Abstract
Important results of a numerical study performed on combined conduction–mixed convection–surface radiation from a vertical channel equipped with three identical flush-mounted discrete heat sources in its left wall are provided here. The channel has walls of identical height with the spacing varied by varying its aspect ratio (AR). The cooling medium is air that is considered to be radiatively transparent. The heat generated in the channel gets conducted along its walls before getting dissipated by mixed convection and radiation. The governing equations for fluid flow and heat transfer are considered without boundary layer approximations and are transformed into vorticity–stream function form and are later normalized. The resulting equations are solved, along with relevant boundary conditions, making use of the finite volume method. The computer code written for the purpose is validated both for fluid flow and heat transfer results with those available in the literature. Detailed parametric studies have been performed and the effects of modified Richardson number, surface emissivity, thermal conductivity and AR on various pertinent results have been looked into. The significance of radiation in various regimes of mixed convection has been elucidated. The relative contributions of mixed convection and radiation in carrying the mandated cooling load have been thoroughly explored.
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Abbreviations
- AR:
-
Aspect ratio (L/W)
- Ar1, Ar2 :
-
Geometric ratios, W/t and W/Lh, respectively
- \({\bar{\text{C}}}_{\text{f}}\) :
-
Mean friction coefficient
- Fik :
-
View factor from the ith element to kth element of an enclosure
- Gr *w :
-
Modified Grashof number [(g β ΔTref W3)/ν 2f ]
- g:
-
Acceleration due to gravity (9.81 m/s2)
- Ji :
-
Radiosity of a given element i of the enclosure (W/m2)
- Ji′:
-
Non-dimensional radiosity of a given element i of the enclosure [Ji/(σ T 4∞ )]
- kf :
-
Thermal conductivity of air (W/m K)
- ks :
-
Thermal conductivity of channel wall as well as heat source (W/m K)
- L, Lh :
-
Heights of channel wall and heat source, respectively (m)
- M1 :
-
Grid number at the top end of the first heat source in the wall
- M2 :
-
Grid number at the bottom end of the second heat source in the wall
- M3 :
-
Grid number at the top end of the second heat source in the wall
- M4 :
-
Grid number at the bottom end of the third heat source in the wall
- M5 :
-
Total number of grids along the wall
- M, N:
-
Total number of grids in X and Y directions, respectively
- n:
-
Total number of elements of the enclosure
- NRF :
-
Radiation-flow interaction parameter [σ T 4∞ /(kf ΔTref/W)]
- Pew :
-
Peclet number based on the width of the channel (RewPr) or (u∞W/α)
- Prf :
-
Prandtl number of air (νf/α)
- qcond,x,in :
-
Conduction heat transfer into an element along the wall (W)
- qcond,x,out :
-
Conduction heat transfer out of an element along the wall (W)
- qconv :
-
Convection heat transfer from an element of the wall (W)
- qgen :
-
Heat generated in an element of the wall (W)
- qrad :
-
Radiation heat transfer from an element of the wall (W)
- qv :
-
Rate of volumetric heat generation in each discrete heat source (W/m3)
- Rew :
-
Reynolds number based on the width of the channel (u∞W/νf)
- \(\hbox{Ri}_{\text{w}}^{\ast}\) :
-
Modified Richardson number based on the width of the channel (gβΔTrefW/u 2∞ ) or (Grw*/Re 2w )
- t:
-
Thickness of channel walls (m)
- T:
-
Local temperature in the computational domain (K or °C)
- Tmax :
-
Maximum temperature in the computational domain (K or °C)
- T∞ :
-
Free stream temperature of air (K or °C)
- u, v:
-
Vertical and horizontal components of velocity, respectively (m/s)
- u∞ :
-
Free stream velocity of air (m/s)
- U:
-
Non-dimensional vertical velocity of air (u/u∞) or (∂ψ/∂Y)
- V:
-
Non-dimensional horizontal velocity of air (v/u∞) or (−∂ψ/∂X)
- W:
-
Width or spacing of the channel (m)
- x, y:
-
Vertical and horizontal distances, respectively (m)
- X, Y:
-
Non-dimensional vertical and horizontal distances, x/W and y/W, respectively
- α:
-
Thermal diffusivity of air (m2/s)
- β:
-
Isobaric cubic expansivity of air [−(1/ρ)(∂ρ/∂T)p] (K−1)
- γ:
-
Thermal conductance parameter [kfW/(kst)]
- ε:
-
Surface emissivity of the walls of the channel
- θ:
-
Non-dimensional local temperature [(T − T∞)/ΔTref]
- θav :
-
Non-dimensional average temperature
- θmax :
-
Non-dimensional maximum temperature
- νf :
-
Kinematic viscosity of air (m2/s)
- σ:
-
Stefan–Boltzmann constant (5.6697 × 10−8 W/m2 K4)
- ψ:
-
Non-dimensional stream function [ψ′/(u∞W)]
- ψ′:
-
Stream function (m2/s)
- ω:
-
Non-dimensional vorticity (ω′W/u∞)
- ω′:
-
Vorticity (s−1)
- Δxhs :
-
Height of the element in the heat source portion of the wall
- ΔXhs :
-
Non-dimensional height of the element in the heat source portion of the wall
- ΔTref :
-
Modified reference temperature difference (qvLht/ks) (K or °C)
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Londhe, S.D., Gururaja Rao, C. Interaction of surface radiation with conjugate mixed convection from a vertical channel with multiple discrete heat sources. Heat Mass Transfer 50, 1275–1290 (2014). https://doi.org/10.1007/s00231-014-1333-1
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DOI: https://doi.org/10.1007/s00231-014-1333-1