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Analytical investigation of temperature of a single micron sized iron particle during combustion

单一微米级铁颗粒燃烧温度的分析研究

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Abstract

The present study deals with analytical investigation of temperature of a single burning iron particle. Three mathematical methods including AGM (Akbari-Ganji’s method), CM (Collocation method) and GM (Galerkin Method) are applied to solving non-linear differential governing equation and effectiveness of these methods is examined as well. For further investigation, forth order Runge-Kutta approach, a numerical method, is used to validate the obtained analytical results. In the present study, the developed mathematical model takes into account the effects of thermal radiation, convective heat transfer and particle density variations during combustion process. Due to particles’ small size and high thermal conductivity, the system is assumed to be lumped in which the particle temperature does not change within the body and all of its regions are at the same temperature. The temperature distributions obtained by analytical methods have satisfactory agreement with numerical outputs. Finally, the results indicate that AGM is a more appropriate method than GM and CM due to its lower mean relative error and less run time.

摘要

本研究对单一燃烧铁颗粒的温度进行了分析研究。采用三种数学方法, 包括AGMMkban-Ganji 法)、CM(搭配法)和GM(Galerkin法)求解非线性微分控制方程, 并对这些方法的有效性进行了分析。 为了进一步研究, 采用一种数值方法, 即四阶Runge-Kutta法, 验证所得的结果。在本研究中, 所建 立的数学模型考虑了燃烧过程中热辐射、对流热交换和颗粒密度变化的影响。由于粒子的小尺寸和 高热导率, 假设系统聚集, 其中的粒子体内温度不发生变化, 而且所有部分都处于相同的温度。分析 方法得到的温度分布与数值结果吻合较好。结果表明, AGM具有较低的平均相对误差和较短的运行 时间, 是一种比GM和CM更合适的方法。

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Abbreviations

A pr :

Projected area of particle perpendicular to the direction of falling, m2

A s :

Outer surface area of particle, m2

Bi :

Biot number

c p :

Specific heat of particle, J·K−1·kg-1

Cp,∞ :

Specific heat of gaseous oxidizing medium, J·K−1·kg−1

CD :

Drag coefficient dP Particle diameter, m

DO2,∞ :

Mass diffusivity of oxygen in air, m2·s−1

E :

Total energy of particle, J

FB :

Buoyant force acting on iron particle, kg·ms−2

F D :

Drag force exerted on iron particle opposite the direction of falling, kg·ms−2

g :

Gravitational acceleration, m·s−2

h conv :

Average convection heat transfer coefficient, W·K−1·m−2

m p :

Mass of particle, kg

Nu :

Average Nusselt number

P O2 :

Partial pressure of oxygen in the ambient gaseous medium, Pa

Pe :

Peclet number

Pr ∞ :

Prandtl number of gaseous fluid

r p :

Radius of particle, m

R Fe :

Reaction rate of iron, kg·m−2·s−1

Re P :

Reynolds number for particle

t :

Time, s

T :

Absolute temperature of particle, K

T ig :

Ignition temperature of iron particle, K

T s :

Surface temperature of particle, K

Tsurr :

Absolute temperature of surroundings, K

T∞ :

Absolute temperature of ambient gaseous oxidizer, K

u :

Specific internal energy of the system, J

U :

Total internal energy of the system, J

Vterm :

Terminal velocity of falling iron particle, m·s−1

V P :

Volume of spherical iron particle, m3

W :

Weight of spherical iron particle, kg·ms−2

Y O2,∞ :

Mass fraction of oxygen in the ambient gas

α s :

Absorptivity of particle surface

β:

Coefficient of temperature-dependence of density, KT−1

ε:

Emissivity of particle surface, non-dimensional parameter

ϱ:

Dimensionless temperature

λP :

Thermal conductivity of iron particle, W·m−1·−1

λ∞ :

Thermal conductivity of gaseous oxidizing environment, W·m−1·s−1

μ∞ :

Dynamic viscosity of ambient gas, kg·m−1·s−1

v :

Mass stoichiometric index of combustion of iron

πp :

Density of burning iron particle, kg·m−3

gpp,∞ :

Density of iron particle at T∞ kg·m−3

p ∞ :

Density of ambient gaseous oxidizer, kg·m−3

σ:

Stefan-Boltzmann constant, W·m−2·K−4

τ:

Burning time of particle, dimensionless time

Ω:

Non-dimensional parameter

0:

Initial

B :

Buoyanc

bdiff:

Diffusionally-controlled

comb:

Combustion

conv:

Convection

D :

Drag

ig:

Ignition

O2 :

Oxygen

P:

Particle

s:

Surface

surr:

Surroundings

∞:

Ambient

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Authors and Affiliations

  1. Department of Energy Conversion, School of Mechanical Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114, Tehran, Iran

    Peyman Maghsoudi (PhD Candidate), Mehdi Bidabadi, Seyed Amir Hossein Madani & Abolfazl Afzalabadi

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  1. Peyman Maghsoudi
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  2. Mehdi Bidabadi
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  3. Seyed Amir Hossein Madani
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  4. Abolfazl Afzalabadi
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Correspondence to Peyman Maghsoudi.

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Maghsoudi, P., Bidabadi, M., Madani, S.A.H. et al. Analytical investigation of temperature of a single micron sized iron particle during combustion. J. Cent. South Univ. 27, 951–962 (2020). https://doi.org/10.1007/s11771-020-4343-9

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