal model of heat conduction were compared with available experimental results.
for a
In the present paper, we modify the model by introducing more realistic approximations of constltutive functions, based on measured heat conductivities and second sound speeds for NaF at low temperatures (10. 20° K). We achieve good accordance between measured second sound pulses and numerical solutions in the temperature range covered by experiments, and reasonable behaviour even beyond this interval. Especially, a passage to the diffusive regime of the classical Fourier law is possible. Keywords-Second time, Hyperbolic
sound waves, Critical temperature, heat conduction, Diffusive regime.
Semi-empirical
temperature scale, Arrival
1. INTRODUCTION The aim of this paper is to calculate numerical solutions to a hyperbolic system of equations modelling heat conduction in a range of temperatures characterized by the occurrence of heat pulses (second sound waves), and to compare these solutions with experimental data. Such heat pulses occur, e.g., in very pure crystalline solids (Bi, NaF) around the critzcal temperature of the material at hand. IJsually, there is a maximum of heat conductivity, and near this point second sound appears. Due to the increasing heat capacity, the pulse speed decreases with increasing temperature. At a certain distance from the maximum, to both sides, the pulses disappear, and the diffusive regime takes over. In recent years, a new theory of heat conduction was developed by Kosiliski et al. [2] which was designed to be simple, consistent with rational thermodynamics, and yet capable of describing heat pulses. Hence, obviously the new theory had to be hyperbolic, but it must also allow a passage to the classical (parabolic) case given by Fourier’s law. Based on elements of approaches given by [3-51, a theory with an additional scalar state variable p and a kinetic equation for it was introduced. Typeset 19
by AM-w
20
V. A. CIMELLIAND K. FRISCHMUTH
By design, at equilibrium, j3 coincides with the classical absolute temperature B, otherwise fl follows after 0 with a certain delay, controlled by a parameter T (called relaxation time).l This delay introduces hyperbolicity and, if chosen small, controls the passage to the classical case. The equations for the rigid heat conductor are B = ~(~,~),
(I)
c,,8 + div q = r.
(2)
By r we denoted heat sources, and by q the heat flux vector. Moreover, c, = t’(e), where Eis the volumic internal energy, means the specific heat. Besides (1) and (2), the constitutive equations q = -tcvp, Ic =
r;(8),
e=
E(B),
(3)
7ii= IN@, VP>, with the Helmholtz free energy $ and the heat conductivity Classical ~sumptions for these functions E= % Ic. =
K have been postulated.
(Debye’s law),
/Qe3
(8)
have been introduced. Finally, the function
b = f (8,P>,
(9)
where
af -l
(>
7-= 33
00)
can be treated as a variable relaxation time. Well-posedness, characteristic wave speeds, and blow-up have been discussed in [2,6]. Recently, classical analytical solutions have been constructed [7]. Numerical solutions to initial-boundary value problems for bismuth and NaF have been shown to reproduce heat pulses at the corresponding critical temperatures [I]. It was observed in [6] that-under the above restrictions (3)-(5)-the Helmholtz free energy $ must be a homogeneous quadratic function in {V/3], with a coefficient being itself a linear homogeneous function in 8. Further, one infers that the kinetic equation is determined up to an integration constant $sO. Hence, the semi-empirical model with (3)-(6) is d et ermined by the constitutive functions E(Q), ~(6’)) and one additional constant $2,. On the other hand, the functions E and K can be identified from experimental data, e.g., in the case of NaF (cf. [8,9]). But moreover, there are also data available for the speed UE of second sound pulses in dependence of the absolute temperature 6 [8,10]. A comparison yields that by choice of $+,, it is possible to obtain the right speed at a certain temperature (e.g., the critical temperature Bc, 8, = 16.5 for NaF)2. In the present paper, we show that heat pulses with the right speed are obtained in a satisfactory range of temperatures, moreover, for increasing temperatures, the solutions tend more and more to the classical behaviour. ‘Observe that the same was true in Maxwell-Cattaneo-Vernotte models, but for the heat flux vector q itself, not for a scalar (temperature-like) parameter. 2Throughout this paper, we wilt use only dimensionless variables. The values of constants and the parameters of functions correspond to the following units: temperature (0 and @) rn OK, length m cm, time in ,US, energy in J.
Determination
2. APPROXIMATION AND
of Material Functions
21
OF HEAT CONDUCTIVITY WAVE SPEED
At very low temperatures for both the specific heat c+,(S) and the heat conductivity usually a third order monomial is assumed, cf. (7),(S) [1,8]. It, is easy to see that in that case finally only the ratio c = Q/Q
K(O),
enters the model equations.
However, an analysis of the experimental data [S] shows that ~(0) reaches a maximum. The coincidence of that maximum with the most clearly visible second sound appearance leads to the conjecture that it could be essential to substitute (8) with a function that fits better to the measured values of K. A family of functions suitable for modelling the behaviour of K is given by
~(0) = e
p+b
In .5+r
I”2
B
(11)
REMARK 1. We found this form thanks to the observation that in a double logarithmic scale, the data for K.are arranged nearly on a parabola.
In the sense of least squares method, we obtained the best fit of the data from [S] with a = -7.150703,
b = 6.530065,
c = -1.204074.
For the minimization of the error sum, we used a quasi-Newton algorithm (cf. [ll]). In Figure 1, a comparison of experimental data and the theoretical curve is shown, while Figure 2 dismantles that the range of validity of (8) IS rather short and all on the left-hand side of the critical temperature.
Figure 1. Approximation
of n
Of course, if we introduce the above form (11) of the heat conductivity, the ratio of K. and d(e) is no longer constant. There is actually no evidence that Debye’s law (7) was not correct.3 Besides the data for heat conductivity and heat capacity, we need some information from which to deduce the most crucial constitutive element of the semi-empirical model, i.e., the right-hand side f of the evolution equation, or equivalently Q-= l/fe. To this end, we use the measured arrival times of the heat pulses in order to identify 7, and hence, f experimentally. C v=
3A negligible dependence
of rc on ~9was reported in [Q]
V. A. CIMELLIAND
22
400
K. FRISCHMUTH
-
300 -
200 -
100 -
Figure 2. Comparisonof 6s. We denote by U,, the speed of a wave running into material at an equilibrium state, i.e., q G 0. As it was observed in [lo], the factor l/U; enters the expression for the relaxation time in their model, and the same is true for ours. A fit of that factor for the range 10.. .20 (data taken again from [S]) with A + BP leads to the parameters
A = 9.09,
B = 0.00222,
n = 3.1.
However, the wave speed UE itself (cakulated from the data or from A + BP)
is almost ideally
linear, we found the best fit with U, = Q:+ 713for ct = 0.41943,
y = -0.0127398.
Figure 3 shows that the difference between the two fitted curves is smaller than the variation of the data. Thus, we use the simpler linear approximation.
Figure 3. Approximation
of UE.
Determination of Material Functions
3. THERMODYNAMICAL
RESULTS
The present section is devoted to investigating the compatibility second law of thermodynamics,
in the form of the Clausius-Duhem 7j +r$J
The following two statements THEOREM 1. Constitutive
23
of equations (2)-(6)
with the
inequality
+;ve
(12)
are easily proved.
assumptions (3),(4).
and (6) are compatible
with (12) if and onlv if (13) (14) (15)
THEOREM 2.
Constitutive
assumptions (3) and (4) are compatible 1c,=
VA(e)
+
with (14) if and only if
+Nvo)2.
(W
Moreover,
1L2(e) := qejT(e)e-l
(17)
and
T(e,p)
(v>‘.
:=
(18)
On the other hand, due to the constitutive assumption (5), T does not depend on ,J, and from (18). it follows a2.f (19) aeap =O. Consequently, 0,
p) = f,(e)
+ f2(0).
(26)
As for the internal energy E, we assumed (4), in particular, Debye’s law (7) is used. By comparison of that equation with W
E=++eq=+eBdB,
(21)
we infer
e(e) =
$04= q - ~~ _ +@&7p)2,
(22)
If we insert the general form of $J, i.e., $ = $1 + $~(0/3)~/2, and observe that the left-hand side of (22) is independent of VP, we finally obtain the conditions
ql - e$$ e2 -
e$$
= E(e), =
(23)
0.
Hence, @Z obviously must be a homogeneous function of degree one, $9 = $&,e, and consequently, by the definition of $2,
K(e)T(e)
= ti2,e2.
(25)
24
V. A. CIMELLI AND K. FRISCHMUTH
So, finally, we obtain kinetic equation4
a thermodynamic
compatibility
T(0) =
From this, fi can be determined
g ( > up to a constant &?, g
We assume
further
f2 = -fi,
hence,
Let us notice that the above expression that the Principle
of Maximum
Entropy
=
-l
condition
for the right-hand
$‘20~2 =m’
(26)
via integration
of
@!r2K(f9).
the integration
constant
(27) finally
of $2 leads to a specific entropy at the equilibrium
and Gurtin [14], we conclude that the equilibrium from the complete set of assumptions
states
is fulfilled.
does not enter the model.5 function
which guarantees
Hence, following Colemann
are asymptotically
stable.
constitutive
-
approximation of K from data (as a function of 8, only), Debye’s law for E with ~0 from [8], (i.e., E is independent of p and VP), dependence of the free energy on 6 and VP (and not on p), f2 = -fi in the additive splitting of the right-hand side of the kinetic equation + fi(P)
Altogether,
law for heat flux: q = -n(@Vp,
-
fl(Q)
side of the
= fl(@
-
f(B, p) =
flU9,
the model is determined up to one scalar parameter ~$2~. Finally, the model contains two constitutive parameters, EO and /~a~, and the constitutive function K : ES++ Ii%++. One should notice that, in particular, due to the compatibility condition (26), the relaxation time T can no longer be assumed to be constant (cf. [15]). Functions ~(0) and fi(0) are plotted in Figure 4.
15
10
5
5
10
15
20
25
30
Figure 4. T fl
Obviously, r is convex in the from (26), it is clear that this for each function K. possessing Now, we observe that until we shall verify the assumptions
range we are interested in, regardless what choice of $9,, . Moreover, result is independent of our approximation of K: the same is true a local maximum. now, we did not use the approximation of UE. In the next section, leading to (26) by comparison of wave speeds.
4Compare this with [12,13]. Note that the relation used there was derived only for K.= constant. 5This assumption is equivalent to the condition that B and p coincide at equilibrium
Determination
4. VERIFICATION
of Material Functions
OF MODEL
One can easily check that the model equations
25
ASSUMPTIONS
can be written
as a hyperbolic
system
in balance
form with the wave speed
for a material at equilibrium, i.e., VP E V0 E 0. From (28), we obtain for the relaxation time T, the following
expression:
We present here a comparison of both functions. Although the qualitative behaviour of T calcula.ted by using the above formula is different from the theoretical one, we observe that in the range of applicability
of our theory,
the curves almost
coincide
(see Figure
5).
3-
2.5
-
2-
1.5
-
l-
o.5p t t........................,.........,
10
11
12
13
14
Figure 5. Comparison
15 of 7s
8.3 Figure 6. Solution under Dilkhlet
conditions.
16
17
V. A. CIMELLI AND K. FRISCHMUTH
26
15.6
15.2
15
8.3 Figure
7. Solution
under
Neumann
conditions.
8.
Figure
5. HEAT
8. Transition
PULSE
to diffusive
regime.
EXPERIMENT
FOR NaF
The experimental setup to be reproduced is the same as described in [l]. The data for the considered IBVP have been chosen to be in accordance with the heat pulse experiment for NaF reported in [8]. At the left-hand side of a specimen of pure NaF, the temperature is raised from the uniform value 80 to the value 8c+ imph (impulse height) for a short time impd (impulse duration). The length of the specimen is 0.83, the impulse height 1.0, and the impulse duration 0.25. The constant eu for NaF is assumed to be equal to 2.3 (cf. [9]). The obtained numerical results are visualized in Figure 6 and Figure 7 both for Dirichlet and Neumann type conditions. It is evident that in the range of temperature observe well-determined second sound signals .
[14,16], i.e., near the critical temperature,
we
Determination
However, dampe?ied We conclude
when
the temperature
is raised
and do not lead to a significant that we observe
Finally, second sound peaks ure 9, this different behaviour the same phenomenon should behaviour of the solution, for
the transition
of Material Functions
to the value
20, then
effect at the right-hand to the diffusive
27
thermal
waves are strongly
side of the specimen
(Figure
8).
regime.
are much more evident in the case of Dirichlet conditions. In Figis shown. Let us notice that, due to the form of heat conductivity, be observed in the range of temperatures below 5, and indeed, the the model considered, is the expected one.
2.
4.
6.
8.
Figure 9. Dirichlet vs. Neumann, right end
6. CONCLUSIONS A phenomenological model of heat conduction, based on the concept of a semi-empirical temperature, has been used to determine the properties of heat pulses at low temperatures. The material functions determining the model have been obtained by fitting experimental data. For the present model, at temperatures near the critical one, second sound waves are made evident, with wave speeds in accordance with the measured arrival times. When we approach to the extremes of the interval of temperatures covered by experiments, second sound signals are dampened and the classical Fourier heat conduction is recovered. Furthermore, there is a different behaviour of heat pulses depending on the boundary conditions. In the case of Nuemann type boundary conditions, i.e., when the heat flux is assigned at the end of the specimen, we observe a much more evident damping of solutions. On the other hand, in the case of Dirichlet type conditions, i.e., when the temperature is assigned on the boundary, then it is possible to observe well-designed heat pulse profiles.
REFERENCES K Frischmuth and V.A. Cimmelli, Numerrcal reconstructron of heat pulse experiments, Int J Eng SC 33 (2). 209-215 (1994) V.A. Cimmelli and W. Kosinski, Nonequilibrium semi-empirical temperature in materrals wrth thermal relaxation, Arch. Mech. 43 (6), 753-767 (1991). C. Cattaneo, Sulla conduzione de1 calore, Attz de1 Semzn. Matem. e Fzs. dell’ Unw. dz Modem 3. 83-101 (1948). M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal. 31, 113-126 (1968). W. Kosidski and P. Perzyna, Analysis of acceleration waves in materials with internal parameters, Arch. Mech. 24 (4), 629-643 (1972).
28 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
V. A. CIMELLI AND K. FIUSCHMUTH V.A. Cimmelli, W. Kosiliski and K. Saxton, A new approach to the theory of heat conduction with finite wave speeds, In Proceedings of VI International Conference on Waves and Stability in Continous Media (Edited by S. Rionero, G. Mulone and F. Salemi), Acireale, Italy, (1991). L. Berg, Private communication, Restock (June 1994). H.E. Jackson and CT. Walker, Thermal conductivity, second sound, and phonon-phonon interactions in NaF, Phys. Rev. Letters 3 (4), 1428-1439, (1971). R. J. Hardy and S.S. Jaswal, Phys. Rev. B 3, 4385-4387 (1971). B.D. Coleman and D.C. Neumann, Implications of a nonlinearity in the theory of second sound in solids, Phys. Rev. B 32, 1492-1498 (1988). J.E. Dennis, Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, (1983). W. Dreyer and H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn. 5, 3-50 (1993). M. Chester, Second sound in solids, Phys. Rev. 131, 2013-2015 (1963). B.D. Coleman and M.E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47, 597-613 (1967). K. Frischmuth and W. Kosiriski, Application of the semi-empirical heat conduction theory, In Proceedings of the 30. Polish Conference on Solid Mechanics, Zakopane, Poland, (1994). M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford Clarendon Press, (1954). W. Kosiriski, Elastic waves in the presence of a new temperature scale, In Proceedings of ICJTAM Symp. on Elastzc Wave Propagation (Edited by M.F. McCarthy and M. Hayes), Galway, Ireland, North Holland, (1989).