 Open Access
 Total Downloads : 39
 Authors : Mamadou Alouma Diallo , Aboubacar Chedikh Beye , Cheikh Mbow
 Paper ID : IJERTV8IS060496
 Volume & Issue : Volume 08, Issue 06 (June 2019)
 Published (First Online): 24062019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Numerical Simulation of a Combined RadiationConduction Heat Transfer in An Electric Furnace
Mamadou Alouma Diallo Department of Physics University Cheikh Anta Diop, Dakar, Senegal
Aboubacar Chedikh Beye Department of Physics University Cheikh Anta Diop, Dakar, Senegal
Cheikh Mbow
Department of Physics University Cheikh Anta Diop, Dakar, Senegal
Abstract – The aim of this article is to present a numerical procedure for the computation of the temperature distribution inside the volume of a heated material and the energy flows at its limits in an electric furnace. For the heat transfer modelling, we use a temperature dependant thermal properties model to get a nonlinear diffusion equation with a uniform radiation from the temperature heat source. The numerical solution of the combined heat transfer problem is obtained through a finite difference discretization in two dimensions with the help SCILAB scripts.
Keywords – Combined radiation and conduction heat transfer, finite difference discretization, Hottles net radiation method
 INTRODUCTION:
Generally, combined radiation and conduction heat transfer takes place in high temperature furnaces to achieve the desired quality of material. Radiation transfer takes place from the source to the boundary of the material while transfer by conduction takes place in the bulk of the material. The goal of solving the problems of combined radiation and conduction heat transfer is the calculation of the temperature distribution in the volume of the medium and the energy flows at its limits.
Three important challenges are faced by the modeler who wishes to study these problems of combined transfers: the complexity and the nonlinearities intrinsic to the radiative transfers, the nonlinear conductive model induced by the properties of the material to be heated, the ignorance of the heat flow and the external surface temperature of the material to be heated.
First, for the radiative transfer, a model that is compatible with resolution techniques and other equations governing the process has to be choosen. The model should also be reliable and able to accurately predict the radiative flux and divergence of radiative flux distributions in the medium. In addition, the model should be efficient from a computational point of view.
The radiative transfer equation is a conservation equation of radiative energy. It is a complex integrodifferential equation. There is no analytical solution available at the moment in its general form. In order to solve it, physical and mathematical approximations must be introduced.
We can consider possible approximations under three different types of categories:
 Simplification of the spectral nature of the properties by means of spectral average radiative properties;
 Use of a radiative heat transfer model based on similar integrated flows or quantities.
These simplifications will then greatly reduce the mathematical complexity required of the model and we will use this approach in this study.
There are various numerical techniques for solving the radiative transfer equation, for example zonal, spherical harmonics, Monte Carlo, flow, discrete ordinate, finite volume and discrete transfer (DTM) methods, finite element method etc. It should be noted, however, that almost all the methods listed above have certain disadvantages. But it is sometimes possible to combine the characteristics of two or more methods to develop a more efficient technique for modeling radiation heat transfer in furnaces.
In this study, we will use the Hottel zonal method. In its most simple version, it consists of decomposing a medium and its boundaries into a large number of isothermal surface element exchange areas and isothermal volume elements in order to calculate the net radiative flux exchanged between all these exchange areas [1] (Goheneche et al. Sacadura, 2002).
For the conduction problem, many practical engineering situations require solving nonlinear transient heat transfer problems. Due to the limitations of analytical solutions for nonlinear heat transfer problems, a number of numerical methods have been developed to solve such problems. The nonlinear resulting heat equation, has applications in various branches of science and engineering, including thermal processing of materials, liquid movement in porous media.
These non linearities in the governing equations and the boundary conditions describing the temperature distribution are due to the fact that most metallic materials have thermal properties (thermal conductivity, specific heat, and density) that are usually temperaturedependent. However, because of the difficulties associated with the solution of these nonlinear heat transfer problems, simplifying assumptions are usually made to linearize such problems. For example, in the case of materials that have thermal conductivity which varies slightly with
temperature, constant thermal conductivity is generally assumed. However, if temperature change is substantial or the thermal conductivity varies greatly with temperature, the assumption of constant thermal conductivity may lead to significant error in the solution. Therefore, when modeling and simulating temperature distribution for such problems, nonlinearities caused by temperature dependent thermal proprieties have to be accounted for by the numerical computation.
Concerning the solution of the combined set of equations, precise and detailed temperature distributions must be obtained so that derivatives can be accurately evaluated.
To calculate the distribution of the temperature over the volume of the material, it is necessary to know the distribution of the intensity of the radiation if it cannot be considered uniform. In addition, according to the radiation transfer equation and its boundary conditions, it follows that in order to calculate the radiation intensity distribution; the temperature distribution must be known for all the points on the beam inside the material and at its limits. This is how the combination of radiative and conductive heat transfer is mathematically manifested.
In combined radiation transfer and conduction calculations, the radiation transfer equation must be iteratively resolved. With the advancement of computers, it is theoretically possible to solve any partial differential equation using an appropriate numerical discretization scheme. The challenge here is to design a scheme that preserves the physics of the problem.
Among the few earlier works on combined conduction radiation problems, Razzaque et al. [2] analyzed the two dimensional coupled conductionradiation problems with finite element method limited to only nonscattering medium with isothermal black walls. The product integration method found its application in the work of Tan [3] for solving combined conductionradiation problem in square enclosure with isothermal walls. Kim et al. [4] employed the discrete ordinates method (DOM) for coupled radiative and conductive heat transfer in rectangular enclosures. Rousse [5] and Rousse and al. [6] used the ControlVolume Finite Element Method (CVFEM) for the solution of combined mode of heat transfer in two dimensional cavities. Collapse dimension method has been used to analyse combined conduction radiation problem by Talukdar et al. [7]. Mahapatra et al.
[8] investigated a new hybrid method where the concepts of modified differential approximation were employed by blending discrete ordinate method and spherical harmonics method on combined heat transfer in twodimensional planar geometry. Recently,Amiri et al. [9] analyzed the problem of combined conduction and radiation heat transfer in 2D irregular geometries by using DOM and blockedoff method with both temperature and heat flux boundary conditions.  MODELLING OF THE CONDUCTION HEAT TRANSFER:
Electric furnaces consist of a heating chamber with electricity as source of heat to achieve high temperatures
for molten metals, alloys, ceramics, refractories and so on. For metalworking, electricity generally does not have an electrochemical effect on the metal but simply heats it.
Although modern electric furnaces are usually arc furnaces or induction furnaces, in this application we will treat a furnace in which heat is produced by resistance elements lining the inside of the furnace.
To model the heat transfer problem, first define a number of parameters.
0() temperature of the radiating source (within the resistors) at time t
[0()] Resistivity in (ohm.m) of the nichrome 8020 corresponding to the temperature (t)() the electrical current flowing through the electrical resistances at time t.
We have the following formula:
[0()]= (1 + [0() ]) (1)With:
– reference temperature in Â° C.
– Resistivity in (ohm.m) of the nichrome 8020 at the reference temperature
temperature coefficient of nichrome 8020
We will choose in this study =20Â°C; = 1.08 106. ; = 58 106
We will express 0() as a function of the current ()
using a linear regression:
We have the following data from a vendor catalog:
Table 1: tabulation of the temperature as function of electric current in a Nichrome 8020 resistance of 7mm diameter.
TempÃ©rature (Â°C) (0) 200 400 600 800 1000 1200 Courant i (A) 54,7 116 184 270 365 473 Table 2: Results of the statistical estimation of I as function of 0
350.3972
Dependent Variable: I(A) Method: Least Squares Date: 12/09/18 Time: 00:50 Sample: 1 6 Included observations: 6 Variable Coefficient Std. Error tStatistic Prob. C 48.66667 17.38394 2.799518 0.0488 0(Â°) 0.417786 0.022319 18.71890 0.0000 Rsquared 0.988713 Mean dependent var 243.7833 Adjusted Rsquared 0.985892 S.D. dependent var 157.2108 S.E. of regression 18.67335 Akaike info criterion 8.953273 Sum squared resid 1394.776 Schwarz criterion 8.883860 Log likelihood 24.85982 Fstatistic DurbinWatson stat 1.055198 Prob(Fstatistic) 0.000048 = 0.41779 0 48.667 (69) (2)
And the correlation coefficient R = 0.99.
The power per furnace resistance unit length () is expressed as follows:
() = (1+[0()]) 2() (3)
The instantaneous power generated by the furnace is then written: () = ()
With = for an oven like the one shown in Figure 4 below.
With
– length of a groove
– Number of grooves inside the oven
The electrical energy consumed during the time T of the heating process is:
k being the thermal conductivity coefficient of Steel steel, 99.2% Fe, 0.2% C.
The conduction or diffusion of heat in the metal is modeled by the following equation:
()() . (()) = 0 (6)
u being the temperature in the metal to be heated,
k is the thermal conductivity coefficient of steel,
the density of steel
the specific heat of steel
So the governing equations of the model are therefore the following set of equations :
()() . (()) = 0, (, ) Ã— ]0, [
(1+[0()]) 2
(, )
() = , (, ) Ã— [0, ] (7)
= 0
() (4)
1
{ (, 0) = 0
The heating process occurs inside an Electric furnace which consist of a heating chamber with electricity as the heat source for achieving the required source temperatures. Because of the lack of accurate radiative properties we limit consideration to gray diffuse surfaces with uniform radiosity
Figure 1: Schematization of the furnace with its load
Heat is transmitted to the object by radiation. The flux transmitted by radiation becomes predominant at temperatures above 400Â°C. This heat transfer is performed according to the following relation:
(,)
In order to obtain a continuous function of the temperature u, a correlation is made between the temperature and the Thermal conductivity. We obtain the following table:
Table 3: Thermal conductivity of steel as function of temperature
TempÃ©rature (Â°C) 0 100 300 500 800 Thermal conductivity (W/m.K) 45,357 45,357 43,031 37,216 30,238 The following figure shows the variations of the thermal conductivity k as a function of the temperature u.
50
45
40
()
()
35
30
25
400 600 800 1000
Figure 2 : Evolution of the thermal conductivity of steel as a function of temperature
()
= 1 (5)
This relationship reflects the fact that the radiated heat is transmitted to the metal by conduction at the boundary.
being the outer unitary normal of the surface boundary;
The below table gives the parameters of the regression made with the Eviews software.
Table 4: Results of the statistical estimation of k as function of temperature
55.06929
Dependent Variable: Method: Least Squares Date: 10/29/18 Time: 21:03 Sample: 1 5 Included observations: 5 Variable Coefficient Std. Error tStatistic Prob. C 52.95365 1.926324 27.48948 0.0001 0.021115 0.002845 7.420869 0.0051 Rsquared 0.948338 Mean dependent var 40.00720 Adjusted Rsquared 0.931117 S.D. dependent var 6.958590 S.E. of regression 1.826325 Akaike info criterion 4.331662 Sum squared resid 10.00638 Schwarz criterion 4.175438 Log likelihood</> 8.829156 Fstatistic DurbinWatson stat 1.384629 Prob(Fstatistic) 0.005063 () = 52.9536539 0.02111466019 (8)
As above, a correlation is made between the temperature and the specific heat. We obtain the following table:
Table 5: specific heat of steel function of temperature
Temp (Â°K) 300 400 500 600 800 1000 Specific heat (J/Kg.K)
450 491 524 555 692 1034 The following figure shows the variations of the specific heat as a function of the temperature u.
1100
1000
900
Table 6: Results of the statistical estimation of as function of temperature
9.098083
83.90058
Dependent Variable: Method: Least Squares Date: 10/29/18 Time: 21:18 Sample: 1 5 Included observations: 5 Variable Coefficient Std. Error t Statistic Prob. C 297.2973 28.18531 10.54795 0.0018 u 0.471351 0.051459 9.159726 0.0028 Rsquared 0.965478 Mean dependent var 542.4000 Adjusted Rsquared 0.953970 S.D. dependent var 92.27296 S.E. of regression 19.79671 Akaike info criterion Sum squared resid 1175.730 Schwarz criterion 8.941859 Log likelihood 20.74521 Fstatistic DurbinWatson stat 1.976317 Prob(Fstatistic) 0.002751 = 297.2972973 + 0.4713513514 (9)
Applying the same procedure as above, we obtain the above table (table 7) and figure (figure 4).
Table 7: Density of steel as a function of temperature
Temp (Â°K) 300 400 500 600 800 Density (Kg/m3) 7860 7830 7800 7760 7690 7900
7850
800
()
()
700
600
7800
7750
500
400
300 400 500 600 700 800 900 1000
Figure 3: Evolution of the specific heat of steel as a function of temperature
The below table gives the parameters of the regression made with the Eviews software.
7700
7650
300 400 500 600 700 800
Figure 4: Evolution of the density of steel as function of temperature
The statistics of the regression are summurized below :
Table 8: Results of the statistical estimation of as L
L
L
Surface 2
Surface 1
Surface 2
Surface 1
Dependent Variable: Method: Least Squares Date: 10/29/18 Time: 21:12 Sample: 1 5 Included observations: 5 Variable Coefficient Std. Error t Statistic Prob. C 7966.486 5.405405 1473.800 0.0000 U 0.343243 0.009869 – 34.78038 0.0001 Rsquared 0.997526 Mean dependent var 7788.000 Adjusted R squared 0.996702 S.D. dependent var 66.10598 S.E. of regression 3.796632 Akaike info criterion 5.795280 Sum squared resid 43.24324 Schwarz criterion 5.639055 Log likelihood 12.48820 Fstatistic 1209.675 DurbinWatson stat 1.652027 Prob(F statistic) 0.000052 Dependent Variable: Method: Least Squares Date: 10/29/18 Time: 21:12 Sample: 1 5 Included observations: 5 Variable Coefficient Std. Error t Statistic Prob. C 7966.486 5.405405 1473.800 0.0000 U 0.343243 0.009869 – 34.78038 0.0001 Rsquared 0.997526 Mean dependent var 7788.000 Adjusted R squared 0.996702 S.D. dependent var 66.10598 S.E. of regression 3.796632 Akaike info criterion 5.795280 Sum squared resid 43.24324 Schwarz criterion 5.639055 Log likelihood 12.48820 Fstatistic 1209.675 DurbinWatson stat 1.652027 Prob(F statistic) 0.000052 function of temperature.
L
= 7966.486486 0.3432432432 (10)
 MODELLING OF THE RADIATION HEAT TRANSFER :
For the calculation of the flux q1 in equation (5), the enclosure boundary is subdivided into areas so that over each such area the following restrictions are met:
 All surfaces are opaque.
 The temperature is uniform.
 The surface properties are uniform.
 The , , and are independent of wavelength and direction so that at any surface temperature, the hemispherical total absorptivity and emissivity are equal and depend only the temperature : = = 1 where is the reflectivity.
 All energy is emitted and reflected diffusely.
 The incident and hence reflected energy flux is uniform over each individual area.
Figure 5: Schematization of the furnace and surfaces 1 and 2
Although no real surface is truly gray, it often happens that is relatively constant over that part of the spectrum where the blackbody emissive power is substantial, making the simplifying assumption of a gray surface warranted.
The below tables (9) and (10) give the values of total emissivity as function of temperature. But thanks to hypothesis 4, we will consider these values as spectral emissivity as well.Furthermore, for simplification purposes and since the furnace will operate at 1000Â°C in this present case, we will choose 1 = 0.85 and 2 = 0.87
Table 9: Emissivity of Oxidized Stainless Steel 303 as function of temperature
Temperature (Â°C) Emissivity 316 0.74 1093 0.87 Table 10: Emissivity of Nichrome 80Ni20Cr, oxidized as function of temperature
Temperature (Â°C) Emissivity 100 0.87 600 0.87 1300 0.89 A complex radiative exchange occurs inside the enclosure as radiation leaves a surface, travels to other surfaces, is partially reflected, and is then rereflected many times within the enclosure with partial absorption at each contact with a surface. It is complicated to follow the radiation as it undergoes this process; fortunately, this is not always necessary. A convenient analysis can be formulated by using the netradiation method. In this method, radiative energy balances are constructed for each surface, and the resulting set of equations is then solved. This method was first devised by Hottel and later developed in a different manner by Poljak [10]. An alternative approach was given by Gebhart in [11] and [12].
In its most simple version, the walls of the enclosure are divided into many uniformproperty, uniformtemperature surface elements. Consider one element of this subdivision which we call the kth inside surface area Ak of the enclosure. The qi,k and qo,k are the rates of incoming and outgoing radiant energy per unit area of Ak.
The outgoing radiation energy flux from a given location
on surface k, qo,k is made up of the emitted and reflected flux from that surface :
, = 4 + ,
(11)
where is the temperature of surface k. Note that all quantities are evaluated at a particular location on surface
 The quantity qi,k is the radiation flux incident at the given location from all other surfaces in the enclosure, including surface k itself, if it is concave. The quantity qo,k is often called the radiosity of the surface. It is derived from the portions of radiant energy leaving the surfaces inside the enclosure that reach the kth surface. The quantity
enclosure are incident to any of the surfaces Ak, the incident energy is then equal to:
, = 1,11 + 2,22 + + , + +
, + + , (14) From configurationfactor reciprocity, we have:
11 = 1; 22 = 2; ; =
; ; = (15)
Then taking into account equation (28), Equation (27) can be written so the only area appearing is :
, = ,11 + ,22 + + ,+ +
, + + , (16)
so that, by dividing both parts of the above equality by ,
the incident flux is:
qi,k, is often referred to as the irradiance. If the kth surface can view itself (is concave), a portion of its outgoing flux
, = 1 ,
(17)
=
=
will contribute directly to its incident flux.
Note that, contrary to the practice in most of heat transfer, these energy fluxes carry a directionality–the radiosity is the portion of the radiant energy flux with the component away from the surface, while the irradiance is the portion
Substituting equation (13) into (12) and (17) into (12) to eliminate , will provide two energy balance equations for in terms of and ,. Therefore, we obtain:
directed toward the surface.The net radiative heat flux
=
=
(4
) (18)
leaving surface k qk is the difference between the radiosity and the irradiance :
1
,
=
=
=
(
)
(19)
=
= (,
,
)
(12)
,
=1 ,
=1
,
,
This flux corresponds to the usual concept used in heat
Application : N=2 ;
transfer, as the net energy flux is taken as positive if in the
= 1
(4
) (20)
direction parallel to the surface normal of the position on k.
1 11 1
,1
The quantity qk is the energy flux supplied to Ak by some
= 2
(4
) (21)
means other than the radiation inside the enclosure, to make up for the net radiative gain or loss and thereby
2 12 2
,2
maintain the specified inside surface temperature. For example, if Ak is the inside surface of a wall of finite thickness, Qk could be the heat conducted through the wall from the outside to Ak.
The final equation for energy transfer quantifies the
irradiance as the sum of the radiant energies reaching a location on surface k from all other areas on the enclosure surface. This relation can have various forms depending on the degree of approximation used in the analysis.
Equation (11) can be rewritten as (noting k = k for a gray surface):
, = 4 + , = 4 + (1 ), =
1 = ,1 ,111 ,212 (22)
2 = ,2 ,121 ,222 (23)
In radiative heat transfer, a view factor is the proportion of the radiation which leaves surface that strikes surface .
The view factor are as follows: 11 = 0, 12 = 1
Introducing the values of 11 and 12 in equation (22), we obtain:
1 = ,1 ,2 (24)
4 + (1 ), (13)
On the other hand, 22 = 1 21 112 = 221
where k = 1 k = 1 k has been used for opaque gray surfaces (refer to assumption 4). Since all the outgoing fluxes from the surfaces A1, A2, , Ak, , AN in the
Which gives the value of 21
: 21
= 1 ;
2
So the view factor 22 can be expressed as follows:
1
(44)
2 1
2 1
=
1 +( 1 1)1
(28)
1
22 = 1
Since 1 = =
1 2 2
2
By replacing geometrical form factors with their
2
2 1
2 1
4 4
expressions, equation (23) can be written as:
1
( )
= 1 1
+( 1)
(29)
= 1 (1 1) = 1 (
) (25)
1 2
2 ,2
,1 2
,2
2
2
,2
,1
And finally,
Eliminating ,1 and ,2 between equations (20), (21) and
=
1 +( 1 1)
(30)
(24), and eliminating ,1 and ,2 between equations (24)
1 2
and (25), we get :
The sign minus in formula (28) means that the flux is in the
= 4 11
4 + 12
direction of ( being the outer unitary normal of the
{ 1 1 1 1
2 2 2
(26)
surface boundary).
= 1
2 2 1
 NUMERICAL RESOLUTION OF THE COMBINED HEAT TRANSFER :
And this will lead to the following system of equations :
12
= 4 4 + 1
This step consists of solving numerically the set of
{ 2
2 2 1
= 1
1 1
(27)
equations (7).
2 2 1
Eliminating 2 between the two above equations leads to the following equation :
12 ( 1 ) = 4 4 + 1

 Discretization of the PDE:
,
,
Let be the solution given by the model over the grid i,j including the boundary at time = and source temperature 0.
FDgrids in our present case are square. The values of the
2
2 1
2 1 1 1
dependent variable are calculated at the nodes, while
And this relation will lead to the following one :
<>parameters are specified for the spacing between the nodes (node centered grid).
Figure 6 : spatial grid for the finite difference method
We obtain the following scheme :
+1 = + [ ( ) ( ) + ( ) ( )]
,
,
2 +1/2,
+1,
,
1/2,
,
1,
,+1/2
,+1
,
,1/2
,
,1
([]4 [ ]4)
,
+1, =
0
,
,
( )
,
, = 0;
([]4 [ ]4)
,
(31)
1, =
0
,
,
( )
,
, = + 1;
([]4 [ ]4)
,
,+1 =
0
,
,
( )
,
, = 0;
([]4 [ ]4)
, ,1
0 ,
= , = + 1;
,
,
( )
= u0
= u0
0
0
{ ,
With :
+1/2, =
+1/2, =
(, +1, ) + (, , ) 2
(, , +1) + (, , ) 2
1/2, =
1/2, =
(, 1, ) + (, , ) 2
(, , 1) + (, , )
2
,
,
The problem is to compute the approximate solution of the PDE (7) at location (, ) and time =
corresponding to the control 0 with the system of equations (31).
Now let us address the problem of computation of boundary temperatures.
Lets consider the second equation of the above system
(31) which we rewrite as below :
Just note that the value = in (33) is known when computing thanks to the first equation of (31).
+1,
+1,
Now, (33) can be solved with the NewtonRaphson
0
0
method to provide the unknown temperatures at the boundary. The initial guess for the unknown temperatures at the boundary at time can be chosen equal to the source temperature .
For example at t = 0, just after power is put on in the furn,ace, we have = 0 = 0. We can then solve the
4 4
+1,
, +
, +
1,
=
([0 ] [,] )
,
,
( )
, = 0 (32)
equation (33) and find = 0 for the points of the
,
,
boundary.
,
,
At = (n = 1) the values 1 that are inside the domain
Note that for this equation, the points on the boundary have index = 0 and the next closest points (that are not in the boundary) have index = 1. Furthermore, the symetry of the problem allows us to deal only with this boundary equation since the others are similar.
In (32), let us set = , = and =
(out of the boundary) will be calculated with the numerical scheme (first equation of the system (31)) but from the values 0 of the points of the boundary for step 1.
,
,
,
,
Then, since all the points inside the domain are known, we calculate = 1 for the points of the boundary for step 1
by solving equation (33).
+1, 0
,
And so on at step n corresponding to the time = , the
Since ( ) = + , (where the coefficients and
values which are inside the domain (outside the
,
,
,
are found by identification with (8)), equation (32) will
boundary) will be calculated with the numerical scheme
yield
=
(44) which will lead to the following
+
(the first equation of the system (31)) but from the values
,
,
1 of the points of the boundary for step n1. Then after
equation in x (which is the unknown boundary temperature to be computed) for every time t: 4 + ( + ) ( + ) 4 = 0 (33)
,
,
Assuming that at time n, all the temperatures inside the heated body are known thanks to the first equation of the system (31), we are looking for = which is on the
boundary.
knowing all the points inside the domain, one calculates
,
,
= for the points of the boundary by solving the equation (33).
 Numerical analysis of the scheme:
 Discretization of the PDE:
Note that we use the same discrÃ©tisation in x as in y. Lets suppose that the solution u is twice differentiable with respect to the space variables and once with respect to the
Vol. 8 Issue 06, June2019
time variable. It is relatively easy to verify that the consistency error associated to the scheme (31) is such that [() + 2] except for the boundary where one has [() + ] and consequently, the scheme is consistent.
For the stability in , the necessary and sufficient condition is:
1
(34)
2 4
Where = 0 () = (0) if is a decreasing function of .
C. Simulation:
C1. Input data:
0 = 20Â° (initial temperature of the load);
0 = 1000Â°; = 7200
1 = 0.85; 2 = 0.87
Stephane Bolzmann coefficient which value is 5.67*108
For the functions (), () and () ; refer to formulas (8), (9) and (10).
= 3; (length of the furnace squared section side)
= 1; (length of the load squared section side) n number of time steps
= /( + 1) (space step);
= 30 (time step);
C2. Results:
921. 921. 885. 861. 846. 842. 846. 861. 885. 921. 921. 921. 817. 741. 692. 663. 654. 663. 692. 741. 817. 921. 885. 741. 640. 573. 536. 524. 536. 573. 640. 741. 885. 861. 692. 573. 496. 453. 440. 453. 496. 573. 692. 861. 846. 663. 536. 453. 407. 393. 407. 453. 536. 663. 846. 842. 654. 524. 440. 393. 377. 393. 440. 524. 654. 842. 846. 663. 536. 453. 407. 393. 407. 453. 536. 663. 846. 861. 692. 573. 496. 453. 440. 453. 496. 573. 692. 861. 885. 741. 640. 573. 536. 524. 536. 573. 640. 741. 885. 921. 817. 741. 692. 663. 654. 663. 692. 741. 817. 921. 921. 921. 885. 861. 846. 842. 846. 861. 885. p>921. 921. Figure 7: distribution of temperatures in the heated metal
Figure 8: temperatures level curves
Level curves are circles centered at the center of the metal. squarred section. The temperatures decrease as one gets far from the boundary. In the boudary, the temperatures are getting bigger as one gets closer to the corners. So the physics of the problem is respected.

 CONCLUSION :
The small time step approximation combined with the finite difference discretization allowed us to compute all the required temperatures in the heated material. And furthermore, the resulting temperatures respect the physics of the problem.
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