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Accurate Solution and Analysis of the Transient Temperature and Stability of Combustible Micron-Sized Iron Particle in Gaseous Oxidizing Environment

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Abstract

Combustible solid state particles are candidates for alternative fuels with zero carbon emissions. This paper presents the transient and thermal stability analysis of combustible micron-sized iron particle in an oxidizing environment. The governing first-order nonlinear differential equation was solved and analyzed using the continuous piecewise linearization method, which enabled very accurate solutions for the strong nonlinear region of the solution domain i.e. up to the steady-state response. Hence, stability analysis of the heat transfer process was conducted to determine the onset of steady-state response and the temperature attained under the thermally-stable condition. The effect of particle size, surrounding temperature, heat realized, heat radiated, and temperature-dependent density on both the transient temperature profile and thermal stability were investigated and discussed. The theoretical solution for the maximum combustion temperature, which was first derived in this study, showed a good agreement with published experiments. The investigations on the thermal stability revealed interesting new results that provide further insight into the heat transfer dynamics of combustible micro-particles. The study shows that the CPLM algorithm is capable of providing simple, fast converging and very accurate results for the entire solution domain of the transient temperature response, especially in the highly nonlinear asymptotic region.

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Abbreviations

\(\dot{E}_{gen}\) :

Rate of energy generated by particle

\(\dot{E}_{in}\) :

Rate of energy entering the particle

\(\dot{E}_{out}\) :

Rate of energy lost by the particle

\(\dot{E}_{p}\) :

Rate of total energy of the particle

\(h_{cv}\) :

Convective heat transfer coefficient

\(A_{rs} ,B_{rs}\) :

Integration constants of the CPLM solution

\(A_{s}\) :

Area of particle surface

\(C_{rs}\) :

Particular solution of the CPLM solution

\(K_{rs}\) :

Linearized stiffness of each discretization

\(T_{\infty }\) :

Temperature of oxidizing environment

\(T_{i}\) :

Ignition or initial temperature

\(T_{surr}\) :

Temperature of surrounding surfaces

\(V_{p}\) :

Volume of the particle

\(c_{p}\) :

Specific heat capacity of the particle

\(d_{p}\) :

Diameter of the particle

\(m_{p}\) :

Mass of the particle

\(u_{p}\) :

Specific internal energy of the particle

\({\varvec{\varepsilon}}_{1}\) :

Temperature-dependent density parameter

\({\varvec{\varepsilon}}_{2}\) :

Heat radiated parameter

\(\theta_{0}\) :

Initial dimensionless temperature

\(\theta_{\infty }\) :

Dimensionless ambient temperature

\(\theta_{\max }\) :

Dimensionless maximum temperature

\(\theta_{r} ,\theta_{s}\) :

Dimensionless temperature at the start and end of a discretization

\(\theta_{sst}\) :

Dimensionless steady-state temperature

\(\theta_{surr}\) :

Dimensionless surrounding temperature

\(\rho_{p,\infty }\) :

Density of the particle at ambient temperature

\(\rho_{p}\) :

Density of the particle

\(\tau_{r} ,\tau_{s}\) :

Dimensionless temperature at the start and end of a discretization

\(\tau_{stab}\) :

Thermal stability time

\(\Delta h_{cb}\) :

Enthalpy of combustion

\(R\) :

Rate of reaction of the combustion

\(T\) :

Temperature of the particle

\(n\) :

Number of discretization

\(\alpha_{s}\) :

Thermal absorptivity of particle surface

\(\beta\) :

Thermal coefficient of variation of density

\(\varepsilon_{s}\) :

Thermal emissivity of particle surface

\(\theta\) :

Dimensionless temperature

\(\sigma\) :

Stefan Boltzmann constant

\(\tau\) :

Dimensionless time

\(\varphi\) :

Heat realized parameter

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Acknowledgements

No financial support was received for the conduct or publication of this research. We are thankful to the anonymous reviewers, whose comments helped to improve the final version of this paper.

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Authors and Affiliations

  1. Applied Mechanics and Design (AMD) Research Group, Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt, Port Harcourt, Nigeria

    Akuro Big-Alabo

  2. Energy and Thermofluid Engineering (ETE) Research Group, Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt, Port Harcourt, Nigeria

    Chidozie Ezekwem

Authors
  1. Akuro Big-Alabo
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  2. Chidozie Ezekwem
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Correspondence to Akuro Big-Alabo.

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Appendix

Appendix

See Fig. 11.

Fig. 11
figure 11

Flowchart showing CPLM solution for the iron micro-particle combustion model

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Big-Alabo, A., Ezekwem, C. Accurate Solution and Analysis of the Transient Temperature and Stability of Combustible Micron-Sized Iron Particle in Gaseous Oxidizing Environment. Int. J. Appl. Comput. Math 7, 57 (2021). https://doi.org/10.1007/s40819-021-00998-4

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