# A Linear Approach Suitable for a Class of Steady-State Heat Transfer Problems with Temperature-Dependent Thermal Conductivity

Authors: R. M. S. Gama; R. Pazetto
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This work presents an useful tool for constructing the solution of steady-state heat transfer problems, with temperature-dependent thermal conductivity, by means of the solution of Poisson equations. Specifically, it will be presented a procedure for constructing the solution of a nonlinear second-order partial differential equation, subjected to Robin boundary conditions, by means of a sequence whose elements are obtained from the solution of very simple linear partial differential equations, also subjected to Robin boundary conditions. In addition, an a priori upper bound estimate for the solution is presented too. Some examples, involving temperature-dependent thermal conductivity, are presented, illustrating the use of numerical approximations.


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## Body

## 1. Introduction

In order to avoid the barriers arising from nonlinear mathematical descriptions, heat transfer problems with temperature-dependent thermal conductivity and/or nonlinear boundary conditions are, usually, represented by means of somewhat inaccurate approximations. Among these approximations, we have the constant thermal conductivity as well as the well-known Dirichlet and Neumann boundary conditions [1].A Dirichlet boundary condition represents a prescribed temperature while the Neumann boundary condition represents a prescribed heat flux. Both are mathematically interesting but lie far from the physical reality. The use of a Robin boundary condition (which involves a relationship between the normal heat flux and the temperature) provides a more realistic description.One of the most usual descriptions for the steady-state conduction heat transfer process in a rigid body, which exchanges energy with the environment by convection, is mathematically represented by(1)k∇•∇T+q˙=0inΩ,−k∇T•n=hT−T∞on∂Ω,in which it is assumed that the thermal conductivity k and the internal heat supply (per unit volume) q˙ are constants. In the boundary condition of equation (1), h represents the convection heat transfer coefficient and T∞ is a temperature of reference (in general, the temperature of the environment) [2]. The boundary condition of (1) is based on Newton’s law of cooling that assumes, roughly, that the energy flux across a fluid-solid interface is proportional to the temperature difference between the surface and the surrounding medium [3]. In reference [4], it is highlighted that the heat transfer coefficient is often assumed to be a constant, despite the fact that this assumption may be far from the reality. Carslaw and Jaeger [5] presented several exact solutions for linear problems like (1). Kreith and Manglik [6] illustrated the use of finite difference schemes for problems involving constant thermal conductivities, like problem (1).Nevertheless, the thermal conductivity of any real body is a function of the local temperature, and each time the importance of this relationship becomes a more relevant issue, since its knowledge allows more accurate and realistic mathematical descriptions [7]. This dependence cannot be neglected in many situations, for instance, when large temperature variations are experienced, as pointed out in [8]. In reference [9], Moitsheki et al. considered a power law relationship between the thermal conductivity and the temperature. References [10–12] are devoted to the determination of temperature-dependent thermal conductivity. In [10], the steady-state nonlinear heat conduction equation is transformed into the Laplace equation by means of the Kirchhoff transformation. The thermal conductivity is modeled as a linear combination of known functions with unknown coefficients. Reference [11] deals with the noniterative inverse determination of the temperature-dependent thermal conductivity in a steady-state heat conduction problem. The thermal conductivity is represented by a polynomial function with unknown coefficients. In [12], Chantasiriwan determined the temperature-dependent thermal conductivity and assumed a polynomial function of the temperature, by means of an algorithm for solving an inverse problem. In reference [13], Afrin et al. provided an estimate for the temperature distribution, in a context of variable thermal conductivity, with the aid of a Kirchhoff transformation, by means of a Hadamard factorization, relating the front and back surface temperature as infinite product of polynomials. Reference [14] provides an a priori upper bound estimate for cases in which the thermal conductivity is a linear function of the temperature.It is remarkable that, even in a context with few temperature variations, bodies at very low temperature levels may present variations of many orders of magnitude in its thermal conductivity [15, 16].For instance, the design of cryogenic thermal links imposes to take into account the influence of the temperature on the thermal conductivity, especially for high-purity aluminium or for high-purity copper [17]. As an illustration, in Figure 1, the thermal conductivity of annealed aluminium (with 99.9999% purity [15]) is presented as a function of the temperature. It is clear the strong relationship between the thermal conductivity and the temperature in this case.Figure 1
Thermal conductivity of high-purity aluminium as a function of the temperature.Table1 presents the thermal conductivities for some materials for eight selected values of the temperature (in Kelvin), obtained from reference [18] (these results can also be found in https://www.efunda.com/materials/elements/element_list.cfm).Table 1
Thermal conductivities for some materials at 1 atm.
TKCopperkWm−1K−1SiliconkWm−1K−1TungstenkWm−1K−1IronkWm−1K−1519500527687083910243002330971014802010800498040501540501250268042740510048288420813420041326418594500386761466110003523111832In this work, we shall take into account that the thermal conductivityk depends on the temperature T (the unknown). In other words, instead of assuming a constant thermal conductivity, we consider that k=k^T. In addition, it will be assumed (as it is in fact) that for any for any given value of T, the thermal conductivity is positive and possesses a constant positive lower bound, which means(2)k=k^T≥k¯=constant>0,for anyT.Whenk=k^T, the steady-state heat transfer process is not represented by problem (1). In order to consider temperature-dependent thermal conductivity, the phenomenon must be mathematically represented by(3)∇•k∇T+q˙=0inΩ,k=k^T,−k∇T•n=hT−T∞on∂Ω,and therefore is inherently nonlinear, even with linear boundary conditions (Robin boundary conditions).It is to be noticed that bothT∞ and h will be allowed to depend on the position on the boundary. So, it will be considered the following representation for the problem:(4)∇•k∇T+q˙=0inΩ,k=k^T,−k∇T•n=fT−gon∂Ω;f=f^x,g=g^x,x∈∂Ω,in which the internal heat supply (per unit volume) q˙ is a known nonnegative function and f is a positive valued function with a positive lower bound.Despite the Dirichlet and the Neumann boundary conditions be far from realistic heat transfer boundary conditions, they can be regarded as limiting cases of the boundary condition presented in (4). The Dirichlet boundary condition [1] can be simulated, for instance, assuming that f=c in which c is a very large positive constant. On the other hand, the usual Neumann boundary condition [1], representing insulated subsets, can be regarded as the limiting case in which the constant c approaches zero and g is zero everywhere.The main objective of this work is to construct the solution of the nonlinear problem (4) by means of the solution of very simple and classical linear problems like (1). In other words, the temperature-dependent thermal conductivity will be treated with the same simple tools used for the classical linear descriptions (constant thermal conductivity). So, the need of gross approximations used for achieving mathematical simplifications vanishes.


## 2. The Kirchhoff Transformation and Its Inverse

Let us defineω as follows (Kirchhoff transformation) [19]:(5)ω=∫0Tk^ξdξ,in which it must be considered k^T=k^TA=kA, for T<TA<TB, and k^T=k^TB=kB, for T>TB>TA, when thermal conductivity data are available only within the interval TA,TB.So, taking into account that∇•k∇T=∇•∇ω, the function ω obeys the following differential equation:(6)∇•∇ω+q˙=0inΩ.The thermal conductivity is a positive valued function for any real material. In fact, for any given material, the thermal conductivity has a positive lower bound as stated in (2). This fact ensures that(7)dωdT=ddT∫TREFTk^ξdξ=k≥k¯>0.Therefore,ω is always an increasing function of T and vice versa. This ensures the existence of the inverse of the Kirchhoff transformation, denoted by T=T˜ω, for any −∞<ω<∞.In terms of the new variableω (Kirchhoff transformation), the problem becomes(8)∇•∇ω+q˙=0inΩ;q˙=q^x,x∈Ω,−∇ω•n=fT−gon∂Ω;T=T˜ω,x∈∂Ω.An interesting relationship betweenω and T, with no limit of accuracy, is given by [20](9)ω=k1T,ifT≤T1,k2T−T1+k1T1,ifT2≥T>T1,⋯⋯kNT−TN−1+kN−1TN−1−TN−2+…+k2T2−T1+k1T1,ifT>TN−1,or in an alternative form(10)ω=12k1+kNT+12∑i=2Nki−ki−1T−Ti−1−Ti−1.The inverse is given by(11)T=T˜ω=121k1+1kNω+∑i=2N1ki−1ki−1ω−ωi−1−ωi−1,in which(12)ωi=∑j=1ikjTj−Tj−1,1≤i≤N−1,T0=0.


## 3. An Upper Bound Estimate

The establishment of an “a priori” upper bound for an unknown may represent an useful information for the kind of problems considered here [21].Define the functionu as follows:(13)u=−Q˙2dx−x02,x∈Ω,d=1,ifu=u⌣x,x≡x∈ℝ,d=2,ifu=u⌣x,y,x≡x,y∈ℝ2,d=3ifu=u⌣x,y,z,x≡x,y,z∈ℝ3,in which(14)Q˙=supΩq˙,and we have that u is the solution of(15)∇•∇u+Q˙=0inΩ,and that(16)∇•∇ω−u≥0inΩ,−∇ω−u•n=fT−g+∇u•non∂Ω;T=T˜ω,x∈∂Ω.Hence,(17)∇•∇ω−u≥0inΩ,−∇ω−u•n=fT−g−Q˙dx−x0•non∂Ω,and the supremum of the difference ω−u on the boundary coincides with the supremum of the difference ω−u in Ω [1, 22]. Therefore,(18)supΩω≥sup∂Ωω−u+supΩu≥sup∂Ωω−inf∂Ωu+supΩu,in which(19)supΩu−inf∂Ωu=Q˙2dR2.The supremum of the differenceω−u on ∂Ω takes place in a subset in which gradω−u•n is nonnegative. In other words, it takes place in a subset ∂Ω¯⊆∂Ω in which(20)fT−g≤Q˙dx−x0•n≤Q˙dsup∂Ωx−x0≤Q˙dR,in which R is any constant (for instance, the smallest one) such that(21)x−x0≤R,x∈∂Ω.Hence, in the subset∂Ω¯, we have that(22)T˜ω≤Q˙/dR+sup∂Ωginf∂Ωf,and thus(23)sup∂Ωω≤Q˙/dR+sup∂Ωginf∂ΩfkMAX,in which kMAX is the maximum considered value for k.So (upper bound estimate), from (18) and (19), we may write(24)supΩω≤Q˙/dR+sup∂Ωginf∂ΩfkMAX+Q˙2dR2.This result ensures the boundedness of the solution of problem (8). In addition, this result may be useful for evaluating an upper bound for the derivative of T with respect to ω.


## 4. Solution Construction

The solution of problem (8), denoted by ω, is the limit of the nondecreasing sequence φ0,φ1,φ2,φ3,φ4,…, with φ0=0 on the boundary, whose elements are obtained from the solution of the linear problem(25)∇·∇φi+q˙=0inΩ;i=1,2,3,…,−∇φi•n=αφi−βi−1on∂Ω;i=1,2,3,…,βi−1=αφi−1−f^xT˜φi−1+g^x,x∈∂Ω;i=1,2,3,….A sufficient condition (not necessary) for ensuring thatφ0,φ1,φ2,φ3,φ4,… is a nondecreasing and convergent sequence is the following one:(26)αb−a≥f^xT˜b−T˜a,x∈∂Ω,for any two constants a and b such that(27)supΩω≥b>a.SinceT˜ω is differentiable with respect to ω, a sufficient (not necessary) condition for inequality (26) is the following one:(28)α≥supx∈∂Ωf^xdTdω,ω∈0,sup∂Ωω.It is to be noticed that the a priori estimate provided by (21) may be used in (25). Nevertheless, it will give rise to greater value for α. The convergence will be reached, since (26) and (28) are sufficient conditions.Once ensuring the convergence of the sequenceφ0,φ1,φ2,φ3,φ4,…, we may represent the solution of (8) as the following limit:(29)ω≡limi⟶∞φiinΩ,and the temperature is obtained from the inverse of the Kirchhoff transformation.It must be highlighted that, for each giveni, problem (25) is a problem with the same mathematical structure of problem (1). This equivalence arises when we define (in problem (1)) h as α, T∞ as βi−1/α, and k=1.It is to be noticed thatq˙≥0 ensures the following:(30)inf∂Ωφ1=infΩφ1≥β0α≥0.


## 5. Proving That the Sequenceφ0,φ1,φ2,φ3,φ4,… Is Nondecreasing

From (28), we may write(31)∇•∇φi+1−φi=0inΩ,−∇φi+1−φi•n=αφi+1−φi−βi−βi−1on∂Ω,βi−βi−1=αφi−φi−1−f^xT˜φi−T˜φi−1,x∈∂Ω.So, there exists a subset∂Ωi+1−⊂∂Ω in which the infimum of φi+1−φi coincides with the infimum of φi+1−φi in Ω. On this subset, we have that ∇φi+1−φi·n must be nonpositive. Hence,(32)αφi+1−φi−βi−βi−1≥0on∂Ωi+1−,and thus from the definition of βi,(33)αφi+1−φi≥αφi−φi−1−f^xT˜φi−T˜φi−1on∂Ωi+1−.Since (28) holds, the right-hand side of the above inequality is nonnegative everywhere, and consequently(34)φi+1≥φion∂Ωi+1−⟹φi+1≥φiinΩ,i=0,1,2,3,….In other words,φ0,φ1,φ2,φ3,φ4,… is a nondecreasing sequence.


## 6. Proving That the Limit Is the Solution of the Problem

The combination of equations (8) and (27) yields(35)∇•∇ω−φi=0inΩ,−∇ω−φi•n=f^xT˜ω−g^x−αφi+βi−1on∂Ω,βi−1=αφi−1−f^xT˜φi−1+g^x,x∈∂Ω.Now, let us consider the subset∂Ωi∗⊂∂Ω in which the infimum of ω−φi coincides with the infimum of ω−φi in Ω. On this subset, we have that ∇ω−φi•n must be nonpositive. Since(36)−∇ω−φi•n=f^xT˜ω−T˜φi−1−αφi−φi−1on∂Ω,we have that(37)f^xT˜ω−T˜φi−1−αφi−φi−1≥0on∂Ωi∗,or alternatively(38)f^xT˜ω−T˜φi≥αφi−φi−1−f^xT˜φi−T˜φi−1on∂Ωi∗.Taking into account (25), we have(39)f^xT˜ω−T˜φi≥0on∂Ωi∗,and thus(40)ω≥φion∂Ωi∗⟹ω≥φiinΩ,i=0,1,2,3,….Therefore, the solutionω is an upper bound for the sequence φ0,φ1,φ2,φ3,…. This ensures the convergence of this sequence and allows us to define φ∞ as the solution of(41)∇·∇φ∞+q˙=0inΩ,−∇φ∞•n=αφ∞−β∞on∂Ω,β∞=αφ∞−f^xT˜φ∞+g^x,x∈∂Ω.This problem is equivalent to(42)∇•∇φ∞+q˙=0inΩ,−∇φ∞•n=f^xT˜φ∞−g^xon∂Ω.Problems (8) and (42) are the same problem. So, φ∞≡limi⟶∞φi≡ω, and (29) is proven.


## 7. An Illustrative Example

In order to illustrate the proposed procedure, it will be considered the heat transfer process in a context with spherical symmetry. The setΩ will be represented by the sphere x2+y2+z2<R2 with boundary x2+y2+z2=R2. Denoting the radial variable x2+y2+z2 by r, the governing partial differential equation reduces to the following ordinary differential equation:(43)1r2ddrkr2dTdr+q˙=0,0≤r<R,q˙=constant≥0.It will be considered the following boundary condition:(44)−kdTdr=γT,r=R,γ=constant>0.Assuming the thermal conductivityk given by [7](45)k=k1,T≤T1,k2,T>T1>0,the Kirchhoff transformation and its inverse are given by [20](46)ω=∫0Tk^ξdξ=12k1+k2T+k2−k1T−T1−T1,(47)T=121k1+1k2ω+1k2−1k1ω−k1T1−k1T1.In this way, the original problem gives rise to the following one:(48)1r2ddrr2dωdr+q˙=0,0≤r<R,−dωdr=γ21k1+1k2ω+1k2−1k1ω−k1T1−k1T1,r=R,which admits the exact solution below:(49)ω=C+q˙6R2−r2=k1+k22q˙R3γ+k2−k12q˙R3γ−T1−T1+q˙6R2−r2.The temperature distribution is obtained inserting (49) into (47).Now, let us employ the procedure proposed in this work. Aiming at this, we have(50)1r2ddrr2dφidr+q˙=0,0≤r<R,−dφidr=αφi−βi−1,r=R,βi−1=αφi−1−γ21k1+1k2φi−1+1k2−1k1φi−1−k1T1−k1T1.From the above differential equation, it is easy to see thatφi may be expressed as(51)φi=Ci+q˙6R2−r2,Ci=constant,i=1,2,3,….Taking into account the boundary condition and the definition ofβi−1, we have(52)q˙R3=αCi−αCi−1+γ21k1+1k2Ci−1+1k2−1k1Ci−1−k1T1−k1T1,which means(53)Ci=Ci−1−γ2α1k1+1k2Ci−1+1k2−1k1Ci−1−k1T1−k1T1+q˙R3α,in which, since φ0≡0 on the boundary, C0=0. So, C1=q˙R/3α.It is quite obvious that in case of convergence,(54)C∞=C∞−γ2α1k1+1k2C∞+1k2−1k1C∞−k1T1−k1T1+q˙R3α⟹⟹12k1+12k2C∞+12k2−12k1C∞−k1T1−k1T1=q˙R3γ⟹⟹C∞=k1+k22q˙R3γ+k2−k12q˙R3γ−T1−T1.Taking (29) into account, it is sufficient (not necessary) to choose α as follows:(55)α=1MINk1,k2.We know that the convergence and the nondecreasing behavior of the sequence depend on the choice ofα. But the speed of convergence depends on α too. In order to illustrate these questions, let us assume selected values for the constants. For instance, consider (in some system of units) R=3.0, q˙=1.0, T1=0.9, k1=2.0, k2=0.1, and γ=1.0. The constant C appearing in (45) is 1.810. In this case, the minimum value of the constant α satisfying (29) (and (55)) is given by(56)α=ddωγTsup∂Ωω=1MINk1,k2=10.1=10.0.Table2 presents the constants Ci obtained with some selected values of α.Table 2
The constantsCi (equation (51)) evaluated with some selected values of α, obtained with R=3.0, q˙=1.0, T0=0.9, k1=2.0, k2=0.1, and γ=1.0.
i=α=0.6α=2.0α=6.0α=10.0α=25.0α=100.011.66670.50000.16670.10000.04000.010021.94440.87500.31940.19500.07920.02003−0.29631.15630.45950.28530.11760.029941.61731.36720.58790.37100.15530.039751.93621.52540.70550.45240.19220.04956−0.16741.64400.81340.52980.22830.059371.63881.73300.91230.60330.26370.069081.93981.79981.00290.67320.29850.07869−0.22351.84981.08600.73950.33250.0882101.62941.65071.16220.80250.36590.0978111.93821.73801.23200.86240.39850.107312−0.19901.80351.29600.91930.43060.1168131.63351.83601.35470.97330.46200.1262141.93891.70611.40851.02470.49270.135515−0.20971.77961.45771.07340.52290.1449201.93871.69961.64901.28300.66480.190830−0.20641.84981.81031.57070.90900.2792401.63231.83621.81001.74301.10860.3634501.93871.73771.81001.81001.27170.44341001.63231.84861.81001.81001.73480.78852001.93871.78331.81001.81001.81001.2661300−0.20651.84681.81001.81001.81001.55544001.63231.66731.81001.81001.81001.73075001.93871.75821.81001.81001.81001.8099It can be noted from Table2 that for α=0.6 and for α=2.0, we do not have convergence. When α=6.0, a convergence is expected, but the sequence is not nondecreasing (see i=30). For α=10.0 and for α=25.0, the convergence is reached. For α=100.0, the convergence will be reached for a larger value of i.This table illustrates the role of the constantα. It must be large enough to ensure the convergence. There is no upper bound for α. If the convergence is reached with α=α¯, then the convergence will be reached with α=10000α¯ too. Nevertheless, if we choose a very large α, the convergence becomes slower. In fact, as α increases, the convergence speed decreases, as shown in Table 3.Table 3
The constantsCi (equation (51)) evaluated with some selected values of α, obtained with R=3.0, q˙=500.0, T1=10.0, k1=8.0, k2=0.5, and γ=100.0.
i=α=6.0α=7.0α=10.0α=13.0α=100.0α=1000.0183.333371.428650.000038.46155.00000.50002−111.111115.306137.500039.94089.37500.99383203.703759.402340.625039.997713.20311.48134Overflow24.755339.843839.999916.55271.96285Overflow51.978040.039140.000019.48362.43836Overflow30.588739.990240.000022.04822.90787Overflow47.394640.002440.000024.29223.37158Overflow34.190039.999440.000026.25563.82939Overflow44.565040.000240.000027.97374.281410Overflow36.413240.000040.000029.47704.727911Overflow42.818240.000040.000030.79245.168812Overflow37.785740.000040.000031.94335.604213Overflow41.739840.000040.000032.95046.034214Overflow38.633040.000040.000033.83166.458715Overflow41.074140.000040.000034.60266.878020Overflow39.678440.000040.000037.23168.897030Overflow39.971240.000040.000039.271712.573340Overflow39.997440.000040.000039.808415.815150Overflow39.999840.000040.000039.949618.6737100Overflow40.000040.000040.000039.999928.6297200Overflow40.000040.000040.000040.000036.7679300Overflow40.000040.000040.000040.000039.0813400Overflow40.000040.000040.000040.000039.7388500Overflow40.000040.000040.000040.000039.9258Now, in order to present another result for the same problem, let us considerR=3.0, q˙=500.0, T1=10.0, k1=8.0, k2=0.5, and γ=100.0. In this case, based on (55), we may calculate α as(57)α=ddωγTsup∂Ωω=100MINk1,k2=1000.5=200.0.It can be noted from Table3 that for α=6.0, we do not have convergence. For α=7.0 and for α=10.0, the convergence is reached, but the sequence is not nondecreasing. For α=13.0 and for α=100.0, the convergence is reached. For α=1000.0, the convergence will be reached for a larger value of i. It is to be noticed that, in fact, (28) holds for α=12.5, since on the boundary we have k=8.0 instead of k=0.5.Tables1 and 2 show that any α, above the minimum necessary for ensuring (28), gives rise to the same solution and a nondecreasing sequence.Figures2 and 3 illustrate the convergence process. In Figure 2, we have the particular case in which (in some system of units) α=2.0, γ=1.0, q˙=1.0, R=6.0, k1=4.0, k2=0.2, and T1=4.0. In Figure 3, we have the particular case in which (in some system of units) α=20.0, γ=0.05, q˙=0.05, R=3.0, k1=0.05, k2=0.8, and T1=2.0. In both the figures, lower values for α are possible, but the convergence becomes very rapid, making impossible a good visual observation.Figure 2
The temperature as a function of the positionr, obtained with α=2.0, for a set of selected values of γ, q˙, R, k1, k2, and T1, associated to the elements φii=1,2,3,4,5,6,8,10 and φ∞(exact).Figure 3
The temperature as a function of the positionr, obtained with α=20.0, for a set of selected values of γ, q˙, R, k1, k2, and T1, associated to the elements φii=1,2,3,4,5,6,7,8,9,10,12,15,20,30,40 and φ∞(exact).Figure4 presents a comparison between the temperature distribution obtained with temperature-dependent thermal conductivity and that with constant thermal conductivity (given as the mean value of the conductivities), considering q˙=200.0, R=3.0, k1=9.0, k2=0.5, and T1=30.0.Figure 4
Temperature distribution obtained withγ=30.0, q˙=200.0, R=3.0, k1=9.0, k2=0.5, and T1=30.0 and with a constant thermal conductivity approach in which k is given by k1+k2/2.It becomes clear that large variations on the thermal conductivity may give rise to temperature distributions quite different from that obtained with the usual constant thermal conductivity assumption.


## 8. An Example Using Numerical Approximations

The procedure presented in this work may be used together with a numerical approximation. In order to illustrate this use, let us consider, for instance, the following problem (representing a conduction heat transfer with convective boundary conditions in a given system of units):(58)ddxkdTdx+10=0,0<x<2,k=1.0,ifT≤20,0.2,ifT>20,kdTdx=4T−4,x=0,−kdTdx=T−50,x=2,whose exact solution is given by(59)T=−5x2+20x+6,ifx<2−65,−25x2+100x−50,ifx≥2−65.With the aid of the Kirchhoff transformation, problem (58) becomes(60)d2ωdx2+10=0,0<x<2,dωdx=4T˜ω−4,x=0,−dωdx=T˜ω−50,x=2,in which(61)T˜ω=3ω+2ω−20−20.So, in order to construct the sequenceφ0,φ1,φ2,φ3,φ4,…, we must solve the following problems:(62)d2φidx2+10=0,0<x<2;i=1,2,3,…,dφidx=αφi−βi−1,x=0;i=1,2,3,…,−dφidx=αφi−βi−1,x=2;i=1,2,3,…,βi−1=αφi−1−f^xT˜φi−1+g^x;f^x=4−32x,g^x=4+23x.Let us approximate the solution by means of the following finite difference scheme:(63)φim+1−2φim+φim−1Δx2+10=0,Δx=2M−1,m=2,3,…,M−1,φi2−φi1Δx=αφi1−βi−11,m=1,−φiM−φiM−1Δx=αφiM−βi−1M,m=M,βi−11=αφi−11−4T˜φi−11+4,βi−1M=αφi−1M−T˜φi−1M+50,which represents, for each i, a linear system of equations whose unknowns are φi1, φi2, φi3, …, φiM. In (63), φim denotes the approximation for φi at the spatial position xm=m−1Δx (corresponding to the node m) and M is the number of nodes (as suggested in Figure 5).Figure 5
The employed finite difference discretization.The linear system may be easily solved with the aid of a Gauss–Seidel scheme [23], as follows:(64)φim=10Δx2+φim+1+φim−12,m=2,3,…,M−1,φi1=φi2+αφi−11−4T˜φi−11+4Δx1+αΔx,T˜φi−11=3φi−11+2φi−11−20−20,φiM=φiM−1+αφi−1M−T˜φi−1M+50Δx1+αΔx,T˜φi−1M=3φi−1M+2φi−1M−20−20.The approximation forT at the point xm is obtained by replacing φim by ω in equation (61).Figure6 depicts a comparison among the exact solution and the approximations obtained with M=51, M=11, and M=6. α=20.0 was used.Figure 6
A comparison between the temperature distribution obtained with three different values ofM and the exact solution.Figure7 shows the evolution of the element φim (for m=26 (corresponding to the spatial position x=1.0)), as i increases from 1 to 500, using four different values of α, assuming M=51. For x=1.0, we have ω=21.0.Figure 7
The elementφi26, obtained with M=51, for four different values of the constant α. Here, φ∞26=21.0.


## 9. Conclusions

The main novelty presented in this work consists of a simple tool which transforms a nonlinear heat transfer problem into a sequence of linear problems.Specifically, the steady-state nonlinear (as consequence of a temperature-dependent thermal conductivity) heat transfer phenomenon is treated as the limit of a nondecreasing sequence of linear problems, with a complete mathematical support ensuring the convergence.With the tool presented here, most of the usual approximations (like constant thermal conductivity or Kirchhoff transformation associated with Dirichlet or Neumann boundary conditions) may be avoided, allowing more precise descriptions without increasing the mathematical difficulties.The main contribution of this work is the presentation of a reliable and very simple method for constructing the solution of a nonlinear heat transfer problem. The proposed procedure requires only basic tools, available for most of engineering students. In other words, there is no more need of gross approximations for achieving a simple mathematical description.The solution is represented, in an exact way, by means of the limit of a sequence whose elements arise from very well-known mathematical problems.It must be highlighted that the proposed procedure may be employed together with any numerical procedure suitable for second-order partial differential equations subjected to Robin boundary conditions, enlarging its range of application.Although the motivating issue of this work is the heat transfer phenomenon, other applications involving the same mathematical structure, like the mass transfer process (second Fick’s law) with concentration-dependent diffusion coefficient, lie within the mathematical scope of this work and one may take advantage of the tools presented here.The process for calculating the constantα, is sometimes tiring. However, as already pointed out, a trial and error procedure can be used. In addition, it is always expected an upper bound for the temperature of a body. For instance, we do not expect (in mechanics) a body with temperature levels beyond 10000 Kelvin. For instance, it is never expected a temperature beyond 10000 Kelvin. So 10000 Kelvin may be used to calculate α



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