Analytical Nusselt number for forced convection inside a porous-filled tube with temperature-dependent thermal conductiv
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Analytical Nusselt number for forced convection inside a porous‑filled tube with temperature‑dependent thermal conductivity arising from high‑temperature applications Maziar Dehghan1 · Milad Tajik Jamalabad1 · Saman Rashidi2 Received: 26 February 2020 / Accepted: 4 April 2020 © Akadémiai Kiadó, Budapest, Hungary 2020
Abstract The convection heat transfer inside a tube filled with a porous material under the constant heat flux thermal boundary condition which is widely used in practical applications is studied in the present article. The Darcy–Brinkman–Forchheimer model is used to cover a wide range of working mediums from clear fluid flow to slug flow (Darcy flow). The case of temperaturedependent thermal conductivity is considered in the present study and the corresponding Nusselt number is analytically obtained using perturbation techniques for the first time. The change in thermal conductivity with respect to temperature occurs at high-temperature applications wherein high-temperature variations exist as well as the radiation heat transfer. A linear model for the thermal conductivity variation with temperature is considered in the present study. The obtained profile for the Nusselt number can be used for quick calculations as well as validation of numerical and experimental studies, especially at high temperatures wherein the experimental studies are accompanied by higher uncertainties. The results show that the Nusselt number increases linearly with the linear increase in the thermal conductivity and as well the heat transfer rate. Furthermore, results show that the Nusselt number (and the heat transfer rate as well) shows more augmentation to the thermal conductivity enhancement due to the temperature-dependent nature of thermal conductivity (especially arising from the radiation heat transfer) in comparison with the clear fluid flow case. Keywords Porous media · Temperature-dependent thermal conductivity · Darcy–Brinkman–Forchheimer model · Perturbation · Radiation–convection · High-temperature applications Abbreviations cp Specific heat at constant pressure (Jkg−1 K−1) CF Inertial constant Da Darcy number, K/R2 F Forchheimer number G Negative of the applied pressure gradient in the flow direction (Pa m−1) K Permeability of the medium (m2) k Thermal conductivity (Wm−1 K−1) M Viscosity ratio Nu Nusselt number q″ Heat flux at the wall (Wm−2) R Tube radius (m) * Maziar Dehghan [email protected] 1
Department of Energy, Materials and Energy Research Center (MERC), Tehran, Iran
Department of Energy, Faculty of New Science and Technologies, Semnan University, Semnan, Iran
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s Porous medium shape parameter T Temperature (K) Tm Bulk mean temperature (K) Tw Wall temperature (K) u Dimensionless velocity u* Velocity (ms−1) û Normalized velocity U* Mean velocity (ms−1) x, r Dimensionless coordinates x*, r* Dimensional coordinates (m Greek letters ε Linear proportionality multiplier of the variable thermal conductivity model θ Dimensionless temperature μ Fluid viscosity (Kgm
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