In this section pure radiative heat transfer model based on unstructured finite vol- ume method is validated for problems with the surface to surface and gas radiation in different types of enclosure. Notably, in all the test cases the size of the spatial
and angular grids have been chosen as specified in the literature unless otherwise stated.
3.2.1 Surface to surface radiation
Surface to surface radiation problems are generally solved using surface models, in this section surface to surface radiation problems are solved using FVM by neglecting the RHS in the RTE i.e RTE does not contain any absorbing, emitting or scattering terms. Three basic problems are solved which can be found in any basic heat transfer books. Radiative heat transfer between infinite long parallel plates, between co-centric cylinders, and in a room. In all the three problems the objective is to calculate the heat flux at the walls, which is compared with the analytical results.
3.2.2 Infinitely long parallel plates
Figure 3.1 shows the grid used for the problem, the length of the plate is 20 m and the width is 1 m, the plate is made infinitely long by applying symmetry condition along the left and right boundary. The top wall is at 350 K, the bottom wall is at 300 K and the walls are assumed black. The same problem is also solved by changing the wall emissivity as 0.8 and 0.9 for the top and bottom walls respectively.
Figure 3.1: Computational domain for parallel plates
The analytical result for the heat flux at the top and bottom walls for this problem can be calculated by using the relation given below.
q12 = Aσ( T14 − T24 )
1
ε1 + ε12 −1 (3.1)
3.2 Validation of Pure Radiation Problems 61 The problem is solved by discretizing the spatial domain in 30 × 10 hexahedral grids with angular discretization of 4 × 6 in θ and φ direction respectively. Table 3.1 shows the wall heat flux for black walls and walls with different emissivity.
Table 3.1: Radiative heat flux at the wall between parallel plates
Temperature in K Wall Emissivity Analytical flux Numerical flux T1=350.0,T2=300.0 ǫ1=1.0, ǫ2=1.0 391.58 mW2 391.58 mW2
T1=350.0,T2=300.0 ǫ1=0.8, ǫ2=0.9 287.71 mW2 287.71 mW2
3.2.3 Infinitely long co-centric cylinders
Figure 3.2 shows the grid used for the problem, the length of the cocentric cylinder is 20 m and the inner and outer radius are 1 m and 2 m respectively. The cylinder is made infinitely long by applying symmetry condition along the top and bottom boundary. The outer cylinder is at 350 K and the inner cylinder is at 300 K and the walls are black, the same problem is also solved by changing the wall emissivity as 0.8 and 0.9 for the outer and inner walls respectively.
Figure 3.2: Computational domain for co-centric cylinders
The analytical result for the heat flux at the top and bottom walls for this problem can be calculated by using the relation.
q12 = Aσ( T14 − T24 )
1
ε1 + 1−ε2ε2 r1r2 (3.2)
The problem is solved by discretizing the spatial domain in 20× 10×5 hexahedral grids wit angular discretization of 4 × 6 inθ and φdirection respectively. Table 3.2 shows the wall heat flux for black walls and walls with different emissivity.
Table 3.2: Radiative heat flux at the wall between co-centric cylinders Temperature in K Wall Emissivity Analytical flux Numerical flux T1=350.0,T2=300.0 ǫ1=1.0, ǫ2=1.0 391.58 mW2 391.58 mW2
T1=350.0,T2=300.0 ǫ1=0.8, ǫ2=0.9 299.93 mW2 299.93 mW2
3.2.4 Rectangular enclosure with opening
The problem is shown in Fig. 3.3 consists of a room with all black walls. The floor of the room is at a temperature of 318 K and all the other walls including ceiling are at 298 K. There is a small window which is maintained at 283 K. The same problem is computed numerically by using 12×10×4 hexahedral grid with angular discretization of 4×20. The net heat loss by the floor is 137.39 mW2 calculated numerically and 137.27 mW2 by taking a view factor as 0.058 between the floor and window.
00 00 00 11 11 11
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000
111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111
00000000000 111111111110011 00
112 m 2 m 4 m
FLOOR 12 m
6 m T
WINDOW T
T =298K
=283 K
=318 K
Figure 3.3: Surface radiation inside a room
From the above three problems it is seen that the present method is capable of computing surface to surface radiation.
3.2 Validation of Pure Radiation Problems 63
3.2.5 Quadrilateral enclosure containing absorbing-emitting medium
The problem consists of a quadrilateral enclosure as shown in Fig. 3.4(a) filled with absorbing-emitting medium. The enclosure medium is maintained at a constant temperature Th = 100 K while the walls are at 0 K. Three different values of ab- sorption coefficients 0.1 m−1, 1.0 m−1 and 10.0 m−1 are used in the present study while the walls of the enclosure are assumed as black. The objective is to study the non-dimensional radiative heat flux at the bottom wall. The study is carried out by employing 20×20 control volumes and 2×8 control angles as also used by Chai et al.[7].
Tmedium κ = 0.1 ε = 1.0
ε = 1.0 κ = 1.0
κ = 10.0 T = 0 K
T =
T = 0 K
0 K m
m m
−1
−1
−1
= 100 K
(0, 0) (2.2, 0)
(0.5,1.0)
(1.5, 1.2)
ε = 1.0
T = 0 K
ε = 1.0 X (m)
Wall Heat Flux (kW/m2 )
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8 1 1.2
Present Chai et al. (1995) κ = 10 m-1
κ = 1 m-1
κ = 0.1 m-1
(a) (b)
Figure 3.4: (a) Quadrilateral enclosure, (b) non-dimensional heat flux at the bottom wall for κa=0.1, 1.0 and 10.0
Figure 3.4(b) represents the comparison of heat fluxes at the base of the enclosure with Chai et al.[7]. From Fig. 3.4(b) an increment in wall heat flux is clearly observed with increasing absorption coefficient of the medium. For κa=10.0 m−1 the radiative heat flux is close to unity, this is primarily due to the large value of intensities incident on the wall from the neighboring hot medium. The reduction in heat fluxes near the edges is due to side cold walls. Furthermore, the reduction in absorption coefficient leads to a gradual decrease in heat flux due to the large extent of the neighboring cold walls and low self-extinction of the medium.
3.2.6 Curved enclosure containing absorbing-emitting medium
The problem consists of radiative heat transfer in an enclosure with a curved bottom wall. The top wall is aty=1.0 m, whereas the bottom wall varies as per the following relation
y= 0.5 [tanh(2−3x)−tanh(2)], 0≤x≤10/3 (3.3) Figure 3.5(a) represents the geometry of the problem where the bottom wall is black and maintained at 1000 K while the other black walls are at 0 K. The medium is kept at 0 K with a absorption coefficient of 1 m−1. The objective is to calculate the heat flux at the top wall. The problem is solved numerically by using a spatial discretization of 40 × 40 grid points while the angular domain is discretized into 4 × 24 control angles in the θ and φ directions, respectively following the work of Chai et al.[7]. Figure 3.5 (b) represents the comparison of radiative heat flux at the top wall with the results of Chai et al.[7]. The results show good agreement with the published literature.
T = Tmedium
ε = 1.0
ε = 1.0
ε = 1.0 κ = 1.0 (0, 0)
T = 0 K
1000 K
T = 0 K (0, 1)
ε = 1.0 T =
= 0 K 0 K
(3.3, 1)
(3.3, −1.96) m−1
X (m) Wall Heat Flux (kW/m2 )
0.5 1 1.5 2 2.5 3
1 2 3 4 5 6 7
Present
Chai et al. (1995)
κ = 1.0 m-1
(a) (b)
Figure 3.5: (a) Curved geometry, (b) heat flux at the top wall forκa=1.0 m−1.
3.2 Validation of Pure Radiation Problems 65
3.2.7 Radiative heat transfer in a rectangular idealized fur- nace
A three-dimensional radiative heat transfer problem is shown in Fig. 3.6. The problem represents a idealized combustion chamber of size 4 m × 2 m × 2 m proposed by Meng¨u¸c and Viskanta [122]. The combustion chamber is filled with a absorbing-emitting and non-scattering medium with κa= 0.5 m−1, with a uniform heat generation of 5.0 kW/m2. The wall boundary conditions are defined as.
X= 0 m ǫw= 0.85 Tw= 1200 K X= 4 m ǫw= 0.70 Tw= 400 K
Rest of the walls are at 900 K with an emissivity of 0.7. The spatial domain is discretized into 25×25×25 hexahedral cells with an angular resolution of of 4 ×20 following the work of Chai et al. [5]. This problem requires the energy equation to be solved iteratively due to the presence of internal heat source term. The temperature can be updated using the relation
T4 = 1 4σ
qgen
κa
+G
(3.4) where G is the irradiation, defined as G= R
4πIdΩ and σ is the Stefan-Boltzmann constant.
y x z
T =
4 m
2 m 2 m
400 K
T = 1200 K ε = 0.85
ε = 0.7 T = ε = 0.7900 K
w
w
w
Figure 3.6: Representation of idealized combustion chamber
Figure 3.7(a) shows the temperature variation at three different planes at X= 0.4 m, 2.0 m and 3.6 m along the center line at y = 1.0 which are compared with the
X (m)
T(Κ)
0 0.2 0.4 0.6 0.8 1
850 900 950 1000 1050 1100
Present
Chai et al. (1994) Z= 0.4 m
Z= 2 m
Z=3.6 m
X (m) Wall Heat Flux ( kW/m2 )
0.5 1 1.5
0 20 40 60 80
Present
Menguc and Viskanta. (1985)
Losses at hot wall
Gain at cold wall
(a) (b)
Figure 3.7: (a) Comparison of temperature variations at planesX=0.4 m, 2.0 m and 3.6 m a y= 1 m, (b) radiative heat flux distribution along the hot and cold walls.
results of Chai et al. [5]. Heat flux distribution along the hot and cold walls is also shown in Fig. 3.7(b), fluxes are compared with the result of Meng¨u¸c and Viskanta [122]. Results from the present solver have a good comparison with the results of literatures.
3.2.8 Three-dimensional kidney shaped combustion cham- ber
A validation of three-dimensional radiative heat transfer in kidney shaped combus- tion chamber as shown in Fig. 3.8(a) proposed by Beak et al [16]. The dimensions of the combustion chamber are 4 m × 2 m × 2 m in X-, Y-, and Z-direction, re- spectively. The problem consists of a combustion chamber filled with an absorbing- emitting and non-scattering medium with a uniform heat generation of 5.0 kW/m2. The wall boundary conditions are defined as.
X=0 m ǫw=0.85 Tw = 1200 K X=4 m ǫw=0.70 Tw= 400 K
Rest of the walls are at 900 K with an emissivity of 0.7. The spatial domain is discretized into 20 × 10 × 10 grids in X-, Y- and Z-direction, respectively with
3.2 Validation of Pure Radiation Problems 67 an angular resolution of 8 × 24 following the work of Beak et al. [16]. The energy equation is solved iteratively with radiative transfer equation until a steady state is achieved. The temperature is updated using the relation as given in Eq. (3.4) The problem is further analyzed forκa=0.1, 1.0 and 5.0. The variation of isotherms at the mid-plane (Z =Z0/2) for various absorption coefficient are presented in Figs.
3.8(b, c, d). Further, the radiative heat fluxes along the cold and hot walls are shown in Fig. 3.9
1080
920
93
0 940 95
0
960
97 98 0 9 0
90 10
00 10
10 1040
(a) (b)
840 86
0 870 88
0 890 900
910
920 930
940
95
0 960 970
980
1000
1030
1060
790 82
0 860 87
0 89
0
900
910 920 930
940
950 960
970 980
990 10
00 10501080
(c) (d)
Figure 3.8: Isotherms at the mid plane z=1.0 for (b) κa=0.1, (c) κa=1.0 and (d) κa=5.0
From Fig. 3.9 it is evident that the heat flux decreases as the extinction coefficient increases for both hot and cold walls. The increase in extinction coefficient dimin- ishes the mean free path which leads to an increase in total radiation absorbed by the medium. Further, from the isotherms, it is observed that for a lower extinction coefficient β = 0.1 m−1 the temperature is higher and more uniformly distributed
Z (m) Wall Heat Flux ( kW/m2 )
0 0.5 1 1.5 2
10 15 20 25 30 35 40
Present
Beak et al. (1998)
β = 0.1
β = 1.0
β = 5.0
Z (m) Wall Heat Flux ( kW/m 2 )
0 0.5 1 1.5 2
30 40 50 60 70 80
Present
Beak et al. (1998)
β = 0.1
β = 1.0
β = 5.0
(a) (b)
Figure 3.9: Radiative heat flux along (a) the cold wall, (b) the hot wall.
throughout the medium. This is due to the large extent of radiation in equalizing the temperature. As the extinction coefficient increases more and more radiation is absorbed by the medium which leads to a reduction in the temperature gradients near the walls. This results in a decreased heat flux at the hot and cold walls.
3.2.9 Enclosure filled with scattering medium
In this section radiative heat transfer in the pure scattering medium is discussed.
The source term in RTE does not have any contribution from absorption and emis- sion. Figure 3.5 (a) shows the geometry of the problem. All the walls are black with the bottom wall at 1000 K, while rest of the walls are at 0 K. The medium is at 0 K with scattering coefficient of 1.0 m−1. The objective is to calculate radiative heat flux at the top wall. The problem is solved numerically by using a spatial resolution of 40×40 grid points while an angular resolution of 4 ×24 control angles is used in the θ andφ directions, respectively following the work of Chai et al.[7]. Figure 3.10 shows the radiative heat flux at the top wall compared with the results of Chai et al.[7]. A similar problem considering absorbing emitting medium was discussed in the previous section, comparing the heat flux from Fig. 3.5 shows that the heat flux is higher in the present case considering only the scattering medium. The increase in
3.3 Variable Density Flows at Large Temperature Difference 69 heat flux is due to the far-reaching effect of radiation and negligible self-extinction of the medium.
X (m) Wall Heat Flux (kW/m2 )
0 0.5 1 1.5 2 2.5 3 3.5
6 8 10 12 14 16 18 20 22 24
Present
Chai et al. (1995) σ =1.0
Figure 3.10: Heat flux at top wall
3.2.10 Enclosure filled with absorbing emitting and scatter- ing medium
In this section, the entire RTE considering the absorbing, emitting and scattering medium is discussed. The validation problem discussed here is similar to the problem discussed in section 3.2.7, the problem is now resolved including the scattering effects with the same spatial and angular grids as previously used in section 3.2.7. The medium has a scattering albedo ω = 0.7 and extinction coefficient of κa = 0.5 m−1. Figure 3.11 shows the temperature variation at three different locations at X = 0.4 m, 2.0 m and 3.6 m along a line at Y = 1 m compared with the results of Truelove [123].
3.3 Variable Density Flows at Large Temperature Difference
In this section validation studies are presented for the developed low-Mach number algorithm for variable density flows at large temperature difference. The developed
Z (m)
T (Κ)
0 0.5 1 1.5 2
850 900 950 1000 1050 1100
Present
Truelove (1988)
X=0.4
X=2.0
X = 3.6
Figure 3.11: Temperature distribution atY = 1 m
low-Mach number solver is validated against a simple differentially heated cavity problem atGa= 1.2,Ra= 106. Contribution from various researchers was collected and published as a benchmark solution for this problem involving low-Mach number flows. The problem is solved for Ra= 106 and P r= 0.71. The initial temperature of the cavity is 600 K and for a temperature difference parameter (Boussinesq pa- rameter, ∆T2T
o=0.6) the hot and cold wall temperatures are obtained as 940 K and 260 K. At this large temperature difference, Boussinesq approximation fails to en- counter the density variation arising due to large temperature gradients, hence this problem serves as a perfect validation for the present algorithm. The geometry of the problem is shown in Fig. 3.12 (a). The problem is solved by considering density variation with constant and variable temperature dependent thermo-physical prop- erties. The property variation is considered using Sutherland’s law. A uniform grid size of 300×300 is employed in the present study following the work of Darbandi and Hosseinizadeh [124] and a time step size of ∆t= 10−2 is used.
Tables 3.3 and 3.4 represent the average Nusselt number variation at the hot and cold walls compared with the results presented by various researchers Le Qu´er´e et al.[125]. Furthermore, the local Nusselt number variation at the hot and cold walls is compared with the results of Darbandi and Hosseinizadeh [124] as presented in Fig.
3.12 (b). The local and average Nusselt number values are in excellent agreement