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Analytical modelling of filtration combustion in inert porous media
Applied Thermal Engineering 27 (2007) 1144–1149 www.elsevier.com/locate/apthermeng
Analytical modelling of filtration combustion in inert porous media Valeri Bubnovich *, Mario Toledo Universidad de Santiago de Chile, Departamento de Ingenierı´a Quı´mica, B. O’Higgins 3363, Casilla 10233, Santiago, Chile Received 20 December 2005; accepted 2 February 2006 Available online 5 May 2006
Abstract A study is made of the self-sustaining combustion waves during the filtration of lean methane–air mixtures in inert porous media using the two-temperature approximation. Such processes are characterized by an intense thermal interaction between the gas and the porous material. Due to interfacial heat transfer, the solid phase is able to redistribute heat absorbed from reaction products to the unburned mixture. The analytical solution is built in three different regions: the pre-heating region, the reaction region and the region occupied by the combustion products. Analytic expressions predicting the temperature and methane mass fraction profiles in the wave, as well as the combustion wave velocity and the longitudinal extension of the reaction region are derived. The results are confirmed by numerical calculations using the finite difference method and a full set of basic equations. 2006 Elsevier Ltd. All rights reserved. Keywords: Combustion; Inert porous media; Analytical solution
1. Introduction Recently, gas combustion in porous media has attracted the attention of many researchers in the fields of environment and combustion due to its interesting industrial applications [1]: oil extraction, infrared burner and heater development, ceramic materials synthesis, porous catalysts, volatile organic compound destruction in the air, diesel engines, and pollution control. Porous media combustion offers exceptional advantages compared with techniques involving free flame burners: higher burning rates [2], increased flame stability and efficiency of combustion systems [3], low calorific fuel combustion and very low pollutant emissions [4]. The process by which the region of exothermic chemical reactions propagates along inert porous media must be viewed within the framework of combustion waves in these types of media as presented schematically in Fig. 1. It is known from the literature [5–7] that during gas mixture combustion in inert porous media, combustion waves that *
Corresponding author. Tel.: +56 2 6812398; fax: +56 2 6817135. E-mail address: [email protected] (V. Bubnovich).
1359-4311/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.02.037
move up-stream or down-stream along the system can be observed. The direction of these movements depends generally on the physical properties of both solid and gas as well as the initial speed, temperature and excess air of the mixture. Conjugating these parameters, movement speeds of said waves are achieved that are much lower to those of gas and the temperature profiles showing a very pronounced maximum in the reaction region. In order to optimize the combustion processes in porous media, obtain low emissions of pollutants, and rapidly adapt porous materials and burner geometries to new applications, numerical codes and analytical solutions are necessary. Much research has been devoted to the theoretical analysis [2,5,8–12], computational modelling [4,13] and experimental studies [3,14,15] of heat transfer and combustion in porous media. Most of the theoretical studies are one-dimensional and use averaged heat transfer properties. In analytic solutions, combustion region thickness is commonly disregarded in order to be able to integrate the energy equation, whose right side has an extremely non-lineal term. Yoshizawa et al. [2] analysed a premixed flame in a porous medium, and the effects of radiative heat transfer and the position of the reaction zone within the porous medium
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
1145
Nomenclature a Cp D dP E h k0 Pr Tg Ts t u uFC w x x z
heat transfer coefficient (W/(m2 K)) specific heat (J/(kg K)) diameter of glass tube (m) diameter of the solid alumina spheres (m) activation energy (K) convection heat transfer coefficient (W/(m2 K)) pre-exponential factor (s1) Prandtl number temperature of gas (K) temperature of porous media (K) time (s) average interstitial gas velocity (m/s) combustion wave velocity (m/s) mass fraction of species, dimensionless spatial distance in moving coordinates (m) dimensionless spatial distance spatial distance (m)
D e e0 e00 u k h q r s
reaction region thickness porosity, dimensionless emissivity, dimensionless transmissivity factor, dimensionless coefficient of excess air, dimensionless thermal conductivity (W/(m K)) dimensionless gas temperature density Stephan–Boltzmann constant dimensionless solid temperature
Subscripts g gas s solid 0 inlet eff effective
Greek symbols bv effective coefficient for heat exchange with surroundings (W/(m3 K))
Products Reaction Region Thickness, Δ
Z Heat Exchanger Region
l
Pre - heating Region
Methane + Air Inlet, z=0 Fig. 1. Schematic diagram of the physical problem considered.
on the flame structure and burning velocity by means a rigorous treatment of the reaction zone in gas–solid two-phase systems. The ability of the energy to be concentrated in the front of a co-flow filtration combustion wave in a porous solid was analyzed by Aldushin et al. [5]. The paper focuses on the filtration combustion (FC) wave propagation mode corresponding to the most pronounced super adiabatic effect. An analytic solution of combustion waves is presented by Foutko et al. [8] assuming that the reaction speed in the combustion region is infinite and that the longitudinal
extension of that region is nil. The theoretical treatment is carried out within a two-temperature model in which external heat losses are taken into account, and neglecting diffusion and thermal conductivity of the gas phase. Akkutlu and Tortsos [9] study the properties of forward combustion fronts propagating at a constant velocity in the presence of heat losses. Heat losses are assumed to be relatively weak and they are expressed using two models: (1) a convective type, using an overall heat transfer coefficient; and (2) a conductive type, for heat transfer by transverse conduction to infinitely large surrounding formations. Like result, they develop expressions for temperature and concentration profiles and the velocity of the combustion front, under both adiabatic and non-adiabatic conditions, in analytical form. Analytical expressions predicting the combustion wave velocity and temperature distributions in the wave are derived. Shkadinsky et al. [10] demonstrate the existence of both one and two stationary reaction zone structures which arise in filtration combustion in a moving porous medium. Using the narrow reaction zone approximation, they derive approximate analytical expressions for the principal combustion characteristics, including the combustion temperature, the temperature and depth of solid conversion at the first reaction zone, the locations of the reaction zones, the inlet and outlet oxidant fluxes, as well as profiles for the spatial distributions of pressure, temperature, and depth of conversion, corresponding to the stationary reaction zone structures. Dynamics of filtration combustion inclination instability is investigated experimentally, analytically and
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V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
numerically by Dobrego et al. [11]. It is found that inclination amplitude growth velocity on the linear stage is proportional to filtration combustion wave velocity, system diameter and inversed diameter of porous media particles. Bubnovich et al. [12] analyzed a premixed flame in a porous medium by means a rigorous treatment of the reaction zone in a one-temperature approximation. However, it is important to consider the situations in which the interfacial heat transfer coefficient has a specific value and affects the characteristics of the combustion wave. Therefore, the main objective of this paper is to provide an adequate analytical solution in a two-temperature approximation that considers the characteristics of the reaction region. 2. Modelling of premixed combustion on inert porous media The physical situation for which the mathematical model is built is the following: • The porous canal is made up of alumina spheres of diameter dP = 5.6 mm, forming a porosity of e = 0.4. Other physical properties of alumina are: mass density qs = 3987 kg/m3, specific heat C ps ¼ 29:567 þ 2:61177 T s 0:00171 T 2s þ 3:382 107 T 3s J=ðkg KÞ, and its thermal conductivity ks ¼ 0:21844539 þ 0:00174653 T s þ 8:2266 108 T 2s W=ðm KÞ. • The premixed methane–air gas enters through section z = 0 (see Fig. 1) with excess air equal to u, initial temperature T0 = 300 K, and gas speed ug0. Other physical properties of the gas are: C p;g ¼ 947 e0:000183T g J=ðkg KÞ, qg0 = 1.13 kg/m3, lg ¼ 3:37 107 T 0:7 g kg=ðm sÞ, kg ¼ 7 0:7 4:82 10 C p;g T g W=ðm KÞ. The gas is incompressible and obeys the perfect gas law. The molecular Lewis number is unity. • The effective coefficient for heat exchange with the surroundings, bv, W/(m3 K), is 4 T 4 T 40 h þ e0 r e00 ð1Þ bv ¼ D T T0 where e 0 = 0.45 is the emissivity of the solid spheres, e00 = 0.38 is a transmissivity factor for the quartz glass tube, D = 7.6 cm is the diameter of the glass tube, h = 10 W/(m2 K) is the convection heat transfer coefficient, and r is the Stefan–Boltzmann constant. • The combustion of the mixture is described by means of the global chemical reaction in a single step: CH4 þ 2ð1 þ uÞðO2 þ 3:76N2 Þ ! CO2 þ 2H2 O þ 2uO2 þ 7:52ð1 þ uÞN2
ð2Þ
where u represents the proportion of excess air in the reactant streams at the system’s inlet. The heat content of the reactive mixture is Dhc = 50150000 J/m3. In order to obey the first-order Arrhenius equation, the reaction rate is considered to have the activation energy E =
129999.97 J/mol and the pre-exponential factor k0 = 2.6 · 108 s1. • The effective thermal conductivity of the porous media can be written in the form: keff ¼ ð1 eÞ ks þ
32 r d P e T 3s 9 ð1 eÞ
ð3Þ
• The interfacial gas–solid heat exchange coefficient [3] is 2 !0:6 3 e q u d 6 ks ð1 eÞ 4 g P g 5 a¼ 2:0 þ 1:1 Pr1=3 lg d 2P ð4Þ • In the initial part of the canal the mixture is ignited and a narrow reaction region appears with a width equal to D which, after reaching section z = 1, starts moving downstream with constant speed uFC ug. The entire region over the z-axis is divided into three parts: the gas preheating region, 0 6 z 6 l; the reaction region, 1 6 z 6 1 + D; and the region occupied by the combustion products, 1 + D < z 6 1. • The reaction region thickness is defined here as the distance in the canal between the section where the temperature is equal to the ignition temperature and the section where the methane mass fraction equals 1/1000 of its value at the inlet. Then, the energy equations for the gas and the solid phases, as well as the continuity equations for the chemical species and the mixture, respectively, are formulated as: e ðq C p Þg
oT g oT g þ ug ot oz
¼ a ðT g T s Þ þ Dhc k 0 w e q eE=RT g oT s o oT s keff ¼ þ aðT g T s Þ bv ðT s T 0 Þ ð1 eÞ ðq C p Þs ot oz oz ow ow o kg ow þ ug ¼ k 0 w eE=RT g ot oz oz C p;g oz
ð5Þ
ð6Þ ð7Þ
where Tg, Ts are the gas and solid temperatures (Cp)g, (Cp)s are their heat capacities, ug is the gas velocity, T0 is the ambient temperature, Dhc is the heat content of the reactive mixture, w is the mass fraction of fuel, and e is the porosity. The following initial and boundary conditions are imposed: t ¼ 0: T g ¼ T s ¼ T 0 ; w ¼ w0 ð8Þ z ¼ 0: T g ¼ T s ¼ T 0 ; ug ¼ u0 ; z ! 1: T 0g ¼ T 0s ¼ w0 ¼ 0
w ¼ w0 ;
ð9Þ
This work has the purpose of obtaining the temperature T(t, z) and methane gas mass fraction w(t, z) profiles as well as the reaction region movement speed values uFC and the width of the reaction region, D. 3. Analytical method for finding the solution Before integrating the mathematical model, there is a shift to the reference system which moves together with
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
the combustion wave: x ¼ z uFC t; uFC ug . Moreover, it is assumed that the diffusion term in Eq. (7) is considerable only in the reaction region. Therefore, the no dimensional mathematical model can be written as: 0 ¼ Ld w 00 Fk w w 0 0 h ¼ g ðs hÞ c w 0
ð10Þ ð11Þ ð12Þ
00
a s ¼ s þ g ðh sÞ b s
x¼
x ; Xc
¼ w
w ; w0
Xc ¼
E
k 0 keff e RT g Fk ¼ ; e qg u2g C pg a¼
b¼
s¼
keff ; e ðq C p Þg ug Ts T0 ; T ig T 0
c¼
Ld ¼
kg e qg keff
w0 Dhc ; C pg ðT ig T 0 Þ
ðq C p Þs ð1 eÞ uFC ðq C p Þg ug e
l d1 ¼ ; Xc
sI ¼ c3 ek3 x þ c4 ek4 x ; eðk3 k4 Þ d1 1 c3 ¼ ; c4 ¼ ; 1 eðk3 k4 Þ d1 1 eðk3 k4 Þ d1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þa 2 1 1 þ 4 b=ð1 þ aÞ k 3;4 ¼ 2 hI ¼ c9 c3 ek3 x þ c10 c4 ek4 x ; c10 ¼
where
Tg T0 h¼ ; T ig T 0
D d¼ ; Xc
g¼
a keff ðqg C pg ug eÞ2
;
ð13Þ
2
In Eq. (13) the Fk coefficient is variable and depends on the temperature. As will be seen below, the temperature at each of the three regions will take a constant characteristic value which allows the integration of Eq. (10) independently from Eqs. (11) and (12). Also, the coefficients (13) will be defined by the same characteristic temperatures in each of the regions. Introducing the hypothesis that h 0 s 0 , Eqs. (11) and (12) can be reduced to only one equation: 0 b s ¼ 0 s00 ð1 þ aÞ s0 c w
ð14Þ
After integration of Eq. (14), the gas temperature can be found by replacing s, s 0 , s00 in Eq. (12). Then Eqs. (10), (12) and (14) are used to build an analytic solution. It can be assumed that the entire integration region is divided into three parts in such a way that the dimensionless boundary conditions are: I ¼ 1; x ¼ 0 : sI ¼ sII ¼ 1; w I ¼ w II x ¼ d1 : sI ¼ 0; w 0 0 ¼ n; x ! 1 : sIII ¼ 0; w ¼ 0. x ¼ d : sII ¼ sIII ; sII ¼ sIII ; w ð15Þ First, the equations are integrated in the pre-heating region. It is assumed that the chemical reaction is negligible and the diffusive term in Eq. (10) is also negligible there; the gas temperature present in the exponential term of Fk is constant and equal to the mean between the initial temperature of the mixture and the ignition temperature. As a result of the integration of Eqs. (10) and (14) with the ¼ 1, s1 = 0 at x ¼ d1 and s1 = 1 boundary conditions, w at x ¼ 0, respectively, we have: I ¼ eFk I ðxþd1 Þ ; w
Fk I ¼
c9 ¼
a k 4 k 24 þ g þ b g
k 0 keff E2 e RðT ig þT 0 Þ 2 e qg ug C pg
ð16Þ
ð17Þ
a k 3 k 23 þ g þ b ; g ð18Þ
In the chemical reaction region, Eq. (10) is integrated assuming that in the exponential term of Fk the temperature is constant and equal to the ignition temperature: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 4 Ld Fk II k 2 xb ; b W II ¼ e ; k 1;2 ¼ 2 Ld
bv keff ðqg e C pg ug Þ
1147
E
k 0 keff e RT ig ¼ Fk I d1 ; Fk II ¼ e C pg qg u2g
ð19Þ
As to the combustion region, it can be said that it starts where the temperature is equal to the ignition temperature, hII = 1, and ends at x ¼ d, where the mass fraction of fuel is ! n. Therefore, from solution (19) one nearly nil: x ! d; w gets d = (ln n + b)/k2. After introducing the corresponding numerical values, and considering that ug = 1.075 m/s, n = 0.001, Tig = 1150 K, it was found that D = 9.156 mm. From this, it is seen that theoretical predictions about the thickness of the combustion region are in agreement with the experimental observations. Now, in order to integrate Eq. (14), the derivative W 0 in that equation is replaced by Eq. (19). The resulting differential equation is solved by means of Lagrange’s method: sII ¼ ek3 x ðc5 þ c5 ðxÞÞ þ ek4 x ðc6 þ c6 ðxÞÞ c k 2 eb eðk2 k4 Þx ; c5 ðxÞ ¼ ðk 3 k 4 Þ ðk 2 k 3 Þ c k 2 eb eðk2 k4 Þx c6 ðxÞ ¼ ðk 4 k 3 Þ ðk 2 k 4 Þ
ð20Þ ð21Þ
Then the gas temperature in the same region is obtained from Eq. (12): c 0 hII ¼ c9 ek3 x ðc5 þ c5 ðxÞÞ þ c10 ek4 x ðc6 þ c6 ðxÞÞ w g ð22Þ ¼w 0 ¼ 0, Finally, in the third region it is assumed that w and the solution to Eqs. (14) and (12), respectively, with the boundary condition sIII ¼ 0; x ! 1 is: sIII ¼ c8 ek4 x ;
hIII ¼ c10 c8 ek4 x
ð23Þ
where the constants c5,c6, c8 are found together from the following system of boundary conditions sII = 1 at x ¼ 0, and s0II ¼ s0III at x ¼ d: c5 ¼ c5 ðdÞ; c6 ¼ 1 c5 ð0Þ c6 ð0Þ þ c5 ðdÞ; c8 ¼ c6 þ c6 ðdÞ ð24Þ
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V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
----- Solid. — Gas
Gas
Solid
1600 1600
1400
1400 1200
1000
T, K
T, K
1200
800
1000 800
600
600
400
400
200 0
0.1
0.2
0.3
0.4
200 0.0
0.1
0.2
0.3
0.4
0.5
Distance, m
Distance, m
Fig. 2. Profiles of the two-temperature analytical (left) and numerical (right) solutions: u = 4.88, ug = 1.075 m/s.
Finally, in order to build the complete analytical solution, the expressions for the combustion wave velocity, uFC, and for the ignition temperature of the mixture, Tig, must be developed. First, from the boundary condition hII = 1 at x ¼ 0, we get 2 3 6 7 qg C pg 1 6 7 uFC ¼ ug 61 7 E=RT ig 5 4 qs C ps 4 bv ðT ig T 0 Þ e 1þ qg w0 Dhc k 0 K ks þ ð25Þ 4 ug qs C ps eE=RT ig Assuming that the fuel consumption is almost completed in the distance of the pore size order, the temperature increase for the gas element may be treated as a thermal explosion. Thus, one can directly apply the corresponding Frank– Kamenetzki theory [16] with the correction for fuel consumption, and derive an approximation for the ignition temperature Tig in the following implicit form: ! E 1 eRT ig Dh2c w20 R5 d P e T 0 k0 ¼ ð26Þ 7=3 ug 2 p2 C 2pg E2 ðE=R T ig Þ
5. Conclusions The theoretical gas combustion study in inert porous media concluded with the analytical construction of a series of simple algebraic formulas which represent the methane mass fraction and temperature profiles in an inert porous medium. Also, two formulas were built, which allow the prediction of the combustion wave velocity in the system and the thickness of the reaction region. Six dimensionless parameters define heat and mass transport in the mathematical model. The analytical solution agrees satisfactorily with the numerical solution, built on the finite differences, of the complete mathematical model. It is shown that the reaction region is not infinitely thin, as is usually considered. Furthermore, it was found that the analytical solution was very sensitive to the value of the wave displacement speed. Consequently, the results of this research can be used in the analysis of combustion waves in porous media for technical applications. Acknowledgements This research was supported by Conicyt-Chile, Fondecyt project 1050241, and by the Academia Polite´cnica Aerona´utica-FACH, Chile.
4. Numerical solution To support and test the analytical solutions obtained here, a computational procedure was carried out using a full set of the basic equations. These were calculated using the implicit finite difference discretisation and the tree-diagonal matrix algorithm. The time step was 0.01 s and 800 grid points were used for the numerical calculations. In Fig. 2 the analytical solution obtained in this paper is compared to the numerical solution built in the same paper. Many similarities between the compared cases are seen. Apart from that, the maximum temperatures reached at the combustion front according to our solution are found within the maximum temperature values in the numerical solution.
References [1] J.R. Howell, M.J. Hall, J.L. Ellzey, Combustion of hydrocarbon fuels within porous inert media, Progress in Energy and Combustion Science, an International Review Journal 22 (1996) 121–145. [2] Y. Yoshizawa, K. Sasaki, R. Echigo, Analytical study of the structure of radiation controlled flame, International Journal of Heat and Mass Transfer 31 (2) (1998) 311–319. [3] W.M. Mathis Jr., J.L. Ellzey, Flame stabilization, operating range, and emissions for methane/air porous burner, Combustion and Science Technology 155 (2003) 825–839. [4] P.H. Bouma, L.P.H. De Goey, Premixed combustion on ceramic foam burners, Combustion and Flame 119 (1999) 133–143. [5] A.P. Aldushin, I.E. Rumanov, J.B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave, Combustion and Flame 118 (1999) 76–90.
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149 [6] A.M. Oliveira, M. Kaviany, Nonequilibrium in the transport of heat and reactants in combustion in porous media, Progress in Energy and Combustion Science 27 (2001) 523–545. [7] V.S. Babkin, Filtration combustion of gases. Present state of affairs and prospects, Pure and Applied Chemistry 65 (2) (1993) 335–344. [8] S.I. Foutko, S.I. Shabunya, S.A. Zhdanok, Superadiabatic combustion wave in a diluted methane–air mixture under filtration in a packed bed, in: Proceedings of Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, p. 222. [9] I.Y. Akkutlu, Y.C. Yortos, The dynamics of in-situ combustion fronts in porous media, Combustion and Flame 134 (2003) 229–247. [10] K.G. Shkadinsky, G.V. Shkadinskaya, B.J. Matkowsky, Filtration combustion in moving media: one and two reaction zone structures, Combustion and Flame 110 (1997) 441–461. [11] K.V. Dobrego, I.M. Kozlov, V.I. Bubnovich, C.E. Rosas, Dynamics of filtration combustion front perturbation in the tubular porous
[12]
[13] [14]
[15]
[16]
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media burner, International Journal of Heat and Mass Transfer 46 (2003) 3279–3289. V. Bubnovich, J. Vergara, C. Rosas, Theoretical analysis of combustion waves in a methane–air mixture under filtration in a packed bed, in: Proceedings of IV International School – Seminar ‘‘Modern Problems of Combustion and its Application’’, Minsk, Belarus, 2001, pp.158–162. M.R. Henneke, J.L. Ellzey, Modelling of filtration combustion in a packed bed, Combustion and Flame 117 (1999) 832–840. J.G. Hoffmann, R. Echigo, H. Yoshida, S. Tada, Experimental study on combustion in porous media with a reciprocating flow system, Combustion and Flame 111 (1997) 32–46. S.A. Zhdanok, L.A. Kennedy, G. Koester, Superadiabatic combustion of methane air mixtures under filtration in a packed bed, Combustion and Flame 100 (1/2) (1995) 221–231. D.A. Frank-Kamenetzki, Diffusia i Teploperedacha v Chimicheskoy Kinetike, Nauka, Moscow, 1987 (in Russian).
Analytical modelling of filtration combustion in inert porous media Valeri Bubnovich *, Mario Toledo Universidad de Santiago de Chile, Departamento de Ingenierı´a Quı´mica, B. O’Higgins 3363, Casilla 10233, Santiago, Chile Received 20 December 2005; accepted 2 February 2006 Available online 5 May 2006
Abstract A study is made of the self-sustaining combustion waves during the filtration of lean methane–air mixtures in inert porous media using the two-temperature approximation. Such processes are characterized by an intense thermal interaction between the gas and the porous material. Due to interfacial heat transfer, the solid phase is able to redistribute heat absorbed from reaction products to the unburned mixture. The analytical solution is built in three different regions: the pre-heating region, the reaction region and the region occupied by the combustion products. Analytic expressions predicting the temperature and methane mass fraction profiles in the wave, as well as the combustion wave velocity and the longitudinal extension of the reaction region are derived. The results are confirmed by numerical calculations using the finite difference method and a full set of basic equations. 2006 Elsevier Ltd. All rights reserved. Keywords: Combustion; Inert porous media; Analytical solution
1. Introduction Recently, gas combustion in porous media has attracted the attention of many researchers in the fields of environment and combustion due to its interesting industrial applications [1]: oil extraction, infrared burner and heater development, ceramic materials synthesis, porous catalysts, volatile organic compound destruction in the air, diesel engines, and pollution control. Porous media combustion offers exceptional advantages compared with techniques involving free flame burners: higher burning rates [2], increased flame stability and efficiency of combustion systems [3], low calorific fuel combustion and very low pollutant emissions [4]. The process by which the region of exothermic chemical reactions propagates along inert porous media must be viewed within the framework of combustion waves in these types of media as presented schematically in Fig. 1. It is known from the literature [5–7] that during gas mixture combustion in inert porous media, combustion waves that *
Corresponding author. Tel.: +56 2 6812398; fax: +56 2 6817135. E-mail address: [email protected] (V. Bubnovich).
1359-4311/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.02.037
move up-stream or down-stream along the system can be observed. The direction of these movements depends generally on the physical properties of both solid and gas as well as the initial speed, temperature and excess air of the mixture. Conjugating these parameters, movement speeds of said waves are achieved that are much lower to those of gas and the temperature profiles showing a very pronounced maximum in the reaction region. In order to optimize the combustion processes in porous media, obtain low emissions of pollutants, and rapidly adapt porous materials and burner geometries to new applications, numerical codes and analytical solutions are necessary. Much research has been devoted to the theoretical analysis [2,5,8–12], computational modelling [4,13] and experimental studies [3,14,15] of heat transfer and combustion in porous media. Most of the theoretical studies are one-dimensional and use averaged heat transfer properties. In analytic solutions, combustion region thickness is commonly disregarded in order to be able to integrate the energy equation, whose right side has an extremely non-lineal term. Yoshizawa et al. [2] analysed a premixed flame in a porous medium, and the effects of radiative heat transfer and the position of the reaction zone within the porous medium
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
1145
Nomenclature a Cp D dP E h k0 Pr Tg Ts t u uFC w x x z
heat transfer coefficient (W/(m2 K)) specific heat (J/(kg K)) diameter of glass tube (m) diameter of the solid alumina spheres (m) activation energy (K) convection heat transfer coefficient (W/(m2 K)) pre-exponential factor (s1) Prandtl number temperature of gas (K) temperature of porous media (K) time (s) average interstitial gas velocity (m/s) combustion wave velocity (m/s) mass fraction of species, dimensionless spatial distance in moving coordinates (m) dimensionless spatial distance spatial distance (m)
D e e0 e00 u k h q r s
reaction region thickness porosity, dimensionless emissivity, dimensionless transmissivity factor, dimensionless coefficient of excess air, dimensionless thermal conductivity (W/(m K)) dimensionless gas temperature density Stephan–Boltzmann constant dimensionless solid temperature
Subscripts g gas s solid 0 inlet eff effective
Greek symbols bv effective coefficient for heat exchange with surroundings (W/(m3 K))
Products Reaction Region Thickness, Δ
Z Heat Exchanger Region
l
Pre - heating Region
Methane + Air Inlet, z=0 Fig. 1. Schematic diagram of the physical problem considered.
on the flame structure and burning velocity by means a rigorous treatment of the reaction zone in gas–solid two-phase systems. The ability of the energy to be concentrated in the front of a co-flow filtration combustion wave in a porous solid was analyzed by Aldushin et al. [5]. The paper focuses on the filtration combustion (FC) wave propagation mode corresponding to the most pronounced super adiabatic effect. An analytic solution of combustion waves is presented by Foutko et al. [8] assuming that the reaction speed in the combustion region is infinite and that the longitudinal
extension of that region is nil. The theoretical treatment is carried out within a two-temperature model in which external heat losses are taken into account, and neglecting diffusion and thermal conductivity of the gas phase. Akkutlu and Tortsos [9] study the properties of forward combustion fronts propagating at a constant velocity in the presence of heat losses. Heat losses are assumed to be relatively weak and they are expressed using two models: (1) a convective type, using an overall heat transfer coefficient; and (2) a conductive type, for heat transfer by transverse conduction to infinitely large surrounding formations. Like result, they develop expressions for temperature and concentration profiles and the velocity of the combustion front, under both adiabatic and non-adiabatic conditions, in analytical form. Analytical expressions predicting the combustion wave velocity and temperature distributions in the wave are derived. Shkadinsky et al. [10] demonstrate the existence of both one and two stationary reaction zone structures which arise in filtration combustion in a moving porous medium. Using the narrow reaction zone approximation, they derive approximate analytical expressions for the principal combustion characteristics, including the combustion temperature, the temperature and depth of solid conversion at the first reaction zone, the locations of the reaction zones, the inlet and outlet oxidant fluxes, as well as profiles for the spatial distributions of pressure, temperature, and depth of conversion, corresponding to the stationary reaction zone structures. Dynamics of filtration combustion inclination instability is investigated experimentally, analytically and
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V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
numerically by Dobrego et al. [11]. It is found that inclination amplitude growth velocity on the linear stage is proportional to filtration combustion wave velocity, system diameter and inversed diameter of porous media particles. Bubnovich et al. [12] analyzed a premixed flame in a porous medium by means a rigorous treatment of the reaction zone in a one-temperature approximation. However, it is important to consider the situations in which the interfacial heat transfer coefficient has a specific value and affects the characteristics of the combustion wave. Therefore, the main objective of this paper is to provide an adequate analytical solution in a two-temperature approximation that considers the characteristics of the reaction region. 2. Modelling of premixed combustion on inert porous media The physical situation for which the mathematical model is built is the following: • The porous canal is made up of alumina spheres of diameter dP = 5.6 mm, forming a porosity of e = 0.4. Other physical properties of alumina are: mass density qs = 3987 kg/m3, specific heat C ps ¼ 29:567 þ 2:61177 T s 0:00171 T 2s þ 3:382 107 T 3s J=ðkg KÞ, and its thermal conductivity ks ¼ 0:21844539 þ 0:00174653 T s þ 8:2266 108 T 2s W=ðm KÞ. • The premixed methane–air gas enters through section z = 0 (see Fig. 1) with excess air equal to u, initial temperature T0 = 300 K, and gas speed ug0. Other physical properties of the gas are: C p;g ¼ 947 e0:000183T g J=ðkg KÞ, qg0 = 1.13 kg/m3, lg ¼ 3:37 107 T 0:7 g kg=ðm sÞ, kg ¼ 7 0:7 4:82 10 C p;g T g W=ðm KÞ. The gas is incompressible and obeys the perfect gas law. The molecular Lewis number is unity. • The effective coefficient for heat exchange with the surroundings, bv, W/(m3 K), is 4 T 4 T 40 h þ e0 r e00 ð1Þ bv ¼ D T T0 where e 0 = 0.45 is the emissivity of the solid spheres, e00 = 0.38 is a transmissivity factor for the quartz glass tube, D = 7.6 cm is the diameter of the glass tube, h = 10 W/(m2 K) is the convection heat transfer coefficient, and r is the Stefan–Boltzmann constant. • The combustion of the mixture is described by means of the global chemical reaction in a single step: CH4 þ 2ð1 þ uÞðO2 þ 3:76N2 Þ ! CO2 þ 2H2 O þ 2uO2 þ 7:52ð1 þ uÞN2
ð2Þ
where u represents the proportion of excess air in the reactant streams at the system’s inlet. The heat content of the reactive mixture is Dhc = 50150000 J/m3. In order to obey the first-order Arrhenius equation, the reaction rate is considered to have the activation energy E =
129999.97 J/mol and the pre-exponential factor k0 = 2.6 · 108 s1. • The effective thermal conductivity of the porous media can be written in the form: keff ¼ ð1 eÞ ks þ
32 r d P e T 3s 9 ð1 eÞ
ð3Þ
• The interfacial gas–solid heat exchange coefficient [3] is 2 !0:6 3 e q u d 6 ks ð1 eÞ 4 g P g 5 a¼ 2:0 þ 1:1 Pr1=3 lg d 2P ð4Þ • In the initial part of the canal the mixture is ignited and a narrow reaction region appears with a width equal to D which, after reaching section z = 1, starts moving downstream with constant speed uFC ug. The entire region over the z-axis is divided into three parts: the gas preheating region, 0 6 z 6 l; the reaction region, 1 6 z 6 1 + D; and the region occupied by the combustion products, 1 + D < z 6 1. • The reaction region thickness is defined here as the distance in the canal between the section where the temperature is equal to the ignition temperature and the section where the methane mass fraction equals 1/1000 of its value at the inlet. Then, the energy equations for the gas and the solid phases, as well as the continuity equations for the chemical species and the mixture, respectively, are formulated as: e ðq C p Þg
oT g oT g þ ug ot oz
¼ a ðT g T s Þ þ Dhc k 0 w e q eE=RT g oT s o oT s keff ¼ þ aðT g T s Þ bv ðT s T 0 Þ ð1 eÞ ðq C p Þs ot oz oz ow ow o kg ow þ ug ¼ k 0 w eE=RT g ot oz oz C p;g oz
ð5Þ
ð6Þ ð7Þ
where Tg, Ts are the gas and solid temperatures (Cp)g, (Cp)s are their heat capacities, ug is the gas velocity, T0 is the ambient temperature, Dhc is the heat content of the reactive mixture, w is the mass fraction of fuel, and e is the porosity. The following initial and boundary conditions are imposed: t ¼ 0: T g ¼ T s ¼ T 0 ; w ¼ w0 ð8Þ z ¼ 0: T g ¼ T s ¼ T 0 ; ug ¼ u0 ; z ! 1: T 0g ¼ T 0s ¼ w0 ¼ 0
w ¼ w0 ;
ð9Þ
This work has the purpose of obtaining the temperature T(t, z) and methane gas mass fraction w(t, z) profiles as well as the reaction region movement speed values uFC and the width of the reaction region, D. 3. Analytical method for finding the solution Before integrating the mathematical model, there is a shift to the reference system which moves together with
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
the combustion wave: x ¼ z uFC t; uFC ug . Moreover, it is assumed that the diffusion term in Eq. (7) is considerable only in the reaction region. Therefore, the no dimensional mathematical model can be written as: 0 ¼ Ld w 00 Fk w w 0 0 h ¼ g ðs hÞ c w 0
ð10Þ ð11Þ ð12Þ
00
a s ¼ s þ g ðh sÞ b s
x¼
x ; Xc
¼ w
w ; w0
Xc ¼
E
k 0 keff e RT g Fk ¼ ; e qg u2g C pg a¼
b¼
s¼
keff ; e ðq C p Þg ug Ts T0 ; T ig T 0
c¼
Ld ¼
kg e qg keff
w0 Dhc ; C pg ðT ig T 0 Þ
ðq C p Þs ð1 eÞ uFC ðq C p Þg ug e
l d1 ¼ ; Xc
sI ¼ c3 ek3 x þ c4 ek4 x ; eðk3 k4 Þ d1 1 c3 ¼ ; c4 ¼ ; 1 eðk3 k4 Þ d1 1 eðk3 k4 Þ d1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þa 2 1 1 þ 4 b=ð1 þ aÞ k 3;4 ¼ 2 hI ¼ c9 c3 ek3 x þ c10 c4 ek4 x ; c10 ¼
where
Tg T0 h¼ ; T ig T 0
D d¼ ; Xc
g¼
a keff ðqg C pg ug eÞ2
;
ð13Þ
2
In Eq. (13) the Fk coefficient is variable and depends on the temperature. As will be seen below, the temperature at each of the three regions will take a constant characteristic value which allows the integration of Eq. (10) independently from Eqs. (11) and (12). Also, the coefficients (13) will be defined by the same characteristic temperatures in each of the regions. Introducing the hypothesis that h 0 s 0 , Eqs. (11) and (12) can be reduced to only one equation: 0 b s ¼ 0 s00 ð1 þ aÞ s0 c w
ð14Þ
After integration of Eq. (14), the gas temperature can be found by replacing s, s 0 , s00 in Eq. (12). Then Eqs. (10), (12) and (14) are used to build an analytic solution. It can be assumed that the entire integration region is divided into three parts in such a way that the dimensionless boundary conditions are: I ¼ 1; x ¼ 0 : sI ¼ sII ¼ 1; w I ¼ w II x ¼ d1 : sI ¼ 0; w 0 0 ¼ n; x ! 1 : sIII ¼ 0; w ¼ 0. x ¼ d : sII ¼ sIII ; sII ¼ sIII ; w ð15Þ First, the equations are integrated in the pre-heating region. It is assumed that the chemical reaction is negligible and the diffusive term in Eq. (10) is also negligible there; the gas temperature present in the exponential term of Fk is constant and equal to the mean between the initial temperature of the mixture and the ignition temperature. As a result of the integration of Eqs. (10) and (14) with the ¼ 1, s1 = 0 at x ¼ d1 and s1 = 1 boundary conditions, w at x ¼ 0, respectively, we have: I ¼ eFk I ðxþd1 Þ ; w
Fk I ¼
c9 ¼
a k 4 k 24 þ g þ b g
k 0 keff E2 e RðT ig þT 0 Þ 2 e qg ug C pg
ð16Þ
ð17Þ
a k 3 k 23 þ g þ b ; g ð18Þ
In the chemical reaction region, Eq. (10) is integrated assuming that in the exponential term of Fk the temperature is constant and equal to the ignition temperature: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 4 Ld Fk II k 2 xb ; b W II ¼ e ; k 1;2 ¼ 2 Ld
bv keff ðqg e C pg ug Þ
1147
E
k 0 keff e RT ig ¼ Fk I d1 ; Fk II ¼ e C pg qg u2g
ð19Þ
As to the combustion region, it can be said that it starts where the temperature is equal to the ignition temperature, hII = 1, and ends at x ¼ d, where the mass fraction of fuel is ! n. Therefore, from solution (19) one nearly nil: x ! d; w gets d = (ln n + b)/k2. After introducing the corresponding numerical values, and considering that ug = 1.075 m/s, n = 0.001, Tig = 1150 K, it was found that D = 9.156 mm. From this, it is seen that theoretical predictions about the thickness of the combustion region are in agreement with the experimental observations. Now, in order to integrate Eq. (14), the derivative W 0 in that equation is replaced by Eq. (19). The resulting differential equation is solved by means of Lagrange’s method: sII ¼ ek3 x ðc5 þ c5 ðxÞÞ þ ek4 x ðc6 þ c6 ðxÞÞ c k 2 eb eðk2 k4 Þx ; c5 ðxÞ ¼ ðk 3 k 4 Þ ðk 2 k 3 Þ c k 2 eb eðk2 k4 Þx c6 ðxÞ ¼ ðk 4 k 3 Þ ðk 2 k 4 Þ
ð20Þ ð21Þ
Then the gas temperature in the same region is obtained from Eq. (12): c 0 hII ¼ c9 ek3 x ðc5 þ c5 ðxÞÞ þ c10 ek4 x ðc6 þ c6 ðxÞÞ w g ð22Þ ¼w 0 ¼ 0, Finally, in the third region it is assumed that w and the solution to Eqs. (14) and (12), respectively, with the boundary condition sIII ¼ 0; x ! 1 is: sIII ¼ c8 ek4 x ;
hIII ¼ c10 c8 ek4 x
ð23Þ
where the constants c5,c6, c8 are found together from the following system of boundary conditions sII = 1 at x ¼ 0, and s0II ¼ s0III at x ¼ d: c5 ¼ c5 ðdÞ; c6 ¼ 1 c5 ð0Þ c6 ð0Þ þ c5 ðdÞ; c8 ¼ c6 þ c6 ðdÞ ð24Þ
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V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149
----- Solid. — Gas
Gas
Solid
1600 1600
1400
1400 1200
1000
T, K
T, K
1200
800
1000 800
600
600
400
400
200 0
0.1
0.2
0.3
0.4
200 0.0
0.1
0.2
0.3
0.4
0.5
Distance, m
Distance, m
Fig. 2. Profiles of the two-temperature analytical (left) and numerical (right) solutions: u = 4.88, ug = 1.075 m/s.
Finally, in order to build the complete analytical solution, the expressions for the combustion wave velocity, uFC, and for the ignition temperature of the mixture, Tig, must be developed. First, from the boundary condition hII = 1 at x ¼ 0, we get 2 3 6 7 qg C pg 1 6 7 uFC ¼ ug 61 7 E=RT ig 5 4 qs C ps 4 bv ðT ig T 0 Þ e 1þ qg w0 Dhc k 0 K ks þ ð25Þ 4 ug qs C ps eE=RT ig Assuming that the fuel consumption is almost completed in the distance of the pore size order, the temperature increase for the gas element may be treated as a thermal explosion. Thus, one can directly apply the corresponding Frank– Kamenetzki theory [16] with the correction for fuel consumption, and derive an approximation for the ignition temperature Tig in the following implicit form: ! E 1 eRT ig Dh2c w20 R5 d P e T 0 k0 ¼ ð26Þ 7=3 ug 2 p2 C 2pg E2 ðE=R T ig Þ
5. Conclusions The theoretical gas combustion study in inert porous media concluded with the analytical construction of a series of simple algebraic formulas which represent the methane mass fraction and temperature profiles in an inert porous medium. Also, two formulas were built, which allow the prediction of the combustion wave velocity in the system and the thickness of the reaction region. Six dimensionless parameters define heat and mass transport in the mathematical model. The analytical solution agrees satisfactorily with the numerical solution, built on the finite differences, of the complete mathematical model. It is shown that the reaction region is not infinitely thin, as is usually considered. Furthermore, it was found that the analytical solution was very sensitive to the value of the wave displacement speed. Consequently, the results of this research can be used in the analysis of combustion waves in porous media for technical applications. Acknowledgements This research was supported by Conicyt-Chile, Fondecyt project 1050241, and by the Academia Polite´cnica Aerona´utica-FACH, Chile.
4. Numerical solution To support and test the analytical solutions obtained here, a computational procedure was carried out using a full set of the basic equations. These were calculated using the implicit finite difference discretisation and the tree-diagonal matrix algorithm. The time step was 0.01 s and 800 grid points were used for the numerical calculations. In Fig. 2 the analytical solution obtained in this paper is compared to the numerical solution built in the same paper. Many similarities between the compared cases are seen. Apart from that, the maximum temperatures reached at the combustion front according to our solution are found within the maximum temperature values in the numerical solution.
References [1] J.R. Howell, M.J. Hall, J.L. Ellzey, Combustion of hydrocarbon fuels within porous inert media, Progress in Energy and Combustion Science, an International Review Journal 22 (1996) 121–145. [2] Y. Yoshizawa, K. Sasaki, R. Echigo, Analytical study of the structure of radiation controlled flame, International Journal of Heat and Mass Transfer 31 (2) (1998) 311–319. [3] W.M. Mathis Jr., J.L. Ellzey, Flame stabilization, operating range, and emissions for methane/air porous burner, Combustion and Science Technology 155 (2003) 825–839. [4] P.H. Bouma, L.P.H. De Goey, Premixed combustion on ceramic foam burners, Combustion and Flame 119 (1999) 133–143. [5] A.P. Aldushin, I.E. Rumanov, J.B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave, Combustion and Flame 118 (1999) 76–90.
V. Bubnovich, M. Toledo / Applied Thermal Engineering 27 (2007) 1144–1149 [6] A.M. Oliveira, M. Kaviany, Nonequilibrium in the transport of heat and reactants in combustion in porous media, Progress in Energy and Combustion Science 27 (2001) 523–545. [7] V.S. Babkin, Filtration combustion of gases. Present state of affairs and prospects, Pure and Applied Chemistry 65 (2) (1993) 335–344. [8] S.I. Foutko, S.I. Shabunya, S.A. Zhdanok, Superadiabatic combustion wave in a diluted methane–air mixture under filtration in a packed bed, in: Proceedings of Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, p. 222. [9] I.Y. Akkutlu, Y.C. Yortos, The dynamics of in-situ combustion fronts in porous media, Combustion and Flame 134 (2003) 229–247. [10] K.G. Shkadinsky, G.V. Shkadinskaya, B.J. Matkowsky, Filtration combustion in moving media: one and two reaction zone structures, Combustion and Flame 110 (1997) 441–461. [11] K.V. Dobrego, I.M. Kozlov, V.I. Bubnovich, C.E. Rosas, Dynamics of filtration combustion front perturbation in the tubular porous
[12]
[13] [14]
[15]
[16]
1149
media burner, International Journal of Heat and Mass Transfer 46 (2003) 3279–3289. V. Bubnovich, J. Vergara, C. Rosas, Theoretical analysis of combustion waves in a methane–air mixture under filtration in a packed bed, in: Proceedings of IV International School – Seminar ‘‘Modern Problems of Combustion and its Application’’, Minsk, Belarus, 2001, pp.158–162. M.R. Henneke, J.L. Ellzey, Modelling of filtration combustion in a packed bed, Combustion and Flame 117 (1999) 832–840. J.G. Hoffmann, R. Echigo, H. Yoshida, S. Tada, Experimental study on combustion in porous media with a reciprocating flow system, Combustion and Flame 111 (1997) 32–46. S.A. Zhdanok, L.A. Kennedy, G. Koester, Superadiabatic combustion of methane air mixtures under filtration in a packed bed, Combustion and Flame 100 (1/2) (1995) 221–231. D.A. Frank-Kamenetzki, Diffusia i Teploperedacha v Chimicheskoy Kinetike, Nauka, Moscow, 1987 (in Russian).