# Conjugate heat transfer in square enclosures

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## Abstract

Building elements represented by square vertical enclosures encircled with finite walls or with centered solid body, could maintain the equivalent fluid volumes through the volume ratio scale. Present work aims to investigate the fluid flow and heat transfer in these two building elements. Complete two-dimensional numerical simulation of the conjugate heat conduction and natural convection occurring in both enclosures is carried out. An analytical expression for the minimum size of the inserted body at which the body begins to suppress the natural convection flow is proposed and validated by the numerical results. The fluid flow and heat transfer characteristics are analyzed through the streamlines, heatlines, and total heat transfer rates across both enclosures. Results reveal that heat transfer rates across both enclosures are complex functions of the volume ratio scale, Rayleigh number, and the relative thermal conductivity.

## Keywords

Heat Transfer Nusselt Number Natural Convection Rayleigh Number Heat Transfer Rate## List of Symbols

*C*_{P}isobaric specific heat (J/K kg)

*g*gravitational acceleration (m/s

^{2})*H*heat function

*k*thermal conductivity (W/m K)

*K*thermal conductivity ratio

- Kr
thermal conductivity ratio of solid to fluid

*L*length of square enclosure (m)

*n*normal direction

*Nu*local Nusselt number

*p*pressure (N/m

^{2})*P*dimensionless pressure

*Pr*Prandtl number, ν/α

*Q*total heat transfer rate (W/m

^{2})*Ra*Rayleigh number,

*g*β(*t*_{h}−*t*_{c})*L*^{3}/να*S*distance (m)

*t*temperature (K)

*T*dimensionless temperature

*u*velocity component in

*x*directions (m/s)*v*velocity component in

*y*directions (m/s)*U*dimensionless velocity component in

*X**V*dimensionless velocity component in

*Y**W*length of inner cavity or body (m)

*x*,*y*Cartesian coordinates (m)

*X*,*Y*dimensionless Cartesian coordinates

## Greek Symbols

- α
thermal diffusivity (m

^{2}/s)- β
volume expansion coefficient (1/K)

- θ
volume ratio scale (°)

- ξ
void fraction of enclosure A

- η
solid-to-fluid volume ratio of enclosure B

- ν
kinematics viscosity (m

^{2}/s)- ρ
density (kg/m

^{3})- Ψ
stream function

## Subscripts

- a
average value

- A
enclosure encircled with finite walls

- B
enclosure inserted with solid body

- c
cold wall

- f
value of the fluid domain

- h
hot wall

- max
maximum

- min
minimum

- s
value of the solid domain

## Superscripts

- *
dimensionless

## Notes

### Acknowledgements

The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (No. 50578059). The authors are also grateful to the anonymous referees who provided detailed and constructive comments.

## References

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