Abstract
The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity are considered. It is found that heterogeneity of permeability and fluid conductivity have a major effect, heterogeneity of interphase heat transfer coefficient and porosity have a lesser effect, while heterogeneity of solid conductivity is relatively unimportant.
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Abbreviations
- \(a\) :
-
Dimensionless horizontal wavenumber
- D :
-
d/dz
- \(h\) :
-
Interface heat transfer coefficient (incorporating the specific surface area) between the fluid and solid particles
- \(\hat{{h}}\) :
-
Parameter defined in Eq. (15)
- \(h_\mathrm{r}\) :
-
Interface heat transfer coefficient ratio, \(h_{2}/h_{1}\)
- \(g\) :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- \(H\) :
-
Dimensional layer depth
- \(k\) :
-
Thermal conductivity of the porous medium
- \(\hat{{k}}_{ f}\) :
-
Parameter defined in Eq. (15)
- \(k_{{ f}\mathrm{r}}\) :
-
Conductivity ratio, \(k_{f2}/k_{f1}\)
- \(\hat{{k}}_{ s}\) :
-
Parameter defined in Eq. (15)
- \(k_{{ s}\mathrm{r}}\) :
-
Conductivity ratio, \(k_{s2}/k_{s1}\)
- \(K\) :
-
Permeability of the porous medium
- \(K_\mathrm{r}\) :
-
Permeability ratio, \(K_{2}/K_{1}\)
- \(\hat{{K}}\) :
-
Parameter defined in Eq. (15)
- \(N\) :
-
Interface heat transfer parameter, \(\frac{h_1 H^{2}}{\phi _1 k_{f1}}\)
- \(P\) :
-
Dimensionless pressure, \(\frac{(\rho c)_{ f} K_1}{\mu k_{f1}}P^*\)
- \(P^{*}\) :
-
Pressure, excess over hydrostatic
- Ra :
-
Rayleigh number, \(\frac{\rho _0 g\beta K_1 H(T_\mathrm{h} -T_\mathrm{c} )}{\mu k_{f1} /(\rho c)_{ f}}\)
- \(t\) :
-
Dimensionless time, \(\frac{k_{f1}}{(\rho c)_{ f} H^{2}}t^{*}\)
- \(t^{*}\) :
-
Time
- \(T\) :
-
Dimensionless temperature, \(\frac{T^{*}-T_\mathrm{c}}{T_\mathrm{h} -T_\mathrm{c}}\)
- \(T^{*}\) :
-
Temperature
- \(T_\mathrm{c}\) :
-
Temperature at the upper boundary
- \(T_\mathrm{h}\) :
-
Temperature at the lower boundary
- \((u,v,w)\) :
-
Dimensionless velocity components, \(\frac{(\rho c)_{ f} H}{k_{f1}}(u^{*},v^*,w^*)\)
- \(\mathbf{u}^{*}\) :
-
Darcy velocity, (\(u^{*},v^{*},w^{*}\))
- \((x,y,z)\) :
-
Dimensionless Cartesian coordinates, \((x^{*},y^{*},z^{*})/H; \,\,z\) is the vertically-upward coordinate
- \((x^{*},y^{*},z^{*})\) :
-
Cartesian coordinates
- \(\alpha \) :
-
Modified thermal diffusivity ratio, \(\frac{(\rho c)_{s1}}{(\rho c)_{f1}}\frac{k_{f1}}{k_{s1}}\)
- \(\beta \) :
-
Volumetric expansion coefficient of the fluid
- \(\gamma \) :
-
Modified thermal conductivity ratio, \(\frac{\phi _1 k_{f1}}{(1-\phi _1 )k_{s1}}\)
- \(\delta \) :
-
Dimensionless layer depth ratio (interface position)
- \(\hat{{\delta }}\) :
-
Parameter defined in Eq. (15)
- \(\delta _\mathrm{r}\) :
-
Inverse solid fraction ratio, \(\frac{1-\phi _1}{1-\phi _2}\)
- \(\varepsilon \) :
-
Dimensionless small quantity
- \(\hat{{\varepsilon }}\) :
-
Parameter defined in Eq. (15)
- \(\varepsilon _\mathrm{r}\) :
-
Solid heat capacity ratio, \(\frac{(\rho c)_{s2}}{(\rho c)_{s1}}\)
- \(\mu \) :
-
Viscosity of the fluid
- \(\rho _{0}\) :
-
Fluid density at temperature \(T_\mathrm{c}\)
- \(\rho _{f}\) :
-
Fluid density
- \(\left( {\rho c} \right) _{ f}\) :
-
Heat capacity of the fluid
- \(\left( {\rho c} \right) _\mathrm{m}\) :
-
Effective heat capacity of the porous medium
- \((\rho c)_{ s}\) :
-
Heat capacity of the solid
- \(\phi \) :
-
Porosity
- \(\phi _\mathrm{r}\) :
-
Porosity ratio, \(\phi _{2}/\phi _{1}\)
- \(\mu \) :
-
Fluid viscosity
- B:
-
Basic state
- c:
-
Critical value
- f :
-
Fluid phase
- m:
-
Effective property for the porous medium
- r:
-
Relative quantity
- s :
-
Solid phase
- 1:
-
The region \(0\le z^{*}<\delta H\)
- 2:
-
The region \(\delta H\le z^{*}\le H\)
- \(^{\prime }\) :
-
Perturbation variable
- \(^{*}\) :
-
Dimensional variable
References
Nield, D.A.: Effects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel. J. Porous Media 1, 181–186 (1998)
Nield, D.A.: General heterogeneity effects on the onset of convection in a porous medium. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media, pp. 63–84. Springer, New York (2008)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, New York (2013)
Patil, P.M., Rees, D.A.S.: Linear instability of a horizontal thermal boundary layer formed by vertical throughflow in a porous medium: the effect of local thermal non-equilibrium. Transp. Porous Media 99, 207–227 (2013)
Postelnicu, A., Rees, D.A.S.: The onset of Darcy–Brinkman convection in a porous layer using a thermal nonequilibrium model—Part I: stress free boundaries. Int. J. Energy Res. 27, 961–973 (2003)
Vadasz, P.: Heat conduction in nanofluid suspensions. ASME J. Heat Transf. 128, 465–477 (2006)
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Nield, D.A., Kuznetsov, A.V. Local Thermal Non-Equilibrium and Heterogeneity Effects on the Onset of Convection in a Layered Porous Medium. Transp Porous Med 102, 1–13 (2014). https://doi.org/10.1007/s11242-013-0224-0
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DOI: https://doi.org/10.1007/s11242-013-0224-0