Thermal characterization of granular materials using a ... - Springer Link

Granular Matter (2010) 12:569–577 DOI 10.1007/s10035-010-0195-6

Thermal characterization of granular materials using a thermal-wave resonant cavity under the dual-phase lag model of heat conduction J. Ordóñez-Miranda · J. J. Alvarado-Gil

Received: 19 February 2010 / Published online: 24 June 2010 © Springer-Verlag 2010

Abstract The process of heat transport in granular materials has generated a great deal of controversy. It has been claimed that the process is affected not only by the thermal conductivity, diffusivity and heat capacity, but also additional parameters in the form of time-phase lags must be considered. These quantities permit to take into account the thermal inertia and the micro-structural interactions of the media in such a way that they establish the non-simultaneity between the temperature and the heat flux. A highly successful model that takes into account these effects is known as the dual-phase lag model of heat conduction. It constitutes an approach that generalizes and overcomes the limitations of the classical Fourier law of heat transport. One of the most sensitive techniques for measuring thermal properties is the thermal-wave resonant cavity, which is formed by three layers. The one in the middle is semi solid, liquid or gas, whose thickness can be changed moving one of the external layers. In order to study the material in the middle, a modulated heat source is applied to one of the external layers, and the changes of temperature are registered at the surface of any of the external layers. This methodology has provided high accuracy results for the thermal properties of liquids, gases and nanofluids in the context of Fourier heat diffusion equation. However results for granular materials using this methodology are scarce and the role of the phase lags in heat transport has not been fully explored. In this work, the theoretical basis for the development of a thermal-wave resonant cavity based on dual-phase lag model is studied. It is shown that this system could be used to measure not only the thermal diffusivity but also the time-phase lags of granular materials, by performing a suitable thickJ. Ordóñez-Miranda · J. J. Alvarado-Gil (B) Departamento de Física Aplicada, Centro de Investigación y de Estudios Avanzados del I.P.N-Unidad Mérida, Carretera Antigua a Progreso km. 6, A.P. 73 Cordemex, 97310 Mérida, Yucatán, México e-mail: [email protected]

ness scan of the cavity. It is shown that the results obtained can be a highly useful in the development of experimental methodologies revealing the possibility of non-Fourier heat transport and how the thermal characterization of granular materials can be performed. Keywords Thermal-wave resonant cavity · Dual-phase lag model · Time delays · Three-layer system · Heat conduction

1 Introduction Granular materials are among the most challenging systems to be studied given that their macroscopic dynamics cannot adequately be described as solids, fluids or gases. In particular heat transport involves a complex process that has been studied applying different models and involves important difficulties from the experimental point of view [1–5]. One of the problems in the development of sophisticated granular models is to find simple and reliable experimental validation methods. These methods should work in in-situ conditions and be as less destructive as possible. Additionally most of the heat transfer models, used for granular materials, are inherited from the study of diluted systems or mixtures that can be considered at a certain scales as homogeneous. However, granular materials are strongly heterogeneous materials requiring the use of well-suited models for the analysis of heat transport. In the case of a two-phase granular system, one of the simplest heat conduction models, that considers the microstructure, is the known as two-equation model [5,6], which has been developed writing the Fourier law of heat conduction [7] for each phase and performing a volume averaging procedure [6]. The average temperature of the solid (T1 ) and fluid (T2 ) phases are

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J. Ordóñez-Miranda, J. J. Alvarado-Gil

∂ T (x, t + τT ) , ∂x

(2)

where x is the spatial coordinate, q[W/m 2 ] is the heat flux, T [K ] is the absolute temperature, k[W/m K ] is the thermal conductivity, τq is the phase lag of the heat flux and τT is the phase lag of the temperature gradient. For the case in which τq > τT , the heat flux (effect) established across the material is a result of the temperature gradient (cause); while for τq < τT , the heat flux (cause) induces the temperature gradient (effect). Note that when τq = τT , the response between the temperature gradient and the heat flux is instantaneous and Eq. (1a) reduces to Fourier law except for a trivial shift in the time scale [1,2]. For the case of granular matter, the phase lag τq has been interpreted as the time delay in heat conduction due to contact thermal resistance among the grains, while τT is interpreted as the time required to travel within the grains [13]. In this way the DPL model provides a comprehensive treatment of the heterogeneous nature of granular media. It is also important to mention that this model has been shown to be admissible by the second law of extended irreversible thermodynamics [1] and by the Boltzmann transport equation [14]. The lagging behavior in the transient process is caused by the finite time required for the microscopic interactions to take place. This time of response has been claimed to be

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Grains

(1b)

where t is the time, k1 and k2 are the effective thermal conductivities of the solid and fluid phase, respectively; k21 is known as the cross thermal conductivity of the two phases, γ1 = (1 − ϕ)ρ1 c1 and γ2 = ϕρ2 c2 , being ρ the density, c the specific heat and ϕ the porosity. Subscripts 1 and 2 refer to the solid and fluid phase, respectively. The porosity-dependent parameters h and av are the film heat transfer coefficient and the interface area per unit volume, respectively; and they come from modeling of the interface flux [5,6]. By comparing Eq. (1a–1b) with the Fourier equation, it can be observed that, the two-phase-system model takes into account the microstructural effects through the coupled thermal conductivity k21 and the factor hav . Notice that, in absence of k21 , Eq. (1a–1b) reduces to the two-equation model commonly-used in the literature [3,4,8,9]. It has been shown that the two-equation model is equivalent to the one-equation model known as the dual-phase lagging (DPL) model, in which the microstructural effects are taken into account by means of two time delays, [1,5,10–12]. This model establishes that either the temperature gradient may precede the heat flux or that the heat flux may precede the temperature gradient and it is based on the hypothesis that the heat flux equation can be written in the form: q(x, t + τq ) = −k

q received at t+τ q

(1a)

Heater

∂ T1 = k1 ∇ 2 T1 + k21 ∇ 2 T2 + hav (T1 − T2 ) , ∂t ∂ T2 = k2 ∇ 2 T2 + k21 ∇ 2 T1 − hav (T1 − T2 ) , γ2 ∂t

γ1

Interstitial matter

T established at t+τ T

Fig. 1 Lagging response induced by the interstitial matter in a granular medium

of the order of several seconds in granular matter, due to the low-conducting pores among the grains and their interface thermal resistance [1,10]. In Fig. 1, it can be observed that when the heat flux precedes the temperature gradient (τq < τT ), due to the presence of pores between the heater and the material volume enclosed by the dashed line, the heat flow produced by the heater at a general time t arrives at the material volume at a later time t + τq . The internal pores within the material volume cause an additional delay in heat transport, prolonging the establishment of the temperature gradient at t + τT . Of course, this type of delayed response depends on the detailed configuration of the grains and the interstitial matter within the material volume. For a time t in the transient larger than process much t >> τq , τT , Eq. (2) can be the time delays τq and τT approximated by a first order Taylor series expansion, as follows: ∂ T (x, t) ∂q(x, t) ∂ 2 T (x, t) = −k + τT q(x, t) + τq . (3) ∂t ∂x ∂t∂ x After combining Eq. (3) with the energy conservation equation in absence of internal heat sources [7] and assuming constant thermal properties, the DPL heat conduction equation is obtained [1,2,15] : ∂ 3 T (x, t) 1 ∂ T (x, t) τq ∂ 2 T (x, t) ∂ 2 T (x, t) − +τ − = 0, T ∂x2 ∂t∂ x 2 α ∂t α ∂t 2 (4) where α is the thermal diffusivity of the involved material [7]. Eq. (4) constitutes a generalized or unified equation that reduces to the hyperbolic heat conduction equation (based on the Cattaneo- Vernotte equation) when τT = 0 [1,2] and to the parabolic heat conduction equation (based on Fourier law) when τq = τT = 0 [1,2,7,16]. For the case when τq < τT , Eq. (4) is of the parabolic type and therefore establishes a non wave-like heat conduction equation which predictions differs from the usual diffusion behavior predicted by Fourier law. On the other hand, for τq > τT , Eq. (4) is of the hyperbolic type and predicts wave-like thermal signals, predominantly [2].

Thermal characterization of granular materials

571

For the case of a two-phase composite, Peter Vadasz [3,4] and Wang and Wei [5], have shown that the Eq. (4) is equivalent with the two-phase-system model. This can be obtained after uncoupling the temperatures T1 and T2 in Eq. (1a–1b). Wang and Wei have demonstrated that this equivalence is possible if the ratio of time delays is given by [5] k1 γ22 + k2 γ12 − 2γ1 γ2 k21 τT , =1+ τq γ1 γ2 (k1 + k2 + 2k21 )

(5)

which indicates that the ratio τT /τq may be larger, equal or smaller than 1 if the quantity k1 γ22 + k2 γ12 − 2γ1 γ2 k21 is positive, zero or negative respectively, depending on the porosity of the fluid phase. Notice that for high (ϕ → 1) and low (ϕ → 0) porosities, τT > τq ; which means that the heat propagates through a diffusion process. On the other hand, for middle values of the porosity (ϕ ≈ 1/2), it is possible that τT < τq , which would imply that the thermal signals propagate by means of waves. Therefore, for constant thermal properties of the two phases, the type of heat conduction in granular matter is determined by their porosity. In this way, the first order approximation of the DPL model (Eq. (3)) is appropriate to undertake the problem of heat transport in granular matter and is more convenient that the twoequation model, because it provides directly the total macroscopic temperature. Both of the time phase lags are treated as intrinsic or structural thermal properties of the material and therefore the application of a heat flux (temperature gradient) at the boundary does not guarantee the precedence of the heat flux (temperature gradient) to the temperature gradient (heat flux). DPL model depends crucially on the time delays involved, however experimental or theoretical methodologies permitting to determine such quantities are scarce [1,10], even though for the case of granular matter for which time delays on the order of seconds are expected. It is therefore important to explore the DPL effects on the temperature and from them to establish methodologies to measure or constrain these parameters [17]. For modulated heat sources, it could be expected that the advantages found in photothermal science in the analysis of thermal depth profiles using Fourier law can also arise when the same kind of heat sources are used with the DPL heat conduction equation [12,16,18–20]. In traditional thermal wave phenomena [20,21], a highly sensitive and accurate methodology, known as thermal wave resonant cavity, has been developed and has provided a great variety of useful results in the study of liquids, soft materials and gases [22–24]. This resonator is formed by three layers with the middle one made of a semi solid or liquid material whose thickness can be changed moving one of the external layers. In order to study the material of the middle layer, a modulated heat source is applied to one of the external layers, and the changes of temperature are registered at the surface of any

of the external layers. However results for granular materials using this methodology are scarce and the role of the phase lags in heat transport has not been fully explored. In this paper, heat transport governed by the DPL model in a thermal waves resonator analyzed [22–24]. On the face of the thermal wave cavity on which the sample is heated, the behavior of the temperature and its oscillations is analyzed as a function of the time delay parameters, showing the conditions on which specific thermal wave modalities dominate. On the opposite face, it is shown that for appropriate thicknesses of the middle layer, the real and imaginary parts of the complex of the temperature present oscillations that depend on the magnitude of the time delays. Analytical expressions for maxima and minima of the amplitude of these oscillations and the thicknesses at which they occurs are obtained. These results are used to find useful formulas for the determination of both time delays as well as additional thermal properties of granular materials. The results obtained can be highly useful in the development of experimental methodologies to show if the non-Fourier behavior is observable and how the thermal characterization of granular matter can be performed.

2 Formulation of the problem and solution The general problem to solve is finding the effective thermal properties of granular materials as well as the conditions, in which the deviations from the traditional heat transport approach, governed by Fourier equation, can be observed. This can be achieved by placing the sample material inside a cavity contained between two layers, onto the first one impinges a periodically intensity modulated laser beam. The third layer is limited by a movable wall as shown in Fig. 2a. The methodology consists on performing a suitable scan of the thickness (l) of the cavity and recoding the temperature on the surface of the first layer or at the inner surface of the movable wall. A schematic diagram of the analyzed system shown in Fig. 2, in which the system is excited at the optically opaque surface x = 0, with a laser beam modulated at frequency f In this case, the conservation of energy establishes that [15,16, 25,26]: (6) q(x = 0, t) = I0 cos(ωt) = Re I0 eiωt , where ω = 2π f , Re() stands for the real part of its argument and the constant factor I0 is proportional to the intensity of the laser beam. For this kind of thermal excitation, the temperature at any point of the system is given by: T (x, t) = Tamb + Tac (x, t),

(7)

where Tamb corresponds to the ambient temperature and Tac (x, t) = Re θ (x) eiωt is the periodic component of

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572


found that for any frequency, χ+ = χ− = 1 and therefore the results predicted by the DPL model must have the same behavior as the ones predicted by Fourier law. Considering that the layers in Fig. 2b are in perfect thermal contact [7,16,18], the boundary conditions obtained from the usual requirement of temperature and heat flux continuity at the interfaces x = l1 , l are given by

(a)

Laser beam

Movable wall l

(b) k1, 1, x 0

q1, T1

k, ,

x l1

q, T

k0 ,

x l1 l

1 + iωτT− k(x − ) 1+iωτq−

0 , q0, T0

x

Fig. 2 (a) Transverse view of the studied system. (b) Schematic representation of the studied layered system. The middle layer of thermal conductivity k, thermal diffusivity α, time delays τq and τT and variable thickness l is surrounded by a finite and semi-infinite layer, at its left and right side, respectively

the temperature, due to the harmonic form of the thermal excitation (Eq. (6)). From now on, the operator Re() will be omitted, taking into account the convention that the real part of the expressions of the temperature must be taken to obtain physical quantities. Our attention will be focused on the spatial component θ (x) of the oscillatory part of the temperature, due to the fact that it is the quantity of interest in lock-in and similar detection techniques. Inserting Eq. (7) into Eq. (4) and considering that there are not any internal heat sources, for x > 0, the general solution of Eq. (4) for θ (x) is given by: θ (x) = Be px + Ce− px ,

(8)

where B and C are two constants which depend on the boundary conditions of the particular problem and P is given by

iω 1 + iωτq χ− + iχ+ , (9a) = p= α 1 + iωτT μ

2α α = , (9b) μ= ω πf

2

1+ ωτq 1+ (ωτT )2 ± ω τq −τT χ± = . (9c) 1+ (ωτT )2 It is important to observe that for low frequencies such that ωτq , ωτT > 1; both parameters and χ− tend to τq /τT and the parameter p → τq /τT pc . These results indicate that the temperature θ (x) predicted by the Fourier and DPL models would be similar for the high and low-frequency regimes. Furthermore for τq = τT , it is

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(10a) θ (x − ) = θ (x + ), + +) 1 + iωτ dθ (x − ) dθ (x T = k(x + ) , (10b) dx 1+iωτq+ d x

where the superscripts “+” and “−” indicate that the limits x → l1 , l1 + l are taken from the right and left of the points x = l1 , l1 + l; respectively. The form of the Eq. (10b) is derived using Eq. (3). Furthermore, according to Eq. (6), the boundary condition at x = 0has the following form: 1 + iωτT 1 dθ (x) = I0 . (11) −k1 1 + iωτq1 d x x=0 Using Eqs. (8), (10a–b) and (11), the solutions for the spatial part of the thermal wave at x = 0, l1 , l1 + l are found to be: 1 + iωτq 1 + Re−2 p1 l1 I0 , θ (x = 0) = √ ε1 iω 1 + iωτT 1 − Re−2 p1 l1 (12a) e − p1 l 1 I0 T02 T21 1 + iωτq θ (x = l1 + l) = √ ε1 iω 1 + iωτT 1 − Re−2 p1 l1 e− pl , (12b) × 1 + R02 R21 e−2 pl where p1 ( p) is defined by Eqs. (9a-c) for the thermal properties of the first (middle) layer and 1 − λmn , 1 + λmn = 1 + Rmn , εm 1 + iωτT m 1 + iωτqn = , εn 1 + iωτqm 1 + iωτT n

Rmn =

(13a)

Tmn

(13b)

λmn

R=

R21 + R02 e−2 pl , 1 + R02 R21 e−2 pl

(13c) (13d)

√ with m, n = 0, 1, 2 and εn = kn / αn being the thermal effusivity of the layer n [7,16,27]. The subscript n = 2corresponds to the middle layer which acts like the cavity of thickness l.

3 Analysis and discussions In this section, the thermal spectra given by Eqs. (12a-b) are going to be analyzed and used to obtain useful formulas for determining the thermal properties of the middle layer.


Notice that the complex amplitude of the temperature in both Eqs. (12a-b) depends on the thickness l and frequency ω, therefore any one of these parameters can be used for the analysis. Under the Fourier approach, it has been established that the cavity-thickness scan can provide more precise and simpler results than the frequency scan [22–24]. This fact could be even more remarkable in the DPL approach due the complicated frequency-dependence of the temperature in Eqs. (12a-b). After writing Eqs. (12a) as a complex function in its polar form, the amplitude A and phase φ of the temperature at θ (x = 0) are obtained. Final expressions are long and complicated, however useful results can be deduced from their √ graphical behavior. The normalized amplitude Aε1 ω/I0 and phase φ as a function of the normalized thickness l/μ of the middle layer are shown in Fig. 3a, b, respectively; for two pair of the products ωτq and ωτT . The corresponding normalized amplitude and phase predicted by Fourier law (τq = τT ) are shown in the same figures by dashed lines. Figure. 3a, b show that when the temperature gradient precedes to the heat flux (τT < τq ), the normalized amplitude and phase of θ (x = 0) present a remarkable oscillatory behavior. The amplitude of the oscillations decreases when the thickness of the cavity increases or when τT → τq , in such a way that for τT = τq the spectra reduces to the ones strongly attenuated predicted by Fourier law (dashed lines). In contrast, when the heat flux is the cause and the temperature gradient the effect (τT > τq ), the normalized amplitude and phase have a monotonous decreasing behavior when the normalized thickness l/μ increases. For a fixed value of the product ωτT = 1/2, Fig. 4a shows that the oscillations of the normalized amplitude become more remarkable and appear at lower thickness when the product ωτq increases. This behavior can be expected, because when the ratio of the time delays τq /τT increases, Eq. (4) predicts an enhancement of the hyperbolic behavior. On the other hand, when ωτq = 1/2 is maintained constant, the normalized amplitude predicted by the DPL model presents a monotonous decreasing behavior, which is stronger for larger values of the product ωτT . This result is due to the fact that Eq. (4) presents a remarkable parabolic behavior when the ratio τq /τT decreases. Similar results as the presented in Fig. 3b can be obtained for the phase of the temperature. These characteristic behaviors of the thermal spectra can be used to accept or reject a DPL behavior of the middle layer by only performing a suitable cavity-thickness scan. For the case τT < τq , the amplitude of the oscillations and the thicknesses at which they occur, depends on the thermal properties of the middle layer, however, it is difficult to find simple analytical formulas to determine them. Therefore their use for measuring the thermal properties of the cavity is very limited. Similar results are obtained if the real and imaginary parts of θ (x = 0) and the amplitude and phase of

573

Fig. 3 Length dependence of the (a) normalized amplitude and (b) phase of the complex amplitude of the temperature predicted by the DPL model at x = 0. The dashed line corresponds to the parabolic model and the solid lines to the DPL one, for two pair of the products ωτq and ωτT . Calculations were performed using l1 /μ1 = 1/10, ε2 /ε1 = 7/10, ε0 /ε2 = 1/20, ωτq1 = 1/2, ωτT 1 = 2/5, ωτq2 = 1/10 and ωτq2 = 1/5

θ (x = l1 + l) are analyzed. However, in what follows, it is shown that the real and imaginary part of θ (x = l1 + l) can give useful analytical formulas, under appropriate conditions over the thickness of the cavity. Considering that the middle layer is thermally thick, in such a way that its thickness l is large enough to satisfy the relationship Re( pl) = lχ− /μ >> 1, the terms of the form exp (−2 pl) can be neglected and therefore Eq. (10) reduces to θ (l) = Ae− pl ,

(14)

where A = |A| eiψ is a complex parameter that depends on the thickness l1 of the first layer and of the modulation frequency, which are kept constant in the present approach. The real θ R and imaginary θ I parts of θ (l) can be written as θ R (x) = |A| e−xχ− cos(xχ+ − ψ),

(15a)

θ I (x) = − |A| e

(15b)

−xχ−

sin(xχ+ − ψ),

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(a) Normalized real part, θR/|A|

0.6

ωτq=1/2, ωτT=5 0.3

0.0

-0.3

ωτq=5, ωτT=1/2 0

1

2

3

4

5

4

5

Normalized thickness, l/

Normalized imaginary part, θI/|A|

(b)

0.3

ωτq=5, ωτT=1/2 0.0

ωτq=1/2, ωτT=5

-0.3

-0.6

0

Fig. 4 Length dependence of the normalized amplitude for (a) ωτT = 1/2 and (b) ωτq = 1/2 predicted by the DPL model at x = 0. The dashed line corresponds to the parabolic model and the solid lines to the DPL one, for three pair of the products ωτq and ωτT . Calculations were performed with the same complementary data used in Fig. 3

where x = l/μ. The normalized real θ R /|A| and imaginary θ I /|A| parts of θ (x) as a function of the normalized thickness l/μ of the middle layer are shown in Fig. 5a, b, respectively; for two pair of the products ωτq and ωτT . The corresponding real and imaginary parts predicted by Fourier law τq = τT are shown in the same figures by dashed lines. In both Fig. 5a, b, it can be observed that the real and imaginary part of the complex amplitude of the temperature present oscillatory behavior when the temperature gradient precedes the heat flux (τT < τq ). The amplitude of the oscillations increases for the case in which τT τq ), the real and imaginary parts of the thermal spectra present a stronger attenuation than the one predicted by Fourier law. These facts indicate that the thermal waves can travel larger distances into the material when τT < τq than in the case

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1

2

3


Fig. 5 Length dependence of the normalized (a) real part and (b) imaginary part of the complex temperature predicted by the DPL model. The dashed line corresponds to the parabolic model and the solid lines to the DPL one for two pair of the products ωτq and ωτT . Calculations were performed taking ψ = −π/4

of τT > τq . The values of the maxima and minima of the amplitude as well as the thicknesses, at which they occur, can be obtained analytically. From these results, the thermal time delays and additional thermal properties of the studied granular material can be determined. For a fixed modulation frequency, it is easy to show that the extremes values of the real and imaginary parts of the thermal signal are given by |A| χ− exp θ Rext (n, ω) = (−1)n χ−2 +χ+2 χ− χ− nπ +ψ− tan−1 , × − χ+ χ+ |A| χ+ θ Iext (n, ω) = (−1)n+1 exp χ−2 +χ+2 χ− −1 χ+ nπ +ψ+ tan , × − χ+ χ−

(16a)

(16b)


575

(a)

π

Normalized real part, θR/|A|

which occurs at the following thicknesses of the middle layer −1 λth n + ψ−tan π(χ− /χ+ ) , θ R = θ Rext (17) ln = −1 2 n + ψ+tan (χ+ /χ− ) , θ I = θ ext I

where λth = 2π μ/χ+ ,

(18)

is the wavelength of the thermal waves predicted by the DPL model [26]. In this way, Eqs. (16a-b) indicate that the extremes values of the real (imaginary) part of the temperature are local maxima (minima) when n is even and minima (maxima) for n odd. Eq. (17) is analogous to the well-known expressions for the nodes (maxima) and antinodes (minima) of the standing waves of a string with fixed extremes. Similar results have been already found under the Fourier law approach [22–24]. According to Eqs. (17) and (18), The difference in length for two successive maxima (minima) of the real or imaginary part of the temperature is given by

From Eqs. (19) and (20), it is obtained that both time delays √ of the middle layer with thermal diffusivity α(μ = α/π f ), can be determined from: πμ , (21) χ+ = l μ ln(B). (22) χ− = l

0.0

-0.3

ωτq=1 -0.6

2

4

6

8

10

8

10


(b)

(19)

In addition, the ratio of two successive maxima and minima of the real or imaginary parts of the PE output is (see Eqs. (18a-b)): θ Rext (n, ω) θ Iext (n, ω) = B ≡ ext θ R (n + 1, ω) θ Iext (n + 1, ω) = exp (π χ− /χ+ ) . (20)

ωτq=5 0.3

0

Normalized imaginary part, θI/|A|

l ≡ ln+1 − ln = π μ/χ+ .

0.6

0.6

ωτq=5

0.3

0.0

-0.3

ωτq=1

-0.6

-0.9 0

2

4

6


Fig. 6 Length dependence of the normalized (a) real part and (b) imaginary part of the complex temperature. The dashed line corresponds to the parabolic model and the solid lines to the Cattaneo-Vernotte one for two values of the product ωτq . Calculations were performed taking φ = −π/4

under the DPL model, the thermal diffusivity of the granular material can be determined by means of Eq. (23) and its corresponding time delays through Eqs. (21) and (22).

Particular cases: •

•

Parabolic case: τq = τT

In this case χ− = χ+ = 1(see Eq. (9c)) and therefore from Eq. (21) it is obtained α=

f (l)2 , π

(23)

which permits to determine the thermal diffusivity of the granular material acting as the middle layer, by performing a suitable thickness scan and measurement the difference in length between two successive extreme values of the real and/or imaginary part of the temperature. It is important to note that Eq. (23) remains valid even when τq = τT , for frequencies low enough such that ωτq , ωτT