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Combined Laminar Mixed Convection and Surface Radiation using Asymptotic Computational Fluid Dynamics (ACFD)

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Abstract

This paper reports the use of the technique of combining asymptotics with computational fluid dynamics (CFD), known as asymptotic computational fluid dynamics (ACFD), to handle the problem of combined laminar mixed convection and surface radiation from a two dimensional, differentially heated lid driven cavity. The fluid under consideration is air, which is radiatively transparent, and all the walls are assumed to be gray and diffuse and having the same hemispherical, total emissivity (ɛ). The computations have been performed on FLUENT 6.2. The full radiation problem (i.e. all the walls are radiatively black corresponding to ɛ = 1) is first taken up and the method of “perturbing and blending” is used wherein, first, limiting solutions of natural and forced convection are perturbed, to obtain correlations for the weighted average convective Nusselt numbers for the full radiation case. These correlations are then blended suitably in order to obtain a composite correlation for the weighted average convective Nusselt number that is valid for the entire mixed convection range, i.e., 0 ≤ Ri ≤ ∞. This correlation is then expanded in terms of ɛ to obtain an expression for the average convective Nusselt number that is valid for any ɛ in the range 0 ≤ ɛ ≤ 1. In so far as radiation heat transfer is concerned, using asymptotic arguments, a new weighted average radiation Nusselt number is defined such that this quantity can be expanded just in terms of ɛ. Hence, by the use of ACFD, the number of solutions required to obtain reasonably accurate correlations for both the convective and radiative heat transfer rates and hence the total heat transfer rate (Nu total = Nu C + Nu R), is substantially reduced. More importantly, the correlations for convection and radiation are asymptotically correct at their ends. The effect of secondary variables like aspect ratio and the case of unequal wall emissivities can also be included without significant additional effort.

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Abbreviations

a, b :

constants in the correlations for Nu C

C 1, C 2 :

constants in the correlations for\({\hat{Nu}}_{\rm C}\)

d :

spacing (or height) of the cavity, m

F ij :

viewfactor from the ith element to the jth element

g :

acceleration due to gravity, 9.81 m/s2

g i :

irradiation on the ith element, W/m2

G i :

dimensionless elemental irradiation, g i /σ T 4H

Gr :

Grashof number based on the spacing of the cavity, g βΔTd 3/ν2

j i :

radiosity of the ith element, W/m2

J i :

dimensionless elemental radiosity, j i /σ T 4H

k :

thermal conductivity of the fluid, W/mK

L :

length of the vertical flat plate, m

m, n :

exponents in the correlation for Nu C

N RC :

radiation convection interaction parameter,\(\frac{{\sigma T_{\rm H} ^{4} d}}{{k(T_{\rm H} - T_{\rm C})}}\)

Nu C :

average convection Nusselt number based on the cavity spacing, Q C/(k ΔT)

Nu R :

average radiation Nusselt number based on the cavity spacing, Q R/(k ΔT)

\({\hat{Nu}}_{\rm C}\) :

weighted average convection Nusselt number defined in Eq. 15

\({\hat{Nu}}_{\rm R}\) :

weighted average radiation Nusselt number defined in Eq. 28

Nu total :

total Nusselt number (Nu C + Nu R)

p :

pressure, Pa

P :

dimensionless pressure, p/ρ u 2∞

Pr :

Prandtl number, ν/α

q :

exponent in the composite correlation for\({\hat{Nu}}_{\rm C}\)

q conv :

convective heat flux, W/m2

q rad :

radiative heat flux, W/m2

Q C :

convective heat transfer rate from the cavity, W/m

Q R :

radiative heat transfer rate from the cavity, W/m

Re :

Reynolds number, u ∞ d/ν

Ri :

Richardson number based on the spacing of the cavity, Gr/Re 2

T C :

temperature of the cold wall, K

T H :

temperature of the hot wall of the cavity, K

T R :

temperature ratio, T C/T H

u :

horizontal velocity, m/s

u ∞ :

velocity of the top (driven) wall, m/s

U :

dimensionless horizontal velocity, u/u ∞

v :

vertical velocity, m/s

V :

dimensionless vertical velocity, v/u ∞

x :

horizontal coordinate, m

X :

dimensionless horizontal coordinate, x/d

y :

vertical coordinate, m

Y :

dimensionless vertical coordinate, y/d

α:

thermal diffusivity, m2/s or absorptivity of the surface

β:

isobaric cubic expansivity, 1/K

ɛ:

hemispherical, total emissivity

ΔT :

temperature difference, (T H − T C)

ν:

kinematic viscosity, m2/s

ρ:

density, kg/m3

σ:

Stefan–Boltzmann constant, 5.67 × 10−8 W/m2 K4

θ:

dimensionless temperature, (T − T C)/(T H − T C)

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Acknowledgments

The first author would like to thank the Alexander Von Humboldt foundation for supporting his research stay at TUHH, Hamburg, Germany.

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Authors and Affiliations

  1. Department of Applied Thermodynamics, Hamburg University of Technology, Denickestrasse 17, 21073, Hamburg, Germany

    C. Balaji, M. Hölling & H. Herwig

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  1. C. Balaji
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  2. M. Hölling
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  3. H. Herwig
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Correspondence to C. Balaji.

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Balaji, C., Hölling, M. & Herwig, H. Combined Laminar Mixed Convection and Surface Radiation using Asymptotic Computational Fluid Dynamics (ACFD). Heat Mass Transfer 43, 567–577 (2007). https://doi.org/10.1007/s00231-006-0145-3

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