Handbook of Thermal Science and Engineering pp 527-602 | Cite as

# Free Convection: External Surface

## Abstract

Some representative recent basic studies involving external free convective heat transfer in situations that are of practical interest are discussed in this chapter. Most of the results considered have been obtained numerically, and a very brief discussion of the methodology used to generate these results is presented. Attention has here first been given to the heat transfer rate from narrow vertical plane surfaces with the effect of the width-to-height ratio of the surface on the heat transfer rate in particular being discussed. Heat transfer from horizontally and vertically spaced pairs of narrow plates has also been considered. Consideration has then been given to the heat transfer rate from horizontal heated surfaces of complex shape, to adjacent pairs of horizontal heated surfaces, and to heat transfer from two-sided circular horizontal surfaces. The effect of a covering surface on the heat transfer from a horizontal heated surface has also been considered. The heat transfer rate from bodies with wavy surfaces has been considered next with attention being given to situations involving two-dimensional flow over vertical and horizontal surfaces and to the effect of surface waviness on the heat transfer rate from cylindrical bodies. The heat transfer from relatively short vertical circular and square cylinders with exposed top surfaces is also considered. Lastly, brief attention is given to external free convective heat transfer to nanofluids.

## Nomenclature

*A*Total surface area

*A*_{bottom}Area of bottom surface

*A*_{c}Area of cylindrical outer surface of cylinder

*A*_{top}Area of top surface

*A*_{t}Area of top surface of cylinder

*A*_{total}Total surface area

*AR*Aspect ratio

*a*Mean side length

*c*_{pf}Specific heat of fluid in which nanoparticles are placed

*c*_{pnf}Specific heat of nanofluid

*c*_{ps}Specific heat of nanoparticles

*D*_{i}Dimensionless diameter of inner adiabatic section of surface

*D*Diameter of circular surface

*d*Heated surface diameter

*d*_{i}Diameter of inner adiabatic section of surface

*G*Gap between heated surfaces

*Gap*Gap between vertically spaced heated surfaces

*g*Gravitational acceleration

*H*_{Gap}Dimensionless size of gap between surfaces

*H*Height of surface and dimensionless recess depth,

*h*/*d**h*Recess depth

*k*Thermal conductivity

*h*Recess depth

*L*l/d

*L*_{out}Dimensionless side length of rectangular adiabatic covering surface,

*L*_{out}*= l*_{out}*/w**l*Reference length and cylinder height

*l*_{out}Side lengths of rectangular adiabatic covering surface

*m*Characteristic length scale of surface

*Nu*Nusselt number

*Nu*_{0}Reference Nusselt number

*Nu*_{a}Mean Nusselt number based on the mean side length,

*a*, of rectangular heated surface*Nu*_{bottom}Mean Nusselt number averaged over bottom surface

*Nu*_{c}Mean Nusselt number averaged over vertical side surface of cylinder

*Nu*_{m}Mean Nusselt number based on

*m**Nu*_{r}Mean Nusselt number based on

*r**Nu*_{rbottom}Mean Nusselt number based on

*r*averaged over bottom surface*Nu*_{rtop}Mean Nusselt number based on

*r*averaged over top surface*Nu*_{t}Mean Nusselt number averaged over horizontal top surface of cylinder

*Nu*_{top}Mean Nusselt number averaged over top surface

*Nu*_{total}Mean Nusselt number averaged over entire surface

*n*Coordinate normal to surface or number of surface waves

*P*Total perimeter of heated surface

*Pr*Prandtl number

- \( {\overline{Q}}^{\prime } \)
Total mean heat transfer rate

- \( {{\overline{Q}}^{\prime}}_{\mathrm{bottom}} \)
Total mean heat transfer rate from bottom surface

- \( {{\overline{Q}}^{\prime}}_{\mathrm{top}} \)
Total mean heat transfer rate from top surface

*q*^{′}Local heat transfer rate per unit area

- \( {\overline{q}}^{\prime } \)
Mean heat transfer rate per unit area from entire cylinder

- \( \overline{q_c^{\prime }} \)
Mean heat transfer rate per unit area from cylindrical outer surface of cylinder

- \( \overline{q_s^{\prime }} \)
Mean heat transfer rate per unit area from vertical side surface of square cylinder

- \( \overline{q_t^{\prime }} \)
Mean heat transfer rate per unit area from top surface of cylinder

- \( \overline{q_w^{\prime }} \)
Mean heat transfer rate per unit area

*R*Dimensionless cylinder radius,

*r*/*l**Ra*Rayleigh number

*Ra*_{a}Rayleigh number based on the mean side length,

*a*, of rectangular heated surface*Ra*_{m}Rayleigh number based on

*m**Ra*_{r}Rayleigh number based on

*r**Ra*^{*}Heat Flux Rayleigh number

*r*Characteristic length scale of surface and the radius of cylinder

*r*_{bottom}Characteristic length scale of bottom surface

*r*_{top}Characteristic length scale of top surface

*s*Arm size of +- shaped and I-shaped heated surfaces

*S*Dimensionless distance between sides of square heated surfaces

*T*Temperature

*T*_{f}Fluid temperature far from surface

*T*_{w}Wall temperature

*T*_{0}Reference fluid temperature

*V*_{Gap}Dimensionless size of gap between surfaces,

*Gap*/*h**W*Dimensionless surface width,

*w*/*h**W*_{out}The dimensionless side lengths of the rectangular surrounding adiabatic surface

*W*_{out}=*w*_{out}/*w**w*Surface width

*w*_{out}Side lengths of the rectangular surrounding adiabatic surface

## Greek Symbols

*α*Thermal diffusivity

*β*Bulk coefficient of thermal expansion

*δ*Measure of the boundary layer thickness

*ϕ*Nanoparticle volumetric fraction

*ν*Kinematic viscosity

*ξ*1/(

*R Ra*^{0.25}) or 1/(*W Ra*^{0.25})*φ*Angle of inclination

*μ*Viscosity

*ρ*Density

*ρ*_{0}Reference density value

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