Application of the similarity theory including variable property effects to a complex benchmark problem

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Abstract

An asymptotic method to account for variable property effects, recently described in this journal, is now applied to a complex benchmark geometry. It is a room which is ventilated by forced convection through inlet and outlet slit nozzles at the top and bottom of the side walls. Four heating elements standing on the ground floor add heat with constant heat flux density of varying strength. CFD solutions with the full coverage of all property temperature dependencies of air and SF6 are compared with asymptotic results (ACFD), applied for these fluids. ACFD results are given as systematic expansions with respect to a heat transfer parameter \({\varepsilon}\) which serves as perturbation parameter. First and second order asymptotic results of the Nußelt number at the surface of the heating elements are shown as well as temperature distributions along the adiabatic walls of the room. Special attention is given to the reference Nußelt numbers of zero order \({(\varepsilon=0)}\) which are those for constant properties only for pure forced convection.

Keywords

76M45 Basic methods in fluid mechanics Asymptotic methods singular perturbations 

Nomenclature

Dimensional quantities

Symbol

Name   Unit

ρ*

Density   (kg/m3)

\({c_p^\ast}\)

Heat capacity   (J/kg/K)

k*

Heat conductivity   (W/mK)

μ*

Dynamic viscosity   (kg/ms)

a*

Variable fluid property –

T*

Temperature   (K)

p*

Pressure   (Pa)

ΔT*

Temperature difference   (K)

L*

Characteristic length   (m)

\({\dot {q}^\ast}\)

Heat flux density   (W/m2)

Nondimensional quantities and groups

Symbol

Name

Nu

Nußelt number

Re

Reynolds number

Pr

Prandtl number

Gr

Grashof number

\({\varepsilon}\)

Temperature difference

Ka, Ka2, Kan

K-values, a = ρ, μ, k, c p

Aa, Aaa, Aab, Aa2

A-values, a = ρ, μ, k, c p

Indices

R

Reference state

cp

Constant properties

0B

Bousinnesq approximation

0i

Ideal gas

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References

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Hamburg University of TechnologyThermo-Fluid Dynamics (M-21)HamburgGermany

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