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Mixed Convection with Conduction and Surface Radiation from a Vertical Channel with Discrete Heating

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Abstract

A numerical investigation into fluid flow and heat transfer for the geometry of a vertical parallel plate channel subjected to conjugate mixed convection with radiation is attempted here. The channel considered has three identical flush-mounted discrete heat sources in its left wall, while the right wall that does not contain any heat source acts as a sink. Air, assumed to be a radiatively non-participating and having constant thermophysical properties subject to the Boussinesq approximation, is the cooling agent. The heat generated in the left wall gets conducted along it and is later dissipated by mixed convection and radiation. The governing equations, considered in their full strength sans the boundary layer approximations, are converted into vorticity-stream function form and are then normalized. These equations along with pertinent boundary conditions are solved through finite volume method coupled with Gauss-Seidel iterative technique. The effects of modified Richardson number, surface emissivity, thermal conductivity and aspect ratio on local temperature distribution along the channel, maximum channel temperature and relative contributions of mixed convection and radiation have been thoroughly studied. The prominence of radiation in the present problem has been highlighted.

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Abbreviations

AR :

Aspect ratio, L/W

A r1 , A r2 :

Geometric ratios, W/t and W/Lh, respectively

C¯ f :

Mean friction coefficient

F ik :

View factor from the ith element to kth element of an enclosure

Gr * w :

Modified Grashof number, [(gβΔTrefW3)/ν 2f ]

g :

Acceleration due to gravity, 9.81 m/s2

J i :

Radiosity of a given element i of the enclosure, W/m2

J i ′ :

Non-dimensional radiosity of a given element i of the enclosure, [Ji/(σT 4∞ )]

k f :

Thermal conductivity of air, W/m K

k s :

Thermal conductivity of channel wall as well as heat source, W/m K

L, L h :

Heights of channel wall and heat source, respectively, m

M 1 :

Grid number at the top end of the first heat source in the wall

M 2 :

Grid number at the bottom end of the second heat source in the wall

M 3 :

Grid number at the top end of the second heat source in the wall

M 4 :

Grid number at the bottom end of the third heat source in the wall

M 5 :

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

N RF :

Radiation-flow interaction parameter, [σT 4∞ /(kfΔTref/W)]

Pe w :

Peclet number based on the width of the channel, [RewPr] or [u∞W/α]

Pr f :

Prandtl number of air, [νf/α]

q cond,x,in :

Conduction heat transfer into an element along the wall, W

q cond,x,out :

Conduction heat transfer out of an element along the wall, W

q conv :

Convection heat transfer from an element of the wall, W

q gen :

Heat generated in an element of the wall, W

q rad :

Radiation heat transfer from an element of the wall, W

q v :

Rate of volumetric heat generation in each discrete heat source, W/m3

Re w :

Reynolds number based on the width of the channel, [u∞W/νf]

Ri * w :

Modified Richardson number based on the width of the channel, [gβΔTrefW/u 2∞ ] or [Gr *w /Re 2w ]

T :

Local temperature in the computational domain, K or °C

T max :

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]

θ av :

Non-dimensional average temperature, \( \left[ {\frac{1}{\text{AR}}\int_{0}^{\text{AR}} {{{\uptheta}}({\text{X}})} {\text{dX}}} \right] \)

θ max :

Non-dimensional maximum temperature

ν f :

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

Δx hs :

Height of the element in the heat source portion of the wall

ΔX hs :

Non-dimensional height of the element in the heat source portion of the wall

ΔT ref :

Modified reference temperature difference, [qvLht/ks], K or °C

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology, Warangal, 506 004, AP, India

    S. D. Londhe & C. G. Rao

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  1. S. D. Londhe
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Correspondence to C. G. Rao.

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Londhe, S.D., Rao, C.G. Mixed Convection with Conduction and Surface Radiation from a Vertical Channel with Discrete Heating. J. Inst. Eng. India Ser. C 94, 213–223 (2013). https://doi.org/10.1007/s40032-013-0076-y

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