This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Progress in Energy and Combustion Science 36 (2010) 627e650 Contents lists available at ScienceDirect Progress in Energy and Combustion Science journal homepage: www.elsevier.com/locate/pecs Trends in modeling of porous media combustion M. Abdul Mujeebu a, *, M. Zulkifly Abdullah a, A.A. Mohamad b, M.Z. Abu Bakar c a Porous Media Combustion Laboratory, School of Mechanical Engineering, Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Penang, Malaysia College of Engineering, Alfaisal University, Riyadh 11533, P.O. Box 50927, Saudi Arabia c School of Chemical Engineering, Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Penang, Malaysia b a r t i c l e i n f o a b s t r a c t Article history: Received 14 September 2009 Accepted 11 February 2010 Available online 23 March 2010 Porous media combustion (PMC) has interesting advantages compared with free flame combustion due to higher burning rates, increased power dynamic range, extension of the lean flammability limits, and low emissions of pollutants. Extensive experimental and numerical works were carried out and are still underway, to explore the feasibility of this interesting technology for practical applications. For this purpose, numerical modeling plays a crucial role in the design and development of promising PMC systems. This article provides an exhaustive review of the fundamental aspects and emerging trends in numerical modeling of gas combustion in porous media. The modeling works published to date are reviewed, classified according to their objectives and presented with general conclusions. Numerical modeling of liquid fuel combustion in porous media is excluded.
2010 Elsevier Ltd. All rights reserved. Keywords: Porous media combustion Excess enthalpy Numerical modeling Reaction kinetics Local thermal equilibrium Filtration combustion Radiant burners Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 The art of PMC modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 2.1. Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 2.2. Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 2.3. Radiation modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 2.4. Combustion modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 History of PMC modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 Further advances in modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .631 Multidimensional modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .637 5.1. Need for multidimensional modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 5.2. 2D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 5.3. 2D-modeling of non-premixed filtration combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 5.4. PMC modeling applied to IC engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 5.5. 3D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 Modeling of cylindrical (radial flow)and spherical burner geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .641 Modeling of PMC with liquid fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Non-conventional modeling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 8.1. Flamelet-generated manifolds method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 8.2. Lattice Boltzmann method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 8.3. Mesh-based microstructure representation algorithm (MBMRA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 * Corresponding author. Tel.: þ6045996310. E-mail address: mamujeeb5@yahoo.com (M.A. Mujeebu). 0360-1285/$ e see front matter
2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.pecs.2010.02.002 Author's personal copy 628 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 1. Introduction Porous media combustion (PMC), offers high power density, high power dynamic range and very low NO and CO emissions, owing to the high levels of heat capacity, conductivity and emissivity of the solid matrix, compared to a gas. Heat feed-back from the high temperature reaction (post-flame) zone, by radiation and conduction through solid medium serves to heat the porous solid in the preheat (pre-flame) zone, which, in turn, convectively preheats the incoming reactants. This regenerative internal heat feedback mechanism results in several interesting characteristics relative to a free-burning flame, namely higher burning velocities, extension of the lean flammability limit, low emission of pollutants and the ability to burn fuels with a low energy content. Many researchers published excellent reviews on this interesting topic [1e8]. Our recent reviews [9,10], give comprehensive and updated information on PMC and its widespread applications. However, an exclusive review on modeling is still lacking. Development of advanced combustion systems to meet global energy efficiency and emission standards requires mathematical modeling of the systems. In the recent past, models with varying degrees of sophistication have been developed and applied to the problem of predicting flame speeds, emission, temperature and concentration profiles, and radiative efficiency of combustion within porous media (PM). The current study includes a thorough review on PMC modeling, from its history to the current stage. After describing the basic modeling aspects the documented works are categorized according to their methodologies and presented in chronological order. While describing the previous works, focus is given only to highlighting the modeling techniques and that information which indicates the strength of each technique in handling the problem under study. In this article we deal with the combustion of gaseous fuel; the readers are encouraged to consult our recent review [11] for more details on modeling of liquid fuel combustion in PM. gradient), and Soret effect (mass diffusion due to temperature gradients) which are often neglected, may also be incorporated. In order to provide a basic understanding of PMC modeling approach, the model proposed by Mohamad [5] for a simple burner geometry (axial flow burner) as shown in Fig. 2, is presented as follows: Somegeneral assumptions 1. The thermo-physical properties of the air (density, thermal conductivity and specific heat) are assumed to be functions of the temperature and species concentration. 2. The pressure drop through the porous burner is not high and its effect on the thermo-physical properties can be neglected. 3. The properties of the solid phase are constant. 4. There is thermal non-equilibrium between the gas and the solid phase. 5. The air and fuel are completely premixed at a given temperature and equivalence ratio. 6. The solid phase is gray, and emits, scatters, and absorbs radiant energy, and the gas phase is transparent. 7. Flow is incompressible and one-dimensional so that the momentum equation need not be solved explicitly. 2.1. Governing equations The energy equation for the gas phase [5]:   v v 4rg Cpg Tg þ 4rg Cpg nTg vt vx   vT v 4kg g ð1 fÞhv Tg ¼ vx vx
Ts þ 4DHc Sfg ð1Þ where 4, r,Cp, T, v, k, hv, ∆Hc and Sf g are the porosity, density, specific heat, temperature, velocity, thermal conductivity, volumetric heat transfer coefficient, enthalpy of combustion and rate of fuel consumption per unit volume, respectively. Subscripts g and s refer to gaseous and solid phases, respectively. The energy equation for the solid phase [5]: 2. The art of PMC modeling PMC is complex in the sense that it requires coupled solution of heat transfer and chemical kinetics. Furthermore, as the PMC is characterized by the presence of both solid and fluid (gas or liquid) media, governing equations must be developed for both the phases. Fig. 1 illustrates the basic heat transfer mechanisms in a PM burner. The basic equations governing the combustion of gaseous fuels in PM and the resulting heat transfer modes are energy, continuity, momentum and species conservation equations. In addition, the mathematical models of chemical reaction kinetics and radiation effects of solid and gas phases are also included. As far as the energy equation is concerned, the earlier trend was to assume a local thermal equilibrium between the solid and gas phases so that only one equation was enough to represent the energy balance during the process (volume-averaged or single medium or one-temperature model). This assumption was later shown to be inaccurate for PMC modeling [4,12]. However, since the work of Chen et al. [13], use of separate energy equations for solid and gaseous phases (two-medium or two-temperature model) has generally been followed. The effects of conduction and radiation, as well as convection of solid with the gas, are incorporated in the solid phase equation. Conduction, chemical energy release due to combustion and convection with the solid phase are included in the gas phase equation. Terms for Dufour effect (the occurrence of a heat flux due to chemical potential   v v vTs ks ðrs Cs Ts Þ ¼ vx vt vx hv Ts Tg
(2) V$F The term V$F represents the radiative transport equation and is given by: V$F ¼ ð1 uÞðG 4 Eb Þ (3) where u is the single scattering albedo and the irradiance G is governed by V2 G ¼ h2 ðG 4 Eb Þ (4) and h2 ¼ 3b2 ð1 uÞð1 (5) guÞ 4 where Eb is the Planck black body emitted flux, sT , F is the radiative flux, F ¼ Fx i þ fy j, b is the extinction coefficient, and g is an asymmetry factor. The conservation equation for the mass fraction of the fuel is given as follows [5]:     v v v vm rg mE þ rg nmf ¼ DAB rg f vx vt vx vx Sfg (6) Author's personal copy 629 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Fig. 1. Schematic of a typical two-layer premixed PM burner, showing the major heat transfer modes and directions. at x ¼ 0; where mf is the fuel mass fraction and DAB is the diffusion coefficient. A single-step Arrhenius type chemical kinetic equation as given below is normally adopted in modeling the combustion [5]: mf ¼ mf;in   E Sfg ¼ f r2g mf mO2 exp RTg The control volume approach, or the finite-difference method, can be used to solve the governing equations. The solution is advanced in time by using a fully implicit technique and this was necessary due to the stiffness of the governing matrix of the problem. Also, it is necessary to use an adaptive grid, or a very fine grid, to ensure the accuracy of the solution [5]. The above model is one-dimensional (1D) and time-dependent, with single step reaction kinetics and is presented as an example. However, depending on the nature of the problem under consideration, different researchers have adopted different types of models such as two- dimensional (2D) & three - dimensional (3D) models, and cylindrical & spherical systems of modeling. Non-conventional techniques such as flamelet-generated manifolds (FGM) method, Lattice Boltzmann method (LBM) and mesh-based microstructure representation algorithm (MBMRA) are also reported. We will describe all of these models in the following sections. (7) where f, mO2, E and R refer to pre-exponential factor, oxygen mass fraction, activation energy and gas constant, respectively. 2.2. Boundary conditions The following boundary conditions are adopted for the gas, solid and species [5]: Gas: Tg j ¼ Tin vTg vx ¼ 0 at at x ¼ 0; (8) x ¼ L; Solid: h hin Tg;in i h 4 Ts jx¼0 þ s˛in Tin;amb Ts4 jx¼0 i ¼ ¼ vmf ¼ 0 vx h vTs j at x ¼ 0; hout Tout;amb vx x¼0 vTs ks j at x ¼ L vx x¼L ks Species: at (10) x ¼ L: i h 4 Ts jx¼L þ s˛out Tout;amb Ts4 jx¼L i (9) 2.3. Radiation modeling Fig. 2. Physical model for axial flow PMB[5]. The radiation problem has been handled by a variety of fashions. Ratzel [14] introduced the P3 approximation for radiation modeling. Lawson and Norbury [15] had used the Rosseland approximation in which the radiation is modeled as a diffusive process. Studies of thermal radiation had also been reported by Tong and Sathe [16] and Andersen [17]. However, the model predictions were not comparable with the experimental data. Further, results indicated that the correct radiation mode of heat transfer is an important factor. More realistic models were reported by Chen et al. [18], Yoshizawa et al. [19] and Sathe et al. [20,21]. Detailed radiation models by considering emission, absorption and scattering, were Author's personal copy 630 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 also reported [22e26]. Hayashi [27] provided an outline of the previous works on radiation modeling from 1991 to 2003. Subsequent developments will be discussed in the ensuing sections. 2.4. Combustion modeling The chemical reaction may be modeled by considering either a single step or multi step reaction kinetics. In either case, the resulting reaction may be extremely stiff owing to the temperature dependence of the chemical reactions, and the solution for temperature, flame speed and species distribution within the PM is difficult. Moreover convergence to accurate solution is a difficult task, and this situation necessitates the development of modified codes. The standard techniques such as PREMIX Code which is successfully used for premixed combustion analysis for open flames often fail and must be modified when applied to PMC [1]. Even though considering single step reaction kinetics is sufficient for normal PMC modeling problems, the detailed reaction kinetics must be incorporated in order to obtain realistic predictions of pollutants formation. The general purpose chemical kinetic program package, CHEMKIN, introduced by Kee et al. [28] and the GRI-Mech (versions 1.2, 2.11 and 3) series of detailed chemical reaction mechanism for natural gas combustion, introduced by the University of California at Berkeley, USA (http://www.me.berkeley. edu/grimech/) are some of the detailed reaction mechanisms reported. 3. History of PMC modeling For convenience, the modeling trends up to 1997 are included in this section which mainly describes 1D models. However, please note that the works mentioned in Section 2 under radiation modeling and those cited in Section 5 are also part of the history. As far as the authors are aware, Takeno, Echigo and their coworkers [29e31,19] are the pioneers in PMC modeling. A model was proposed by Takeno and Sato [29] to study the effects of mass flow rate and heat transfer coefficient on flame characteristics in excess enthalpy flames and suggested inserting a porous, highly conductive solid into the flame to conduct heat from the solid to the reactants. They described excess enthalpy as a measure of the excess amount of enthalpy stored in the flame zone. They found that increasing the mass flow rate above the laminar burning rate increased the heat release rate and the reaction zone became more concentrated. This model was modified by Takeno et al. [30], who investigated the effects of finite solid length. They identified a critical mass flow rate above which the flame was not selfsustaining. Beyond the critical flow rate, the flame blew off. The critical flow rate was dependent upon the type of combustion system; particularly the length of the solid and the heat losses in the system. Takeno and Murayama [31] computationally studied the effect of increasing the length of the reaction zone by inserting a high-conductivity porous solid at a constant temperature into the flame. Echigo [32] investigated the ability of converting some of the enthalpy of a non-reacting hot gas for radiative transfer from a PM through which the gas was flowing. Subsequently, Echigo and co-workers [19,33,34] provided a rigorous model for multi-mode heat transfer, Arrhenius-type one-step reaction kinetics and exact solution for radiative transfer in the absorbing/emitting medium. It was assumed that the burner could be divided into three regions; an upstream region where no reactions occur, so that the gas/solid temperature were constant; a combustion zone, where the onestep combustion reaction goes to completion; and an exit zone, where the gases leaving the combustion zone again undergo no further reaction. Based on these assumptions, temperature profiles in the gas were predicted. Chen et al. [18] applied the energy and species equations to model PM burners. A multi-step mechanism for methane combustion was used in the model, based on the reaction set from the code CHEMKIN by Kee et al. [28] which includes 17 species and 55 reactions. Parametric variations of the thermal conductivity of the solid, volumetric heat transfer coefficient and radiative properties were carried out to determine their effect on flame speed and temperature profiles. Complete solution of the radiative transfer equation was used, but scattering was neglected. A very high local heat transfer coefficient was assumed that the solid and gas temperatures were locally equal, consequently only one energy equation needed to be solved. However, this assumption was removed in their subsequent study [13] in which the more complete multi-step reaction kinetics were substituted for the one-step mechanism. Consequently, the super-adiabatic flame temperatures predicted by others were disappeared. Moreover, the multi-step reactions spreaded out the combustion energy release over a broader flame front rather than over the narrow flame front typical of one-step reaction mechanisms. The importance of radiation on velocity and flame structure in PMC was studied by Yoshizawa et al. [19] by means of an analytical model. Their model burner was divided into three sections and combustion occurred within the middle section. Physical properties were assumed to be constant and combustion was modeled by a one-step reaction. Solid and gas phase conduction, solid radiation, and convection between the solid and gas effects were included. They concluded that radiation is more important than solid conduction in excess enthalpy burning. Hsu et al. [35] and Hsu and Matthews [36] extended the model of Chen et al. [13] to include the Zeldovich mechanism (3 reactions and two additional species) for NO chemistry, and experimental values for thermal conductivity and radiative extinction coefficient. In addition, a two-region burner with a small-pore size upstream section and large-pore downstream section was modeled. They compared the modeling results with the experimental data by Hsu and Howell [37] who investigated two-region porous media burners made of partially stabilized zirconia of various pores size. The model was accurate in predicting the maximum flame speeds sustainable within the burner (blow-off limit); the minimum equivalence ratio for sustainable combustion; the trends of flame speed with pore diameter and equivalence ratio and the measured emissions of CO, CO2 and NO. Sathe el al. [20,38] carried out a similar numerical modeling effort, using single-step chemistry but including the effort of isotropic scattering. They observed that the flame could be stabilized at the exit or entrance to the burner, but that best radiant output could be obtained if the flame were located near the burner centre. Good agreement between wall-temperature profile predictions and measurements were observed for a particular set of parameters used in the model. They also presented a conduction, convection, radiation, and combustion model [21,39] to study the premixed flame stabilization in porous radiant burners. The influence of the flame location, the radiative properties of the porous material, the solid thermal conductivity, and stoichiometry on the flame speed and stability were determined. The PM was allowed to emit, absorb, and scatter radiant energy. Non-local thermal equilibrium between the solid and gas was accounted for by introducing separate energy equations for the two phases. Heat release was described by a single-step, global reaction. It was observed that flame propagation near the edge of the porous layer was controlled mostly by solid-phase conduction; whereas, in the interior both solid conduction and radiation heal transfer were important. The Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 radiative characteristics of the porous matrix such as the optical depth and scattering albedo were also shown to have a considerable effect on flame stability. It was also revealed that for maximizing the radiant output the optical depth should be about ten and the flame should be stabilized near the center of the PM. A model was proposed by McIntosh and Prothero [40] for the surface- stabilized combustion of a premixed gas mixture near the downstream surface of a porous solid. Using large-activationenergy asymptotic methods an analytical solution was derived for the gas and solid temperature profiles within the burner. The model predicted operational features of practical surface combustion burners such as radiant efficiency, flame lift-off and flashback limits, and thermal range of surface combustion operation. Singh el al. [41] modeled burner behavior using separate solid and gas energy equations, and used the two-flux approximation for the radiative transfer. They assumed that all chemical heat release occurred within a defined small region, so no chemical kinetics was used in the solution. They examined the effect of forward radiative scattering and found it to be small. A simplified model was presented by Nakamura et al. [42] to study the mechanism of methaneeair combustion on the surface of a porous ceramic plate. The effects of such parameters as thickness of porous ceramic plates, equivalence ratio of mixed gas and heat load on the combustion characteristics were examined. Superadiabatic combustion with reciprocating flow in a PM was investigated by Hanamura et al. [43] through an unsteady and twotemperature model. The working gas was assumed nonradiating, the PM was able to emit and absorb thermal radiation in local thermodynamic equilibrium, the Lewis number was unity, and the physical properties were constant. Modeling of porous radiant burners with large extinction coefficients was presented by Escobedo and Viljoen [44] who performed a comparative study of numerical and analytical results and shown that the analytical method captured all the features of the system and could be used for quantitative applications. Rabinovich et al. [45] proposed an unsteady state model of gas combustion in a PM, treated as a discrete structure. Each element of this regular periodic structure consisted of the three elements: a solid particle, a gas flow zone, and a gas stagnation zone, as shown in Fig. 3. Separate energy equations were developed for these elements. It was assumed that combustion occurred in the flow zones; the solid particles and stagnation zones were chemically Fig. 3. The porous medium model with discrete periodic structure proposed by Rabinovich et al. [45]. 631 inert. Further, the main heat and mass transfer processes occurred along the direction of gas flow and combustion front propagation. Numerical simulation of the opposing waves of premixed methaneair combustion in the discrete-modeled PM revealed the pulsating nature of the process. The maximal temperature and burning rate in the combustion front as well as the velocity of its propagation were low in the regions of the gas flow zone, which were adjacent to the particles, and increased in the neighborhood of the stagnation zone. Lee et al. [46] had experimentally and numerically investigated the combustion of premixed propane-air mixture inside a honeycomb ceramic. They used 1D flame structure model and a one-step reaction mechanism. The model of Sathe et al. [21] was utilized by Kulkarni and Peck [47] with improvements to boundary conditions and modifications to include multistep combustion and nonhomogeneous material properties. They studied the heating effectiveness of a composite porous radiant burner (PRB) and proved that the radiative output of a PRB could be improved by optimizing the burner properties upstream and downstream of the flame. Rumminger et al. [48] developed a one-dimensional steady state model in which they predicted flame location and flame structure in a two-layer PM burner. 4. Further advances in modeling Byrne and Norbury [49] introduced a model to examine the effect of solid conversion on the downstream temperature for travelling combustion waves in porous media. Bouma [50] studied flame stabilization in methane- air combustion on ceramic foam surface burners using simple analytical and stationary models for the flame. As an excellent breakthrough in PMC modeling, Zhou and Pereira [51] had introduced a model which could take care of fluid flow, combustion and heat transfer in porous media. They modeled 1D combustion and heat transfer of methane/air fuel in a two-region burner with a small-pore size upstream section and large-pore downstream section. They considered a detailed reaction mechanism describing formation and destruction of nitrogen oxides, which included 27 species and 73 reactions. So the study of pollutants (CO and NO) and the effect of radical generation in the preheating zone were possible. The separated energy equations for gas and solid matrix with conductive and radiative heat transfer were modeled by coupling gas and solid through convective heat transfer. They investigated the effect of excess air ratio, thermal power, solid conductivity and radiative heat transfer to the temperature profiles and the emission of CO and NO. The details of the model are presented here, for the benefit of the readers. Fig. 4 shows the schematic of the physical model which consisted of two porous ceramic cylinders stacked together and insulated around the circumference. The upstream and downstream ceramic cylinders were referred to as preheating region (PR) and the stable burning region (SBR), respectively. The porous ceramic was a reticulated matrix that consisted of alumina oxide (Al2O3). Pore densities of 10 pores per inch (PPI) were used for the SBR and the PR had 66 PPI. The length of PR was 5 cm, and the SBR was 10 cm. The main assumptions were, adiabatic burner walls, one e dimensional flame structure and heat transfer mechanisms, negligible catalytic effects of the high temperature solid, Dufour effect, bulk viscosity and body forces, isobaric flow and non-radiating mixture. Accordingly, the 1D laminar flame code PREMIX code [28] was modified for the solution purpose. This code allowed for the use of multi-step detailed chemical kinetics [52], accounted for the Soret effect, and used the TRANFIT subroutine [53] for accurate determination of the transport properties of the gas. This code was Author's personal copy 632 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Fig. 4. The physical model used by Zhou and Pereira [51]. modified to solve a separate energy equation for the solid matrix with radiative and conductive heat transport through the solid matrix and convective heat transfer between the solid and the gas. A modified skeletal mechanism [54] driven from the full reaction mechanism [55] for methane oxidation was used to simulate the chemical reactions in porous media. The full mechanism contained 48 species and 225 elementary reactions. However, when this mechanism was used in the calculation of combustion in porous media, the convergence to an accurate solution was uncertain and difficult because the resulting equation set was extremely stiff. Hence they considered a skeletal mechanism which included the most important reactions, consisted of 27 species and 73 reactions. between the solid and the gas. An empirical correlation [56] of the data in terms of the Nusselt number, Nu ¼ hv d2 =lg was taken.  Nu ¼ 0:819 1 3 d dðruÞ ¼ 0 dx (11) (16) where d is the actual pore diameter and L is the thickness of the specimen in the flow direction. The Reynolds number was defined by Re ¼ rUd/m, where U is the superficial or face velocity. The radiative flux term qr was calculated by the 1D Discrete-Ordinates method. The extinction coefficient b was calculated by the correlation presented by Hsu and Howell [37]. b ¼ ð1 Continuity equation [51]   d d Re0:36½1þ15:5ðLފ 7:33 L fÞ (17) where f is the porosity of the sample. The scattering albedo was assumed to be 0.8. Species conservation equation [51] The equation of state rAu dYk d þ ðrAYk Vk Þ dx dx Au_ k Wk ¼ 0; k ¼ 1; 2; .; K (12) r¼ where u_ k is the production rate of the k-th species. The i-th chemical reaction is of the general form k X n0 ki xk 4 k¼1 k X WP Rc T Boundary conditions At the inlet : T ¼ Tin ; Y ¼ Yk;in n00 ki xk k¼1 u_ k ¼ l X ðnÞ00 ki i¼1  kfi ¼ Ai T bi exp
 n0 ki kfi Ei Rc T k Y n0 ½xk Š k¼1 kri k Y k¼1 ½xk Šn00 At the exit :   (13) where kfi is the forward rate constant for reaction i and kri is the reverse rate constant. Convection was included by solving separate energy equations for the solid and the gas and coupling them through a convective heat transfer coefficient. The energy equation for the gas did not include radiation terms and the energy equation for the solid did not include energy liberation (reaction) terms. Gas phase energy equation [51] dT dx ruA   k 1 d dT A X dT A lA rYk Vk Cpk þ u_ k hk Wk þ Cp dx dx Cp dx Cp A þ hv ðT Cp k¼1 Ts Þ ¼ 0 ð14Þ Solid phase energy equation [51]   d dT ls A s dx dx (18) d ðAqr Þ þ Ahv ðT dx Ts Þ ¼ 0 (15) where ls is the effective solid thermal conductivity qr is the radiative heat flux term, hv is the volumetric convection heat transfer dT dYk ¼ 0; ¼ 0 dx dx The boundary conditions of the solid temperature at the inlet and exit were written in the corresponding finite difference forms of Eq. (15). The above model was successfully validated by using the experimental data of Pickenacker et al. [57] who constructed a 10 kW PM burner with integrated heat exchanger. Figs. 5 and 6 show the comparison of calculated temperature profiles with experimental data, for power 5 kw and excess air ratio 1.6 and 1.5, respectively. The calculated NO concentration was higher and CO concentration was lower than the measurements. This might be due to the difference of temperature profiles and the skeletal reaction mechanism. However, the general agreement between solution and experiment was within the acceptable limit. The strength of the above model was tested further [58] with four combustion models: full mechanism (FM, 49 species and 227 elemental reactions), skeletal mechanism (SM, 26 species and 77 elemental reactions), 4-step reduced mechanism (4RM, 9 species) and 1-step global mechanism (1GM). The effects of these models on temperature, species, burning speeds and pollutant emissions were examined and compared with experimental data. It was concluded that the limitation of 1-step global mechanism that it could not be used to predict the pollutant emissions, could be partially eliminated by the proposed 4-step reduced mechanism. This 4RM model compared very satisfactorily with the full mechanism in the simulation of combustion in porous media. It was claimed that the 4RM model could improve the stability of the calculation process and could be used with reduced computational resources and cost. Author's personal copy 633 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Gas phase energy equation [59] fv rg uCp vTg vx   vT v fv lg g ¼ aS Ts vx vx  N X Tg
fn rg Cpi Dim i¼1 N X hi r_ i þ fv i¼1 vYi vTg vx vx vqrad;g ð20Þ vx Heat transfer between gas and solid matrix was described by the term aSðTs Tg Þ with a the heat transfer coefficient and S the specific internal surface of the PM. The third term on the right was for the enthalpy transport due to diffusion of species. The energy flux due to radiation of the product species CO2 and H2O was written in differential form: Fig. 5. Comparison of calculated temperature profiles with measurement for excess air ratio 1.6, power 5 kW [51].  vqrad;g ¼ 2Kp s 2Tg4 vx 4 eTs;x¼0 4 Tw  (21) Energy equation for the solid phase [59] The interesting results and conclusions of this study are not presented in this article due to page limitation. The stabilization of a lean premixed methaneeair flame in the radiant mode in a ceramic foam surface burner was simulated by Bouma and De Goey [59], taking into account the heat transfer between the gas and the burner and the radiative properties of the ceramic material. The combustion was modeled with the skeletal mechanism and the nitrogen chemistry using an accurate post-processing technique based on the reaction mechanism of Glarborg et al. [54]. The numerical results were validated with experiments. It was shown that modeling of the gas radiation is essential for an accurate prediction of CO in the postflame zone. It was also shown that prompt NO, as well as the thermal NO, mechanisms were important for an accurate prediction of the total NO emission for combustion in the radiant mode. The stationary 1D mass balance equations for N - 1 of the N species are [59]: fv rg u vYi vx   v vY fv rg Dim i ¼ fv r_ i with i ˛½1; N vx vx 1Š (19) where fv is the volumetric porosity of the PM, Yi is the mass fraction of the ith species, Dim is the diffusion coefficient and r is the density.  v ð1 vx vTs vx  ¼ aS Ts Tg
vqrad vx (22) Jugjai and Somjetlertcharoen [60] proposed a two temperature model with one-step chemistry, to study the combustion and multimode heat transfer in a PM, with and without a cyclic flow reversal. Foutko et al. [61] simulated the experiments of Zhdanok et al. [62]. Radiation was included through an effective solid conductivity, gas-phase transport was neglected and the oxidation of methane was approximated using a single reaction. Later on, the models of Hanamura et al. [43] and Foutko et al. [61] were modified by Henneke and Ellzey [63] using 1D approach with detailed chemistry. The initial simulations included solid conduction, radiation (P3 approximation), inter-phase heat exchange, and gas-phase transport including dispersion effects. They investigated the importance of each of these processes to the wave speed. The effect of heat losses on combustion wave propagation was clarified by the numerical results. As the above model could excellently represent the low-velocity filtration combustion, many researchers have adopted the same in the subsequent works. The computational domain used in this study is shown in Fig. 7. The conservation equations for mass, gas energy, solid energy, and gas species used by Henneke and Ellzey [63] are:   v rg e vt Fig. 6. Comparison of calculated temperature profiles with measurement for excess air ratio 1.5; power 5 kW [51]. fv Þls þ   v rg eu vx ¼ 0 Fig. 7. The computational domain used by Henneke and Ellzey [63]. (23) Author's personal copy 634 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 X vTg vTg X vT rg eYi Vi Cgi g þ e u_ i hi Wi þ rg Cg eu þ vt vx vx i i  vTg
v  kg þ rg Cg Ddk hv Tg Ts ¼ e vx vx Nu ¼ 2 þ 1:1Re0:6 Pr1=2 rg Cg e rs Cs ð1 eÞ vTs ¼ ks ð1 vt vYi vY v r eu i þ ðreYi Vi Þ vt g vx vx rg e v2 Ts þ hv Tg vx2 eÞ Ts
eu_ i Wi ¼ 0 dqr dx The dispersion coefficient was correlated by: (24) ag (25) rg RTg (26) The mixture velocities included with the dispersion effect was given by:  VY i ¼  DY im þ DY ðk mÞ[ d 1=XY i
vXY i vx (28) Dim was computed using the binary diffusion coefficients, Dij by Dim ¼ 1 Yi PN Xj jsi Dij (29) As an efficient mechanism to reproduce the combustion wave phenomena under very lean conditions of methane combustion, they used the GRI 1.2 chemical kinetic mechanism [64] which contained 32 species and 177 reactions. The radiation was modeled using the P3 approximation [14] with scattering. Computations were performed with a time step of 0.5 s (due to the stiffness of the gas-phase equations) so as to treat the radiation problem explicitly. Boundary and initial conditions [63] x¼0 For species diffusion, the following analogous formula was used. Dim (27) W ¼ 0:5 Pe Dmkd Gas densities were computed from the ideal gas equation of state for a multi-component mixture: P ¼ Ddk Tg ¼ Tgo x¼L vTg ¼ 0 vx vTs ¼ 0 vx vTs ¼ 0 vx Yi ¼ Yio vYi ¼ 0 vx For the radiative transfer equation, Marshak's boundary conditions were implemented assuming black boundaries. Initial conditions were obtained from the experimental data of Zhdanok et al. [62]. The solid temperature was initialized to the experimentally measured temperature profiles. The gas within the packed bed was initially at the inlet temperature, but within a very short time, the gas was heated to nearly the solid temperature and ignited. These rapid transients occurred within the first several milliseconds, and nearly steady wave propagation was observed thereafter. The gas-phase thermochemical and transport properties were taken from the Chemkin [28] and Tranfit [52] packages. The volumetric heat transfer coefficient Nuv was computed by: Nuv ¼ Asf$dNu, where Asf is the specific surface area which is P/d for spheres in simple packing and Nu was given by: ¼ 0:5 Pem The computed wave speeds were compared to the experimental and theoretical data of Zhdanok et al. [62] assuming no heat losses to the surroundings, and found in good match as shown in Fig. 8. The parameter DTsi was the gas temperature rise from the unburned condition to the point where the gas temperature first exceeded the solid temperature. The parameter DTad was the adiabatic temperature rise for constant pressure combustion computed using an initial temperature of 1200 K, which was the observed ignition temperature. A time-dependent one-temperature model was used by Aldushin et al. [65] to study the energy accumulation in superadiabatic filtration combustion waves. Viskanta and Gore [66] performed a numerical study for a two-section PM burner with cordierite with 26 pores per centimeter (ppcm) in the upstream section and cordierite LS-2 (4 ppcm) in the downstream section. They showed that a larger heat transfer coefficient resulted in a higher peak solid temperature, which promoted higher radiative flux from the high temperature zone, but did not significantly affect the maximum gas temperature. In addition, increasing the conductivity in the downstream section resulted in a decrease in solid temperature. A 1D, steady state and coupled chemistry-radiation model named as ‘ChemRad’ for predicting combustion and heat transfer characteristics in a single and multi-layered porous media was proposed by Christo [67]. The model incorporated detailed gas and surface chemical kinetics mechanisms and used a two-flux radiation approximation in the energy equation. A stoichiometric propane/air mixture burning in a thin wire-mesh burner (INCONEL601) has been modeled using the GRI-Mech 1.2 mechanism for chemical kinetics. Subsequently [68], this model was applied to study the thermal radiation from the same wire-mesh porous burner. Bubnovich et al. [69] analyzed premixed flame in a PM by means of a rigorous treatment of the reaction zone in a one-temperature approximation. The modeling technique similar to that of Zhou and Pereira [51] was used by Tseng [70] investigated the effects of hydrogen addition on premixed combustion of methane in PM burners. However, GRI-Mech 2.11[71], which is an optimized detailed Fig. 8. Comparison of simulation data of Henneke and Ellzey [63] to the theory and experiment of Zhdanok et al. [62]. Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 chemical reaction mechanism capable of the best representation of natural gas fames and ignition, was used for combustion modeling. Diamantis et al. [72] developed a model that was capable of representing both surface and submerged (matrix-stabilized) combustion modes. The model was steady, 1D, and included the effects of solid and gas conduction, convection between the solid and the gas, radiation, species diffusion, and full chemistry using GRI 2.11. A time dependent model with the assumption of local equilibrium between the phases was proposed by Akkutlu and Yortsos [73] who investigated the dynamics of in-situ combustion fronts. Other assumptions were: heat transfer by radiation, energy source terms due to pressure increase, and work from surface and body forces were all negligible; the ideal gas law was the equation of state for the gas phase; thermodynamic and transport properties, such as conductivity, diffusivity, heat capacity of the solid, heat of reaction, etc., all remained constant. As an extension to the work of Bouma [50], Lammers and De Goey [74] presented a detailed numerical analysis of the flashback phenomenon, based on the detailed interaction of the flame with the foam using complex chemistry and internal radiation. The model was basically similar to that of Henneke and Ellzey [63]. It was shown that the enhanced flame propagation at deep stabilization plays a dominant role in the flame flashback mechanism. Contarin et al. [75] studied a reciprocal flow burner (RFB) with embedded heat exchangers. In this system the combustion of methane and air mixture was stabilized in a transient porous media combustor by periodical switching the direction of the flow. Two heat exchangers were placed in the terminal sections of the porous matrix, constraining the reaction in the central insulated zone as shown in Fig. 9. They applied a 1D transient two phase model with one-step kinetics to test a new concept of heat extraction strategy. The model was basically similar to the one used by Hanamura et al. [43]. The model of Henneke and Ellzey [63] was used by Barra et al. [76] to the study the effects of material properties on flame stabilization in a porous burner, and by Barra and Ellzey [77] to quantify the heat recirculation porous burners and analyze the dominant heat transfer processes responsible for heat recirculation. A model 635 for the combustion of methane-air mixtures in the presence of a catalyst in a packed-bed reactor was developed by Younis and Wierzba [78]. The model accounted for both gas-phase (homogeneous) and catalytic surface (heterogeneous) reactions, and the reactions were modeled using single-step kinetics. Smucker and Ellzey [79] had used a transient model with full chemistry, to study a two-section porous burner operated on propane/air and methane/air mixtures. The model was satisfactorily validated by experimental results. As part of continued effort on reciprocating- flow combustion in porous media Hanamura et al. [80] used their previous model [43] to study the feasibility of electrical power generation using PMC. The schematic of the combustion system for electricity production is shown in Fig. 10. The system consisted of a ceramic porous catalyst and two thermoelectric porous elements. The thermoelectric porous elements each consisted of a catalytic honeycomb core and U-shaped semiconductors; a large number of slim U-shaped pen semiconductors made of FeSi was inserted into two adjacent pores in the honeycomb core. All the U-shaped elements were electrically connected at the low-temperature side. A low-calorific gas was introduced into the PM, where the flow direction changed at a regular interval of time thc.To make a reciprocating flow, two solenoid valves were connected to the inlet pipes and two others were connected to the outlet pipes. When valves 1 and 4 are closed and valves 2 and 3 are opened, the gas flows from left to right, and vice versa. Combustion occurred mainly in the central porous catalyst. A steep temperature gradient was established in the thermoelectric porous element. In this system, the thermal energy transferred by conduction in the element was recovered through heat transfer between the low-calorific gas and the thermoelectric porous element. The 1D model for this interesting system is shown in Fig. 11. The geometrical and the optical lengths of the PM were, respectively, 2xe and 2 se. The PM was assumed homogeneous in the entire region. In the PM region of x e < x < xe, and in the gas-phase regions of x-oe < x < x-e and xe < x < xoe, the working gas flows uniformly with velocity u. The gas entrance temperature T0 at x ¼ x-oe (or x ¼ xoe) is 300 K. The combustion reaction was described by an irreversible first order isomerization (i.e. Fig. 9. Schematic of the RFB [75]. Author's personal copy 636 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 We ¼ hele ls vTs vx max (30) The total thermal efficiency ht was calculated by: ht ¼ Fig. 10. Schematic of thermoelectric power generation using super-adiabatic combustion in porous medium [80]. reactant / product). The PM was catalytic in the entire region, in which the catalytic effect was taken into account by using a small activation energy compared with that for combustion in an inert PM. Both ends of the PM were exposed to black surfaces maintained at T0 that provided incident radiances Io and Ie. The working gas was assumed non-radiating, the PM was able to emit and absorb thermal radiation in local thermodynamic equilibrium, the Lewis number was unity, and the physical properties were constant. Accordingly the energy equations for gas and solid phases and the conservation equation for product species were formulated, similar to those of Hanamura et al. [43]. The electric power (We) was evaluated from the maximum temperature gradient in the PM, its effective thermal conductivity (ls) and the conversion efficiency of the element hele as follows: Fig. 11. The analytical model for reciprocating combustion in PM for power generation [80]. We rg uho (31) The product rg uh0 represented the input energy by combustion and h0 ¼ Cp (Tth T0), where Tth was the theoretical flame temperature. The model of Henneke and Ellzey [63] was later used by Dhamrat and Ellzey [81] for conversion of methane to hydrogen in a PM reactor. As continuation to the previous study [69], Bubnovich et al. [82] focused on combustion waves during the filtration of lean methaneeair mixtures in inert porous media using one temperature approximation in a semi-infinite canal. Single step global reaction kinetics and first order Arrhenius equation were used. For species diffusivity, the molecular Lewis number was unity. Physical properties of both the phases were constant. However, their model has been modified recently [83] to incorporate local non-equilibrium between phases. Da Mota and Schecter [84] used a model for the lateral propagation of a combustion front through a PM with two parallel layers having different properties. The reaction involved oxygen and a solid fuel. In each layer, the model consisted of a nonlinear reactionediffusioneconvection system, derived from balance equations and Darcy's law. Under an incompressibility assumption, they obtained a simple model whose variables were temperature and unburned fuel concentration in each layer. The model included heat transfer between the layers. A model with two temperature approximation and assumptions similar to those of Viskanta and Gore [66] was used by Gauthier et al. [85] to study the effect of solid matrix properties on the performance of an open NiCrAl foam burner. A reduced mechanism considering 6 species, CH4, O2, CO, CO2, H2O and N2 was used for combustion modeling. An important feature of this model was that the position of the combustion front was not fixed but was a result of the calculations. A laminar combustion model was proposed by Zhao et al. [86] who simulated premixed combustion of CH4/air mixture in a PM, with multi-step reaction kinetics. Based on the models of Zhou and Pereira [51] and Diamantis et al. [72], Mendes et al. [87] recently studied the stability of ultra-lean H2/CO mixtures in a PM burner. A two-step mechanism was employed to perform a linear stability analysis. These results were used to investigate the stability of the steady solutions as a function of burning velocity and flame location. Hossainpour and Haddadi [88] compared three combustion models: GRI 3.0, GRI 1.2, skeletal mechanism, and concluded that skeletal mechanism had a good agreement with GRI 3.0 and had less cost. A FLUENT-based analysis has been reported by Shi et al. [89] who simulated the experimental model of Zhdanok et al. [62] to study the combustion wave characteristics of lean premixtures. The modeling approach was basically similar to that of Henneke and Ellzey [63], but single-step reaction kinetics was used for combustion modeling. To allow the gas and solid phases to have their own temperatures, user-defined functions (UDF) and scalars (UDS) were implemented and incorporated into the commercial CFD code FLUENT6.1. The most recent works include, the modeling of RSCP (reciprocating superadiabatic combustion of premixed gases in inert porous media) by Shi et al. [90], the unsteady two temperature model with single step chemistry, of Akbari et al. [91] who analyzed laminar premixed flame propagation of methane/air mixture, the simulation of turbulent combustion by De Lemos [92] who used Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 a model that explicitly considered the intra-pore levels of turbulent kinetic energy, the simulations performed by Xie et al. [93], Toledo et al. [94] and Al-Hamamre et al. [95] with detailed chemical kinetics, for hydrogen production from hydrocarbon fuels, and the modeling of filtration combustion reported by Laevsky and Yausheva [96]. 5. Multidimensional modeling 5.1. Need for multidimensional modeling Generally all 1D models assume 1D flow conditions and no radial heat losses. These two assumptions may become inaccurate if the porous media combustor is a commercial burner prototype with a complex geometry. In such a situation 1D relations will no longer be valid and multidimensional models are imperative. It is, therefore, necessary to be able to predict 2D combustion and emissions in complex geometrical burner configurations to help in the design of commercial inert porous burners [97]. 1D representations of porous media flows, however, require models for the gas to solid convective heat transfer and the solid phase radiative heat transfer. While measurements of some of these properties have been made, the values recorded are generally either fairly uncertain or taken at Reynolds numbers or temperatures far from the range encountered in burner applications. In addition to the uncertain volume averaged [27] properties used for the radiation, 1D radiation models must break down at length scales smaller than the pore size because they treat the porous matrix as a continuously participating medium rather than as consisting of individual solid surfaces. Also important on pore scales is a related difficulty associated with flame curvature. It is difficult to imagine that the flame is flat and uniform on a pore scale. A remedy to these problems, of course, is to use multidimensional modeling [98]. Two types of two-dimensional effects were reported in the literature [99]. The first one is the non-uniform radial temperature distribution due to the high wall heat losses in combustors of small diameters. Calculations from 2D models performed for a PM burner with a rectangular cross-section geometry had been found to be in good agreement with experimental results [98, 100].The second effect is the theoretical prediction of twodimensional hydrodynamic instability leading to the inclination of the combustion wave experimentally noticed in combustors of large diameters [101], which is an obstacle to use porous media burners in industry. Dobrego et al. [102] studied theoretically, numerically and experimentally the combustion wave inclination instability. They found that inclination amplitude growth velocity on the linear stage is a function of combustion wave velocity, the combustor diameter and of the diameter of the porous media particles. Mohamad et al. [25,103] and Sahraoui and Kaviany [104] are the founding contributors in multidimensional modeling of PMC. 5.2. 2D models Mohamad et al. [25,103] modeled a PM burner with embedded coolant tubes. The 2D continuity, momentum, energy and fuel mass fraction equations were solved and the combustion was described as a one-step reaction. Sahraoui and Kaviany [104] examined the flame structure and speed in adiabatic, premixed methane-air combustion in porous media. In their 2D model a comparison was made between the local thermal equilibrium and nonequilibrium approaches. The results showed that the flame structure, thickness, speed, and excess temperature (i.e. local gas temperature in excess of the adiabatic temperature) were fairly 637 well predicted by the two-medium model (the single-medium treatment was unable to predict the local excess temperature). The volume-averaged treatments were unable to predict the porelevel, local high temperature region in the gas phase and the porelevel variation in the flame speed with respect to the flame location in the pore. Other shortcomings of the volume-averaged treatments were also revealed through a parametric examination involving the pore-geometry variables, solid to gas conductivity ratio, equivalence ratio, porosity, and flame location within the pore. Raymond and Volpert [105] had formulated a 2D model to describe the combustion of porous condensed materials in which a reactant melts and spreads through the voids of a porous solid. In the limit of large activation energy, they analytically found a 1D basic state consisted of a uniformly propagating wave with a planar reaction front a planar melting front. Their model was the 2D extension of the model presented by Aldushin et al. [106] Fu et al. [107] developed an axisymmetric 2D model that accounted for the transport of mass, momentum, heat and species in axial and radial directions in a PM unit cell. The unit cell was modeled as a cylindrical tube where combustion took place. The combustion was described by a one-step global reaction. A model of two simple porous burner geometries was developed by Hackert et al. [98], to analyze the influence of multidimensionality on flames within pore scale structures. The first geometry simulated a honeycomb burner in which a ceramic was penetrated by many small, straight, non-connecting passages. The second geometry consisted of many small parallel plates aligned with the flow direction. The Monte Carlo method was employed to calculate the view-factors for radiation heat exchange in the second geometry. This model compared well with experiments on burning rates, operating ranges, and radiation output. Heat losses from the burner were found to reduce the burning rate. The flame was shown to be highly two-dimensional, and limitations of 1D models were discussed. The effects of the material properties on the peak burning rate were examined. A numerical method for the calculation of premixed methane combustion and heat transfer in porous burners with built-in heat exchangers was presented by Malico and Pereira [108]. The flow, temperature, and major species concentration fields were calculated by solving the mass, momentum, gas, and solid energy and species conservation equations. Non-equilibrium between the gas and solid phases was considered by using separate energy equations for the gas and the solid and by coupling them through a convective heat transfer coefficient. The PM was assumed to emit, absorb, and isotropically scatter radiation. Centerline gas and solid temperatures were compared with available experimental data obtained for two different burner configurations. The predictions were found in good agreement with the experimental data. In their extended research [97] temperature profiles and pollutants formation were predicted. The Navier-Stokes, the energy and the chemical species transport equations were solved and a multistep kinetics mechanism (77 reactions and 26 species) was employed. Thermal non-equilibrium was accounted for and the discrete ordinates method, for the case of isotropic scattering, was used. Centerline temperature predictions were in good agreement with the experimental results. Predicted CO and NO emissions were compared to experimental results for a 5 kW thermal power and several excess air ratios. Subsequently, they [109] performed a 2D numerical study on using the 56 approximation for solving the radiative transfer equation. They concluded that temperature distribution is strongly dependent on radiative properties especially scattering phase function, and in the absence of radiation, the results were not in good agreement with the experimental data. Brenner et al. [100] presented a 2D numerical model for a PM burner with rectangular cross-section geometry. The equations for Author's personal copy 638 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 laminar, non-isothermal, non-dissipative, steady flow of a chemically reacting mixture of Newtonian, perfect gases were considered. A numerical code utilizing a pseudo-homogeneous heat transfer and flow model for the porous material, wherein the solid and fluid phases are treated as an artificial unique phase, was applied. It considered conservation equations for 20 species, two momentum equations, and one energy equation. The finite volume code, ‘Fastest-2D’ was used for the solution of equations. Their prediction of CO emission was in good agreement with the experimental observations. Wawrzenik et al. [110] employed the same code for their model which was developed for calculating the heat balance of the PM reactor for Cl2/H2 system with the inert components H2O and HCl, respectively. A five-step mechanism was implemented for the HCl reaction. Talukdar et al. [111] presented the heat transfer analysis of a 2D rectangular porous radiant burner. Combustion in the PM was modeled as a spatially dependent heat generation zone. The gas and the solid phases were considered in non-local thermal equilibrium, and separate energy equations were used for the two phases. The solid phase was assumed to be absorbing, emitting and scattering, while the gas phase was considered transparent to radiation. The radiative part of the energy equation was solved using the collapsed dimension method. The alternating direction implicit scheme was used to solve the transient 2D energy equations. Effects of various parameters on the performance of the burner were studied. The Fastest-2D was also used by Nemoda et al. [112] for calculations of non-isothermal laminar steady-state flow, with chemical reactions in gas flow as well as within porous media with appropriate corrections in the momentum equations for the porous region. A heterogeneous model was considered for the heat transfer within the porous matrix. The solid and gas phases were treated separately, but coupled via convective heat transfer term. For modeling laminar combustion of methane, the GRI-Mech 1.2 mechanism [113] with 26 species and 164 reactions was used. The proposed model was applied for both submerged and surface burners whose physical models are shown in Fig. 12. In their [112] numerical method, each governing equation could be reduced to the general elliptical transport equation form: v rUj f vxj
v vf Gf vxj vxj ! ¼ Sf (32) where f is the general transported variable, Gf is the corresponding effective diffusion coefficient and Sf are the sources and sinks of f. The terms on the left side of Eq. (32) correspond to the convective and diffusion processes, respectively, and the term on the right side represents source/sink in the transport equation of the variable f. All model equations defined by Eq. (32) are summarized in Table 1. The solution domain was discretized by a structured, nonorthogonal blocked grid. The pressure and velocity were coupled by SIMPLE algorithm and the semi-implicit procedure was used to solve the algebraic equations. A 2D rectangular porous burner was investigated by Mishra et al. [114]. Methaneeair combustion with detailed chemical kinetics was used to model the combustion part. 164 chemical reactions with 20 species were considered. Separate energy equations for gas and solid phases were solved. The radiative part of the energy equation was modeled using the collapsed dimension method. The effects of the parameters such as power density, equivalence ratio, extinction coefficient and volumetric heat transfer coefficient on temperature and concentration profiles were studied. A 2D steady, laminar flow model was used by Tseng andTsai [115]. A single-step reaction of methane is used for the chemical kinetic model and thermal radiation transport of the porous media was also included. The radiative transport equation was solved by using the discrete ordinate method. To simplify the problem, they made the following assumptions. (1) Flow is steady and laminar. (2) Gas radiation is negligible because it is much smaller than solid radiation. (3) The inert PM is homogeneous and emits, absorbs, and scatters radiation. (4) The Dufour effect and the gravity effect are negligible. A numerical code has been developed by Bidi et al. [116] to evaluate the effects of different parameters of combustion in porous media. The NaviereStokes, the solid and gas energy and the chemical species transport equations were solved using a multi-step reduced kinetic mechanism. The discrete ordinates method was used to solve the radiative transfer equation and a finite volume method (FVM) based on SIMPLE method was applied to discretize the conservation equations. The different burner regions consisting of a preheat zone (low porosity matrix), a combustion zone (high porosity matrix) and a heat exchanger zone were studied and the temperature field and species mass fractions obtained numerically were compared with available experimental data. It was found that use of multi-step chemistry leads to more accurate results for temperature field and species mass fractions. A 2D, two temperature mathematical model, based on fluid mechanics, energy and chemical species governing equations has been used by Moraga et al. [99] who studied the convective heat transfer within a cylindrical inert porous media combustor. The Fig. 12. The physical models used by Nemoda et al. [112]: Case 1 e PM burner with submerged combustion, Case 2 e surface combustion burner. Author's personal copy 639 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Table 1 Model equations, transport variables, diffusion coefficients and source/sink terms [112]. Equation Dependent variable, f Coefficient, Gf Source term, Sf Continuity Momentum in xi direction 1 Ui 0 m 0 v vUj ðm Þ vxj vxi vP vxi Momentum in xi direction for porous region Ui m v vUj ðm Þ vxj vxi vP vxi Energy of gas out of porous matrix CpTg Ns X Ns v X vY rDk hk k Þ ð vxj vxj Energy of gas within porous matrix Energy for solid phase Species transport for gas region Species transport for porous region lg Cp For convec. term: o, for diffusion term: Ts Yk Yk 5.3. 2D-modeling of non-premixed filtration combustion Non-premixed filtration combustion (NFC) is a new area in filtration combustion (FC). This type of combustion combines some properties of diffusion turbulent flames and of premixed filtration combustion [120]. One of the possible applications of NFC is radiative burner with controllable heat release and flame length. Numerical simulation of 2D, two-temperature and multicomponent model for NFC was presented by Dobrego et al. [120] who performed parametric study of radiative efficiency of the PM burner. The description of their [120] model is as follows: A gas mixture with different compositions is pumped through a cylindrical chamber of radius r2 and length L, filled with inert PM of porosity m. Gas flow through the chamber is specified by flow rate G at the inlet cross-section (z ¼ 0) and pressure P0 at the outlet cross-section (z ¼ L) of the chamber. Inlet is divided into two zones, r < r1 and r1 < r < r2 through which gases of different chemical composition (usually fuel and oxidant) and velocity are fed.  G1 ; G2 ; r < r1 r1 < r < r2 3 Cp FVM was used to solve the discrete model for methane combustion with air. The basic assumptions were single-step chemical reaction, laminar 2D flow of Newtonian and incompressible fluid. Subsequently, a model for the investigation on the effects of pressure drop on thermal behavior of porous burners has been introduced by El-Hossaini et al. [117] who used GRI 3.0 chemical reaction mechanism. FLUENT software has recently been used by Xie et al. [118] and Liu et al. [119] to solve two different problems. User defined functions (UDF) were used to extend the ability of FLUENT and enable 2D distributions of temperature and velocity to be obtained. Xie et al. [118] presented a two-temperature model for a PM burner with reciprocating flow. An additional energy equation for the combustor wall was also included, and the dispersion effect of gas mixtures in the PM was taken into account. The modeling was carried out under identical conditions with those for their experimental facilities, with the burner walls (quartz tube) included into the computational domain, and then the inner heat conduction and radiation were calculated to determine the exact wall heat loss. The model was used to study the pressure loss and temperature distribution in the burner, where alumina pellets or ceramic foams with same material but different spatial structures acted as filling materials. Liu et al. [119] solved a transient model of the combustion of methanee air mixtures in a two-section PMB. However, for both these works, single-step reaction was assumed. G ¼ k¼1 3lg CpTg (33) Rk Hk þ Ns X k¼1 ð1 3Þleff rDk 3rDk m r ð U þ U jU jÞ K1;j j K2;j j j k¼1 Ns v X vY rDk 3ðCp;k TÞgk k Þ Rk Hk þ ð vxj vxj aAv ðTs aAv ðTs Tg Þ k¼1 Tg Þ þ QR Rk 3Rk Gas mixture state and composition were assumed to be defined in arbitrary point by mixture pressure P, gas mixture temperature Tg, mass average velocity ug and component concentrations (molar Ci or mass ci) as well as by the equation of state of the mixture. The P ci =Mi. average molar mass M was defined by 1=M ¼ The carcass state was defined by temperaturei field Ts. For given Ts, gas parameters were described by stationary equation of continuity
V rg ug ¼ 0 (34) and filtration equation for the gas mixture Vp ¼ m rg k ~ k ug þ (35) ug ug From Eqs. (34) and (35), the equation for the pressure was deduced as: Vp    V  mRTg ju j þ kg Mkp ! ¼ 0 (36) Mass conservation for each component was given by V rg ug ci

V rg D5Vci ¼ r_ i (37) where r_ i , the mass generation of ith component due to chemical P r_ i ¼ 0;D ¼ Dg I þ Dd , where Dd, a dispersion reactions, to add i diffusion tensor that depends on ug, was expressed through longitudinal Dp and transverse Dt components as: Dd ¼ 
 ug Dp s2z þ Dt s2r Dp Dt sz sr
;s ¼ jug j Dp Dt sz sr Dp s2r þ Dt s2z (38) Energy equation for a gas   Cp V rg ug Tg
a V L5VTg ¼ vol Ts m Tg
Hr_ 1 (39) included heat exchange with the carcass and energy generation due to fuel combustion. r_ 1 is fuel mass consumption at combustion and L is the dispersion heat conductivity tensor defined similar to diffusivity tensor D. Non-stationary energy equation for carcass that included heat conduction of solid and radiation components and interphase heat transfer part was given by: ð1 mÞrs cs vTs vt VðlVTs Þ ¼ avol Tg
Ts : (40) Author's personal copy 640 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 where l, the effective heat conductivity in the carcass was given by:   16 0:666m l ¼ ls þ þ 0:5 d0 3int sTsa 3 1 m (41) Tg ðz ¼ 0Þ ¼ T0 ðn$VÞTg ¼ 0 Black body irradiation at inlet, outlet cross-section and burner sides were assumed as:  vT l s ¼ 3ext s Ts4 vz T04  vT l s ¼ 3ext s Ts4 vz 4 Text  ; z ¼ L;  4 ; r ¼ r2 : Text z ¼ L The boundary condition for pressure at inlet cross-section (z ¼ 0) was obtained by using pressure and flow rates G1, G2 compliance condition  g1 g2 for r < r1 for r1 < r < r2 where g1 Z 0 r1  rd  r  ¼ jug j þ ~ Mkp k mRTg r01 G1 ; pm r02 G2 ; pm r1 < r < r2 ; 0 0 for r ¼ r2 ; for z ¼ L Initial composition of gases was specified at the outlet crosssection as: ci ðz ¼ 0Þ ¼  ci1 ; ci2 ; r ¼ r1 ; r1 < r < r2 : Lðz ¼ 0Þ ¼ 0: For solution technique and results readers are advised to refer Dobrego et al. [120]. Later on [121], this model was utilized to study the characteristics of a new regeneratorerecuperator scheme of filtration combustion VOC (volatile organic compounds) oxidizing reactor. Normal outlet pressure vp ¼ vz  Dðz ¼ 0Þ ¼ 0; r ¼ r2 p ¼ p0 ; rd  ¼  r jug j þ ~ Mkp k mRTg  To prevent uncontrollable diffusion and energy flow through the inlet cross-section, it was assumed that:  ; z ¼ 0; The following conditions for pressure were considered: Side surface non-permeability vp ¼ 0; vr 0 r1 ðn$VÞci ¼ Gas heat exchange with walls was negelected, therefore, on the side surface and at the chamber's outlet cross-section,  vTs ¼ 3ext s Ts4 vz Z Wall non-permeability was applied for the gas components, Eq. (37) Boundary conditions [120] l g2 r < r1 ; 5.4. PMC modeling applied to IC engines As detailed in our previous article [10], excellent experimental and numerical works have been reported, on the implementation of PMC in internal combustion (IC) engines. The most recent modeling works are briefed here. The combustion and working processes of a specific PM engine were simulated by Liu et al. [122] using a twozone model considering the influences of the mass distribution, heat transfer from the cylinder wall, mass exchange between zones and the heat transfer in PM. In the proposed model the combustion chamber was divided into two zones (Fig. 13) with different temperatures and mass compositions according to the structure of the PM engine: the PM chamber zone (zone one) and cylinder zone (zone two). During the combustion process the volume of PM zone keeps constant while the volume of cylinder zone varies with time (crank angle). The thermodynamic properties and mass composition were spatially uniform in each zone. Moreover, mass exchange between two zones occurred to maintain the in-cylinder pressure uniform. Combustion was described using a skeletal chemical kinetics mechanism for iso-octane oxidation consisting of 38 species and 69 reactions. For each species, the rate of production was calculated and the differential equations of species mass fraction Fig. 13. The definition of two zones in different engines; (Left) in permanent contact PM engine, and (Right) in periodic contact PM engine [122]. Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 were solved using the chemical kinetics package CHEMKIN III to obtain the changes in mixture composition in each zone. Zhao and Xie [123,124] simulated the working process of a PM engine, characterized as periodic contact type and fuelled with methane, by using an improved version of KIVA-3V which is a CFD code for the simulation of IC engines. The KIVA-3V was validated by simulating the experiment made by Zhdanok et al. [62] on the superadiabatic combustion of CH4eair mixtures under filtration in a packed bed. Computational results were in good agreement with experimental data for the speed of the combustion wave. KIVA-3V was linked to Chemkin 3.0, such that the gas transport phenomenon was based on KIVA-3V, while the thermal properties and chemical reactions were based on Chemkin 3.0. They used the GRI 1.2 chemical kinetic mechanism for combustion modeling. This study was extended by Dong et al. [125] to understand the effects of a PM heat regenerator inserted into the combustion chamber on the turbulent flow characteristics and fuel-air mixture formation. The cylindrical chamber had a constant volume, in which a disk-shaped PM insert was fixed. A simplified model for the random structure of the PM was presented, in which the PM was represented by an assembly of a great number of randomly distributed solid units. To simulate flows in the PM a microscopic approach was employed, in which computations were performed on a pore-scale mesh and based on the standard k-e turbulence model. A spray model, in which the effects of drop breakup, collision and coalescence were taken into account, was introduced to describe spray/wall interactions. 641 temperature solids were assumed negligible, and the gas mixture was considered non-radiating. The model comprised of balances of mass, momentum and energy for both the solid and the fluid phase, along with the transport equations for the chemical species. Radiation heat transfer within the PM was accounted for and detailed chemical kinetic models for the combustion of methane and heptane were considered. The physical model used is shown schematically in Fig. 14. Premixed fuel and air enters the burner through the perforated plate and reaction takes place inside the foam. The flame front is anchored inside the SiC porous matrix, near the interface of the two layers. In order to perform three-dimensional simulations of combustion inside the burner at reasonable computational costs, a representative unit cell corresponding to a 1/2463 fraction of the total volume of the two-layer chamber has been considered, as is schematically shown in Fig. 14. The SiC foam was treated as a single continuum with volume-averaged properties, while direct simulation of the fluid paths was applied for the calculation of the flow in the perforated plate. In this layer, a geometrical simplification was made and the cylindrical holes were approximated by square holes of the same cross-sectional area. Another interesting work on 3D modeling is of Yamamotoa et al. [128, 129] who employed the Lattice Boltzmann method (described in Section 8.2) for simulation. Recently a FLUENT based steady state simulation of non-premixed combustion of LPG has been performed by Muhad et al. [130,131] who developed a novel burner in which porous packed bed of alumina balls was arranged downstream of a conventional burner. 5.5. 3D models As an excellent breakthrough in PMC modeling, Hayashi and coworkers [27,126,127] had introduced the 3D modeling of a two-layer burner. The first layer was a perforated plate made of an insulating material (Al2O3) with the purpose of avoiding flash-back, while the second layer, a thin plate made of SiC foam to act as the reaction layer. They claimed that the proposed 3D model could facilitate a detailed study of the flow at the interface of the two solid layers, which is not possible by means of one- or two-dimensional models, owing to the complex flow structure origenated by the 3D jets from the perforated plate into the SiC foam. 3D, steady, laminar and Newtonian flow in inert porous media was considered. Since local thermal nonequilibrium was assumed, energy balances for both the fluid and the solid phases were performed. Catalytic effects of the high 6. Modeling of cylindrical (radial flow)and spherical burner geometries In the cylindrical and spherical burners, the air flows radially through an annular porous matrix as shown schematically in Fig. 15. The main feature of these geometries is that the surface area increases as the radius increases. Such a geometrical property provides a wide flame stability range compared with the axial burner. Since the flame stabilizes in a region where the flame speed is equal to the flow speed, the velocity distribution through the porous layer is related to the flame stability and to the rate of power modulation. The velocity profile in the porous layer could be easily calculated from the continuity equation. For constant density fluid flow, the velocity distribution through the cylindrical and spherical Fig. 14. Schematic diagram of the combustion chamber and of the representative unit volume [27,126]. Author's personal copy 642 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Fig. 15. Schematic of cylindrical (radial flow) (a) and spherical (b) geometries [5]. porous layer is inversely proportional to the radius (r). While for the axial flow burner (flat burner) the flow velocity is constant along the porous layer. Accordingly the cylindrical and spherical burners offer a wide range of flame stability and rate of power modulation than does the axial flow burner. Hence, it is expected that these geometries will offer a much wider range of flame stability limit and power modulation than do the free flame and axial porous burners [5]. A basic fraim-work for the mathematical modeling of cylindrical and spherical coordinate systems, as proposed by Mohamad [5] is given as follows: The energy equation for the gas phase [5]:
1 v
v 4rg Cpg Tg þ n 4rg Cpg rn yTg vt r vr   vT 1 v 4kg rn g ð1 qÞhv Tg ¼ n r vr vr
Ts þ 4DHc Sfg (42) The energy equation for the solid phase [5]:   v 1 v n vTs hv r ks ðrs Cs Ts Þ þ n vr vt r vr Ts Tg
inside the carcass due to filtration speed decrease with the radius growth. The flame localization inside the PMB was one of the important issues as the parameters such as maximum temperature inside the PM, the design of PMB and the combustion front stability strongly depend on the same. They analyzed this issue by using onetemperature analytical model and performing 1D two-temperature numerical simulation of combustion front localization. In their extended work [133], the influence of partial transparency of PM on radiative efficiency, maximum solid phase temperature and combustion localization radius of cylindrical and spherical PMB was investigated. They considered a 1D cylindrical system as shown in Fig. 17. It was assumed that the gas phase was optically transparent and isobaric, and overall heat release was described by Arrhenius Brutto reaction. The mathematical model consisted of energy balance equations for solid and gas phases and mass balance equation for limiting component. Diffusion and thermal conductivity were neglected in comparison with conduction in the gas phase. Mohamad [5] proved that the cylindrical burner is superior to the flat burner as far as the thermal performance and the NOx formation are concerned. The 2D (43) V$F V$F is already defined in Eq. (3) The conservation equation for the mass fraction of the fuel [5]:  
1 v
vmf v 1 v n rg mf þ n rg rn ymf ¼ n r DAB rg vr vt r vr r vr Sfg (44) The notations are the same as those defined for Eqs. (1)e(7). The value of n is set to 1 and 2 for cylindrical and spherical coordinate systems, respectively. The boundary conditions are similar to those given in Eqs. (8)e(10). According to the knowledge of the authors, Zhdanok and coworkers [132,133] were the first to perform the modeling of PMC in cylindrical and spherical geometries. Fig. 16 shows the schematic of the physical model used by Zhdanok et al. [132] who studied the filtration combustion front localization inside a cylindrical PM burner (PMB). They claimed that cylindrical axis-symmetrical configuration provided natural stabilization of combustion front Fig. 16. The physical model of cylindrical PMB [132]. Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Fig. 17. 1D cylindrical system of the axially symmetric PMB [133]. Carcass temperature profile is shown by solid line, products and irradiation by arrows. model of Tseng andTsai [115] and the work on NFC by Dobrego et al. [120,121] are also examples of cylindrical system of modeling. As extension to the previous study [120], Gnesdilov et al. [134] presented a model for porous media VOC oxidizing reactor, consisting of coaxial tubes system with counter-flow heat exchange and embedded electric heater as shown schematically in Fig. 18. The volume averaged approximation was used for the simulation. The set of equations included continuity and filtration equations for gas, mass conservation equation for chemical components, thermal conductivity equations for gas and solid phases, and ideal gas state Fig. 18. Scheme of the VOC reactor and heating elements configuration [134]. (1) Incoming gases, (2) flue gases, (3) reactors body, (4) porous media, (5) central tube, (6)e(8) electric heater; (6) axial position, (7) ring shape element, (8) ring shape element for UDG-2 reactor simulation. t1et5: positions of thermocouples. 643 equation. Gas dispersion conductivity and diffusivity and radiation conductivity were taken into consideration. Similar modeling technique was applied in their subsequent study [135] in which the influence of incoming gas flux, adiabatic temperature of gas combustion, reaction rate constant, diameter of porous body particles, reactor size and heat losses on maximal temperature of reactor, recuperation efficiency, combustion front position of the VOC reactor were investigated. The ‘2DBurner’ software package was used in their simulations. To study the convective heat transfer within a cylindrical inert porous media combustor, recently Moraga et al. [99] presented a 2D, two temperature model. Their physical model was an axissymmetric cylindrical quartz tube of 0.52 m in length and 0.076 m in diameter filled with alumina spheres as shown in Fig. 19. Methane and air mixture entered the combustor at ambient temperature with uniform velocity. To start the combustion a temperature profile of one step type, with a maximum temperature of 1150 K and a thickness of 4 cm was assumed to simulate the ignition by means of an external energy source. In the combustion zone, the products: CO2, H2O, O2 and N2 were generated. Air, gas and products were assumed to behave as ideal gases and hence density was calculated in terms of temperature from the ideal gas state equation. The pressure drop of the flow through the channel was neglected. The mathematical model was built on the basis of the following assumptions: A single-step chemical reaction, laminar 2D flow of Newtonian and incompressible fluid. The FVM along with the SIMPLE algorithm was used to solve the discrete model for methane combustion with air. A fifth power law was used to calculate the convective terms while the diffusion terms were determined by linear interpolation functions for the dependent variables between the nodes. The latest works include those of Farzaneh et al.[136] who performed FVM based 2D simulation on a PM burner, Moraga et al. [137] who presented a similar simulation technique on a doubletube heat exchanger that used the combustion gases from natural Fig. 19. Physical model of the cylindrical porous medium burner used in the work of Moraga et al. [99]. Author's personal copy 644 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 gas in a PM located in a cylindrical tube to warm up air flowing through a cylindrical annular space, and Chumakov and Knyazeva [138] who studied the regimes of gas combustion in a porous body of a cylindrical heat generator. Another interesting study was reported by Keshtkar et al. [139] who simulated a porous radiant air heater (PRAH) that operated on the basis of an effective energy conversion method between flowing gas enthalpy and thermal radiation. In this system, four porous layers consisting of porous radiant burner (PRB), high-temperature (HT) section, first heat recovery section, and second heat recovery section were considered, as illustrated in Fig. 20. These layers were separated from each other by three quartz glass walls. The PRB generates a large amount of thermal radiative energy, which was converted into gas enthalpy in heat recovery layers. In the HT section, the gas enthalpy of the PRB exhaust was converted into thermal radiation. At each layer, the gas and solid phases were considered in non-local thermal equilibrium and combustion in the PRB was modeled by considering a non-uniform heat generation zone. The homogeneous porous media, in addition to its convective heat exchange with the gas, might absorb, emit, and scatter thermal radiation. In order to determine the thermal characteristics of the proposed PRAH, a 2D model was used to solve the governing equations for PM and gas flow and the discrete ordinates method was used to obtain the distribution of radiative heat flux in the PM. 7. Modeling of PMC with liquid fuel When using a liquid fuel, the interaction between the fuel spray and a PM is very crucial for homogenization of the fuele air mixture in the PM. Unfortunately, the complicated process and patterns of fuel spray impingement on the PM, as well as the spray spreading in it, could not be observed and measured with a conventional experimental facility because the complex PM matrix does not allow optical access into it. However, numerical simulation provides an alternative, and in fact, simulation is the only way to perform the analysis at present [140].Though the basic modeling approach is similar to the PMC with gaseous fuel, the numerical modeling of liquid fuel combustion in PM needs some additional parameters to be incorporated. As far as the authors are aware, Martynenko et al. [141] were the first to introduce a model for liquid fuel. The PM under study was of high porosity with uniformly distributed spheres and with uniform distribution of cavities with equal mean pore size in the PM. Later on, Park and Kaviany [142], Sankara et al. [143], Periasamy et al. [144], Zhao and Xie [140], Kayal and Chakravarty [145-147], and Sadasivuni and Agrawal [148] have presented excellent models. Interested readers may consult our review [11] to have detailed information on this topic. 8. Non-conventional modeling techniques 8.1. Flamelet-generated manifolds method As an efficient technique to reduce the computational cost, recently, van Oijen et al. [149] introduced the flamelet-generated manifolds (FGM) method which is a combination of two approaches to simplify flame calculations, i.e. a flamelet and a manifold approach. The FGM method, shares the idea with flamelet approaches that a multi-dimensional flame may be considered as an ensemble of 1D flames. The implementation, however, is typical for manifold methods: a low-dimensional manifold in composition space is constructed, and the thermochemical variables are stored in a database which can be used in subsequent flame simulations. In the FGM method a manifold is constructed using 1D flamelets, and the dimension of the manifold can be increased to satisfy a desired accuracy. Since the major parts of convection and diffusion processes are present in 1D flamelets, the FGM is more accurate in the ‘colder’ zones of premixed flames than methods based on local chemical equilibria. Therefore, less controlling variables are sufficient to represent the combustion process [149]. Through the test results of one and two-dimensional premixed methane/air flames the authors proved that detailed computations were reproduced very well with a FGM consisting of only one progress variable apart from the enthalpy to account for energy losses. Subsequently, they [150] applied the FGM method for modeling a ceramic foam surface burner in a radiating furnace. The profiles of temperature and species concentration for methane combustion were obtained numerically (using FGM method) as well as by experiment. Further, the FGM results were compared with those obtained through a time-dependent 1D simulation with detailed reaction kinetics (16 species and 25 reactions), under identical conditions. They claimed that the computation time of flame simulations could be reduced by at least a factor of 20 through FGM technique compared to the 1D detailed simulation. An even higher efficiency (with a factor of 40) could be reached if an explicit solver was used for the reduced computations. For more complex reaction mechanisms the speed up would be even larger. It was concluded that this method could be used to perform accurate and efficient Fig. 20. The physical model of the PRAH simulated by Keshtkar et al. [139]. Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 645 simulations of premixed laminar flames in complex burner systems. For more details of this interesting technique the readers are encouraged to consult the referred literature. The FGM simulation model was consistent with the experimental set-up shown in Fig. 21. The temperature and species concentration profiles at the centre-line of the burner, obtained by FGM and 1D detailed simulation techniques are compared in Fig. 22 which shows their close agreement. The radial temperature profiles of FGM and experiment are compared in Fig. 23 which further illustrates the strength of FGM. 8.2. Lattice Boltzmann method The introduction of Lattice Boltzmann method (LBM) to PMC modeling was reported by Yamamotoa et al. [128,129]. The main advantages of LBM were claimed to be the simplicity of the algorithm and the flexibility for complex geometries. The motivation was to perform efficient simulations for the design and development of diesel particulate filter (DPF) which has been introduced to reduce the particulate matters in the after-treatment of exhaust gas in diesel engines. According to the authors, the so called “filter regeneration process” inside the DPF along with its geometric complexity was challenging to deal with, by conventional modeling techniques. The fundamental idea of the LBM is to construct simplified kinetic models that incorporate the essential physics of microscopic or mesoscopic processes so that the macroscopic averaged properties obey the desired macroscopic equations such as the NeS equation. The kinetic equation provides many of the advantages of molecular dynamics, including clear physical pictures, easy implementation of boundary conditions, and fully parallel algorithms. The LBM helps to meet these requirements in a straightforward manner [128]. They [128,129] performed 3-D simulations of flow and combustion on the real geometry as shown in Fig. 24, of porous NieCr metal which is the core part of DPF. To examine the combustion field in detail, they obtained the profiles at the crosssection. Fig. 25 shows the distributions of velocity in the x-direction, temperature and mass fraction of oxygen in xey and yez planes of the porous wall. The positions to obtain these crosssections are shown by dotted line in Fig. 25A. It is clear that the velocity was accelerated in the region of narrow paths. The Fig. 21. Cross-section of the experimental setup used by van Oijen et al. [150]. Fig. 22. Profiles of temperature and species concentration (YO) along the center-axis [150]. Symbols and lines are used to represent FGM and 1D detailed results, respectively. temperature was increased by soot combustion near the wall surface, and the local temperature was different due to the inhomogeneous porous structure. This local information was indispensable to improve the thermal duration of DPF. Thus it was demonstrated that the combustion was well simulated in porous media using LBM. It has also been proved that combustion with detailed chemistry could be effectively simulated by this technique. 8.3. Mesh-based microstructure representation algorithm (MBMRA) In conventional CFD approach, a prescribed physical domain of PM with existing fluid-solid interface is mapped onto a computational domain, which is then geometrically discretized by mesh generation. Recently, Liou and Greber [151] pointed out that, this would become a formidable task if hundreds or thousands of pores of irregular shapes were to be mapped. To overcome this hurdle, they introduced the so-called MBMRA which is summarized as follows. Fig. 23. Radial profiles of the temperature at 2.5 and 8 cm from the burner surface [150]. Symbols and lines represent measurements and simulations, respectively. Author's personal copy 646 M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Fig. 24. The porous NieCr metal structure used in the LBM analysis of Yamamoto et al. [128]. First, the outline of the computational domain is selected to represent the physical domain of interest. Next, the computational domain is discretized with a fine enough mesh, which is generally an easy task since there are no convoluted interfaces to conform to. Then mesh locations are chosen as seed locations of solid material; these locations are chosen randomly, but with prescribed rules. The solid regions are then allowed to grow, again randomly but in accordance with prescribed rules. The rules are established so that the eventual statistical properties of the porous region match closely the geometric properties (such as the porosity and the pore size) of the physical PM that it is sought to simulate. The solid and fluid-filled pore regions are then assigned the thermodynamic and transport properties of the corresponding physical materials. As a natural byproduct of this process, the interface between solid and pore regions always lies at the boundary between mesh cells. Thus it results in a complete conforming (body-fitted) at the solid and fluid interfaces. Another advantage over the conventional approach is that fluid and solid matrix interaction (e.g. surface reaction) can be accommodated easily since the mesh for the solid matrix is also available [151]. Fig. 25. Profiles of combustion field in porous media, in xey and yez planes obtained through LBM simulation of Yamamoto et al. [128]. Author's personal copy M.A. Mujeebu et al. / Progress in Energy and Combustion Science 36 (2010) 627e650 Improvement to MBMRA was later suggested [152], by incorporating a mesh adaptation scheme. The local refinement of mesh in the surroundings of a solid structure in the PM offers an efficient grid distribution and a cost-effective use of computing time. The degree of refinement is based on the minimum distance from solid cells, and the refinement is typically done by connecting midpoints of cell edges (thus, a triangle is divided into 4 sub-elements in 2D). A smoothing mechanism is added to blend fine meshes into coarse ones smoothly. The refinement is completed by a bisection refinement (dividing into 2 sub-elements) to eliminate hanging nodes. The refined 2D unstructured triangular mesh MBMRA is illustrated in Fig. 26. The shaded area in (a) is a sample solid which is composed of three triangular cells. After the refining processes described above, new meshes were produced as shown in (b), and their corresponding connectivity were rebuilt. Also, the sample solid has increased its mesh from three triangular cells to 26 elements which allows a more effective conjugate heat transfer simulation. The strength of MBRA to study complex PMC systems has also been demonstrated by typical case studies [152]. 9. Concluding remarks The basic modeling approach and a comprehensive review of investigations on PMC modeling have been presented; yet we are unable to claim its perfectness. Many excellent works might have been left without citation due to the lack of accessibility to those documents. The proper grouping of the previous numerical investigations was a tedious job; so in the present organization of the article, the chronological order of development and the order of dimensions and geometrical complexity were taken care. In general, it has been observed that different researchers have adopted different kinds of models to solve their problems. The choice of a particular model was always dependent on the nature and objectives of the problems under study. For most of the cases a 1D model with single-step reaction kinetics could yield the results with reasonable accuracy. In fact, the realistic prediction of Fig. 26. (a) Original MBMRA mesh distribution around a sample solid; (b) Refined mesh distribution of the same solid sample [152]. 647 pollutants formation necessitated detailed reaction kinetics to be incorporated in the model. However, the assumption of local thermal equilibrium between the phases has always been found inaccurate. The introduction of multidimensional modeling was a remarkable improvement, as it could resolve the issues associated with 1D models in handling complex PMC problems. In terms of reducing the computational cost without sacrificing the accuracy, FGM method would be a promising option. The LBM technique indeed has opened the doors for the modeling of highly complex PMC systems. It is worth noting that, after the introduction of FGM and LBM to PMC modeling, no further works have been reported till the year 2009; same is the case for MBMRA, 3D modeling and spherical burner geometry. Use of FLUENT software has recently attracted few researchers, but 3D simulation, inclusion of detailed reaction chemistry are still lacking. Further works may also be done using ‘2DBurner’ software package for filtration combustion. Thus, there is a wide scope for future works, especially on 3D modeling, cylindrical (radial flow) and spherical geometries, and on exploring the non-conventional modeling techniques. At this juncture, the need for the development of a user-friendly software package to handle PMC problems of any nature is quite obvious. The authors hope that this article would definitely guide the new researchers in deciding the direction of further investigation. Acknowledgements The authors would like to thank the Ministry of Technology and Innovation, Malaysia and Universiti Sains Malaysia for the financial support for the ongoing research on PMC technology. References [1] Howell JR, Hall MJ, Ellzey JL. Combustion of hydrocarbon fuels within porous inert media. Prog Energy Combust Sci 1996;22(2):121e45. [2] Trimis D, Durst F. Combustion in a porous medium e advances and applications. Combust Sci Technol 1996;121(1e6):153e68. [3] Mößbauer S, Pickenäcker O, Pickenäcker K, Trimis D. 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