Two-Dimensional C-V Heat Conduction Investigation of an FG-Finite Axisymmetric Hollow Cylinder

Abstract

In the present work, we implement a graded finite element analysis to solve the axisymmetric 2D hyperbolic heat conduction equation in a finite hollow cylinder made of functionally graded materials using quadratic Lagrangian shape functions. The graded FE method is verified, and the simple rule of the mixture with power-law volume fraction is found to enhance the effective thermal properties’ gradation along the radial direction, including the thermal relaxation time. The effects of the Vernotte numbers and material distributions on temperature waves are investigated in depth, and the results are discussed for Fourier and non-Fourier heat conductions, and homogeneous and inhomogeneous material distributions. The homogeneous cylinder wall made of SUS304 shows faster temperature wave velocity in comparison to the ceramic-rich cylinder wall, which demonstrates the slowest one. Furthermore, the temperature profiles along the radial direction when n = 2 and n = 5 are almost the same in all Ve numbers, and by increasing the Ve numbers, the temperature waves move slower in all the material distributions. Finally, by tuning the material distribution which affects the thermal relaxation time, the desirable results for temperature distribution can be achieved.

1. Introduction

FGMs feature inhomogeneous constituents and microstructures, which result in spatial gradation in the material properties [1]. Pressure vessels and thermal barrier components are only two examples of the many uses for them because of their unique ability to withstand fracture brought on by mechanical and thermal loads [2]. The manufacturing processes for FGMs are developing, and among them, additive manufacturing is being considered nowadays [3].

In many different applications, heat transfer analysis is crucial. Accurate estimation of thermal shock resistance is essential when evaluating the lifetime and performance of piezoelectric [4] or cracked [5] structures under thermal shock loading. To address the heat conduction issue in FGM cylinders utilizing Fourier heat conduction, some analytical and numerical solutions have been carried out [6,7]. In general, many researchers used classical Fourier heat conduction to study thermal stresses in FG hollow cylinder architectures [8,9,10,11,12,13].

The basis for a thermal stress analysis is heat conduction analysis, which is generally divided into being based on Fourier or non-Fourier theory. A thermal wave has an infinite velocity according to the Fourier theory; as a result, the hyperbolic heat conduction method is suggested [14,15], which takes into account the heat wave propagation in the domain with a thermal relaxation time parameter. The Cattaneo–Vernotte (C-V) model has been discussed in the literature [16,17,18,19,20,21] as an alternative to the diffusion model among these models.

Fourier effects are becoming increasingly attractive in practical engineering problems due to the widespread use of heat sources with very short duration or very high frequency, such as in lasers and microwaves [22] or in nano-fluid flow [23,24,25]. Recent research has focused on non-Fourier heat conduction in structures, such as FGMs, with a heterogeneous material composition. Working in very-high-temperature environments and sustaining sudden temperature gradients are some of these materials’ key uses. For instance, the thermal barrier-covering FGMs for a Japanese space shuttle project must withstand a high surface temperature (2000 K) and a high temperature gradient (1000 K) that is less than 10 nm, where non-Fourier heat conduction is more pronounced [26].

Thermal stress analysis of FG solid and hollow cylinders utilizing the C-V heat conduction theory has been investigated [27,28,29,30]. Malekzadeh et al. used DQM to solve the axisymmetric C-V problem while considering constant thermal relaxation time, and the influence of the relaxation time on the moving boundary flux was not taken into account [28].

The hyperbolic heat conduction for a slab exposed to a fluctuating thermal loading was studied analytically by Tang and Araki [22]. An analytical double-series solution has been suggested by Singh et al. for heat conduction in multi-layer polar coordinates [31].

A numerical analysis of the laminar flow of nano-fluids and the impact of their thermo-physical properties on convective heat transfer was studied in [32]. The flow of magnetite-water nano-fluid and mixed convection inside a duct with an open trapezoidal cavity, two distinct heat sources, and moving walls had also been studied [33,34].

Moosaie employed a superposition approach to analyze the Fourier integral coefficients for one-dimensional non-Fourier heat conduction for one layer of homogenous material [35]. Mishra and Sahai compared the lattice Boltzmann method (LBM) with the finite volume technique (FVM) for the study of non-Fourier heat conduction in 1D spherical and cylindrical geometries [36].

Analytical investigations on the hyperbolic heat conduction in FG anisotropic materials [37] and spherical substrate in a semi-infinite FGM medium [38] had been conducted. To solve the governing hyperbolic equations for 1D [28,39,40,41,42] and 2D [15,43,44,45] heat conductions, researchers have mostly employed Laplace transformation and numerical inverse LT among the analytical solution methods described in the literature.

The method of separation of variables (SOV) has been performed by some authors for 1D and 2D hyperbolic heat conduction cylindrical coordination [46,47,48,49].

Other solution procedures for the hyperbolic heat conduction equation, such as linear inverse Laplace methods [50], have been employed in an electronic device and a 1D infinitely wide plane slab [51].

Hyperbolic heat conduction for non-homogeneous materials has become a debate in some experiments [52]. One group of researchers has detected that the thermal relaxation times are in the order of tens of seconds [53,54]. In contrast, another group claimed that their experiments have not depicted such a thermal relaxation time [55,56]. However, to resolve this argument, [52] showed that there is a certain hyperbolic effect in the “bulk” conduction in non-homogeneous materials. In addition, a wide range of experiments for studying different parameters have been conducted. These studies have presented the necessity of hyperbolic heat conduction for inhomogeneous media.

The diffusion heat conduction hypothesis, on the other hand, has been cited as a solid theory in other investigations since there is not enough experimental support for the hyperbolic theory [57,58]. In conclusion, the discussion over many non-Fourier forms of heat conduction is still up for debate despite the fact that there is no conclusive proof that Fourier heat conduction can be applied universally to all physical problems [59]. FGMs are a type of heterogeneous materials in which porosity is a necessary component during manufacture. In this regard, the relaxation time and its gradation along the spatial direction should be taken into account, and its values require reliable experimental data.

The problem of axisymmetric heat conduction for an axisymmetric hollow finite FG cylinder has never been investigated using graded FEM, and the majority of scholars have modeled the gradation of thermal characteristics along a particular direction using a simple material scheme. As a result, in their investigations, the thermal relaxation time was assumed to be constant, which is incorrect for FGMs. The elements’ shape functions were assumed to be linear in the work that used FEM to solve the hyperbolic heat conduction equation, which meant that the temperature gradient was also constant and the material gradation effect could not be shown using a single element. Furthermore, the effect of material distributions on temperature has not been investigated in depth. The following sections present the mathematical and material models of this study, the numerical schemes and numerical results, and, finally, the discussions.

2. Mathematical and Material Models

In this section, the mathematical modelling and material scheme are presented. Cattaneo and Vernotte [16,17] independently proposed a constitutive relation that connects the heat flux vector q to the temperature gradient in solids in order to resolve the paradox of the infinite speed of heat propagation:

$$\mathbb{Q}+\tau \frac{\partial \mathbb{Q}}{t}=-k\nabla T\Rightarrow \mathbb{Q}=-\tau \frac{\partial \mathbb{Q}}{t}-k\nabla T$$

(1)

When Equation (1) is inserted into the energy conservation equation (Equation (2)) without the source term, the hyperbolic heat transport equation has the following form:

$$-\nabla .\mathbb{Q}=\rho c\frac{\partial T}{\partial t}\Rightarrow \nabla .(-k\nabla T)=\rho c\left(\frac{\partial T}{\partial t}+\tau \frac{{\partial }^{2}T}{\partial t^2}\right)$$

(2)

Considering a hollow FG axisymmetric cylinder with non-uniform transient inner surface temperature boundary condition, the thermal relaxation time $\tau$ and the hyperbolic heat conduction equation with the boundary and initial conditions are similar to Equations (1)–(6):

$$\frac{1}{r}\frac{\partial }{\partial r}\left(rk(r)\frac{\partial T(r,z,t)}{\partial r}\right)+\frac{\partial }{\partial z}\left(k(r)\frac{\partial T(r, z , t)}{\partial z}\right)=\rho (r)c(r)\left(\frac{\partial T(r,z,t)}{\partial t}+\tau (r)\frac{{\partial }^{2}T(r,z,t)}{\partial t^2}\right)$$

(3)

where $\mathbb{Q}$, $k\left(r\right)$, $\rho \left(r\right)$, $c\left(r\right)$, and $\tau \left(r\right)$ are the thermal heat flux, thermal conductivity coefficient, mass density, specific heat capacity, and relation time, respectively.

$$T(a,z,t)=700(1-e^{-2t})\sin (\frac{\pi z}{L})+25 °\mathrm{C}$$

(4)

$$T(r,0,t)=T(r,L,t)=25 °\mathrm{C}; \frac{\partial T(b,z,t)}{\partial r}=0$$

(5)

$$T(r,z,0)=25 °\mathrm{C}; \frac{\partial T}{\partial t}(r,z,0)=0$$

(6)

This arbitrary temperature distribution (Equation (4)) prescribes an exponentially increase in the values of the temperature of the inner surface of the cylinder of 725 °C until the moment of t = 2 s, and it is subsequently constant over time to reach a final value of 725 °C.

2.1. FGM Material Distribution

It is assumed that the 1D FGM cylinder is composed of ceramic and metal. The inner and the outer surfaces consist of pure ceramic and metal, respectively. The materials’ volume fraction distributions can be explained as in [60]:

$$V_c=1-{(\frac{r-a}{b-a})}^n, V_m=1-V_c$$

(7)

where $V_m$ and $V_c$ are the volume fraction of the metal and ceramic cylinders, respectively, and $r, n$ are their radial coordinates and power-law exponents. The radial direction-dependent properties of FG cylinders are considered to be following the simple rule of mixture. The effective material properties $P$ are explained as follows [61]:

$$P={\displaystyle \sum _{j=1}^2{P}_j V_j}$$

(8)

where $P_j$ is the material constituents and $V_j$ is the volume fraction of the material constituents $\left(j\right)$.

The relaxation time varies between ${10}^{-10}$ and ${10}^{-14}$ seconds for various solids and between ${10}^{-8}$ and ${10}^{-10}$ seconds for gases. For some biological and non-homogeneous materials, this range increases by ${10}^2$ [52,62].

The non-dimensional Vernotte number (Ve) in the C-V heat conduction equation can be explained as follows:

$$\tau (r) ={\frac{(b.Ve)}{\alpha (r)}}^2; \alpha (r)=\frac{k(r)}{\rho (r)c(r)}$$

(9)

where $\alpha (r)$ is the coefficient of the thermal diffusivity. The second sound (temperature) wave velocity $C_T(r)$ can be measured using the following equation:

$$C_T(r)=\sqrt{\frac{\alpha (r)}{\tau (r)}}$$

(10)

2.2. Numerical Scheme

This section shows the graded finite element formulation for two-dimensional hyperbolic heat transfer based on the graded thermal relaxation time in an FG axisymmetric cylinder, in which the material property gradations are assigned to the integration points. In contrast to traditional homogeneous elements, where material properties are constants, a graded element offers a spatial dependency of the material characteristics. This can be achieved by evaluating the integrals across the element’s area using Gauss quadrature. Due to the fact that material constants vary depending on where they are located within the element, the numerical integration applied to the entire area of the element assigns different values of material properties to the various integration points. As a result, by integrating the finite element matrices, the true gradients of the material properties may be defined within the element [63]. According to [64], graded elements outperform conventional elements when simulating FGMs.

The governing equations of complex problems can be solved numerically with the help of FEM [65]. As a result, FEM has been used to solve Equation (3) with respect to the boundary conditions (4 and 5) and initial condition (6). By obtaining the weak formulations of the governing equations, the typical FEM approach for non-Fourier heat transfer can be produced. Once the governing equation has been multiplied by the independent test function $\partial T\left(r,z,t\right)$, the boundary conditions are employed to be integrated over the spatial domain.

The unknown temperature $T$ and the accompanying test function are calculated using the same shape function based on the Galerkin methods. In order to discretize the cylinder wall, an axisymmetric 9-node square ring element with a rectangular cross section is assumed [66]:

$$T\left(r,z,t\right)={\displaystyle \sum _{i=1}^9(\left[N_i\left(r,z\right)\right]\left\{q_i(t)\right\}}). i=1,2\dots 9$$

(11)

where $\left[N_i\left(r,z\right)\right]$ and $\left\{q_i(t)\right\}$ are the matrix of the quadratic interpolation functions and the nodal temperature, respectively. In the Galerkin methods, the test functions can be determined as follows:

$$\partial T\left(r,z,t\right)={\displaystyle \sum _{i=1}^9(\left[N_i\left(r,z\right)\right]\partial \left\{q_i(t)\right\}}). i=1,2\dots 9$$

(12)

Accordingly, the weak form of Equation (3) is derived as follows:

$${\displaystyle {\iiint }_{\mathsf{\Omega }}\left[\frac{1}{r}\left[\frac{\partial \delta T}{\partial r}\left(k.r\frac{\partial T}{\partial r}\right)\right]+\frac{\partial \delta T}{\partial z}\left(k\frac{\partial T}{\partial z}\right)+\rho c\delta T\left(\frac{\partial T}{\partial t}+\tau \frac{{\partial }^{2}T}{\partial t^2}\right)\right]}d\mathsf{\Omega }={\displaystyle {\iint }_s\delta T.f_n}ds=0$$

(13)

where $f_n$ is the boundary flux, $\mathsf{\Omega }$ is the volume, $S$ is the boundary surfaces of the cylinder, $k$ is the thermal conductivity coefficient, $c$ is the specific heat capacity, $\rho$ is the density, and $\tau$ is the thermal relaxation time.

After the assembly of the element matrixes, Equation (14) is derived as follows:

$$\left[M_1\right]\left\{\stackrel{\dot{}\dot{}}{Q}\right\}+\left[M_2\right]\left\{\dot{Q}\right\}+\left[K\right]\left\{Q\right\}=0$$

(14)

where $\left\{\stackrel{}{Q}\right\}$ is the nodal temperature vector in which

$$[M_1]={\displaystyle \sum _{e=1}^m{{\displaystyle \int }}_{\Omega }\rho c\tau {\left[N\right]}^T\left[N\right]\mathrm{d}\mathsf{\Omega }}$$

(15)

$$[M_2]={\displaystyle \sum _{e=1}^m{{\displaystyle \int }}_{\mathsf{\Omega }}\rho c{\left[N\right]}^T\left[N\right]\mathrm{d}\mathsf{\Omega }}$$

(16)

$$[K]={\displaystyle \sum _{e=1}^m{{\displaystyle \int }}_{\mathsf{\Omega }}{\left[B\right]}^{T}k\left[B\right]\mathrm{d}\mathsf{\Omega }}$$

(17)

where $\left[B\right]=[L]\left[N\right]$ in which $[L]$ is the differential operator matrix.

The minimum number of 3200 quadrilateral elements after performing the mesh dependency analysis is applied (40 elements along the radial direction and 80 elements along the axial direction) using quasi-linearization and the unconditionally stable implicit backward difference formula (BDF) approach with a stabilizing step-time of 0.01 s [66].

3. Numerical Results and Discussions

In this section, two examples are solved using the graded FEM. The first example is the verification example that shows the reliability of the graded FEM in the current work. The second example is the main numerical example of the current study.

3.1. Example 1: Verification

According to the authors’ knowledge, there is no study on two-dimensional C-V heat conduction for an FG axisymmetric hollow cylinder that compares the efficiency of the present method. Therefore, a one-dimensional C-V heat conduction analytical solution for an FG long cylinder is chosen to find the reliability of the FEM. Figure 1 demonstrates the FG cylinder configuration with the inner and outer temperatures boundary conditions.

In reference [15], the continuous material properties along the r-direction follow a power law, except for constant thermal relaxation time:

$$k(\eta )=k_o{\eta }^{n_1}, \rho (\eta )={\rho }_o{\eta }^{n_2}, c_p(\eta )=c_{po}{\eta }^{n_3}$$

(18)

where $\eta =r/r_o$ and $k, \rho , c_p$ are the thermal conductivity coefficient, density, and specific heat capacity, and $n_1, n_2, n_3$ are the power exponents for each of them, respectively. After some mathematical manipulations and normalizations, the governing equation of hyperbolic heat conduction is considered as follows [15]:

$$\eta {\theta }_{,\eta \eta }+\left(n_1+1\right){\theta }_{,\eta }-{\eta }^{n_2+n_3-n_1+1}\left({\theta }_{,\xi }+{\epsilon }_0{\theta }_{,\xi \xi }\right)=0$$

(19)

The B.Cs and I.Cs are

$${\theta (\eta ,\xi )|}_{\eta =r_{\gamma }}=T_{\gamma },{ \theta (\eta ,\xi )|}_{\eta =1}=1{, \theta (\eta ,\xi )|}_{\xi =0}={{\theta }_{,\xi }(\eta ,\xi )|}_{\xi =0}=0$$

(20)

where

$$\begin{array}{l}\xi =k_{o}t/r_o^2, {\epsilon }_o=k_o\tau /r_o^2, r_{\gamma }=r_i/r_o, \theta =\left(T-T_{\infty }\right)/\left(T_{wo}-T_{\infty }\right),\\ T_{{}_{\gamma }}=\left(T_{wi}-T_{\infty }\right)/\left(T_{wo}-T_{\infty }\right)\end{array}$$

(21)

Equation (19) is solved using the graded FEM while taking the initial temperature (300 K) and the boundary conditions into consideration in order to compare the current work with reference [15]. The temperature of the inner surface is $T=0 \mathrm{K}$, and the outer surface temperature is $T_{wo}=500 \mathrm{K}$. The outer radius of the cylinder is 1 m, and the inner radius is 0.6 m.

According to Figure 2. the present graded FEM with a higher-order shape function has good agreement with the analytical solution obtained by Babaei and Chen [15].

3.2. Example 2: Graded Finite Element Analysis of FG Axisymmetric Cylinder

The FGM cylinder composed of two materials ($\mathrm{Z}\mathrm{r}{\mathrm{O}}_2$ and SUS304) with the inner radius, outer radius, and length of $a=0.125, b=0.15$, and $L=0.1 \mathrm{m}$, respectively, is demonstrated in Figure 3.

It is assumed that the inner wall of the cylinder is made of 100% ceramic and the outer wall is made of 100% metal, and the material distribution is governed by the simple rule of the mixture according to the power-law volume fraction (

Table 1. The cylinder constituents’ thermal properties.

Materials/Thermal Properties	$k\left[\frac{\mathrm{W}}{\mathrm{m}.\mathrm{K}}\right]$	$c\left[\frac{\mathrm{J}}{\mathrm{K}\mathrm{g}.\mathrm{K}}\right]$	$\rho \left[\frac{\mathrm{K}\mathrm{g}}{{\mathrm{m}}^3}\right]$
$\mathrm{Z}\mathrm{r}{\mathrm{O}}_2$	1.71	491	5670
SUS304	14.91	483	7790

).

Figure 4 depicts the temperature wave velocity along with the normalized r for different values of n. The temperature velocity is not constant nor linear for the FG cylinder, and the highest velocity is related to the homogeneous (SUS304) cylinder (0.53 mm/s). The lowest temperature velocity is detected for the inner wall of the cylinder (0.08 mm/s).

In the homogeneous cylinder wall, the relaxation time $\tau$ is constant like other properties, and the temperature wave moves faster because of the higher value of the thermal conductivity coefficient $k$ for the metallic cylinder wall.

The temperature waves that travel through the uniform (metal-rich) cylinder wall at various time frames are shown in Figure 5. Due to the symmetrical boundary conditions and 1D material distributions, the heat flux direction is parallel to the r-direction through the cylinder wall. The temperature wave is more similar to the temperature loading situation at the left side of the cylinder wall at time t = 15 s. The wave has advanced but not yet crossed to the other side of the cylinder wall at time t = 30 s. Additionally, the temperature profile is impacted by the isolated boundary conditions on the right side of the cylinder wall at t = 45 s, and by t = 60 s, the wave has virtually reached the opposite side of the cylinder wall.

The propagation of temperature waves through the FG cylinder wall is shown in Figure 6. At t = 15 s, the temperature wave is in its first propagation phase. After t = 30 s, the wave moves further due to small fluctuations. At t = 45 s, the wave advances and the small fluctuations spread further, and at t = 60 s, the wave spreads further and its main small fluctuations cover more areas of the cylinder wall. In comparison with Figure 5, however, the Ve number is smaller due to the lower value of the thermal conductivity (k), and the temperature waves propagate slower. The comparison between the plots (c) for these figures depicts that in the homogeneous cylinder, the temperature waves almost travel the entire cylinder wall, but in the hyperbolic material distribution one, they travel just 20% of the cylinder wall.

Figure 7 depicts three plots in one figure, including the contour, the radial and axial temperature wave profile in the specific material distribution, and the time frame, which means 2D temperature distribution through the cylinder wall.

The non-dimensional time named Fourier number (Fo) is the measure of the heat conducted through a domain in relation to the heat stored in it. Thus, a higher value of Fo indicates faster temperature propagation through the cylinder wall. The effect of Ve number on the temperature wave for different values of material distributions is illustrated in Figure 8. The homogeneous cylinder, which is made of 100% metal, shows lower resistance to the stored heat in comparison to other FG cylinders. When Fo = 0.002, the temperature wave in plot a has propagated faster than the temperature wave in plot b. It can also be extended to other plots. It should be considered that for n = 2 and 5, there are no significant differences between temperature wave propagations. In all plots, the Fourier heat conduction (Ve = 0) curves are not wavy.

In Figure 9, the time history of temperature waves on the homogeneous and inhomogeneous cylinder walls for various Ve values is shown. For the Fourier heat conduction case (plot a), the temperature gradient moves faster when the cylinder wall is fully metallic, and it experiences a higher value of temperature. In the case of the inhomogeneous cylinder wall, the lowest temperature is related to the linear material distribution. When Ve = 0.05, the results of n = 2 and n = 5 curves are the same, and for the homogenous case, the temperature wave peak passes the point P1 at 6.4 s, at 30 s when n = 1, and at 37 s when n = 2 and 5. When Ve = 0.1, the temperature wave peak almost reaches the point P1 at 60 s for the FG cylinder walls. Finally, when Ve = 0.15, the temperature wave peak does not reach the point P1 for the FG cylinder walls.

For the other non-Fourier cases, by elevating the value of n, the temperature waves propagate slower, and the difference between n = 2 and 5 is not significant because there is not much difference in the coefficient of thermal diffusivity ($\alpha$) at this specific point. By increasing the Ve number, the temperature wave peaks are delayed. For example, in the case of linear material distribution (plot b), the temperature wave peak has reached the point P1 after 30 s, but in (plot c), it just reaches the point P1 after 60 s.

Figure 10 demonstrates the effects of Vernotte number on the temperature wave along the radial direction of the cylinder wall. In all plots, by increasing the Ve number, the temperature wave propagates slower. In the homogeneous cylinder wall (n = 0), the temperature wave moves faster at lower values of Ve number.

Figure 11 reveals that the effects of the power exponent parameter (n) on the temperature wave along the radial direction of the cylinder wall at a specific time. In all plots, there is no significant difference between FG material distributions; moreover, by increasing the Ve number, this difference becomes smaller (plot d). In addition, by increasing the Ve numbers and the parameter “n”, the temperature wave propagates slower. In the case of Ve = 0.05 and homogeneous cylinder, the temperature wave travels faster than the other FG cylinder. When Ve = 0.1, the temperature wave peak has almost traveled 30% of the homogeneous cylinder wall, and there is no significant difference for the FG cylinders’ wave profile. When Ve = 0.15, the temperature wave peak has almost traveled 20% of the homogeneous cylinder wall, and the wave profile is the same for the FG cylinder walls (n = 1, 2, and 5).

The temperatures below the initial temperature, which are demonstrated in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, can be justified by the literature. Certainly, in contrast to parabolic heat conduction, hyperbolic heat conduction is in contradiction with the local equilibrium hypothesis, so the principles of thermodynamics are not violated in these figures [29,67].

4. Conclusions

In order to solve the two-dimensional C-V heat conduction equation for an axisymmetric hollow finite-length cylindrical structure composed of 1D FGM, graded FEM has been used. The analytical study has proven the solution approach, and a comprehensive discussion of temperature distributions has been offered. Additionally, the following novelty and outcomes might be taken into consideration:

1-

The effective material thermal properties’ gradation, including the thermal relation time, is distributed along the radial direction by the simple rules of the mixture and power-law volume fractions.

2-

Faster temperature wave velocity is related to the homogeneous cylinder wall made of SUS304 in contrast with the ceramic-rich cylinder wall, which is demonstrated to be the slowest one.

3-

For n = 2 and n = 5, there is no difference in the temperature distributions along the radial direction for all Ve numbers.

4-

By increasing the Ve numbers, the temperature waves move slower for all material distributions.

5-

The tuning of the material distribution and, consequently, thermal relaxation time can lead to desirable results for the temperature distribution.