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Similarity Solutions for Local Thermal Non-equilibrium Boundary-Layer Flows in a Fluid-Saturated Porous Medium

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Abstract

A general boundary-layer coordinate transformation was proposed in order to find the necessary conditions under which similarity solutions exist for free, forced and mixed convective Darcian flows subject to local thermal non-equilibrium. The aim of this study is to present the effects of local thermal non-equilibrium on the boundary-layer heat transfer in porous media, identifying each of the parameters controlling the thermal boundary-layer development. The findings of this systematic study are quite beneficial to those dealing with thermal applications using porous media. Similarity solutions are found to exist for such local thermal non-equilibrium cases as forced convection around the stagnation regions of horizontal cylinder and sphere heated according to a power function of the distance from the stagnation point, free convection over a vertical plate with its temperature increasing linearly, and free and mixed convection around the stagnation regions of isothermally heated horizontal cylinder and sphere. The study clearly indicates that the local thermal equilibrium assumption results in substantial overestimation of heat transfer rates. The local non-equilibrium effects must be considered when the modified interstitial Stanton number is less than its criterion value around 10.

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Abbreviations

\(c_{\text{pf}}\) :

Specific heat of fluid at constant pressure (J/kg K)

\(f\) :

Dimensionless streamfunction

\(g, g_{x}\) :

Acceleration of gravity and its tangential component (m/s2)

\(h_{v}\) :

Interstitial volumetric heat transfer coefficient (W/m3 K)

i :

i = 0 for plane flow and i = 1 for axisymmetric flow

I :

Function accounting for the geometrical and thermal effects on the boundary layer

\(k_{\text{fe}}\) :

Effective thermal conductivity for the fluid phase (W/mK)

\(k_{\text{se}}\) :

Effective thermal conductivity for the solid phase (W/mK)

\(K\) :

Permeability (m2)

\(m_{\text{T}}\) :

Function describing the wall temperature variation

p :

Pressure (Pa)

\(q_{\text{w}}\) :

Wall heat flux (W/m2)

\(r\left( x \right)\) :

Geometrical function (m)

\(r^{*}\) :

\(r^{*} = 1\) for plane body and \(r^{*} = r\) for axisymmetric body (m)

R :

Radius (m)

\({\text{St}}_{v}\) :

Interstitial Stanton number

T :

Temperature

\(u, v\) :

Darcian velocity (apparent velocity) components in x and y directions (m/s),

\(x\) :

Streamwise boundary-layer coordinate

\(y\) :

Coordinate normal to the wall surface (m)

\(z_{g}\) :

Elevation measured from the front stagnation point (m)

\(\alpha_{\text{fe}}\) :

Effective thermal diffusivity of the fluid (m2/s)

\(\beta\) :

Thermal expansion coefficient (1/K)

\(\gamma\) :

Ratio of the effective fluid thermal conductivity to its solid-phase counterpart

\(\delta_{\text{f}}\) :

Fluid-phase thermal boundary-layer thickness (m)

\(\delta_{\text{s}}\) :

Solid-phase thermal boundary-layer thickness (m)

\(\zeta\) :

Ratio of the fluid-phase boundary-layer thickness to the solid-phase counterpart

\(\eta\) :

Transformed normal coordinate

\(\theta\) :

Temperature profile function

\(\kappa\) :

Ratio of the buoyancy-induced velocity to the externally forced velocity

\(\lambda\) :

Shape parameter for the temperature profile function

\(\mu_{\text{f}}\) :

Fluid viscosity (Pa s)

\(\nu_{\text{f}}\) :

Fluid kinematic viscosity (m2/s)

\(\xi\) :

Transformed streamwise coordinate

\(\rho_{\text{f}}\) :

Fluid density (kg/m3)

\(\psi\) :

Stream function (m2/s)

e:

Boundary-layer edge

f:

Fluid phase

s:

Solid phase

w:

Wall

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Authors and Affiliations

  1. Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan

    Yuan Yi

  2. Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu, 432-8561, Japan

    Fujio Kuwahara & Akira Nakayama

  3. School of Power and Energy, Northwestern Polytechnical University, 127 West Youyi Road, Xi’an, 710072, Shaanxi, People’s Republic of China

    Xiaohui Bai & Akira Nakayama

Authors
  1. Yuan Yi
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  2. Fujio Kuwahara
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  3. Xiaohui Bai
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  4. Akira Nakayama
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Correspondence to Akira Nakayama.

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Yi, Y., Kuwahara, F., Bai, X. et al. Similarity Solutions for Local Thermal Non-equilibrium Boundary-Layer Flows in a Fluid-Saturated Porous Medium. Transp Porous Med 134, 117–137 (2020). https://doi.org/10.1007/s11242-020-01439-4

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