Three-Phase-Lag Heat Conduction in a Functionally Graded Hollow Cylinder · THREE-PHASE-LAG HEAT...

THREE-PHASE-LAG HEAT CONDUCTION IN A FUNCTIONALLY GRADED HOLLOWCYLINDER

Abdolhamid Akbarzadeh1,2, Jiawei Fu2,3 and Zengtao Chen2

1Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada2Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

3Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, P.R. ChinaE-mail: [email protected]; [email protected]

Received August 2013, Accepted November 2013No. 13-CSME-142, E.I.C. Accession Number 3600

ABSTRACTHeat conduction in a functionally graded, infinitely-long hollow cylinder is studied based on the three-phase-lag model. Material properties except the phase-lags vary according to a power-law within the cylinder. Thephase-lag heat conduction equation is written in a form in which various models of heat conduction theoriescan be generated. The governing differential equations in the Laplace domain are solved exactly and anumerical Laplace inversion technique is employed for restoring results in the time domain. The effects ofdifferent heat conduction theories, phase-lags, geometries, and non-homogeneity indices are studied on thespatial distribution and time-history of temperature.

Keywords: dual-phase-lag model; functionally graded material; heat conduction; hollow cylinder; thermalwave; three-phase-lag model.

LA CONDUCTION THERMIQUE D’UN CYLINDRE CREUX FONCTIONNELLEMENTGRADUÉ AVEC UN DÉCALAGE TRIPHASÉ

RÉSUMÉLa conduction thermique dans un cylindre creux fonctionnellement gradué et infiniment long, est étudiéeselon le modèle de décalage triphasé. Les propriétés des matériaux à l’exception des décalages de phasevarient selon une loi de puissance à l’intérieur du cylindre. L’équation de la conduction thermique avecdécalage de phase est écrite dans un format qui permet la génération de différents modèles théoriques dela conduction thermique. Les équations différentielles gouvernantes dans le domaine de Laplace sont réso-lues exactement et une technique numérique d’inversion de Laplace est employée pour la restauration desrésultats dans le domaine temporel. Les effets des différentes théories de conduction thermique, le retardde phase, des géométries et des indices de la non-homogénéité sont étudiés en considérant la distributionspatiale et l’historique temporel de la température.

Mots-clés : décalage biphasé ; matériaux fonctionnellement gradué ; conduction thermique ; cylindre creux ;onde thermique ; décalage triphasé.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014 155

1. INTRODUCTION

The relation between heat flux and temperature gradient is given by the constitutive relation of heat fluxwhich along with the energy conservation equation or first law of thermodynamics establishes the heat con-duction equation. The earliest constitutive relation of heat flux was proposed by Fourier in 1807 which leadsto the classical parabolic heat conduction equation with an infinite thermal wave speed [1]. It means thatthermal disturbances are felt instantaneously at all points far from the thermal source, which is physicallyunrealistic for applications involving very low temperature, ultrafast laser heating, and micro temporal andspatial scales [2]. Consequently, different non-Fourier constitutive relations and heat conduction equationshave been introduced to remove this inherent drawback of the classical Fourier heat conduction.

The C-V constitutive heat flux relation with hyperbolic-type heat conduction equation was proposed byCattaneo [3] and Vernotte [4]. In this model, the time-lag needed for the establishment of heat flux was char-acterized by the thermal relaxation time. Although the C-V model predicts a finite thermal wave speed, ithas been observed that some of its predictions are not compatible with the experimental results due to over-looked microstructural effects in the heat conduction process. To consider the microstructural effects suchas phonon-electron interaction and phonon scattering, the phase-lag of temperature gradient was consideredin the heat flux constitutive relation by Tzou [5, 6] along with the phase-lag of heat flux which is interpretedas the time delay due to the fast transient effect of thermal inertia. The constitutive model results in thedual-phase-lag (DPL) heat conduction equation. Later, Choudhuri [7] established the three-phase-lag (TPL)constitutive model by introducing the phase-lag of heat flux, temperature gradient, and thermal displace-ment gradient in heat conduction equation. Contrary to temperature gradient which serves as a constitutivevariable in Fourier, C-V, and DPL heat conduction theories, temperature displacement gradient is consideredas a constitutive variable in TPL model. The introduction of TPL model provides a general theoretical heatconduction model with different microstructural considerations in order to enable scientists in the field ofheat conduction with a multiscale model to predict accurately the thermal behavior of structures. Recently,the new generalized fractional model for heat conduction has also been introduced [8]. Since the phase-lagtheories could lead to parabolic-type or hyperbolic-type differential equations based on the order of Taylorseries expansion of the phase-lags, analytical and numerical investigations on thermal responses of differenttypes of DPL and TPL models are significant for application of phase-lag theories in thermal analysis.

Laminated and multi-layered composites are suffering from the structural failure caused by the mismatchof material properties along the interfaces. To avoid abrupt changes of material properties across conven-tional laminated composites, functionally graded materials (FGMs) with continuous spatial transition ofmaterial properties have been introduced which experience the reduced stress concentration, stress intensityfactor, and thermal stresses [9, 10]. Subsequently, thermal analysis of functionally graded (FG) structures iscrucial for accurate design of FG components.

Frankel et al. [11] presented general one-dimensional temperature and heat flux formulations for hyper-bolic heat conduction in composite media and obtained the distribution from the heat flux formulation usinga generalized finite integral transform technique. Two-dimensional hyperbolic heat conduction problemswere solved by Chen and Lin [12] with the hybrid numerical scheme. Babaei and Chen [13] analyzed thehyperbolic heat conduction in an FG hollow cylinder. Similar investigation was done by Keles and Conker[14] to show the non-Fourier heat conduction in FG hollow cylinders and spheres with exponentially-varyingmaterial properties. Darabseh et al. [15] studied numerically the uncoupled thermoelasticity problem in anorthotropic cylinder using the hyperbolic heat conduction theory. Furthermore, Abd-Alla and Mahmoud [16]discussed the magneto-thermoelastic behavior in a rotating non-homogeneous orthotropic hollow cylinderwith the hyperbolic heat conduction model.

The heat conduction based on the DPL theory in different FG cylindrical panels was investigated by Ak-barzaheh and Chen [17]. They also considered the heat conduction in one-dimensional FG media with DPL

156 Transactions of the Canadian Society for Mechanical Engineering, Vol. 38, No. 1, 2014

theory for different geometries in Cartesian, cylindrical, and spherical coordinate systems [18]. Ramadan[19] obtained the semi-analytical solution of transient heat conduction in multilayered media with DPL the-ory. Two-dimensional DPL thermal behavior in multilayered structures was studied by Chou and Yang [20]using the space-time conservation element and solution element method, in which the temperature and heatflux can be solved simultaneously at each time-step. Liu [21] numerically analyzed the DPL heat conductionin two-layered cylinders with non-linear interface boundary conditions because of the difference in relax-ation time of two dissimilar materials. Akbarzadeh et al. [22, 23] studied the coupled thermopiezoelectricbehavior of a one-dimensional, functionally graded piezoelectric rod based on the classical uncoupled, clas-sical coupled, and generalized coupled thermoelasticity. The DPL model was used by Abouelregal et al.[24] to clarify the problem of an infinite non-homogeneous solid having a spherical cavity in the magneticfield. Liu and Chen [25] analyzed the inverse non-Fourier bioheat transfer problem in the bi-layer sphericalgeometry to explore the existence of DPL thermal behavior in tissues. Moreover, Tabrizi and Andarwa [26]proposed a method based on the frequency response to measure the time lag constants in heat conduction.

Quintanilla and Racke [27] discussed the stability conditions of the TPL heat conduction model; thefundamental solutions of this model were also derived by [28]. Mukhopadhyay and Kumar [29] analyzedthe effect of phase-lags on the thermoelastic wave propagation in a two-dimensional, thick plate usingdifferent thermoelasticity theories. The thermoelastic and thermo-visco-elastic problems of a spherical shellunder thermal shock condition, using the TPL model, were studied by Kar and Kanoria [30,31]. Banikand Kanoria [32] considered the thermoelastic interactions in an FG isotropic unbounded medium due toperiodically varying heat sources with TPL heat conduction theory.

This paper investigates the three-phase-lag heat conduction in an FG infinitely-long hollow cylinder witha power-law distribution of material properties along the radial direction. The most generalized, three-phase-lag heat conduction equation in a non-homogeneous cylinder is derived which includes the wave-likeand diffusive-like TPL and DPL models as well as C-V and classical Fourier heat conduction theories. Asemi-analytical solution procedure is used for obtaining the transient temperature field for different temporalboundary conditions. The thermal wave speeds based on different heat conduction theories in the cylindricalcoordinate are also compared. Finally, numerical results are depicted for various heat conduction theories,phase-lag of heat flux, phase-lag of temperature gradient, and phase-lag of thermal displacement gradient.

2. GOVERNING EQUATIONS

The three-phase-lag heat conduction model was developed by Choudhuri [7] and can be expressed as:

q(r, t+ τq

)=−

[k∇T (r, t+ τT )+k∗∇v(r, t+ τv)

](1)

in which q, k, k∗, T , v, τq , τT , τv, and t are the heat flux vector, thermal conductivity, rate of thermalconductivity, absolute temperature, thermal displacement which satisfies v = T , phase-lag of heat flux,phase-lag of temperature gradient, phase-lag of thermal displacement gradient, and time, respectively; r isthe position vector and ∇ is the gradient operator. Taylor’s series expansion of Eq. (1) up to the second orderfor τq , τT , and τv leads to the following heat conduction equation:(

1+ τq∂

∂t+τ 2q

2∂2

∂t2

)q=−

[(k+k∗τv

)∇T +

(kτT +k

∗τ 2v

2

)∇T +k

τ 2T

2∇T +k∗∇v

](2)

The overdots indicate time derivatives. The energy conservation equation can also be written as

−∇ ·q+R = ρcp∂T

∂t(3)

where R, ρ, and cp are, respectively, the internal heat generation, density, and specific heat.


By eliminating q using Eqs. (2) and (3), the heat conduction equation based on the TPL model for non-homogeneous materials can be obtained as(

1+ τq∂

∂t+τ 2q

2∂2

∂t2

)(ρcpT − R

)=∇ ·

[(k+k∗τv)∇T

]+∇ ·

[(kτT +k

∗τ 2v

2

)∇T

]+∇ ·

(kτ 2T

2∇T

)+∇ · (k∗∇T ) (4)

It is worth noting that all phase-lags are assumed to be constant for simplicity. For an infinitely-long hollowcylinder with inner radius ri and outer radius ro, Eq. (4) can be expressed in the cylindrical coordinate as(

1+ τq∂

∂t+τ 2q

2∂2

∂t2

)(ρcpT − R

)=

1r

∂

∂r

{r

[k∗∂T

∂r+ (k+k∗τv)

∂T

∂r+

(kτT +k

∗τ 2v

2

)∂T

∂r+kτ 2T

2∂T

∂r

]}(5)

Equation (5) provides a general heat conduction equation based on the TPL model. Setting k∗ = 0 leads tothe diffusive-like DPL heat conduction in which both τq and τT have been expanded to the second order. Wecould further provide the wave-like DPL model by setting k∗ = 0 and omitting the second-order expansionτ 2T in Eq. (5). Moreover, discarding the second-order time derivative from the left hand side of Eq. (5)

and letting k∗ = τT = 0 result in the C-V or hyperbolic heat conduction theory. The classical Fourier heatconduction theory can also be obtained by vanishing τq in the C-V model. Furthermore, the TPL modelcould also simulate the situations in which thermal conductivity k is much smaller than the rate of thermalconductivity k∗ that cannot be conducted by the other non-Fourier heat conduction theories [7].

The material properties of an infinitely-long, axisymmetric FG hollow cylinder are assumed to vary alongthe radial direction according to the power-law formulation as follows [33, 34]:

k(η)= koηn1, k∗(η)= k∗oη

n1, ρ(η)= ρoη

n2, cp(η)= cpoηn3 (6)

where ko, k∗o , ρo, and cpo are constants; nj (j = 1,2,3) are non-homogeneity indices and η = r/ro. Thefollowing non-dimensional parameters are also used in the solution procedure:

ζ =k′ot

r2o

, εo =k′oτq

r2o

, δo =k′oτT

r2o

, αo =k′oτv

r2o

, η =r

ro, rγ =

ri

ro

θ =T −T∞

T∞, θwi =

Twi−T∞

T∞, θwo =

Two−T∞

T∞, Qr =

roqr

koT∞(7)

in which T∞ is the ambient temperature, Twi and Two are temperature on the inner and outer surfaces, qr isthe heat flux in the radial direction, and k′o = ko/ρocpo.

3. SOLUTION PROCEDURE

The solution procedure for the transient heat conduction analysis in a one-dimensional FG hollow cylinderis given in this section. The inner and outer surfaces of the cylinder could be subjected to different types oftransient thermal loading.


Substituting Eqs. (6) and (7) into Eq. (5) and omitting the internal heat generationR result in the followinggoverning equation:

ηn2+n3−n1+2(

1+ εo∂

∂ζ+ε2o

2∂2

∂ζ 2

)∂2θ

∂ζ 2

=

[C2T +

(1+C2

T αo) ∂∂ζ+

(δo+

C2T

2α2o

)∂2

∂ζ 2+δ2o

2∂3

∂ζ 3

][η2 ∂

2θ

∂η2+ (n1+1)η

∂θ

∂η

](8)

in which the TPL parameter is defined as

C2T=

ρocpok∗

or2o

k2o

(9)

The following Laplace transform is used to deal with the transient response of the FG cylinder:

f (s)=

∫∞

0f (ζ )exp(−sζ )dζ (10)

where f denotes the Laplace transform of functionf (ζ ) and s is the Laplace variable. Due to the time-derivative terms up to the fourth-order in Eq. (8), the following non-dimensional initial conditions, compat-ible with the stability of initial ambient temperature, are considered in this analysis:

θ(η,ζ )|ζ=0 = 0,∂

∂ζθ(η,ζ )

∣∣∣∣ζ=0= 0,

∂

∂ζ 2θ(η,ζ )

∣∣∣∣ζ=0= 0,

∂

∂ζ 3θ(η,ζ )

∣∣∣∣ζ=0= 0 (11)

Applying the Laplace transform to Eq. (8) results in

η2 ∂2θ

∂η2+ (n1+1)η

∂θ

∂η−D(s)ηn2+n3−n1+2θ = 0 (12)

in which

D(s)=

(1+ εos+

ε2o

2 s2)s2

C2T +

(1+C2

T αo)s+

(δo+

C2T

2 α2o

)s2+

δ2o

2 s3

(13)

The solution of the ordinary differential equation (12) can be expressed as

θ (η,s)=

{η−n1/2

[A1JG(Iη

H )+A2YG(IηH )], n1−n2−n3 6= 2

A1ηλ1+A2η

λ2, n1−n2−n3 = 2(14)

whereG=

n1

n2+n3−n1+2(15a)

H =n2+n3−n1+2

2(15b)

I =2√−D

n2+n3−n1+2(15c)

λ1,2 =−n1±

√n2

1+4D

2(15d)


Table 1. Time-dependent functions of transient thermal loadings.

Thermal load type Temporal functiong(ζ ) Laplace transform g(s)

I H(ζ) 1s

II(1− e−ζ/ζ0

)H(ζ) 1

s(1+ζ0s)

III H(ζ)−H(ζ − ζ0)1−e−ζ0s

s

and A1and A2 are unknown constants to be determined by satisfying thermal boundary conditions; JG andYG represent, respectively, the Gth-order Bessel function of the first and second kind. The heat flux can alsobe obtained in the Laplace domain using Eqs. (2), (6), (7), (10), and (14) as follows:

Qr(η,s)=

P(s)

2ηn1/2−1{A1[MJG(Iη

H )−2IHηHJG+1(IηH )]+

+A2[MYG(IηH )−2IHηHYG+1(Iη

H )]}, n1−n2−n3 6= 2

P(A1λ1η

λ1+n1−1+A2λ2η

λ2+n1−1), n1−n2−n3 = 2

(16)

whereM = 2GH −n1 (17a)

P(s)=−C2T +

(1+C2

T αo)s+

(δo+

12C

2T α

2o

)s2+

12δ

2os

3

s+ εos2+ 12ε

2os

3(17b)

The non-dimensional boundary conditions are specified as follows:

θ(η,ζ )|η=rγ = θwig(ζ ), θ(η,ζ )|η=1 = θwog(ζ ) (18)

in which g(ζ ) is the temporal function for the applied thermal boundary conditions as defined in Table 1along with their Laplace transform g(s) for different types of applied transient thermal loadings. In Table 1,H(ζ) and ζ0 are the Heaviside step function and non-dimensional removal time of the thermal disturbance,respectively.

The unknown constants A1 and A2 in Eqs. (14) and (16) can be obtained by satisfying the thermal bound-ary condition (18) in the Laplace domain.

A1 =

[rn1/2γ θwiYG (I )− θwoYG

(IrHγ

)]JG(IrHγ

)YG (I )−JG (I )YG

(IrHγ

) g(s)A2 =

[rn1/2γ θwiJG(I )− θwoJG(I r

Hγ )]

YG(IrHγ

)JG (I )−JG

(IrHγ

)YG (I )

g(s)

(n1−n2−n3 6= 2) (19a)

A1 =θwi− r

λ2γ θwo

rλ1γ − r

λ2γ

g(s)

A2 =rλ1γ θwo− θwi

rλ1γ − r

λ2γ

g(s)

(n1−n2−n3 = 2) (19b)


Then the fast Laplace inversion technique (FLIT) is used to convert the transient temperature and heat fluxfrom the Laplace domain to the time domain [18, 35].

The heat conduction equation (8), in which all phase-lags have been expanded up to the second-orderin Taylor series expansion, does not show the wave-like behavior for temperature and heat flux. The ther-mal behavior based on the heat conduction equation (8) is diffusive-like instead. However, thermal wavebehavior could be observed when the phase-lags of heat flux τq and thermal displacement gradients τv areexpanded to the second-order and the phase-lag of temperature gradient τT is expanded to the first-order inTaylor series expansion. The wave-like behavior for this case could be observed by isolating Eq. (8) in thefollowing wave form:

∂2

∂ζ 2

(∂2θ

∂ζ 2−C2

TPL1∂2θ

∂η2

)= F

(η,∂θ

∂η,∂2θ

∂ζ 2,∂2θ

∂η∂ζ,∂2θ

∂η2,∂3θ

∂ζ 3,∂3θ

∂ζ 2∂η,∂3θ

∂ζ∂η2

)(20)

in which

F =2ηn1−n2−n3+2

ε2o

[C2T (n1+1)η

∂θ

∂η+ηn2+n3−n1+2 ∂

2θ

∂ζ 2+ (1+αoC2

T )∂2θ

∂η∂ζ+C2

T η2 ∂

2θ

∂η2

+ ηn2+n3−n1+2εo∂3θ

∂ζ 3+

(δo+

12C2T α

2o

)(n1+1)η

∂3θ

∂ζ 2∂η+ (1+αoC2

T )η2 ∂3θ

∂ζ∂η2

](21)

and

CTPL1 =

√2δo+C2

T α2o

εoη(n1−n2−n3)/2 (22)

where CTPL1 represents the thermal wave speed within the FG cylinder based on the first-type TPL model,in which τq and τv are expanded up to the second-order and τT is expanded to the first-order. Furthermore,a second-type TPL model can be developed by expanding τq to the second order and τT and τv to thefirst-order. The thermal wave speed based on the second-type TPL model is expressed as

CTPL2 =

√2δoεo

η(n1−n2−n3)/2 (23)

For wave-like DPL and C-V heat conduction models, a similar approach can be used to obtain the thermalwave speed CDPL and CC−V, respectively [18]. Consequently, the relation among CTPL2, CDPL, and CC−V

can be obtained as

CDPL = CTPL2 = CC−V

√2δoεo

(24)

4. NUMERICAL RESULTS

In this section, the effects of different heat conduction theories, phase-lags, non-homogeneity indices, andgeometries on the thermal behavior of an infinitely-long hollow cylinder are depicted. The initial tem-perature of the cylinder and the outer radius of FG cylinders are assumed to be T∞ = 300 K and ro = 1,respectively, for all of the numerical results. The non-dimensional phase-lags of heat flux, temperature gra-dient, and thermal displacement gradient are adopted to be εo = 0.35, δo = 0.25, and αo = 0.15; it is alsoassumed that C2

T = 2.5.The effect of heat conduction theories on predicted temperature distribution and temperature time-history

are illustrated in Fig. 1. The thermal responses of first- and second-type TPL, wave-like DPL, and C-V


(a)

(b)

Fig. 1. Effects of different wave-like heat conduction theories on (a) temperature distribution at non-dimensional timeζ = 0.14; (b) temperature time-history at non-dimensional coordinate η = 0.8.

models are compared with each other. The inner radius of the cylinder are assumed to be ri = 0.6 andall non-homogeneity indices are assumed the same as n1 = n2 = n3 = n = 1. A sudden temperature riseTwo = 600 K, according to the thermal load type I in Table 1, is applied to the outer surface of the cylinderwhile the inner surface is kept at the initial temperature. Accordingly, Fig. 1(a) shows the temperaturedistribution at non-dimensional time ζ = 0.14 and Fig. 1(b) depicts the temperature time-history at non-dimensional radial coordinate η = 0.8. As seen in Fig. 1(a), a thermal wave based on all of these wave-likeheat conduction theories propagates from the outer surface and comes toward the inner surface; it is thenreflected from the inner surface and travels toward the outer surface and continues this back and forthmovement to reach the steady state condition. Furthermore, the thermal wave speed based on the first-type


(a)

(b)

Fig. 2. Effects of different diffusive-like heat conduction theories on (a) temperature distribution at non-dimensionaltime ζ = 0.14; (b) temperature time-history at non-dimensional coordinate η = 0.75.

TPL is greater than the second-type TPL model, as shown in Fig. 1(a). Independent of material properties,the thermal wave-speed based on the second-type TPL and wave-like DPL models are the same; however, theC-V thermal wave could precede the TPL and DPL thermal waves depending on the material properties, astheoretically predicted in Eqs. (22) through (24). In addition, Fig. 1(b) shows that the temperature oscillatesaround the steady-state solution for different wave-like heat conduction theories. It is worthwhile to mentionthat the results given in Figs. 1(a–b), based on the wave-like DPL and C-V models, are the same as those in[13, 18] which verify the solution procedure.

Figure 2 shows the thermal behavior based on the different diffusive-like heat conduction models. Twodiffusive-like TPL models as well as two diffusive-like DPL models are considered along with the classical


(a)

(b)

Fig. 3. Effect of non-dimensional phase-lag of heat flux εo on (a) temperature distribution at non-dimensional timeζ = 0.056; (b) temperature time-history at non-dimensional coordinate η = 0.6.

Fourier heat conduction theory. The diffusive-like TPL and DPL models include either cases of Taylor seriesexpansion up to the first-order or up to the second-order for all the phase-lags. The material properties andboundary conditions are assumed the same as those mentioned in Fig. 1 except the inner radius which isassumed to be ri = 0.5. The temperature distribution at the non-dimensional time ζ = 0.14 and temperaturetime-history at the non-dimensional coordinate η = 0.75 are shown in Figs. 2(a–b). Fig. 2(a) obviously re-veals that the temperature propagates with an infinite thermal wave speed for all these models. Furthermore,the maximum values of the transient temperature based on the TPL models are greater than those based onthe DPL and Fourier models, as seen in Fig. 2(b).


(a)

(b)

Fig. 4. Effect of non-dimensional phase-lag of temperature gradient δo on (a) temperature distribution at non-dimensional time ζ = 0.14; (b) temperature time-history at non-dimensional coordinate η = 0.6.

The effects of non-dimensional phase-lags of heat flux εo, temperature gradient δo, and thermal displace-ment gradient αo as well as the TPL parameter C2

T on the thermal responses of the cylindrical medium areillustrated in Figs. 3 through 6. The exact values of the phase lags of heat flux and temperature gradientfor typical metals can be found in [36]. The inner radius of the cylinder is assumed ri = 0.2 and the non-homogeneity indices are n1 = n2 = n3 = n = 1. The inner surface of the cylinder experiences a dynamic,type I temperature rise Twi = 600 K while the outer surface is kept at initial temperature Two = 300 K.The second-type TPL model is used to study the effects of phase-lags on the thermal behavior, while thefirst-type TPL model is employed to investigate the TPL parameter’s effect.


(a)

(b)

Fig. 5. Effect of non-dimensional phase-lag of thermal displacement gradientαo on (a) temperature distribution atnon-dimensional timeζ = 0.196; (b) temperature time-history at non-dimensional coordinate η = 0.6.

Figure 3(a) shows the temperature distribution at the non-dimensional time ζ = 0.056 while Fig. 3(b) de-picts the temperature time-history at the non-dimensional coordinate η= 0.6. The effect of non-dimensionalphase-lag of heat flux is depicted while other parameters are taken as: δo = 0.25, αo = 0.15, and C2

T = 2.5.Figure 3(a) shows that thermal wave speed increases by decreasing the phase-lag of εo which is compatiblewith Eq. (23). Furthermore, a higher value of εo results in a greater maximum value of transient temperatureand longer duration time for reaching the steady-state situation.

Figure 4 depicts the effect of phase-lag of temperature gradient δo on the thermal behavior of the FGcylinder. To investigate the effect of δo, all other phase-lags and the TPL parameter are kept constant asεo = 0.35, αo = 0.15, and C2

T = 2.5. Figure 4(a) shows the temperature distribution at the non-dimensionaltime ζ = 0.14, while Fig. 4(b) depicts the temperature time-history at the non-dimensional coordinate η =


(a)

(b)

Fig. 6. Effect of TPL parameter C2T on (a) temperature distribution at non-dimensional time ζ = 0.196; (b) temperature

time-history at non-dimensional coordinate η = 0.6.

0.6. As seen in these figures, increasing non-dimensional phase-lag of δo increases the thermal wave speed,decreases the maximum transient temperature, and shortens the oscillation duration time around the steady-state solution.

According to Eq. (23), the thermal wave speed based on the second-type TPL model is independent ofnon-dimensional phase-lag of thermal displacement gradient, αo. This phenomenon can be observed inFig. 5(a) which presents the temperature distribution at the non-dimensional time ζ = 0.14. The phase-lags of heat flux and temperature gradient as well as the TPL parameter are kept constant as εo = 0.35,δo = 0.25, and C2

T = 2.5, while the effect of αo is illustrated in Fig. 5. The temperature time-history at thenon-dimensional coordinate η = 0.6 is also shown in Fig. 5(b). Although the thermal wave speed basedon the second-type TPL model is independent of phase-lag of thermal displacement gradient, increasing αo


Fig. 7. Effect of inner radius ri on temperature distribution at non-dimensional time ζ = 0.14.

enhances the magnitude of thermal wave-front and increases the maximum transient temperature, as seen inFig. 5.

Since the TPL parameter C2T affects the thermal wave speed based on Eq. (22), Fig. 6 reveals its influence

on the thermal behavior of the FG cylinder. The influence of C2T on the temperature distribution at the non-

dimensional time ζ = 0.196 and the temperature time-history at the non-dimensional coordinate η = 0.6are depicted, respectively, in Figs. 6(a) and 6(b). The three non-dimensional phase-lags are assumed to beεo = 0.35, δo = 0.25, and αo = 0.25. As seen in Fig. 6, a greater value of C2

T leads to a higher thermal wavespeed and a greater maximum value of the transient temperature.

In order to analyze the effect of geometry on the temperature field, the temperature distribution at thenon-dimensional time ζ = 0.196 is shown in Fig. 7. The outer radius of the cylinder is kept constant ro = 1,while the inner radius changes for different geometries. The non-dimensional phase-lags and TPL parameterare assumed to be: εo = 0.35, δo = 0.25, αo = 0.15, and C2

T = 2.5; and the non-homogeneity indices arealso assumed to be the same as n1 = n2 = n3 = n= 1. The outer surface of the cylinder is subjected to thetransient temperature rise of type III of Two = 600 K. As depicted in Fig. 7, the inner radius does not affectthe thermal wave speed and magnitude of the transient temperature.

At last, Fig. 8 shows the effect of non-homogeneity indices on temperature distribution at the non-dimensional time ζ = 0.14 and temperature time-history at the non-dimensional coordinate η = 0.8. Thesecond-type TPL model is adopted for the investigation. To focus on the effect of non-homogeneity, allnon-homogeneity indices are assumed the same n1 = n2 = n3 = n. The outer surface of the FG cylinder issubjected to a dynamic thermal load of type II with the magnitude Two = 600 K, while the inner surface ofthe cylinder is kept at the initial temperature Twi = 300 K. As seen in Fig. 8(a), a higher non-homogeneityindex n results in a greater thermal wave speed. Furthermore, increasing the non-homogeneity index n leadsto a higher maximum transient and steady-state temperature as shown in Fig. 8(b).

5. CONCLUSIONS

The thermal behavior in an FG, infinitely-long, hollow cylinder is studied using the TPL heat conductiontheory which also covers the DPL, C-V, and classical Fourier heat conduction theories. Among different


(a)

(b)

Fig. 8. Effect of non-homogeneity indices n on (a) temperature distribution at non-dimensional time ζ = 0.14; (b)temperature time-history at non-dimensional coordinate η = 0.8.

types of TPL model, two could result in the wave-like behavior for temperature. Taylor series expansionof the phase-lag of heat flux τq up to the second-order and the phase-lag of temperature gradient τT up tothe first-order could lead to the first- or second-type of TPL model providing the Taylor series expansion ofthe phase-lag of thermal displacement gradient τv up to the second- or first-order, respectively. All materialproperties are assumed to change continuously according to a power-law formulation in the radial directionexcept the phase-lags which are kept constant. Using Laplace-transform, the time-dependent governingdifferential equation is reduced to a Bessel differential equation. The temperature and heat flux are thenconverted to the time domain using the fast Laplace inversion technique. Using the semi-analytical solutions,the effects of different heat conduction theories including the TPL, DPL, C-V, and Fourier models as well as


the influence of phase-lags, TPL parameter, the geometry, and non-homogeneity indices of the cylinder onthe thermal response are graphically depicted. The following conclusions can be drawn from our theoreticaland numerical results:

(a) The first- and second-type TPL, wave-like DPL, and C-V models reveal a finite thermal wave speedin the heat conduction process in contrast with the classical Fourier heat conduction theory. Thethermal wave speed based on the first-type TPL model is larger than the second-type TPL modelwhich predicts the same thermal wave speed as the wave-like DPL model.

(b) The effects of phase-lags of heat flux τq , temperature gradient τT , and thermal displacement gradientτv as well as TPL parameter C2

T on the thermal behavior are significant in the TPL heat conductionmodels. For the first-type TPL model, thermal wave speed increases by increasing non-dimensionalphase-lag of temperature gradient δo, phase-lag of thermal displacement gradient αo, and the TPLparameter C2

T ; nonetheless, increasing non-dimensional phase-lag of heat flux εo decreases the ther-mal wave speed. On the other hand, thermal wave speed based on the second-type TPL model isindependent of the phase-lag of thermal displacement gradient and TPL parameter.

(c) For cylinders with the same outer radius, the inner radius of the cylinder, and accordingly, the ge-ometry of the cylindrical medium does not affect the thermal wave speed and magnitude of transienttemperature.

(d) The effect of non-homogeneity indices on the thermal behavior is remarkable. Increasing the non-homogeneity indices increases the thermal wave speed and the maximum transient temperature.

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