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Time variation of combustion temperature and burning time of a single iron particle
International Journal of Thermal Sciences 65 (2013) 136e147
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Time variation of combustion temperature and burning time of a single iron particle Mehdi Bidabadi, Majid Mafi* Combustion Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 April 2011 Received in revised form 26 October 2012 Accepted 26 October 2012 Available online 4 December 2012
In the present study, combustion of a single iron particle is modeled by virtue of a novel thermophysical procedure. The temperature of iron particle during combustion is studied analytically and numerically. In the proposed model, the effect of thermal radiation from the external surface of burning particle, and alterations of density of iron particle with temperature are considered. Iron particle burns heterogeneously in air. Because of high thermal conductivity and micro size of iron particle, the Biot number is negligibly small, and the lumped system analysis can be utilized for the combustion modeling. The nonlinear energy equation resulted from modeling is solved by using homotopy perturbation method. The assumptions applied in the modeling are such that do not violate the actual combustion phenomenon. Also, the numerical solution of nonlinear differential equation is presented and compared with the analytical solution obtained from homotopy method. It is the first model presented for analyzing the combustion of single iron particle. 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Single iron particle Combustion modeling Nonlinear differential equation Homotopy perturbation method Heterogeneous reaction Iron particle burning time
1. Introduction Combustion of metallic materials is a challenging scientific subject that also has significant practical applications. Since the combustion of metallic particles is a high exothermic chemical reaction, use of them in the industrial operations is significant and efficient. Combustion of iron particle cloud experimentally investigated in some works. Iron is regarded as a non-volatile metallic fuel and the oxidation process takes place as a heterogeneous surface reaction. The major characteristic feature of iron combustion is that iron burns heterogeneously in air. It means that combustion reaction occurs at the iron particle surface and no flame is observed in the gaseous oxidizer phase. Iron particles do not evaporate during the combustion process and also combustion product (iron oxide) remains in the condensed phase. Because of high energy density of iron, the combustion of iron can be used as an alternative fuel in automobile engines and solid propellant rocket motors in the future. Recently, Bidabadi et al. [1] analytically investigated the distribution of iron dust particles through unburned zone across the flame propagation in a vertical duct by using homotopy analysis method. Also, Bidabadi et al. [2] theoretically proposed a mathematical modeling of velocity and number * Corresponding author. Tel.: þ98 21 77 240 197; fax: þ98 21 77 240 488. E-mail address: majid.mafi@gmail.com (M. Mafi). 1290-0729/$ e see front matter 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.10.019
density profiles of particles across the flame propagation through a micro-iron dust cloud. Both studies focused on the dynamic behavior of iron particle clouds and various forces exerted on the particles in the preheat zone, prior to the flame zone. In addition, Bidabadi et al. studied the combustion of various type of particle cloud in several works [3e14]. Furthermore, Tang et al. [15] experimentally performed the so-called argon/helium test to determine the modes of particle combustion in iron dust flames. Experiments were accomplished in a reduced-gravity environment. Also, Tang et al. [16] experimentally studied the quenching and laminar flames propagating in fuel-rich suspension of iron dust in air in a reduced-gravity environment provided by a parabolic flight aircraft. Moreover, Sun et al. [17] measured the particle velocity and particle number density profiles across upward and downward flame propagating through iron particle clouds by using the highspeed photomicrographs. Sun et al. [18] experimentally examined the concentration profile of particles across a flame propagating through an iron particle cloud. Sun et al. [19] experimentally evaluated the temperature profile across the combustion zone propagating through an iron particle cloud. Also, Sun et al. [20] experimentally investigated the combustion behavior of iron particles suspended in air. In addition, Steinberg et al. [21e32] have carried out many experiments on the burning iron rods. Homotopy is a mathematical technique for solving nonlinear differential equations [33,34]. Homotopy method has been used for
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
solving various nonlinear differential equations raised in mathematical modeling of physical phenomena [35e43]. In the previous works, combustion of iron cloud has been experimentally investigated, or the dynamic behavior of iron particle in preheat zone has been studied. This paper proposed a novel analytical approach to model the combustion of single iron particle burning in the gaseous oxidizing medium. By using this model, the variation of iron particle temperature during the combustion process as a function of time can be obtained. This model includes the thermal radiation effect and, also, variation of specific heat of iron particle by temperature. This model applied for the combustion of iron particle for the first time, and can be improved in the future studies to include more complicated parameters that influence the circumstance of combustion. Because of nonlinearity of the governing differential equation, the homotopy perturbation method is utilized for solving the energy equation of iron particle. 2. Homotopy perturbation method Consider the following nonlinear differential equation:
A ðuÞ f ðrÞ ¼ 0;
r˛U
(1)
with boundary conditions
vu B u; ¼ 0; vn
r˛L
(2)
where A ðuÞ is a general differential operator, B is a boundary operator, f(r) is a known analytic function, and L is the boundary of domain U. The operator A ðuÞ can be divided to two parts L ðuÞ and N ðuÞ, where L is linear, and N is nonlinear.
A ðuÞ ¼ L ðuÞ þ N ðuÞ
(3)
Hence Eq. (1) can be written as
L ðuÞ þ N ðuÞ f ðrÞ ¼ 0;
r˛U
(4)
by means of homotopy technique [33,34], a homotopy function can be constructed, which satisfies the following equation:
H ðv; pÞ ¼ ð1 pÞ½L ðvÞ L ðu0 Þ þ p½A ðvÞ f ðrÞ ¼ 0;
p ˛½0; 1
(5)
or
H ðv; pÞ ¼ L ðuÞ L ðu0 Þ þ pL ðu0 Þ þ p½N ðvÞ f ðrÞ ¼ 0 where
vðr; pÞ : U ½0; 1/R
parameter”, applying the classical perturbation technique, we can assume that the solution of Eq. (6) can be expressed as a power series in p, i.e.,
v ¼ v 0 þ pv 1 þ p 2 v 2 þ /
(7)
where ˛½0; 1 is an embedding parameter, and u0 is the initial approximation of Eq. (1) which satisfies the boundary conditions. Obviously, we have
H ðv; 0Þ ¼ L ðvÞ L ðu0 Þ ¼ 0
(8)
H ðv; 1Þ ¼ A ðvÞ f ðrÞ ¼ 0
(9)
The process of changing p from zero to unity results in changing vðr; pÞ from u0 to u(r). This is called deformation, and also L ðvÞ L ðu0 Þ and A ðvÞ f ðrÞ are called homotopic in topology. If the embedding parameter p˛½0; 1 is considered as a “small
(10)
and setting p ¼ 1 results in the approximate solution of Eq. (1) as
u ¼ lim v ¼ v 0 þ v 1 þ v 2 þ / p/N
(11)
It is worth to note that the major advantage of He’s homotopy perturbation method is that the perturbation equation can be freely constructed in many ways (therefore is problem dependent) by homotopy in topology and the initial approximation can also be freely selected.
3. Modeling of combustion of single iron particle As the thermal diffusivity of substance is large and Biot number is very small (BiH 0.1), it is assumed that the particle is isothermal. In this state, a lumped system analysis is applicable. When this criterion is satisfied, the variation of temperature with location within the body will be slight and can be reasonably be approximated as being uniform. It means that the iron particle has a spatially uniform temperature, thus implying very large thermal conductivity of iron particle. Therefore, the temperature of particle is a function of time only, T ¼ T(t), and is not a function of radial coordinate, T s T(r). The assumptions used in this modeling are: 1. The spherical iron particle burns in a quiescent, infinite ambient medium that contains only oxygen and an inert gas, such as nitrogen. There are no interactions with other particles, and the effects of forced convection are ignored. 2. Constant thermophysical properties for the iron particle and ambient gaseous oxidizer are used, except for the specific heat of the iron particle which varies by temperature. 3. The particle is of uniform temperature and radiates as a gray body to the surroundings without participation of the intervening medium. First, iron particle is considered as a thermodynamic system, and by using of principle of conservation of energy (first law of thermodynamics), the energy balance equation for this particle can be written as
E_ in E_ out þ E_ gen ¼ (6)
137
dE dt
(12) p
where E_ in is the rate of energy entering the system which is owing to absorption of entire radiation incident on the particle surface from the surrounding surfaces, E_ out is the rate of energy leaving the system by mechanisms of convection on the particle surface to the ambient oxidizer gas (air), and thermal radiation that emits from the outer surface of particle, E_ gen is the rate of generation of energy inside the particle due to the combustion process and equals to the heat released from the chemical reaction, and (dE/dt)p is the rate of change in total energy of iron particle that is specified to the rate of change in the temperature of the particle. 4 E_ in ¼ as q_ incident ¼ as sAs Tsurr
(13)
where as is the absorptivity of the iron particle surface, q_ incident is the flux of thermal radiation incident to the particle surface from the surrounding surfaces which enclose the particle,
138
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
s ¼ 5.67 108 W/m2 K4 is the StefaneBoltzmann constant, As ¼ 4pr2p is the outer surface area of the iron particle, and Tsurr is the absolute temperature of the surrounding surfaces.
rp Vp cp
dT dt
p
4 þ hconv As ðT TN Þ þ 3 s sAs T 4 Tsurr
_ Fe As Dh ℛ comb ¼ 0 E_ out ¼ hconv As ðTs TN Þ þ 3 s sAs Ts4
(14)
where hconv is the average convective heat transfer coefficient at the entire surface of iron particle, Ts is the absolute temperature of the particle that is only a function of time, TN is the absolute temperature of the ambient oxidizer gas (air) around the particle, 3 s is the emissivity of the surface of the iron particle.
_ Fe As Dh E_ gen ¼ Q_ comb ¼ ℛ comb
(15)
where Q_ comb is the rate of heat released within the particle due to _ Fe is the reaction rate of iron with oxygen, Dhcomb the combustion, ℛ is the enthalpy of reaction of iron which is a negative quantity. The rate of change in total energy of iron particle is composed of three terms and is expressed as
dE dt
¼ p
dU dðK:E:Þ dðP:E:Þ þ þ dt p dt dt p p
dE dt
p
dU du dT ¼ ¼ mp ¼ rp Vp cp dt p dt p dt p
h
4 as sAs Tsurr hconv As ðT TN Þ þ 3 s sAs T 4
(17)
i
dT _ Fe As Dh r þ ℛ ¼ V c p p p comb dt p
Tð0Þ ¼ Tig
(19b)
Also, the density of iron particle is a function of temperature of particle, and thus, it can be considered as a linear function of particle temperature
rp ¼ rp ðTÞ ¼ rp;N ½1 þ bðT TN Þ
(18)
In general, both absorptivity and emissivity of the surface depend on the temperature and the wavelength of radiation. Kirchhoff’s law of radiation states that the absorptivity and the emissivity of a surface at a given temperature and wavelength are equal, i.e., (as y 3 s). Therefore, the governing energy equation of burning iron particle can be expressed as
(20)
where rp,N is the density of particle at reference temperature of TN, and b is the coefficient of variation of density of iron with temperature. Hence, the governing equation of burning iron particle can be written as
rp;N Vp cp ½1 þ bðT TN Þ
dT dt
p
þ hconv As ðT TN Þ
4 _ Fe As Dh ℛ þ 3 s sAs T 4 Tsurr comb ¼ 0
where mp is the mass of the iron particle, u is the specific internal energy of the particle, rp is the density of the burning iron particle, Vp is the volume of the particle, and cp is the specific heat of the iron particle. Iron is an incompressible substance, therefore, the cp,p and cv,p values of iron particle are identical and both specific heats can be represented by a single symbol which is denoted by cp. Because the temperature of particle is uniform at each instant of time, thus the surface temperature of particle equals to the temperature of particle, Ts ¼ T. By substituting of (13)e(15) and (17) in (12), we have
The iron particle temperature is taken to be uniform. The temporal variation of temperature of iron particle can be obtained by solving of above equation. This equation is a nonlinear nonhomogeneous first order differential equation that cannot be solved by conventional methods in mathematics, and exact solution of this equation is not obtained easily. Obviously, for solving this first order differential equation, an initial condition is required. The initial temperature of the iron particle at the beginning of combustion can be regarded as the initial condition. This temperature is known as ignition temperature.
(16)
where (dU/dt)p is the rate of change in the total internal energy of the particle, (d(K.E.)/dt)p is the rate of change in the total kinetic energy of the particle, and (d(P.E.)/dt)p is the rate of change in the total potential energy of the particle. Assuming particle falling freely due to the gravity with constant terminal velocity, the rate of change in total kinetic energy is zero, (d(K.E.)/dt)p ¼ 0. On the other hand, the rate of change in total potential energy of particle during short burning time of particle is negligibly small and it can be neglected compared to the considerable amount of the rate of change in internal energy of the particle, (d(P.E.)/dt)p y 0. Consequently
ð19aÞ
ð21Þ
This equation has two nonlinear terms. For solving this nonlinear differential equation, all the terms must be converted to the dimensionless form. The following dimensionless variables are defined to solve the equation:
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <
T Tig T qN ¼ N Tig Tsurr qsurr ¼ Tig t s ¼ rp;N Vp cp
q ¼
> > > > hconv As > > > _ Fe As Dh > ℛ > comb > > J ¼ > > > hconv As Tig > > > > 31 ¼ bTig > > > > 3 > 3 s sT > ig > > 32 ¼ > > > hconv > > > :
(22)
Consequently, the nonlinear differential equation and its initial condition can be expressed in the dimensionless form
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
8 > < > :
½1 þ 31 ðq qN Þ
dq 4 4 þ ðq qN Þ þ 32 q qsurr J ¼ 0 ds qð0Þ ¼ 1
(23a)
The governing equation can be rearranged as
8 > < > :
ð1 31 qN Þ
dq dq 4 4 þ q ¼ 32 q qsurr J þ 31 q qN ds ds qð0Þ ¼ 1 (23b)
In above equation, 31 and 32 are the small parameters compared to the other terms appeared in equation. This equation is solved by the homotopy perturbation method.
p 1 : ð1 31 qN Þ p 2 : ð1 31 qN Þ p 3 : ð1 31 qN Þ p 4 : ð1 31 qN Þ p 5 : ð1 31 qN Þ
dq0 p 0 : ð1 31 qN Þ þ q0 ¼ 0 ; q0 ð0Þ ¼ 1 ds dq1 dq 4 4 þ q1 ¼ 32 q0 qsurr J þ 31 q0 0 qN ds ds dq2 dq 4 4 þ q2 ¼ 32 q1 qsurr J þ 31 q1 1 qN ds ds dq3 dq2 4 4 qN þ q3 ¼ 32 q2 qsurr J þ 31 q2 ds ds dq4 dq 4 4 þ q4 ¼ 32 q3 qsurr J þ 31 q3 3 qN ds ds dq5 dq 4 4 þ q5 ¼ 32 q4 qsurr J þ 31 q4 4 qN ds ds
139
dv 0 dv 1 dv 2 þ pð1 31 qN Þ þ p 2 ð1 31 qN Þ ds ds ds q d 0 þ v 0 þ pv 1 þ p 2 v 2 ð1 31 qN Þ q0 ds dq þ p ð1 31 qN Þ 0 þ q0 þ ds dq 4 p 32 v 40 qsurr J þ 31 v 0 qN ds 4 þ L ðvÞ L ðq0 Þ þ pL ðq0 Þ þ p 32 v 4 qsurr dv J þ 31 v qN ¼ 0 ds
ð1 31 qN Þ
ð26Þ
Therefore
; q1 ð0Þ ¼ 0 ; q2 ð0Þ ¼ 0 (27)
; q3 ð0Þ ¼ 0 ; q4 ð0Þ ¼ 0 ; q5 ð0Þ ¼ 0
4. Homotopy perturbation solution
By solving above differential equation system, the following results can be achieved:
According to homotopy perturbation method, the following homotopy is constructed:
q0 ðsÞ ¼ 1
(28)
then
dv J þ 31 v qN L ðvÞL ðq0 Þ þ pL ðq0 Þ þ p ds dq dq0 ¼ 0ð1 31 qN Þ þ q ð1 31 qN Þ q0 ds d s dq 4 4 qN ¼ p 32 q qsurr J þ 31 q ds dq ð1 31 qN Þ 0 þ q0 ds
4 4 32 v qsurr
(24) where
L ðvÞ ¼ ð1 31 qN Þ
dv þv ds
dq0 L ðq0 Þ ¼ ð1 31 qN Þ þ q0 ds
(25)
By substituting Eqs. (24) and (25) in Eq. (23) and equating corresponding coefficient of same power terms, the following differential equation system can be obtained:
q1 ðsÞ ¼
h 32
i 4 1 qsurr J qN 1 exp
s
1 31 qN
(29)
and by substituting this function into the third differential equation in system, we have
ð1 31 qN Þ
h i dq2 4 þ q2 ¼ 32 32 1 qsurr J qN ds 4 s 4 qsurr 1 exp 1 31 qN h i 4 J þ 31 32 1 qsurr J qN s 1 exp 1 31 qN i (h 4 32 1 qsurr J qN exp 1 31 qN )
s
1 3 1 qN
qN ; q2 ð0Þ ¼ 0
(30)
The solution of this differential equation is
q2 ðsÞ ¼ cexp
*
s
"
1
!
exp
s
#
2 ð1 31 qN Þ ð31 qN 1Þ 4 4 2s 1 3s 4 2 4 2 232 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ exp 32 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ exp þ 3 ð1 31 qN Þ ð1 31 qN Þ 13 20 3 2 0 1 4 4 4 3 B 6B C 4 1 þ 432 qsurr 1 1232 qsurr 1 ðj 1Þþ C7 6B 4 4 4 3 4 4 A C7 j q q q j ð þ 4 1 þ 3 1Þ 3 3 1 þ 6 B 32 qN þ 352 qsurr 1 þqN @ 2 2 surr N 2 surr C7 6B 4 2 3 2 C7 6B q j j 3 1 ð 1Þ 4 3 ð 1Þ 12 2 surr 2 C7 6B C7 6B 2 3 2 2 C7 þ 6@ 2 4 4 4 4 A7 2 3 7 6 4 3 2 q q j q j q j q j j þ6 N 32 1 þ 32 surr 1 432 surr 1 ð 1Þ þ 632 surr 1 ð 1Þ 632 surr 1 ð 1Þ þ 7 6 7 6 7 6 7 6 5 4 s 2 ð31 qN 1Þ exp ð1 31 qN Þ ð1 31 qN Þ
þ
+
(31)
3 0 1 2 7 6 2 2 3 2 4 4 @ 43 1 j þ q þ 3 q4 1 6 1 j þ qN þ 32 q4surr 1 þ31 1 þ 4qN 32 þ 8qN 32 1 þ 32 qsurr 1 j þ 4qN 32 32 1 32 qsurr þ j As 7 N 2 2 surr 5 4 2
2
0
2
6 4 3 2 þ31 1 þ 6qN 32 þ 12qN 32 1 j þ 32 632 1 j þ qN þ 32 qsurr 1 6 6 ð3 q 1Þ 1 j þ q þ 3 q4 1 2 B B N 6 1 N 2 surr @ 6 4 þ6qN 32 j 1 þ 32 32 qsurr 4
3 1 7 q4surr 1 7 C s 7 Cexp 7 A ð1 31 qN Þ 7 5
The unknown factor c can be determined by using the initial condition
h i 1 4 4 4 2 4 2 232 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ 32 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ þ 3 20 13 4 2 3 4 4 4 4 4 5 2 3 4 3 2 6 B 32 qN þ 32 qsurr 1 þqN 1 þ 43 qsurr 1 123 qsurr 1 ðj 1Þ þ 123 qsurr 1 ðj 1Þ 432 ðj 1Þ C 7 2 2 2 6B C7 6B 2 3 C7 7 6B 4 3 4 2 4 4 4 4 6 B þ32 ðj 1Þ qsurr þ 4qN 32 1 þ 32 qsurr 1 j þ 6qN 32 1 þ 32 qsurr 1 j 432 qsurr 1 ðj 1Þ C !# * " + C7 7 6B C 2 2 1 6@ A7 4 4 2 3 3 2 7 6 c ¼ þ632 qsurr 1 ðj 1Þ 632 qsurr 1 ðj 1Þ þj 2 7 6 ð31 qN 1Þ 5 4 2 ð31 qN 1Þ 3 2 0 2 1 4 3 2 4 2 B 632 1 j þ qN þ 32 qsurr 1 þ31 1 þ 6qN 32 þ 12qN 32 1 j þ 32 qsurr 1 6 C7 6 ð31 qN 1Þ 1 j þ qN þ 32 q4surr 1 @ A7 5 4 4 þ6qN 32 j 1 þ 32 32 qsurr (32)
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Therefore, the final solution of the governing equation describing the temperature changes of the burning iron particle versus time, can be expressed as
qðsÞ ¼ q0 ðsÞ þ q1 ðsÞ þ q2 ðsÞ þ /
(33)
5. Results and discussion 5.1. Chemical kinetics of iron combustion The chemical reaction formula of combustion of iron at stoichiometric condition is expressed as
nFe Fe þ nO2 O2 /nFeO FeO mFe Fe þ mO2 O2 /mFeO FeO
(34)
141
particle can attain when all the forces balance each other and the net force acting on the particle (and thus its acceleration) is zero. The forces exerted on the falling particle are usually the weight of the particle, the buoyant force, and the drag force. At terminal velocity the particle will not be accelerating so the forces acting on the free falling particle are at the equilibrium, i.e., the weight of the particle equals the buoyant force plus the drag force ðV p ¼ V term Þ. The particle terminal velocity can be estimated as follows:
W ¼ FB þ FD 4 4 pr gr3 ¼ prN gr3p þ 12 CD rN Apr V 3 p p 3
2 term
(41)
Thus where
8 < mFe ¼ nFe MFe m ¼ nO2 MO2 : O2 mFeO ¼ nFeO MFeO
(35)
V
term
¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u u8p rp rN gr3 p t 3CD rN Apr
(42)
where nFe is molar stoichiometric coefficient of iron in chemical reaction formula, nO2 is molar stoichiometric coefficient of oxygen in chemical reaction formula, nFeO is molar stoichiometric coefficient of iron oxide in chemical reaction formula, MFe is atomic weight of iron, MO2 is molecular weight of oxygen, MFeO is molecular weight of iron oxide, mFe is mass stoichiometric coefficient of iron in chemical reaction formula, mO2 is mass stoichiometric coefficient of oxygen in chemical reaction formula, mFeO is mass stoichiometric coefficient of iron oxide in chemical reaction formula. Dividing the mass stoichiometric coefficients in the chemical reaction formula to the mass stoichiometric coefficient of iron changes the form of chemical reaction formula of combustion of iron to
where W is the weight of iron particle, FB is buoyant force, FD is drag force, g is gravitational acceleration, rN is density of gaseous medium, CD is drag coefficient, Apr is projected area of particle perpendicular to the direction of falling, V term is the terminal velocity of falling iron particle. Assuming the iron particle burns in the quiescent medium and the relative velocity between particle and ambient gas is very small. It means that the Reynolds number for fluid flow around the particle is infinitesimally small (Rep < 1). At low Reynolds numbers (it is called a Stokes flow if Rep < 1), the drag coefficient for a spherical particle is given by
Fe þ n O2 /ð1 þ nÞ FeO
CD ¼
(36)
24 24 24mN ¼ ¼ rN V p dp rN V term dp Rep
mN
where n is mass stoichiometric index which is defined as
n ¼
mO2 mFe
(37)
For combustion of iron, the chemical formula can be written as
(38)
Mass stoichiometric index, n, for iron combustion equals n ¼ (31.99 kg) O2/(111.69 kg) Fe ¼ 0.2864. And the rate of reaction of iron with oxygen per surface area of iron particle is experimentally obtained [24]
4 1 g _ Fe ¼ 7:05 103 exp 2:9684 10 P 2 ℛ O2 T cm2 s
Dhcomb ¼ 365:2
kJ kJ kJ ¼ 6:54 ¼ 6540 mol Fe g Fe kg Fe
V
term
(39)
(40)
¼
9mN
hconv dp
(44)
¼ 2 þ Pe þ Pe2 lnPe þ f ðPrN ÞPe2 :
lN
Rep 1 ;
PrN w1
(45)
where Nu is the average Nusselt number for iron particle, hconv is the convective heat transfer coefficient, dp is the diameter of the iron particle, lN is the thermal conductivity of the gaseous oxidizing environment, Pe ¼ RepPrN is the Peclet number, PrN is the Prandtl number of ambient gaseous oxidizing medium, and
f ðPrN Þ ¼ 5.2. Dynamics of a single iron particle and convective heat transfer coefficient
2gr2p rp rN
On the other hand, in the case of particle falling in the motionless oxidizer medium, the average convective heat transfer coefficient on the particle surface can be obtained by a relation for the Nusselt number for low Reynolds number flow [44]
Nu ¼
where PO2 is the partial pressure of oxygen in the ambient gaseous medium (air). Also, the specific enthalpy of combustion of iron is [24]
therefore
sto
ð2 kmolÞ Fe þ ð1 kmolÞO2 /ð2 kmolÞ FeO ð111:69 kgÞ Fe þ ð31:99 kgÞ O2 /ð143:68 kgÞ FeO
(43)
1 2
213 þ 2 Pr3N 3PrN 2 40 lnPrN 2ðPrN þ 1Þ2 ðPrN 2ÞlnðPrN þ 1Þ 2Pr2N PrN 4g
(46) It is assumed that the iron particle is dropped into the air; it falls due to the gravity. During this free fall, iron particle reaches its terminal velocity which is the maximum velocity that a free falling
where g ¼ 0.5772156. is the Euler number. Prandtl number of ambient gas is defined as
142
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Table 1 Summary of the thermophysical properties used in the calculations. Property
Symbol
Value
Unit
Density of atmospheric air at 1000 K Density of burning iron particle at TN ¼ 1000 K Specific heat of burning iron particle Coefficient of temperaturedependence of density of iron particle Thermal conductivity of atmospheric air at 1000 K Mass diffusivity of oxygen in air at 1000 K Mass fraction of oxygen in air far from particle surface Mass stoichiometric index Partial pressure of oxygen in atmospheric air Emissivity of burning iron particle at 2300 K Ignition temperature of iron particle Ambient temperature Surrounding temperature Prandtl number of atmospheric air at 1000 K Dimensionless parameter defined in Eq. (22) Dimensionless parameter defined in Eq. (22) Dimensionless parameter defined in Eq. (22)
rN
0.3482
kg/m3
[46]
3
PrN ¼
Ref.
rp,N
7650
kg/m
[46]
cp
748.5
J/kg K
[46]
b
5.62 105
1/K
[46]
lN
66.7 103
W/m C
[47]
D O2 N
15.2 105
m2/s
[47]
YO2 ;N
0.233
e
e
n PO2
0.2864 21.278
e kPa
e e
3s
0.505
e
[48]
Tig
850
K
16
TN Tsurr PrN
1000 300 0.7107
K K e
e e [47]
31
0.051595
e
e
qN
1.17647
e
e
qsurr
0.35294
e
e
mN cp;N lN
(47)
where cp,N is specific heat capacity at constant pressure of ambient gaseous fluid. For the iron particle with diameter of 20 micron and temperature of 800 K falling in the air at temperature of 1000 K, the particle terminal velocity calculated from the above equation is 0.04036 m/ s and thus the Reynolds number equals 0.006848 which is much lesser than unity. And convective heat transfer coefficient obtained from Eq. (45) is equal to 6685.77 W/m2 K. Moreover, it is assumed that the Biot number for the burning iron particle is a negligible value, because of the high thermal conductivity of iron particle and small diameter of the particle ðBiH ¼ ðhconv dp =lp Þ 0:1Þ. Hence, the lumped system analysis can be applied for iron particle combustion, and it means that the temperature of the iron particle is a function of time only and not a function of location in the particle (T ¼ T(t)). 5.3. Burning time of a single iron particle On the other hand, the mode of combustion of micro sized iron particle is diffusionally-controlled regime, and diffusion of oxygen
Fig. 1. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 20 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
in iron particle is a slower phenomenon. Iron burns in oxygen as heterogeneous combustion. The burning time of iron particles in diffusionally-controlled regime can be obtained from the following relation [45]:
tbdiff ¼
rp d2p;0
8rN D O2 N ln 1 þ nYO2 ;N
(48)
where tbdiff is the burning time of iron particle in diffusionallycontrolled regime, rp is the iron particle density, dp,0 is the initial diameter of the iron particle, rN is the ambient gas density, D O2 N is the mass diffusivity of oxygen into the air, n is the mass stoichiometric index of combustion of iron, and YO2 ;N is the mass fraction of oxygen in the ambient gas far from the particle surface. 5.4. Results In this section, the diagrams of temperature of iron particle as a function of time and burning time of particle are plotted. The summary of the thermophysical properties used in the calculations is given in Table 1. Also, the detailed values of parameters used in the plotted figures for the various particle diameters are given in Table 2. Fig. 1 shows the variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 20 micron. The variation of absolute temperature of iron particle versus time for an iron particle with the diameter of 20 micron is illustrated in Fig. 2. Fig. 3 represents the variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 40 micron. The variation of absolute temperature of iron particle versus time for an iron particle with the diameter of 40 micron is shown in Fig. 4. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 60 micron is plotted in Fig. 5. Fig. 6 represents the variation of absolute
Table 2 Values of parameters used in the plotted figures for the various particle diameters. dp [mm]
tb [s]
vterm ½m=s
Rep [e]
Pe [e]
Nu½e
hconv ½w=m2 k
s [e]
32 ½e
J [e]
20 40 60 80 100
0.009 0.038 0.082 0.150 0.232
0.04036 0.16144 0.36324 0.64576 1.009
0.006848 0.054784 0.184896 0.438272 0.856
0.004869 0.038934 0.131406 0.311479 0.608359
2.00473 2.03348 2.09027 2.16408 2.29384
6685.77 3390.83 2323.68 1804.3 1529.99
0.3187 0.3412 0.3364 0.3584 0.3760
0.002630 0.005186 0.007567 0.009745 0.011493
0.98579 1.94371 2.83636 3.65282 4.30774
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Fig. 2. Absolute temperature of burning iron particle versus time for particle diameter of 20 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
temperature of iron particle versus time for an iron particle with the diameter of 60 micron. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 80 micron is depicted in Fig. 7. The variation of absolute temperature of iron particle as a function of time for an iron particle with the diameter of 80 micron is sketched in Fig. 8. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 100 micron is illustrated in Fig. 9. The variation of absolute temperature of iron particle as a function of time for an iron particle with the diameter of 100 micron is plotted in Fig. 10. The convective heat transfer coefficient as a function of iron particle diameter is sketched in Fig. 11. The burning time of iron particle increases quadratically with increasing of particle size, as depicted in Fig. 12. The fulllogarithmic chart of iron burning time as a function of particle diameter is shown in Fig. 13. Fig. 14 demonstrates that particle burning time decreases as ambient temperature (air temperature) raises. For comparison of results, the numerical solution of the nonlinear differential equation defined in Eq. (23) for iron particle with diameter of 40 micron is sketched in Fig. 15. As seen, this curve is qualitatively identical to the theoretical result depicted in Fig. 3;
Fig. 3. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
143
Fig. 4. Absolute temperature of burning iron particle versus time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
and quantitatively differs from the values of the theoretical result only about 2.5 percent. Fig. 16 demonstrates that the maximum temperature of combustion of iron particle increases with increasing particle diameter. 5.5. Discussion It is seemed from the figures that at the beginning of the combustion, the temperature of the particle is equal to the ignition temperature and then the temperature of the iron particle increases. The figures indicate that the temperature of iron particle with any size reaches to the maximum value at the end of the burning time. After the temperature reaches to its highest value, the chemical reaction finishes and heat generation inside the particle vanishes. The final temperature of iron particle is a function of particle size. Larger particles have higher maximum temperature than that of smaller particles. This result is similar to the experimental result presented by Tang et al. [15,16], that by increasing the particle size, the flame temperature increases in iron dust flames. Although, there is no available experimental study that presents the temperature of single iron particle during its combustion. But, there are several experimental studies that demonstrate the
Fig. 5. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 60 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
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Fig. 6. Absolute temperature of burning iron particle versus time for particle diameter of 60 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
temperature of iron clouds. Therefore, the only possible way is that the acquired results of this paper are compared to the experimental results for iron particle clouds. By comparison to the experimental data, it is observed that the maximum temperature of iron particle
Fig. 7. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 80 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 8. Absolute temperature of burning iron particle versus time for particle diameter of 80 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 9. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 100 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
obtained from presented theoretical model is in good agreement with the adiabatic flame temperature. However, the adiabatic flame temperature expresses the maximum temperature of stoichiometric mixture of iron particles and air, but, since in this state, the
Fig. 10. Absolute temperature of burning iron particle versus time for particle diameter of 100 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 11. The convective heat transfer coefficient as a function of iron particle diameter.
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
145
Fig. 12. Burning time of spherical iron particle versus particle diameter in air at atmospheric pressure and 1000 K.
Fig. 15. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from numerical solving of nonlinear differential equation defined in Eq. (23).
Fig. 13. Logarithmic chart of burning time of spherical iron particle as a function of particle diameter burning in air at atmospheric pressure and 1000 K.
Fig. 16. Maximum temperature of combustion of iron particle as a function of particle diameter.
adiabatic condition exists, thus, air reaches to the burning iron particle and thermal equilibrium between air and particles is established. It is noted that the particle temperature must be always greater than that of the air in the combustion zone. Because, heat is generated in the particle due to chemical reaction and then,
heat is transferred to the ambient air. Also, the numerical result is very close to the analytical result obtained from the homotopy perturbation method.
Fig. 14. Burning time of spherical iron particle as a function of ambient temperature for particle with diameter of 20 micron burning in air at atmospheric pressure.
6. Conclusion It is deduced that by increasing the particle size, the maximum attained temperature of iron particle becomes higher. Because, by increasing the particle size, the ratio of surface area to the volume of iron particle (As/Vp ¼ 3/rp) decreases and it causes that the heat of combustion generated inside the particle releases from the smaller area and much amount of heat generated in the particle accumulates in the particle and the particle temperature is raised much more. The rate of heat generation inside the particle is the rate of heat liberated by exothermic reaction and this is a volumetric phenomenon. Also, heat loss due to convection and thermal radiation from the particle surface is a surface phenomenon. It means that in the smaller particles, the heat loss from the particle surface is greater than the energy storage in the particle. Also, in the larger particle, the accumulation of energy in the particle is dominant. The convective heat transfer coefficient is nonlinearly proportional to the particle diameter and by increasing the particle size, the convective heat transfer coefficient decreases. Equilibrium thermodynamic calculation estimates that stoichiometric adiabatic flame temperature of iron in air is equal to
146
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2230 K. This temperature is below the boiling point of iron of 3130 K at atmospheric pressure. Also the volatilization temperature of iron oxide formed from the combustion is 3400 K that is well higher than adiabatic flame temperature. This proved that iron burns as a pure heterogeneous flame in air. The analytical solution obtained from the homotopy perturbation method is verified with the numerical solution of the nonlinear energy equation of the iron particle. Obviously, the analytical and numerical results are satisfactorily identical to each other, as depicted in Fig. 10. It indicates that the mathematical solving technique for combustion modeling of iron particle is efficiently acceptable. Iron particle burning time increases quadratically with increasing the iron particle diameter. On the other hand, Because of temperature dependency of mass diffusivity of oxygen in air 3=2 1 Þ, iron particle burning ðD O2 N fTN Þ and density of air ðrN fTN time decreases with increasing the temperature of the gaseous 1=2 oxidizing medium ðtbdiff fðrN D O2 N Þ1 fTN Þ. Nomenclature Apr As BiH cp cp,N CD dp D O2 N E FB FD g hconv mFe mO2 mFeO mp MFe MO2 MFeO nFe nO2 nFeO Nu PO2 Pe PrN rp _ Fe ℛ Rep t tbdiff T Tig Ts Tsurr TN u U V term
projected area of particle perpendicular to the direction of falling outer surface area of particle Biot Number in heat transfer specific heat of iron particle specific heat of gaseous oxidizing medium drag coefficient particle diameter mass diffusivity of oxygen in air total energy of particle buoyant force acting on iron particle drag force exerted on iron particle opposite the direction of falling gravitational acceleration average convection heat transfer coefficient mass stoichiometric coefficient of iron mass stoichiometric coefficient of oxygen mass stoichiometric coefficient of iron oxide mass of particle atomic weight of iron molecular weight of oxygen molecular weight of iron oxide molar stoichiometric coefficient of iron molar stoichiometric coefficient of oxygen molar stoichiometric coefficient of iron oxide average Nusselt number partial pressure of oxygen in the ambient gaseous medium Peclet number Prandtl number of gaseous fluid radius of iron particle mass reaction rate of iron Reynolds number for iron particle time burning time of particle based on diffusionally-controlled regime absolute temperature of iron particle ignition temperature of iron particle surface temperature of iron particle absolute temperature of surroundings absolute temperature of ambient gaseous oxidizer specific internal energy of the system total internal energy of the system terminal velocity of falling iron particle
Vp W YO2 ;N
volume of spherical iron particle weight of spherical iron particle mass fraction of oxygen in the ambient gas
Greek symbol as absorptivity of particle surface b coefficient of temperature-dependence of density g Euler number Dhcomb enthalpy of combustion of iron 31 non dimensional parameter defined in Eq. (22) 32 non dimensional parameter defined in Eq. (22) 3p emissivity of particle surface q dimensionless temperature lp thermal conductivity of iron particle lN thermal conductivity of gaseous oxidizing environment mN dynamic viscosity of ambient gas n mass stoichiometric index of combustion of iron rp density of burning iron particle rp,N density of iron particle at TN rN density of ambient gaseous oxidizer s StefaneBoltzmann constant s dimensionless time defined in Eq. (22) J non dimensional parameter defined in Eq. (22) Subscripts 0 initial B buoyancy diffusionally-controlled combustion bdiff comb combustion conv convection D drag ig ignition Oxygen O2 p particle pr projected s surface surr surroundings term terminal N ambient References [1] M. Bidabadi, A. Haghiri, A. Rahbari, M. Broumand, Distribution of iron dust particles through unburned zone across the flame propagation in a vertical duct by using homotopy analysis method, Combust. Explosion Shock Waves, submitted for Publication. [2] M. Bidabadi, A. Haghiri, A. Rahbari, Mathematical modeling of velocity and number density profiles of particles across the flame propagation through a micro-iron dust cloud, J. Hazard. Mater. 176 (1e3) (2010) 146e153. [3] M. Bidabadi, J. Fereidooni, R. Tavakoli, M. Mafi, Premixed filtration combustion of micron and sub-micron particles in inert porous media: a theoretical analysis, Korean J. Chem. Eng. 28 (2) (2011) 461e469. [4] M. Bidabadi, A. Shakibi, A. Rahbari, The radiation and heat loss effects on the premixed flame propagation through lycopodium dust particles, J. Taiwan Inst. Chem. Eng. 42 (1) (2011) 180e185. [5] A. Haghiri, M. Bidabadi, Modeling of laminar flame propagation through organic dust cloud with thermal radiation effect, Int. J. Therm. Sci. 49 (8) (2010) 1446e1456. [6] M. Bidabadi, A. Haghiri, A. Rahbari, The effect of Lewis and Damkohler numbers on the flame propagation through micro-organic dust particles, Int. J. Therm. Sci. 49 (3) (2010) 534e542. [7] M. Bidabadi, A. Fanaee, A. Rahbari, Investigation over the recirculation influence on the combustion of micro organic dust particles, Appl. Math. Mech. 31 (6) (2010) 685e696. [8] M. Bidabadi, S. Montazerinejad, A. Fanaee, An analytical study of radiation effect on the ignition of magnesium particles using perturbation theory, Lat. Am. Appl. Res. 40 (2010) 351e357. [9] M. Bidabadi, N. Moallemi, A. Shabani, M.A. Abdous, Analysis of size distribution and ignition temperature effects on flame speeds in aluminum dust clouds, Proc. Inst. Mech. Eng. G J. Aero. Eng. 224 (1) (2010) 113e119.
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Time variation of combustion temperature and burning time of a single iron particle Mehdi Bidabadi, Majid Mafi* Combustion Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology (IUST), Narmak, 16846-13114 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 April 2011 Received in revised form 26 October 2012 Accepted 26 October 2012 Available online 4 December 2012
In the present study, combustion of a single iron particle is modeled by virtue of a novel thermophysical procedure. The temperature of iron particle during combustion is studied analytically and numerically. In the proposed model, the effect of thermal radiation from the external surface of burning particle, and alterations of density of iron particle with temperature are considered. Iron particle burns heterogeneously in air. Because of high thermal conductivity and micro size of iron particle, the Biot number is negligibly small, and the lumped system analysis can be utilized for the combustion modeling. The nonlinear energy equation resulted from modeling is solved by using homotopy perturbation method. The assumptions applied in the modeling are such that do not violate the actual combustion phenomenon. Also, the numerical solution of nonlinear differential equation is presented and compared with the analytical solution obtained from homotopy method. It is the first model presented for analyzing the combustion of single iron particle. 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Single iron particle Combustion modeling Nonlinear differential equation Homotopy perturbation method Heterogeneous reaction Iron particle burning time
1. Introduction Combustion of metallic materials is a challenging scientific subject that also has significant practical applications. Since the combustion of metallic particles is a high exothermic chemical reaction, use of them in the industrial operations is significant and efficient. Combustion of iron particle cloud experimentally investigated in some works. Iron is regarded as a non-volatile metallic fuel and the oxidation process takes place as a heterogeneous surface reaction. The major characteristic feature of iron combustion is that iron burns heterogeneously in air. It means that combustion reaction occurs at the iron particle surface and no flame is observed in the gaseous oxidizer phase. Iron particles do not evaporate during the combustion process and also combustion product (iron oxide) remains in the condensed phase. Because of high energy density of iron, the combustion of iron can be used as an alternative fuel in automobile engines and solid propellant rocket motors in the future. Recently, Bidabadi et al. [1] analytically investigated the distribution of iron dust particles through unburned zone across the flame propagation in a vertical duct by using homotopy analysis method. Also, Bidabadi et al. [2] theoretically proposed a mathematical modeling of velocity and number * Corresponding author. Tel.: þ98 21 77 240 197; fax: þ98 21 77 240 488. E-mail address: majid.mafi@gmail.com (M. Mafi). 1290-0729/$ e see front matter 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2012.10.019
density profiles of particles across the flame propagation through a micro-iron dust cloud. Both studies focused on the dynamic behavior of iron particle clouds and various forces exerted on the particles in the preheat zone, prior to the flame zone. In addition, Bidabadi et al. studied the combustion of various type of particle cloud in several works [3e14]. Furthermore, Tang et al. [15] experimentally performed the so-called argon/helium test to determine the modes of particle combustion in iron dust flames. Experiments were accomplished in a reduced-gravity environment. Also, Tang et al. [16] experimentally studied the quenching and laminar flames propagating in fuel-rich suspension of iron dust in air in a reduced-gravity environment provided by a parabolic flight aircraft. Moreover, Sun et al. [17] measured the particle velocity and particle number density profiles across upward and downward flame propagating through iron particle clouds by using the highspeed photomicrographs. Sun et al. [18] experimentally examined the concentration profile of particles across a flame propagating through an iron particle cloud. Sun et al. [19] experimentally evaluated the temperature profile across the combustion zone propagating through an iron particle cloud. Also, Sun et al. [20] experimentally investigated the combustion behavior of iron particles suspended in air. In addition, Steinberg et al. [21e32] have carried out many experiments on the burning iron rods. Homotopy is a mathematical technique for solving nonlinear differential equations [33,34]. Homotopy method has been used for
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
solving various nonlinear differential equations raised in mathematical modeling of physical phenomena [35e43]. In the previous works, combustion of iron cloud has been experimentally investigated, or the dynamic behavior of iron particle in preheat zone has been studied. This paper proposed a novel analytical approach to model the combustion of single iron particle burning in the gaseous oxidizing medium. By using this model, the variation of iron particle temperature during the combustion process as a function of time can be obtained. This model includes the thermal radiation effect and, also, variation of specific heat of iron particle by temperature. This model applied for the combustion of iron particle for the first time, and can be improved in the future studies to include more complicated parameters that influence the circumstance of combustion. Because of nonlinearity of the governing differential equation, the homotopy perturbation method is utilized for solving the energy equation of iron particle. 2. Homotopy perturbation method Consider the following nonlinear differential equation:
A ðuÞ f ðrÞ ¼ 0;
r˛U
(1)
with boundary conditions
vu B u; ¼ 0; vn
r˛L
(2)
where A ðuÞ is a general differential operator, B is a boundary operator, f(r) is a known analytic function, and L is the boundary of domain U. The operator A ðuÞ can be divided to two parts L ðuÞ and N ðuÞ, where L is linear, and N is nonlinear.
A ðuÞ ¼ L ðuÞ þ N ðuÞ
(3)
Hence Eq. (1) can be written as
L ðuÞ þ N ðuÞ f ðrÞ ¼ 0;
r˛U
(4)
by means of homotopy technique [33,34], a homotopy function can be constructed, which satisfies the following equation:
H ðv; pÞ ¼ ð1 pÞ½L ðvÞ L ðu0 Þ þ p½A ðvÞ f ðrÞ ¼ 0;
p ˛½0; 1
(5)
or
H ðv; pÞ ¼ L ðuÞ L ðu0 Þ þ pL ðu0 Þ þ p½N ðvÞ f ðrÞ ¼ 0 where
vðr; pÞ : U ½0; 1/R
parameter”, applying the classical perturbation technique, we can assume that the solution of Eq. (6) can be expressed as a power series in p, i.e.,
v ¼ v 0 þ pv 1 þ p 2 v 2 þ /
(7)
where ˛½0; 1 is an embedding parameter, and u0 is the initial approximation of Eq. (1) which satisfies the boundary conditions. Obviously, we have
H ðv; 0Þ ¼ L ðvÞ L ðu0 Þ ¼ 0
(8)
H ðv; 1Þ ¼ A ðvÞ f ðrÞ ¼ 0
(9)
The process of changing p from zero to unity results in changing vðr; pÞ from u0 to u(r). This is called deformation, and also L ðvÞ L ðu0 Þ and A ðvÞ f ðrÞ are called homotopic in topology. If the embedding parameter p˛½0; 1 is considered as a “small
(10)
and setting p ¼ 1 results in the approximate solution of Eq. (1) as
u ¼ lim v ¼ v 0 þ v 1 þ v 2 þ / p/N
(11)
It is worth to note that the major advantage of He’s homotopy perturbation method is that the perturbation equation can be freely constructed in many ways (therefore is problem dependent) by homotopy in topology and the initial approximation can also be freely selected.
3. Modeling of combustion of single iron particle As the thermal diffusivity of substance is large and Biot number is very small (BiH 0.1), it is assumed that the particle is isothermal. In this state, a lumped system analysis is applicable. When this criterion is satisfied, the variation of temperature with location within the body will be slight and can be reasonably be approximated as being uniform. It means that the iron particle has a spatially uniform temperature, thus implying very large thermal conductivity of iron particle. Therefore, the temperature of particle is a function of time only, T ¼ T(t), and is not a function of radial coordinate, T s T(r). The assumptions used in this modeling are: 1. The spherical iron particle burns in a quiescent, infinite ambient medium that contains only oxygen and an inert gas, such as nitrogen. There are no interactions with other particles, and the effects of forced convection are ignored. 2. Constant thermophysical properties for the iron particle and ambient gaseous oxidizer are used, except for the specific heat of the iron particle which varies by temperature. 3. The particle is of uniform temperature and radiates as a gray body to the surroundings without participation of the intervening medium. First, iron particle is considered as a thermodynamic system, and by using of principle of conservation of energy (first law of thermodynamics), the energy balance equation for this particle can be written as
E_ in E_ out þ E_ gen ¼ (6)
137
dE dt
(12) p
where E_ in is the rate of energy entering the system which is owing to absorption of entire radiation incident on the particle surface from the surrounding surfaces, E_ out is the rate of energy leaving the system by mechanisms of convection on the particle surface to the ambient oxidizer gas (air), and thermal radiation that emits from the outer surface of particle, E_ gen is the rate of generation of energy inside the particle due to the combustion process and equals to the heat released from the chemical reaction, and (dE/dt)p is the rate of change in total energy of iron particle that is specified to the rate of change in the temperature of the particle. 4 E_ in ¼ as q_ incident ¼ as sAs Tsurr
(13)
where as is the absorptivity of the iron particle surface, q_ incident is the flux of thermal radiation incident to the particle surface from the surrounding surfaces which enclose the particle,
138
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
s ¼ 5.67 108 W/m2 K4 is the StefaneBoltzmann constant, As ¼ 4pr2p is the outer surface area of the iron particle, and Tsurr is the absolute temperature of the surrounding surfaces.
rp Vp cp
dT dt
p
4 þ hconv As ðT TN Þ þ 3 s sAs T 4 Tsurr
_ Fe As Dh ℛ comb ¼ 0 E_ out ¼ hconv As ðTs TN Þ þ 3 s sAs Ts4
(14)
where hconv is the average convective heat transfer coefficient at the entire surface of iron particle, Ts is the absolute temperature of the particle that is only a function of time, TN is the absolute temperature of the ambient oxidizer gas (air) around the particle, 3 s is the emissivity of the surface of the iron particle.
_ Fe As Dh E_ gen ¼ Q_ comb ¼ ℛ comb
(15)
where Q_ comb is the rate of heat released within the particle due to _ Fe is the reaction rate of iron with oxygen, Dhcomb the combustion, ℛ is the enthalpy of reaction of iron which is a negative quantity. The rate of change in total energy of iron particle is composed of three terms and is expressed as
dE dt
¼ p
dU dðK:E:Þ dðP:E:Þ þ þ dt p dt dt p p
dE dt
p
dU du dT ¼ ¼ mp ¼ rp Vp cp dt p dt p dt p
h
4 as sAs Tsurr hconv As ðT TN Þ þ 3 s sAs T 4
(17)
i
dT _ Fe As Dh r þ ℛ ¼ V c p p p comb dt p
Tð0Þ ¼ Tig
(19b)
Also, the density of iron particle is a function of temperature of particle, and thus, it can be considered as a linear function of particle temperature
rp ¼ rp ðTÞ ¼ rp;N ½1 þ bðT TN Þ
(18)
In general, both absorptivity and emissivity of the surface depend on the temperature and the wavelength of radiation. Kirchhoff’s law of radiation states that the absorptivity and the emissivity of a surface at a given temperature and wavelength are equal, i.e., (as y 3 s). Therefore, the governing energy equation of burning iron particle can be expressed as
(20)
where rp,N is the density of particle at reference temperature of TN, and b is the coefficient of variation of density of iron with temperature. Hence, the governing equation of burning iron particle can be written as
rp;N Vp cp ½1 þ bðT TN Þ
dT dt
p
þ hconv As ðT TN Þ
4 _ Fe As Dh ℛ þ 3 s sAs T 4 Tsurr comb ¼ 0
where mp is the mass of the iron particle, u is the specific internal energy of the particle, rp is the density of the burning iron particle, Vp is the volume of the particle, and cp is the specific heat of the iron particle. Iron is an incompressible substance, therefore, the cp,p and cv,p values of iron particle are identical and both specific heats can be represented by a single symbol which is denoted by cp. Because the temperature of particle is uniform at each instant of time, thus the surface temperature of particle equals to the temperature of particle, Ts ¼ T. By substituting of (13)e(15) and (17) in (12), we have
The iron particle temperature is taken to be uniform. The temporal variation of temperature of iron particle can be obtained by solving of above equation. This equation is a nonlinear nonhomogeneous first order differential equation that cannot be solved by conventional methods in mathematics, and exact solution of this equation is not obtained easily. Obviously, for solving this first order differential equation, an initial condition is required. The initial temperature of the iron particle at the beginning of combustion can be regarded as the initial condition. This temperature is known as ignition temperature.
(16)
where (dU/dt)p is the rate of change in the total internal energy of the particle, (d(K.E.)/dt)p is the rate of change in the total kinetic energy of the particle, and (d(P.E.)/dt)p is the rate of change in the total potential energy of the particle. Assuming particle falling freely due to the gravity with constant terminal velocity, the rate of change in total kinetic energy is zero, (d(K.E.)/dt)p ¼ 0. On the other hand, the rate of change in total potential energy of particle during short burning time of particle is negligibly small and it can be neglected compared to the considerable amount of the rate of change in internal energy of the particle, (d(P.E.)/dt)p y 0. Consequently
ð19aÞ
ð21Þ
This equation has two nonlinear terms. For solving this nonlinear differential equation, all the terms must be converted to the dimensionless form. The following dimensionless variables are defined to solve the equation:
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <
T Tig T qN ¼ N Tig Tsurr qsurr ¼ Tig t s ¼ rp;N Vp cp
q ¼
> > > > hconv As > > > _ Fe As Dh > ℛ > comb > > J ¼ > > > hconv As Tig > > > > 31 ¼ bTig > > > > 3 > 3 s sT > ig > > 32 ¼ > > > hconv > > > :
(22)
Consequently, the nonlinear differential equation and its initial condition can be expressed in the dimensionless form
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
8 > < > :
½1 þ 31 ðq qN Þ
dq 4 4 þ ðq qN Þ þ 32 q qsurr J ¼ 0 ds qð0Þ ¼ 1
(23a)
The governing equation can be rearranged as
8 > < > :
ð1 31 qN Þ
dq dq 4 4 þ q ¼ 32 q qsurr J þ 31 q qN ds ds qð0Þ ¼ 1 (23b)
In above equation, 31 and 32 are the small parameters compared to the other terms appeared in equation. This equation is solved by the homotopy perturbation method.
p 1 : ð1 31 qN Þ p 2 : ð1 31 qN Þ p 3 : ð1 31 qN Þ p 4 : ð1 31 qN Þ p 5 : ð1 31 qN Þ
dq0 p 0 : ð1 31 qN Þ þ q0 ¼ 0 ; q0 ð0Þ ¼ 1 ds dq1 dq 4 4 þ q1 ¼ 32 q0 qsurr J þ 31 q0 0 qN ds ds dq2 dq 4 4 þ q2 ¼ 32 q1 qsurr J þ 31 q1 1 qN ds ds dq3 dq2 4 4 qN þ q3 ¼ 32 q2 qsurr J þ 31 q2 ds ds dq4 dq 4 4 þ q4 ¼ 32 q3 qsurr J þ 31 q3 3 qN ds ds dq5 dq 4 4 þ q5 ¼ 32 q4 qsurr J þ 31 q4 4 qN ds ds
139
dv 0 dv 1 dv 2 þ pð1 31 qN Þ þ p 2 ð1 31 qN Þ ds ds ds q d 0 þ v 0 þ pv 1 þ p 2 v 2 ð1 31 qN Þ q0 ds dq þ p ð1 31 qN Þ 0 þ q0 þ ds dq 4 p 32 v 40 qsurr J þ 31 v 0 qN ds 4 þ L ðvÞ L ðq0 Þ þ pL ðq0 Þ þ p 32 v 4 qsurr dv J þ 31 v qN ¼ 0 ds
ð1 31 qN Þ
ð26Þ
Therefore
; q1 ð0Þ ¼ 0 ; q2 ð0Þ ¼ 0 (27)
; q3 ð0Þ ¼ 0 ; q4 ð0Þ ¼ 0 ; q5 ð0Þ ¼ 0
4. Homotopy perturbation solution
By solving above differential equation system, the following results can be achieved:
According to homotopy perturbation method, the following homotopy is constructed:
q0 ðsÞ ¼ 1
(28)
then
dv J þ 31 v qN L ðvÞL ðq0 Þ þ pL ðq0 Þ þ p ds dq dq0 ¼ 0ð1 31 qN Þ þ q ð1 31 qN Þ q0 ds d s dq 4 4 qN ¼ p 32 q qsurr J þ 31 q ds dq ð1 31 qN Þ 0 þ q0 ds
4 4 32 v qsurr
(24) where
L ðvÞ ¼ ð1 31 qN Þ
dv þv ds
dq0 L ðq0 Þ ¼ ð1 31 qN Þ þ q0 ds
(25)
By substituting Eqs. (24) and (25) in Eq. (23) and equating corresponding coefficient of same power terms, the following differential equation system can be obtained:
q1 ðsÞ ¼
h 32
i 4 1 qsurr J qN 1 exp
s
1 31 qN
(29)
and by substituting this function into the third differential equation in system, we have
ð1 31 qN Þ
h i dq2 4 þ q2 ¼ 32 32 1 qsurr J qN ds 4 s 4 qsurr 1 exp 1 31 qN h i 4 J þ 31 32 1 qsurr J qN s 1 exp 1 31 qN i (h 4 32 1 qsurr J qN exp 1 31 qN )
s
1 3 1 qN
qN ; q2 ð0Þ ¼ 0
(30)
The solution of this differential equation is
q2 ðsÞ ¼ cexp
*
s
"
1
!
exp
s
#
2 ð1 31 qN Þ ð31 qN 1Þ 4 4 2s 1 3s 4 2 4 2 232 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ exp 32 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ exp þ 3 ð1 31 qN Þ ð1 31 qN Þ 13 20 3 2 0 1 4 4 4 3 B 6B C 4 1 þ 432 qsurr 1 1232 qsurr 1 ðj 1Þþ C7 6B 4 4 4 3 4 4 A C7 j q q q j ð þ 4 1 þ 3 1Þ 3 3 1 þ 6 B 32 qN þ 352 qsurr 1 þqN @ 2 2 surr N 2 surr C7 6B 4 2 3 2 C7 6B q j j 3 1 ð 1Þ 4 3 ð 1Þ 12 2 surr 2 C7 6B C7 6B 2 3 2 2 C7 þ 6@ 2 4 4 4 4 A7 2 3 7 6 4 3 2 q q j q j q j q j j þ6 N 32 1 þ 32 surr 1 432 surr 1 ð 1Þ þ 632 surr 1 ð 1Þ 632 surr 1 ð 1Þ þ 7 6 7 6 7 6 7 6 5 4 s 2 ð31 qN 1Þ exp ð1 31 qN Þ ð1 31 qN Þ
þ
+
(31)
3 0 1 2 7 6 2 2 3 2 4 4 @ 43 1 j þ q þ 3 q4 1 6 1 j þ qN þ 32 q4surr 1 þ31 1 þ 4qN 32 þ 8qN 32 1 þ 32 qsurr 1 j þ 4qN 32 32 1 32 qsurr þ j As 7 N 2 2 surr 5 4 2
2
0
2
6 4 3 2 þ31 1 þ 6qN 32 þ 12qN 32 1 j þ 32 632 1 j þ qN þ 32 qsurr 1 6 6 ð3 q 1Þ 1 j þ q þ 3 q4 1 2 B B N 6 1 N 2 surr @ 6 4 þ6qN 32 j 1 þ 32 32 qsurr 4
3 1 7 q4surr 1 7 C s 7 Cexp 7 A ð1 31 qN Þ 7 5
The unknown factor c can be determined by using the initial condition
h i 1 4 4 4 2 4 2 232 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ 32 1 j þ qN þ 32 qsurr 1 ð31 qN 1Þ þ 3 20 13 4 2 3 4 4 4 4 4 5 2 3 4 3 2 6 B 32 qN þ 32 qsurr 1 þqN 1 þ 43 qsurr 1 123 qsurr 1 ðj 1Þ þ 123 qsurr 1 ðj 1Þ 432 ðj 1Þ C 7 2 2 2 6B C7 6B 2 3 C7 7 6B 4 3 4 2 4 4 4 4 6 B þ32 ðj 1Þ qsurr þ 4qN 32 1 þ 32 qsurr 1 j þ 6qN 32 1 þ 32 qsurr 1 j 432 qsurr 1 ðj 1Þ C !# * " + C7 7 6B C 2 2 1 6@ A7 4 4 2 3 3 2 7 6 c ¼ þ632 qsurr 1 ðj 1Þ 632 qsurr 1 ðj 1Þ þj 2 7 6 ð31 qN 1Þ 5 4 2 ð31 qN 1Þ 3 2 0 2 1 4 3 2 4 2 B 632 1 j þ qN þ 32 qsurr 1 þ31 1 þ 6qN 32 þ 12qN 32 1 j þ 32 qsurr 1 6 C7 6 ð31 qN 1Þ 1 j þ qN þ 32 q4surr 1 @ A7 5 4 4 þ6qN 32 j 1 þ 32 32 qsurr (32)
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Therefore, the final solution of the governing equation describing the temperature changes of the burning iron particle versus time, can be expressed as
qðsÞ ¼ q0 ðsÞ þ q1 ðsÞ þ q2 ðsÞ þ /
(33)
5. Results and discussion 5.1. Chemical kinetics of iron combustion The chemical reaction formula of combustion of iron at stoichiometric condition is expressed as
nFe Fe þ nO2 O2 /nFeO FeO mFe Fe þ mO2 O2 /mFeO FeO
(34)
141
particle can attain when all the forces balance each other and the net force acting on the particle (and thus its acceleration) is zero. The forces exerted on the falling particle are usually the weight of the particle, the buoyant force, and the drag force. At terminal velocity the particle will not be accelerating so the forces acting on the free falling particle are at the equilibrium, i.e., the weight of the particle equals the buoyant force plus the drag force ðV p ¼ V term Þ. The particle terminal velocity can be estimated as follows:
W ¼ FB þ FD 4 4 pr gr3 ¼ prN gr3p þ 12 CD rN Apr V 3 p p 3
2 term
(41)
Thus where
8 < mFe ¼ nFe MFe m ¼ nO2 MO2 : O2 mFeO ¼ nFeO MFeO
(35)
V
term
¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u u8p rp rN gr3 p t 3CD rN Apr
(42)
where nFe is molar stoichiometric coefficient of iron in chemical reaction formula, nO2 is molar stoichiometric coefficient of oxygen in chemical reaction formula, nFeO is molar stoichiometric coefficient of iron oxide in chemical reaction formula, MFe is atomic weight of iron, MO2 is molecular weight of oxygen, MFeO is molecular weight of iron oxide, mFe is mass stoichiometric coefficient of iron in chemical reaction formula, mO2 is mass stoichiometric coefficient of oxygen in chemical reaction formula, mFeO is mass stoichiometric coefficient of iron oxide in chemical reaction formula. Dividing the mass stoichiometric coefficients in the chemical reaction formula to the mass stoichiometric coefficient of iron changes the form of chemical reaction formula of combustion of iron to
where W is the weight of iron particle, FB is buoyant force, FD is drag force, g is gravitational acceleration, rN is density of gaseous medium, CD is drag coefficient, Apr is projected area of particle perpendicular to the direction of falling, V term is the terminal velocity of falling iron particle. Assuming the iron particle burns in the quiescent medium and the relative velocity between particle and ambient gas is very small. It means that the Reynolds number for fluid flow around the particle is infinitesimally small (Rep < 1). At low Reynolds numbers (it is called a Stokes flow if Rep < 1), the drag coefficient for a spherical particle is given by
Fe þ n O2 /ð1 þ nÞ FeO
CD ¼
(36)
24 24 24mN ¼ ¼ rN V p dp rN V term dp Rep
mN
where n is mass stoichiometric index which is defined as
n ¼
mO2 mFe
(37)
For combustion of iron, the chemical formula can be written as
(38)
Mass stoichiometric index, n, for iron combustion equals n ¼ (31.99 kg) O2/(111.69 kg) Fe ¼ 0.2864. And the rate of reaction of iron with oxygen per surface area of iron particle is experimentally obtained [24]
4 1 g _ Fe ¼ 7:05 103 exp 2:9684 10 P 2 ℛ O2 T cm2 s
Dhcomb ¼ 365:2
kJ kJ kJ ¼ 6:54 ¼ 6540 mol Fe g Fe kg Fe
V
term
(39)
(40)
¼
9mN
hconv dp
(44)
¼ 2 þ Pe þ Pe2 lnPe þ f ðPrN ÞPe2 :
lN
Rep 1 ;
PrN w1
(45)
where Nu is the average Nusselt number for iron particle, hconv is the convective heat transfer coefficient, dp is the diameter of the iron particle, lN is the thermal conductivity of the gaseous oxidizing environment, Pe ¼ RepPrN is the Peclet number, PrN is the Prandtl number of ambient gaseous oxidizing medium, and
f ðPrN Þ ¼ 5.2. Dynamics of a single iron particle and convective heat transfer coefficient
2gr2p rp rN
On the other hand, in the case of particle falling in the motionless oxidizer medium, the average convective heat transfer coefficient on the particle surface can be obtained by a relation for the Nusselt number for low Reynolds number flow [44]
Nu ¼
where PO2 is the partial pressure of oxygen in the ambient gaseous medium (air). Also, the specific enthalpy of combustion of iron is [24]
therefore
sto
ð2 kmolÞ Fe þ ð1 kmolÞO2 /ð2 kmolÞ FeO ð111:69 kgÞ Fe þ ð31:99 kgÞ O2 /ð143:68 kgÞ FeO
(43)
1 2
213 þ 2 Pr3N 3PrN 2 40 lnPrN 2ðPrN þ 1Þ2 ðPrN 2ÞlnðPrN þ 1Þ 2Pr2N PrN 4g
(46) It is assumed that the iron particle is dropped into the air; it falls due to the gravity. During this free fall, iron particle reaches its terminal velocity which is the maximum velocity that a free falling
where g ¼ 0.5772156. is the Euler number. Prandtl number of ambient gas is defined as
142
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Table 1 Summary of the thermophysical properties used in the calculations. Property
Symbol
Value
Unit
Density of atmospheric air at 1000 K Density of burning iron particle at TN ¼ 1000 K Specific heat of burning iron particle Coefficient of temperaturedependence of density of iron particle Thermal conductivity of atmospheric air at 1000 K Mass diffusivity of oxygen in air at 1000 K Mass fraction of oxygen in air far from particle surface Mass stoichiometric index Partial pressure of oxygen in atmospheric air Emissivity of burning iron particle at 2300 K Ignition temperature of iron particle Ambient temperature Surrounding temperature Prandtl number of atmospheric air at 1000 K Dimensionless parameter defined in Eq. (22) Dimensionless parameter defined in Eq. (22) Dimensionless parameter defined in Eq. (22)
rN
0.3482
kg/m3
[46]
3
PrN ¼
Ref.
rp,N
7650
kg/m
[46]
cp
748.5
J/kg K
[46]
b
5.62 105
1/K
[46]
lN
66.7 103
W/m C
[47]
D O2 N
15.2 105
m2/s
[47]
YO2 ;N
0.233
e
e
n PO2
0.2864 21.278
e kPa
e e
3s
0.505
e
[48]
Tig
850
K
16
TN Tsurr PrN
1000 300 0.7107
K K e
e e [47]
31
0.051595
e
e
qN
1.17647
e
e
qsurr
0.35294
e
e
mN cp;N lN
(47)
where cp,N is specific heat capacity at constant pressure of ambient gaseous fluid. For the iron particle with diameter of 20 micron and temperature of 800 K falling in the air at temperature of 1000 K, the particle terminal velocity calculated from the above equation is 0.04036 m/ s and thus the Reynolds number equals 0.006848 which is much lesser than unity. And convective heat transfer coefficient obtained from Eq. (45) is equal to 6685.77 W/m2 K. Moreover, it is assumed that the Biot number for the burning iron particle is a negligible value, because of the high thermal conductivity of iron particle and small diameter of the particle ðBiH ¼ ðhconv dp =lp Þ 0:1Þ. Hence, the lumped system analysis can be applied for iron particle combustion, and it means that the temperature of the iron particle is a function of time only and not a function of location in the particle (T ¼ T(t)). 5.3. Burning time of a single iron particle On the other hand, the mode of combustion of micro sized iron particle is diffusionally-controlled regime, and diffusion of oxygen
Fig. 1. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 20 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
in iron particle is a slower phenomenon. Iron burns in oxygen as heterogeneous combustion. The burning time of iron particles in diffusionally-controlled regime can be obtained from the following relation [45]:
tbdiff ¼
rp d2p;0
8rN D O2 N ln 1 þ nYO2 ;N
(48)
where tbdiff is the burning time of iron particle in diffusionallycontrolled regime, rp is the iron particle density, dp,0 is the initial diameter of the iron particle, rN is the ambient gas density, D O2 N is the mass diffusivity of oxygen into the air, n is the mass stoichiometric index of combustion of iron, and YO2 ;N is the mass fraction of oxygen in the ambient gas far from the particle surface. 5.4. Results In this section, the diagrams of temperature of iron particle as a function of time and burning time of particle are plotted. The summary of the thermophysical properties used in the calculations is given in Table 1. Also, the detailed values of parameters used in the plotted figures for the various particle diameters are given in Table 2. Fig. 1 shows the variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 20 micron. The variation of absolute temperature of iron particle versus time for an iron particle with the diameter of 20 micron is illustrated in Fig. 2. Fig. 3 represents the variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 40 micron. The variation of absolute temperature of iron particle versus time for an iron particle with the diameter of 40 micron is shown in Fig. 4. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 60 micron is plotted in Fig. 5. Fig. 6 represents the variation of absolute
Table 2 Values of parameters used in the plotted figures for the various particle diameters. dp [mm]
tb [s]
vterm ½m=s
Rep [e]
Pe [e]
Nu½e
hconv ½w=m2 k
s [e]
32 ½e
J [e]
20 40 60 80 100
0.009 0.038 0.082 0.150 0.232
0.04036 0.16144 0.36324 0.64576 1.009
0.006848 0.054784 0.184896 0.438272 0.856
0.004869 0.038934 0.131406 0.311479 0.608359
2.00473 2.03348 2.09027 2.16408 2.29384
6685.77 3390.83 2323.68 1804.3 1529.99
0.3187 0.3412 0.3364 0.3584 0.3760
0.002630 0.005186 0.007567 0.009745 0.011493
0.98579 1.94371 2.83636 3.65282 4.30774
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Fig. 2. Absolute temperature of burning iron particle versus time for particle diameter of 20 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
temperature of iron particle versus time for an iron particle with the diameter of 60 micron. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 80 micron is depicted in Fig. 7. The variation of absolute temperature of iron particle as a function of time for an iron particle with the diameter of 80 micron is sketched in Fig. 8. The variation of dimensionless temperature of iron particle versus dimensionless time for an iron particle with the diameter of 100 micron is illustrated in Fig. 9. The variation of absolute temperature of iron particle as a function of time for an iron particle with the diameter of 100 micron is plotted in Fig. 10. The convective heat transfer coefficient as a function of iron particle diameter is sketched in Fig. 11. The burning time of iron particle increases quadratically with increasing of particle size, as depicted in Fig. 12. The fulllogarithmic chart of iron burning time as a function of particle diameter is shown in Fig. 13. Fig. 14 demonstrates that particle burning time decreases as ambient temperature (air temperature) raises. For comparison of results, the numerical solution of the nonlinear differential equation defined in Eq. (23) for iron particle with diameter of 40 micron is sketched in Fig. 15. As seen, this curve is qualitatively identical to the theoretical result depicted in Fig. 3;
Fig. 3. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
143
Fig. 4. Absolute temperature of burning iron particle versus time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
and quantitatively differs from the values of the theoretical result only about 2.5 percent. Fig. 16 demonstrates that the maximum temperature of combustion of iron particle increases with increasing particle diameter. 5.5. Discussion It is seemed from the figures that at the beginning of the combustion, the temperature of the particle is equal to the ignition temperature and then the temperature of the iron particle increases. The figures indicate that the temperature of iron particle with any size reaches to the maximum value at the end of the burning time. After the temperature reaches to its highest value, the chemical reaction finishes and heat generation inside the particle vanishes. The final temperature of iron particle is a function of particle size. Larger particles have higher maximum temperature than that of smaller particles. This result is similar to the experimental result presented by Tang et al. [15,16], that by increasing the particle size, the flame temperature increases in iron dust flames. Although, there is no available experimental study that presents the temperature of single iron particle during its combustion. But, there are several experimental studies that demonstrate the
Fig. 5. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 60 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
144
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
Fig. 6. Absolute temperature of burning iron particle versus time for particle diameter of 60 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
temperature of iron clouds. Therefore, the only possible way is that the acquired results of this paper are compared to the experimental results for iron particle clouds. By comparison to the experimental data, it is observed that the maximum temperature of iron particle
Fig. 7. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 80 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 8. Absolute temperature of burning iron particle versus time for particle diameter of 80 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 9. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 100 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
obtained from presented theoretical model is in good agreement with the adiabatic flame temperature. However, the adiabatic flame temperature expresses the maximum temperature of stoichiometric mixture of iron particles and air, but, since in this state, the
Fig. 10. Absolute temperature of burning iron particle versus time for particle diameter of 100 micron burning in air at atmospheric pressure and 1000 K obtained from homotopy perturbation analysis.
Fig. 11. The convective heat transfer coefficient as a function of iron particle diameter.
M. Bidabadi, M. Mafi / International Journal of Thermal Sciences 65 (2013) 136e147
145
Fig. 12. Burning time of spherical iron particle versus particle diameter in air at atmospheric pressure and 1000 K.
Fig. 15. Dimensionless temperature of burning iron particle versus dimensionless time for particle diameter of 40 micron burning in air at atmospheric pressure and 1000 K obtained from numerical solving of nonlinear differential equation defined in Eq. (23).
Fig. 13. Logarithmic chart of burning time of spherical iron particle as a function of particle diameter burning in air at atmospheric pressure and 1000 K.
Fig. 16. Maximum temperature of combustion of iron particle as a function of particle diameter.
adiabatic condition exists, thus, air reaches to the burning iron particle and thermal equilibrium between air and particles is established. It is noted that the particle temperature must be always greater than that of the air in the combustion zone. Because, heat is generated in the particle due to chemical reaction and then,
heat is transferred to the ambient air. Also, the numerical result is very close to the analytical result obtained from the homotopy perturbation method.
Fig. 14. Burning time of spherical iron particle as a function of ambient temperature for particle with diameter of 20 micron burning in air at atmospheric pressure.
6. Conclusion It is deduced that by increasing the particle size, the maximum attained temperature of iron particle becomes higher. Because, by increasing the particle size, the ratio of surface area to the volume of iron particle (As/Vp ¼ 3/rp) decreases and it causes that the heat of combustion generated inside the particle releases from the smaller area and much amount of heat generated in the particle accumulates in the particle and the particle temperature is raised much more. The rate of heat generation inside the particle is the rate of heat liberated by exothermic reaction and this is a volumetric phenomenon. Also, heat loss due to convection and thermal radiation from the particle surface is a surface phenomenon. It means that in the smaller particles, the heat loss from the particle surface is greater than the energy storage in the particle. Also, in the larger particle, the accumulation of energy in the particle is dominant. The convective heat transfer coefficient is nonlinearly proportional to the particle diameter and by increasing the particle size, the convective heat transfer coefficient decreases. Equilibrium thermodynamic calculation estimates that stoichiometric adiabatic flame temperature of iron in air is equal to
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2230 K. This temperature is below the boiling point of iron of 3130 K at atmospheric pressure. Also the volatilization temperature of iron oxide formed from the combustion is 3400 K that is well higher than adiabatic flame temperature. This proved that iron burns as a pure heterogeneous flame in air. The analytical solution obtained from the homotopy perturbation method is verified with the numerical solution of the nonlinear energy equation of the iron particle. Obviously, the analytical and numerical results are satisfactorily identical to each other, as depicted in Fig. 10. It indicates that the mathematical solving technique for combustion modeling of iron particle is efficiently acceptable. Iron particle burning time increases quadratically with increasing the iron particle diameter. On the other hand, Because of temperature dependency of mass diffusivity of oxygen in air 3=2 1 Þ, iron particle burning ðD O2 N fTN Þ and density of air ðrN fTN time decreases with increasing the temperature of the gaseous 1=2 oxidizing medium ðtbdiff fðrN D O2 N Þ1 fTN Þ. Nomenclature Apr As BiH cp cp,N CD dp D O2 N E FB FD g hconv mFe mO2 mFeO mp MFe MO2 MFeO nFe nO2 nFeO Nu PO2 Pe PrN rp _ Fe ℛ Rep t tbdiff T Tig Ts Tsurr TN u U V term
projected area of particle perpendicular to the direction of falling outer surface area of particle Biot Number in heat transfer specific heat of iron particle specific heat of gaseous oxidizing medium drag coefficient particle diameter mass diffusivity of oxygen in air total energy of particle buoyant force acting on iron particle drag force exerted on iron particle opposite the direction of falling gravitational acceleration average convection heat transfer coefficient mass stoichiometric coefficient of iron mass stoichiometric coefficient of oxygen mass stoichiometric coefficient of iron oxide mass of particle atomic weight of iron molecular weight of oxygen molecular weight of iron oxide molar stoichiometric coefficient of iron molar stoichiometric coefficient of oxygen molar stoichiometric coefficient of iron oxide average Nusselt number partial pressure of oxygen in the ambient gaseous medium Peclet number Prandtl number of gaseous fluid radius of iron particle mass reaction rate of iron Reynolds number for iron particle time burning time of particle based on diffusionally-controlled regime absolute temperature of iron particle ignition temperature of iron particle surface temperature of iron particle absolute temperature of surroundings absolute temperature of ambient gaseous oxidizer specific internal energy of the system total internal energy of the system terminal velocity of falling iron particle
Vp W YO2 ;N
volume of spherical iron particle weight of spherical iron particle mass fraction of oxygen in the ambient gas
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