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Conjugate mixed convection with surface radiation from a vertical electronic board with multiple discrete heat sources

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Abstract

The problem of combined conduction-mixed convection-surface radiation from a vertical electronic board provided with three identical flush-mounted discrete heat sources is solved numerically. The cooling medium is air that is considered to be radiatively transparent. The governing equations for fluid flow and heat transfer are converted from primitive variable form to stream function-vorticity formulation. The equations, thus obtained, are normalised and then are converted into algebraic form using a finite volume based finite difference method. The resulting algebraic equations are then solved using Gauss–Seidel iterative method. An optimum grid system comprising 151 grids along the board and 111 grids across the board is chosen. The effects of various parameters, such as modified Richardson number, surface emissivity and thermal conductivity on temperature distribution along the board, maximum board temperature and relative contributions of mixed convection and radiation to heat dissipation are studied in detail. Further, the contributions of free and forced convection components of mixed convection to board temperature distribution and peak board temperature are brought out. The exclusive roles played by surface radiation and buoyancy in the present problem are clearly elucidated.

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Abbreviations

A r1, A r2 :

geometric ratios, L/t and L/L h, respectively

g :

acceleration due to gravity (9.81 m/s2)

Gr *L :

modified Grashof number, gβΔT ref L 3/ν 2f

k s :

thermal conductivity of the board material and heat source (W/m K)

k f :

thermal conductivity of air (W/m K)

L, t :

height and thickness of the board, respectively (m)

L h :

height of each of the discrete heat sources (m)

M, N :

number of grids in horizontal and vertical directions, respectively

N 1 :

number of grids up to the trailing edge of the board

N 2 :

total number of grids up to the end of the first heat source

N 3 :

total number of grids up to the start of the second heat source

N 4 :

total number of grids up to the end of the second heat source

N 5 :

total number of grids up to the start of the third heat source

N RF :

radiation–flow interaction parameter, \( \raise0.7ex\hbox{${\sigma T_{\infty } ^{4} }$} \!\mathord{\left/ {\vphantom {{\sigma T_{\infty } ^{4} } {{\left[ {{k_{{\text{f}}} \Updelta T_{{{\text{ref}}}} } \mathord{\left/ {\vphantom {{k_{{\text{f}}} \Updelta T_{{{\text{ref}}}} } {L}}} \right. \kern-\nulldelimiterspace} {L}} \right]}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\left[ {{k_{{\text{f}}} \Updelta T_{{{\text{ref}}}} } \mathord{\left/ {\vphantom {{k_{{\text{f}}} \Updelta T_{{{\text{ref}}}} } {L}}} \right. \kern-\nulldelimiterspace} {L}} \right]}}$} \)

P :

pressure at any location in the computational domain (Pa)

Pe L :

Peclet number, Re L Pr

Pr :

Prandtl number of air

q v :

volumetric heat generation in each of the discrete heat source (W/m3)

Re L :

Reynolds number, u ∞ L/ν f

Ri *L :

modified Richardson number, gβΔT ref L/u 2∞ or Gr *L /Re 2L

T :

temperature at any location in the computational domain (K or °C)

T max :

maximum temperature in the board (K or °C)

T ∞ :

free stream temperature of air (K or °C)

u, v :

vertical and horizontal components of velocity of air (m/s)

u ∞ :

velocity of air (m/s)

U :

non-dimensional vertical velocity components of air, u/u ∞ or ∂ψ/∂Y

V :

non-dimensional horizontal velocity components of air, v/u ∞ or −∂ψ/∂X

W :

width of the computational domain (m)

x :

vertical distance (m)

X :

non-dimensional vertical distance, x/L

y :

horizontal distance (m)

Y :

non-dimensional horizontal distance, y/L

α :

thermal diffusivity of air (m2/s)

β :

isobaric cubic expansivity of air, \( - \frac{1} {\rho }{\left( {\frac{{\partial \rho }} {{\partial T}}} \right)}_{{\text{p}}} \) (K−1)

γ :

thermal conductance parameter, k f L/k s t

δ c :

convergence criterion, in percentage, \( {\left| {{{\left( {\xi _{{{\text{new}}}} - \xi _{{{\text{old}}}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {\xi _{{{\text{new}}}} - \xi _{{{\text{old}}}} } \right)}} {\xi _{{{\text{new}}}} }}} \right. \kern-\nulldelimiterspace} {\xi _{{{\text{new}}}} }} \right|} \times 100\% \)

ΔT ref :

modified reference temperature difference, q v L h t/k s (K or °C)

Δx f :

height of the board element chosen for energy balance in non-heat source portion (m)

Δx h :

height of the board element chosen for energy balance in the heat source portion (m)

ε :

surface emissivity

θ :

non-dimensional temperature, (T − T ∞)/ΔT ref

θ max :

non-dimensional maximum board temperature

ν f :

kinematic viscosity (m2/s)

ξ :

any dependent variable (ψ, ω or θ) over which convergence is being tested for

ρ, ρ ∞ :

local and characteristic values of fluid density, respectively (kg/m3)

σ :

Stefan–Boltzmann constant (5.6697 × 10−8 W/m2 K4)

ψ :

non-dimensional stream function, ψ′/u ∞ L

ψ′:

stream function (m2/s)

ω :

non-dimensional vorticity, ω′L/u ∞

ω′:

vorticity (s−1)

cond, x, in:

conduction heat transfer into an element along the board

cond, x, out:

conduction heat transfer out of an element along the board

conv:

convection heat transfer from an element

new, old:

values of any variable from the current and previous iterations, respectively

rad:

heat transfer by surface radiation from an element

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Authors and Affiliations

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

    S. M. Sawant & C. Gururaja Rao

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  1. S. M. Sawant
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  2. C. Gururaja Rao
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Correspondence to C. Gururaja Rao.

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Sawant, S.M., Gururaja Rao, C. Conjugate mixed convection with surface radiation from a vertical electronic board with multiple discrete heat sources. Heat Mass Transfer 44, 1485–1495 (2008). https://doi.org/10.1007/s00231-008-0395-3

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