- Email: [email protected]
Energy principles for temperature varied with time
International Journal of Thermal Sciences 120 (2017) 80e85
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Energy principles for temperature varied with time Zhen-Bang Kuang Shanghai Jiaotong University, Shanghai, 200240, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 November 2016 Received in revised form 28 February 2017 Accepted 24 March 2017
This paper discusses some extensions of energy principles in classical and continuum thermodynamics. The modified energy principles will extend the first and second laws in classical continuum thermodynamics to the case where the temperature is varied with time. it is shown that the modified energy principles are consistent with the statistical mechanics. The inertial heat and inertial entropy concepts play the kernel roles in the modified energy principles. The temperature wave equation with finite phase velocity is a natural result of the inertial entropy theory. The effect of the temperature inertia in multifield coupling case is discussed shortly. © 2017 Published by Elsevier Masson SAS.
Keywords: Modified energy principles Thermodynamic laws Inertial heat and inertial entropy Statistical mechanics Temperature wave
1. Introduction When the temperature is varied in space, the irreversible heat conduction is discussed extensively [1e3]. When the temperature varies with time, in the classical heat conduction theory the temperature propagates subjected to the parabolic differential equation and with infinite phase speed. However the second sound speed observed by experiments in liquid helium [4] and later in solids [5] at low temperature urges the development of the hyperbolic temperature wave equation with finite phase speed. The applications of short pulse lasers in the fabrication of sophisticated microstructures, syntheses of advanced materials and measurements of the thin-film properties [6] and applications in bioengineering [7] are the new active research areas. In the problems with microscopic size and time, especially at low temperature, the temperature satisfies the wave equation. Therefore many generalized thermodynamic theories, especially the generalized heat conduction theories, were proposed to allow a finite propagation speed for the temperature [7,8]. In this paper the following notations are used: a comma followed by index i indicates partial differentiation with respect to xi , such as ui;j ¼ vui =vxj ; a superimposed dot indicates partial differentiation with respect to time, such as u_ ¼ du=dt. For the small deformation case vðÞ=vt ¼ dðÞ=dt. In this paper the Einstein summation convention, i.e. repeated subscripts of a variable imply
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijthermalsci.2017.03.030 1290-0729/© 2017 Published by Elsevier Masson SAS.
summation over the range of that subscript, such as ui;i ¼ u1;1 þ u2;2 þ u3;3 in three dimensional space, is also used. For the isotropic media, the first temperature wave theory was proposed by Cattaneo [9,10] and then by Vernotte [11]. They proposed the following C-V heat conduction equation:
qi þ t0 q_ i ¼ lw;i
(1)
where q is the heat flow vector, t0 is a relax time constant, w ¼ T T0 , where T is the Kelvin temperature and T0 is a reference temperature, such as the environment temperature, l is the heat conduction coefficient. In usual pure heat conduction theory we have
u_ ¼ g_ qi;i ¼ C w_
(2)
where u is the internal energy, C is the specific heat per volume and g is the internal heat source strength. Combing Eqs. (1) and (2) we get a hyperbolic equation for temperature:
€ ¼ g_ þ t0 g € þ lw;ii C w_ þ t0 w
(3)
C-V theory is applied extensively. From Eqs. (1) and (2) it can be seen that the entropy density is not a state function which is not consistent with the classical thermodynamics. Jou et al. [12] proposed the extended irreversible thermodynamics to mend this difficulty and other problem in nonequilibrium thermodynamics by modifying the classical entropy formula and introducing the
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
81
nonequilibrium temperature. Green and Laws [13] used a new temperature function _ T ¼ fðT; 0Þ instead of the Kelvin temperature T, but the f ¼ fðT; TÞ; concrete relation is related to the other assumptions. They use a new free energy g ¼ u fs and modified the Clausius-Duhem R R R _ inequality as V sdV V ðg=fÞdV þ a ðqi =fÞni da 0, where n is the outer normal of a surface a of the volume V, s is the entropy density per volume. On this basis Green and his coworker discussed the thermoelasticity under finite deformation. After simplification they proposed a temperature wave equation with two parameters. Obviously the free energy g in the Green- Laws theory is not a state function and is also not consistent with the classical thermodynamics. Guo et al. [14] proposed the microscopic thermomass theory based on the assumption that the thermal energy is equivalent to a small “ thermomass ” and the Einstein's mass-energy equivalence relation is used. In dielectric materials they treated the thermomass as the phonon gas and the heat transport is considered as the motion of phonon gas. Using this theory they derived the generalized heat conduction law and the nonequilibrium temperature. There are also several other theories for the temperature wave equation. These theories in most of literature were mainly established through modified Fourier heat conduction equation or introduced new temperature function and modified the ClausiusDuhem inequality [7,8]. In Eq. (1) the term containing t0 is a viscous term for the heat flow-temperature equation, however in the temperature equation Eq. (3) the term containing t0 is obviously an inertial term. Such being the case, why we do not directly introduce the concept of the temperature inertia to simplified the problem? In literature [15e17] and some other papers Kuang proposed a new inertial entropy theory (IET), directly introducing the temperature inertia to derive the temperature wave equation. This method is based on the universal modified classical thermodynamic principles.
€ is the inertial force according to the D'Alembert's prinwhere ru _ ðaÞ is the work rate done by the inertial force. ciple and W In literature [15e17] the author considered that the variation of temperature with time should be supplied extra heat from the environment to overcome the heat or temperature inertia. From this idea the temperature wave equation will be derived easily and the definition of usual temperature and the Fourier's law are still kept. Though many literature have been pointed out that “heat may have inertia”, such as Nernst [18], Onsager [19], Kaliski [20], but these authors only used this concept indirectly through the generalized heat conducting equation. When the temperature varies with time, in literature [15e17] the author proposed to modify the first thermodynamic law expressed in Eq. (7) to
2. Modified energy principles and inertial entropy
ðaÞ ¼ Q_
The classical thermodynamics discusses thermal process in an equilibrium state, i.e. the temperature is constant in the four dimensional (space-time) space. The first classical thermodynamic law is
_ þ Q_ ; W _ ¼ U_ ¼ W U_ ¼
Z
Z
Z f $vdV þ
aZ s
VZ
_ udV; Q_ ¼
q$nda þ aq
V
T$vda
g_ dV; v ¼ u_
(4)
Zb Sb Sa
Z dQ =T; S ¼
a
sdV
(5)
V
where S is the entropy in the equilibrium state, Sb and Sa are the values of the entropy at states a and b, s is the entropy density per volume. In Eq. (5) the symbol “ ¼ ” is used for the reversible process and the symbol “>” is for the irreversible process. In classical thermodynamics for a moving body Eq. (4) is changed to
rv$vdV
K ¼ ð1=2Þ
(6)
V
where K is the kinetic energy. Eq. (6) can also be written as
_ ðaÞ ¼ _ W _ ðaÞ þ Q_ ; W U_ ¼ W
Z
€ $vdV ru
(7)
V
_ W _ ðaÞ þ Q_ Q_ ðaÞ U_ ¼ W
(8)
where Q ðaÞ is the inertial heat. Eq. (8) shows that the supplied heat dQ by the surrounding cannot be entirely absorbed by the system immediately, but only dQ dQ ðaÞ, where dQ ðaÞ is used to overcome the temperature inertia. dQ ðaÞ may be positive or negative. In Refs. [15e17] author considered that the inertial heat is induced by the temperature inertia, so assumed
Q ðaÞ ¼
Z V Z
€ C rs0 TdV ¼
Z
€ rs TdV
V
(9)
_ C rs0 TdV
V
where rs0 is an inertial entropy time constant and rs ¼ C rs0 . Eq. (9) shows that, when T_ > 0, Q ðaÞ > 0, i.e. when the temperature increases, the inertial heat is positive; on the contrary, T_ < 0, Q ðaÞ < 0. Correspondingly we can introduce the concept of the inertial ðaÞ entropy rate S_ as
V
where U; W and Q are internal energy, work done by the external force and heat supplied by the environment respectively, v is the velocity, u is the displacement vector. f is the body force, T is the surface traction, as and aq are respectively the parts giving external force and heat flow of a. The second classical thermodynamic law is
Z
_ þ Q_ ; U_ þ K_ ¼ W
ðaÞ ðaÞ ¼ T S_ ¼ Q_
Z
T s_ðaÞ dV;
V
€ T s_ðaÞ ¼ C rs0 T;
sðaÞ ¼
Zt
(10) s_ðaÞ dt
0
where sðaÞ is the inertial entropy density per volume, which is only related to the temperature variation with time and does not connected with any dissipative process. The second thermodynamic law can now be modified to
Zb Sb Sa a
dQ dQ ðaÞ T
(11)
Correspondingly the classical entropy equation is modified to [15e17]
82
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
T s_ðtÞ ¼ g_ ðT h_ i Þ;i ; T h_ i ¼ qi ; s_ðtÞ ¼ s_ þ s_ðaÞ ; s_ ¼ s_ðrÞ þ s_ðiÞ g_ ðrÞ s_ þ s_ðaÞ ¼ h_ i;i ; T ðiÞ T s_ ¼ T s ¼ T s_ s_ðrÞ ¼ h_ i T;i 0
(12)
In Eq. (12) sðtÞ is the total entropy density per volume, s is the usual entropy density in the thermodynamic equilibrium state and it is still a “state function”, sðrÞ and sðiÞ ¼ s are the reversible part and the irreversible part of s. T s_ðaÞ is the inertial heat rate per volume; h is the entropy displacement vector, h_ is the entropy flow vector. If the inertial entropy is omitted, Eq. (12) is reduced to the entropy equation of the usual irreversible thermodynamics. In the inertial entropy theory, T s_ þ T s_ðaÞ in Eq. (12) is used to instead of T s_ in the usual irreversible thermodynamics. Using Ti ¼ sji nj and the Gauss divergence theorem, from Eqs. (4), (8) and (10) we get
Z V
_ udV ¼
Z
€ i Þvi dV þ ðfi ru
Z
sji ni vj da
Z Z Z qi ni da þ g_ dV T s_ðaÞ dV as
V
aZ
¼
V
€ i Þvi dV þ ðfi ru
ZV
sji;i vj dV
(13)
V Z ZV _ þ sji vj;i dV þ T sdV V
V
In Eq. (13) sij is the stress, ε_ ij ¼ ðvi;j þ vj;i Þ=2 is the strain rate tensor and εij ¼ ðui;j þ uj;i Þ=2 is the strain tensor. By using the motion equation
sji;j þ fi ru€ i ¼ 0
(14)
From Eq. (13) the energy equation is
u_ ¼ sji vj;i þ T s_
(15)
Eq. (15) shows that the internal energy density u is the function of the state variables εij and s only and is not related to the inertial entropy density. The conjugate variables of ðεij ; sÞ are ðsij ; TÞ. Eq. (15) is just the same as that in the usual continuum thermodynamics. Using the theory of the linear irreversible thermodynamics [1,21] from Eq. (12) we can get
T h_ i ¼ qi ¼ kij ðx; TÞT;j
(16a)
where k is the heat conductive coefficient. Eq. (16a) is just the Fourier's law. The linear relation between the irreversible force and flow can be extended to a general expression:
h_ i ¼ f x; t; T; T;i ; T_
(16b)
From Eq. (16b) we can also get a similar heat conduction equation with the C-V equation. _ so from Eqs. (12) For pure heat conduction we have u_ ¼ T s_ ¼ C w, and (16a) we get
€ ¼ g_ þ kij ðx; tÞw;ji C w_ þ rs0 w
(17)
Eq. (17) is the result of IET. For the isotropic heat conduction kij ¼ kdij . Let g ¼ 0, Eq. (17) is identical with Eq. (3); however when gs0, they are different. As an example, we discuss a case that the temperature is homogeneous in space, but varies with time [17]. In _ ¼ this case Eqs. (3) and (17) are respectively reduced to Cðw þ t0 wÞ _ ¼ g. The former is an algebraic g þ t0 g_ or Cw ¼ g and Cðw þ rs0 wÞ equation and its response to external reaction is instantaneous, but the later is a differential equation in first order and its response to external reaction is delayed. It is also noted that when we use the IET, the generalized heat conduction theories can be simultaneously used to instead of Fourier theory, because the IET is a universal energy principle and the generalized heat conduction theories are the state equations or the constitutive equations. From Eqs. (8) and (12) it can be seen that if we introduce the inertial heat Q ðaÞ and subtract it from the external supplied heat, or we introduce the inertial entropy density rate s_ðaÞ and use s_ðrÞ þ s_ðaÞ to instead of s_ðrÞ in the usual entropy equation, the IET can deal with the local nonequilibrium thermodynamics where the temperature varied with time, by the framework of the classical equilibrium thermodynamics [17]. This is similar to the D'Alembert principle in the mechanical dynamics. The modified energy principles can be extended to more complex cases, such as to the thermodiffusion problem [16,22], where the concentration inertia is introduced, etc.
3. The statistical mechanics foundation of the inertial heat and inertial entropy theories The inertial heat and inertial entropy theories are consistent with the statistical mechanics from the material microscopic theory. In this paper we give a simple discussion and the more rational microscopic theory need studied in the future. The kinetic theory of molecules [2,3] shows that, the statistical average value 2 of the kinetic energy 2 of gas molecules at time t is 2 ¼ ð1=2Þmv2 , where m is the mass of a molecule, v is its random motion velocity, v2 is the mean value of v2 . For the gas we have
Z dn dv1 dv2 dv3 2 ¼ ð1=2Þmv2 ¼ ð1=2Þm v2 n m 3=2 Z∞ Z∞ Z∞ 2 ¼ 2pm emv =2kT v4 dv 2pkT 0
0
0
3 kT; v2 ¼ v$v 2 dn m 3=2 mv2 =2kT ¼ e dv1 dv2 dv3 n 2pkT 3 E ¼ N2 ¼ NkT 2 ¼
(18)
where k is the Boltzmann constant, n ¼ N=V, N is the total number of the gas molecules in V, n is the molecule number per volume, dn is the molecule number with velocity v, v is the velocity of a molecule, E is the total energy of molecules in V. Eq. (18) shows that the Kelvin temperature is a physical quantity representing the average kinetic energy of molecules. In the first line of Eq. (18) the second equality is an expression in the Cartesian coordinate, but the third equality is an expression in the sphere coordinate [2,3]. Now we discuss a varying isothermal system, i.e. all the molecules in the system have same temperature, but it may be varied with time. Let a molecule be in an equilibrium state “0” with temperature T0 , velocity v0 at time t0 . From time t0 to t, the system absorbs heat DQ from the environment, so the molecule is accelerated to a new state with temperature T and acceleration v_ at time
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
t. If DQ is infinite small, the acceleration is also infinite small, or _ v/0, the system is quasi-static continuously transited from one state to the neighboring with equilibrium temperature Teq at teq /∞ (on the theoretical meaning). If DQ is finite, the process from one state at t0 with temperature T0 to another state at t with instantaneous temperature T. In this case the increment of its in· Rt ternal energy t0 N2 dt is different with DQ because the system is in the accelerated state and it may also take some time to transmit the heat from the heat source to the system by the molecule motion. The difference between DQ and the internal energy will be called the inertial heat denoted by Q ðaÞ. So at time t we have
energy rate need expend a part of the work rate supplied by the external force. The inertial energy rate in Eq. (21) is expressed by the acceleration, the second derivative of displacement, but in Eq. (20) it is expressed by the second derivative of temperature, because the former discusses the ordered motion of the centroid of a microscopic element, but the latter discusses the molecular random motion. If we adopt the modified energy principle Eq. (8) or (15), the Boltzmann relation in the regular distribution theory of statistical mechanics may be still expressed as
S ¼ k ln W Zt
DQ ¼
·
N2 dt þ Q ðaÞ
(19)
t0
After t, we move the heat source and after infinite time Q ðaÞ is fully transformed to the internal energy and the system reaches a new equilibrium state with temperature Teq > T if T_ > 0 or Teq < T if T_ < 0 at time teq /∞. In Refs. [12,14] authors introduced the nonequilibrium temperature related to q,q; in Ref. [13] authors introduced _ The instantaneous a new temperature function f related to T. temperature T in this paper is automatically related to T_ as T ¼ T0 þ Rt _ 0 Tdt and a few numerical examples can be seen in literature [17]. T is consistent with the temperature measured in practice. But the difference between T and Teq is small due to small molecular inertia. Though we are lack of experimental data to determine Q ðaÞ , but experience shows that Q ðaÞ has the· same sign and the similar varying speed trends with DQ and 2 , so we can assume Q ðaÞ · proportional to 2 . Because the average kinetic energy of molecule motion is proportional to Kelvin temperature, we get ðaÞ _ € Obviously Q_ ðaÞ is related to the acceleration Q ðaÞ fT; Q_ fT. distribution function of molecules and the real situation is more complex. The more exact theory should be continuously studied in the future. According to the above discussions we can assume
Q Q ðaÞ ¼ N 2 ¼ U; · ðaÞ _ ¼ N 2 ¼ U; Q_ Q_
or ðaÞ Q_ fT€ ðt Þ
(20)
Eq. (20) is a macroscopic theory and consistent with Eq. (8). It shows that the inertial heat and inertial entropy theories are consistent with the molecule kinetic theory in the statistical mechanics in some aspect. Though Eqs. (19) and (20) are inferential, but there are also some similar theories in statistical mechanics, such as in Smoluchowski- Einstein fluctuation theory they generalized the relation between the entropy and the thermodynamic probability in equilibrium state to fluctuation case [2,3]; In discussion the reciprocal theorem Onsager also assumed that the attenuation of the fluctuation at equilibrium can be used to the nonequilibrium case where the state is slightly departure from the equilibrium state [1,19]. At the last part of this section we discuss some similarity with the continuum dynamics. In the continuum dynamics the energy equation can be written as
Z
Z dudV ¼
V
Z fk duk dV þ
V
Z Tk duk da
as
ru€ k duk dV
(21)
V
Eq. (21) shows that internal energy rate of a system is equal to the work rate supplied by the environment minus the work rate to overcome the inertial force. This phenomenon is similar to the heat conduction. In Eq. (20) The inertial energy rate need expend a part of heat rate supplied from the environment. In Eq. (21) the inertial
83
(22)
but where S is equal to the total entropy minus the inertial entropy, as shown in Eqs. (10) and (12). In Eq. (22) W is the thermodynamic probability [2,3]. W is proportional to the distribution fal g, in fal g al molecules have the same energy and satisfies the following conditions:
X al ¼ N; l
X 2l al ¼ E
(23)
l
4. Effect of the temperature inertia in multi-field coupling case Now we discuss the effect of the temperature inertia in multifield coupling case. As a simple coupling case we discuss the thermo-elasticity. In practical applications the free energy is more convenient, so in the following part we use the free energy gðεkl ; wÞ ¼ uðεkl ; sÞ Ts. In the thermo-elasticity the elastic behavior is reversible, only the heat behavior is irreversible. The thermo-elastic theory was studied in many literature, such as in Refs. [16,17,23]. In Refs. [16,17] the free energy gðεkl ; wÞ and the constitutive equations of ðsij ; εij Þ and ðs; TÞ are assumed:
1 Cw2 gðεkl ; wÞ ¼ Cijkl εji εlk aij εij w 2 2T vg sij ¼ ¼ Cijkl εkl aij w; vεij vg Cw C w 2 s ¼ ¼ aij εij þ vw T 2 T
(24)
where Cijkl ¼ Cjikl ¼ Cijlk ¼ Cklij is the elastic coefficient, aij ¼ aji is the stress-temperature coefficient. From Eqs. (12), (16) and (24) and sðaÞ in (10) we get:
€ ¼ g_ qi;i ; qi ¼ lij w;j ; T s_ þ C rs0 w w w 1 s ¼ aij εij þ C T 2T w w 1 s_ ¼ aij ε_ ij þ a_ ij εij þ C_ T 2T $ $ w w w w þC C 1 T 2T 2T T
(25)
w_ w zaij ε_ ij þ a_ ij εij þ C_ þ C T T For the pure heat conduction case with constant material coefficients, Eq. (25) is reduced to (17). For an isotropic material we have
84
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
Cijkl ¼ ldij dkl þ G dik djl þ dil djk
aij ¼ adij Cijkl ; kij ¼ kdij (26)
sij ¼ lεkk dij þ 2Gεij adij w s_ ¼ aε_ jj þ a_ εjj þ C_
w w_ þC T T
where l and G are Lame constants. When the variation of temperature w is not large and the material behavior is independent to temperature the complete governing equation system without external force and internal heat source is
€ þ w_ þ aT0 ε_ ii kw;ii ¼ C rs0 w ru€ i ¼ sji;j ¼ luj;ij þ 2Gui;jj aw;i
(27)
The second in Eq. (27) is the motion equation. For simplicity, we discuss 1-D isotropic problem [16,17]:
ru€ Yu;11 þ aw;1 ¼ 0 € þ w_ kw;11 þ aT0 u_ ;1 ¼ 0 C rs0 w s ¼ Yε aw; s ¼ aε þ Cw=T0
(28)
where Y is the elastic modulus, a is the stress-temperature coefficient, u and s are the displacement and stress along x1 respectively. For a plane traveling wave let
u ¼ U exp½iðkx utÞ; q ¼ Q exp½iðkx utÞ;
k ¼ k1 þ ik2 c ¼ u=k1
(29)
where U and Q are the amplitudes of u and w respectively, k is the wave number and u is the circular frequency, c is the wave speed. Substituting Eq. (29) into (28), we have
2 Yk ru2h U þ iakQ ¼ 0
i
(30)
aT0 kuU þ kk2 C rs0 u2 þ iu Q ¼ 0 In order to U and Q have nontrivial solutions, it must
2 Yk ru2 aT ku 0
¼ 0; 2 2 kk C rs0 u þ iu iak
Ak4 Bk2 þ M ¼ 0; k2 ¼
(31)
M ¼ M1 þ iM2 ¼ rCðrs0 u þ iÞu3 B ¼ B1 þ iB2 ¼ ðYC rs0 þ rkÞu2 þ i YC þ a2 T0 u
A ¼ Yk
For the plane temperature wave the wave speed is very slowly, so the sign “þ” in Eq. (31) is used for the temperature wave and the temperature wave is always attenuated. Sign “” in Eq. (31) is for the elastic wave, so does the elastic wave be attenuated or amplified, it is determined by the sign of k2 , the imaginary part of k. For a coupling thermo-elastic wave when the temperature is varied, the inertial effect of the heat or temperature will hold back the change of the temperature and elastic waves. In the plane wave problem under following condition the amplitude of the elastic wave may be magnified:
k2 ¼ Imk ¼ Im
i 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B± B2 4AM 2A
1 2
<0
5. Conclusions In this paper we show that when the temperature is varied with time, the heat or the temperature inertia should be considered. The concepts of the heat inertia, temperature inertia and the inertial entropy are consistent with the statistical mechanics. The modified laws of the thermodynamics are the natural extension of the classical thermodynamics and should be checked by experiments in the future. The effect of the temperature inertia in multi-field coupling problems are also discussed shortly. It is also shown that for a general thermodynamic problem we can use the inertial entropy theory to discuss the variation of the temperature with time and use the Fourier or generalized Fourier law to discuss the variation of the temperature in the space. References
or
i 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B± B2 4AM 2A
inertia is similar. For a coupling thermo-elastic wave when the temperature is varied with time, the inertial effect of the temperature (or heat) will hold back the change of the temperature and elastic waves. Under some conditions the amplitude of the elastic wave may be magnified [24,25]. If we consider the material coefficients are the functions of the temperature, we can obtain a nonlinear equation system and the temperature wave and elastic wave will combined an elastic-temperature wave propagated with one wave speed. How to understand the temperature wave is still a problem and will be discussed in the future. In general the engineering composites are viscous materials, so the viscosity should be considered [16,17,25,27,28]. According to the irreversible thermodynamics, in Refs. [16,17,21] the author divided the stress s into reversible part sðrÞ and irreversible part sðiÞ : s ¼ sðrÞ þ sðiÞ , sðrÞ is determined by the state function, the free energy g, as shown in Eq. (24); sðiÞ is determined by the theory of the irreversible flow and force. Zhou and Yang [25] discussed the thermo-viscoelastic effect of temperature wave in detail. The amplified effect of the elastic wave due to the inertia of temperature wave may be small comparing with the viscous effect. So if we consider viscosity the elastic wave may also be always attenuated in engineering.
(32)
The thermo-electroelasticity was also discussed in many literature, such as in Refs. [15,23e26] and the effect of the temperature
[1] De Groot SR, Mazur P. Nonequilibrium thermodynamics. New York: Dover Publications; 1984. [2] Pathria RK, Beale PD. Statistical mechanics. third ed. Academic Press; 2011. [3] Wang Z-C. Thermodynamics,Statistical physics. fifth ed. Beijing: Higher Education Press; 2013 (in Chinese). [4] Landau L. The theory of superfluidity of helium II. J Phys 1941;5:71. [5] Jackson HE, Walker CT. Thermal conductivity, second sound and phononphonon interactions in NaF. Phys Rev B 1971;3:1428. [6] Marciak-kozlowska J, Mucha Z, Kozlowski M. Picosecond thermal pulses in thin gold films. Int J Thermophys 1995;16:1489. [7] Straughan B. Heat waves. Berlin: Springer; 2011. [8] Joseph DD, Preziosi L. Heat waves. Rev Mod Phys 1989;61:41e73. [9] Cattaneo C. Sulla conduzione de calore. Atti Del Semin Mat E Fis Univ Modena 1948;3:83. quation de la chaleur e liminant le paradoxe [10] Cattaneo C. Sur une forme de 10 e mie Sci Paris d' une propagation instantanee. Comptes Rendus 10 Acade 1958;247:431. orie continue de l’equation de la chaleeur. [11] Vernotte P. Les paradoxes de la the mie Sci Paris 1958;246:3154. Comptes Rendus 10 Acade [12] Jou D, Casas-Vazquez J, Lebon G. Extended irreversible thermodynamics. third ed. Berlin, Heidelberg: Springer; 2001. [13] Green AE, Laws N. On the entropy production inequality. Arch Ration Mech Anal 1972;45:47. [14] Dong Y, Cao B-Y, Guo Z-Y. Temperature in nonequilibrium states and nonFourier heat conduction. Phys Rev E 2013;87:032150. [15] Kuang Z-B. Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mech 2009;203:1. [16] Kuang Z-B. Some thermodynamic problems in continuum mechanics. In: n CM, editor. Thermodynamics - kinetics of dynamic systems. Croatia: Piraja Intech open access publisher; 2011. [17] Kuang Z-B. Discussions on the temperature wave equation. Int J Heat Mass Transf 2014;71C:424.
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85 [18] Nernst W. Die theoretischen und experimentellen grundlagen des n warmestatzes,Knapp, Halle. 1917. [19] Onsager L. Reciprocal relations in irreversible processes. Phys Rev 1931;37: 405e26. mie Polonaise [20] Kaliski S. Wave equation of heat conduction. Bulletin de l’Acade rie Des Sci Tech 1965;13:211e9. des Sciences. Se [21] Kuang Z-B. Nonlinear continuum mechanics. Shanghai: Shanghai Jiaotong university press; 2002 (in Chinese). [22] Kuang Z-B. Variational principles for generalized thermodiffusion theory in pyroelectricity. Acta Mecha 2010;214:275e89. [23] Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity.
85
J Mech Phys Solids 1967;15:299e309. [24] Yuan X-G, Kuang Z-B. Waves in pyroelectrics. J Therm stress 2008;31:1190. [25] Zhou Z-D, Yang F-P. Plane waves in pyroelectrics with viscous effect. Acta Mech 2014;225:509. [26] Kuang Z-B. Theory of electroelasticity. Shanghai, Berlin: Shanghai Jiaotong University Press, Springer; 2013. [27] Ezzat MA, Othman MI, El-Karamany AS. State space approach to generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 2002;40:283. [28] Lionetto F, Montagna F, Maffezzoli A. Ultrasonic dynamic mechanical analysis of polymers. Appl Rheol 2005;15:326.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Energy principles for temperature varied with time Zhen-Bang Kuang Shanghai Jiaotong University, Shanghai, 200240, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 November 2016 Received in revised form 28 February 2017 Accepted 24 March 2017
This paper discusses some extensions of energy principles in classical and continuum thermodynamics. The modified energy principles will extend the first and second laws in classical continuum thermodynamics to the case where the temperature is varied with time. it is shown that the modified energy principles are consistent with the statistical mechanics. The inertial heat and inertial entropy concepts play the kernel roles in the modified energy principles. The temperature wave equation with finite phase velocity is a natural result of the inertial entropy theory. The effect of the temperature inertia in multifield coupling case is discussed shortly. © 2017 Published by Elsevier Masson SAS.
Keywords: Modified energy principles Thermodynamic laws Inertial heat and inertial entropy Statistical mechanics Temperature wave
1. Introduction When the temperature is varied in space, the irreversible heat conduction is discussed extensively [1e3]. When the temperature varies with time, in the classical heat conduction theory the temperature propagates subjected to the parabolic differential equation and with infinite phase speed. However the second sound speed observed by experiments in liquid helium [4] and later in solids [5] at low temperature urges the development of the hyperbolic temperature wave equation with finite phase speed. The applications of short pulse lasers in the fabrication of sophisticated microstructures, syntheses of advanced materials and measurements of the thin-film properties [6] and applications in bioengineering [7] are the new active research areas. In the problems with microscopic size and time, especially at low temperature, the temperature satisfies the wave equation. Therefore many generalized thermodynamic theories, especially the generalized heat conduction theories, were proposed to allow a finite propagation speed for the temperature [7,8]. In this paper the following notations are used: a comma followed by index i indicates partial differentiation with respect to xi , such as ui;j ¼ vui =vxj ; a superimposed dot indicates partial differentiation with respect to time, such as u_ ¼ du=dt. For the small deformation case vðÞ=vt ¼ dðÞ=dt. In this paper the Einstein summation convention, i.e. repeated subscripts of a variable imply
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijthermalsci.2017.03.030 1290-0729/© 2017 Published by Elsevier Masson SAS.
summation over the range of that subscript, such as ui;i ¼ u1;1 þ u2;2 þ u3;3 in three dimensional space, is also used. For the isotropic media, the first temperature wave theory was proposed by Cattaneo [9,10] and then by Vernotte [11]. They proposed the following C-V heat conduction equation:
qi þ t0 q_ i ¼ lw;i
(1)
where q is the heat flow vector, t0 is a relax time constant, w ¼ T T0 , where T is the Kelvin temperature and T0 is a reference temperature, such as the environment temperature, l is the heat conduction coefficient. In usual pure heat conduction theory we have
u_ ¼ g_ qi;i ¼ C w_
(2)
where u is the internal energy, C is the specific heat per volume and g is the internal heat source strength. Combing Eqs. (1) and (2) we get a hyperbolic equation for temperature:
€ ¼ g_ þ t0 g € þ lw;ii C w_ þ t0 w
(3)
C-V theory is applied extensively. From Eqs. (1) and (2) it can be seen that the entropy density is not a state function which is not consistent with the classical thermodynamics. Jou et al. [12] proposed the extended irreversible thermodynamics to mend this difficulty and other problem in nonequilibrium thermodynamics by modifying the classical entropy formula and introducing the
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
81
nonequilibrium temperature. Green and Laws [13] used a new temperature function _ T ¼ fðT; 0Þ instead of the Kelvin temperature T, but the f ¼ fðT; TÞ; concrete relation is related to the other assumptions. They use a new free energy g ¼ u fs and modified the Clausius-Duhem R R R _ inequality as V sdV V ðg=fÞdV þ a ðqi =fÞni da 0, where n is the outer normal of a surface a of the volume V, s is the entropy density per volume. On this basis Green and his coworker discussed the thermoelasticity under finite deformation. After simplification they proposed a temperature wave equation with two parameters. Obviously the free energy g in the Green- Laws theory is not a state function and is also not consistent with the classical thermodynamics. Guo et al. [14] proposed the microscopic thermomass theory based on the assumption that the thermal energy is equivalent to a small “ thermomass ” and the Einstein's mass-energy equivalence relation is used. In dielectric materials they treated the thermomass as the phonon gas and the heat transport is considered as the motion of phonon gas. Using this theory they derived the generalized heat conduction law and the nonequilibrium temperature. There are also several other theories for the temperature wave equation. These theories in most of literature were mainly established through modified Fourier heat conduction equation or introduced new temperature function and modified the ClausiusDuhem inequality [7,8]. In Eq. (1) the term containing t0 is a viscous term for the heat flow-temperature equation, however in the temperature equation Eq. (3) the term containing t0 is obviously an inertial term. Such being the case, why we do not directly introduce the concept of the temperature inertia to simplified the problem? In literature [15e17] and some other papers Kuang proposed a new inertial entropy theory (IET), directly introducing the temperature inertia to derive the temperature wave equation. This method is based on the universal modified classical thermodynamic principles.
€ is the inertial force according to the D'Alembert's prinwhere ru _ ðaÞ is the work rate done by the inertial force. ciple and W In literature [15e17] the author considered that the variation of temperature with time should be supplied extra heat from the environment to overcome the heat or temperature inertia. From this idea the temperature wave equation will be derived easily and the definition of usual temperature and the Fourier's law are still kept. Though many literature have been pointed out that “heat may have inertia”, such as Nernst [18], Onsager [19], Kaliski [20], but these authors only used this concept indirectly through the generalized heat conducting equation. When the temperature varies with time, in literature [15e17] the author proposed to modify the first thermodynamic law expressed in Eq. (7) to
2. Modified energy principles and inertial entropy
ðaÞ ¼ Q_
The classical thermodynamics discusses thermal process in an equilibrium state, i.e. the temperature is constant in the four dimensional (space-time) space. The first classical thermodynamic law is
_ þ Q_ ; W _ ¼ U_ ¼ W U_ ¼
Z
Z
Z f $vdV þ
aZ s
VZ
_ udV; Q_ ¼
q$nda þ aq
V
T$vda
g_ dV; v ¼ u_
(4)
Zb Sb Sa
Z dQ =T; S ¼
a
sdV
(5)
V
where S is the entropy in the equilibrium state, Sb and Sa are the values of the entropy at states a and b, s is the entropy density per volume. In Eq. (5) the symbol “ ¼ ” is used for the reversible process and the symbol “>” is for the irreversible process. In classical thermodynamics for a moving body Eq. (4) is changed to
rv$vdV
K ¼ ð1=2Þ
(6)
V
where K is the kinetic energy. Eq. (6) can also be written as
_ ðaÞ ¼ _ W _ ðaÞ þ Q_ ; W U_ ¼ W
Z
€ $vdV ru
(7)
V
_ W _ ðaÞ þ Q_ Q_ ðaÞ U_ ¼ W
(8)
where Q ðaÞ is the inertial heat. Eq. (8) shows that the supplied heat dQ by the surrounding cannot be entirely absorbed by the system immediately, but only dQ dQ ðaÞ, where dQ ðaÞ is used to overcome the temperature inertia. dQ ðaÞ may be positive or negative. In Refs. [15e17] author considered that the inertial heat is induced by the temperature inertia, so assumed
Q ðaÞ ¼
Z V Z
€ C rs0 TdV ¼
Z
€ rs TdV
V
(9)
_ C rs0 TdV
V
where rs0 is an inertial entropy time constant and rs ¼ C rs0 . Eq. (9) shows that, when T_ > 0, Q ðaÞ > 0, i.e. when the temperature increases, the inertial heat is positive; on the contrary, T_ < 0, Q ðaÞ < 0. Correspondingly we can introduce the concept of the inertial ðaÞ entropy rate S_ as
V
where U; W and Q are internal energy, work done by the external force and heat supplied by the environment respectively, v is the velocity, u is the displacement vector. f is the body force, T is the surface traction, as and aq are respectively the parts giving external force and heat flow of a. The second classical thermodynamic law is
Z
_ þ Q_ ; U_ þ K_ ¼ W
ðaÞ ðaÞ ¼ T S_ ¼ Q_
Z
T s_ðaÞ dV;
V
€ T s_ðaÞ ¼ C rs0 T;
sðaÞ ¼
Zt
(10) s_ðaÞ dt
0
where sðaÞ is the inertial entropy density per volume, which is only related to the temperature variation with time and does not connected with any dissipative process. The second thermodynamic law can now be modified to
Zb Sb Sa a
dQ dQ ðaÞ T
(11)
Correspondingly the classical entropy equation is modified to [15e17]
82
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
T s_ðtÞ ¼ g_ ðT h_ i Þ;i ; T h_ i ¼ qi ; s_ðtÞ ¼ s_ þ s_ðaÞ ; s_ ¼ s_ðrÞ þ s_ðiÞ g_ ðrÞ s_ þ s_ðaÞ ¼ h_ i;i ; T ðiÞ T s_ ¼ T s ¼ T s_ s_ðrÞ ¼ h_ i T;i 0
(12)
In Eq. (12) sðtÞ is the total entropy density per volume, s is the usual entropy density in the thermodynamic equilibrium state and it is still a “state function”, sðrÞ and sðiÞ ¼ s are the reversible part and the irreversible part of s. T s_ðaÞ is the inertial heat rate per volume; h is the entropy displacement vector, h_ is the entropy flow vector. If the inertial entropy is omitted, Eq. (12) is reduced to the entropy equation of the usual irreversible thermodynamics. In the inertial entropy theory, T s_ þ T s_ðaÞ in Eq. (12) is used to instead of T s_ in the usual irreversible thermodynamics. Using Ti ¼ sji nj and the Gauss divergence theorem, from Eqs. (4), (8) and (10) we get
Z V
_ udV ¼
Z
€ i Þvi dV þ ðfi ru
Z
sji ni vj da
Z Z Z qi ni da þ g_ dV T s_ðaÞ dV as
V
aZ
¼
V
€ i Þvi dV þ ðfi ru
ZV
sji;i vj dV
(13)
V Z ZV _ þ sji vj;i dV þ T sdV V
V
In Eq. (13) sij is the stress, ε_ ij ¼ ðvi;j þ vj;i Þ=2 is the strain rate tensor and εij ¼ ðui;j þ uj;i Þ=2 is the strain tensor. By using the motion equation
sji;j þ fi ru€ i ¼ 0
(14)
From Eq. (13) the energy equation is
u_ ¼ sji vj;i þ T s_
(15)
Eq. (15) shows that the internal energy density u is the function of the state variables εij and s only and is not related to the inertial entropy density. The conjugate variables of ðεij ; sÞ are ðsij ; TÞ. Eq. (15) is just the same as that in the usual continuum thermodynamics. Using the theory of the linear irreversible thermodynamics [1,21] from Eq. (12) we can get
T h_ i ¼ qi ¼ kij ðx; TÞT;j
(16a)
where k is the heat conductive coefficient. Eq. (16a) is just the Fourier's law. The linear relation between the irreversible force and flow can be extended to a general expression:
h_ i ¼ f x; t; T; T;i ; T_
(16b)
From Eq. (16b) we can also get a similar heat conduction equation with the C-V equation. _ so from Eqs. (12) For pure heat conduction we have u_ ¼ T s_ ¼ C w, and (16a) we get
€ ¼ g_ þ kij ðx; tÞw;ji C w_ þ rs0 w
(17)
Eq. (17) is the result of IET. For the isotropic heat conduction kij ¼ kdij . Let g ¼ 0, Eq. (17) is identical with Eq. (3); however when gs0, they are different. As an example, we discuss a case that the temperature is homogeneous in space, but varies with time [17]. In _ ¼ this case Eqs. (3) and (17) are respectively reduced to Cðw þ t0 wÞ _ ¼ g. The former is an algebraic g þ t0 g_ or Cw ¼ g and Cðw þ rs0 wÞ equation and its response to external reaction is instantaneous, but the later is a differential equation in first order and its response to external reaction is delayed. It is also noted that when we use the IET, the generalized heat conduction theories can be simultaneously used to instead of Fourier theory, because the IET is a universal energy principle and the generalized heat conduction theories are the state equations or the constitutive equations. From Eqs. (8) and (12) it can be seen that if we introduce the inertial heat Q ðaÞ and subtract it from the external supplied heat, or we introduce the inertial entropy density rate s_ðaÞ and use s_ðrÞ þ s_ðaÞ to instead of s_ðrÞ in the usual entropy equation, the IET can deal with the local nonequilibrium thermodynamics where the temperature varied with time, by the framework of the classical equilibrium thermodynamics [17]. This is similar to the D'Alembert principle in the mechanical dynamics. The modified energy principles can be extended to more complex cases, such as to the thermodiffusion problem [16,22], where the concentration inertia is introduced, etc.
3. The statistical mechanics foundation of the inertial heat and inertial entropy theories The inertial heat and inertial entropy theories are consistent with the statistical mechanics from the material microscopic theory. In this paper we give a simple discussion and the more rational microscopic theory need studied in the future. The kinetic theory of molecules [2,3] shows that, the statistical average value 2 of the kinetic energy 2 of gas molecules at time t is 2 ¼ ð1=2Þmv2 , where m is the mass of a molecule, v is its random motion velocity, v2 is the mean value of v2 . For the gas we have
Z dn dv1 dv2 dv3 2 ¼ ð1=2Þmv2 ¼ ð1=2Þm v2 n m 3=2 Z∞ Z∞ Z∞ 2 ¼ 2pm emv =2kT v4 dv 2pkT 0
0
0
3 kT; v2 ¼ v$v 2 dn m 3=2 mv2 =2kT ¼ e dv1 dv2 dv3 n 2pkT 3 E ¼ N2 ¼ NkT 2 ¼
(18)
where k is the Boltzmann constant, n ¼ N=V, N is the total number of the gas molecules in V, n is the molecule number per volume, dn is the molecule number with velocity v, v is the velocity of a molecule, E is the total energy of molecules in V. Eq. (18) shows that the Kelvin temperature is a physical quantity representing the average kinetic energy of molecules. In the first line of Eq. (18) the second equality is an expression in the Cartesian coordinate, but the third equality is an expression in the sphere coordinate [2,3]. Now we discuss a varying isothermal system, i.e. all the molecules in the system have same temperature, but it may be varied with time. Let a molecule be in an equilibrium state “0” with temperature T0 , velocity v0 at time t0 . From time t0 to t, the system absorbs heat DQ from the environment, so the molecule is accelerated to a new state with temperature T and acceleration v_ at time
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
t. If DQ is infinite small, the acceleration is also infinite small, or _ v/0, the system is quasi-static continuously transited from one state to the neighboring with equilibrium temperature Teq at teq /∞ (on the theoretical meaning). If DQ is finite, the process from one state at t0 with temperature T0 to another state at t with instantaneous temperature T. In this case the increment of its in· Rt ternal energy t0 N2 dt is different with DQ because the system is in the accelerated state and it may also take some time to transmit the heat from the heat source to the system by the molecule motion. The difference between DQ and the internal energy will be called the inertial heat denoted by Q ðaÞ. So at time t we have
energy rate need expend a part of the work rate supplied by the external force. The inertial energy rate in Eq. (21) is expressed by the acceleration, the second derivative of displacement, but in Eq. (20) it is expressed by the second derivative of temperature, because the former discusses the ordered motion of the centroid of a microscopic element, but the latter discusses the molecular random motion. If we adopt the modified energy principle Eq. (8) or (15), the Boltzmann relation in the regular distribution theory of statistical mechanics may be still expressed as
S ¼ k ln W Zt
DQ ¼
·
N2 dt þ Q ðaÞ
(19)
t0
After t, we move the heat source and after infinite time Q ðaÞ is fully transformed to the internal energy and the system reaches a new equilibrium state with temperature Teq > T if T_ > 0 or Teq < T if T_ < 0 at time teq /∞. In Refs. [12,14] authors introduced the nonequilibrium temperature related to q,q; in Ref. [13] authors introduced _ The instantaneous a new temperature function f related to T. temperature T in this paper is automatically related to T_ as T ¼ T0 þ Rt _ 0 Tdt and a few numerical examples can be seen in literature [17]. T is consistent with the temperature measured in practice. But the difference between T and Teq is small due to small molecular inertia. Though we are lack of experimental data to determine Q ðaÞ , but experience shows that Q ðaÞ has the· same sign and the similar varying speed trends with DQ and 2 , so we can assume Q ðaÞ · proportional to 2 . Because the average kinetic energy of molecule motion is proportional to Kelvin temperature, we get ðaÞ _ € Obviously Q_ ðaÞ is related to the acceleration Q ðaÞ fT; Q_ fT. distribution function of molecules and the real situation is more complex. The more exact theory should be continuously studied in the future. According to the above discussions we can assume
Q Q ðaÞ ¼ N 2 ¼ U; · ðaÞ _ ¼ N 2 ¼ U; Q_ Q_
or ðaÞ Q_ fT€ ðt Þ
(20)
Eq. (20) is a macroscopic theory and consistent with Eq. (8). It shows that the inertial heat and inertial entropy theories are consistent with the molecule kinetic theory in the statistical mechanics in some aspect. Though Eqs. (19) and (20) are inferential, but there are also some similar theories in statistical mechanics, such as in Smoluchowski- Einstein fluctuation theory they generalized the relation between the entropy and the thermodynamic probability in equilibrium state to fluctuation case [2,3]; In discussion the reciprocal theorem Onsager also assumed that the attenuation of the fluctuation at equilibrium can be used to the nonequilibrium case where the state is slightly departure from the equilibrium state [1,19]. At the last part of this section we discuss some similarity with the continuum dynamics. In the continuum dynamics the energy equation can be written as
Z
Z dudV ¼
V
Z fk duk dV þ
V
Z Tk duk da
as
ru€ k duk dV
(21)
V
Eq. (21) shows that internal energy rate of a system is equal to the work rate supplied by the environment minus the work rate to overcome the inertial force. This phenomenon is similar to the heat conduction. In Eq. (20) The inertial energy rate need expend a part of heat rate supplied from the environment. In Eq. (21) the inertial
83
(22)
but where S is equal to the total entropy minus the inertial entropy, as shown in Eqs. (10) and (12). In Eq. (22) W is the thermodynamic probability [2,3]. W is proportional to the distribution fal g, in fal g al molecules have the same energy and satisfies the following conditions:
X al ¼ N; l
X 2l al ¼ E
(23)
l
4. Effect of the temperature inertia in multi-field coupling case Now we discuss the effect of the temperature inertia in multifield coupling case. As a simple coupling case we discuss the thermo-elasticity. In practical applications the free energy is more convenient, so in the following part we use the free energy gðεkl ; wÞ ¼ uðεkl ; sÞ Ts. In the thermo-elasticity the elastic behavior is reversible, only the heat behavior is irreversible. The thermo-elastic theory was studied in many literature, such as in Refs. [16,17,23]. In Refs. [16,17] the free energy gðεkl ; wÞ and the constitutive equations of ðsij ; εij Þ and ðs; TÞ are assumed:
1 Cw2 gðεkl ; wÞ ¼ Cijkl εji εlk aij εij w 2 2T vg sij ¼ ¼ Cijkl εkl aij w; vεij vg Cw C w 2 s ¼ ¼ aij εij þ vw T 2 T
(24)
where Cijkl ¼ Cjikl ¼ Cijlk ¼ Cklij is the elastic coefficient, aij ¼ aji is the stress-temperature coefficient. From Eqs. (12), (16) and (24) and sðaÞ in (10) we get:
€ ¼ g_ qi;i ; qi ¼ lij w;j ; T s_ þ C rs0 w w w 1 s ¼ aij εij þ C T 2T w w 1 s_ ¼ aij ε_ ij þ a_ ij εij þ C_ T 2T $ $ w w w w þC C 1 T 2T 2T T
(25)
w_ w zaij ε_ ij þ a_ ij εij þ C_ þ C T T For the pure heat conduction case with constant material coefficients, Eq. (25) is reduced to (17). For an isotropic material we have
84
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85
Cijkl ¼ ldij dkl þ G dik djl þ dil djk
aij ¼ adij Cijkl ; kij ¼ kdij (26)
sij ¼ lεkk dij þ 2Gεij adij w s_ ¼ aε_ jj þ a_ εjj þ C_
w w_ þC T T
where l and G are Lame constants. When the variation of temperature w is not large and the material behavior is independent to temperature the complete governing equation system without external force and internal heat source is
€ þ w_ þ aT0 ε_ ii kw;ii ¼ C rs0 w ru€ i ¼ sji;j ¼ luj;ij þ 2Gui;jj aw;i
(27)
The second in Eq. (27) is the motion equation. For simplicity, we discuss 1-D isotropic problem [16,17]:
ru€ Yu;11 þ aw;1 ¼ 0 € þ w_ kw;11 þ aT0 u_ ;1 ¼ 0 C rs0 w s ¼ Yε aw; s ¼ aε þ Cw=T0
(28)
where Y is the elastic modulus, a is the stress-temperature coefficient, u and s are the displacement and stress along x1 respectively. For a plane traveling wave let
u ¼ U exp½iðkx utÞ; q ¼ Q exp½iðkx utÞ;
k ¼ k1 þ ik2 c ¼ u=k1
(29)
where U and Q are the amplitudes of u and w respectively, k is the wave number and u is the circular frequency, c is the wave speed. Substituting Eq. (29) into (28), we have
2 Yk ru2h U þ iakQ ¼ 0
i
(30)
aT0 kuU þ kk2 C rs0 u2 þ iu Q ¼ 0 In order to U and Q have nontrivial solutions, it must
2 Yk ru2 aT ku 0
¼ 0; 2 2 kk C rs0 u þ iu iak
Ak4 Bk2 þ M ¼ 0; k2 ¼
(31)
M ¼ M1 þ iM2 ¼ rCðrs0 u þ iÞu3 B ¼ B1 þ iB2 ¼ ðYC rs0 þ rkÞu2 þ i YC þ a2 T0 u
A ¼ Yk
For the plane temperature wave the wave speed is very slowly, so the sign “þ” in Eq. (31) is used for the temperature wave and the temperature wave is always attenuated. Sign “” in Eq. (31) is for the elastic wave, so does the elastic wave be attenuated or amplified, it is determined by the sign of k2 , the imaginary part of k. For a coupling thermo-elastic wave when the temperature is varied, the inertial effect of the heat or temperature will hold back the change of the temperature and elastic waves. In the plane wave problem under following condition the amplitude of the elastic wave may be magnified:
k2 ¼ Imk ¼ Im
i 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B± B2 4AM 2A
1 2
<0
5. Conclusions In this paper we show that when the temperature is varied with time, the heat or the temperature inertia should be considered. The concepts of the heat inertia, temperature inertia and the inertial entropy are consistent with the statistical mechanics. The modified laws of the thermodynamics are the natural extension of the classical thermodynamics and should be checked by experiments in the future. The effect of the temperature inertia in multi-field coupling problems are also discussed shortly. It is also shown that for a general thermodynamic problem we can use the inertial entropy theory to discuss the variation of the temperature with time and use the Fourier or generalized Fourier law to discuss the variation of the temperature in the space. References
or
i 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B± B2 4AM 2A
inertia is similar. For a coupling thermo-elastic wave when the temperature is varied with time, the inertial effect of the temperature (or heat) will hold back the change of the temperature and elastic waves. Under some conditions the amplitude of the elastic wave may be magnified [24,25]. If we consider the material coefficients are the functions of the temperature, we can obtain a nonlinear equation system and the temperature wave and elastic wave will combined an elastic-temperature wave propagated with one wave speed. How to understand the temperature wave is still a problem and will be discussed in the future. In general the engineering composites are viscous materials, so the viscosity should be considered [16,17,25,27,28]. According to the irreversible thermodynamics, in Refs. [16,17,21] the author divided the stress s into reversible part sðrÞ and irreversible part sðiÞ : s ¼ sðrÞ þ sðiÞ , sðrÞ is determined by the state function, the free energy g, as shown in Eq. (24); sðiÞ is determined by the theory of the irreversible flow and force. Zhou and Yang [25] discussed the thermo-viscoelastic effect of temperature wave in detail. The amplified effect of the elastic wave due to the inertia of temperature wave may be small comparing with the viscous effect. So if we consider viscosity the elastic wave may also be always attenuated in engineering.
(32)
The thermo-electroelasticity was also discussed in many literature, such as in Refs. [15,23e26] and the effect of the temperature
[1] De Groot SR, Mazur P. Nonequilibrium thermodynamics. New York: Dover Publications; 1984. [2] Pathria RK, Beale PD. Statistical mechanics. third ed. Academic Press; 2011. [3] Wang Z-C. Thermodynamics,Statistical physics. fifth ed. Beijing: Higher Education Press; 2013 (in Chinese). [4] Landau L. The theory of superfluidity of helium II. J Phys 1941;5:71. [5] Jackson HE, Walker CT. Thermal conductivity, second sound and phononphonon interactions in NaF. Phys Rev B 1971;3:1428. [6] Marciak-kozlowska J, Mucha Z, Kozlowski M. Picosecond thermal pulses in thin gold films. Int J Thermophys 1995;16:1489. [7] Straughan B. Heat waves. Berlin: Springer; 2011. [8] Joseph DD, Preziosi L. Heat waves. Rev Mod Phys 1989;61:41e73. [9] Cattaneo C. Sulla conduzione de calore. Atti Del Semin Mat E Fis Univ Modena 1948;3:83. quation de la chaleur e liminant le paradoxe [10] Cattaneo C. Sur une forme de 10 e mie Sci Paris d' une propagation instantanee. Comptes Rendus 10 Acade 1958;247:431. orie continue de l’equation de la chaleeur. [11] Vernotte P. Les paradoxes de la the mie Sci Paris 1958;246:3154. Comptes Rendus 10 Acade [12] Jou D, Casas-Vazquez J, Lebon G. Extended irreversible thermodynamics. third ed. Berlin, Heidelberg: Springer; 2001. [13] Green AE, Laws N. On the entropy production inequality. Arch Ration Mech Anal 1972;45:47. [14] Dong Y, Cao B-Y, Guo Z-Y. Temperature in nonequilibrium states and nonFourier heat conduction. Phys Rev E 2013;87:032150. [15] Kuang Z-B. Variational principles for generalized dynamical theory of thermopiezoelectricity. Acta Mech 2009;203:1. [16] Kuang Z-B. Some thermodynamic problems in continuum mechanics. In: n CM, editor. Thermodynamics - kinetics of dynamic systems. Croatia: Piraja Intech open access publisher; 2011. [17] Kuang Z-B. Discussions on the temperature wave equation. Int J Heat Mass Transf 2014;71C:424.
Z.-B. Kuang / International Journal of Thermal Sciences 120 (2017) 80e85 [18] Nernst W. Die theoretischen und experimentellen grundlagen des n warmestatzes,Knapp, Halle. 1917. [19] Onsager L. Reciprocal relations in irreversible processes. Phys Rev 1931;37: 405e26. mie Polonaise [20] Kaliski S. Wave equation of heat conduction. Bulletin de l’Acade rie Des Sci Tech 1965;13:211e9. des Sciences. Se [21] Kuang Z-B. Nonlinear continuum mechanics. Shanghai: Shanghai Jiaotong university press; 2002 (in Chinese). [22] Kuang Z-B. Variational principles for generalized thermodiffusion theory in pyroelectricity. Acta Mecha 2010;214:275e89. [23] Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity.
85
J Mech Phys Solids 1967;15:299e309. [24] Yuan X-G, Kuang Z-B. Waves in pyroelectrics. J Therm stress 2008;31:1190. [25] Zhou Z-D, Yang F-P. Plane waves in pyroelectrics with viscous effect. Acta Mech 2014;225:509. [26] Kuang Z-B. Theory of electroelasticity. Shanghai, Berlin: Shanghai Jiaotong University Press, Springer; 2013. [27] Ezzat MA, Othman MI, El-Karamany AS. State space approach to generalized thermo-viscoelasticity with two relaxation times. Int J Eng Sci 2002;40:283. [28] Lionetto F, Montagna F, Maffezzoli A. Ultrasonic dynamic mechanical analysis of polymers. Appl Rheol 2005;15:326.