Table 6.3: Average Nusselt number values at the hot and cold walls with an inline configuration of front-back partitions.
Hot Wall Cold Wall
Nusselt number Nuc Nur Nut Nuc Nur Nut
Pure Convection 9.392 NA 9.392 9.392 NA 9.392 Surface Radiation 4.867 35.524 40.392 12.877 27.508 40.385 Gas Radiation 3.944 35.897 39.841 13.825 26.008 39.833 Table 6.4: Average Nusselt number values at the hot and cold walls with offset configuration of front-back partitions.
Hot Wall Cold Wall
Nusselt number Nuc Nur Nut Nuc Nur Nut
Pure Convection 9.363 NA 9.363 9.363 NA 9.363 Surface Radiation 4.972 36.378 41.350 13.062 28.265 41.328 Gas Radiation 4.124 36.168 40.293 13.823 26.463 40.286 cold fluids at the partitions. Similarly, the radiation Nusselt number distribution in Figs. 6.14 (b) and (d) demonstrates clustering of lower values of the radiation Nusselt number near Z=1 for cold wall and Z=0 for the hot wall. The decrease in radiation Nusselt number is a consequence of the presence of partitions acting as partial radiation barriers.
6.4 Entropy Generation 155 and radiation. The velocity and temperature gradients are accountable for the en- tropy production by conduction, convection, and viscous dissipation. On the other hand, the long-range nature of radiation compels the radiative entropy generation rates to be independent of the local temperature gradients. The radiation entropy production rates are dependent on the gross temperature field of the entire system.
The entropy production rate for buoyancy-induced convection under the influence of radiative heat transfer is shown in Eq. (6.3), following the work of Slimi [131]
SG= k T02
"
∂T
∂xj
2# + 1
T0
τij
∂ui
∂xj
+ κ
T0
4πIb− Z
4π
IdΩ
(6.3) where
τij =µ ∂ui
∂xj + ∂uj
∂xi − 2 3
∂ui
∂xi
(6.4) The terms on the right hand side of Eq. (6.3) represent the rate of entropy pro- duction by convection, fluid friction, and radiative heat transfer. The dimensionless form of Eq. (6.3) is obtained by rescaling using Eq. (2.66) as done previously for obtaining the non-dimensional form of governing equations. The dimensionless form of entropy generation is described in Eq. (6.5) by dropping the superscripts.
SG =
"
∂T
∂xj
2# +φ0
∂ui
∂xj
+ ∂uj
∂xi − 2 3
∂ui
∂xi
∂ui
∂xj
+φ1
4πIb − Z
4π
IdΩ
(6.5) In the above equation for entropy generation, φ represents the irreversibility ratio defined as
φ0 = µ T0u20
k(∆T)2, φ1 = τ I0H T0
k(∆T)2 (6.6)
The values ofφ0 ≈10−6, φ1 ≈2.2 are obtained based on order of magnitude analysis at initial condition of T0 = 600 K, µ = 5.34×10−5, k = 0.07 with u20 ≈ 1.08 and τ = 0.20 are used for a ∆T = 720 K. Figure 6.15 represents the isosurface of en- tropy generation for partitions protruding from the top-bottom walls. Figures 6.15 (a) and (d) correspond to the local heat transfer irreversibility (HT I), Figs. 6.15 (b) and (e) show the fluid friction irreversibility (F F I) and Figs. 6.15 (c) and (f) express the radiative heat transfer irreversibility (RT I). The local variation of en- tropy generation in Fig. 6.15 reveals that the heat transfer (conduction, radiation)
is the dominant mechanism of irreversibilities. On the other hand, fluid friction irreversibility is insignificant in comparison to heat transfer as the strength of the convection currents are rather weak in a typical natural convection driven flow. The dominance of entropy generation by heat transfer is also recognized from the average and maximum values of entropy production as seen from Tables (6.5, 6.6). From the heat transfer irreversibilities in Figs. 6.15 (a) and (d) it is observed that the maximum entropy production by heat transfer is obtained near the base and head of hot and cold isothermal walls, respectively. On analyzing the average entropy production between the inline and offset configurations from Tables 6.5, 6.6 it is observed that the mean value of entropy generation by inline configuration exceeds that of an offset configuration of partition. The maximum value of heat transfer irreversibilities is higher for offset configuration of top-bottom and front-back par- titions. The offset arrangement of separating walls offers lesser obstruction to flow and radiation. The enhanced heat transfer due to secondary obstruction increases the temperature gradients near the isothermal walls. Higher temperature gradients are accountable for preeminent values of irreversibility by heat transfer. The signif- icance of entropy production by fluid friction is realized near the isothermal walls at X=0, 1 and adiabatic front and back walls at Z= 0, 1. However, the contribu- tion of viscous dissipation in the total entropy generation is almost negligible. The isosurface of entropy generation by radiation reveals the appearance of maximum entropy generation near the isothermal hot wall at X=0. Additionally, radiative entropy production also contributes along the top wall, posterior to the partition.
Figure 6.16 depicts the entropy generation for partitions along the front and back walls. The primary distribution of entropy production by convection, fluid friction and radiation remain same as observed previously in the case of partitions along the top and bottom walls. The maximum value of heat transfer irreversibilities is obtained near the cold isothermal wall for both inline and offset configuration. It is important to notice that these values are marginally higher than those obtained for partitions along the top and bottom walls. This study demonstrates that in- line configuration depicts higher values of entropy generation as compared to offset configuration for both top-bottom and front-back partitions. Maximum entropy production is obtained for inline configuration of top-bottom partitions.
6.4 Entropy Generation 157
(a) (b) (c)
(d) (e) (f)
Figure 6.15: Isosurface of local entropy generation in an inline and offset configura- tion of top-bottom partitions: (a, d) entropy generation due to heat transfer, (b, e) entropy generation due to fluid friction, (c, f) entropy generation due to radiative heat transfer at an optical thickness of τ = 0.20.
Table 6.5: Maximum and average values of entropy generation for inline and offset configuration of top-bottom partitions.
Average values Maximum values Entropy generation HT I F F I RT I HT I F F I RT I Inline Configuration 11.91 8.5×10−6 6.15 590 0.0004 13.8 Offset Configuration 7.16 6.0×10−6 3.16 680 0.0004 11.80
(a) (b) (c)
(d) (e) (f)
Figure 6.16: Isosurface of local entropy generation in an inline and offset configura- tion of front-back partitions: (a, d) entropy generation due to heat transfer, (b, e) entropy generation due to fluid friction, (c, f) entropy generation due to radiative heat transfer at an optical thickness of τ = 0.20.
Table 6.6: Maximum and average values of entropy generation for inline and offset configuration of front-back partitions.
Average values Maximum values Entropy generation HT I F F I RT I HT I F F I RT I Inline Configuration 8.92 8.5×10−6 3.41 606 0.0005 17.34 Offset Configuration 8.90 8.1×10−6 3.22 695 0.0005 15.93