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Two-Temperature Model and Simulation of Induced Electric Field During Combustion Synthesis of Zinc Sulfide in Argon

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Abstract

A two-temperature (2T) model is employed to analyze an electric field induced during the combustion synthesis of zinc sulfide under the two opposite gravitational conditions of the ascending and descending temperature combustion front motion. This model undertakes situations when the gas is always in a massive abundance, and therefore, the temperatures of the solid and gas phases do not have sufficient time to equilibrate thermally each other during the combustion front motion and generate conditions under which the assumption of one-temperature model of thermal quasi-equilibrium between the phases is no longer valid. The numerical study conducted in a comparison with the experimental results and 1T model shows that 2T model predicts more accurately quantitative values of electric charge density, gas pressure as well as generated voltage during the combustion at both descending and ascending directions of thermal front motions. This demonstrates the key role of non-equilibrium heat transfer for the combustion synthesis, while the temperatures of gas and solid are not equal. The predicted results of characteristic features of induced electric field are in a good agreement with the experimental data.

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Acknowledgements

KM would like to acknowledge the financial support of this research in part of NSF PREM (Award DMR-1523577: UTRGV-UMN Partnership for Fostering Innovation by Bridging Excellence in Research and Student Success), while AM and IF would like to acknowledge the financial support of their participation in this research in the frames of the RAS budget financial support.

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Correspondence to Karen S. Martirosyan.

List of Symbols

List of Symbols

Name

Dimension SI

\( t_{*} = \frac{{\exp \left( {E_{*} /RT_{*} } \right)}}{{k_{*} }} \)—Reference time scale

s

E*—Reference activation energy of chemical reaction

J·mol−1

k*—Reference reaction rate constant

1/s

\( x_{*} = \sqrt {\lambda_{*} t_{*} /(c_{*} \rho_{*} )} \)—Reference length scale

m

\( T^{*} = T_{boil}^{S} \)—Reference temperature

K

\( \frac{T}{{T_{*} }} = 1 + \beta \tilde{T} \)—Dimensionless temperature marked by tildes

1

\( \beta = \frac{{RT_{*} }}{E},\gamma = \frac{{c_{P} T_{*} \beta }}{Q} \)—Similarity main parameters

1

\( \tilde{T}_{0} = \frac{1}{\gamma } \)—Dimensionless initial temperature

1

\( \tilde{x} = x_{n} /x_{*} ,\,\tilde{u}_{n} = u_{n} t_{*} /{\text{x}}_{*} ,n = 1,2,3;\tilde{t} = t/t_{*} ,\tilde{p} = p/p_{*} \)—The dimensionless variables (space coordinate, the gas velocity and pressure correspondingly) are marked by tilde

1

\( p_{*} = \frac{{R\rho_{*} T_{*} }}{{M_{0} }} \) and \( \rho_{*} = n_{S} \bar{\rho }_{1S}^{0} \)—Reference pressure and density

N·m−2, kg·m−3

\( \frac{1}{{M_{0} }} = \frac{1}{{M_{1g} }} + \frac{1}{{M_{2g} }},M_{1g} ,M_{2g} \)—Molar mass of gas species

kg·mol−1

\( \bar{\rho }_{1g} = \bar{\rho }_{{S_{2}^{2 - } (g)}} ,\;\bar{\rho }_{2g} = \bar{\rho }_{e(g)} \)—Gas density of sulfur ions and electrons, correspondingly

kg·m−3

\( \bar{\rho }_{1S}^{0} \)—Initial density of sulfur

kg·m−3

\( \bar{\rho }_{1S} = \bar{\rho }_{{S_{2} }} ,\bar{\rho }_{2S} = \bar{\rho }_{{h^{ + } }} \,\bar{\rho }_{3S} = \bar{\rho }_{Zn} \,\bar{\rho }_{4S} = \bar{\rho }_{{Zn^{2 + } }} ,\bar{\rho }_{5S} = \bar{\rho }_{ZnS} \)—Densities of solid components

kg·m−3

\( \tilde{\rho }_{jS} = \bar{\rho }_{jS} /\rho_{*} ,\tilde{\rho }_{ig} = \bar{\rho }_{ig} /\rho_{*} ,j = 1, \ldots ,5;i = 1,2 \)—The dimensionless densities

1

\( \tilde{c}_{jS} = c_{jS} /c_{*} \)—Specific heat capacity of solid substance j at a constant pressure

J·(kg−1·K−1)

\( \tilde{c}_{ig} = c_{ig} /c_{*} \)—Specific heat capacity of gas substance i at a constant pressure

J·(kg−1·K−1)

\( D_{j} ,D_{ki} \)—Diffusivity

m2·s−1

\( E_{i} \)—Activation energy for ith reaction

J·mol−1

h = cpT—Overall specific enthalpy of the a gas mixture

J·kg−1

\( \kappa \)—Heat transfer coefficient

J·(m−3·s−1·K−1)

\( Q_{V} ,Q_{Ts} \)—Flux of heat release for chemical reactions

J·(m−3·s−1)

\( J_{{j{\text{g}}}}^{\text{macro}} ,J_{{s \to {\text{g}}}}^{\text{macro}} \)—Macroscopic material fluxes caused by chemical reactions and phase conversions

kg·(s−1·m−3)

R—Universal molar gas constant

J·(mol−1·K−1)

\( ({\mathbf{S}}_{{\mathbf{v}}} )_{i} = - u_{i} R_{i} ,R_{i} = \alpha_{i} |{\mathbf{u}}| + \zeta_{i} ,i = 1,2,3 \)—Porous resistance

kg·(m−2·s−2) = N·m−3

χ—Porosity coefficient, dimensionless

1

\( \rho_{ig} = \chi \bar{\rho }_{ig} \), i = 1, 2. Densities of gas components

kg·m−3

\( \rho_{jS} = (1 - \chi )\bar{\rho }_{jS} \), j = 1, …, 5. Densities of solid components

kg·m−3

λ—Thermal conductivity

J·(m−1·s−1·K−1)

\( \vec{B} \), Magnetic induction, Tesla

kg·A−1·s−2

\( \bar{E} \) Electric intensity, V·m−1

kg·m·A−1·s−3

\( \mu_{*} \) Reference gas viscosity

kg·(m−1·s−1)

\( Re = \frac{{\rho_{*} x_{*}^{{}} u_{*} }}{{\mu_{*} }} \) Reynolds number

1

\( \Pr = \frac{{c_{*} \mu_{*} }}{{\lambda_{*} }} \) Prandtl number

1

Electric current I

A

Electric voltage V

kg·m2·A−1·s−3

Intensity of electric field E

kg·m·A−1·s−3

Electric current density j

A·m−2

Electric charge Q, q, e

A·s

Electric charge linear density L

(A·s)·m−1

Electric charge volume density CV

(A·s)·m−3

Electric charge surface density σ

(A·s)·m−2

Specific electric conductivity σEC

\( {\text{A}}^{2} \cdot {\text{s}}^{3} \cdot {\text{kg}}^{ - 1} \cdot {\text{m}}^{ 2} =\Omega ^{ - 1} \)

Electric resistance R

\( \Omega = {\text{kg}} \cdot {\text{m}}^{- 2} \cdot {\text{A}}^{ - 2} \cdot {\text{s}}^{ - 3} \)

Specific electric resistance Rsp

\( \Omega \cdot {\text{m}} \), \( \Omega \cdot {\text{m}} = {\text{kg}} \cdot {\text{m}}^{ - 1} \cdot {\text{A}}^{ - 2} \cdot {\text{s}}^{ - 3} \)

Electric capacity, Ce in farads, microfarads and\or picofarads

F, or μF, πF; \( 1{\text{F}} \equiv {\text{A}}^{2} \cdot {\text{s}}^{4} \cdot {\text{kg}}^{ - 1} \cdot {\text{m}}^{ - 2} \)

ε0—Electric constant, ε0 = 8.854 187 817 × 10−12 F·m−1 (precisely)

\( {\text{F}} \cdot {\text{m}}^{ - 1} \equiv {\text{A}}^{2} \cdot {\text{s}}^{4} \cdot {\text{kg}}^{ - 1} \cdot {\text{m}}^{ - 3} = {\text{s}} \cdot\Omega ^{ - 1} \cdot {\text{m}}^{ - 1} \)

Specific electric resistance of Zn

\( R_{sp}^{Zn} \cong 5.65 \times 10^{ - 8} \,\Omega \cdot {\text{m}} \)

Light velocity

c = 2.99 × 108 m·s−1

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Markov, A.A., Filimonov, I.A. & Martirosyan, K.S. Two-Temperature Model and Simulation of Induced Electric Field During Combustion Synthesis of Zinc Sulfide in Argon. Int J Thermophys 40, 6 (2019). https://doi.org/10.1007/s10765-018-2469-x

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