Maximum superadiabatic temperature for stabilized flames within porous inert media q Fernando M. Pereira a,⇑ , Amir A.M. Oliveira b , Fernando F. Fachini c a Departamento de Engenharia Mecânica, Universidade Federal do Rio Grande do Sul, 90050-170 Porto Alegre, RS, Brazil b Departamento de Engenharia Mecânica, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil c Instituto Nacional de Pesquisas Espaciais, 12630-000 Cachoeira Paulista, SP, Brazil article info Article history: Received 19 January 2011 Received in revised form 18 March 2011 Accepted 1 April 2011 Available online 27 April 2011 Keywords: Superadiabatic temperature Porous inert media Stabilized flames abstract This work analyzes the superadiabatic temperature for laminar stationary lean premixed flames within porous inert media. The analysis is based on the excess enthalpy function applied to the one-dimensional volume-averaged equations. This formulation, with results obtained in a previous work, allows for the construction of an analytical solution valid over a large range of equivalence ratios. The model reveals the existence of a maximum non-dimensional superadiabatic temperature at a precisely determined equivalence ratio and connects previous works for near-stoichiometric and ultra-lean mixtures. Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved. 1. Introduction In combustion processes, the recirculation of thermal energy from the burned gases to the incoming reactants without dilution can be used to increase the reaction zone temperature beyond the adiabatic flame limit [1]. At sufficiently high recirculation rates, reactants of very low heat content become flammable due to the excess enthalpy at the reaction zone [2,3]. However, usually an external device is required for this recirculation. Alternatively, the insertion of a highly conductive porous medium at the flame region was shown to change the flame structure, creating a heat recirculation mechanism at the flame scale [4]. Nevertheless, the limit for the excess flame temperature for a stabilized premixed flame within a porous inert medium as a function of the equiva- lence ratio has not yet been determined. Here we show this limit, clarifying its relation to the magnitude of the heat recirculation and to the porous medium properties. Some asymptotic analysis have been proposed to the problem of premixed gas combustion within inert porous media for semi- infinite and finite length burners [5–11]. All these models divide the flame in a preheating region, a thin combustion region and a pos-combustion region. Expressions for the gas and solid tempera- tures and flame position are obtained as a function of an imposed flame velocity. In a previous work [12], an asymptotic solution for adiabatic stationary planar premixed flames within porous inert media is proposed taking advantage of the large difference between the thermal conductivity of the solid and gas-phases. In that solution, a two equation model for the energy conservation is used with effective properties. The flame structure is divided in three charac- teristic length scales. The large region of thermal non-equilibrium between the phases is related to a characteristic solid-phase length scale defined as the ratio of the solid-phase conduction to the gas- phase advection, l S  (1  e)k s /(eq n s F c p ), where k s is an effective so- lid-phase thermal conductivity, e is the solid matrix volumetric porosity, q n is the unburnt gas-phase density, s F is the flame veloc- ity, measured in the unburned stream, and c p is the gas-phase heat capacity. Similarly, a characteristic gas-phase length scale is de- fined as the ratio of the gas-phase conduction to the gas-phase advection, l G  ek g /(eq n s F c p ), where k g is an effective gas-phase thermal conductivity. Finally, based on the simplest kinetic mech- anism of one global step, modeled by an Arrhenius reaction rate model, the reaction length scale is determined as l R /l G  d, where d is the inverse of a modified Zel’dovich number. The two innermost length scales (l R and l G ) are the same scales defined in the classical premixed flame structure analysis. The outermost scale l S is responsible for the particular characteristics of flames within porous media. The solution is obtained for the condition N  O(1), where N is the interphase heat transfer parameter de- fined as N  k s h v /(q n s F c p ) 2 , where h v is the volumetric heat trans- fer coefficient. The parameter C is the thermal conductivities ratio defined as C = k s /k g . With the condition N  O(1), the interphase heat transfer at the gas-phase and at the reaction length scales is 0010-2180/$ - see front matter Ó 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2011.04.001 q Presented partially at the 12th Brazilian Congress of Thermal Engineering and Sciences, Belo Horizonte, 10–14 November, 2008. ⇑ Corresponding author. Fax: +55 51 33083222. E-mail address: fernando@mecanica.ufrgs.br (F.M. Pereira). Combustion and Flame 158 (2011) 2283–2288 Contents lists available at ScienceDirect Combustion and Flame journal homepage: www.elsevier.com/locate/combustflame