J. Cent. South Univ. (2020) 27: 951-962

DOI: https://doi.org/10.1007/s11771-020-4343-9

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

Peyman MAGHSOUDI, Mehdi BIDABADI, Seyed Amir Hossein MADANI, Abolfazl AFZALABADI

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

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

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.

Key words: iron particle combustion; analytical solutions; thermal radiation; convective heat transfer

Cite this article as: Peyman MAGHSOUDI, Mehdi BIDABADI, Seyed Amir Hossein MADANI, Abolfazl AFZALABADI. Analytical investigation of temperature of a single micron sized iron particle during combustion [J]. Journal of Central South University, 2020, 27(3): 951-962. DOI: https://doi.org/10.1007/s11771-020-4343-9.

1 Introduction

The possibility of a dust explosion in the exposed industrial sites has made that the scientists recognize this danger and devise precautions in order to prevent this incident. The specific accidents have been continuously reported since 150 years ago which shows not only a historic issue, but also a challenging problem, because it has not been solved yet [1].

A low concentration of solid particles is exposed to the risk of being exploded if the particles are dispersed properly in air and ignited by an unintentional external source. In this condition, the flame rapidly passes through the disposed environment and causes severe human and financial damages. This phenomenon involves a wide variety of sites and factories working with flour, plastics, wood, coal and metals [2-4]. The most iterated explosions are the ones where metal particles are involved, encompassing 24% of all [5].

With the development of industrial processes and progressive usage of powder in technology, it is more important than ever to recognize the threats and establish preventive actions in order to maintain safety. Therefore, better understanding of the details is necessary to decrease the probability of occurrence of dust explosions.

Moreover, the combustion of particles has drawn a great amount of attention from the researchers which is caused by the inevitable shortage of fossil fuels and the increasing urgent need for energy in near future [6]. In spite of the efforts conducted to analyze the combustion characteristics of metal particles, there are still unknown issues to be clarified [7].

Among all materials, iron is widely studied in the context due to its abundancy. Iron is utilized for synthesizing of liquid hydrocarbons as a catalyst [8] and NO_x control [9]. The byproduct of iron combustion, iron oxide, can be collected and recycled using existing metal smelters or novel technological processes powered by green power sources [10]. The kinetics of the reaction occurring between iron and air is so challenging due to its complicated reaction chains and the mechanism in which iron burns from the surface. Also, providing proper experimental conditions is another obstacle.

Many researches which have been carried out on iron particles combustion are concerned about flame propagation through a dispersed dust of particles both experimentally and analytically and few specific studies are executed to solely discuss the combustion of a single iron particle. TANG et al [11] performed experiments in a reduced gravity condition during a flight in order to estimate the quenching distance, flame speed and temperature of a rich iron dust suspension in air. TANG et al [12] also conducted the so-called argon/helium experiment to distinguish the combustion modes of iron particles. In a series of studies, SUN et al [13-16] investigated the combustion characteristics of the iron particles such as maximum and flame temperature, flame propagation speed. They examined the effect of particle diameter on the burning time as well as temperature profile and flame speed as a function of dust concentration. BROUMAND et al [17] analytically studied the burning velocities of micro particles of the iron regarding their concentration across a vertical duct. Their results showed that the concentration of particles at the leading edge of combustion zone influences the flammability limits more than the ones which are far from the flame front.

In the mathematical modeling of thermal science problems, the governing equations consist of ordinary and partial differential equations which in some cases are nonlinear and cannot be solved by analytical solutions. To overcome this problem, researchers have proposed different approaches to solve non-linear differential equations [18-22]. Some of the suggested methods are Homotopy Perturbation method [23, 24], differential transformation method [25], homotopy analysis method [26], Adomian’s decomposition method [27], variational iteration method [28], Akbari- Ganji’s method [29, 30], least square method [31], Galerkin method [32] and collocation method [33]. GHADIKOLAEI et al [34] studied an unsteady two dimensional squeezing non-Newtonian flow in the presence of magnetic field and thermal radiation using AGM [34]. Non-Newtonian fluid over a linear stretching sheet in boundary layer flow was analytically investigated by RAHIMI et al [35]. In this research, CM was applied to solving governing equation. SAEDI et al [36] utilized two different methods of AGM and DTM to examine the influence of adding nanoparticle in blood flow. They considered the effects of magnetic field and porosity in their investigations. HPM, CM and finite element method(FEM) were applied to investigating the effect of variable magnetic field on nanofluid flow in movable parallel plates [37]. ATOUEI et al [38] scrutinized heat transfer of semi-spherical fins with the aid of LSM and CM. The effects of heat generation, convection and radiation heat transfers and temperature dependent properties were taken into account in their study.

In this paper, analysis of combustion of single iron particle is developed by considering thermal radiation effect, convective heat transfer and particle density variations. By considering these items, the governing equation becomes non-linear and requires special mathematical methods to solve.

At the first step of this study, the energy equation for combustion of single iron particle is derived. Then, the dimensionless form of governing equation is obtained. Afterwards, the Akbari-Ganji’s, Galerkin and collocation techniques are utilized to solve this equation. In order to validate the accuracy of these methods, a numerical approach, the forth order Runge-Kutta method, is applied. Furthermore, a comparative study is conducted to investigate the efficiency of implemented methods based on the mean relative errors and run times. Finally, the effects of non-dimensional parameters on temperature distribution are examined.

2 Mathematical modeling

In this paper, the system is assumed to be lumped, i.e., particle temperature does not change within the body and all of its regions are at the same temperature. To satisfy this condition, the material diffusivity needs to be large enough which is applicable for iron particle. Hence, the temperature within the particle body is only a function of time. The model assumptions are presented below.

The ambient medium consists of oxygen and nitrogen as inert gas and spherical particle only reacts with the oxygen gas.

1) Thermophysical properties of the ambient medium and particle are considered to be constant except for the particle density which is a function of temperature.

2) The iron particle is spherical and its diameter remains constant during combustion process as well as its shape [39, 40].

Considering a control volume surrounding the particle, the first law of thermodynamics is expressed as follows [41]:

                  (1)

where is the rate of energy absorbed by the particle by means of radiation mechanism from the medium, is rate of energy leaving the particle surface to the environment by convection and radiation mechanisms, is the energy generated by combustion within the particle and is the rate of change in the energy of control volume which is related to the alterations of the particle temperature. is calculated as follows [42]:

                             (2)

where α_s is the radiative absorption coefficient, σ is the Stefan-Boltzmann constant, A_s is the particle outer surface area and T_surr is the surrounding temperature of control volume.is computed as follows [42]:

               (3)

where is the mean convective heat transfer coefficient, T_s is the time dependent temperature of the particle,is the medium temperature and ε_s is the emissivity coefficient of the iron surface. is defined as follows [43]:

                     (4)

is the specific enthalpy of reaction and is the reaction rate per surface area of iron particle combustion with oxygen which is given as follows [43]:

=-7.05×        (5)

where P_O2 is the partial pressure of the oxygen in the surrounding medium.

The specific enthalpy of reaction is presented as below[43]:

                       (6)

is expressed as follows [41]:

       (7)

where is the rate of change in total energy of iron particle and is the rate of change in internal energy of the particle.

Since it is assumed that the particle gains a final constant velocity, the rate of changes in kinetic energybecomes zero. Also, due to small reaction time, the rate of changes in potential energyis considered to be zero.

The rate of changes in internal energy is calculated as follows [41]:

     (8)

where m_p is the particle mass, u is specific internal energy, ρ_p is the density of particle and V_p is the particle volume.

By substituting Eqs. (a)-(4) and (8) in Eq. (1), below relation is obtained:

                   (9)

According to the Kirchhoff’s law of radiation, at a given temperature and wavelength, the values of absorption and emission coefficients are equal. Particle density is a function of temperature which is given as follows [41]:

               (10)

where β is the coefficient of temperature- dependence of density. Consequently, the general relation for energy equation is given as follows:

       (11)

For solving the governing equation, an initial condition is required which is the temperature of iron particle at the beginning of combustion process.

T(0)=T_ig                                                  (12)

In order to simplify the solving process, the non-dimensional form of the energy equation is obtained using the following parameters:

, , , ,

, ε₁=βT_ig,      (13)

By substituting the non-dimensional parameters in Eqs. (11) and (12), the energy equation and its initial condition are given as follows:

                      (14)

2.1 Convective heat transfer coefficient derivation

As mentioned previously, the particle falls freely because of the gravity, and gains a final velocity which remains constant due to the forces balance on the particle. The considered forces are buoyant (F_b), drag (F_d) and weight (W). The forces balance is given as follows [44]:

,

         (15)

                     (16)

where C_d is the drag coefficient, A_pr is the projected area of the particle perpendicular to the direction of falling and V_f is the final velocity of particle.

The medium is assumed to be quiescent which means the Reynolds number for the flow in the vicinity of particle is relatively small.

For the mentioned flow (Re<1), stokes flow, the drag coefficient for a spherical particle is calculated as follows [44]:

                      (17)

Substituting of Eq. (17) into Eq. (16) results in Eq. (18):

                    (18)

Furthermore, for low Reynolds numbers, the average Nusselt number can be calculated as follows [45]:

                     (19)

where d_p is the particle diameter, λ_∞ is the thermal conductivity of medium and Pe is the Peclet number of the medium.

2.2 Determining maximum flame temperature

In order to determine the maximum flame temperature, the combustion time for single particle should be known. According to the thermodynamic specifications of iron particle, its adiabatic flame temperature, boiling point and the volatilization temperature are 2230, 3130 and 3400 K, respectively which is the proof of heterogeneous combustion of iron particle in air. For a diffusion- controlled regime, the burning time of an iron particle in oxygen was obtained by Glassman as follows [46]:

              (20)

where τ_bdiff is the burning time, D_O2,∞is the mass diffusivity of the oxygen, v is the mass stoichiometric index of combustion of iron and Y_O2,∞ is the mass fraction of oxygen in the ambient gas.

3 Solving procedure

In this paper, AGM, GM and CM are chosen to solve Eq. (14) with respect to initial condition. In order to verify the obtained results, a numerical calculation is performed using the forth order Runge-Kutta method [47].

3.1 Akbari-Ganji’s method (AGM)

Boundary and initial conditions are always required to solve linear or non-linear differential equations. In AGM, a simple analytical solution is presented using the governing boundary and initial conditions. The differential equation (P) for u and its derivatives is defined as follows [48]:

              (21)

The boundary (or initial) conditions of Eq. (21) are given as follows [48]:

          (22)

The polynomial solution is considered as follows [48]:

         (23)

It is clear that the more values of n, the more precise of the solution. In order to obtain the n+1 unknown coefficients of polynomial, n+1 equations are needed. The unknown coefficients are computed with the aid of following items [48].

The approximate answer should satisfy the boundary (or initial) conditions. First, boundary condition at x=0 is applied to Eq. (23) as follows [48]:

      (24)

Then, the boundary condition at x=l is performed as follows [48]:

       (25)

By substituting Eq. (23) into Eq. (21), a function is obtained which should be satisfied at x=0, l as follows [48]:

             (26)

The derivatives of Eq. (21) should be satisfied at x=0, l. Here is the example for the first order derivative as follows [48]:

         (27)

3.2 Galerkin and collocation

GM and CM, which are called the weighted residuals methods (WRMs), have many common points. The differential equation is written as follows [33]:

D(u(x))=p(x)                              (28)

where D is differential operator.

The function u is approximated by which is a linear combination of a set of functions [33].

                          (29)

The error or residual is calculated as follows [33]:

              (30)

The main idea of CM and GM is to minimize the average residual of Eq. (30) [33].

i=1, 2, …, n                 (31)

The definition of W_i(x) is the difference between CM and GM which are presented as follows [32, 33]:

In Galerkin,

i=1, 2, …, n                     (32)

Collocation,

i=1, 2, …, n                 (33)

where δ is the Dirac Delta function.

4 Results and discussion

In this study, combustion of single iron particle that falls into oxidizer medium is examined. To solve the nonlinear differential equation, the analytical methods including AGM, GM and CM are utilized and the temperature of iron particle is determined. In addition, the effects of parameters such as particle diameter, ε₂ and ψ as dimensionless parameters on the temperature distributions are investigated and discussed. Furthermore, the mean relative error and run times of analytical methods are compared to each other and results are validated with the data obtained from numerical solution.

The initial temperature and pressure of the medium are considered to be 1000 K and 101 kPa, respectively. The summary of the thermo-physical properties of iron particle and the coefficients used in the equations are presented in Table 1.

Table 1 Thermo-physical properties of iron particle and coefficients used in mathematical model

Figure 1 compares the dimensionless temperatures of the iron particle with the diameter of 80 mm obtained by three analytical methods with numerical outputs. The accuracy of the proposed methods is also depicted. The obtained correlations for each method are as follows:

For AGM,

θ=1+3.8581τ-2.4081τ²+1.0327τ³-0.9186τ⁴_;

For GM,

θ=1+3.7342τ-1.8129τ²_;

For CM,

θ=1+3.7976τ-1.9677τ²_.

The relative errors of each analytical method with respect to the numerical results are presented in Table 2. It is clear that the AGM method has the lowest relative errors among the implemented methods.

For further discussion of the competency of the analytical methods, the effects of polynomial degree on relative error and process time are investigated. The mean relative errors and run times of AGM, GM and CM methods for various degrees of polynomials are given in Table 3.

Figure 1 Comparison of dimensionless temperatures of iron particle with diameter of 80 mm obtained by analytical methods with numerical outputs

Table 2 Relative errors of analytical methods with respect to numerical results

The increase in the degree of polynomial causes the increase in the run time and decrease in the relative error. In order to reach a better approximation of the solution of differential equation, one can increase the polynomial degree. Consequently, the number of algebraic equations for obtaining polynomial constants is increased which requires more run time to be solved.

Tables 4-7 are presented in order to gain a better comprehension of the dependency of the relative error and run time on the implemented methods and particle diameters. The results indicate that with rising particle diameter, both the relative error and run time increase. Inversely, by increasing the degree of polynomial for the AGM method, the relative error diminishes and run time increases.

Table 3 Mean relative errors and run time for various degrees of proposed polynomials for particle with diameter of 80 mm

Table 4 Mean relative errors and run time for various degrees of proposed polynomials for particle with diameter of 20 mm

Table 5 Mean relative errors and run time for various degrees of proposed polynomials for particle with diameter of 40 mm

According to the data presented so far, it can be concluded that AGM method is more appropriate than the CM and GM for analytical modeling of single particle combustion due to its less runtime and mean relative error. In order to study the effects of particle characteristics and medium conditions, AGM is applied as superior model.

Table 6 Mean relative errors and run times for various degrees of proposed polynomials for particle with diameter of 60 mm

Table 7 Mean relative errors and run times for various degrees of proposed polynomials for particle with diameter of 100 mm

The effect of particle diameter on temperature is investigated and presented in Figure 2. As it is illustrated, with increasing the particle diameter, the temperature of particle rises. Heat storage in the particle and heat loss from its surface depend on the particle’s volume and surface area, respectively. By increasing the particle diameter, the ratio of its volume to its surface increases which causes more heat to be accumulated in the particle, leading to high temperature. On the other hand, based on Eq. (19), with rising particle diameter, the convection heat transfer coefficient reduces leading to less heat loss which justifies the increase in the particle temperature.

Figure 3 represents the variations of the particle non-dimensional temperature with different values of dimensionless parameter ψ. The dimensionless parameter ψ is proportional to the ratio of heat released by combustion to convection heat loss. According to the figure, increasing the value of ψ results in more particle temperature. Based on the definition of ψ, the value of this parameter can be increased with increasing the amount of heat released by combustion or decreasing in the convection heat loss, which causes higher particle temperature.

Figure 2 Dimensionless temperature of particle as function of particle diameter and dimensionless time

Figure 3 Dimensionless temperature of particle as function of dimensionless parameter ψ and time

Figure 4 shows the effect of non-dimensional parameter ε₂ on the dimensionless temperature of particle. As shown in the figure, with rising the value of ε₂, particle temperature decreases. According to Eq. (14), by altering this parameter (ε₂), the amount of radiative heat loss changes. Moreover, increasing in ε₂ results in more radiative heat loss leading to lower particle temperature.

Figure 4 Dimensionless temperature of particle as a function of dimensionless parameter ε₂ and time

One of the major outcomes of the present study is the calculation of particle maximum temperature which has significant applications in dust propagation simulation. In order to measure the maximum temperature, the combustion time should be calculated to determine when the particle will be quenched. By assuming that the reaction occurs in diffusion-controlled regime, Eq. (20) can be utilized to determine quenching time. The calculated maximum temperature as a function of diameter is demonstrated in Figure 5. As can be seen, particle maximum temperature increases by rising its diameter. Heat storage in the particle and its surface heat loss are proportional to the particle volume and surface area, respectively. The ratio of particle surface area to its volume is 3/r_p, therefore, with increasing the particle diameter, surface heat loss is less than particle heat storage, which results in more temperature.

Figure 5 Maximum temperature of particle as a function of particle diameter

5 Conclusions

In the present study, combustion of single iron particle falling into the oxidizer medium was investigated. The developed mathematical model was obtained with respect to the facts that iron particle reacts heterogeneously in air and the Biot number is small. The effects of thermal radiation, convective heat transfer and variations of particle density during combustion process were investigated and discussed. In order to find the answer for nonlinear energy equation, the dimensionless form of governing equation was solved with the aid of analytical approaches including AGM, CM and GM. Afterwards, for checking the accuracy of analytical methods, the energy equation was solved by forth order Runge-Kutta method. The comparisons showed that the analytical results had good agreements with the numerical outputs. According to the obtained results, AGM was the more appropriate method compared to CM and GM due to its less mean relative error and run time. Subsequently, the effects of particle diameter and dimensionless parameters on temperature distribution are examined. Based on the gained results, with increasing particle diameter, its temperature increases. Finally, the maximum temperature of particle is calculated and then it is shown that its amount increases with rising particle diameter.

Nomenclature

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

c_p,∞

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

C_D

Drag coefficient

d_P

Particle diameter, m

D_O2,∞

Mass diffusivity of oxygen in air, m²·s^-1

E

Total energy of particle, J

F_B

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

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

m_p

Mass of particle, kg

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

T_surr

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

V_term

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

Greek symbols

α_s

Absorptivity of particle surface

β

Coefficient of temperature-dependence of density, K^-1

ε

Emissivity of particle surface, non- dimensional parameter

θ

Dimensionless temperature

λ_P

Thermal conductivity of iron particle, W·m^-1·s^-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

ρ_p

Density of burning iron particle, kg·m^-3

ρ_p,∞

Density of iron particle at T_∞, kg·m^-3

ρ_∞

Density of ambient gaseous oxidizer, kg·m^-3

σ

Stefan-Boltzmann constant, W·m^-2·K^-4

τ

Burning time of particle, dimensionless time

ψ

Non-dimensional parameter

Subscripts

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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(Edited by FANG Jing-hua)

中文导读

单一微米级铁颗粒燃烧温度的分析研究

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AGM(Akbari-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。

关键词：铁粒子燃烧；解析解；热辐射；对流热交换

Received date: 2018-02-18; Accepted date: 2019-03-12

Corresponding author: Peyman MAGHSOUDI, PhD Candidate; Tel: +98-21-77240540; E-mail: maghsoudi_peyman@mecheng.iust. ac.ir; ORCID: 0000-0003-0018-9344