Yang Liu,Yangbo Deng,,*,Junrui Shi,Rujie Xiao,Houping Li

1 Marine Engineering College,Dalian Maritime University,Dalian 116026,China

2 Naval Architecture and Ocean Engineering College,Dalian Maritime University,Dalian 116026,China

3 School of Transportation and Vehicle Engineering,Shandong University of Technology,Zibo 255049,China

Keywords: Numerical simulation Porous media Thermal non-equilibrium Heat transfer Thermal conductivity

ABSTRACT A simplified two-dimensional model of two-layer porous burner based on pore level is developed.The heat transfer of solid phase in porous burner is seen as the synergistic effects of conduction through connecting bridges and surface radiation between the solid particles in the model.A numerical simulation study on the characteristics of flow,combustion and heat transfer in the two-layer porous burner is carried out using the pore level model,and the effects of the control parameters such as the inlet velocity and solid thermal conductivity on thermal non-equilibrium are investigated.The results show that the flame structure is highly two-dimensional based on pore level.Obvious thermal non-equilibrium in the burner for the two phases and solid phase are observed,the largest temperature difference between the gas and solid phases is observed in combustion zone,while the temperature difference inside the solid particles is largest near the flame front.The results also reveal that thermal non-equilibrium of porous burner is much affected by the inlet velocity and solid thermal conductivity.

1.Introduction

Combustion in porous burners has been extensively studied in past decades.Compared to free-flame burners,the advantages of porous burners can be attributed to the heat recirculation of solid matrix.The heat is recirculated by solid matrix from combustion region to the preheat region by radiation and conduction,and then the incoming mixture is preheated by convection.Consequently,the flame temperature is increased,i.e.the combustion characteristics is enhanced [1–3].

Experimental and numerical studies are the two major approaches commonly employed to investigate combustion characteristics in porous media.Gaoet al.conducted a series of experimental studies on combustion performance in a two-layer burner[4–6].They have focused on extending flammability limit and reducing emission products.The behavior of premixed combustion of lean methane–air mixtures in a two-layer porous media combustor were studied by Bubnovichet al.[7].The results indicated that the flame is stabilized near the interface of the two sections.Zhenget al.[8]have made outstanding contributions in this regard.They used bare and coated thermocouple junctions to register gas and solid temperatures simultaneously.But the measured gas temperature needs to be checked because of the heat transfer between the bare junction and its surroundings.They found that the temperature distribution of gas and solid phases is different as the flame propagates upstream or downstream,and the temperature gap between the gas and solid phases has maximum and minimum points in the reaction zone and preheating zone,respectively.

Historically the numerical research on combustion characteristics of porous media burners has two aspects,macro level and pore level.The volume averaged model which based on macro level considered the transport and non-equilibrium between phase states by the empirical formula.Most researchers used volume averaged model to perform numerical studies in porous media burners [9–12].Coupled chemistry-hydrodynamics simulations about syngas production in a packed bed of uniform 3 mm diameter alumina spherical particles were performed by Liet al.[13].It was found that syngas products can be obtained vastly from fuel-rich and it can be obtained up to a mole fraction of 23%.De Lemoset al.[14]numerically studied turbulent combustion characteristics in porous combustor based on volume-and-time double averaged concept.Wenet al.[15]modeled premixed flames in the porous media with the use of the zonal hybrid Reynolds-averaged Navier-Stokes (RANS)/Large Eddy Simulations (LES) model,and flame propagating process and quenching behaviors were analyzed.They found that the flame quenching phenomenon is controlled by a competition between heat release rate from chemical reactions and heat losses rate to the solid phase.Djordjevicet al.[16]presented a one-dimensional numerical model of the combustion in ceramic sponge-like structures,and the important feature of the suggested model is that it is able to predict the flame stability behavior in different sponge structures.Zenget al.[17]presented an experimental and numerical study on syngas production in a two-layer burner.In their numerical study,a two-dimensional two-temperature model based on the volume averaged model was conducted.The results showed that the temperature profiles of gas and solid were inhomogeneous.

In recent decades,pore level numerical simulations were applied by many researches to study the combustion characteristics in a porous burner.Jianget al.[18]modeled propane-air premixed combustion in randomly packed beds,and the flame characteristics at various time points with different inlet velocities were analyzed.Shiet al.[19]performed two-dimensional pore level simulation of the low-velocity filtration combustion in a single-layer porous burner with staggered cylindrical particles.The solid surface radiation was considered by discrete ordinates(DO) model,but the contact thermal resistance between particles was ignored.The obvious thermal non-equilibrium for the intraphase and heterogeneous phase were observed except for the inlet and outlet regions of the burner.A two-scale method was proposed by Chenet al.[20]to study the dispersion effect and turbulent premixed flame characteristics in a porous burner at the pore level and system level.They analyzed the thermal non-equilibrium of low-velocity reaction flows in isotropic porous media.Lu [21]and Xinet al.[22]reported a multi-scale model that the temperature differences in solid particles were considered by the coupling of gas energy equation at representative elementary volume(REV)level with the heat conduction equation of solid particles at pore level.The effects of particle diameters,mass flow rates and solid materials on the characteristics of local thermal non-equilibrium(LTNE) were investigated when cryogenic nitrogen flowing through a porous packed bed.It was shown that larger particle diameter and mass flow rate result in greater temperature differences between the fluid and solid particle centers,and the time reaching local thermal equilibrium (LTE) was affected by particle diameter,mass flow rate,and the heat capacity of the materials.

Recently,Yakovlevet al.[23]carried out pore level simulation on combustion characteristics of lean and fuel-rich CH4-Air mixtures in a three-dimensional model.They simulated particles falling under the action of gravity to generate the geometric model of the packed bed and used the root mean squares (RMS) temperature to quantitatively analyzed the thermal non-equilibrium between the gas and solid phases.It was shown that the thermal non-equilibrium is placed near the heat release region and the volume averaged approach underestimated the thermal nonequilibrium compared with the pore level study.But the thermal non-equilibrium of same phase has not been studied in the research.

It is important to note that we should differentiate between the LTNE and thermal non-equilibrium mentioned above.When we are talking about the LTNE of porous media,we solve two thermal equations for each and every single location/point of the geometry.Pore-level solves one thermal equation for each location in the geometry.However,the thermal non-equilibrium which also some researches talk about it and is driving force of the solution and simulation of thermal phenomena.

As reviewed above,although the combustion characteristics in porous burners has been carried out by some researches,most of them were macroscopically studied,it was difficult to obtain the temperature,species distribution,and the true structure of the flame in the pores because it ignored the effect of porous structures on combustion and heat transfer.It should be pointed out that the complex geometric structure and high computational cost of the 3D model are still huge challenges at present.So,the establishment of a simplified two-dimensional model for researching in a porous media burner at the pore level is the more feasible method at present.It makes up the shortage of the volume averaged model and provides a reference for numerical researches at the 3D pore level in the future.

In the present work,we numerically examine the combustion characteristics in porous media within the framework of the pore-level simulation approach.The novelty of the presented study is developing a simplified two-dimensional pore-level model of two-layer porous burner.The DO model is used to simulate the radiative heat transfer between solid particles.The heat conduction between particles is considered.Meanwhile,the chemical reaction is described by a single-step chemical reaction mechanism.The objective of this work is to obtain flame structure in pore level and explore the thermal non-equilibrium characteristics under different inlet velocity and solid thermal conductivity.And the thermal non-equilibrium in the burner is analyzed quantitatively with root mean squares (RMS) temperature.

2.Problem Description and Mathematical Model

2.1.Problem description

The physical model of this research is a two-layer porous media designed by Gaoet al.[4].The burner consists of a quartz tube of internal diameter 50 mm.The upstream filled with 3 mm Al2O3particles with length of 50 mm,and the porosity is 0.4,which is used as the flame diffusion layer.The Al2O3particles with different diameters are filled in the downstream with 50 mm long,porosities vary from 0.43 to 0.52,as a flame support layer.In this paper,3 mm Al2O3particles in the upstream section and 9 mm Al2O3particles in the downstream section are selected.

For simplification,the three-dimensional random packed bed[4]is simplified to a two-dimensional structured packed bed.We select a representative intermediate part as computational domain because of the assumption of structured packed bed,it extends 9 mm in the upstream and 27 mm in the downstream.At first,the Al2O3particle diameters are reduced by 0.98 times,the particles in the upstream are assumed to be arranged in equilateral triangles,and then the length and width of the upstream particle unit are obtained based on the assumption of same porosity for the random and structured packed bed,thus the arrangement position of particles are determined.The width of downstream large particle unit is equal to that of the upstream.According to the principle of equal porosity again and the length of the downstream large particle unit is obtained.The specific structure is shown in Fig.1.The conduction of the neighboring particles is considered by bridge approach,and the diameter of the short cylinder is 0.2 times of the Al2O3particles as recommended by Dixonet al.[24].It is assumed that the short cylinders are highly permeable media,and its physical properties are similar to those of Al2O3particles [25].The porosity of the packed bed is computed as:

where ε is porosity;dandDare diameters of the particle and burner,respectively.

Fig.1.Schematic of the two-layer burner.

2.2.Governing equations

Considering the complexity of heat transfer between the gas and solid,the model is simplified and the following assumptions are made:

(1) The gas radiation is ignored.

(2) The porous media are isotropic.The surface to surface radiation is taken into account and computed by DO model,solid surface scattering is ignored.

(3) The flow of gas mixture in the packed bed is assumed to be laminar.

(4) The ‘‘bridge” sections are assumed to be highly permeable media,and its physical properties are the same as those of Al2O3particles.

(5) Pressure loss in the burner is neglected.

Heat loss to the surrounding is considered to be proportional to the temperature difference between the local and ambient temperature by the factor of heat loss coefficient β [26].

Under the above assumptions and considerations,the following governing equations are established.

Continuity equation

where ρgrepresents density of gas mixture;v denotes the velocity vector.

Axial momentum equation

whereuis the axial velocity,μ is dynamic viscosity,andpis the pressure.

Radial momentum equation

where v is the radial velocity.

Gas phase energy equation

whereTg,cg,λgare the gas temperature,specific heat and thermal conductivity,respectively.ωi,hi,Wiare the chemical reaction rate,molar enthalpy and molar weight of speciesi.β is the heat loss coefficient.

Species conservation equation

whereDiandYiare diffusion coefficient and mass fraction of speciesi,respectively.

Ideal gas equation of state

whereRis the general constant of the gas.

Solid phase energy equation

whereTsand λsare the solid temperature and thermal conductivity,respectively.

2.3.Boundary conditions

The boundary conditions are imposed as follows:

(1) Inlet

where εrand σ are solid surface emissivity and Stefan-Boltzmann constant,respectively.

(2) Outlet

(3) Symmetry

At the particle walls,the non-slip boundary condition is imposed for gas velocity.

3.Numerical Model

The numerical calculation is based on ANSYS 15.0.We use unstructured mesh and add prism layers to the particle walls,the mesh is generated by Gambit software.The chemistry is treated with single-step mechanism provided by ANSYS 15.0.The velocity and pressure coupling uses the SIMPLE algorithm,a second-order upwind scheme is used for the equations of momentum,energy,and species transport.In order to simulate the methane ignition process,a solid high-temperature region of 2200 K is set near the interface between two sections,the temperatures of other calculation domains are set to ambient temperature.

4.Mesh and Model Validation

4.1.Mesh independence study

The reliability of the numerical results is closely associated with the mesh convergence study.The mesh must be fine enough to meet the needs of the calculation.Three mesh configurations are analyzed during the convergence study,as shown in Table 1.Non-uniform mesh is used in the computation.The mesh is finer near the particle walls in fluid side and extremely narrow regions,and it is coarser inside particles.Fig.2 shows the three mesh configurations near the interfaces of upstream and downstream sections.

Table 1Mesh configurations

To estimate the influence of the mesh resolution,we analyze the averaged solid temperature and averaged axial velocity in theburner.The averaged solid temperature and axial velocity are defined as averaged values perpendicular to the flow direction.The calculations are performed for three mesh configurations at the equivalent ratio of 0.65 and the mixture inlet velocity of 0.3 m·s-1,all parameters and solution methods are consistent with the full problem.In Fig.3 the averaged solid temperature and axial velocity are shown.In order to facilitate later analysis,the starting position of the upstream particles is set to coordinate origin.It is seen from Fig.3a,in the preheating zone,the averaged solid temperatures predicted by the three mesh configurations are almost same.There are differences among the three mesh configurations in the combustion and post-flame zone,but the differences are very small,the largest deviation of 2.12% between the Mesh 2 and Mesh 3 is observed.Good agreement between the predictions by Mesh 2 and Mesh 3 is observed in the entire zone shown in Fig.3b.However,the predicted averaged axial velocity by Mesh 1 deviates significantly from those by Mesh 2 and Mesh 3 near the combustion zone.Therefore,the calculation of this problem is carried out by Mesh 2.

4.2.Model verification

The averaged temperature of the gas and solid phases were obtained from the array of vertical line segments intersecting the flow direction with 2 mm step (50 line segments in total),and it is calculated using the following formula:

whereTg/s，aveis averaged temperature of gas or solid phase,Tiis local temperature of gas or solid at each mesh node,nis the number of mesh nodes of gas or solid phase in the vertical line segments.

Fig.4 illustrates the predictedTg，ave,Ts，aveand the experimental values by Gaoet al.[4].The values are computed under the conditions of φ=0˙65,u0=0.3 m·s-1and the solid thermal conductivity λsis taken as the polynomial function of the temperature by Munroet al.[27].The selected thermal conductivity values of Al2O3are summarized in Table 2.For comparison,we select experimental values that upstream filled with 3 mm Al2O3particles and downstream filled with 8 or 10 mm Al2O3particles.TheTg，aveandTs，avepredicted by the model shows the same trend as experiments.It indicates that the simplified two-dimensional model can precisely predict combustion characteristics in a twolayer porous burner at the pore level.The predicted averaged temperatures are slightly higher than the experimental values,which may be attributed to the use of single-step mechanism in the calculation.The 3D random packed bed is simplified to 2D structured packed bed,such simplification may be another influencing factor to temperature divergence.In addition,the particle diameter in the downstream used in this study is different from that of experiment[4].

5.Results and Discussion

Several series of simulations are run with the purpose of investigating the influence of the most essential parameters:inlet velocityu0and solid thermal conductivity λs.All cases calculated in this paper are summarized in Table 3.

Table 2Selected thermal conductivity values for Al2O3

Table 3Simulation cases carried out in this work

5.1.Flame structure

Fig.5 shows the contour of the mass fraction of CH4and CO2,temperature and reaction heat distributions for φ=0˙65,u0=0.2 m·s-1and the solid thermal conductivity of λs.It can be seen from Fig.5a–c that the species and temperature distributions are highly two-dimensional.Significant decreases inYCH4is observed in pores near the interface of two sections and it means that extensive chemical reaction is occurred.Then,significant increases inYCO2is observed,this indicates that CO2is formed.The combustion takes place in pores and generates a lot of heat,the flame stabilizes at the interface of the two sections.There is a significant temperature difference between the gas and solid phases in the combustion zone,as shown in Fig.5c.The fresh mixture absorbs heat from solid particles through heat convection and is preheated,this process is mainly completed in the zone of the first three layers of Al2O3particles upstream the flame.It can be seen in Fig.5d that the flame is affected severely by the porous structure at the pore level and it is highly curved and surrounded by solid particles.The volume averaged model cannot capture detailed information within the pores.

5.2.Influence of inlet velocity on thermal non-equilibrium

Fig.2.Three meshing configurations near the two sections (a) Mesh 1,(b) Mesh 2,(c) Mesh 3.

Fig.3.Averaged solid temperature and axial velocity for three mesh configurations (a) Averaged solid temperature,(b) Averaged axial velocity.

Fig.4.Averaged temperatures of gas and solid and experiment results by Gao et al.for φ=0˙65 and u0=0.3 m·s-1 (3–8 and 3–10 mm represent upstream filled with 3 mm Al2O3 particles and downstream filled with 8 or 10 mm Al2O3 particles).

A series of simulations are performed varying theu0from 0.2 to 0.45 m·s-1.The solid thermal conductivity is kept constant λs.The averaged temperature gapTgapbetween gas and solid phases is defined asTgap，ave=Tg，ave-Ts，ave.The obtainedTg，ave,Ts，aveandTgapare shown in Fig.6.One can see in Fig.6a that theTg，aveandTs，aveincreases withu0due to the increase of the power load.The thermal non-equilibrium of gas and solid phases is more obvious in the combustion zone,it is smaller and theTis almost 0 K in the preheating and post-flame zone,as clearly seen from Fig.6b.The maximumTgapin the combustion zone increases withu0,for example,the maximumTgapis 332 K foru0=0.2 m·s-1,whereas the maximumTgapis 566 K foru0=0.45 m·s-1.The reason is explained as follows.On the one hand,the increase of theu0leads to an increase in heat release per unit time.Since the heat capacity of the gas phase is smaller than that of the solid phase,when the heat is redistributed,the increment of gas temperature is higher than that of the solid phase,hence theTgapis increased in the combustion zone.On the other hand,the increase of theu0leads to an increase in combustion temperature,gas phase flow rate in the combustion zone is raised and heat convection is strengthened,which results a decrease inTgap.The two factors compete with each other,obviously,the former is dominant here.

Fig.7 shows the effect ofu0on temperature distributions of solid particles,the working conditions are the same as above.For simplification,in the following sections we analyze the contour of solid temperature distributions near combustion zone.The temperature gapTs，gapinside a solid particle is defined asTs，gap=Ts，max-Ts，min,whereTs，maxandTs，minare the maximum and minimum temperature of same solid particle.It should be noted that along thex-axis direction,the left layer of the small particles in the figure is defined as the first layer,and so on.The thermal non-equilibrium also exists in solid particles based on the pore level.The flame is stable at the interface between two sections.It is seen from Fig.7a that the largestTs，gapis observed before the flame front,as the distance from the flame increases,theTs，gapgradually decreases.The reason is that in the combustion and post-flame zones,sufficient heat convection occurs,and the temperature tends to be uniform inside solid particles.In the preheating zone,the heat transfer of neighboring solid particles includes heat radiation and conduction.The solid radiation is transferred layer by layer from the high temperature zone to low temperature zone[23].For the same particle,theTsnear the flame is high,while theTsnear the preheating zone is decreased because of preheating the fresh mixture.In the preheating zone away from flame,both solid radiation and preheating effect are weakened due to lower solid temperatures.For this reason that the larger thermal nonequilibrium of the solid particles in the preheating zone near the flame is predicted.

Fig.5.Predicted mass fractions of CH4,CO2,temperature,and heat of reaction for φ=0˙65 and u0=0.2 m·s-1 (a) YCH4,(b) YCO2,(c) Temperature,(d) Heat of reaction.

Fig.6.Effect of inlet velocity on temperature distributions for φ=0˙65 (a) Tg， ave and Ts， ave profiles (solid line is Tg， ave; dashed line is Ts， ave),(b) Tgap profiles.

It can be seen in Fig.7a that the thermal non-equilibrium inside the solid particles increases as theu0increases.TheTs，gapof the third layer particles grows from 141 K atu0=0.2 m·s-1to 294 K atu0=0.45 m·s-1.This is because with the increase of theu0,the power of the porous burner is increased,so theTg，aveandTs，aveare increased in the combustion zone.Radiation between neighboring solid particles is enhanced because of the increased solid temperature,meanwhile,the preheating effect on the fresh mixture is strengthened.So the temperature of solid particles is non-uniform in the preheating zone,the closer to the flame the larger is theTs，gap.

RMS is used to quantitatively analyze the temperature inside solid particles in this study.The RMS temperature is defined by the following formula.

Fig.7b shows theTs，RMSprofiles,the trends ofTs，RMSare consistent for differentu0.At the inlet,theTs，RMSare very small for allu0and thenTs，RMSincreases slightly forx<0.03 m,this means that the extent of thermal non-equilibrium increases slowly along the flow direction.However,theTs，RMSincreases rapidly forx> 0.03 m and reaches the maximum value upstream the flame,the thermal nonequilibrium is the largest.The maximumTs，RMSincreases withu0,for example,the maximumTs，RMSis 39 K foru0=0.2 m·s-1,whereasTs，RMSis 85 K foru0=0.45 m·s-1.In the post-flame zone,theTs，RMSdecreases dramatically and exists a minimum point,then theTs，RMSincreases again due to the radiative heat loss near the exit of the burner,the thermal non-equilibrium intensifies once again.

5.3.Influence of solid thermal conductivity on thermal nonequilibrium

Fig.7.Effect of inlet velocity on temperature distributions of solid particles for φ=0˙65 (a) Temperature contours,(b) Ts，RMS profiles.

Fig.8 presents the effect of λson the temperature distribution of the gas and solid phases for φ=0˙65 andu0=0.3 m·s-1.It is noted that as the λsincreases,theTg，aveandTs，aveincrease in the preheating zone and decrease in the combustion and post-flame zone,as shown in Fig.8a.TheTgapin the combustion zone also increases with the λs,as shown in Fig.8b.Increasing the λsserves to rise the heat recirculation through the solid from the post-flame to the preheating zone; which,in turn increases theTg，aveandTs，avein the preheating zone.TheTg，aveandTs，avein the combustion and post-flame zone is reduced due to the enhanced heat transfer,so theTgapis increased.

Fig.9 shows the effect of λson temperature distributions of solid particles.TheTs，gapdecreases with the λs.It is shown in Fig.9a that the maximum value ofTs，gapreaches 340 K for case 1,while it is only 110 K for case 3.It is means that the change of the λsseriously affects the thermal non-equilibrium inside solid particles.The larger the λsthe better is the heat transfer by heat conduction,and the more uniform is the temperature distribution inside solid particles.Fig.9b describes theTs，RMSprofiles of solid particles.Ts，RMSis less than 10 K on entire burner except reaction zone and burner exit.The maximum value ofTs，RMSappears in the preheating zone near the flame,the smaller the λsthe larger theTs，RMSis.For instance,the maximumTs，RMSis 102 K,60 K and 31 K for 0.5λs,λsand 2λs,respectively.TheTs，RMSnear the burner exit decreases with the λs.This is because burner exit exists radiative heat loss,which causes theTsto decrease.Meanwhile the heat conduction is weakened with the smaller λs,the temperature of same particles is non-uniform.

Fig.8.Effect of thermal conductivity on temperature distributions for φ=0˙65 and u0=0.3 m·s-1(a)Tg， ave and Ts， ave profiles(solid line is Tg， ave;dashed line is Ts， ave),(b)Tgap profiles.

Fig.9.Effect of thermal conductivity on temperature distributions of solid particles for φ=0˙65 and u0=0.3 m·s-1(a) Temperature contours,(b) Ts，RMS profiles.

6.Conclusions

Two-dimensional pore level numerical simulations of combustion characteristics in a simplified two-layer porous burner are conducted.The flame structure such as species,temperature and reaction heat distribution in pores are analyzed.Besides,thermal characteristics of the two-layer porous burner operating under different inlet velocities and solid thermal conductivities are carried out.The major conclusions are as follows.

(1) The simplified two-dimensional pore level model can accurately predict the temperature distribution,the flame structure is highly two-dimensional based on pore level.

(2) The thermal non-equilibrium between the gas and solid phases is most obvious in combustion zone,the thermal non-equilibrium inside the solid particles is the most obvious near the flame front.

(3) The thermal non-equilibrium between the gas and solid phases and inside the solid particles is getting larger with increasing in the inlet velocity.

(4) As the thermal conductivity increases,the thermal nonequilibrium between gas and solid phases aggravates,but it weakens inside the solid particles.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors wish to acknowledge the support to this work by the National Natural Science Foundation of China (No.51876107).

Nomenclature

cspecific heat,kJ·kg-1·K-1

Ddiameter of burner,m

Didiffusion coefficient of speciesi,cm2·s-1

ddiameter of spheres,m

hithe molar enthalpy of speciesi,kJ·kg-1

ppressure,Pa

Rgeneral constant of the gas,J·mol-1·K-1

Ttemperature,K

T0ambient temperature,K

uaxial velocity,m·s-1

v velocity vector,m·s-1

vradial velocity,m·s-1

Wimolecular weight of speciesi,kg·kmol-1

Xaxial coordinate,m

Yradial coordinate,m

Yimass fraction of speciesi

β heat loss coefficient,W·m-3·K-1

ε porosity

εrsolid surface emissivity

λ thermal conductivity,W·m-1·K-1

μ dynamic viscosity,kg·m-1·s-1

ρ density,kg·m-3

σ Stephan-Boltzmann constant,W·m-2·K-4

φ equivalent ratio

ωireaction rate of speciesi,kmol·m-3·s-1

Subscripts

g gas

in burner inlet

out burner outlet

s solid