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Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions
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International Communications in Heat and Mass Transfer 35 (2008) 103 – 111 www.elsevier.com/locate/ichmt
Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions☆ Amin Moosaie Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Available online 14 September 2007
Abstract The non-Fourier transient heat conduction in a finite medium with insulated boundaries under the influence of arbitrary initial conditions is investigated analytically. The solution is expressed in the form of Fourier cosine series. The obtained analytical closed-form solution can be used for validation of numerical solutions. The resulting solution is valid for any choice of integrable initial conditions. The significant feature of proposed solution is its generality. In order to show the applicability of the presented solution, two numerical examples are solved. © 2007 Elsevier Ltd. All rights reserved. Keywords: Non-Fourier conduction; Hyperbolic conduction; Insulated boundaries; Finite medium; Arbitrary initial conditions
1. Introduction During the past few years there has been research concerned with departures from the classical Fourier heat conduction law. The motivation for this research was to eliminate the paradox of an infinite thermal wave speed which is in contradiction with Einstein's theory of relativity and thus provide a theory to explain the experimental data on ‘second sound’ in liquid and solid helium at low temperatures [1,2]. In addition to low temperature applications, nonFourier theories have attracted more attention in engineering sciences, because of their applications in high heat flux conduction, short time behavior as found, for example, in laser-material interaction, etc. The classical Fourier conduction law relates the heat flux vector q to the temperature gradient ▿ϑ, by the relation q ¼ kj#;
ð1Þ
where the material constant λ is the thermal conductivity. Eq. (1) along with the conservation of energy gives the classical parabolic heat equation aD# ¼
∂# ; ∂t
ð2Þ
where a = λ/ρc, ρ, c and Δ are thermal diffusivity, mass density, specific heat capacity and Laplace's differential operator, respectively. Eq. (2) yields temperature solutions which imply an infinite speed of heat propagation. ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.08.001
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A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Nomenclature a an An Bn c F(x) Fo G(x) L q Q t ν V Ve x X
thermal diffusivity, m2/s Fourier coefficients integration constants defined by Eqs. (13b) and (16) integration constants defined by Eqs. (15a) and (15b) specific heat capacity, J/kg K initial temperature, K Fourier number, at/L2 initial rate of temperature change, K/s thickness of medium, m heat flux vector, W/m2 heat source term, K/s time, s propagation speed of heat waves, m/s dimensionless temperature, pffiffiffiffiffiffiffiffiffiffiffiffi ϑ/ϑ0 Vernotte number, as0 =L spatial variable, m dimensionless spatial variable, x/L
Greek symbols αn auxiliary variable defined by Eq. (10b) κn auxiliary variable defined by Eq. (9b) λ thermal conductivity, W/m K ϑ temperature, K ϑ0 amplitude of initial temperature, K ρ mass density, kg/m3
In order to eliminate this paradox, Cattaneo [3] and Vernotte [4] independently postulated a time-dependent relaxation model for the heat flux in solids: q þ s0
∂q ¼ kj#; ∂t
ð3Þ
where τ0 is the so-called relaxation time (a non-negative constant). If the relaxation time vanishes, i.e. τ0 = 0, the heat flux law of the non-Fourier model defined by Eq. (3) reduces to the classical Fourier's model for heat conduction, i.e., Eq. (1). However, this model was first proposed by Maxwell [5]. Inserting Eq. (3) into the energy conservation equation, the hyperbolic heat transport equation, including source term, takes the form aD# ¼
∂# ∂2 # þ s0 2 þ Qðx; t Þ; ∂t ∂t
ð4Þ
pffiffiffiffiffiffiffiffiffi where Q is the source term and v ¼ a=s0 denotes the propagation speed of temperature wave. Various solution schemes for Eq. (4) with different initial and boundary conditions can be browsed in literature. Baumeister and Hamill [6] have given the solution for a semi-infinite problem. Sadd and Cha [7] have solved Eq. (4) for axisymmetric conduction in cylindrically bounded domains. Several solutions for finite medium have been given in [8–14]. Tang and Araki [15,16] have solved the problem of non-Fourier heat conduction in a finite medium under periodic surface disturbance. The periodic condition in their works was a harmonic one. Mikhailov and Cotta [17] solved the steady-periodic hyperbolic heat conduction in a finite slab by the use of computer algebra system
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
105
Mathematica®. Abdel-Hamid [18] solved non-Fourier heat conduction under periodic surface disturbance using the finite integral transform approach. Tang and Araki [19,20] investigated the non-Fourier heat conduction in a finite medium under pulse surface heating. Glass et al. [21] presented an explicit analytical solution for the hyperbolic heat conduction in a semi-infinite solid subjected to a periodic on-off surface heat flux and used a finite-difference scheme to solve the non-linear cases with surface radiation. Yen and Wu [22] gave a semi-analytical numerical solution of hyperbolic heat conduction in a finite medium with periodic thermal disturbance and surface radiation. Their periodic condition was an on-off one. Lewandowska and Malinowski [23] gave an analytical solution of the hyperbolic heat conduction for a finite medium symmetrically heated on both sides. Moosaie solved the problem of hyperbolic heat conduction in a finite medium subjected to arbitrary periodic surface disturbance [24]. In this paper, we are going to solve analytically the (1 + 1)-dimensional non-Fourier conduction equation for a finite medium with insulated boundaries and arbitrary initial conditions. The method of solution is the well-known separation of variables method and the solution is expressed in terms of Fourier cosine series. Finally, the solutions for the two special choices of initial temperatures are presented as two examples that demonstrate the applicability of the proposed solution procedure. 2. Analysis Let us consider a slab of thickness L composed of an isotropic heat conducting material with insulated boundaries in which one-dimensional heat conduction and constant thermal properties prevail. The governing Eq. (4) in this case and with the assumption of vanishing source term reduces to a
∂2 # ∂# ∂2 # þ s0 2 : ¼ 2 ∂x ∂t ∂t
ð5Þ
In this paper, we are interested in the non-Fourier heat conduction in a finite medium with insulated boundary conditions and arbitrary initial conditions. The boundary and initial conditions for such a problem are as follows. ∂ ∂ #ð0; t Þ ¼ 0; #ð L; t Þ ¼ 0; ∂x ∂x #ð x; 0Þ ¼ F ð xÞ;
ð6aÞ
∂ #ð x; 0Þ ¼ Gð xÞ; ∂t
ð6bÞ
where F and G are arbitrary functions of the spatial variable x. In order to obtain an analytical solution, we assume the following form for the temperature distribution. #ð x; t Þ ¼ a0 ðt Þ þ
l X
an ðt Þcos
n¼1
npx : L
ð7Þ
This temperature distribution fulfills the boundary conditions (6a) automatically. By substituting this solution into Eq. (5) we come up to the following ordinary differential equation for a0(t). s0 dda0 ðt Þ þ ad 0 ðt Þ ¼ 0;
ð8aÞ
which has the following general solution. a0 ðt Þ ¼ A0 þ B0 e
st
0
:
ð8bÞ
Moreover, the unknown functions an(t) must fulfill the following equation. s0 ddan ðt Þ þ ad n ðt Þ þ j2n an ðt Þ ¼ 0;
ð9aÞ
where j2n ¼ a
np2 : L
ð9bÞ
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A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
The general solution to this homogeneous ordinary differential equation is given by an ð t Þ ¼ e
2st
0
ðAn cos an t þ Bn sin an t Þ;
ð10aÞ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s0 j2n 1 an ¼ : 2s0
ð10bÞ
Therefore, the temperature distribution becomes #ð x; t Þ ¼ A0 þ B0 e
st
þ
0
l X
e
2st
0
ðAn cos an t þ Bn sin an t Þcos
n¼1
npx : L
ð11Þ
The integration constants An and Bn (n = 0, 1,…) can be calculated by enforcing initial conditions (6b). #ð x; 0Þ ¼ A0 þ B0 þ
l X
An cos
n¼1
npx ¼ F ð xÞ: L
ð12Þ
Utilizing the orthogonality of cosine functions, one deduces that 1 L
A0 þ B0 ¼
Z
L
F ð xÞdx;
ð13aÞ
0
R F ð xÞcos ¼ R L
An
0
npx L dx
L
:
ð13bÞ
cos2 npx L dx 0
The initial rate of change of temperature implies that l X ∂ 1 #ð x; 0Þ ¼ B0 þ ∂t s0 n¼1
1 npx ¼ Gð xÞ; An þ an Bn cos 2s0 L
ð14Þ
which yields s0 B0 ¼ L
Z
L
Gð xÞdx;
ð15aÞ
0
8 > > 1< Bn ¼ an > > :
R Gð xÞcos R L
0
npx L dx
L
þ
cos2 npx L dx 0
9 > > =
1 An : 2s0 > > ;
ð15bÞ
Eqs. (13a) and (15a) give the following explicit relation for A0. 1 A0 ¼ L
Z
L 0
½F ð xÞ þ s0 Gð xÞdx:
ð16Þ
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Fig. 1. Triangular initial temperature distribution.
Fig. 2. Dimensionless temperature at X = 0 for Ve2 = 0.1 due to triangular initial temperature distribution.
Fig. 3. Dimensionless temperature at X = 0 for Ve2 = 1 due to triangular initial temperature distribution.
Fig. 4. Dimensionless temperature at X = 0 for Ve2 = 5 due to triangular initial temperature distribution.
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3. Numerical examples In order to show how the presented solution works, the obtained general solution is reduced to two special cases. For the first example, the initial temperature distribution is defined as 2#0 L L x #ð x; 0Þ ¼ F ð xÞ ¼ ; ð17Þ 2 L 2
j
j
which is plotted in Fig. 1. The initial rate of temperature change is assumed to be zero, i.e. ∂ #ð x; 0Þ ¼ Gð xÞ ¼ 0: ∂t
ð18Þ
The temperature distribution can be readily obtained by computing coefficients An and Bn via Eqs. (13b), (15), and (16). Furthermore, we introduce the following dimensionless quantities. V ¼
# ; #0
Fo ¼
ð19aÞ
at ; L2
Ve2 ¼
ð19bÞ
as0 ; L2
ð19cÞ
x X ¼ : L
ð19dÞ
The dimensionless temperature at X = 0 as a function of Fourier number Fo for different Vernotte numbers Ve2 is plotted in Figs. 2, 3, 4, and 5. The graph at X = 1 is the same, because of the symmetry of boundary and initial conditions. The dimensionless temperature at X = 0.5 is presented in Fig. 6 for Ve2 = 1. As another numerical example, we consider the initial temperature in the form of a finite step function given by #ð x; 0Þ ¼ F ð xÞ ¼
#0 ; L=4VxV3L=4; 0; otherwise:
ð20Þ
The temperature distribution (20) is depicted in Fig. 7. The initial rate of temperature change is assumed to be zero which is stated in Eq. (18). The temperature distribution due to this choice of initial conditions can be calculated by introducing these initial conditions to Eqs. (13b), (15), and (16). The obtained results are presented in Figs. 8, 9, 10, and 11. The dimensionless temperature at X = 0.5 is also presented in Fig. 12 for Ve 2 = 1. 4. Concluding remarks In this paper, the (1 + 1)-dimensional non-Fourier heat conduction equation was solved analytically for a finite medium under the influence of arbitrary initial conditions and insulated boundaries. The solution is expressed in terms of Fourier cosine series. The important feature of this solution is its generality, namely, it is valid for any choice of initial conditions and it can be used for validation of numerical recipes. In order to show how the proposed method of solution works, two numerical examples have been solved. The graphs demonstrate that the time needed for reaching steady-state situation is increased with increase of the relaxation time τ0. This result could be p expected, ffiffiffiffiffiffiffiffiffi because the speed of propagation of heat waves is inversely proportional to relaxation time via the relation v ¼ a=s0 . Another conclusion from the graphs is that the shape of the heat wave is inherited from the shape of initial temperature distribution within the medium.
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Fig. 5. Dimensionless temperature at X = 0 for Ve2 = 100 due to triangular initial temperature distribution.
Fig. 6. Dimensionless temperature at X = 0.5 for Ve2 = 1 due to triangular initial temperature distribution.
Fig. 7. Stepwise initial temperature distribution.
Fig. 8. Dimensionless temperature at X = 0 for Ve2 = 0.1 due to stepwise initial temperature distribution.
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Fig. 9. Dimensionless temperature at X = 0 for Ve2 = 1 due to stepwise initial temperature distribution.
Fig. 10. Dimensionless temperature at X = 0 for Ve2 = 5 due to stepwise initial temperature distribution.
Fig. 11. Dimensionless temperature at X = 0 for Ve2 = 100 due to stepwise initial temperature distribution.
Fig. 12. Dimensionless temperature at X = 0.5 for Ve2 = 1 due to stepwise initial temperature distribution.
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References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M. Chester, Second sound in solids, Phys. Rev. 131 (1963) 2013–2015. J.B. Brown, D.Y. Chung, P.W. Matthews, Heat pulses at low temperatures, Phys. Lett. 21 (1966) 241–243. C. Cattaneo, Sur une forme de l'equation de la chaleur elinant le paradoxe d'une propagation instantance, C. R. Acad. Sci. 247 (1958) 431–432. P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur, C. R. Acad. Sci. 246 (1958) 3154–3155. J.C. Maxwell, On the dynamic theory of gases, Phil. Trans. R. Soc. Lond. 157 (1867) 49–88. K.J. Baumeister, T.D. Hamill, Hyperbolic heat conduction equation — a solution for the semi-infinite body problem, J. Heat Transfer 91 (1969) 543–548; K.J. Baumeister, T.D. Hamill, Hyperbolic heat conduction equation — a solution for the semi-infinite body problem, J. Heat Transfer 93 (1971) 126–127. M.H. Sadd, C.Y. Cha, Axisymmetric non-Fourier temperatures in cylindrically bounded domains, Int. J. Non-Linear Mech. 17 (1982) 129–136. V. Taitel, On the parabolic, hyperbolic and discrete formulation of the heat conduction equation, Int. J. Heat Mass Transfer 15 (1972) 369–371. G.F. Carey, M. Tsai, Hyperbolic heat transfer with reflection, Numer. Heat Transf. 5 (1982) 309–327. M.N. Özisik, B. Vick, Propagation and reflection of thermal waves in a finite medium, Int. J. Heat Mass Transfer 27 (1984) 1845–1854. J.I. Frankel, B. Vick, M.N. Özisik, Flux formulation of hyperbolic heat conduction, J. Appl. Phys. 58 (1985) 3340–3345. D.E. Glass, M.N. Özisik, B. Vick, Hyperbolic heat conduction with surface radiation, Int. J. Heat Mass Transfer 28 (1985) 1823–1830. D.E. Glass, M.N. Özisik, D.S. McRae, B. Vick, Hyperbolic heat conduction with temperature-dependent thermal conductivity, J. Appl. Phys. 59 (1986) 1861–1865. J. Gembarovic, V. Majernik, Non-Fourier propagation of heat pulse in finite medium, Int. J. Heat Mass Transfer 31 (1988) 1073–1081. D.W. Tang, N. Araki, Non-Fourier heat conduction in a finite medium under periodic surface thermal disturbance, Int. J. Heat Mass Transfer 39 (1996) 1585–1590. D.W. Tang, N. Araki, Non-Fourier heat conduction in a finite medium under periodic surface thermal disturbance-II. Another form of solution, Int. J. Heat Mass Transfer 39 (1996) 3305–3308. M.D. Mikhailov, R.M. Cotta, Steady-periodic hyperbolic heat conduction in a finite slab, Int. Commun. Heat Mass Transf. 24 (1997) 725–731. B. Abdel-Hamid, Modelling non-Fourier heat conduction with periodic thermal oscillation using the finite integral transform, Appl. Math. Model. 23 (1999) 899–914. D.W. Tang, N. Araki, Analytical solution of non-Fourier temperature response in a finite medium under laser-pulse heating, Heat Mass Transfer 31 (1996) 359–363. D.W. Tang, N. Araki, Non-Fourier heat condution behavior in finite mediums under pulse surface heating, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 292 (2000) 173–178. D.E. Glass, M.N. Özisik, B. Vick, Non-Fourier effects on transient temperature resulting from periodic on-off heat flux, Int. J. Heat Mass Transfer 30 (1987) 1623–1631. C.-C. Yen, C.-Y. Wu, Modelling hyperbolic heat conduction in a finite medium with periodic thermal disturbance and surface radiation, Appl. Math. Model. 27 (2003) 397–408. M. Lewandowska, L. Malinowski, An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides, Int. Commun. Heat Mass Transf. 33 (2006) 61–69. A. Moosaie, Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance, Int. Commun. Heat Mass Transf. 34 (2007) 996–1002.
International Communications in Heat and Mass Transfer 35 (2008) 103 – 111 www.elsevier.com/locate/ichmt
Non-Fourier heat conduction in a finite medium with insulated boundaries and arbitrary initial conditions☆ Amin Moosaie Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Available online 14 September 2007
Abstract The non-Fourier transient heat conduction in a finite medium with insulated boundaries under the influence of arbitrary initial conditions is investigated analytically. The solution is expressed in the form of Fourier cosine series. The obtained analytical closed-form solution can be used for validation of numerical solutions. The resulting solution is valid for any choice of integrable initial conditions. The significant feature of proposed solution is its generality. In order to show the applicability of the presented solution, two numerical examples are solved. © 2007 Elsevier Ltd. All rights reserved. Keywords: Non-Fourier conduction; Hyperbolic conduction; Insulated boundaries; Finite medium; Arbitrary initial conditions
1. Introduction During the past few years there has been research concerned with departures from the classical Fourier heat conduction law. The motivation for this research was to eliminate the paradox of an infinite thermal wave speed which is in contradiction with Einstein's theory of relativity and thus provide a theory to explain the experimental data on ‘second sound’ in liquid and solid helium at low temperatures [1,2]. In addition to low temperature applications, nonFourier theories have attracted more attention in engineering sciences, because of their applications in high heat flux conduction, short time behavior as found, for example, in laser-material interaction, etc. The classical Fourier conduction law relates the heat flux vector q to the temperature gradient ▿ϑ, by the relation q ¼ kj#;
ð1Þ
where the material constant λ is the thermal conductivity. Eq. (1) along with the conservation of energy gives the classical parabolic heat equation aD# ¼
∂# ; ∂t
ð2Þ
where a = λ/ρc, ρ, c and Δ are thermal diffusivity, mass density, specific heat capacity and Laplace's differential operator, respectively. Eq. (2) yields temperature solutions which imply an infinite speed of heat propagation. ☆
Communicated by W.J. Minkowycz. E-mail address: [email protected].
0735-1933/$ - see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2007.08.001
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A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Nomenclature a an An Bn c F(x) Fo G(x) L q Q t ν V Ve x X
thermal diffusivity, m2/s Fourier coefficients integration constants defined by Eqs. (13b) and (16) integration constants defined by Eqs. (15a) and (15b) specific heat capacity, J/kg K initial temperature, K Fourier number, at/L2 initial rate of temperature change, K/s thickness of medium, m heat flux vector, W/m2 heat source term, K/s time, s propagation speed of heat waves, m/s dimensionless temperature, pffiffiffiffiffiffiffiffiffiffiffiffi ϑ/ϑ0 Vernotte number, as0 =L spatial variable, m dimensionless spatial variable, x/L
Greek symbols αn auxiliary variable defined by Eq. (10b) κn auxiliary variable defined by Eq. (9b) λ thermal conductivity, W/m K ϑ temperature, K ϑ0 amplitude of initial temperature, K ρ mass density, kg/m3
In order to eliminate this paradox, Cattaneo [3] and Vernotte [4] independently postulated a time-dependent relaxation model for the heat flux in solids: q þ s0
∂q ¼ kj#; ∂t
ð3Þ
where τ0 is the so-called relaxation time (a non-negative constant). If the relaxation time vanishes, i.e. τ0 = 0, the heat flux law of the non-Fourier model defined by Eq. (3) reduces to the classical Fourier's model for heat conduction, i.e., Eq. (1). However, this model was first proposed by Maxwell [5]. Inserting Eq. (3) into the energy conservation equation, the hyperbolic heat transport equation, including source term, takes the form aD# ¼
∂# ∂2 # þ s0 2 þ Qðx; t Þ; ∂t ∂t
ð4Þ
pffiffiffiffiffiffiffiffiffi where Q is the source term and v ¼ a=s0 denotes the propagation speed of temperature wave. Various solution schemes for Eq. (4) with different initial and boundary conditions can be browsed in literature. Baumeister and Hamill [6] have given the solution for a semi-infinite problem. Sadd and Cha [7] have solved Eq. (4) for axisymmetric conduction in cylindrically bounded domains. Several solutions for finite medium have been given in [8–14]. Tang and Araki [15,16] have solved the problem of non-Fourier heat conduction in a finite medium under periodic surface disturbance. The periodic condition in their works was a harmonic one. Mikhailov and Cotta [17] solved the steady-periodic hyperbolic heat conduction in a finite slab by the use of computer algebra system
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
105
Mathematica®. Abdel-Hamid [18] solved non-Fourier heat conduction under periodic surface disturbance using the finite integral transform approach. Tang and Araki [19,20] investigated the non-Fourier heat conduction in a finite medium under pulse surface heating. Glass et al. [21] presented an explicit analytical solution for the hyperbolic heat conduction in a semi-infinite solid subjected to a periodic on-off surface heat flux and used a finite-difference scheme to solve the non-linear cases with surface radiation. Yen and Wu [22] gave a semi-analytical numerical solution of hyperbolic heat conduction in a finite medium with periodic thermal disturbance and surface radiation. Their periodic condition was an on-off one. Lewandowska and Malinowski [23] gave an analytical solution of the hyperbolic heat conduction for a finite medium symmetrically heated on both sides. Moosaie solved the problem of hyperbolic heat conduction in a finite medium subjected to arbitrary periodic surface disturbance [24]. In this paper, we are going to solve analytically the (1 + 1)-dimensional non-Fourier conduction equation for a finite medium with insulated boundaries and arbitrary initial conditions. The method of solution is the well-known separation of variables method and the solution is expressed in terms of Fourier cosine series. Finally, the solutions for the two special choices of initial temperatures are presented as two examples that demonstrate the applicability of the proposed solution procedure. 2. Analysis Let us consider a slab of thickness L composed of an isotropic heat conducting material with insulated boundaries in which one-dimensional heat conduction and constant thermal properties prevail. The governing Eq. (4) in this case and with the assumption of vanishing source term reduces to a
∂2 # ∂# ∂2 # þ s0 2 : ¼ 2 ∂x ∂t ∂t
ð5Þ
In this paper, we are interested in the non-Fourier heat conduction in a finite medium with insulated boundary conditions and arbitrary initial conditions. The boundary and initial conditions for such a problem are as follows. ∂ ∂ #ð0; t Þ ¼ 0; #ð L; t Þ ¼ 0; ∂x ∂x #ð x; 0Þ ¼ F ð xÞ;
ð6aÞ
∂ #ð x; 0Þ ¼ Gð xÞ; ∂t
ð6bÞ
where F and G are arbitrary functions of the spatial variable x. In order to obtain an analytical solution, we assume the following form for the temperature distribution. #ð x; t Þ ¼ a0 ðt Þ þ
l X
an ðt Þcos
n¼1
npx : L
ð7Þ
This temperature distribution fulfills the boundary conditions (6a) automatically. By substituting this solution into Eq. (5) we come up to the following ordinary differential equation for a0(t). s0 dda0 ðt Þ þ ad 0 ðt Þ ¼ 0;
ð8aÞ
which has the following general solution. a0 ðt Þ ¼ A0 þ B0 e
st
0
:
ð8bÞ
Moreover, the unknown functions an(t) must fulfill the following equation. s0 ddan ðt Þ þ ad n ðt Þ þ j2n an ðt Þ ¼ 0;
ð9aÞ
where j2n ¼ a
np2 : L
ð9bÞ
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A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
The general solution to this homogeneous ordinary differential equation is given by an ð t Þ ¼ e
2st
0
ðAn cos an t þ Bn sin an t Þ;
ð10aÞ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s0 j2n 1 an ¼ : 2s0
ð10bÞ
Therefore, the temperature distribution becomes #ð x; t Þ ¼ A0 þ B0 e
st
þ
0
l X
e
2st
0
ðAn cos an t þ Bn sin an t Þcos
n¼1
npx : L
ð11Þ
The integration constants An and Bn (n = 0, 1,…) can be calculated by enforcing initial conditions (6b). #ð x; 0Þ ¼ A0 þ B0 þ
l X
An cos
n¼1
npx ¼ F ð xÞ: L
ð12Þ
Utilizing the orthogonality of cosine functions, one deduces that 1 L
A0 þ B0 ¼
Z
L
F ð xÞdx;
ð13aÞ
0
R F ð xÞcos ¼ R L
An
0
npx L dx
L
:
ð13bÞ
cos2 npx L dx 0
The initial rate of change of temperature implies that l X ∂ 1 #ð x; 0Þ ¼ B0 þ ∂t s0 n¼1
1 npx ¼ Gð xÞ; An þ an Bn cos 2s0 L
ð14Þ
which yields s0 B0 ¼ L
Z
L
Gð xÞdx;
ð15aÞ
0
8 > > 1< Bn ¼ an > > :
R Gð xÞcos R L
0
npx L dx
L
þ
cos2 npx L dx 0
9 > > =
1 An : 2s0 > > ;
ð15bÞ
Eqs. (13a) and (15a) give the following explicit relation for A0. 1 A0 ¼ L
Z
L 0
½F ð xÞ þ s0 Gð xÞdx:
ð16Þ
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Fig. 1. Triangular initial temperature distribution.
Fig. 2. Dimensionless temperature at X = 0 for Ve2 = 0.1 due to triangular initial temperature distribution.
Fig. 3. Dimensionless temperature at X = 0 for Ve2 = 1 due to triangular initial temperature distribution.
Fig. 4. Dimensionless temperature at X = 0 for Ve2 = 5 due to triangular initial temperature distribution.
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A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
3. Numerical examples In order to show how the presented solution works, the obtained general solution is reduced to two special cases. For the first example, the initial temperature distribution is defined as 2#0 L L x #ð x; 0Þ ¼ F ð xÞ ¼ ; ð17Þ 2 L 2
j
j
which is plotted in Fig. 1. The initial rate of temperature change is assumed to be zero, i.e. ∂ #ð x; 0Þ ¼ Gð xÞ ¼ 0: ∂t
ð18Þ
The temperature distribution can be readily obtained by computing coefficients An and Bn via Eqs. (13b), (15), and (16). Furthermore, we introduce the following dimensionless quantities. V ¼
# ; #0
Fo ¼
ð19aÞ
at ; L2
Ve2 ¼
ð19bÞ
as0 ; L2
ð19cÞ
x X ¼ : L
ð19dÞ
The dimensionless temperature at X = 0 as a function of Fourier number Fo for different Vernotte numbers Ve2 is plotted in Figs. 2, 3, 4, and 5. The graph at X = 1 is the same, because of the symmetry of boundary and initial conditions. The dimensionless temperature at X = 0.5 is presented in Fig. 6 for Ve2 = 1. As another numerical example, we consider the initial temperature in the form of a finite step function given by #ð x; 0Þ ¼ F ð xÞ ¼
#0 ; L=4VxV3L=4; 0; otherwise:
ð20Þ
The temperature distribution (20) is depicted in Fig. 7. The initial rate of temperature change is assumed to be zero which is stated in Eq. (18). The temperature distribution due to this choice of initial conditions can be calculated by introducing these initial conditions to Eqs. (13b), (15), and (16). The obtained results are presented in Figs. 8, 9, 10, and 11. The dimensionless temperature at X = 0.5 is also presented in Fig. 12 for Ve 2 = 1. 4. Concluding remarks In this paper, the (1 + 1)-dimensional non-Fourier heat conduction equation was solved analytically for a finite medium under the influence of arbitrary initial conditions and insulated boundaries. The solution is expressed in terms of Fourier cosine series. The important feature of this solution is its generality, namely, it is valid for any choice of initial conditions and it can be used for validation of numerical recipes. In order to show how the proposed method of solution works, two numerical examples have been solved. The graphs demonstrate that the time needed for reaching steady-state situation is increased with increase of the relaxation time τ0. This result could be p expected, ffiffiffiffiffiffiffiffiffi because the speed of propagation of heat waves is inversely proportional to relaxation time via the relation v ¼ a=s0 . Another conclusion from the graphs is that the shape of the heat wave is inherited from the shape of initial temperature distribution within the medium.
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Fig. 5. Dimensionless temperature at X = 0 for Ve2 = 100 due to triangular initial temperature distribution.
Fig. 6. Dimensionless temperature at X = 0.5 for Ve2 = 1 due to triangular initial temperature distribution.
Fig. 7. Stepwise initial temperature distribution.
Fig. 8. Dimensionless temperature at X = 0 for Ve2 = 0.1 due to stepwise initial temperature distribution.
109
110
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
Fig. 9. Dimensionless temperature at X = 0 for Ve2 = 1 due to stepwise initial temperature distribution.
Fig. 10. Dimensionless temperature at X = 0 for Ve2 = 5 due to stepwise initial temperature distribution.
Fig. 11. Dimensionless temperature at X = 0 for Ve2 = 100 due to stepwise initial temperature distribution.
Fig. 12. Dimensionless temperature at X = 0.5 for Ve2 = 1 due to stepwise initial temperature distribution.
A. Moosaie / International Communications in Heat and Mass Transfer 35 (2008) 103–111
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