Numerical investigations on the difference between aiding and opposing flows in the developing regime of laminar mixed convection in vertical tubes

Abstract

The present article discusses the numerical simulation results of laminar mixed convection flow in vertical tubes. A comparative analysis of the thermal and hydrodynamic features of both buoyancy-assisted and opposed flows was performed for Reynolds number ($100 \le \mathit{\text{ Re }}\le 2300$), Grashof number (${10}^3 \le \mathit{\text{ Gr }}\le 7.935\times {10}^6$) and Richardson number ($0.1 \le \mathit{\text{ Ri }}\le 1.5$) with uniform heat flux boundary condition. 2-D axisymmetric steady state simulations were carried out for a length-to-diameter ($L/D$) ratio of $\le 1000$ with water as the working fluid. Numerical simulations were performed by employing SIMPLE scheme for pressure–velocity coupling in momentum equations and second order UPWIND scheme for solving energy equations. In case of assisting flow ($Re\sim 250$), at fully developed state the centerline velocity decreases, and velocity is increased near the tube wall due to heat flux induced free convection. With increasing heat flux, the decrement in centerline velocity compared to the no-heat flux condition increases. Further, with increasing heat flux, the increase in friction factor and Nusselt number was observed. Therefore, at the same $Re,$ the variation of heat flux led to unique velocity profiles and temperature gradients. The variation of centerline velocity and temperature in developing region was also studied. While centerline temperature was monotonically increasing with length, the centerline velocity increased up to a maximum in the developing region and then attained the steady state at a lower value in the fully developed state. Subsequently we studied the dependence of $Ri$ on the hydrodynamic and thermal features. For constant $Re,$ friction factor and Nusselt number was observed to increase with increase in $Ri$ from 0.1 to 1.5 range. At fixed $Re,$ variation of heat flux leads to variation of $Gr$ or $Ri.$ Hence, the buoyancy effect has a significant role in the entrance length development. The hydrodynamic entry length after an initial increase, attained an almost constant value with increasing $Ri.$ On the other hand, the thermal entry length exhibited a decreasing trend with increasing $Ri.$ Similarly, at constant $Ri,$ Nusselt number increased with increasing $Re$ for a range of 100–1000. It was evident that for a given $Re$ and $Ri,$ the heat transfer in the developing flow is always higher as that of the fully developed flow. Contrasting observations were observed for buoyancy-opposed flow. Velocity is accelerated at the center as compared to the tube wall in case of opposing flow for same $Re$ and $Ri.$ For a fixed $Re,$ the friction factor and Nusselt number decreased with increasing the $Ri.$ Both the hydrodynamic and thermal entry length increases for opposing flows. Finally, we have developed correlations for fully developed friction factor ($f$) with $Re$ as well as $Ri$ and Nusselt number ($Nu$) with $Ri$ for buoyancy-assisting and buoyancy-opposing flows. Two independent correlations are also produced for $Nu$ with Graetz number ($Gz$) for developing and fully developed regimes in both assisting as well as opposing flows.

Keywords:

• Buoyancy-aiding and -opposing flows
• developing flow
• entry length
• mixed convection
• vertical tube

Acknowledgments

SG would like to thank Ministry of Education, Govt. of India. SKD and DS would like to thank IIT Ropar for providing the computational resources.