Abstract
An analytical and numerical study was made on thermally developing forced convective flow in a channel filled with a fluid-saturated porous medium, subject to constant heat flux, under local thermal non-equilibrium. The Brinkman-extended Darcy model was employed to investigate the combined effects of the Darcy number, Biot number, effective thermal conductivity ratio and Graetz number on the longitudinal thermal development in a channel filled with a highly porous medium. 3D convective regime maps were constructed for wide ranges of the parameters, using a 3D coordinate system based on the Biot, Darcy and Graetz number coordinates. The 3D convective regime maps proposed in this study enable one to identify the three distinctive thermal regions possible along the longitudinal coordinate in a channel, namely the entrance developing region, transition region and fully developed region, depending on the relative thicknesses of the fluid and solid thermal boundary layers. It has been found that the Darcy number strongly influences the heat transfer rate in the developing region, yielding a higher Nusselt number for a lower Darcy number, whereas both Biot number and the effective thermal conductivity ratio markedly influence the level of the fully developed Nusselt number. The heat transfer performance evaluation carried out under equal pumping power for the aluminum foam embedded channel saturated with air reveals that a substantial heat transfer argumentation is possible by filling the channel with an aluminum foam.
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Abbreviations
- \(Bi\) :
-
\(\frac{{h_{v} H^{2} }}{{k_{se} }}\) Biot number
- \(c_{pf}\) :
-
Specific heat of fluid at constant pressure (J/kg K)
- \(Da\) :
-
\(\frac{K}{{H^{2} }}\) Darcy number
- \(d_{m}\) :
-
Pore diameter (m)
- \(H\) :
-
Half of the channel height (m)
- \(h_{}\) :
-
\(\frac{{q_{w} }}{{\left( {T_{w} - T_{fB} } \right)}}\) Heat transfer coefficient (W/m2K)
- \(h_{f}\) :
-
Heat transfer coefficient of an empty channel (W/m2K)
- \(h_{v}\) :
-
Interstitial volumetric heat transfer coefficient (W/m3K)
- \(K\) :
-
Permeability (m2)
- \(k_{fe}\) :
-
Effective thermal conductivity for the fluid phase (W/mK)
- \(k_{se}\) :
-
Effective thermal conductivity for the solid phase (W/mK)
- \(Nu_{H}\) :
-
\(\frac{{q_{w} H}}{{\left( {T_{w} - T_{e} } \right)k_{fe} }}\) Local Nusselt number
- \(Nu_{Nield}\) :
-
Local Nusselt number defined by Nield et al.
- \(p\) :
-
Intrinsic pressure (Pa)
- \(PP\) :
-
\(\left( {\left( {\frac{{\rho_{f} u_{B} H}}{{\mu_{f} }}} \right)^{3} \left( { - \frac{dp}{{dx}}} \right)\frac{H}{{\rho_{f} u_{B}^{2} }}} \right)^{\frac{1}{3}}\) Dimensionless pumping power
- \(q_{w}\) :
-
Wall heat flux (W/m2)
- \(T\) :
-
Temperature (K)
- \(T_{0}\) :
-
Inlet fluid temperature (K)
- \(T_{fB}\) :
-
Fluid-phase bulk mean temperature (K)
- \(T_{fB}^{*}\) :
-
\(\frac{{\left( {T_{fB} - T_{0} } \right)}}{{\left( {T_{w} - T_{0} } \right)}}\) Dimensionless fluid-phase bulk mean temperature
- \(T_{fc}^{*}\) :
-
\(\frac{{\left( {T_{fc} - T_{0} } \right)}}{{\left( {T_{w} - T_{0} } \right)}}\) Dimensionless fluid-phase temperature at the channel center
- \(T_{sc}^{*}\) :
-
\(\frac{{\left( {T_{sc} - T_{0} } \right)}}{{\left( {T_{w} - T_{0} } \right)}}\) Dimensionless solid-phase temperature at the channel center
- \(u, v\) :
-
Darcian velocity (apparent velocity) components (m/s)
- \(u_{B}\) :
-
Bulk mean velocity (m/s)
- \(x\) :
-
Streamwise boundary layer coordinate (m)
- \(x_{1} , x_{2}\) :
-
Coordinate point for the convective regime transition (m)
- \(x^{*}\) :
-
Dimensionless streamwise coordinate \(x^{*} = x/H\)
- \(y\) :
-
Coordinate normal to the wall surface (m)
- \(y^{*}\) :
-
Dimensionless vertical coordinate \(y^{*} = y/H\)
- \({\upgamma }\) :
-
Ratio of the effective fluid thermal conductivity to its solid-phase counterpart \({ }\gamma = \frac{{k_{fe} }}{{k_{se} }}\)
- \(\delta\) :
-
Thermal boundary layer thickness (m)
- \(\delta_{f}^{*}\) :
-
Dimensionless fluid-phase thermal boundary layer thickness \(\delta_{f}^{*} = \delta_{f} /H\)
- \(\delta_{s}^{*}\) :
-
Dimensionless solid-phase thermal boundary layer thickness \(\delta_{s}^{*} = \delta_{s} /H\)
- \(\varepsilon\) :
-
Porosity
- \(\zeta\) :
-
\(\frac{{1 + \sqrt {9 + 4\varepsilon /Da} }}{2}\) Exponent for the velocity profile
- \(\mu_{f}\) :
-
Fluid viscosity (Pa s)
- \(\xi\) :
-
\(\frac{{x^{*} }}{{\left( {\frac{{{\uprho }_{f} {\text{c}}_{{p_{f} }} uH}}{{k_{fe} }}} \right)}}\) Graetz number
- \(\xi_{1} , \xi_{2}\) :
-
Dimensionless coordinate point for the convective regime transition
- \(\rho_{f}\) :
-
Fluid density (kg/m3
- B :
-
Bulk mean
- C :
-
Channel center
- \(f\) :
-
Fluid phase
- \(s\) :
-
Solid phase
- \(w\) :
-
Wall
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Yi, Y., Bai, X., Kuwahara, F. et al. Analytical and Numerical Study on Thermally Developing Forced Convective Flow in a Channel Filled with a Highly Porous Medium Under Local Thermal Non-Equilibrium. Transp Porous Med 136, 541–567 (2021). https://doi.org/10.1007/s11242-020-01524-8
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DOI: https://doi.org/10.1007/s11242-020-01524-8