Solving Non-Fourier Heat Conduction-Radiation Problem, a Semi-Analytical Approach

Objectives: In this study, the thermal analysis of a thin film exposed to laser heating, was investigated using Cattaneo Vernotte non-Fourier heat conduction model. Methods/Statistical Analysis: Nonfourier equations were derived based on radiative heat loss and also variable thermal conductivity. Then Variational Iteration Method (VIM) and Reduced Differential Transformation Method (RDTM) were used to solve these equations. Findings: VIM is an efficient choice to find an approximate solution of nonlinear non-Fourier heat conduction and its accuracy is higher than RDTM. Considering the thermal conductivity vary with temperature creates a significant difference in the temperature profiles, which shows the importance of nonlinear analysis of the problem. In certain values of relaxation time, temperature variations take place and out of that certain domain, variations of relaxation time doesn’t effect on temperature at all. Application/Improvements: There is a remarkable difference between the Fourier and non-Fourier solutions of heat conduction equations that demonstrate the importance of non-Fourier analysis of this type of equations. *Author for correspondence Indian Journal of Science and Technology, Vol 11(15), DOI: 10.17485/ijst/2018/v11i15/121417, April 2018 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645


Introduction
Nowadays, using of energetic laser rays has more applications in medicine, manufacturing and material processes. In fact, the ability of the laser in controlling of depth and width of the heating of the specified place and the controlling of the power of the irradiated beam, make using of these tools interesting in the industry and medical applications 1 . The lasers are powerful sources of heat generation and the produced heat in the certified body by them should be analyzed properly and carefully, especially when radiative heat loss has a significant role in such cases. In fact, studying of heat transfer in some cases like the processes of melting and freezing in the furnaces 2 , heat pipes and combustion chambers 3,4 and laser heating 5 , should be done in the combined mode of conduction-radiation.
One of the cases that the classic law of Fourier heat conduction confronts failure is transferring high heat flux in short period of time. Laser heating is a good example of better understanding this process. Therefore, it is better to use non-Fourier models in the laser heating problems. First 6,7 developed a new macroscopic heat conduction model according to a Fourier approach. Since this model showed a wave-like behavior for heat conduction, they called it heat wave model or in honor of these scientists, it was known as (C-V) model.
In the recent years, using of non-Fourier equations for investigating laser heating is reported in a large number of studies. In 8 used a numerical method for analyzing of heat wave propagation on a body under laser irradiation. In 9 used the non-Fourier model for the problem of laser heating of a thin film and solved it analytically. In 10 investigated parabolic and hyperbolic heat equation of skin under short and long periods of laser irradiation. Non-Fourier analysis of the thermal damage caused by laser heating was the subject of a studying done 11,12 analyzed the laser heating of living tissues by the non-Fourier model in a newer study.
Keywords: Conduction-Radiation, Laser Heating, Non-Fourier, Semi-Analytical, RDTM, VIM Since in the present work, thermal conductivity is variable and the thermal radiation is also taken into account, we are facing a non-linear problem (with the high degree of nonlinearity). Therefore it is the semi-analytical methods which are built for this purpose. VIM which is used in this paper is one of the most flexible and most powerful of these tools that are used for solving of linear or nonlinear equations in a large number of investigations. This method was invented 13 and then many researchers emphasized on its suitable converging [14][15][16][17][18][19] . Recently many applications of this method have been reported. In 20 developed three new equations of forth order of Emdenfowler type by VIM. A solution of the transverse forced vibrations in a thermo-elastic thin beam by VIM was the subject of a work done 21,22 used VIM to present an analytic solution for particle breakage equation. In 23 solved the problem of flap wise bending vibration of rotating tapered beams by VIM.
RDTM is another method which is used in this study. Although this method is not as famous as VIM, it has a lot of abilities in solving PDE problems. In 24 introduced RDTM to solve PDE problems. Then they solved both non-linear and linear wave-like equations by RDTM in another study 25 . Solving of two dimensional Volterra integral equations by RDTM was the subject of study done 26 . Recently RDTM is used to solve partial differentialalgebraic equations 27 .
Only a few authors have presented semi-analytical solutions for non-linear non-Fourier heat equations. Homotopy perturbation method was utilized 28 to derive explicit solutions for a nonlinear Fourier heat transfer problem. In 29 solved the same problem by variational iteration method (VIM). In 30 in another work used Differential Transformation Method (DTM) to investigate convective-radiative lumped systems with variable surface emissivity and specific heat. The last three papers, mentioned above, have been used semi-analytical techniques to solve ordinary differential equations and to the best of the author's knowledge, semi analytical methods have not been utilized to solve partial differential equations of Fourier heat conduction problems.
In this study, the temperature distribution in a thin film under laser irradiation is determined using a Fourier approach. Non-linear equations were derived due to variable thermal conductivity and also radiative heat loss. Derived equations were solved by VIM and RDTM. Utilizing semi-analytical techniques to solve nonlinear PDE of non-Fourier heat conductionradiation equation is the originality and novelty and of this paper. Figure 1 indicates a simplified schematic of the problem. A film with the thickness of L and with the initial temperature of T 0 is heated by laser heat flux from the left side and left and right boundaries are kept at T 0 and zero temperatures, respectively. Moreover, radiative heat loss is considered. Energy equation assuming the presence of a heat source is as follows:

Mathematical Modeling
ρc The conduction heat flux equation dominating on the concern problem based on the heat wave model (6, 7) is: The radiative heat flux can be used as follows 31 : A variation in thermal conductivity is considered as a function of the film temperature, as follows 32 : The heat flux of laser heat source is modeled as follows 33,34 : The following dimensionless parameters are introduced: Thus equations (1) and (2) are written the dimensionless form: Ve The dimensionless form of the initial and boundary are:

Basic Concepts of VIM
In this section, the fundamental ideas of VIM are presented. Consider the following nonlinear differential equation in its general form: Where L and N are linear and nonlinear operators respectively and g(x, t) is the non-homogeneous term, which in turn is an analytic continuous function. According to VIM, the correction functional can be defined as follows: Where, λ is Lagrange multiplier that can be determined using various ways. The subscript of n represents the n th approximate of the restricted variation, where δ  u n = 0 . The one is free to choose the initial guess (u 0 ), but it can be achieved by solving the following equation: Finally, the converged solution that is the exact solution of the problem can be obtained by the successive limit of approximations:

Basic Concepts of RDTM
In this section, the basic ideas of RDTM are presented. Suppose that u(x, t) is analytic and continuously differentiable with respect to t. Then: Then the inverse of U k (x) is as follows: By combining (14) and (15): Some functions and their transformation are presented in Table 1. Now, consider a general PDE as follows: Where L is the highest order of differentiation which is invertible. R is the linear differential operator which is containing of remaining terms, Nu is containing of nonlinear terms and g is the non-homogeneous term. An initial condition is supposed as follows: Finally, with general equation of (17) and initial condition of (18) and also with transformed functions of Table 1, it can be achieved to an iterative formula. For the problem of this paper L t = ∂ ∂ and the iterative formula is as follows:  Nu(x, t) and g(x, t) respectively. Transformed function of initial condition is also as follows: Substituting equation (20) in equation (19) and sweeping an iterative process, finally the values of Uk(x) will be determined.

VIM Application
In this section, it is solved Equations (7) and (8) based on VIM. To continue the solution process, heat source term in Equation (7) should be evaluated. One assumption for this term is taken from 35 To solve Equation (7), the correctional function is written as follows: The correctional function of the Equation (8) is of the following form: Ve q x q x n n q n n To determine Lagrange multipliers by the new method introduced 36 , we do as follows: we solve the linear part of equations by the Laplace transform method and set it equals to (−1) n , where n is the highest order of the available derivative in the linear part of equation. Boundary conditions appeared in the solution process should be set equal to zero. Finally, everywhere the variable t was appeared, it should be replaced by ( ) ( ) − − 1 n t τ  . By doing all the aforementioned stages, the Lagrange multipliers for the problem can be obtained: The only thing remaining is determination of the initial functions. For this purpose, Equation (12) is used with the following initial conditions: This way, VIM formula to determine the desired functions is obtained:

RDTM Application
The transformed form of equation (7) according to iterative equation (19) is as follows: The transformed form of equation (8) based on iterative equation (20) is as follows: Initial conditions of equation (9) are transformed in the following form: By substituting equation (31) in equations (29) and (30), the values of U k (x -) and V k (x -) are obtained. The final solution is also obtained based on equations (21) and (22).

Results and Discussion
For validating the solution methods, they are compared with 31 which is a numerical study that is shown in Figure 2. As the figure shows, there are significant differences among the diagram of numerical solution and the diagrams corresponding to VIM and RDTM. Especially the diagram of RDTM doesn't show a proper prediction of the related trend of numerical solution. Quantitatively, the average error for RDTM is 26.42 % and for VIM is 11.56 %. Although it seems that the mentioned error is too high for an analytical solution, it should be noted that an analytical solution for a problem similar to the solved problem in the present study (with the same level of nonlinearity) can't be found in the literature. In fact the semi-analytical methods are not able to produce a diagram with such sharp front and if the readers know semi-analytical methods, they will definitely remember the smooth diagrams with soft curves resulted from these methods. Of course, it is necessary to mention that VIM has a less error compared with RDTM and it follows the trend of numerical method somewhat in producing a progressive thermal wave. Furthermore, the reason of existence of high errors in boundaries is in fact the lack of imposing the boundary conditions in the process of solving. The linear operator is considered in the direction of dimensionless time, so it is lost the effect of boundary conditions to get to the solution.
An analytical researcher may believe that the mentioned error is so much for solving an analytical solution, but it should be noted that advantages of semi-analytical methods such as VIM compared to the conventional numerical methods make the use of these tools highly desirable for experimental scientists. Determining the approximate solution of the problem helps these researchers to choose their equipment properly and it reduces the cost and the quantity of their experiments.
VIM is used to obtain the rest of results. Figure 3 shows the effect of dimensionless absorption coefficient (B) on the profile of temperature inside the body. As it is seen, with the increase of B in a certain point inside the film, the values of dimensionless temperature decreases, this is natural because according to the equation (7), with the increase of B, the dissipation of laser energy becomes steeper and it doesn't have enough time to increase the temperature of the film. Furthermore, with the increase of xin a given B, the same behavior is seen that is predictable according to equation (6).  In the (Figure 5), the effect of γ on the temperature profiles is studied. As it is shown in the figure, increasing of γ causes increase of temperature of the body. In some references such as 31 , a different result has been achieved. In fact these references have reported decreasing of temperature with the increase of γ. There is an obvious reason for the distinction between the results of present paper and the results of others; different heat source has been chosen. Figure 5 also shows the important effect of variable thermal conductivity. In fact in the sensitive applications that little temperature changes is important, nonlinear analysis has a significant role.

Conclusion
On this paper the effect of laser heating of a metal film temperature was studied. The thermal wave model of Cattaneo-Vernotte was used. Thermal conductivity was considered temperature-dependent and the effect of thermal radiation was also considered and then the nonlinear equations were obtained. To solve the equations, semi-analytical methods of RDTM and VIM were used and following results were obtained, briefly: 1) The accuracy of VIM is higher than RDTM. 2) Although VIM has somewhat error, but it can be a good and efficient choice for the researchers who are seeking to determine a relatively good solution of the problem as quick as possible. 3) Considering the thermal conductivity vary with temperature creates a significant difference in the temperature profiles, which shows the importance of nonlinear analysis of the problem. 4) Using laser as a heat source with thermal radiation, remarkable difference between the Fourier and the non-Fourier solutions is evident that must be taken into account in the analysis of similar problems. 5) Only in certain values of relaxation time, temperature variations take place and out of that certain domain, variations of relaxation time doesn't effect on temperature at all.