Effect of surface radiation on RBC in cavities heated from below

1462 M.A. Gad, C. Balaji / International Communications in Heat and Mass Transfer 37 (2010) 1459–1464
Fig. 2. Change in Nusselt number with respect to Rayleigh number for a square cavity.
variation <strong>of</strong> convection and <strong>radiation</strong> Nusselt numbers with Rayleigh
number for different values <strong>of</strong> the emissivity <strong>of</strong> side wall. Fig. 4 (a)
shows that <strong>radiation</strong> results in a reduction in the convection
component. With an increase in the emissivity <strong>of</strong> the sidewalls, the
convection component decreases except for Rayleigh number in the
range 10,000–50,000. However, in general <strong>surface</strong> <strong>radiation</strong> leads to
an overall increase in the heat transfer across the cavity. Three
different regimes are observed in the range <strong>of</strong> the Rayleigh number
(Ra) from 5×10 3 to 10 5 . The emissivity <strong>of</strong> the side walls has a
negligible effect on the convective heat transfer in the range <strong>of</strong>
Rayleigh number 4000bRab10,000. It is observed that flow structure
changes from unicellular to bicellular for 10,000bRab50,000 and a
corresponding increase in the convective heat transfer occurs for a
higher emissivity <strong>of</strong> the horizontal sidewall. For RaN50,000, the
convection is suppressed significantly with an increase in the
emissivity <strong>of</strong> the side walls. For the case where <strong>surface</strong> <strong>radiation</strong> is
not considered, unicellular solutions are found to exist. So, Rayleigh–
Benard convection has a tendency to produce unicellular flow
structure. When <strong>surface</strong> <strong>radiation</strong> is considered, for Rayleigh number
10,000 to 50,000 a bicellular flow structure is obtained. Figs. 5 and 6
show the streamlines and isotherms for Rayleigh number 30,000 and
90,000 when ε=0.85 (the maximum value <strong>of</strong> the stream function is
mentioned at the appropriate streamline in the figure).
4.2.2. Heat transfer for higher aspect ratio cavities
Fig. 7 (a) and (b) show the variation <strong>of</strong> the convection and
<strong>radiation</strong> Nusselt numbers for a cavity <strong>of</strong> aspect ratio 2, for different
Rayleigh number, Ra
4400
4100
3800
3500
3200
2900
2600
2300
2000
1700 1 2 3 4 5 6 7 8 9 10
Aspect ratio
= 0
= 0.1
= 0.5
= 0.85
Fig. 3. <strong>Effect</strong> <strong>of</strong> the emissivity <strong>of</strong> the sidewalls on the onset <strong>of</strong> convection for cavities <strong>of</strong>
different aspect ratios.
Convection Nusselt number, NuC
b
Radiation Nusselt number, NuR
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Rayleigh Number, Ra x10-4 5.00
4.75
4.50
4.25
4.00
3.75
3.50
0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
Rayleigh Number, Ra x10-4 Rayleigh numbers and emissivities <strong>of</strong> the sidewalls. For the case <strong>of</strong> a
cavity with AR=2, only bicellular solution is found to exist
irrespective <strong>of</strong> the emissivity <strong>of</strong> the sidewalls. The observations
made in the case <strong>of</strong> bicellular flow structure for a square cavity are
also valid for this case i.e. an increase in the convective heat transfer
and a decrease in the radiative heat transfer with an increase in the
emissivity <strong>of</strong> the sidewalls. Fig. 8 (a) and (b) show the stream function
and isotherms respectively when the emissivity <strong>of</strong> the sidewall is 0.85
and the Rayleigh number is 75,000. For the case <strong>of</strong> aspect ratio 3, for
the pure convection case, tri-cellular flow structure is found to exist
while for the case where <strong>surface</strong> <strong>radiation</strong> is considered for Rayleigh
number greater than 10,000, tetra-cellular flow structure is found to
= 0
= 0.10
= 0.50
= 0.85
= 0.10
= 0.50
= 0.85
Fig. 4. Variation <strong>of</strong> (a) convection and (b) <strong>radiation</strong> Nusselt number with Rayleigh
number for different values <strong>of</strong> the emissivity <strong>of</strong> the horizontal side walls (AR=1).
1.5 x10 -4
303 K
315 K
Fig. 5. (a) Streamlines and (b) isotherms for Ra =30,000 and ε=0.85.