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Analysis of the Effect of the Orthotropic Thermal Conductivity Tensor During Microwave-Based Heat Treatment of a Core–Shell Spherical System

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Abstract

In this article, we propose a numerical analysis of the effect of the orthotropic tensor of thermal conductivity during microwave heating of a heterogeneous core–shell morphology. The core is made of a material with high thermal conductivity, whose dielectric loss coefficient guarantees high microwave energy to heat conversion. This type of morphology has a high potential for use in the ablation of tumors, chemotherapy, drug release, and enhancing nano-catalysis, among other applications. Nonetheless, the effect of orthotropic thermal conductivity has not been extensively studied. The system under analysis is a core surrounded by two shells, which are made of materials whose thermal conductivities vary orthogonally. The thermal model consists of a system of three time-dependent coupled parabolic partial differential equations. Such a model is numerically solved using finite elements, and assuming a thermal conductivity tensor for each layer. A strong effect of this type of anisotropy was observed on temperature profiles compared to traditional isotropic materials. Besides, the symmetric release of its internally generated energy was seriously affected. Selected simulated experimental scenarios are presented.

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Abbreviations

\(\alpha \left[ {{\text{J}} \cdot {\text{m}}^{ - 3} \cdot {\text{K}}^{ - 1} } \right]\) :

Volumetric coefficient of thermal expansion

\(A_{s} \left[ {{\text{m}}^{2} } \right]\) :

Surface area for convection heat transfer

\(c \left[ {{\text{W}} \cdot {\text{m}}^{ - 3} \cdot {\text{K}}^{ - 1} } \right]\) :

Specific heat capacity

\(\varepsilon_{0} \left[ {{\text{F}} \cdot {\text{m}}^{ - 1} } \right]\) :

Vacuum permittivity

\(\varepsilon_{ef}^{''}\) :

Imaginary part of effective permittivity

\(E_{rms} \left[ {{\text{V}} \cdot {\text{m}}^{ - 1} } \right]\) :

Electrical field strength

\(f \left[ {\text{GHz}} \right]\) :

Microwave frequency

\({\mathfrak{F}}_{i} \left[ {\text{K}} \right]\) :

Initial temperature distribution function

\(\dot{g} \left[ {{\text{W}} \cdot {\text{m}}^{ - 3} } \right]\) :

Internal volumetric heat generation

\(h\) \(\left[ {{\text{W}} \cdot {\text{m}}^{ - 2} \cdot {\text{K}}^{ - 1} } \right]\) :

Convection heat transfer coefficient

\(H_{rms} \left[ {{\text{A}} \cdot {\text{m}}^{ - 1} } \right]\) :

Magnetic field strength

\(i\) :

Index corresponding to the \(i\)th sphere or shell

\(i:j\) :

Interface between bodies \(i\) and \(j\)

\(\varvec{K }\left[ {{\text{W}} \cdot {\text{m}}^{ - 1} \cdot {\text{K}}^{ - 1} } \right]\) :

Thermal conductivity tensor

\(\varvec{k}_{ij} \left[ {{\text{W}} \cdot {\text{m}}^{ - 1} \cdot {\text{K}}^{ - 1} } \right]\) :

ijth component of \(\varvec{K}\)

\(\mu\) :

Variable change of \(\theta ,\,\mu = \cos \theta\)

\(\mu_{0} \left[ {{\text{H}} \cdot {\text{m}}^{ - 1} } \right]\) :

Vacuum magnetic permeability

\(\mu_{ef}^{''}\) :

Imaginary part of the effective permeability

\(\varvec\nabla\) :

Del operator

\({\varvec{\Omega}}\) :

Domain limited to a plane or to a region

\(\omega\) [rad·s−1]:

Angular microwave frequency

\(\partial /\partial \xi\) :

Partial differential operator with respect to \(\xi\)

\(P_{av}^{'''} \left[ {{\text{W}} \cdot {\text{m}}^{ - 3} } \right]\) :

Average volumetric power dissipated

\(\phi \left[ {\text{rad}} \right]\) :

Azimuthal angle in the spherical coordinates

\(q^{\prime\prime}\) [\({\text{W}} \cdot {\text{m}}^{ - 2}\)]:

Heat flow per unit area dissipated

\(\dot{Q}_{conv} \left[ {\text{W}} \right]\) :

Convection heat power

\(r \left[ {\text{m}} \right]\) :

Radius in the spherical coordinate system

\(\rho \left[ {{\text{kg}} \cdot {\text{m}}^{ - 3} } \right]\) :

Density

\(r_{i} \left[ {\text{m}} \right]\) :

Radius of the \(i\)-th sphere or shell

\(t \left[ {\text{s}} \right]\) :

Temporal variable

\(T\) [K] or [°C]:

Temperature

\(\theta \left[ {\text{rad}} \right]\) :

Polar angle in the spherical coordinate system

\(T_{\infty }\) [K] or [°C]:

Temperature of the fluid away from the surface

\(T_{s}\) [K] or [°C]:

Temperature at the surface

\(x \left[ {\text{m}} \right]\) :

Cartesian coordinate

\(y \left[ {\text{m}} \right]\) :

Cartesian coordinate

\(z \left[ {\text{m}} \right]\) :

Cartesian coordinate

\(\xi\) :

Dummy variable

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Correspondence to Rodrigo Correa.

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Cruz-Duarte, J.M., Amaya, I. & Correa, R. Analysis of the Effect of the Orthotropic Thermal Conductivity Tensor During Microwave-Based Heat Treatment of a Core–Shell Spherical System. Int J Thermophys 41, 82 (2020). https://doi.org/10.1007/s10765-020-02664-1

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