Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method

Computers and Mathematics with Applications xxx (xxxx) xxx

Contents lists available at ScienceDirect

Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method ∗

Yousef Kazemian a , Saman Rashidi e , Javad Abolfazli Esfahani b,c , , Omid Samimi-Abianeh d a

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Department of Mechanical Engineering, Wayne State University, Detroit, MI, USA e Department of Energy, Faculty of New Science and Technologies, Semnan University, Semnan, Iran b

article

info

Article history: Available online xxxx Keywords: Combustion Porous media Grains shapes Lattice Boltzmann method Flame

a b s t r a c t The Lattice Boltzmann method is used to simulate the propane–air mixture combustion in a porous media with different grain shapes of triangular, elliptical, rectangular, and star. The effects of these shapes on the flow and temperature fields and the flame characteristics were investigated. In order to simulate the flow in the porous medium, the method of creating barriers against the flow is used. The black and white photos are transformed into the matrix of 0 and 1, written in the form of the lattice Boltzmann code for solving the momentum, energy, and concentration equations. The results show that the porous media made by grains with sharp corners provide the highest reverse flow. The porous media with grain shapes of rectangular and triangular provide the highest and least pressure drop, respectively. The length of the flame decreases by using the porous media and the star shape has the largest length of the flame, while the rectangular one has the smallest flame length. The shapes of flames obtained for the elliptical and triangular grains are very similar to each other, while, the shape of flame for the star grain is close to the non-porous medium case. In general, using the porous media increases the heat transfer rate from the walls and also creates the fluctuations in the heat transfer along the wall. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction There are many studies about combustion in porous media due to the higher burning rates, lower emissions, and the wider flammability limits (e.g., [1,2]. In addition, the stability and propagation of the combustion waves can be controlled by using the porous materials (e.g., [3,4]. Some of these works which are relevant to the current research are reviewed in the section. Shi and Wang [5] studied the technologies of regenerative combustion and porous medium to have a steady combustion using low calorific value fuel. They found that combining these technologies create an applicable burner with better heat load. Tarokh and Mohamad [6] used the Lattice Boltzmann method to study the influences of porous materials at the exit of counterflow combustor. Their results indicated that using porous materials decreases the ∗ Corresponding author at: Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Viet Nam. E-mail address: [email protected] (J.A. Esfahani). https://doi.org/10.1016/j.camwa.2019.10.015 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

2

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Nomenclature am A Á C C3 H 8 CO2 CO2 C3 H 8 Cp cs c dp Df Da ei e21 a e21 ext E F eq Fs,i fi eq fi H2 O H h´ I i K kov k L Lem m M Mm N2 Nu O2 P

Stoichiometric coefficients (-) Surface (m2 ) Surface of grains (m2 ) Molar concentration of propane (kmol/m3 ) Molar concentration of oxygen (kmol/m3 ) Carbon dioxide (-) Propane (-) Heat capacity (J/kg K), = 1.01 × 103 J/kg K Speed of sound (-) Lattice streaming speed (-), = 1 Diameter of grains (m) Diffusion coefficient (m2 /s), DC3H8 = 1 .1 × 10 −5 m2 /s, DO2 = 2.1× 10 10 −5 m2 /s, DH2O = 2.2 × 10 −5 m2 /s Darcy number (-) Discrete particle velocity in LBE model (-) Approximate relative error (-) Extrapolated relative error (-) Activation energy (kcal/mol), = 30 kcal/mol Distribution function of temperature and concentration (-) Equilibrium distribution function of temperature and concentration (-) Distribution function of flow (-) Equilibrium distribution function (-) Water vapor (-) Channel height (m) Environment of grains (m) Unit matrix (-) Direction index in LBE model (-) Permeability coefficient (m2 ) Reaction coefficient (cm3 /mol s), = 9.9 × 1013 cm3 /mol.s Thermal conductivity (J/m s K) Channel length (m) Lewis number (-) Species (-) Number of node (-) molecular weight of species m (-) Nitrogen (-) Nusselt number (-) Oxygen (-) p Non-dimensional pressure (-) = 2

P Pr p Q Re r R T t T0 t∗ u Uin

Apparent order Prandtl number (-) Pressure (kg/m s2 ) Heat of overall reaction (J/mol), = 2.05 × 106 J/mol U H Reynolds number (-) = 0ν Ratio of the number of nodes (-) Universal gas constant (J/mol K) Temperature (K) Time (s) Wall and ambient temperature (K) ν t Non-dimensional time (-)= Hf 2 Vector velocity (m/s) Velocity inlet of lattice (-)

−5

m2 /s, DCO2 = 1 .6 ×

U0 ρ0

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Uout U U U0 u V

ν wi x X Y Ym

Velocity outlet of lattice Non-dimensional Vector velocity (-) Non-dimensional fluid velocity in x direction (m/s) = Inlet velocity (m/s) Fluid velocity of lattice (-) Non-dimensional fluid velocity in y direction (m/s) Fluid velocity in y direction (m/s) Weighting coefficient Axial coordinate (m) Non-dimensional Axial coordinate (-) Non-dimensional Transversal coordinate (-) mass fraction of species (-)

3

u U0

Greek symbols

δx δt ∇ ∇. ρ0 ρ τυ ω θ αf ωow vf ωm ϕ θmean ρout ρin ε ϵ 21 ϕext ψ ∀void ∀total

Space step (-) Time step (-) Gradient (-) Divergence (-) Ambient density of lattice (-) Density of propane–air mixture (Kg/m3 ), = 1.1 kg/m3 Relaxation time (-) Reverse of relaxation time (-) Non-dimensional temperature (-) Thermal diffusivity coefficient for fluid (m2 /s), = 2.2 × 10 Overall reaction rate (mol/m3 s) Kinematical viscosity for fluid (m2 /s), = 1.6 × 10 −5 m2 /s Source of equation of concentration (kg/m3 s) Equivalence ratio (-) Non-dimensional mean temperature (-) Lattice Density at outlet (-) Lattice Density at inlet (-) Porosity (-) Difference of data (-) Extrapolated values (-) Constant parameter (-) Void volume of porous medium (m3 ) Total volume of porous medium (m3 )

−5

m2 /s

carbon monoxide and unburned fuel concentrations at the exit. Liu et al. [7] investigated the lean premixed combustion in the porous medium numerically and experimentally. They considered the packed bed of aluminum oxide pellets as the porous medium. They concluded that the porosity of the packed bed increases by increasing the diameter of pellets in the range of 6 to 10 mm; this leads to the increase in the mean velocity of the flame. Moghadasi et al. [8] simulated the counterflow, non-premixed combustion of porous fuel particles. They considered the influences of thermal radiation in their simulations. Their results showed that the flame temperature increases by increasing the dust concentration and decreasing the particles radius and their porosity. Lutsenko [9] simulated the heterogeneous combustion in the porous materials under forced filtration or free convection. The combustion under free convection, when there is any forced gas input in porous materials, can be observed in a large number of natural or anthropogenic disasters such as burning of peatlands, coal dumps, etc. Accordingly, their simulation is important. Kim et al. [10] investigated the instability in combustion of a lab-scale gas turbine combustor with a sponge porous material. They observed that the porous material has the damping influences on the unstable flame dynamics. Wang et al. [11] investigated the premixed combustion in a partially filled micro porous combustor. They concluded that using larger values of wall thermal conductivity can increase the chemical reaction within porous material. Some researchers have focused on the porous burners. Dehaj et al. [12] performed an experimental study on the natural gas combustion of the porous burner. Their results showed that the maximum efficiency can be achieved for the excess air ratio of half as the combustion products have their maximum temperature at this ratio. Hashemi et al. [13] investigated Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

4

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 1. Computational domain.

Fig. 2. Different grain shapes used in this study.

numerically the influences of the thermal boundary conditions and porous material characteristics on the combustion of a porous-free flame burner. They found that the flame stability limit can extend as the pore density increases. Banerjee and Saveliev [14] focused on the high temperature thermal energy extracted from the counterflow porous burner. The temperature of heat extraction was in the range of 300 to 1300 K in this study. They showed that it is possible to extract about 60% of the thermal energy at 1300 K by porous burner with alumina separation wall. The discussed literature shows that although there are some studies about the combustion in porous media but there is no research work with a focus on the effects of grains shapes of porous media and their structures on flame shape, length of flame, and heat transfer. In addition, most of the attentions were focused on the investigation of some parameters in the porous media such as porosity, permeability, thermal conductivity. The numerical methods are successfully used by the researchers in the fields of biomass and combustion [15–20]. In this paper, the Lattice Boltzmann method is used to simulate the combustion onset of the propane–air mixture in the porous media with different grain shapes including triangular, elliptical, rectangular, and star shapes. The effects of these shapes on the flow and temperature fields, the shape, lift of length are studied. Moreover, the results obtained for the cases of using porous media are compared along with the results for the case of without using the porous media. 2. Physical model and assumptions The schematic view of the investigated model is shown in Fig. 1. The propane–air mixture flows through a twodimensional channel with uniform inlet velocity and constant temperature. The channel is filled with the porous media with different grain shapes including triangular, elliptical, rectangular, and star shapes. These shapes are shown in Fig. 2. The walls of the channel have the constant temperature. Moreover, the walls of the channel and solid boundaries in the porous structure for the concentration equation are no heat-flux. At the outlet of the channel, the output pressure is known and the flow has the zero temperature and concentration gradients. The simulations are performed at the constant Reynolds number of 30 and Prandtl number of 0.727. The equivalence ratio of 0.6 is considered for all cases. The channel length to width ratio is 2.5 and the macroscopic magnitude of the channel’s width is 0.01 m. The following assumptions were used for the simulation:

• The chemical reaction has no effect on the flow as the incompressible fluid is assumed. • The buoyancy effect is assumed to be negligible. Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

5

• The transport properties of fluid are constant. • The state of the flow is laminar and steady. 3. Governing equations 3.1. Lattice Boltzmann equation The D2Q9 model is used to solve the momentum, energy, and concentration equations in two-dimensional channel with the uniform grid size. The discrete speeds, {ei | i = 0, 1, . . . , 8}, are defined for all equations as follows [21]:

ei =

⎧ , ⎪ ⎨ 0 (

i=0

[ (i−1)π ]

[ (i−1)π ])

, sin c, 4 [ (i−1)π ]) [ (i−1)π ] , sin c 2 cos 4 4

cos

⎪ ⎩ √ (

4

i = 1, . . . , 4

(1)

i = 5, . . . , 8

δx δt

where c = is the Lattice velocity and it is equal to 1 in this work. The weight function in different directions can be written for all equations in the following form:

⎧ 4 ⎪ ⎪ , ⎪ ⎪ 9 ⎪ ⎨ 1 wi = , ⎪ 9 ⎪ ⎪ ⎪ ⎪ ⎩ 1, 36

i=0 i = 1, . . . , 4

(2)

i = 5, . . . , 8

3.2. Flow field equations The macroscopic equations of flow, including continuity and momentum equations, are:

∇.U = 0 (3) ∂U + Re (U .∇) U = −Re∇ (P ) + ∇ 2 U (4) ∂t∗ where t ∗ , Re, U , P, ν , and U0 are non-dimensional time, Reynolds number, non-dimensional velocity vector, nondimensional pressure, kinematic viscosity, and inlet velocity, respectively. Depending on the Knudsen number, the governing equations of the flow regime are different. For the Knudsen number lower than 0.001, the Navier–Stokes equations can be used (an Eulerian viewpoint in the flow with a finite volume assumption). However, for the Knudsen number in the range of 0.1 and 10, the Lattice Boltzmann equations can be employed. (A Lagrangian viewpoint considering particle clusters.) The Lattice Boltzmann equation, LBGK equation, with the single relaxation time can be written as [22]: fi (x + ei δ t , t + δ t ) − fi (x, t ) = −

) 1 ( eq fi (x, t) − fi (x, t)

(5)

τυ

This equation can be reformulated in the following form: eq

fi (x + ei δ t , t + δ t ) = ωfi (x, t ) + (1 − ω)fi (x, t)

(6) eq

In this equation, ω = , which τυ is the relaxation time of the fluid. fi , ei , and fi are the particle distribution function, the eq particle streaming velocity, and local equilibrium distribution function, respectively. fi for D2Q9 model can be presented in the following form [21]: 1 τυ

fi

eq

] [ uu: (ei ei − cs2 I) ei .u = wi ρ 1 + 2 + 4 cs

2cs

(7)

In this equation, ρ is the fluid density and u is the macroscopic fluid flow velocity in lattice units. Moreover, in the Lattice Boltzmann method, the distribution function, fi , can be employed to calculate the macroscopic properties∑ of fluid such ∑8 8 as the fluid density, ρ , and also√flow momentum, ρ u.; as they are calculated using ρ = f and ρ u = i=0 i i=0 ei fi . cs is sound speed and is equal to c / 3. 3.3. Energy and species conservation equations The dimensionless form of energy equation with the source term is:

( ) ( 2 ) ( ) ∂θ ∂θ ∂θ 1 ∂ θ ∂ 2θ 1 H2 Q .ωow + Re U + V = + + ∂t∗ ∂X ∂Y Pr ∂ X 2 ∂Y 2 Pr αf T0 ρ.Cp f

(8)

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

6

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

where θ = TT and T0 are the non-dimensional temperature and ambient temperature, respectively. U = Uu and V = Uv 0 0 0 y are the non-dimensional velocities and X = Hx and Y = H are the non-dimensional coordinates. Q , Cp , and k are the heat of overall reaction, the heat capacity, and the thermal conductivity, respectively, which have constant values. αf is the thermal diffusivity and is equal to ρkC . p The dimensionless form of concentration equation is:

( ) ( ) ( ) ∂Y m ∂Y m ∂Y m Lem ∂ 2 Y m ∂ 2Y m 1 H 2 ωm + Re U +V = + + ∂t∗ ∂X ∂Y Pr ∂X2 ∂Y 2 Pr αf ρ f m = C3 H8 , O2 , CO2 , H2 O

(9)

(D )m

where Lem = αf , Df , and Ym are the Lewis number, the fluid diffusion coefficient, the concentration of each species, f respectively. ωi and Dm are the mass rate of production and the diffusion coefficient of species, respectively. The summation of above mentioned species and Nitrogen to be equal one:

∑

Ym = 1

(10)

m=5

The concentration equation in mesoscopic scale for the Lattice Boltzmann is: Fs,i (x + ei δ t , t + δ t ) − Fs,i (x, t ) = − s = T, Ym (m = C3 H8 , O2 , CO2 , H2 O)

1 [

τs

eq

Fs,i (x, t ) − Fs,i (x, t ) + wi .Qs

]

(11)

The source term of chemical reaction, Qs , is given by the similarity in dimensionless forms of the energy and concentration equations. The equilibrium distribution function is defined:

{

eq

Fs,i = wi s 1 + 3

(ei .u)

+

c2

9 (ei .u)2 c4

2

−

3 u2

}

2 c2

(12)

The temperature and mass fraction of species can be determined in terms of the distribution function as follows: T =

∑

FT ,i

(13)

i

Ym =

∑

FYm,i

(14)

i

The mass rate of species production is as follows:

ωm = am .Mm .ωov = am .Mm .kov CC3 H8 CO2 exp(−E /RT )

(15)

aO2 = −5, aC3H8 = −1, aH2O = 4, and aCO2 = 3 are stoichiometric coefficients. All reaction parameters used in this simplified reaction equation were reported by Yamamoto [23] and Westbrook and Dryer [24]. Accordingly, the correction is required for heat expansion even for the incompressible model; otherwise the high overestimation occurs in the reaction rate. The following molar concentrations’ equation is used for obtaining the reaction rate: Cm =

ρ0 Ym Mm

.

(

T0

) (16)

T

The reaction rate in the Lattice Boltzmann method is defined as:

(ωm )LBM =

ωm · ρ0 U0 /L

(

ρ0 U 0 L

) (17) LBM

The thermal diffusivity and diffusion coefficient are defined as:

αf =

2τT − 1 6

c 2 δ t , Df =

2τY − 1 6

c 2 δt

(18)

3.4. Nusselt number definition The mean temperature is employed to define the Nusselt number:

θmean =

∫

ρ U θ dA/

∫

ρ UdA

(19)

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

7

In this equation, θmean , θ , and u indicate the mean temperature, the local temperature, and the local velocity, respectively. The Nusselt number on the wall of the channel is defined as follows: ∂θ H )(− ) (20) Nu = ( θ0 − θmean ∂y where H and θ0 are the channel width and the non-dimensional wall temperature, respectively. 3.5. Darcy number and porosity definitions The Darcy law, which is another method for modeling the flow in a porous medium, involves two important nondimensional parameters, including the Darcy number and porosity coefficient, in the problem. In this section, the result of current study is compared with the Darcy method. According to experimental results of Vafai [25] for the porous media with a homogeneous and uniform structure and with a bed of spherical beads with diameter of dp , the Darcy number and the porosity coefficient are calculated by:

ε=

∀void ∀total

(21)

K (22) H2 where ε , ∀v oid , ∀total , Da, K , and H are the porosity coefficient, void volume of the porous medium, total volume of the porous medium, Darcy number, permeability coefficient, characteristic length, respectively. Note that the channel width is considered as the characteristic length in this study. According to the experimental data, K can be calculated as follows: Da =

K =

d2p ε 3

(23)

180(1 − ε )2

where dp is the diameter of the beads of the porous medium. Since in the present work, the shape of the beads is not spherical, in order to obtain the mean diameter for all structures, the following relation is used: dp =

4A´

h´ where A and h are the surface area and circumference of each bead, respectively.

(24)

3.6. Boundary conditions The boundary conditions for the lattice Boltzmann method used in this problem are presented in this section. For the channel inlet, the velocity profile is assumed to be uniform:

ρin = (f0 + f2 + f4 + 2(f3 + f6 + f7 ))/(1 − Uin )

(25)

f1 = f3 + 2/3ρin Uin

(26)

f5 = f7 + 1/6ρin Uin − 0.5(f2 − f4 )

(27)

f8 = f6 + 1/6ρin Uin − 0.5(f4 − f2 )

(28)

where Uin and ρin are mesoscopic input velocity and macroscopic input density, respectively. The constant temperature and concentration are assumed: Fs,8 = 1/6 s − Fs,0 + Fs,2 + Fs,3 + Fs,4 + Fs,6 + Fs,7

))

(29)

Fs,5 = 1/6 s − Fs,0 + Fs,2 + Fs,3 + Fs,4 + Fs,6 + Fs,7

))

(30)

Fs,1 = 4/6 s − Fs,0 + Fs,2 + Fs,3 + Fs,4 + Fs,6 + Fs,7

))

(31)

(

(

(

(

(

(

The bounce back boundary conditions are used as the boundary conditions for the flows on the channel walls and outer surfaces of the grains in the porous media. For example, for the bottom wall, these boundary conditions are: f2 = f4

(32)

f6 = f8

(33)

f5 = f7

(34)

Moreover, for the walls, the constant temperature is used as the boundary condition for the energy equation, while the no-flux boundary is considered for the concentration equation. The constant pressure is used as the boundary condition of flow equation at the channel output: Uout = −1 + (f0 + f2 + f4 + 2∗(f1 + f5 + f8 ))/ρout

(35)

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

8

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

f3 = f1 − 2/3ρout Uout

(36)

f7 = f5 − 1/6ρout Uout + 0.5(f2 − f4 )

(37)

f6 = f8 − 1/6 ρout Uout + 0.5(f4 − f2 )

(38)

In addition, the zero gradients are considered for the temperature and concentration at the channel output. These boundary conditions can be presented in the following forms: Fsn,3 = Fns,−3 1

(39)

=

Fns,−6 1

(40)

=

Fns,−7 1

(41)

Fsn,6 Fns,7

4. Grid dependency and validation 4.1. Grid dependency investigation A structured square mesh is used for the entire solution area to investigate the dependency of the results to the mesh resolution as shown in Fig. 3. In order to perform a grid dependency investigation, the Grid Convergence Index, GCI, criterion is used. In this criterion, the less GCI means that results are more independent of the grid sizes. The GCI criterion is determined for the flow, energy, and concentration equations for three different grid sizes and conditions. The results of this criterion for the case of porous medium with the star grain shape are presented in Table 1. The data in this table are presented at X = 0.5 and Y = 0.5. r is the ratio of the space step of each grid size to the space step of former grid and the apparent order P of the method is determined as follows:

ϵ32 P = ln ln(r21 ) ϵ21 1

( (

)

( + ln

P −ψ r21

))

P r32 −ψ

ψ = 1.sgn(ϵ32 /ϵ21 )

(42) (43)

ψ is a constant parameter, which is used for simplifying Eq. (38). In this equation, ϵ21 = Data2 − Data1 and ϵ32 = Data3 − Data2. The extrapolated values are determined by: P P ∅21 ext = (r21 Data1 − Data2)/(r21 − 1)

(44)

The approximate relative error is: e21 a

⏐ ⏐ ⏐ Data1 − Data2 ⏐ ⏐ ⏐ =⏐ ⏐ Data1

(45)

The extrapolated relative error can be defined as follows:

⏐ 21 ⏐ ⏐ ∅ext − Data1 ⏐ ⏐ ⏐ ∅21

⏐ e21 ext = ⏐

(46)

ext

Eventually, the fine-grid convergence index can be determined by: 21 GCIfine =

1.25e21 a P r21 −1

(47)

According to the GCI results, the differences between the velocity, temperature, and concentration calculated by the grid numbers of 80 × 200 and 160 × 400 are negligible. Accordingly, in order to reduce the solution time and achieve the results independent from the grid size, the grid number of 80 × 200 has been chosen for this study. 4.2. Validation To validate the results and benchmark the accuracy of the numerical solution, the results of the numerical study are compared with the results of Yamamoto et al. [26] for two opposed flows, and counter flows of fuel and air through a two-dimensional empty geometry without using porous materials. Note that these flows can provide almost a flat flame that the flame properties, such as temperature and concentration, change only in the direction of the gas flow. In other words, such flame can be considered one-dimensional. The schematic of the modeled case is shown in Fig. 4. The results of the distributions of temperature, concentration, and velocity are disclosed in Fig. 5. A good agreement was reached between simulations and measurements. Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

9

Fig. 3. The grid used inside the computational domain.

Fig. 4. The schematic of molded case for validation (Figure is taken from [26] with permission from the publisher).

Table 1 The results of GCI criterion for the case of porous medium with the star grain shape, M is the number of grids in the solution area. M1 , M2 , M3 r21 r32 Data1 Data2 Data3 P ∅21 ext

U(0.5,0.5)

θ (0.5, 0.5)

YO2 (0.5, 0.5)

160×400, 80×200, 40×100 2 2 0.658 0.655 0.591 4.415 0.658

160×400, 80×200, 40×100 2 2 5.953 5.944 5.751 4.422 5.953

160×400, 80×200, 40×100 2 2 0.106 0.105 0.100 3.807 0.106 0.943

e21 a (%)

0.455

0.151

e21 ext (%)

0.021

0.007

0.072

21 GCIfine (%)

0.001

0.009

0.090

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

10

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 5. The comparison between the results of the present numerical study and the results of Yamamoto et al. [26].

5. Results and discussion The effects of porous media with different grain shapes on the streamlines, pressure drop, temperature contours, and flame shape and length are studied and discussed in this section. The effects of porous media with different grain shapes on the streamlines are shown in Fig. 6. The streamlines for the case of channel without using porous medium are parallel, as expected. The porous material with rectangular structure provides the streamlines with the lowest reverse flow and deviation as well as the least changes in terms of fluid redirection, while the streamlines for the case of the porous material with star structure have most disorderly among all cases. Note that for a constant value of Reynolds number, the case of porous material with star structure provides more reverse flow and vortices. After that, the porous material with triangular structure has high reverse flow and vortices but they are slighter than that for the case of star structure. However, for rectangular and elliptical structures, the reverse flow can be seen rarely and the vortices do not exist at all. As a result, the porous media with structures with sharper corners have the highest reverse flows and vortices, while the structures with flatter surfaces, rectangular, elliptical, and circular, rarely generate the reverse flow and vortices. Flow pressure drops of different cases are reported in Table 2. The pressure drop increases by using the porous materials as the solid matrix of these materials act as barrier and exerts the viscous drag against the flow. The minimum pressure drop is related to the case of triangular shape, while the rectangular shape has the maximum one among all porous structures considered in this study. The triangular and star shapes have almost similar pressure drops. The porous media with solid matrixes with sharper corners, corner angles lower than 90 deg., provide less pressure drop penalty as compared with the structures with corner angles greater than 90 deg. The values of the Darcy number and porosity coefficient shown in Table 3 are used in this study. As shown in Table 3, the porosity coefficients are approximately equal for all of the cases, and also the Darcy numbers of different structures Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

11

Fig. 6. The effects of porous media with different grain shapes on the streamlines. Table 2 The pressure drops of different cases considered in this study. Cases

Without porous medium

Star shape

Triangular shape

Rectangular shape

Elliptical shape

Non-dimensional pressure drop

0.044

0.831

0.761

1.599

1.249

Table 3 The values of the Darcy number and porosity coefficient for different porous structures. Different grain shapes

Da∗10−4

dp ∗10−3 (m)

ε

Star grain shape Rectangular grain shape Triangular grain shape Elliptical grain shape

8.71 3.30 9.90 5.96

1.13 2.06 1.39 1.87

0.76 0.61 0.77 0.67

have the same orders. This indicates that despite the close values of the porosity coefficients and the Darcy numbers of different structures, porous media can have very different structures and shapes, which may affect the combustion. The effects of porous media with different grain shapes on the non-dimensional temperature field are shown in Fig. 7. The maximum non-dimensional temperature of 5.85579 can be observed for the case of channel without using the porous media. The non-dimensional temperature is defined as the ratio of flow temperature inside the channel to the ambient temperature. The maximum non-dimensional temperature increases by using the porous media inside the channel, but this maximum temperature can be observed locally at just one point. The flames have irregular and asymmetric shapes for the cases of channel enhanced with porous media with different particle structures as compared with the case of empty channel. This causes an irregular and non-uniform heat transfer from the channel walls for these cases. The results also show that the mean temperature decreases in the channel by using all porous media with different particle structures, which thus can improve the combustion rate. Moreover, the flame length decreases by using all porous media with different structures as compared with the case of empty channel, which thus causes the instability in the flame and increases the likelihood of its extinction. These reductions are about 92.64%, 72.92%, 46.36%, and 59.68% for the Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

12

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 7. The effects of porous media with different grain shapes on the non-dimensional temperature field.

structures of star, triangular, rectangular, and elliptical, respectively. By comparing the results of different porous media with different structures, it can be concluded that the smallest flame length and the most unstable flame is related to the rectangular structure, while the maximum flame length, and the most stable flame is observed for the star structure. The maximum and minimum temperatures are observed for the rectangular and elliptical structures, respectively. It should be noted that the maximum temperatures (θmax = 6.11) of the elliptical and star structures are very close to each other. In terms of the flame shapes, the flame shapes of elliptical and triangular structures are similar to each other and their flames have a circular shape. Moreover, the flame shapes of star structure and case of empty channel are very similar and their flames have a triangular shape. In the meantime, only the flame of rectangular structure is not similar to the flames of other cases. Finally, the flame for the star structure has the least distance from the entrance of the channel (X = 0.07) as compared with other cases, while the flame of rectangular structure has the highest distance (X = 0.31) from the entrance. The elliptic and triangular structures provide similar results from the flame location viewpoint. In Fig. 7, the local temperature distributions of different cases were shown. However, in order to have a more accurate comparison of the flame temperature in the porous media, the mean temperatures should be compared between various cases. Hence, Fig. 8 shows the mean non-dimensional temperatures along the channel length for the cases of channel enhanced with porous media with different grain shapes and empty channel. As shown in this figure, the mean temperatures for all cases of channel enhanced with porous media with different structures are equal to or less than the case of empty channel at X ≥ 0.7. Accordingly, in general, the porous media reduces overall temperature and produces a combustion with a lower temperature. Among different porous structures, only rectangular structure has a higher temperature in the combustion region than the case of empty channel. The temperature for this case, rectangular structure, decreases along the channel length and reaches to the values lower than the temperature for the case of empty channel. The best case among different structures belongs to the elliptical structure, which provides the lowest mean temperature along the channel length. The star and triangular structures have the closest results to each other and in the combustion region, the mean temperature of star structure is very close to the mean temperature of the empty channel. Fig. 9 shows the average non-dimensional temperatures along the channel length for the cases of channel enhanced with porous media with different grain shapes and empty channel. As shown in this figure, the maximum average for the empty channel is 5.119, which occurs at X = 0.189. Among all cases, the star structure has the maximum average temperature of 5.428. For this case, the maximum average temperature is observed at a distance of X = 0.138 from the input of the channel, which is the closest distance from the input between different cases. The average temperature can increase about 6.03% by using the porous medium with the star structure as compared with the empty channel. Moreover, the elliptical structure has the minimum average temperature of 4.959 located at X = 0.387. The temperature at the channel output is very important. Generally, the output temperature decreases by using the porous medium inside the channel. The maximum reduction is about 17.10%, which is observed for the elliptical structure. Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

13

Fig. 8. The mean non-dimensional temperatures along the channel length for the cases of channel enhanced with porous media with different grain shapes and empty channel.

Fig. 9. The average non-dimensional temperatures along the channel length for the cases of channel enhanced with porous media with different grain shapes and empty channel.

Fig. 10 shows the Nusselt number on the bottom wall of the channel for the cases of channel enhanced with porous media with different grain shapes and empty channel. It can be seen that the Nusselt numbers for the cases of channel enhanced with different porous media are oscillating. The heat transfer rate increases by using porous media with different grain shapes as compared with the case of the empty channel. Moreover, the heat transfer rate in the combustion zone is higher for the case of star structure as compared with other cases, while the minimum heat transfer rate in the flame region is observed for the triangular structure among all cases. The values of the average Nusselt number on the bottom wall of the channel for different porous structures are presented in Table 4. As presented in this table, among all cases considered in this study, the maximum value of Nusselt number is observed for the star structure, while the minimum value is related for the empty channel. Among the cases of channel enhanced with different porous media, the lowest average Nusselt number is for the porous medium with triangular structure. The results show that the porous medium can increase the average Nusselt number up to 38.21%, which indicates the good abilities of this material for heat transfer enhancement. 6. Conclusions In this study, the lattice Boltzmann method was employed to simulate the premixed combustion of propane–air mixture in the porous media with different grain shapes including triangular, elliptical, rectangular, and star shapes. The Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

14

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 10. The Nusselt number on the bottom wall of the channel for the cases of channel enhanced with porous media with different grain shapes and empty channel.

Table 4 The values of the average Nusselt number on the bottom wall of the channel for different porous structures. Different grain shapes

Average Nusselt number on the bottom wall of the channel

Star grain shape Rectangular grain shape Triangular grain shape Elliptical grain shape Empty channel

5.870 5.566 5.379 5.612 4.247

effects of these shapes on the flow and temperature fields and the shape and length of the flame were studied. The results obtained for the cases of using porous media were compared along with the results for the case of without using the porous media. The main results of this study are listed as follows:

• The case of porous medium with star structure provides the most reverse flow and vortices. After that, the porous

• • • • •

• •

material with triangular structure has the most reverse flow and vortices. However, for the other two structures, the reverse flow can be seen rarely and the vortices do not exist at all. The minimum pressure drop is related to the case of triangular shape (∆P = 0.761), while rectangular shape (∆P = 1.599) has the maximum one among all porous structures considered in this study. The mean temperature decreases in the channel by using all porous media with different structures, which thus can improve the combustion rate. The smallest flame length (∆X = 0.658) and the most unstable flame is related to the rectangular structure, while the maximum flame length (∆X = 1.314), and the most stable flame is observed for the star structure. The flame for the star structure has the least distance (X = 0.07) from the entrance of the channel as compared with other cases, while the flame of rectangular structure has the highest distance (X = 0.31) from the entrance. In general, the porous media reduce overall temperature and provides a combustion with lower temperature. Among different porous structures, only rectangular structure has a higher temperature (θmax = 7.41) in the combustion region than the case of empty channel. The best case among different structures belongs to the elliptical structure, which provides the lowest mean temperature (θmax = 6.11) along the channel length. The heat transfer rate in the combustion zone is higher for the case of star structure (Nuav e = 5.87) as compared with other cases, while the minimum heat transfer rate in the flame region is observed for the rectangular structure (Nuav e = 5.37) among all cases.

Finally, this work discloses the effect of grain shapes on pre-combustion with the powerful lattice Boltzmann method. This needs more research to understand the dynamics of flow in each case and may modify them. Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.

Y. Kazemian, S. Rashidi, J.A. Esfahani et al. / Computers and Mathematics with Applications xxx (xxxx) xxx

15

References [1] M.A. Mujeebu, M.Z. Abdullah, M.A. Bakar, A.A. Mohamad, R.M.N. Muhad, M.K. Abdullah, Combustion in porous media and its applications–a comprehensive survey, J. Environ. Manage. 90 (8) (2009) 2287–2312. [2] F. Al-Malki, Influence of transverse temperature gradient on the propagation of triple flames in porous channels, Comput. Math. Appl. 76 (2018) 406–418. [3] G. de Soete, Stability and propagation of combustion waves in inert porous media, Symp. Int. Combust. 11 (1) (1967) 959–966. [4] V.S. Kakutkina, Propagation of spherical gas filtration combustion waves in inert porous media, Combust. Explosion Shock Waves 35 (1) (1999) 53–59. [5] W. Shi, D. Wang, Combined application of regenerative combustion technology and porous medium combustion technology, Energy Procedia 66 (2015) 209–212. [6] A. Tarokh, A.A. Mohamad, Investigation of the effects of porous media at the exit of counterflow combustion using the lattice Boltzmann method, Spec. Top. Rev. Porous Media Int. J. 6 (3) (2015). [7] H. Liu, D. Wu, M. Xie, H. Liu, Z. Xu, Experimental and numerical study on the lean premixed filtration combustion of propane/air in porous medium, Appl. Therm. Eng. 150 (2019) 445–455. [8] H. Moghadasi, N. Malekian, Khoeini Poorfar A., M. Bidabadi, Thermal radiative study of counterflow combustion of porous particles, Chem. Eng. Process.-Process. Intensif. 185 (2018) 163–173. [9] N.A. Lutsenko, Numerical model of two-dimensional heterogeneous combustion in porous media under natural convection or forced filtration, Combust. Theory Model. 22 (2) (2018) 359–377. [10] Y.J. Kim, D.K. Lee, Y. Kim, Experimental study on combustion instability and attenuation characteristics in the lab-scale gas turbine combustor with a sponge-like porous medium, J. Mech. Sci. Technol. 32 (4) (2018) 1879–1887. [11] W. Wang, Z. Zuo, J. Liu, Numerical study of the premixed propane/air flame characteristics in a partially filled micro porous combustor, Energy 167 (2019) 902–911. [12] M.S. Dehaj, R. Ebrahimi, M. Shams, M. Farzaneh, Experimental analysis of natural gas combustion in a porous burner, Exp. Therm. Fluid Sci. 84 (2017) 134–143. [13] S.A. Hashemi, M. Nikfar, S.A. Ghorashi, Numerical study of the effect of thermal boundary conditions and porous medium properties on the combustion in a combined porous-free flame burner, Proc. Inst. Mech. Eng. Part A J. Power Energy 232 (7) (2018) 799–811. [14] A. Banerjee, A.V. Saveliev, High temperature heat extraction from counterflow porous burner, Int. J. Heat Mass Transfer 127 (2018) 436–443. [15] S. Gur, T. Danielson, Q. Xiong, C. Hin, S. Pannala, G.N. Frantziskonis, A. Savara, C.S. Daw, Wavelet-based surrogate time series for multiscale simulation of heterogeneous catalysis, Chem. Eng. Sci. 144 (2016) 165–175. [16] Q. Xiong, S.C. Kong, High-resolution particle-scale simulation of biomass pyrolysis, ACS Sustain. Chem. Eng. 4 (2016) 5456–5461. [17] Q. Xiong, Y. Yang, F. Xu, Y. Pan, J. Zhang, K. Hong, G. Lorenzini, S. Wang, Overview of computational fluid dynamics simulation of reactor-scale biomass pyrolysis, ACS Sustain. Chem. Eng. 5 (2017) 2783–2798. [18] M. Bidabadi, M. Harati, Q. Xiong, E. Yaghoubi, M.H. Doranehgard, P. Aghajannezhad, Volatization & combustion of biomass particles in random media: Mathematical modeling and analyze the effect of lewis number, Chem. Eng. Process. -Process. Intensif. 126 (2018) 232–238. [19] M. Bidabadi, Q. Xiong, M. Harati, E. Yaghoubi, M.H. Doranehgard, A. Rahbari, Study on the combustion of micro organic dust particles in random media with considering effect of thermal resistance and temperature difference between gas and particles, Chem. Eng. Process.-Process. Intensif. 126 (2018a) 239–247. [20] H. Zhong, Q. Xiong, Y. Zhu, S. Liang, J. Zhang, B. Niu, X. Zhang, CFD modeling of the effects of particle shrinkage and intra-particle heat conduction on biomass fast pyrolysis, Renew. Energy 141 (2019) 236–245. [21] Y.H. Qian, D. d’Humières, P. Lallemand, Lattice BGK models for Navier–Stokes equation, Europhys. Lett. 17 (6) (1992) 479. [22] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (3) (1954) 511. [23] K. Yamamoto, Pressure change and transport process on flames formed in a stretched, rotating flow, Combust. Flame 118 (3) (1999) 431–444. [24] C.K. Westbrook, F.L. Dryer, Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames, Combust. Sci. Technol. 27 (1–2) (1981) 31–43. [25] K. Vafai, Convective flow and heat transfer in variable-porosity media, J. Fluid Mech. 147 (1984) 233–259. [26] K. Yamamoto, X. He, G.D. Doolen, Simulation of combustion field with lattice Boltzmann method, J. Stat. Phys. 107 (1–2) (2002) 367–383.

Please cite this article as: Y. Kazemian, S. Rashidi, J.A. Esfahani et al., Effects of grains shapes of porous media on combustion onset—A numerical simulation using Lattice Boltzmann method, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.10.015.