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Transport in Porous Media

, Volume 95, Issue 3, pp 507–534 | Cite as

Thermal Resonance in Hyperbolic Heat Conduction in Porous Media due to Periodic Ohm’s Heating

  • Peter VadaszEmail author
  • Milan Carsky
Article

Abstract

The effect of periodic Ohm’s heating on hyperbolic heat conduction in porous media is studied analytically with the objective of identifying the thermal resonance conditions. Local thermal equilibrium conditions are assumed to apply. The paper focuses initially on the temperature solution and looks at the conditions required for resonating the temperature signal. The heat flux solution is then evaluated. While a discrete infinite set of modes can be resonated, it is shown that in practice the resonance in the temperature signal is felt starting from moderately small values of Fourier numbers and becomes too small to be noticed if the Fourier number is extremely small. The temperature solution is shown to represent a standing wave the amplitude of which is strongly affected by the Fourier number. While the heat flux solution is shown to differ from the one obtained for the temperature, it also shows similar features such as the standing wave behavior the amplitude of which is strongly affected by the Fourier number.

Keywords

Porous media Hyperbolic conduction Relaxation time Thermal resonance Ohm’s heating 

Nomenclature

Latin Symbols

cs

Specific heat of the solid phase (dimensional)

cf

Specific heat of the fluid phase (dimensional)

ce

Effective specific heat of porous medium equals φc f + (1 − φ)c s (dimensional)

c*

The speed of thermal wave propagation, equals \({\sqrt{{\alpha _{\rm e}}/\tau}}\) (dimensional)

c

Dimensionless speed of thermal wave propagation, equals Fo − 1/2, Eq. (38)

Fo

Fourier number, equals \({{\alpha_{\rm e}\tau}/{L_*^2}}\)

Go

Amplitude of the Ohm’s heating source Eq. (11) (dimensional)

H*

The height of the slab (dimensional)

kf

Thermal conductivity of the fluid phase (dimensional)

ks

Thermal conductivity of the solid phase (dimensional)

ke

Effective thermal conductivity of the porous medium equals φk f + (1 − φ)k s (dimensional)

i*

AC electric current (dimensional)

I*

Amplitude of the AC electric current (dimensional)

L*

The length of the slab (dimensional)

q*

Heat flux vector (dimensional)

S*

Rate of heat generated per unit volume (dimensional)

t*

Time (dimensional)

T*

Temperature (dimensional)

T

Dimensionless temperature, equals \({(T_* - T_{\rm C})k_{\rm e}/{G_{\rm o} L_*^2}}\)

TC

Coldest (ambient) temperature (dimensional)

W*

The depth of the slab (dimensional)

x*

Horizontal co-ordinate (dimensional)

Greek Symbols

αe

Effective thermal diffusivity, equals k e/ρ e c e (dimensional)

γ

A parameter, equals Fo −1 (dimensionless)

φ

Porosity

ρf

Density of the fluid phase (dimensional)

ρs

Density of the solid phase (dimensional)

ρe

Effective density of the porous medium equals φρ f + (1 − φ)ρ s (dimensional)

ηn

Damping strength equals γ/ω n , Eq. (44)

τ

Relaxation time associated with the lag in the heat flux, defined by Eq. (1) (dimensional)

ω

Imposed forcing frequency due to the Ohm’s heating

ωn

Dimensionless natural thermal frequency defined by Eq. (38)

ωr

Frequency ratio equals 2 ω/ω n , Eq. (62)

Subscripts

*

Corresponding to dimensional values, except where there is no ambiguity

cr

Associated with a critical value

h

Homogeneous part of solution

n

Corresponding to the nth Fourier mode

p

Particular part of solution

pt

Post-transient solution

t

Time-dependent part of solution

Superscript

*

Resonant value

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNorthern Arizona UniversityFlagstaffUSA
  2. 2.Faculty of EngineeringUniversity of KZ NatalDurbanSouth Africa

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