# Mixed Convection with Thermal Radiation in a Vertical Pipe with Partially Heated or Cooled Wall

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## Abstract

The effect of partial heating/cooling of the wall on the mixed convection with thermal radiation in incompressible laminar pipe flow has been investigated. The gas is assumed to be gray, emitting and absorbing with constant thermophysical properties except the density variation in the buoyancy term. The partial heating/cooling of the wall has significant effect on the Nusselt number. The radiation parameter increases the heat transfer, but reduces the effect of buoyancy. The heat transfer also increases with the optical thickness until a certain value, beyond which it decreases.

## Keywords

Mixed convection Thermal radiation Vertical pipe Partially heated or cooled wall## List of Symbols

*a*Radius of the pipe

*e*_{w}Wall emissivity

*g*Acceleration due to gravity

*G*Dimensionless total irradiation

*G*^{*}Total irradiation

- Gr
Grashof number

*k*Thermal conductivity

- Nu
_{conv} Local Nusselt number due to convection

- Nu
_{rad} Local Nusselt number due to radiation

- Nu
_{T} Local total Nusselt number

*p*Dimensionless pressure

*p*^{*}Pressure

- Pr
Prandtl number

- \( q_{\text{w}}^{*} \)
Wall heat flux

*r*Dimensionless radial distance

*r*^{*}Radial distance

- RD
Radiative-conduction parameter

- Re
Reynolds number

*S*_{R}Dimensionless optical thickness

- \( S_{\text{R}}^{*} \)
Total volumetric absorption coefficient

*T*Temperature

*T*_{b}Bulk temperature

- \( T_{\text{w}} \left( {x^{*} } \right) \)
Temperature distribution in a section of the wall

*T*_{w0}Uniform wall temperature

*T*_{0}Inlet temperature

*u*,*v*Dimensionless axial and radial velocities, respectively

*u*^{*},*v*^{*}Axial and radial velocities, respectively

*u*_{0}Inlet axial velocity

*x*Dimensionless axial distance

*x*^{*}Axial distance

## Greek Symbols

*α*Thermal diffusivity

*β*Thermal expansion coefficient

*ε*Dimensionless constant

*θ*Dimensionless wall temperature

*λ*Dimensionless buoyancy parameter

*ν*Kinematic viscosity

*ρ*Density

*σ*Stefan-Boltzmann constant

## Subscripts

- b
Bulk quantity

- conv
Convective

- 0
Inlet condition

- rad
Radiation

- T
Total

- w
Condition at the wall

## Notes

### Acknowledgments

One of the authors (MK) is thankful to the University Grants Commission, India, for the financial support under the Research Scientist Scheme.

## References

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