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Thermal and dynamic optimization of the convective flow in Trombe Wall shaped channels by numerical investigation

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Abstract

Two-dimensional, laminar, transitional and turbulent simulations were obtained by solving the fully-elliptic governing equations of the motion established by natural convection in channels, with Trombe Wall configuration, for different geometrical parameters and symmetrical heating. In transitional and turbulent cases, the low-Re k−ω turbulence model has been employed. To validate the numerical results, some comparisons with experimental results taken from literature have been carried out. Numerical results for the average Nusselt number and the non-dimensional induced mass-flow rate have been obtained for a wide and not yet covered range of the Rayleigh number varying from 105 to 1012. Correlations for the thermal and the mass-flow optimum wall-to-wall spacing have been presented. Finally, additional configurations including discrete heat sources have been studied, in order to obtain thermal and dynamic improvements. These intermediate devices were tested as turbulence generators, in the transitional range of Rayleigh numbers.

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Abbreviations

A1, A2:

Bottom and top opening heights, respectively (Fig. 1) (m)

A :

A1 or A2

b :

Wall-to-wall spacing of the vertical channel (m) (Fig. 1)

c p :

Specific heat at constant pressure (J kg−1K−1)

E :

Thickness of the Trombe Wall (Fig. 1) (m)

\(f_{\beta^*},\;f_{\beta}\) :

Turbulent factors depending on ξ k and ξω, Eqs. 10 and 13, respectively

g :

Gravitational acceleration (m−2)

Gr H :

Grashof number for isothermal cases, gβ(T w−T ∞) H 3/ν2

Gr H :

Grashof number for heat flux cases, gβq H 4/ν2κ

h x :

Local heat transfer coefficient, −κ(∂T/∂n)w/(T w−T ∞) (W m−2 K−1)

I :

Turbulence intensity, Eq. 14

k :

Turbulent kinetic energy, Eq. 6 (m2 s−2)

H :

Height of the wall 1 (Fig. 1) (m)

L :

Length of the room (Fig. 1) (m)

M :

Non-dimensional mass-flow rate (m/ρν)

m :

Two-dimensional mass-flow rate (kg m−1s−1)

n :

Coordinate perpendicular to wall (m)

Nu H :

Average Nusselt number based on H, isothermal cases, Eq. 17

Nu H :

Average Nusselt number based on H, heat flux cases, Eq. 18

Nu x :

Local Nusselt number, h x H/κ

P :

Average reduced pressure (N m−2)

Pr :

Prandtl number, μc p/κ

q :

Wall heat flux (W m−2)

R β, R k , R ω :

Constants of the turbulence model

Ra H :

Rayleigh number based on H, (Gr H ) (Pr)

Re t :

Turbulent Reynolds number, k/νω

S ij :

Mean-strain tensor (s−1)

T, T′:

Average and turbulent temperatures, respectively (K)

\(-\overline{T^{\prime}u_{j}}\) :

Average turbulent heat flux (K m s−1)

U j , u j :

Average and turbulent components of velocity, respectively (m s−1)

\(-\overline{u_{i}u_{j}}\) :

Turbulent stress (m2 s−2)

u τ :

Friction velocity, u τ = (τw/ρ)1/2 (m s−1)

x, y :

Cartesian coordinates in the vertical and horizontal directions (m)

y + :

ρy 1 u τ/μ, with y 1 the distance between the wall and the first grid point

α, β*:

Turbulent damping functions, Eqs. 12 and 11, respectively

\(\alpha_0^\ast,\) \(\alpha_0^\ast, \alpha_{\infty}, \alpha_0\) :

Constants of the turbulence model

β:

Coefficient of thermal expansion, 1/T ∞ (K−1)

δ ij :

Krönecker delta

κ:

Thermal conductivity (W m−1 K−1)

Λ:

Comparison thermal parameter, Eq. 25

μ:

Viscosity (kg m−1 s−1)

ν:

Kinematic viscosity, μ/ρ (m2s−1)

ξ k , ξω :

Parameters of the turbulence model, Eqs. 10 and 13, respectively

ρ:

Density (kg m−3)

σ k , σω :

Constants of the turbulence model

τw :

Wall shear stress (N m−2)

Ψ:

Comparison mass-flow rate parameter, Eq. 25

Ω ij :

Mean-rotation tensor (s−1)

ω:

Specific dissipation rate of k (or turbulent frequency) (s−1)

1:

Outer wall (corresponding to glazing)

2:

Inner wall (corresponding to masonry wall)

max:

Maximum

opt:

Optimum

ref:

Reference mesh

t :

Turbulent

w:

Wall

∞:

Ambient or reference conditions

–:

Averaged value

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Acknowledgments

This research was supported by the ‘Dirección General de Investigación’ of ’Ministerio de Educación y Ciencia’ of Spanish Government, through DPI 2003-02719 Project. The computations were carried out on a HP-Compaq HPC160 platform, in the ’Servicio de Apoyo a la Investigación Tecnológica’, SAIT, of the Polythecnic University of Cartagena.

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  1. Dpto. Ingeniería Térmica y de Fluidos, Universidad Politécnica de Cartagena, Doctor Fleming s/n, 30202, Cartagena, Spain

    Blas Zamora & Antonio S. Kaiser

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  1. Blas Zamora
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  2. Antonio S. Kaiser
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Correspondence to Blas Zamora.

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Zamora, B., Kaiser, A.S. Thermal and dynamic optimization of the convective flow in Trombe Wall shaped channels by numerical investigation. Heat Mass Transfer 45, 1393–1407 (2009). https://doi.org/10.1007/s00231-009-0509-6

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