Effective properties of a thermoelectric composite containing an elliptic inhomogeneity

Song, Kun; Song, Haopeng; Li, M.; Schiavone, Peter; Gao, Cun‐Fa

doi:10.1016/j.ijheatmasstransfer.2019.02.088

Cited by 24 publications

(5 citation statements)

References 26 publications

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“…The material parameters include the electric conductivity δ , thermal conductivity κ and Seebeck coefficient ε . The governing equations for the thermoelectric material in the absence of electric charges and heat sources can be given by [19, 20]

- J = δ \nabla ϕ + δ ε \nabla T,

{boldJ}_{q} = ε T J - κ \nabla T,

where J is the electric current density vector and

J_{q}

is the thermal flux vector. The thermoelectric equilibrium equations are represented as [17]

\nabla \cdot J = 0,

\nabla \cdot {boldJ}_{u} = 0,

in which

J_{u}

is the energy flow vector expressed by

{boldJ}_{u} = {boldJ}_{q} + ϕ J .

…”

Section: Governing Equations Of Thermoelectric Materialsmentioning

confidence: 99%

“…On the basis of the theory of plane thermoelasticity, the governing equation coupling the Airy stress function Θ and the temperature field T can be expressed as [21]

\nabla^{4} Θ + E λ \nabla^{2} T = 0,

where λ and E are the thermal expansion coefficient and Young's modulus, respectively. The complete solution for Equation () can be given as [20]

Θ = \frac{1}{2} false[\overset{z}{true} φ (z) + z \bar{φ false(z false)} + θ (z) + \bar{θ false(z false)} false] + \frac{E λ δ}{16 κ} F false(z false) \bar{F (z)},

where

φ (z)

and

θ (z)

are two analytic complex potential functions of z , and

F (z) = \int f false(z false) d z

. Introducing a new function

ψ false(z false) = θ^{'} false(z false)

, the components of stress and displacement can be derived as

σ_{y} + σ_{x} = \frac{E λ δ}{4 κ} f false(z false) \bar{f (z)} + 2 false[φ^{'} (z) + \bar{φ^{'} (z)} false],

σ_{y} - σ_{x} + 2 i τ_{x y} = E

…”

Section: Governing Equations Of Thermoelectric Materialsmentioning

confidence: 99%

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Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

Xing

Gao

2021

Z Angew Math Mech

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Hollow fibers are commonly introduced into flexible thermoelectric materials for certain engineering applications. However, significant stress concentration may be caused in the vicinity of the interface between the fibers and thermoelectric matrix threatening the reliability of the composite structure. In this paper, we study the plane deformation problem of a rigid hollow fiber embedded in a thermoelectric material. By utilizing the complex variable method, the closed‐form solutions for electric, thermal and elastic fields in both of the fiber and the matrix are obtained. The effects of the thermal resistance and geometric parameters of the hollow fiber on the temperature and stress distributions are numerically examined. The results show that the temperature difference (amplitude of variation in temperature in the matrix side around the interface) and the stress field around the interface are proportion to the inner radius of the hollow fiber and the thermal resistance, respectively. Meanwhile, we find that the temperature and the stress distributions are more sensitive to the geometric parameters than to the thermal resistance. The results in this paper may serve as guidelines for design of fibers‐filled thermoelectric structures.

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- J = δ \nabla ϕ + δ ε \nabla T,

{boldJ}_{q} = ε T J - κ \nabla T,

where J is the electric current density vector and

J_{q}

is the thermal flux vector. The thermoelectric equilibrium equations are represented as [17]

\nabla \cdot J = 0,

\nabla \cdot {boldJ}_{u} = 0,

in which

J_{u}

is the energy flow vector expressed by

{boldJ}_{u} = {boldJ}_{q} + ϕ J .

…”

Section: Governing Equations Of Thermoelectric Materialsmentioning

confidence: 99%

“…On the basis of the theory of plane thermoelasticity, the governing equation coupling the Airy stress function Θ and the temperature field T can be expressed as [21]

\nabla^{4} Θ + E λ \nabla^{2} T = 0,

where λ and E are the thermal expansion coefficient and Young's modulus, respectively. The complete solution for Equation () can be given as [20]

Θ = \frac{1}{2} false[\overset{z}{true} φ (z) + z \bar{φ false(z false)} + θ (z) + \bar{θ false(z false)} false] + \frac{E λ δ}{16 κ} F false(z false) \bar{F (z)},

where

φ (z)

and

θ (z)

are two analytic complex potential functions of z , and

F (z) = \int f false(z false) d z

. Introducing a new function

ψ false(z false) = θ^{'} false(z false)

, the components of stress and displacement can be derived as

σ_{y} + σ_{x} = \frac{E λ δ}{4 κ} f false(z false) \bar{f (z)} + 2 false[φ^{'} (z) + \bar{φ^{'} (z)} false],

σ_{y} - σ_{x} + 2 i τ_{x y} = E

…”

Section: Governing Equations Of Thermoelectric Materialsmentioning

confidence: 99%

Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

Xing

Gao

2021

Z Angew Math Mech

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“…Referring to Eqs. (3a), (15a), (27), and (32), the H function is immediately obtained when the applied energy density u 0 is equal to zero, and given as…”

Section: Solutions For Thermoelectric Materials With a Spherical Incl...mentioning

confidence: 99%

“…A theoretical model to analyze the energy conversion efficiency of a cracked thermoelectric material with finite height and width based on the nonlinearly coupled transport equations of electricity and heat was developed by Zhang et al [34] The three-dimensional ellipsoidal in-clusion problem in thermoelectric materials and the effective material properties of the matrix-inclusion system were studied by Wang et al [35] From Refs. [30,[32][33][34][35], it is also found that the effective figure of merit can exceed that of each constituent. A continuum linear theory for thermoelectric materials was developed by Liu under the conditions of small variations of temperature, electric potential, and their gradients, and this linear theory is further used to predict effective properties of thermoelectric composites.…”

Section: Introductionmentioning

confidence: 96%

“…[30,31] The closed-form solutions of effective properties of a thermoelectric composite containing an elliptic inhomogeneity or an arbitrarily shaped hole were provided in Refs. [32,33]. A theoretical model to analyze the energy conversion efficiency of a cracked thermoelectric material with finite height and width based on the nonlinearly coupled transport equations of electricity and heat was developed by Zhang et al [34] The three-dimensional ellipsoidal in-clusion problem in thermoelectric materials and the effective material properties of the matrix-inclusion system were studied by Wang et al [35] From Refs.…”

Section: Introductionmentioning

confidence: 99%

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Influence of spherical inclusions on effective thermoelectric properties of thermoelectric composite materials

Yan

Zhang

et al. 2020

Chinese Phys. B

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A homogenization theory is developed to predict the influence of spherical inclusions on the effective thermoelectric properties of thermoelectric composite materials based on the general principles of thermodynamics and Mori–Tanaka method. The closed-form solutions of effective Seebeck coefficient, electric conductivity, heat conductivity, and figure of merit for such thermoelectric materials are obtained by solving the nonlinear coupled transport equations of electricity and heat. It is found that the effective figure of merit of thermoelectric material containing spherical inclusions can be higher than that of each constituent in the absence of size effect and interface effect. Some interesting examples of actual thermoelectric composites with spherical inclusions, such as insulated cavities, inclusions subjected to conductive electric and heat exchange and thermoelectric inclusions, are considered, and the numerical results lead to the conclusion that considerable enhancement of the effective figure of merit is achievable by introducing inclusions. In this paper, we provide a theoretical foundation for analytically and computationally treating the thermoelectric composites with more complicated inclusion structures, and thus pointing out a new route to their design and optimization.

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Conversion efficiency and effective properties of particulate-reinforced thermoelectric composites

Song

Yin

Schiavone

2020

Z. Angew. Math. Phys.

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Effective properties of a thermoelectric composite containing an elliptic inhomogeneity

Cited by 24 publications

References 26 publications

Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

Analysis of a hollow fiber in thermoelectric materials considering interfacial thermal resistance

Influence of spherical inclusions on effective thermoelectric properties of thermoelectric composite materials

Conversion efficiency and effective properties of particulate-reinforced thermoelectric composites

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