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Buoyancy-aided convection flow in a heated straight pipe: comparing different asymptotic models

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Abstract

A vertical straight circular adiabatic vertical long tube, open at its lower and upper ends, is heated at its base on a short portion. The flow is studied with the hypothesis of no pressure drop between the entrance and the exit. Direct resolution of Navier Stokes equations is done by finite volumes. The numerical solutions are then compared to a one dimensional model and to two asymptotic models. The first asymptotic model is inspired from boundary layer approximations whereas the second one is more a linear perturbation of the Navier Stokes Boussinesq equations. For moderate values of the Grashof number, pressure, starting from zero decreases over the heated part to a minimum and increases on the adiabatic tube to zero. For larger values of Grashof, a local maximum in pressure appears, this pressure hump may even be positive. The four model agree, for moderate Grashof. When increasing the Grashof, only the two asymptotic models recover the behavior obtained from the numerical simulations.

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Abbreviations

Gr :

Grashof (equivalent to Re due to the choice of scales)

g :

Gravitational acceleration

h :

Heat transfer coefficient

k :

Thermal conductivity

\(\ell\) :

Length of the heated part

\(L_c\) :

Length of the adiabatic channel

\({\bar{p}}\) :

Deviation of pressure from hydrostatic pressure without dimension

Pe :

Péclet number

r :

Radial variable

R :

Radius of the channel

Re :

Reynolds number

S :

Section of the channel

\({\bar{T}}_m\) :

Mean value of the temperature across a section ponderated by the Poiseuille solution

\(T_0\) :

Reference temperature

\(T_w\) :

Temperature of the heated part of the channel

u :

Velocity along the tube

\({\bar{U}}_{max}\) :

Maximum of the velocity without dimension

\(\bar{U}\) :

Mean value of the velocity across a section

\(U_0= (R^2\rho g \alpha (T_w-T_0))/\mu\) :

Velocity scale

v :

Radial velocity across the tube

x :

Axial direction along the tube

\(\alpha\) :

Coefficient of thermal expansion

\(\Delta T=(T_x-T_0)\) :

Driving difference of temperature

\(\lambda\) :

A thermal length for the 1D model

\(\mu\) :

Dynamic viscosity

\(\nu\) :

Kinematic viscosity

\(\rho _0\) :

Reference density of the fluid

\(-\,\hat{}\,\sim\) :

Quantity without dimension

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Acknowledgments

The authors wish to thank ENSTA Paristech (aka. Techniques Avancées) to provide links between them trough a series of courses of heat transfer at ENIT Tunis.

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Authors and Affiliations

  1. Ecole Nationale d’Ingénieurs de Tunis (ENIT), Le Belvédère 1002, BP 37, Tunis, Tunisia

    Walid Arfaoui & Mohamed Jomaa Safi

  2. CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, UPMC Univ Paris 06, Boîte 162, 4 place Jussieu, 75005, Paris, France

    Pierre-Yves Lagrée

  3. Sorbonne Universités, UPMC Univ Paris 06, Institut Jean Le Rond d’Alembert, 4 place Jussieu, 75005, Paris, France

    Walid Arfaoui & Pierre-Yves Lagrée

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  1. Walid Arfaoui
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Correspondence to Pierre-Yves Lagrée.

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Arfaoui, W., Safi, M.J. & Lagrée, PY. Buoyancy-aided convection flow in a heated straight pipe: comparing different asymptotic models. Heat Mass Transfer 52, 1515–1527 (2016). https://doi.org/10.1007/s00231-015-1677-1

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