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Numerical Investigation of the Transient Radiative Heat Transfer inside a Hexagonal Furnace Filled with Particulate Medium
Elham Khademi Moghadam1,
, Rasool Nasr Isfahani2, Arash Azimi3
1Department of Physics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
2Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
3Department of Mechanical and Aerospace Engineering, Islamic Azad University, Tehran 14778-93855, Iran
Abstract
The transient radiative heat transfer in two 2D irregular geometry filled with particulate media is numerically investigated. The radiative transfer equation is solved with FTn finite volume method. Non-orthogonal mesh is used to discretize the computational domain and the high resolution CLAM scheme is utilized to relate the facial intensities to the nodal values. Various cases of scattering in media with real indices of refraction (dielectric particles) and complex indices of refraction (absorbing particles) are considered. Both Mie theory and equivalent isotropic approximation are used to account for the scattering behavior of the media. The difference between these two methods is found insignificant, especially for the steady state solutions and for media with complex indices of refraction, while the complexity and computational cost of the equivalent isotropic approximation is much lower than those of Mie theory.
Keywords: transient radiative heat transfer, Mie Theory, equivalent isotropic approximation, finite volume method, non-orthogonal grid
Copyright © 2016 Science and Education Publishing. All Rights Reserved.Cite this article:
- Elham Khademi Moghadam, Rasool Nasr Isfahani, Arash Azimi. Numerical Investigation of the Transient Radiative Heat Transfer inside a Hexagonal Furnace Filled with Particulate Medium. American Journal of Mechanical Engineering. Vol. 4, No. 2, 2016, pp 42-49. https://pubs.sciepub.com/ajme/4/2/1
- Moghadam, Elham Khademi, Rasool Nasr Isfahani, and Arash Azimi. "Numerical Investigation of the Transient Radiative Heat Transfer inside a Hexagonal Furnace Filled with Particulate Medium." American Journal of Mechanical Engineering 4.2 (2016): 42-49.
- Moghadam, E. K. , Isfahani, R. N. , & Azimi, A. (2016). Numerical Investigation of the Transient Radiative Heat Transfer inside a Hexagonal Furnace Filled with Particulate Medium. American Journal of Mechanical Engineering, 4(2), 42-49.
- Moghadam, Elham Khademi, Rasool Nasr Isfahani, and Arash Azimi. "Numerical Investigation of the Transient Radiative Heat Transfer inside a Hexagonal Furnace Filled with Particulate Medium." American Journal of Mechanical Engineering 4, no. 2 (2016): 42-49.
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At a glance: Figures
1. Introduction
The transient radiative transfer finds applications in many emerging technologies such as short pulsed laser in turbid media, remote sensing, nondestructive biomedical diagnosis and combustion product analysis. The transient radiative transfer equation (TRTE) must be solved to find the radiative intensity field.
Thus far, most of the proposed methods for solving the radiative transfer equation (RTE) are extended to solve TRTE. Among these various methods, finite volume method (FVM) has been a favorite to the researchers due to its merits such as being easily programmable, fairly accurate and computationally cheap [1-17][1]. Chai [1, 10] and Chai et al. [2] have applied FVM to solve TRTE. They numerically investigated radiative transfer in 1-, 2- and 3D systems containing the emitting, absorbing and anisotropically scattering media. They also compared the results obtained by STEP scheme with those of CLAM scheme and concluded that CLAM scheme yield more accurate results especially in the early stages of the transient radiative transfer. They also could solve the transient radiative transfer in 2D irregular geometries by using non-orthogonal mesh (which is also known as body fitted coordinates). Ruan et al. [3] used FVM to solve the TRTE in 1D homogeneous and inhomogeneous media. They found that for the two-layer media, changing in the scattering albedo and optical thickness does not alter the location of the local minimum moment. Moghadasian and Moghadassian [5] and Moghadassian and Kowsary [6] numerically investigated conjugated heat transfer problems in 2- and 3-D cavities. Kim et al. [7] applied FVM to solve TRTE in 1D slab subjected to the radiative equilibrium condition. They utilized and compared various schemes for calculating convective terms of the discretized FVM. They found that FVM is an accurate method for solving TRTE when the medium is cold or it is subjected under the radiative equilibrium condition.
One of the major difficulties in solution of RTE or TRTE is the modeling of radiative scattering. In many cases in both nature and science, there are tiny particles in media that make the scattering phenomena even more complicated. Such media is known as particulate media. Some examples are the scattering of sunshine by the atmosphere which results in blue sky, red sunset and colorful rainbow, scattering of light in interstellar dust, pulverized coal furnaces, combustion systems, etc. The most famous and general rule that governs the radiative scattering in particulate media is Mie theory that is named after Gustav Mie who solved the Maxwell equations to relate the scattering phase function to the medium properties like index of refraction and particle size. Mie theory founded a base for further studies. Some researchers in the field of radiative heat transfer have utilized it. Trivic et al. [8, 12] considered Mie scattering in the square and cubic enclosures. They successfully coupled the finite volume method (FVM) with Mie theory to perform a numerical study of the effect of anisotropic scattering on the radiative heat transfer. Steward and Trivic studied heat transfer of a convective-radiative flow [15]. They used zonal method for solving radiative heat transfer (RTE) and Mie theory for calculating the scattering phase function in a three dimensional furnace. The comparison with the experimental data showed the accuracy of their applied solution method.
In most cases, thermal engineers have preferred to use simplified relations to account radiative scattering, due to complexity of Mie theory. The Henyey-Greenstein, delta-Eddington and Legendre polynomial approximations as well as formulation of equivalent isotropic scattering are some of these relations [18].
Menguç and Viscanta [19] approximated the scattering phase function by delta-Eddington relation and solved RTE in a rectangular cube by the first and third-order spherical harmonics. In the work of Kim and Lee [20], the coupling between discrete ordinates method (DOM) and Legendre polynomial approximation was done to find the average radiative intensity and radiative heat flux in a 2D rectangular enclosure of an emitting, absorbing and anisotropically scattering medium. Guo and Kumar [21] considered a slab containing scattering media. They compared the results of applying the Henyey-Greenstein phase function with the equivalent isotropic scattering approximation. They compared the temporal distribution of transmittance and reflectance and observed that the error between the two models is more significant for the cases of backward scattering.
The aim of present study is to solve the transient radiative heat transfer in a 2D irregular geometry containing particulate media with uniform particle diameter. The TRTE is solved by FVM. The scattering phase function is calculated by applying Mie theory. The validation checks are done to make sure of the numerical code. The results are reported for media with different particle sizes, particle densities and indices of refraction. In addition, the heat transfer parameters (i.e. the heat flux and the incident irradiation) calculated by applying Mie theory are compared with those obtained by the isotropic equivalent approximation.
2. Mathematical Formulation
2.1. TRTE Solution MethodThe transient radiative transfer equation in a grey, emitting, absorbing and scattering medium at any position r along a path s is given by
 | (1) |
Where the source function can be defined as
 | (2) |
The boundary condition for a diffusely emitting and reflecting wall can be written as follows
 | (3) |
In the above equations I is the radiative intensity, 
is the light speed in the medium; 
, 
and 
are extinction, absorption and scattering coefficients, respectively. 
is the wall emissivity and 
is the scattering phase function. The discretization process of eq. (1) is well explained in [1, 2] is not repeated here.
After discretization (see [2] for example), the source term in eq. (2), can be calculated as
 | (4) |
If the analytical expression for scattering phase function 
exists, the average scattering phase function in eq. (8) can be obtained by the following relation
 | (5) |
When the integration is not feasible or 
is an unknown function but its value for different s and s′ can be obtained, its average can be estimated by the following formula [8]
 | (6) |
For particulate media of this study, the analytical expression for scattering phase function does not exist and therefore eq. (6) must be used. Scattering phase function in any combination of s and s′ is obtained by applying Mie theory [18].
The details of complicated Mie theory are beyond the scope of this work. They may be found in the books written by van de Hulst [22] and Kerker [23]. A very compact formulation is presented subsequently.
The scattering phase function is
 | (7) |
The parameters of eq. (7) are
 | (8) |
and 
are complex functions that can be expressed in terms of infinite summation of specific functions as follows
 | (9) |
Where 
and 
are related to Legendre polynomials by the following relations
 | (10) |
and Mie scattering coefficients 
and 
are complex functions of 
and 
.
 | (11) |
Where 
and 
are known as Riccati-Bessel functions and can be related to Bessel and Hankel functions by
 | (12) |
On the other hand the 
is also can be expressed as a function of Mie scattering coefficients as follows
 | (13) |
If the particles have the same size, the scattering and the extinction coefficients of the particles can be expressed as
 | (14) |
Although the value of 
is dependent on wavelength, for the sake of analysis in this study, it is calculated for the typical wavelength of 
=3.1415 μm as suggested by Modest [18].
Table 1. media properties for N T=105 (particles/cm3) and λ=3.1415 μm (the values for index of refraction are brought from [8,18])
 and λ=3.1415 μm (the values for index of refraction are brought from [8,18]))
Table 2. Dimensionless G (=G/Ebw) along the centerline of the square enclosure (x=0.5 m) predicted by Kim et al. [20]
 along the centerline of the square enclosure (x=0.5 m) predicted by Kim et al. [20])
In addition, the evolution of infinite series in the above equation is terminated by the following relation of Deirmendjian et al. [24].
 | (15) |
There are recursive formulae and series for calculation of the Bessel and Hankel as well as Legendre polynomials and their derivatives [24] that are used in this study to write the numerical FORTRAN 90 code.
2.3. Equivalent Isotropic ApproximationThe equivalent scattering and extinction coefficients as well as the equivalent single scattering albedo can be related to the original values according to the following relations [21],
 | (16) |
 | (17) |
 | (18) |
where the dimensionless asymmetry factor is
 | (19) |
and 
.
The integration of eq. (19) is calculated numerically using the trapezoid rule with 1000 points in the range of 
.
3. Validation Check
First the numerical code for the Mie scattering is validated by the results of Trivic et al. [8] and Kim et al. [20]. Consider a square enclosure of length L=1 m. The bottom wall is black and hot (Eb,ref=1 W/m2) and other walls are cold and black at 0K. The medium is purely scattering with 
=1 m-1. The indices of refraction are assumed real and are brought in Table 1. Trivic et al. [8] and Kim et al. [20] considered this problem but just the results of Kim et al. [20] are brought here for the sake of brevity. Their predicted values for G/Eb,ref along the centerline of enclosure (x=0.5 m) are reported in Table 2. The results of this study are brought in Table 3. The identical 2525 spatial grid is used for better point-to-point comparison. The Ls=33 is used as suggested in [8]. The finer sub-control angle is seen to produce insignificant difference in results.
In addition, the numerical code is checked for the cases of complex indices of refraction of Table 1. The scattering phase function for different values of 
is obtained and is shown in Figure 1a and Figure 1b. They are exactly the same as the diagrams obtained by Trivic et al. [12] that are not shown here.
Table 3. Dimensionless G (=G/Ebw) along the centerline of the square enclosure (x=0.5 m) predicted in this study
 along the centerline of the square enclosure (x=0.5 m) predicted in this study)
Then the ability of the code is examined for the case of non-orthogonal mesh. The transient radiative heat transfer in 2D quadrilateral enclosure of Figure 2a is considered. The enclosure is filled with an emitting, absorbing and non-scattering medium at Tref. The 2525 grid system is shown in Figure 2a. All walls are black at Tw=0K. The extinction coefficient is 1 m-1. The characteristic time step is 
where 
m for this case. It is found that further decrease in 
does not change the results significantly. The dimensionless radiative heat flux on the bottom wall is compared with previously published results of Chai et al. [10]. The comparison is shown in Figure 2b.
As it can be seen in all cases a good agreement between the results of present study and previously reported ones is seen. This can secure the accuracy of the results of subsequent sections.
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4. Results and Discussions
The transient radiative heat transfer is numerically investigated in a quadrilateral of Figure 2a. It should be stated that the radiative equilibrium condition is applied to all cases. All the walls are black. The wall number 3 is hot 
. Other walls are cold at 0K. The medium is initially at 0K. The characteristic time step is 
.
First the dielectric particles (those with real index of refraction) of Table 1 in medium with 
=1 m-1 are considered. For the sake of analysis, the medium is assumed purely scattering and eq. (14) is not used to find the values of 
and 
.
The temporal dimensionless heat flux on the bottom wall of Figure 2a is illustrated in Figure 3a and Figure 3b for F1 and B2. It is seen that for forward scattering medium, in each time step the heat flux on the bottom surface is higher because the value of radiative intensity is larger in comparison with backward scattering. In these figures Mie calculations are compared with the equivalent isotropic approximation. It should be noted that both of the media are assumed purely scattering. It is seen that the equivalent isotropic approximation underpredicts the radiative flux for backward scattering medium and overpredicts it for forward scattering medium. The reason is that the equivalent optical thickness is higher than the original value for the backward scattering. Therefore the medium absorbs greater amount of energy and weaker radiative intensity reaches at the cold surface. Conversely, the equivalent optical thickness is lower than the original value for forward scattering. The maximum relative difference between the results of the equivalent isotropic approximation and those of Mie theory is 9.0% which happens for B2 at 
. The maximum relative difference for the steady state results 
is 3.6%.
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Then the absorbing particles (those with complex index of refraction) of Table 1 Are considered. The particle density is 
particles/cm3. The scattering phase function for different materials at 
is shown in Figure 4a. The scattering phase function for different sizes of the carbon particles is also shown in Figure 4b. It is easily noticed that all particles have the forward scattering character because their scattering phase functions have higher values for acute angles of scattering. The peak of the function happens at 
. This forward characteristic becomes more significant for higher size parameters as can be seen in Figure 4b where the peak of the scattering phase function at 
becomes more pronounced at higher values of 
.
Figure 5 shows the dimensionless radiative flux on the bottom surface. As shown in this figure, the difference in the scattering phase function for various particles is not much. Therefore the difference in the values of radiative flux of Figure 5 is mainly attributed to the values of 
and 
. Besides the fact that the radiative intensity travels with lower changes in a media with lower extinction coefficient, the value of single scattering albedo has a significant impact on radiative transfer. In cold regions such as neighborhood of the bottom wall, the first term on the right of eq. (4) tends to zero and the source term reduces. But the nodal coefficient 
remains unchanged. Therefore the ratio of 
to 
becomes important and higher 
leads to higher radiative intensity. As a result the heat flux in a medium such as ash with lower value of 
and higher value of 
must be stronger.
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Another noticeable issue in Figure 5 is the influence of cold walls on the radiative heat flux of the bottom wall. The heat flux is low at smaller values of x because this region of wall 1 sees more of wall 4 that is a cold wall.
When x/h increases, the influence of wall 4 diminishes and in the midsection of the bottom wall (around x/h=1), the heat flux reaches its peak under the influence of hot wall 3. Then due to the presence of another cold wall (wall 2) the decrement of q starts and at the end of the bottom wall, it again reaches minimum.
The comparison of Mie calculations with the equivalent isotropic approximation for media of carbon and lignite particles is displayed in Figure 6a and Figure 6b, respectively. Again the heat flux is slightly overpredicted by the equivalent isotropic approximation. The larger relative difference between Mie theory and the equivalent isotropic calculations is 4.3% which occurs for lignite particles at 
. The maximum difference has reduced since in these cases the medium is not purely scattering and by using the equivalent approximation the optical thickness does not change as significant as in the cases of a purely isotropic medium. Also this difference is larger for medium containing lignite particles because it has a higher 
as reported in Table 1. As a result, applying isotropic approximation changes its equivalent 
and 
more abruptly relative to the media such as carbon particles that have lower value of 
.
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5. Conclusions
In this work the transient radiative heat transfer in particulate media with the radiative equilibrium condition is considered. The solution is performed numerically by coupling of Mie theory and FVM with CLAM scheme. It is assumed that the medium itself (without the particles) does not participate in radiative transfer. After validating the code, different types of media are considered. In addition, the results of applying Mie theory are compared with the equivalent isotropic approximation. Based on the results, some conclusions can be drawn:
• Radiative heat flux on the cold surface is higher for forward scattering media;
• The transient and steady state temperature fields tend to be more uniform for forward scattering;
• At a constant particle diameter ash particles have the highest value of 
among absorbing particles because they have the lowest absorptive index among other materials of Table 1 and radiative flux is higher for medium of fly ash particles;
• The effect of cold walls is important in predicted values of heat flux;
• The equivalent isotropic approximation underpredicts the heat flux on the cold surface for the backward scattering medium. It overpredicts the heat flux for the forward scattering medium;
• The radiative flux and integrated intensity are higher for media with smaller particles;
• The equivalent isotropic approximation is much simpler than the Mie theory and its predictions are accurate especially in the cases of forward scattering and absorbing particles. Also it is more accurate for the steady state solutions than the transient ones. However, its key parameter, g, is found when 
is obtained. Therefore an analytical expression or numerical data of 
is required.
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 different materials of Table 1 at xp=1 and (b) for Ash at different particle size parameters){kind=link}
 the geometry and grid and (b) the dimensionless heat flux on the bottom wall){kind=link}
 F1 (the forward scattering medium) and (b) B2 (the backward scattering medium)){kind=link}
 different materials of at xp=2 and (b) for carbon particles at different size parameters){kind=link}
 carbon particles and (b) lignite particles){kind=link}
