Effective Thermal Properties of Layered Systems ...

J. Ordoń˜ez-Miranda J. J. Alvarado-Gil1

Effective Thermal Properties of Layered Systems Under the Parabolic and Hyperbolic Heat Conduction Models Using Pulsed Heat Sources

e-mail: [email protected] Departamento de Fı´sica Aplicada, Centro de Investigacioń y de Estudios Avanzados del I.P.N.-Unidad Me´rida, Carretera Antigua a Progreso km. 6, A.P. 73 Cordemex, Me´rida, Yucatań, Me´xico C.P. 97310

In this work, transient heat transport in a flat layered system, with interface thermal resistance, is analyzed, under the approach of the Cattaneo–Vernotte hyperbolic heat conduction model using the thermal quadrupole method. For a single semi-infinite layer, analytical formulas useful in the determination of its thermal relaxation time as well as its thermal effusivity are obtained. For a composite-layered system, in the long time regime and under a Dirichlet boundary condition, the well-known effective thermal resistance formula and a novel expression for the effective thermal relaxation time are derived, while for a Neumann problem, only a heat capacity identity is found. In contrast in the short time regime, under both Dirichlet and Neumann conditions, an expression that involves the effective thermal diffusivity and relaxation time as a function of the time is derived. In this time regime and under the Fourier approach, a formula for the effective thermal diffusivity in terms of the time, the thermal properties of the individual layers and its interface thermal resistance is obtained. It is shown that these results can be useful in the development of experimental methodologies to perform the thermal characterization of materials in the time domain. [DOI: 10.1115/1.4003814] Keywords: effective thermal properties, thermal quadrupoles, parabolic and hyperbolic heat conduction

1

Introduction

Effective models have provided a useful basis for the interpretation of experimental data and understanding of heat transport in nonhomogeneous systems [1–7]. Most of these models have been based on Fourier law [8–11]. This law is supported by a vast quantity of useful and successful results that show a very good agreement with experimental data for a great variety of experimental conditions [8,10–14]. However, it is well-known that Fourier law does not take into account the speed of heat propagation and therefore it cannot provide a general description of the heat conduction [2,4–7,15–23]. One of the simplest and accepted models to solve the drawbacks of Fourier law was suggested by Cattaneo [24] and independently by Vernotte [25]. These authors incorporate the finite propagation speed of heat while retaining the basic nature of Fourier law, modifying it as follows ~ x; tÞ ~ x; tÞ þ s @ Jð~ ¼ krTð~ x; tÞ Jð~ @t

(1)

where J~ (W/m2) is the vector heat flux, T (K) is the absolute temperature, k (W/mK) is the thermal conductivity and s (s) is a thermal property of the medium, known as the thermal relaxation time, which represents the time necessary for the initiation of the heat flux after a temperature gradient has been imposed at the boundary of the medium. Conversely, s represents the time necessary for the disappearance of the heat flux after the removal of temperature gradient [16,17]. The relaxation time s has been reported to be of the order of microseconds (106 s) to picosec1 Corresponding Author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 2, 2010; final manuscript received March 1, 2011; published online July 8, 2011. Assoc. Editor: Patrick E. Phelan.

Journal of Heat Transfer

onds (1012 s) for metals, superconductors, and semiconductors [26]. These small values of the thermal relaxation time indicate that its effects will not be significant if the physical time scales are larger than a few microseconds, in which cases the Fourier law provide adequate results. However, in modern applications such as analysis and processing of materials using ultrashort laser pulses and high speed electronic devices, the finite value of the thermal relaxation time is necessary to be considered [16,27–33]. After combining Eq. (1) with the energy conservation equation in absence of internal heat sources [34], the hyperbolic Cattaneo– Vernotte (CV) heat conduction equation is obtained [14,24,25,35] x; tÞ r2 Tð~

1 @Tð~ x; tÞ s @ 2 Tð~ x; tÞ ¼0 a @t a @t2

(2)

where a (m2/s) is the thermal diffusivity of the medium [36]. On the left hand side of this equation, the second order time derivative term establishes pffiffiffiffiffiffiffi that heat propagates as a wave with a characteristic speed a=s. Note that the first order time derivative term corresponds to a diffusive process, which is damping spatially the heat wave. Equation (2) reduces to the parabolic heat diffusion equation (based on Fourier law) for s ! 0 or in steady-state con~ x; tÞ @t ¼ 0 (Ref. [15]). ditions @ Jð~ Fourier [8–11,13] and hyperbolic models [2,3,14,18– 21,29,37,38] have been used to study heat transport in layered systems. These kind of physical systems constitute one of the basic configurations for the development of models to study the effective thermal properties of heterogeneous systems [1]. Most of the models have been derived in stationary conditions or when a modulated heat source is applied to the studied material, providing useful and meaningful results for the interpretation of heat transport [8–11,13]. In the case of modulated heat sources, it is wellknown that the prediction for the effective thermal properties in terms of the thermal properties of the composing layers is

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restricted to well-defined modulation frequency intervals. In fact, only for very low or very high frequency regimes, the obtained equations are usable. It is therefore important to explore, in the time domain, the form that the effective thermal properties equations would take in layered systems. In this work, the heat transport in a two-layer system with interface thermal resistance is analyzed using the hyperbolic CV equation and considering Dirichlet and Neumann boundary conditions. It is shown that, in general, the thermal properties of an effective layer cannot be determined. However, in the long time domain, analytical expressions for the effective thermal properties as a function of the thermal properties of the individual layers are obtained. On the other hand, in the short time regime, a formula for the effective thermal diffusivity as a function of time is derived under the approach of Fourier law. All of these results provide useful formulas for the characterization of layered systems in the time domain. The fundamental role of the thermal relaxation time in addition to the other thermal properties and interface thermal resistance of the composing layers is discussed.

2

p¼

Tðx; 0Þ ¼ T0 @Tðx; tÞ ¼0 @t t¼0 @Jðx; tÞ ¼0 @t t¼0

k dhðxÞ 1 þ ss dx

d2 hðxÞ p2 hðxÞ ¼ 0 dx2

B A

hðxÞ /ðxÞ

(8)

where A ¼ coshðpxÞ

(9a) (9b)

C ¼ v sinhðpxÞ

(9c)

(3a)

v ¼ kp=ð1 þ ssÞ

(9d)

(3b)

(4)

Equation (8) represents the fundamental result of the thermal quadrupole method [39], which links linearly the input vector (temperature, heat flux) to its corresponding output vector by means of a square matrix. This matrix is called the thermal quadrupole matrix [39] and is unitary, and its diagonal coefficients are equal. Since the formulation of this elegant method by Carslaw and Jaeger [34], it has been successfully applied to study heat transport, under the framework of the Fourier law. In the present case, its applicability is extended to consider the predictions of the CV model. Notice that in the limit of s ! 0, Eqs. (9a)–(9c) reduce to the well-known results obtained under the Fourier law. This method is suitable to study the heat transport along layers of finite thickness, as the ones shown in Fig. 1(b). However, to treat with a semi-infinite layer, as the one shown in Fig. 1(a), it is useful to take into account the Laplace transform of the flux is given by

(5)

/ðxÞ ¼ vhðxÞ

(3c)

where s > 0 is the Laplace parameter. The solution of Eq. (5) is well-known [34] and is given by hðxÞ ¼ a sinhðpxÞ þ b coshðpxÞ

hð0Þ A ¼ /ð0Þ C

B ¼ sinhðpxÞ=v

Note that under the change T ! T T0 , these initial conditions become homogeneous and Eq. (2) does not change. Therefore, without loss of generality, from now on, the initial temperature is going to be taken as zero (T0 ¼ 0). After taking the Laplace transform in both Eqs. (1) and (2), and using Eqs. (3a)–(3c), it is obtained that in the Laplace domain, the heat flux /ðxÞ L½Jðx; tÞ and temperature hðxÞ L½Tðx; tÞ satisfy /ðxÞ ¼

(7)

As is known, in Laplace transform formalism, the limit t ! 1 (t ! 0) in the time domain, corresponds to the limit s ! 0 (s ! 1) in the Laplace domain and vice versa [39]. Therefore, according to Eq. (7), in the long time domain, the parameter p tends to its classical value when ss 1 [39], which means that the effects of the thermal relaxation time are negligible in this time domain. On the other hand, pffiffiffiffiffiffiffiin the short time domain (s ! 1), the parameter p ! s s=a depends on the thermal relaxation time and therefore the hyperbolic effects are expected to be present in the temperature. After combining Eqs. (4) and (6), it is found that the temperature and heat flux at x ¼ 0 are related to hðxÞ and uðxÞ for x 0, as follows

Mathematical Formulation and Solutions

Considering the configuration shown in Fig. 1(a), in which the system is excited externally in such a way that the heat propagates along the x direction, the heat flux and the temperature of the medium satisfy the one-dimensional form of Eqs. (1) and (2). Assuming that the initial temperature of the layer is uniform and equal to T0 , the following initial conditions are established

rffiffiffi s pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ss a

(6)

where a and b are two constants that depend on the boundary conditions of the corresponding problem and p is defined by

(10)

which is valid for x 0 and can be derived by writing Eq. (6) in terms of exponentials, requiring that hðx ! 1Þ ! 0; in order to obtain a solution with physical meaning, and combining the result with Eq. (4). According to the thermal quadrupole method [39], the corresponding matrix equation for the temperature and heat flux at the points x ¼ 0; l; of the layered system shown in Fig. 1(b) is

Fig. 1 Schematic diagram of the studied layered systems. (a) A semi-infinite layer of thermal conductivity j, thermal diffusivity a and thermal relaxation time s. (b) The layer of thermal conductivity ki , thermal diffusivity ai , thermal relaxation time si , and thickness li is in thermal contact with the layer of conductivity ki11, diffusivity ai11, relaxation time si11 , and thickness li11 ; for i51; 2: The third layer is considered as a semi-infinite one (l3 ! ‘).

091301-2 / Vol. 133, SEPTEMBER 2011

Transactions of the ASME


where A C

hð0Þ A ¼ /ð0Þ C

B D

hðlÞ /ðlÞ

(11)

n The front surface is excited by a Dirac heat pulse of the form

B A1 ¼ C1 D

B1 A1

1 R12 0 1

A2 C2

B2 A2

1 R23 0 1

hð0Þ ¼ ðA þ v3 BÞhðlÞ

(13a)

/ð0Þ ¼ ðC þ v3 DÞhðlÞ

(13b)

Note that Eqs. (13a) and (13b) permit to find the temperature at x ¼ l if the temperature or the heat flux at x ¼ 0 are given.

Results

In this section, useful formulas to determine the thermal properties of a layered medium, under the framework of the CV model of heat conduction are obtained and analyzed. 3.1 Thermal Properties of a Semi-Infinite Layer. According to Eqs. (7), (9d), and (10), the temperature at x ¼ 0 of the semiinfinite medium shown in Fig. 1(a) is given by rffiffiffiffiffiffiffiffiffiffiffiffiffi /ð0Þ 1 þ ss (14) hð0Þ ¼ e s pffiffiffi where e ¼ k= a is the thermal effusivity of the medium [34]. Under the parabolic approach considering that the inverse pffiffiffiffi pffiffi (s ¼0), Laplace transform L1 ½1= s ¼ 1 pt (Ref. [42]) and applying the convolution theorem [42], it is obtained that the temperature in the time domain at x ¼ 0 is given by 1 Tð0; tÞ ¼ pffiffiffi pe

ðt 0

Jð0; t uÞ pffiffiffi du u

(15)

On the other hand, under the hyperbolic model (s 6¼ 0), taking into account that the inverse Laplace transform of the square root in Eq. (14) is given by Ref. [42] "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# s þ s1 1 ¼ dðtÞ þ ½I0 ðt=2sÞ þ I1 ðt=2sÞet=2s (16) L1 2s s where d is the Dirac delta function, and I0 and I1 are the modified Bessel functions of the first kind of order zero and one, respectively [43]; the convolution theorem establishes that temperature in the time domain is given by # pffiffiffi " ð t=2s s u Jð0; tÞ þ Jð0; t 2suÞ½I0 ðuÞ þ I1 ðuÞe du Tð0; tÞ ¼ e 0 (17) which shows that for t ! 0, the temperature is determined only by the contribution of its first term. In this way, Eqs. (15) and (17) permit to find the temperature at the front surface (x ¼ 0) of the semi-infinite medium (see Fig. 1(a)) for any profile of heat flux Journal of Heat Transfer

Jð0; tÞ ¼ QdðtÞ;

(12)

in which the coefficients Ai ; Bi ; andCi are given by Eqs. (9a)–(9d) for the layer i ¼ 1; 2 and Ri;iþ1 is the interface thermal resistance of the neighboring layers i and i þ 1 (Refs. [40 and 41]). Equation (11) determines both the temperature and heat flux (in the Laplace domain) at x ¼ l when they are given at x ¼ 0 and vice versa. For the case in which the third layer in Fig. 1(b) is semi-infinite, according to Eq. (10), /ðlÞ ¼ v3 hðlÞ; where v3 is defined by Eq. (9d) for the thermal properties of this layer; therefore Eq. (11) can be split as follows

3

applied on the same surface. To get a deeper insight in the predictions of these equations, two particular profiles of the heat flux are considered:

(18)

where Q (J/m2) is the applied energy density. This kind of boundary condition is typical of the heat pulse method to measure thermal properties of materials [34,39]. After inserting Eq. (18) into Eqs. (15) and (17), it is obtained that the temperature predicted by the parabolic (Tp ) and hyperbolic (Th ) models are given by Q Tp ð0; tÞ ¼ pffiffiffiffi e pt

Th ð0; tÞ ¼

pffiffiffi sQ Q dðtÞ þ pffiffiffi ½I0 ðt=2sÞ þ I1 ðt=2sÞet=2s e 2e s

(19a) (19b)

respectively. In the long time regime (t 2s), by using the asymptotic expressions of I0 and I1 [44], it is found that Eq. (19b) reduces to Eq. (19a). This shows that the effect of the thermal relaxation time s is negligible after a time much longer than twice the relaxation time and that Eq. (19b) can be used to determine the thermal effusivity of the medium under the hyperbolic and parabolic models of heat conduction. On the other hand, in the short time regime (0 6¼ t 2s), Eq. (19b) reduces to Q t (20) Tð0; tÞ ¼ pffiffiffi 1 4s 2e s which has a weak dependence on time and can be used to determine the thermal relaxation time s, after the thermal effusivity have been determined in the long time limit. Equation (20) shows that the effects of the thermal relaxation time are strongly present in the short time regime and they vanish in the long time regime pffiffiffi (see Eq. (19a)). The behavior of the normalized temperature e sT=Q as a function of the normalized time t=s > 0 is shown in Fig. 2(a). The solid line corresponds to the prediction of the hyperbolic model and the dashed ones to the Fourier parabolic model. Note that the temperature predicted by the CV equation is smaller than the one predicted by the Fourier law in the whole interval of time and their difference decreases when the time increases, which indicates that the hyperbolic effects vanish in the long time regime. n If the front surface is excited (for t > 0;) by a constant heat flux given by Jð0; tÞ ¼ I

(21)

the right-hand sides of Eqs. (15) and (17) can be integrated in a closed form, and the final results for the parabolic and hyperbolic model are given by rffiffiffi 2I t (22a) Tð0; tÞ ¼ e p pffiffiffi I s Tð0; tÞ ¼ ½ð1 þ t=sÞI0 ðt=2sÞ þ ðt=sÞI1 ðt=2sÞet=2s (22b) e respectively. In the long time regime (t 2s), the temperature predicted by the hyperbolic model (Eq. (22b)) reduces to the one predicted by the parabolic one (Eq. (22a)). On the other hand, in the short time regime (0 6¼ t 2s), Eq. (22b) takes the form pffiffiffi I s t 1þ (23) Tð0; tÞ ¼ e 2s which is nearly constant inside its time domain. In this case, the time behavior of the temperatures predicted by the Fourier and SEPTEMBER 2011, Vol. 133 / 091301-3


Fig. 2 Time dependence on the surface temperature at x 5 0, for an excitation of a (a) Dirac heat pulse and (b) constant heat flux. The solid lines correspond to the hyperbolic model and the dashed lines to the parabolic one.

CV models are shown in Fig. 2(b) by dashed and solid lines, respectively. After determining the time dependence on the temperature at the surface x ¼ 0, Eq. (22b) can be used to determine both the thermal effusivity and the thermal relaxation time, considering its long and short time regimes, respectively, or by means of a fitting procedure, which can be applied in general to Eq. (17), for a given heat flux at x ¼ 0. 3.2. Effective Thermal Properties of a Two-Layer System. After calculating explicitly the matrix coefficients A and D in Eq. (12), it is found that they are not equal, even for the case in which both interface thermal resistances R12 and R23 are neglected. Therefore, although the left hand side matrix in Eq. (12) has a unit determinant, because the square matrixes on its right hand side are unitary [43], it does not have the diagonal form of the square matrix involved in Eq. (8). This fact establishes that in general, there is not an effective medium that can be equivalent to the layered system shown in Fig. 1(b) (see Eq. (8)). However, there are two particular cases in which the effective homogeneous medium does exist. n If the first two layers thermal effusivities (e1 ¼ e2 ) and the thermal relaxation times (s1 ¼ s2 ) are equal. In absence of thermal interface resistance, the elements of the thermal quadrupole matrix (Eq. (12)) of the layered system, shown in Fig. 1(b), reduce to A ¼ D ¼ coshðp1 l1 þ p2 l2 ÞZ

(24a)

B ¼ sinhðp1 l1 þ p2 l2 Þ=v1

(24b)

C ¼ v1 sinhðp1 l1 þ p2 l2 Þ

(24c)

After comparing Eqs. (24a)–(24c) with Eqs. (9a)–(9c) for an effective layer of thermal diffusivity a, thermal effusivity e, relaxation time s, and thickness l ¼ l1 þ l2 , it is found that e ¼ e1 ¼ e2 , s ¼ s1 ¼ s2 and l l1 l2 pffiffiffi ¼ pffiffiffiffiffi þ pffiffiffiffiffi a1 a2 a

(25)

which are valid for any value of the Laplace parameter s and therefore for any time t > 0 (Ref. [39]). Under the Fourier parabolic approach, Eq. (25) has been derived previously in the book of Maillet et al. [39], in the time domain, for the case in which e1 ¼ e2 and by Tominaga and Ito [13] in the frequency domain, in the limit of high modulation frequencies. Under the hyperbolic approach, Eq. (25) has been also derived by Ordoń˜ez-Miranda and Alvarado-Gil, in the frequency domain [14]. Considering that the thermal properties of the given material are related by Ref. [34] 091301-4 / Vol. 133, SEPTEMBER 2011

e¼

pffiffiffiffiffiffiffiffi pffiffiffi k kqc ¼ pffiffiffi ¼ qc a a

(26)

where q and c are its density and specific heat, respectively, Eq. (25) with e ¼ e1 ¼ e2 are equivalent to the following pair of equations l l1 l2 ¼ þ k k1 k2 lqc ¼ l1 q1 c1 þ l2 q2 c2

(27a) (27b)

which are the well-known formulas for the effective properties under steady-state conditions [39]. This interesting fact indicates that in the transient process of heat conduction, the two-layer system shown in Fig. 1(b) behaves as they were under steady-state conditions if their thermal effusivities and relaxations times of each layer are the same. n When the Laplace parameter is very small, in such a way that pi li 1, then under a first order approximation in pi li , the elements of the thermal quadrupole matrix in Eq. (12) take the form AD1 l1 l2 B ð1 þ ss1 Þ þ ð1 þ ss2 Þ þ R12 þ R23 k1 k2 C ¼ ðl1 q1 c1 þ l2 q2 c2 Þs

(28a) (28b) (28c)

Now the thermal properties of the effective layer of the two-layer system shown in Fig. 1(b) can be derived from two different and equivalent ways. The first one stems from comparing Eqs. (28a)– (28c) with the corresponding results obtained using Eqs. (9a)–(9c) for the equivalent effective layer and an approximation of the same order and making R23 ¼ 0 in Eq. (28b). This last assumption was made considering that this resistance is a parameter of interface not belonging to the two-layer system whose effective thermal properties we want to find. The second way stems from rewriting Eqs. (28a)–(28c) for the effective homogeneous layer by assuming that the thermal properties of the layers are equal in such a way that the quantity P ¼ P1 ¼ P2 ; for P ¼ k; s; q; c; l ¼ l1 þ l2 and R12 ¼ 0, and comparing the results with the corresponding Eqs. (28a)–(28c). From now on, this last method is going to be used to obtain the thermal properties of the effective layer. The results obtained for the effective thermal conductivity and relaxation time are given by l l1 l2 ¼ þ þ R12 k k1 k2

(29a)



s¼

l1 s1 l2 s2 þ k1 k2

l1 l2 þ þ R12 k1 k2

(29b)

and for the effective heat capacity, Eq. (27b) is obtained. The remaining effective thermal properties can also be found by means of Eq. (26). Equation (29a) has been previously obtained by Dramicanin et al. [45], in absence of an interface thermal resistance, it reduces to the formula determined by Lucio et al. [8], and it is widely used in thermal characterization [8,9,12] under the framework of the parabolic and hyperbolic models of heat conduction. Here, it is proved that Eq. (29a) is also true in the time domain for hyperbolic heat conduction. For the case in which Ri;iþ1 li =ki , i ¼ 1; 2; Eq. (27a) is derived rather than Eq. (29a). Equation (29b) establishes that the effective thermal relaxation time is a weighted average of the thermal relaxation times of each composing layer, being the weights the corresponding thermal resistances. When the Laplace parameter is small (pi li 1, i ¼ 1; 2), it can be shown that Eqs. (29a) and (29b) are valid for

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 si 1þ 1 (30) s 2si tmi where tmi ¼ l2i 4ai is proportional and very close to the thermalization time, obtained in the analysis of thermal transients [46]. Ordoń˜ez -Miranda and Alvarado-Gil [14] have shown that the hyperbolic model is valid under the constraint si < tmi , where si is the thermal relaxation time, measuring the time needed to start the heat flux and the thermalization time tmi is associated to the required time to reach the maximum value of the heat flux. In the time domain, Eq. (30) is associated to [39]

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi si (31) t 2tmi 1 þ 1 þ tmi which under the parabolic model reduces to t 4tmi . Equation (31) shows that Eqs. (27b) and (29a)–(29b) are valid for times much longer than four times the thermalization time of the component layers. The fact that Eqs. (27b) and (29a) are also valid under steady-state conditions indicates that, under the hyperbolic approach, this condition can be reached approximately after a time satisfying the inequality (31). It is important to mention that for large Laplace parameter domain (pi li 1, i ¼ 1; 2) (short time domain), the elements of the thermal quadrupole matrix (Eq. (12)) does not give coherent results for the effective thermal properties of the two-layer system. This is due to the fact that, in the first instants of the transient process of heat transport, the effective thermal properties depend on the time as will be shown later in this paper, in analogy to the case in which the effective thermal properties depend on the modulation frequency, in the frequency domain [8,14]. To get a further insight in the thermal properties of the effective layer, the form of the temperature at x ¼ l, for two kind of boundary conditions, is going to be analyzed. 3.2.1. Dirichlet Problem. In this case, this problem is defined by specifying the temperature at x ¼ 0; in such a way that its Laplace transform hð0Þ is an arbitrary known function, and therefore the temperature at x ¼ l is given by Eq. (13a). The limiting cases of small and large Laplace parameter of this equation can be considered n Long Time Results: s ! 0. After expanding the matrix elements A and D in powers of s, Eq. (13a) takes the form

pffiffi hð0Þ l1 l2 ¼1þ þ þ R12 þ R23 e3 s hðlÞ k k

2 1 22 l l e2 l1 l2 l2 s ::: þ (32) þ 1 þ 2 þ 2 pffiffiffiffiffi pffiffiffiffiffi þ 2R12 e2 pffiffiffiffiffi a2 2 a1 a2 e1 a1 a2 Journal of Heat Transfer

where it has been assumed that the thermal relaxation time s3 of the semi-infinite layer is small enough in such a way that ss3 ! 0: For the effective medium, the equivalent expression to Eq. (32) can be obtained as it was done previously in this work. The result is

pffiffi l2 s hð0Þ l ¼1þ þ R23 e3 s þ þ ::: (33) hðlÞ k a2 Comparing Eqs. (32) and (33), it can be noted that for an approximation up to s1=2 (long time domain), the effective thermal conductivity obtained coincides with the value previously derived for the case of steady-state conditions (Eq. (29a)). This shows that this approximation is associated to the case of steady-state conditions, and the approximation up to s1 (the next higher order) corresponds to the thermal conditions close to the steady-state conditions. Under this last approximation, the effective thermal diffusivity is given by l2 l21 l22 e2 l1 l2 l2 ¼ þ þ 2 pffiffiffiffiffi pffiffiffiffiffi þ 2R12 e2 pffiffiffiffiffi a2 a a1 a2 e1 a1 a2

(34)

Note that in both Eqs. (29a) and (34), the effective thermal properties depend on the interface thermal resistance of the two component layers and are independent of their thermal relaxation time. In addition, Eq. (29a) establishes that in the effective thermal conductivity, the order of the layers is not important, given that R12 ¼ R21 ; however, this is not true for the effective thermal diffusivity which, according to Eq. (34), depends on the order of the component layers. Considering that under steady-state conditions, the thermal diffusivity as of the effective layer is given by l2 as ¼ ðl=kÞðlqcÞ, where l=k is defined in Eq. (29a) and lqc by Eq. (27b), observe that the thermal effusivity given in Eq. (34) differs from as as follows

l2 l2 l1 l2 2 e e21 þ R12 ðl2 q2 c2 l1 q1 c1 Þ ¼ a as k1 k2 2

(35)

where Eq. (26) has been used. Equation (35) indicates that a as (a as ) when e2 e1 (e2 e1 ) and l2 q2 c2 l1 q1 c1 (l2 q2 c2 l1 q1 c1 ), respectively. This difference shows that Eq. (34) is not valid in steady-state conditions but rather in near steady-state (quasisteady-state) conditions, because Eq. (34) arises from the coefficient of the second no null power of s in Eq. (32). Note that when e2 ¼ e1 and in absence of the interface thermal resistance (R12 ¼ 0), a ¼ as , which agrees with our previous result (see the paragraph just after Eq. (27b)). For a higher order expansion in powers of the Laplace parameters, Eqs. (32) and (33) do not provide coherent results for the effective thermal properties of the effective layer. This is because of higher order expansions mean that s is not so short and therefore the time is not so long to guarantee a steady or quasi-steady state condition in the process of heat transport. Note that, if the labels of layer 1 and 2 are interchanged in Eq. (34), and if the order of the layers were ignored, in such a way that the effective thermal diffusivity is the same before and after the interchange of the layers (which, in general, is not true), the following equation for the effective thermal diffusivity would be obtained

l2 l21 l22 l1 l2 e2 e1 l1 l2 þ R12 e1 pffiffiffiffiffi þ e2 pffiffiffiffiffi þ ¼ þ þ pffiffiffiffiffi pffiffiffiffiffi a1 a2 e1 e2 a1 a2 a a1 a2 (36) which surprisingly coincides with the results obtained under steady-state conditions, (a ¼ as ), and it has been used extensively in the literature [1,9]. It can be inferred that the only form of obtaining this equation is ignoring the order of the layers or going pffiffi to very long times, such that only an approximation up to s is required to be considered in the expansion presented in Eq. (32). SEPTEMBER 2011, Vol. 133 / 091301-5


n Short time results: s ! 1 Considering that the Laplace parameter is high enough to make that pi li 1 for i ¼ 1; 2; in such a way that sinhðpi li Þ coshðpi li Þ expðpi li Þ=2, Eq. (13a) can be written as

2 2 hðlÞ ¼ hð0Þ 1 þ v21 þ R12 v2 1 þ v32 þ R23 v3

exp½ðp1 l1 þ p2 l2 Þ (37) where vij ¼ vi vj being i; j ¼ 1; 2; 3: For the effective homogeneous layer of thickness l ¼ l1 þ l2 , Eq. (37) reads

2 expðplÞ (38) hðlÞ ¼ hð0Þ 1 þ v32 þ R23 v3 where the interface parameter has been kept unchanged, because they do not belong to the two-layer system, whose effective properties are being analyzed. After equating Eqs. (37) and (38), the effective thermal diffusivity as a function of the Laplace parameter is obtained, which is not of practical interest, because of the transient process of heat conduction is studied in the time domain. To determine this effective thermal property in this domain, let us define L½f ðtÞ ¼

hð0Þ 1 þ v32 þ R23 v3

2 exp½ðp1 l1 þ p2 l2 Þ L½gðtÞ ¼ 1 þ v21 þ R12 v2 L½hðtÞ ¼ expðplÞ

(39b) (39c)

from which, it is obtained the following effective relation (41)

which, after solving the condition of long Laplace parameter (pi li 1, i ¼ 1; 2), is found to be valid for a short time satisfying the inequality [Eq. (31)] with the symbols of the inequality in opposite direction. Equation (41) represents a general relation, where both the thermal diffusivity and relaxation time of the effective layer, are involved. Under our approach, it is not possible to obtain an expression for aðtÞ or sðtÞ separately but only a coupled equation of these thermal properties. This fact together with the long and complicated problem of determining the inverse Laplace transform of Eq. (39b), even in absence of the interface thermal resistance (R12 ¼ 0), imply that Eq. (40) would not be of practical interest, under the hyperbolic model of heat conduction. In contrast, in the domain of validity of the parabolic model, Eq. (41) can provide a useful expression for the effective thermal diffusivity as a function of time. This is the reason why, from now on, the explicit expression of aðtÞ predicted by Fourier law is going to be obtained and analyzed. Under the Fourier parabolic approach (si ¼ 0), the inverse Laplace transform of Eq. (39c) is well-known and given by Ref. [42]

1 l l2 (42) hðtÞ ¼ pffiffiffiffiffiffi pffiffiffi exp 4at 2 pt3 a In order to determine the inverse Laplace transform of Eq. (39b), two different cases are going to be considered: * For a perfect thermal contact of the two component layers (R12 ¼ 0), the inverse Laplace transform of Eq. (39b) is given by Ref. [42] 091301-6 / Vol. 133, SEPTEMBER 2011

where T21 ¼ 2=ð1 þ e2 =e1 Þ is the transmission coefficient from layer 1 to layer 2 (Ref. [47]). After inserting Eqs. (42) and (43) into Eq. (41) yields "

#

l l2 l1 l2 1 l1 l2 2 pffiffiffi exp ¼ T21 pffiffiffiffiffi þ pffiffiffiffiffi exp pffiffiffiffiffi þ pffiffiffiffiffi 4t a1 a2 a1 a2 4at a (44) which determines the behavior of aðtÞ in the first instants of the transient process of heat conduction determined by t 4tmi ¼ l2i ai , for i ¼ 1; 2: Note that if e1 ¼ e2 , the effective thermal diffusivity is independent of time and is given by Eq. (25), as it was established before. In general, Eq. (44) can be solved for aðtÞ in terms of the Lambert W-function [48,49] as follows "

!# 2 l2 l1 l2 2 1 l1 l2 2 T21 ¼ 2tWg pffiffiffiffiffi þ pffiffiffiffiffi exp pffiffiffiffiffi þ pffiffiffiffiffi 2t a1 a2 a1 a2 aðtÞ 2t (45)

0

hðtÞ ¼ gðtÞ

(43)

(39a)

then the convolution theorem establishes that the temperature in the time domain and at x ¼ l is given by ðt ðt (40) Tðl; tÞ ¼ f ðt uÞhðuÞdu ¼ f ðt uÞgðuÞdu 0

"

# l1 l2 1 l1 l2 2 T21 gðtÞ ¼ pffiffiffiffiffiffi pffiffiffiffiffi þ pffiffiffiffiffi exp pffiffiffiffiffi þ pffiffiffiffiffi 4t a1 a2 a1 a2 2 pt3

where, in order to obtain real positive values with physical meaning for aðtÞ, the branch with g ¼ 1 have to be used [48,49]. In order to see the graphical behavior of this effective thermal diffusivity, it is convenient to define the following normalized nondimensional variables

a l1 l2 2 þ p ffiffiffiffi ffi p ffiffiffiffi ffi l2 a1 a2

l1 l2 2 tN 2t pffiffiffiffiffi þ pffiffiffiffiffi a1 a2 aN

which transform Eq. (45) as follows

1 1 T2 ¼ tN W1 21 exp aN ðtN Þ tN tN

(46a) (46b)

(47)

Equation (47) reveals that the transmission coefficient T21 (and therefore the ratio e2 =e1 of thermal effusivities) plays a determinant role in the time behavior of the aN . Taking into account that Eq. (45) is valid for a time t l2i ai , by using Eq. (46b), it can be deduced that Eq. (47) is valid for tN 1=2. The normalized effective thermal diffusivity aN as a function of the normalized time tN is shown in Fig. 3(a), for three values of the ratio e2 =e1 of thermal effusivities. Note that aN increases with the time when e2 =e1 ¼ 2=3(T21 ¼ 6=5). On the other hand, for e2 =e1 ¼ 3=2 (T21 ¼ 4=5) aN decreases with the time, and it remains constant for e2 ¼ e1 , with a value that was established in Eq. (25). This fact points out that in the first instants of the transient process of heat conduction, the effective thermal diffusivity increases (decreases) compared with its value at steady-state conditions, when the transmission coefficient T21 is larger (shorter) than the unit. For an imperfect thermal contact of the two component layers (R12 6¼ 0), the inverse Laplace transform of Eq. (39b) may be determined using the convolution theorem, after expressing its right-hand side as the product of two functions, being one of them its exponential part, in such a way that their inverse Laplace transform are known [42]. After following this procedure, inserting the result into Eq. (41) and making a suitable change of variable, yields

pffiffiffiffi l l2 pffiffiffi exp (48) ¼ 2tqðtN Þ 4at a Transactions of the ASME


where

" pffiffiffiffi ð tN 1 1 T21 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp exp r 2utN r 0 pð1 uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#

tN ð1 uÞ tN ð1 uÞ du pffiffiffiffiffi (49) erfc

r2 r u3

qðtN Þ ¼

being erfc() the error function complement [43] and r the normalized nondimensional interface thermal resistance defined by pffiffiffi 2R 12 (50) r¼

1 1 l1 l2 þ pffiffiffiffiffi þ pffiffiffiffiffi e1 e2 a1 a2 The fact that Eq. (48) involves a transmission coefficient different of the unity, due to the difference of the thermal effusivities of the layers, and their interfacial thermal resistance, indicates that the effective thermal diffusivity a takes into account that the thermal energy is not only transmitted but also reflected at the interface of the layers. Note that Eq. (48) can be solved for the effective thermal diffusivity in terms of the Lambert W-function [48,49] as follows 1 ¼ tN W1 q2 ðtN Þ aN ðtN Þ

(51)

where tN and aN are defined by Eqs. (46a) and (46b). The complicated Eq. (49) shows that the transmission coefficient T21 as well as the normalized interface thermal resistance r play an important role in determining the time behavior of the normalized effective thermal diffusivity. For a constant r and different ratios e2 =e1 , aN has a similar behavior than the one presented in Fig. 3(a). On the other hand, Fig. 3(b) shows the time-dependence on the effective thermal diffusivity for three values of the normalized interface thermal resistance r and a constant ratio e2 =e1 ¼ 2=3. Note that in the time domain, the effective thermal diffusivity presents a monotonous decreasing behavior and takes lower values when the normalized interface thermal resistance increases, agreeing with the physical meaning of the interface thermal resistance [34], which acts as a barrier for the heat transport through the component layers. The complex time-dependence on the thermal diffusivity of the effective layer (see Eqs. (44) and (48)) shows explicitly why most of the research works dedicated to the thermal characterization of materials in the time domain have been performed under steady or quasisteady conditions (long time domain) [34,45]. In Addition, Eqs. (44) and (50) could be useful in the thermal characterization of a two-layer system and in the understanding of the heat transport, in the short time domain.

3.2.2. Neumann Problem. This problem is defined by specifying the heat flux at x ¼ 0; in such a way that its Laplace transform /ð0Þ is an arbitrary known function, and therefore the temperature at x ¼ l is given by Eq. (13b). The following two limiting cases of this equation are going to be considered n Long time results: s ! 0 After expanding the matrix elements B and C in powers of s and following a similar pffiffi procedure to obtain Eq. (34), it can be shown that at order s the effective heat capacity is given by Eq. (27b) and at order s, the effective thermal diffusivity is given by l2 l21 l22 e1 l1 l2 l1 ¼ þ þ 2 pffiffiffiffiffi pffiffiffiffiffi þ 2R12 e1 pffiffiffiffiffi a1 a a1 a2 e2 a1 a2

(52)

In this way, by combining Eqs. (27b) and (52), the thermal characterization of the two-layer system can be performed, under a Neumann boundary condition and quasisteady-state conditions. Note that Eq. (34) as well as Eq. (52) establish that the effective thermal diffusivity depends on the order of the layers and that after interchanging the subscripts 1 and 2, Eq. (52) takes the form of Eq. (34). This indicates that, to determine the effective thermal diffusivity, it is equivalent to perform an experiment under a Dirichlet boundary condition with the layers in the order 1 ! 2 and another one under a Neumann boundary condition with the component layers in the order 2 ! 1. In the same way that for the Dirichlet problem if the order of the layers is ignored Eq. (36) is obtained. n Short time results: s ! 1 Under the same considerations made to obtain Eq. (37) for the Dirichlet problem, but under a Neumann boundary condition, the same effective relation found in Dirichlet problem (Eq. (41)) is obtained. This establishes that in an experiment dedicated to measure the effective thermal diffusivity of a two-layer system, in the first instants of the transient process of heat conduction, it is equivalent to establish a temperature or a heat flux boundary condition as a thermal excitation source at the surface x ¼ 0 of the first layer. Similar results have been found in the frequency domain for the effective thermal properties [14].

4

Conclusions

The transient heat transport in a flat layered system with interface thermal resistance has been analyzed, under the Cattaneo– Vernotte hyperbolic heat conduction model and using the thermal quadrupole method. For a single semi-infinite layer, analytical formulas useful in the determination of its thermal relaxation time and effusivity have been derived by using a heat flux as a thermal

Fig. 3 Time dependence on the normalized parabolic effective thermal diffusivity, for three values of the (a) ratio e2 =e1 of thermal effusivities with R12 5 0 and (b) normalized interface thermal resistance r , for a constant ratio e2 =e1 5 2=3


SEPTEMBER 2011, Vol. 133 / 091301-7


excitation. For a two-layer system in thermal contact with a semi-infinite layer, in the long time regime and under a Dirichlet boundary condition, the well-known effective thermal resistance formula and a novel expression for the effective thermal relaxation time are derived, while for a Neumann problem, only a heat capacity identity is found. In addition, in a quasi-steadystate condition, a formula for the effective thermal diffusivity has been presented. In contrast in the short time regime, under the Fourier approach and both Dirichlet and Neumann conditions, a formula for the effective thermal diffusivity in terms of the time, the thermal properties of the individual layers and its interface thermal resistance was derived. These results can be very useful in the development of experimental methodologies to perform the thermal characterization of materials in the time domain.

Nomenclature a¼ A¼ b¼ c¼ B¼ C¼ D¼ erfc ¼ f¼ g¼ h¼ I¼ I0 ¼ I1 ¼ J¼ k¼ l¼ L¼ L1 ¼ p¼ q¼ Q¼ R¼ r¼ s¼ t¼ T¼ T ¼ x¼ W¼

constant, K matrix coefficient constant, K specific heat, J/kg K matrix coefficient, K m2/W matrix coefficient, W/K m2 matrix coefficient error function complement time-dependent function, K time-dependent function time-dependent function external energy flux, W/m2 modified Bessel function of the first kind and order zero modified Bessel function of the first kind and order one heat flux, W/m2 thermal conductivity, W/m K thickness, m Laplace operator inverse Laplace operator real parameter, m1 time dependent function energy density, J/m2 interface thermal resistance, m2 K/W normalized interface thermal resistance laplace parameter, 1/s time, s temperature, K transmission coefficient spatial coordinate, m Lambert W-function

Greek Symbols a¼ d¼ e¼ h¼ q¼ s¼ u¼ v¼

thermal difussivity, m2/s dirac delta function thermal effusivity, Ws1/2/m2K laplace transform of the temperature, K s density, kg/m3 thermal relaxation time, s Laplace transform of the heat flux, W s/m2 real parameter, W/K m2

Subscripts g¼ s¼ m¼ N¼ 0¼ 1¼ 2¼ 3¼

relative to the branches of the Lambert W-function relative to the steady-state condition relative to the thermalization time relative to a normalized quantity relative to an initial value relative to the first layer relative to the second layer relative to the third layer

091301-8 / Vol. 133, SEPTEMBER 2011

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