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Lean flammability limits for stable performance with a porous burner
Applied Energy 86 (2009) 2635–2643
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Lean flammability limits for stable performance with a porous burner M.H. Akbari *, P. Riahi, R. Roohi Department of Mechanical Engineering, Shiraz University, Molla-Sadra Ave., Shiraz 71348-51154, Iran
a r t i c l e
i n f o
Article history: Received 11 October 2008 Received in revised form 9 April 2009 Accepted 16 April 2009 Available online 17 May 2009 Keywords: Porous burner Lower flammability limit Flash-back Blow-out Turn–down ratio Matrix porosity
a b s t r a c t Applications of porous burners are of high interest due to many advantages such as extended lean flammability limit in comparison with free flame structures. In this work, laminar premixed flame propagation of methane/air mixture in a porous medium is numerically investigated. An unsteady onedimensional physical model of a porous burner is considered, in which the flame location is not predetermined. The computational domain is extended beyond either side of the porous medium to accurately model reactions close to the edges of the solid matrix. After validation of the model and performing a baseline simulation, a parametric study is carried out to investigate the lean flammability limits of the burner and the unstable flash-back/blow-out phenomena. Stable performance diagrams are given for two controlling parameters of turn–down ratio and porous medium porosity. The simulation results indicate that the stable performance range of the burner is extended when the equivalence ratio increases; however, the blow-out region expands with an increase in the firing rate. For constant values of porosity and firing rate, increasing the equivalence ratio can change the operating regime of the burner from blow-out to a stable condition. It is observed that by the variation of porosity in the range of 0.6–0.9, and for the equivalence ratios of more than 0.6, the flame flash-back cannot occur. An equivalence ratio of 0.43 is found to be the lower limit at which the flame stabilizes in the matrix. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Combustion in porous media has become a recent subject of interest since it enhances the efficiency of a combustion system and offers higher power densities, higher dynamic power ranges, high compactness and less emission compared to free flame combustion [1–6]. The enhanced characteristics of combustion in porous media are due to high heat recirculation through solid conduction and solid-to-solid radiation from the matrix downstream of the flame to the matrix upstream. Convective heat transfer between the solid phase and gas phase causes an effective preheating of the reactants, which results in a considerable increase in the flame speed and temperature. The flame flash-back and blow-out are two undesirable phenomena in the operation of a porous burner. On the other hand, extension of lean flammability limit is one of the main characteristics of porous burners. Hanamura and Echigo [7] developed a onedimensional mathematical model to study flame stabilization mechanism in porous surface radiant burners. Their domain covered three regions consisting of two free space regions and a porous medium in the middle. All the thermophysical properties were assumed to be constant. A single-step global chemical
* Corresponding author. Tel.: +98 917 308 8424; fax: +98 711 647 3511. E-mail address: [email protected] (M.H. Akbari). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.04.019
kinetics in which only the product species are considered was used. Three critical limits for flame stability in the vicinity of the porous surface were reported, i.e., blow-out under the condition of higher mixture velocity than the burning velocity of the flame in free space, flame extinction under extremely low mixture velocity, and flash-back into the porous medium. The flame was stabilized in the porous medium only in the first half of the layer, and was not sustained in the second half. An experimental and theoretical investigation of premixed methane/air combustion within a non-homogeneous porous ceramic was carried out by Hsu et al. [8]. Detailed chemical kinetics was utilized, with a 58-reaction mechanism involving 17 species. A non-scattering gray medium with constant absorption coefficient was used for radiation heat transfer; however, the flame location was arbitrarily specified. The burner consisted of two porous ceramic cylinders of equal length and diameter that were stacked together and insulated around the circumferences. Four series of experiments were performed to determine the lean limit using three different pore sizes in the downstream ceramic cylinder (10, 30, and 45 pores per inch (PPI)); the pore size in the upstream ceramic cylinder was maintained constant (65 PPI). The range of equivalence ratio used was between 0.41 and 0.68 in the experiments, and between 0.43 and 1.0 in the simulations. The results demonstrated that porous ceramic burners provide a range of stable burning rates at a constant equivalence ratio and that the maximum flame speed inside the burners was much higher than the
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Nomenclature A C Cp d D Ea FR h H L MW n N Nu p PPC Pr Re Ru t T u V Y x X
pre-exponential factor (m3/kg s) specific heat (kJ/kg K) constant pressure specific heat of gas mixture (kJ/kg K) diameter (m) diffusion coefficient (m2/s) activation energy (kJ/kmol) inlet firing rate (kW/m2) Enthalpy (kJ/kg) volumetric convective heat transfer coefficient (kW/m3 K) solid matrix length (m) molecular weight (kg/kmol) refractive index total number of species Nusselt number pressure (kN/m2) pore density (1/m) Prandtl number Reynolds number universal gas constant (8.315 kJ/kmol K) time (s) temperature (K) velocity (m/s) diffusion velocity (m/s) mass fraction axial distance (m) igniter length (m)
premixed, freely-burning adiabatic laminar flame speed. The lean limits in the porous burners were lower than those of the free flame. The numerical model in this study predicted with reasonable accuracy the combustion phenomena within porous ceramics. Mital et al. [9] determined the stability range of a bilayer porous medium burner-radiant heater. Their burner was composed of a diffuser layer (DL) and a flame support layer (FSL). The DL was made of cordierite with a porosity of 0.73 and 19 mm thickness. The FSL was 2-mm thick with a porosity of 0.83. They showed that the stability limit was generally between an equivalence ratio of 0.6 and 0.7. Lift off occurred first from the edges of the burner because of higher heat losses in this region which reduced the flame speed compared to the central region of the burner. With further decrease of the equivalence ratio, the entire flame lifted off and resulted in complete extinction. The lean limit was at an equivalence ratio of between 0.5 and 0.6 depending on the firing rate. There was no flashback below a firing rate of 300 kW/m2; however, above this firing rate the burner showed flash-back at equivalence ratios close to the stoichiometric. Qiu and Hayden [10] investigated oxygen-enriched combustion of natural gas in porous radiant burners. A reticulated ceramic burner and a ceramic fiber burner were used in this study. The experimental results indicated an increase in both the radiation output and the radiant efficiency with an increase in oxygen concentrations in the combustion air. The effects of oxygen enrichment on combustion mode, and flame stability on the porous media were also studied. Two modes of combustion were observed in the burners they studies: radiant mode, and convective mode. Their experiments showed that the upper limit of radiant mode combustion increased as the combustion air was enriched with oxygen. This was attributed to the increase in flame speed with oxygen concentration. Bizzi et al. [11] analyzed the stability and flash-back resistance of catalytic and non-catalytic premixed metal fiber burners using a one-dimensional methane combustion model. They discussed the sensitivity of the burner performance to some relevant design
Greek symbols a excess air fraction b extinction coefficient (1/m) / solid porosity k thermal conductivity (kW/m K) q density (kg/m3) r Boltzmann constant (5.672 1011 kW/m2 K4) x_ reaction rate (kg/m3 s) U equivalence ratio Subscripts air air cond conduction eff effective g gas phase in inlet k kth species km kth species in gas mixture l solid matrix trailing edge 0 solid matrix leading edge out outlet p pore rad radiation s solid phase surr surrounding v volumetric
and physical parameters such as porosity, emissivity, thermal conductivity and fiber diameter. The numerical simulations showed that a low thermal conductivity increases flash-back resistance, but tends to push the flame front outside the porous medium. Finally, they suggested a bi-layer solid structure to improve the flash-back resistance of the burner. Despite all the investigations that have been carried out on this subject, simultaneous study of unstable flash-back/blow-out phenomena and the flammability limits has received little attention in the literature. In this work, laminar premixed flame propagation of methane/ air mixture in a porous medium is numerically investigated. For this purpose, an unsteady one-dimensional physical model of the porous burner is considered. All the thermophysical properties of the gas mixture are taken as functions of the temperature. The flame location is not predetermined, thus the flame can submerge in the solid matrix or it can blow out or flash back from the matrix. Radiative heat transfer in the burner is modeled using a diffusion approximation. The computational domain is extended beyond either side of the porous region to accurately model reactions close to the edges of the porous region. The scope of this study is to investigate simultaneously the unstable flash-back/blow-out phenomena and the flammability limits at realistic operating conditions of a porous burner.
2. Numerical model Numerical simulation of flames in porous media has progressed to a satisfactory degree, providing guidance for future designs. It helps to understand flame stabilization mechanism inside a porous matrix, and allows prediction of the stable operating conditions of a burner. Nonetheless, models for premixed combustion within inert porous media are complicated by the highly non-linear equations and boundary conditions, which result in the stiffness of the equations.
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@Y k @Y k @ _k þ qg /u þ ðq /Y k V k Þ ¼ /x @t @x @x g
2.1. Model geometry
qg /
The model schematic and geometric dimensions are shown in Fig. 1. The computational domain is divided into three distinct regions A, B and C. The solid matrix is confined in region B, and regions A and C contain only the fluid phase.
All variables are defined in the nomenclature. Similar equations have been used in previous numerical investigations, such as [2,4,5]. Gas densities are computed from the ideal gas equation of state for a multicomponent mixture.
2.2. Model assumptions
qg ¼
p 1 Ru T g PN
Yk k¼1 MW k
In this work the following assumptions are made: (1) Transport processes in the burner are one-dimensional and heat losses from the sides of the burner are negligible in comparison to chemical energy release rate. (2) The solid matrix is isotropic, homogeneous and inert. (3) Dufour and Soret effects (mass diffusion due to pressure and temperature gradients, respectively) are neglected. (4) Gas radiation is neglected. (5) Radiative heat transfer in the solid matrix is modeled using a diffusion approximation. The effect of radiative heat flux is accounted for in the effective thermal conductivity of the porous solid. This model can be applied in optically thick porous media, and has been used by other researchers in porous burners; see for example [3,12,13]. (6) Due to high solid porosity, low gas velocity, and low matrix length, the pressure drop through the solid matrix is negligible; thus, the momentum equation is automatically satisfied. (7) The flow regime is laminar, uniform and incompressible. (8) The gas mixture is treated as an ideal gas. It is noted here that similar assumptions have been made in previous investigations of this type of flame, for example in [2,8].
To derive the conservation equations in the porous solid, spatial averaging method has been used [14]. In this approach, a macroscopic variable is defined as an appropriate mean over a sufficiently large representative elementary volume (r.e.v.); this operation yields the value of that variable at the centroid of the r.e.v. It is assumed that the result is independent of the size of the representative elementary volume. The length scale of the r.e.v. is much larger than the pore scale, but considerably smaller than the length scale of the macroscopic flow domain. The set of the governing equations, which are written bellow, includes continuity, and conservations of gas phase energy, solid phase energy and gas species.
@ðqg /Þ @ðqg /uÞ þ ¼0 @t @x @T @T @ @T g qg C p;g / g þ qg C p;g /u g /kg @t @x @x @x N X _ k Hv ðT g T s Þ ¼ / hk x k¼1
qs C s ð1 /Þ
ð1Þ
ð2Þ
@T s @ @T s ð1 /Þks;eff ¼ Hv ðT g T s Þ @t @x @x
Reactants
A
xin
B
x0 0. 02 m
C
Products
xout
xl L=0.04 m
ð3Þ
0.02 m
Fig. 1. Burner geometry and the computational domain.
x-axis
ð5Þ
The ordinary diffusion velocity of species k is calculated by the following [15]:
Vk ¼
Dkm @Y k Y k @x
ð6Þ
In the above equation Dkm is the effective binary diffusion coefficient for species k in mixture m. The separate energy equations are coupled through a volumetric convective heat transfer coefficient, Hv. The correlation for volumetric Nusselt number is given by [16] as follows:
dp Nuv ¼ 0:0426 þ 1:236 Re 2 Re 836 L
ð7Þ
The Reynolds number is based on the mean pore diameter, dp, which for the reticulated ceramics can be approximated by the following equation [17]:
dp ¼
1 100PPC
rffiffiffiffiffiffi 4/
ð8Þ
p
The volumetric heat transfer coefficient is thus calculated from the following:
Hv ¼ 2.3. Governing equations
ð4Þ
Nuv kg
ð9Þ
2
dp
2.4. Chemical kinetics The results of all previous studies suggest that the temperature distribution near the reaction zone in combustion within a porous inert medium is much wider than that in a free combustion. Therefore, it is evident that the global chemical reaction rate constants obtained under adiabatic free combustion are not applicable to nonadiabatic combustion occurring within porous media. Furthermore, it is clear that, in general, a global chemical mechanism correlated for combustion in a specific porous medium is not applicable for another one. Fu [17] derived a suitable chemical kinetics for simulation of methane combustion within reticulated ceramics using a singlestep mechanism. In his work, the rate of fuel consumption was assumed to have the form suggested by Kou [18], and five chemical parameters (i.e. pre-exponential factor, activation energy, and exponents for the temperature and fuel and oxidizer concentrations) were either taken from the work of Westbrook and Dryer [19] or adjusted to match the predicted species concentrations of Mital [20]. This chemical kinetics is used in the present study. The combustion is assumed to occur according to the following chemical reaction:
CH4 þ 2ð1 þ aÞðO2 þ 3:76N2 Þ ) CO2 þ 2H2 O þ 2aO2 þ 7:52ð1 þ aÞN2
ð10Þ
The rate of consumption of methane is given by a single-step Arrhenius rate equation as follows [17]:
kG ¼ Aq2g exp
Ea Y CH4 Y O2 Ru T g
ð11Þ
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The pre-exponential factor and global activation energy are A = 1 109 m3/kg s, Ea = 1.3 105 kJ/kmol. This simple, one-step global mechanism is in contrast to detailed mechanisms used in some other numerical investigations of the subject, such as [5,8]. However, as shown in Section 3.2, since this one-step global mechanism is adjusted for this specific type of flame, acceptable results are obtained despite the simplicity of the model. 2.5. Initial and boundary conditions A successful numerical modeling of this type of flame often depends much on proper specification of boundary conditions (see, for example [21]). The conditions specified at the inlet are the temperature, species concentrations, and velocity of the inlet mixture, while the axial derivatives of these quantities are assumed to vanish at the domain outlet. The temperature of the solid at the leading and trailing edges are calculated accounting for the radiative and convective heat transfer between the porous matrix and its surroundings. These boundary conditions are as follows:
h
i @T s hin ðT g;in T s;0 Þ þ re T 4surr T 4s;0 ð1 /Þ ¼ ks;eff @x
at x ¼ x0 ð12Þ
h
hout ðT g;out T s;l Þ þ re T 4surr T 4s;l
i
ð1 /Þ ¼ ks;eff
@T s @x
at x ¼ xl ð13Þ
The initial conditions for all dependent variables are set by their values at the initial time. Similar boundary conditions have been used in related numerical studies, such as [2,5]. It is noted here that by increasing the inlet temperature of the mixture, the limiting conditions for stable performance of the burner will be affected; more specifically, such an increase will reduce the lower firing rate for stable operation of the burner (flashback). However, in this study we limit the inlet temperature to only one particular value of 298 K. 2.6. Property data All gas mixture thermophysical properties are assumed to be functions of the temperature. The specific heat is determined from the existing curve fits given in [15], while the Sutherland’s law is used for viscosity calculation [15]. Mixture conductivity value is approximated using a constant Prandtl number (Pr = 0.7), and the binary diffusion coefficients are predicted by an approach presented in [15]. The effective thermal conductivity of the porous medium is calculated under the assumption that the medium is a continuum and its optical thickness is infinitely large. This approximation is accurate enough, since the extinction coefficient of the modeled reticulated ceramic (cordierite) is sufficiently large [22]. Therefore, the solid phase radiation has been taken into account by means of a radiation conductivity.
ks;eff ¼ ks;cond þ ks;rad
ð14Þ
The above equation states that the effective thermal conductivity is the sum of the conductivities due to conduction and radiation. Following the diffusion approximation equation, one obtains the following [22]:
ks;rad ¼
16n2 rT 3s 3bð1 /Þ
Table 1 Porous solid properties and baseline parameters data. Material
Cordierite
Porosity Pore density Specific heat, Cs Conductivity, ks,cond Density, qs Extinction coefficient, b Refractive index, n Equivalence ratio, U Inlet firing rate, FR Inlet gas temperature, Tg,in Surrounding temperature, Tsurr
0.83 26 PPC 1300 J/kg K 2.6 103 W/m K 2.5 103 kg/m3 3100 1/m 1.0 0.9 750 kW/m2 298 K 298 K
by others in order to improve the overall accuracy of the simulations. For example, radiation modeling was done in [17] using the two-flux model, and in [21] by solving the radiation transfer equation inside the porous medium without any simplifying assumptions. However, as the results presented in Section 3 suggest, the present model can provide sufficient accuracy to produce acceptable predictions. Porous solid properties and designated baseline parameters are summarized in Table 1. 2.7. Solution method The governing equations are discretized using a fully implicit finite volume method, and a segregated solution scheme is used. The upwind scheme was applied in the discretization of the convective terms, while diffusive terms were discretized using central differences. The resulting algebraic equations are solved by a point-bypoint iteration procedure which employs a tri-diagonal matrix algorithm. A relative convergence criterion for numerical computation of all variables is set to 105. A steady-state solution is achieved when the normalized residuals of all dependent variables become 1010. The grid used is generally nonuniform. Numerical simulation tests indicated that a total number of 300 grid nodes were sufficient to obtain grid-independent results. The grid is strongly clustered in region B and adjacent to the flame to model the reaction zone with desirable accuracy. 3. Results and discussion 3.1. Ignition simulation An igniter is the starter of chemical reactions in a combustion system. In this work, a heat source term in region A is used to simulate the ignition power. This term is activated only for a short time, and supplies the following power:
Ignition power ¼ 0:5
FR X
ð16Þ
In this equation FR and X indicate the firing rate and the extent of the ignition region, respectively. The ignition region is assumed to extend over 20 computational nodes. When the average temperature of the solid matrix exceeds 700 K, the igniter power is reduced to zero over a few time steps using an exponential function. However, the numerical results are independent of both the magnitude and the location of the ignition mechanism.
ð15Þ
This method for the calculation of solid-phase radiative heat transfer has been applied previously in similar investigations, for example in [3,12,13]. More complicated radiation models have been used
3.2. Model validation For the validation of the model, first a comparison with the experimental data for a two-stage burner was performed. This bur-
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ner was used by Mital [20] in his experiments, and is composed of two sections of cordierite with different thicknesses, porosities, pore densities and thermophysical properties. Detailed characteristics of this burner are given in [20]. Since all the mixture properties are dependent on temperature, the steady-state gas temperature profile is chosen to validate the present analysis. The results are shown in Fig. 2. Also shown in this figure is the corresponding profile as predicted by Fu [17]. As seen in Fig. 2, the present prediction overestimates the gas temperature profile in the flame support layer in comparison to both Mital’s experiments [20] and Fu’s numerical computations [17]. This discrepancy is attributed to the simulation of radiative heat flux by radiation conductivity approximation in the present investigation as opposed to the two-flux model used in [17]. To improve the accuracy of the simulation results, one would resort to more advanced models for the radiative heat transfer, such as that used in [21]. It is observed in this figure that the present numerical calculations are in better agreement with the experimental data upstream of the flame location, while Fu’s calculations show better agreement in close proximity to the flame. Thus, we may not claim superiority of one set of numerical data over the other. To further verify the accuracy of the current model near flash back and blow out conditions, a comparison is made between the present results at such conditions and the corresponding numerical results of [5] and experimental data of [23]. For this purpose, all the dimensions and thermophysical properties of the mixture and solid matrix are set to their corresponding values presented in [5]. The solid matrix consists of two sections with different properties. These properties are given in Table 2. For these conditions, the stable range of flame speed at different equivalence ratios are calculated by the present model and compared with those given in [5,23]. These comparisons are made in Figs. 3 and 4 for the maximum and minimum stable flame speeds, respectively. It is seen in Fig. 3 that the maximum stable flame speed (blow-out limit) is overestimated in both the present study and [5]. However, the flash back limit (minimum stable flame speed) is predicted by the current simulations more accurately than in [5] (except for the case with U = 0.8). The discrepancy between the numerical results presented in Figs. 3 and 4 is believed to be mostly due to the difference between the chemical kinetic models used in these two studies. It is noted that the GRI 1.2 chemical kinetics with 32 species and 177 reactions was used in [5],
Table 2 Porous solid properties used for model validation.
Pore density Pore diameter Porosity Conductivity Density Specific heat
Upstream
Downstream
25.6 PPC 0.00029 m 0.835 0.2 W/m K
3.9 PPC 0.00152 m 0.87 0.1 W/m K 510 kg/m3 824 J/kg K
while a single-step global mechanism is implemented in the present calculations. Considering this fact, one can observe that the overall accuracy of the present data is acceptable when compared with the previous data. This acceptable level of accuracy is due to the fact that the present kinetics is adjusted specifically for this type of flame. 3.3. Baseline simulation As stated previously, baseline parameters are summarized in Table 1. Gas and solid phase temperature profiles are the main thermal characteristics of a porous burner. Transient variations of these profiles, from t = 1 s to steady-state condition, are given in Figs. 5 and 6 for the baseline case. As seen in Fig. 5, the igniter effect disappears as the flame enters the solid matrix. The flame location is approximately fixed at about time t = 30 s; afterwards the temperature of combustion products increases until it reaches a steady value. The gas temperature profile is quite flat at the matrix exit (i.e. region C). A small fracture in the temperature profile can be seen as the combustion products leave the porous region. This is expected as a result of sudden release of gas mixture from the pores of the solid matrix; this effect was also reported in [21]. Fig. 6 shows that the solid temperature profile tends to become more uniform after the reaction zone as steady conditions are approached; the same behavior is also observed in Fig. 5 for the gas temperature. Variation of volumetric heat transfer rate and chemical heat release rate from the combustion reaction along the solid matrix are shown in Fig. 7. It is observed that the chemical energy is released just at the reaction zone, with a peak value of 7.9 105 kW/m2 at the flame location.
2200 DL
FSL
2000
Gas temperature (K)
1800 1600
Computational data [17] Experiment data [20] Present study
1400 1200 1000 800 600 400 200 0.01 0
0.01 5
0.02 0
0.02 5
0.030
x (m) Fig. 2. Comparison of the present numerical results with other studies at FR = 315 kW/m2, and equivalence ratio of 0.9.
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1.4 Present study
1.2 Numerical results [5]
Vmax (m/s)
1
Exprimental results [23]
0.8 0.6 0.4 0.2 0 0.55
0.60
0.65
0.70
0.75
0.80
Equivalence ratio Fig. 3. Comparison of the maximum stable flame speed at different equivalence ratios as predicted by the present study, [5,23].
0.8 Present study
0.7 Numerical results [5]
0.6
Vmin (m/s)
Exprimental results [23]
0.5 0.4 0.3 0.2 0.1 0 0.55
0.60
0.65
0.70
0.85
0.80
0.75
Equivalence ratio Fig. 4. Comparison of the minimum stable flame speed at different equivalence ratios as predicted by the present study, [5,23].
B
A
C
Gas temperature (K)
3100 2600 2100 steady state
1600 t=50(s)
1100
t=30(s) t=10(s) t=1(s)
600 100 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
x (m) Fig. 5. Transient variations of the gas phase temperature from t = 1 s to steady-state condition.
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2200 2000
Solid temperature (K)
1800
steady state
1600 t=50(s)
1400 1200 t=30(s)
1000 800 t=10(s)
600 400 t=1(s)
200 0.0202
0.0252
0.0302
0.0352
0.0402
0.0452
0.0502
0.0552
x (m) Fig. 6. Transient variations of the solid phase temperature from t = 1 s to steady-state condition.
The amount of volumetric heat transfer rate is a function of the solid and gas phase temperatures and the volumetric heat transfer coefficient. At the flame location the local temperature difference between the two phases is a maximum, and the local volumetric heat transfer coefficient also reaches its peak value; hence the volumetric heat transfer rate hits its maximum value at the flame location. A positive value indicates the transfer of heat from the gas mixture to the solid matrix, and vice versa. At the flame location there is a considerable jump in the value of the volumetric heat transfer rate from its minimum value (a negative quantity) to its maximum. This is due to a large gradient in the temperature profiles of both phases. At either side of the porous medium the volumetric heat transfer rate assumes a local peak due to a considerable temperature difference between the two phases.
3.4. Effect of turn–down ratio on the lower flammability limit One of the main advantages of porous burners in comparison with a free flame structure is their extended flammability limits. Several design and operating parameters influence the flammability limits of a porous burner. Inlet firing rate and matrix porosity are two of these parameters. The former is an operating parameter which determines the inlet flow rate, while the latter is a property of the burner material. The simulation results regarding stable performance of the burner under consideration are summarized in Fig. 8, where the flammability limits and turn–down ratio (range of admissible firing rates) are studied. Three distinct regions are marked on this diagram consisting of flash-back, stable and blow-out regions. The burner conditions must be set such that it
900000
60000
B 50000
Convective heat trasfer rate between two phases (kW/m^3)
40000 700000 30000 600000 20000 500000
10000 0 0.015 -10000
400000 0.025
0.035
0.045
0.055 300000
-20000 200000 -30000 100000
-40000
0
-50000
x (m) Fig. 7. Variation of volumetric heat transfer and chemical heat release rates along the solid matrix.
Chemical heat release rate (kW/m^3)
800000
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Flash-back line
Blow-out line
1 0.95 0.9 0.85
Equivalence ratio
Stable region
Flashback region
0.8 0.75 Blow-out region
0.7 0.65 0.6 0.55 0.5 0
200
400
600
800
1000
Firing rate(kW/m^2) Fig. 8. Stability diagram with respect to flammability limits and turn–down ratio.
operates in the stable region, far away from both flash-back and blow-out regions. As shown in the figure, the extent of the stable performance of the burner is increased when the equivalence ratio increases. Generally, with an increase in the equivalence ratio, flash-back and blow-out phenomena occur at higher values of the firing rate. An equivalence ratio of 0.43 is the lower limit at which the flame stabilizes in the matrix at a firing rate of 50 kW/m2. Reduction of the lower limit of flammability in the stable operating regime is one of the main advantages of porous burners that allows economizing the system with respect to fuel consumption. 3.5. Effect of matrix porosity on the lower flammability limit Porosity of the solid matrix is another parameter that controls the burner stability. The matrix porosity influences the location where the flame can form and stabilize. Reticulated ceramics are
FR=400
categorized in the group of high-porosity porous materials [24]. Practically, most of these ceramics have porosities in the range of 0.6–0.9. Shown in Fig. 9 is the stability diagram for the burner operation with respect to the lower flammability limit for different values of matrix porosity in a practical range, and for three different firing rates. The left side of each line identifies the stable region, while the blow-out region is located on the right of the line. It is observed that the blow-out region expands with an increase in the firing rate. Only conditions of the left side of each line must be considered in the burner operation. With fixed values of porosity and firing rate, increasing the equivalence ratio can change the operating regime of the burner from blow-out to stable condition. On the other hand, with fixed values of equivalence ratio and firing rate, an increase in the matrix porosity can cause the flame to blow-out, which is very undesirable. No flash-back line can be observed on this diagram. It means that by varying the porosity in the range of 0.6–0.9, and for the equivalence ratios of
FR=750
FR=1000
1 0.95
Stable region
Equivalence ratio
0.9 0.85 0.8 0.75 0.7 0.65 0.6
Blow-out region
0.55 0.5 0.6
0.65
0.7
0.75
0.8
0.85
0.9
Porosity Fig. 9. Stability diagram with respect to the lower flammability limit and the matrix porosity for three distinct values of firing rate.
M.H. Akbari et al. / Applied Energy 86 (2009) 2635–2643 Table 3 Porosity at blow-out limit for different equivalence rations and firing rates. Equivalence ratio
FR = 400 (kW/m2)
FR = 750 (kW/m2)
FR = 1000 (kW/m2)
0.6 0.7 0.8 0.9 1.0
0.67 ± 0.01 0.85 ± 0.01 / > 0.9 / > 0.9 / > 0.9
/ < 0.6 0.61 ± 0.01 0.75 ± 0.01 0.87 ± 0.01 / > 0.9
/ < 0.6 / < 0.6 0.63 ± 0.01 0.75 ± 0.01 0.85 ± 0.01
Table 4 Stability state of the flame for three firing rates and equivalence ratio of 0.5. FR
Flame situation
400 750 1000
0.6 < / < 0.78 ) extinction / = 0.79 ± 0.01 ) blow-out 0.6 < / < 0.9 ) extinction 0.6 < / < 0.9 ) extinction
more than 0.6, the flame flash-back from the matrix cannot occur. Detailed descriptions of these critical limits for each of the three firing rates are given in Table 3. 3.6. Analysis of the least value of the lower flammability limit As observed in Fig. 8, the least value of the lower flammability limit for all the situations studied is an equivalence ratio of about 0.5. In fact, the more exact value of this limit, not shown in Fig. 8, is an equivalence ratio of 0.43. On the other hand, for the baseline porosity of 0.83 and firing rate of 25 ± 25 kW/m2 the flame is either extinguished or tends to flash-back, while the firing rate of 125 ± 25 kW/m2 causes the flame to blow-out. The near-critical value of U = 0.5 is studied more. Given in Table 4 are the stability state of the flame for three firing rates with U = 0.5. It is seen that for U = 0.5 and firing rates of 400, 750 and 1000 kW/ m2, stabilization of the reaction zone into the porous matrix is impossible. In these conditions the flame extinction or blow-out cannot be avoided even for a low medium porosity of 0.6. 4. Conclusions In this paper a one-dimensional model of a porous burner was employed to numerically investigate the laminar premixed combustion of methane/air mixture using a suitable single-step chemical kinetics. The flame location was not arbitrarily fixed, and the flame was set free to submerge within the solid matrix or blow out or flash back. One of the main advantages of porous burners in comparison with a free flame structure is their extended flammability limit. Simultaneous study of the lower flammability limit and flame stability shows that the inlet firing rate and matrix porosity are two main parameters which influence the lean flammability limit of a porous burner. These quantities must be adjusted such that the burner operating regime is within the stable region. The stable range of the burner is extended with an increase in fuel-lean values of equivalence ratio. When the mixture composition approaches the stoichiometric value, the widest stable range of the flame appears. In other words, by increasing the equivalence ratio, flash-back and blow-out phenomena occur at larger values of
2643
the firing rate, but the difference in these two limiting values increases considerably. The results indicate that the matrix porosity can influence the flame location. The simulation results show that with fixed values of equivalence ratio and firing rate, an increase in the matrix porosity can lead to flame blow-out. It is also observed that by the variation of porosity in the range of 0.6–.9, and for the equivalence ratios greater than 0.6, flame flash-back cannot occur. An equivalence ratio of 0.43 is the lower flammability limit at which the flame stabilizes in the matrix at a firing rate of 50 kW/m2. It is shown that for an equivalence ratio of 0.5 and at firing rates of 400, 750 and 1000 kW/m2, stabilization of the reaction zone into the porous matrix is impossible. References [1] Abdul Mujeebu M, Abdullah MZ, Abu Bakar MZ, Mohamad AA, Abdullah MK. Applications of porous media combustion technology – a review. Appl Energy 2009;86:1365–75. [2] Barra AJ, Ellzey JL. Heat recirculation and heat transfer in porous burners. Combust Flame 2004;137:230–41. [3] Bouma PH, de Goey LPH. Premixed combustion on ceramic foam burners. Combust Flame 1999;119:133–43. [4] Brenner G, Pickenacker K, Pickenacker O, Trimis D, Wawrzinek K, Weber T. Numerical and experimental investigation of matrix-stabilized methane/air combustion in porous inert media. Combust Flame 2000;123:201–13. [5] Barra AJ, Diepvens G, Ellzey JL, Henneke MR. Numerical study of the effects of material properties on flame stabilization in a porous burner. Combust Flame 2003;134:369–79. [6] Hayashi TC, Malico I, Pereira JCF. Three-dimensional modeling of a two-layer porous burner for household applications. Comput Struct 2004;82:1543–50. [7] Hanamura K, Echigo R. An analysis of flame stabilization mechanism in radiation burners. Wärme-und Stoffübertragung 1991;26:377–83. [8] Hsu PF, Evans WD, Howell JR. Experimental and numerical study of premixed combustion within non-homogeneous porous ceramics. Combust Sci Technol 1993;90:149–72. [9] Mital R, Gore JP, Viskanta R. A study of submerged reaction zone in porous ceramic radiant burners. Combust Flame 1997;111:175–84. [10] Qiu K, Hayden ACS. Increasing the efficiency of radiant burners by using polymer membranes. Appl Energy 2009;86:349–54. [11] Bizzi M, Saracco G, Specchia V. Improving the flashback resistance of catalytic and non-catalytic metal fiber burners. Chem Eng J 2003;95:123–36. [12] Bubnovich V, Henriquez L, Gnesdilov N. Numerical study of the effect of the diameter of alumina balls on flame stabilization in a porous-medium burner. Numer Heat Transfer, Part A 2007;52:275–95. [13] Mohamad AA, Viskanta R, Ramadhyani S. Numerical prediction of combustion and heat transfer in a packed bed with embedded coolant tubes. Combust Sci Technol 1994;96:387–407. [14] Nield DA, Bejan A. Convection in porous media. 3rd ed. New York: Springer Science + Business Media; 2005. [15] Turns SR. An introduction to combustion. 2nd ed. New York: McGraw-Hill; 2000. [16] Fu X, Viskanta R, Gore JP. Measurement and correlation of volumetric heat transfer coefficients of cellular ceramics. Exp Thermal Fluid Sci 1998;17: 285–93. [17] Fu X. Modeling of a submerged flame porous burner/radiant heater. Ph.D. Diss., Purdue University, USA; 1997. [18] Kuo K. Principles of combustion. New York: John Wiley & Sons; 1986. [19] Westbrook CK, Dryer FL. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combust Sci Technol 1981;27:31–4. [20] Mital R. An experimental and a theoretical investigation of combustion and heat transfer characteristics of reticulated ceramic burner. Ph.D. Diss., Purdue University, USA; 1996. [21] Diamantis DJ, Mastorakos E, Goussis A. Simulation of premixed combustion in porous media. Combust Theor Model 2002;6:383–411. [22] Modest MF. Radiative heat transfer. 2nd ed. New Jersey: McGraw-Hill; 2003. [23] Khanna V. Experimental analysis of radiation for methane combustion within a porous medium burner. M.S. Diss., University of Texas, Austin, USA; 1992. [24] Kaviany M. Principles of heat transfer in porous media. 2nd ed. New York: Springer-Verlag; 1995.
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Lean flammability limits for stable performance with a porous burner M.H. Akbari *, P. Riahi, R. Roohi Department of Mechanical Engineering, Shiraz University, Molla-Sadra Ave., Shiraz 71348-51154, Iran
a r t i c l e
i n f o
Article history: Received 11 October 2008 Received in revised form 9 April 2009 Accepted 16 April 2009 Available online 17 May 2009 Keywords: Porous burner Lower flammability limit Flash-back Blow-out Turn–down ratio Matrix porosity
a b s t r a c t Applications of porous burners are of high interest due to many advantages such as extended lean flammability limit in comparison with free flame structures. In this work, laminar premixed flame propagation of methane/air mixture in a porous medium is numerically investigated. An unsteady onedimensional physical model of a porous burner is considered, in which the flame location is not predetermined. The computational domain is extended beyond either side of the porous medium to accurately model reactions close to the edges of the solid matrix. After validation of the model and performing a baseline simulation, a parametric study is carried out to investigate the lean flammability limits of the burner and the unstable flash-back/blow-out phenomena. Stable performance diagrams are given for two controlling parameters of turn–down ratio and porous medium porosity. The simulation results indicate that the stable performance range of the burner is extended when the equivalence ratio increases; however, the blow-out region expands with an increase in the firing rate. For constant values of porosity and firing rate, increasing the equivalence ratio can change the operating regime of the burner from blow-out to a stable condition. It is observed that by the variation of porosity in the range of 0.6–0.9, and for the equivalence ratios of more than 0.6, the flame flash-back cannot occur. An equivalence ratio of 0.43 is found to be the lower limit at which the flame stabilizes in the matrix. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Combustion in porous media has become a recent subject of interest since it enhances the efficiency of a combustion system and offers higher power densities, higher dynamic power ranges, high compactness and less emission compared to free flame combustion [1–6]. The enhanced characteristics of combustion in porous media are due to high heat recirculation through solid conduction and solid-to-solid radiation from the matrix downstream of the flame to the matrix upstream. Convective heat transfer between the solid phase and gas phase causes an effective preheating of the reactants, which results in a considerable increase in the flame speed and temperature. The flame flash-back and blow-out are two undesirable phenomena in the operation of a porous burner. On the other hand, extension of lean flammability limit is one of the main characteristics of porous burners. Hanamura and Echigo [7] developed a onedimensional mathematical model to study flame stabilization mechanism in porous surface radiant burners. Their domain covered three regions consisting of two free space regions and a porous medium in the middle. All the thermophysical properties were assumed to be constant. A single-step global chemical
* Corresponding author. Tel.: +98 917 308 8424; fax: +98 711 647 3511. E-mail address: [email protected] (M.H. Akbari). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.04.019
kinetics in which only the product species are considered was used. Three critical limits for flame stability in the vicinity of the porous surface were reported, i.e., blow-out under the condition of higher mixture velocity than the burning velocity of the flame in free space, flame extinction under extremely low mixture velocity, and flash-back into the porous medium. The flame was stabilized in the porous medium only in the first half of the layer, and was not sustained in the second half. An experimental and theoretical investigation of premixed methane/air combustion within a non-homogeneous porous ceramic was carried out by Hsu et al. [8]. Detailed chemical kinetics was utilized, with a 58-reaction mechanism involving 17 species. A non-scattering gray medium with constant absorption coefficient was used for radiation heat transfer; however, the flame location was arbitrarily specified. The burner consisted of two porous ceramic cylinders of equal length and diameter that were stacked together and insulated around the circumferences. Four series of experiments were performed to determine the lean limit using three different pore sizes in the downstream ceramic cylinder (10, 30, and 45 pores per inch (PPI)); the pore size in the upstream ceramic cylinder was maintained constant (65 PPI). The range of equivalence ratio used was between 0.41 and 0.68 in the experiments, and between 0.43 and 1.0 in the simulations. The results demonstrated that porous ceramic burners provide a range of stable burning rates at a constant equivalence ratio and that the maximum flame speed inside the burners was much higher than the
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Nomenclature A C Cp d D Ea FR h H L MW n N Nu p PPC Pr Re Ru t T u V Y x X
pre-exponential factor (m3/kg s) specific heat (kJ/kg K) constant pressure specific heat of gas mixture (kJ/kg K) diameter (m) diffusion coefficient (m2/s) activation energy (kJ/kmol) inlet firing rate (kW/m2) Enthalpy (kJ/kg) volumetric convective heat transfer coefficient (kW/m3 K) solid matrix length (m) molecular weight (kg/kmol) refractive index total number of species Nusselt number pressure (kN/m2) pore density (1/m) Prandtl number Reynolds number universal gas constant (8.315 kJ/kmol K) time (s) temperature (K) velocity (m/s) diffusion velocity (m/s) mass fraction axial distance (m) igniter length (m)
premixed, freely-burning adiabatic laminar flame speed. The lean limits in the porous burners were lower than those of the free flame. The numerical model in this study predicted with reasonable accuracy the combustion phenomena within porous ceramics. Mital et al. [9] determined the stability range of a bilayer porous medium burner-radiant heater. Their burner was composed of a diffuser layer (DL) and a flame support layer (FSL). The DL was made of cordierite with a porosity of 0.73 and 19 mm thickness. The FSL was 2-mm thick with a porosity of 0.83. They showed that the stability limit was generally between an equivalence ratio of 0.6 and 0.7. Lift off occurred first from the edges of the burner because of higher heat losses in this region which reduced the flame speed compared to the central region of the burner. With further decrease of the equivalence ratio, the entire flame lifted off and resulted in complete extinction. The lean limit was at an equivalence ratio of between 0.5 and 0.6 depending on the firing rate. There was no flashback below a firing rate of 300 kW/m2; however, above this firing rate the burner showed flash-back at equivalence ratios close to the stoichiometric. Qiu and Hayden [10] investigated oxygen-enriched combustion of natural gas in porous radiant burners. A reticulated ceramic burner and a ceramic fiber burner were used in this study. The experimental results indicated an increase in both the radiation output and the radiant efficiency with an increase in oxygen concentrations in the combustion air. The effects of oxygen enrichment on combustion mode, and flame stability on the porous media were also studied. Two modes of combustion were observed in the burners they studies: radiant mode, and convective mode. Their experiments showed that the upper limit of radiant mode combustion increased as the combustion air was enriched with oxygen. This was attributed to the increase in flame speed with oxygen concentration. Bizzi et al. [11] analyzed the stability and flash-back resistance of catalytic and non-catalytic premixed metal fiber burners using a one-dimensional methane combustion model. They discussed the sensitivity of the burner performance to some relevant design
Greek symbols a excess air fraction b extinction coefficient (1/m) / solid porosity k thermal conductivity (kW/m K) q density (kg/m3) r Boltzmann constant (5.672 1011 kW/m2 K4) x_ reaction rate (kg/m3 s) U equivalence ratio Subscripts air air cond conduction eff effective g gas phase in inlet k kth species km kth species in gas mixture l solid matrix trailing edge 0 solid matrix leading edge out outlet p pore rad radiation s solid phase surr surrounding v volumetric
and physical parameters such as porosity, emissivity, thermal conductivity and fiber diameter. The numerical simulations showed that a low thermal conductivity increases flash-back resistance, but tends to push the flame front outside the porous medium. Finally, they suggested a bi-layer solid structure to improve the flash-back resistance of the burner. Despite all the investigations that have been carried out on this subject, simultaneous study of unstable flash-back/blow-out phenomena and the flammability limits has received little attention in the literature. In this work, laminar premixed flame propagation of methane/ air mixture in a porous medium is numerically investigated. For this purpose, an unsteady one-dimensional physical model of the porous burner is considered. All the thermophysical properties of the gas mixture are taken as functions of the temperature. The flame location is not predetermined, thus the flame can submerge in the solid matrix or it can blow out or flash back from the matrix. Radiative heat transfer in the burner is modeled using a diffusion approximation. The computational domain is extended beyond either side of the porous region to accurately model reactions close to the edges of the porous region. The scope of this study is to investigate simultaneously the unstable flash-back/blow-out phenomena and the flammability limits at realistic operating conditions of a porous burner.
2. Numerical model Numerical simulation of flames in porous media has progressed to a satisfactory degree, providing guidance for future designs. It helps to understand flame stabilization mechanism inside a porous matrix, and allows prediction of the stable operating conditions of a burner. Nonetheless, models for premixed combustion within inert porous media are complicated by the highly non-linear equations and boundary conditions, which result in the stiffness of the equations.
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@Y k @Y k @ _k þ qg /u þ ðq /Y k V k Þ ¼ /x @t @x @x g
2.1. Model geometry
qg /
The model schematic and geometric dimensions are shown in Fig. 1. The computational domain is divided into three distinct regions A, B and C. The solid matrix is confined in region B, and regions A and C contain only the fluid phase.
All variables are defined in the nomenclature. Similar equations have been used in previous numerical investigations, such as [2,4,5]. Gas densities are computed from the ideal gas equation of state for a multicomponent mixture.
2.2. Model assumptions
qg ¼
p 1 Ru T g PN
Yk k¼1 MW k
In this work the following assumptions are made: (1) Transport processes in the burner are one-dimensional and heat losses from the sides of the burner are negligible in comparison to chemical energy release rate. (2) The solid matrix is isotropic, homogeneous and inert. (3) Dufour and Soret effects (mass diffusion due to pressure and temperature gradients, respectively) are neglected. (4) Gas radiation is neglected. (5) Radiative heat transfer in the solid matrix is modeled using a diffusion approximation. The effect of radiative heat flux is accounted for in the effective thermal conductivity of the porous solid. This model can be applied in optically thick porous media, and has been used by other researchers in porous burners; see for example [3,12,13]. (6) Due to high solid porosity, low gas velocity, and low matrix length, the pressure drop through the solid matrix is negligible; thus, the momentum equation is automatically satisfied. (7) The flow regime is laminar, uniform and incompressible. (8) The gas mixture is treated as an ideal gas. It is noted here that similar assumptions have been made in previous investigations of this type of flame, for example in [2,8].
To derive the conservation equations in the porous solid, spatial averaging method has been used [14]. In this approach, a macroscopic variable is defined as an appropriate mean over a sufficiently large representative elementary volume (r.e.v.); this operation yields the value of that variable at the centroid of the r.e.v. It is assumed that the result is independent of the size of the representative elementary volume. The length scale of the r.e.v. is much larger than the pore scale, but considerably smaller than the length scale of the macroscopic flow domain. The set of the governing equations, which are written bellow, includes continuity, and conservations of gas phase energy, solid phase energy and gas species.
@ðqg /Þ @ðqg /uÞ þ ¼0 @t @x @T @T @ @T g qg C p;g / g þ qg C p;g /u g /kg @t @x @x @x N X _ k Hv ðT g T s Þ ¼ / hk x k¼1
qs C s ð1 /Þ
ð1Þ
ð2Þ
@T s @ @T s ð1 /Þks;eff ¼ Hv ðT g T s Þ @t @x @x
Reactants
A
xin
B
x0 0. 02 m
C
Products
xout
xl L=0.04 m
ð3Þ
0.02 m
Fig. 1. Burner geometry and the computational domain.
x-axis
ð5Þ
The ordinary diffusion velocity of species k is calculated by the following [15]:
Vk ¼
Dkm @Y k Y k @x
ð6Þ
In the above equation Dkm is the effective binary diffusion coefficient for species k in mixture m. The separate energy equations are coupled through a volumetric convective heat transfer coefficient, Hv. The correlation for volumetric Nusselt number is given by [16] as follows:
dp Nuv ¼ 0:0426 þ 1:236 Re 2 Re 836 L
ð7Þ
The Reynolds number is based on the mean pore diameter, dp, which for the reticulated ceramics can be approximated by the following equation [17]:
dp ¼
1 100PPC
rffiffiffiffiffiffi 4/
ð8Þ
p
The volumetric heat transfer coefficient is thus calculated from the following:
Hv ¼ 2.3. Governing equations
ð4Þ
Nuv kg
ð9Þ
2
dp
2.4. Chemical kinetics The results of all previous studies suggest that the temperature distribution near the reaction zone in combustion within a porous inert medium is much wider than that in a free combustion. Therefore, it is evident that the global chemical reaction rate constants obtained under adiabatic free combustion are not applicable to nonadiabatic combustion occurring within porous media. Furthermore, it is clear that, in general, a global chemical mechanism correlated for combustion in a specific porous medium is not applicable for another one. Fu [17] derived a suitable chemical kinetics for simulation of methane combustion within reticulated ceramics using a singlestep mechanism. In his work, the rate of fuel consumption was assumed to have the form suggested by Kou [18], and five chemical parameters (i.e. pre-exponential factor, activation energy, and exponents for the temperature and fuel and oxidizer concentrations) were either taken from the work of Westbrook and Dryer [19] or adjusted to match the predicted species concentrations of Mital [20]. This chemical kinetics is used in the present study. The combustion is assumed to occur according to the following chemical reaction:
CH4 þ 2ð1 þ aÞðO2 þ 3:76N2 Þ ) CO2 þ 2H2 O þ 2aO2 þ 7:52ð1 þ aÞN2
ð10Þ
The rate of consumption of methane is given by a single-step Arrhenius rate equation as follows [17]:
kG ¼ Aq2g exp
Ea Y CH4 Y O2 Ru T g
ð11Þ
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The pre-exponential factor and global activation energy are A = 1 109 m3/kg s, Ea = 1.3 105 kJ/kmol. This simple, one-step global mechanism is in contrast to detailed mechanisms used in some other numerical investigations of the subject, such as [5,8]. However, as shown in Section 3.2, since this one-step global mechanism is adjusted for this specific type of flame, acceptable results are obtained despite the simplicity of the model. 2.5. Initial and boundary conditions A successful numerical modeling of this type of flame often depends much on proper specification of boundary conditions (see, for example [21]). The conditions specified at the inlet are the temperature, species concentrations, and velocity of the inlet mixture, while the axial derivatives of these quantities are assumed to vanish at the domain outlet. The temperature of the solid at the leading and trailing edges are calculated accounting for the radiative and convective heat transfer between the porous matrix and its surroundings. These boundary conditions are as follows:
h
i @T s hin ðT g;in T s;0 Þ þ re T 4surr T 4s;0 ð1 /Þ ¼ ks;eff @x
at x ¼ x0 ð12Þ
h
hout ðT g;out T s;l Þ þ re T 4surr T 4s;l
i
ð1 /Þ ¼ ks;eff
@T s @x
at x ¼ xl ð13Þ
The initial conditions for all dependent variables are set by their values at the initial time. Similar boundary conditions have been used in related numerical studies, such as [2,5]. It is noted here that by increasing the inlet temperature of the mixture, the limiting conditions for stable performance of the burner will be affected; more specifically, such an increase will reduce the lower firing rate for stable operation of the burner (flashback). However, in this study we limit the inlet temperature to only one particular value of 298 K. 2.6. Property data All gas mixture thermophysical properties are assumed to be functions of the temperature. The specific heat is determined from the existing curve fits given in [15], while the Sutherland’s law is used for viscosity calculation [15]. Mixture conductivity value is approximated using a constant Prandtl number (Pr = 0.7), and the binary diffusion coefficients are predicted by an approach presented in [15]. The effective thermal conductivity of the porous medium is calculated under the assumption that the medium is a continuum and its optical thickness is infinitely large. This approximation is accurate enough, since the extinction coefficient of the modeled reticulated ceramic (cordierite) is sufficiently large [22]. Therefore, the solid phase radiation has been taken into account by means of a radiation conductivity.
ks;eff ¼ ks;cond þ ks;rad
ð14Þ
The above equation states that the effective thermal conductivity is the sum of the conductivities due to conduction and radiation. Following the diffusion approximation equation, one obtains the following [22]:
ks;rad ¼
16n2 rT 3s 3bð1 /Þ
Table 1 Porous solid properties and baseline parameters data. Material
Cordierite
Porosity Pore density Specific heat, Cs Conductivity, ks,cond Density, qs Extinction coefficient, b Refractive index, n Equivalence ratio, U Inlet firing rate, FR Inlet gas temperature, Tg,in Surrounding temperature, Tsurr
0.83 26 PPC 1300 J/kg K 2.6 103 W/m K 2.5 103 kg/m3 3100 1/m 1.0 0.9 750 kW/m2 298 K 298 K
by others in order to improve the overall accuracy of the simulations. For example, radiation modeling was done in [17] using the two-flux model, and in [21] by solving the radiation transfer equation inside the porous medium without any simplifying assumptions. However, as the results presented in Section 3 suggest, the present model can provide sufficient accuracy to produce acceptable predictions. Porous solid properties and designated baseline parameters are summarized in Table 1. 2.7. Solution method The governing equations are discretized using a fully implicit finite volume method, and a segregated solution scheme is used. The upwind scheme was applied in the discretization of the convective terms, while diffusive terms were discretized using central differences. The resulting algebraic equations are solved by a point-bypoint iteration procedure which employs a tri-diagonal matrix algorithm. A relative convergence criterion for numerical computation of all variables is set to 105. A steady-state solution is achieved when the normalized residuals of all dependent variables become 1010. The grid used is generally nonuniform. Numerical simulation tests indicated that a total number of 300 grid nodes were sufficient to obtain grid-independent results. The grid is strongly clustered in region B and adjacent to the flame to model the reaction zone with desirable accuracy. 3. Results and discussion 3.1. Ignition simulation An igniter is the starter of chemical reactions in a combustion system. In this work, a heat source term in region A is used to simulate the ignition power. This term is activated only for a short time, and supplies the following power:
Ignition power ¼ 0:5
FR X
ð16Þ
In this equation FR and X indicate the firing rate and the extent of the ignition region, respectively. The ignition region is assumed to extend over 20 computational nodes. When the average temperature of the solid matrix exceeds 700 K, the igniter power is reduced to zero over a few time steps using an exponential function. However, the numerical results are independent of both the magnitude and the location of the ignition mechanism.
ð15Þ
This method for the calculation of solid-phase radiative heat transfer has been applied previously in similar investigations, for example in [3,12,13]. More complicated radiation models have been used
3.2. Model validation For the validation of the model, first a comparison with the experimental data for a two-stage burner was performed. This bur-
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ner was used by Mital [20] in his experiments, and is composed of two sections of cordierite with different thicknesses, porosities, pore densities and thermophysical properties. Detailed characteristics of this burner are given in [20]. Since all the mixture properties are dependent on temperature, the steady-state gas temperature profile is chosen to validate the present analysis. The results are shown in Fig. 2. Also shown in this figure is the corresponding profile as predicted by Fu [17]. As seen in Fig. 2, the present prediction overestimates the gas temperature profile in the flame support layer in comparison to both Mital’s experiments [20] and Fu’s numerical computations [17]. This discrepancy is attributed to the simulation of radiative heat flux by radiation conductivity approximation in the present investigation as opposed to the two-flux model used in [17]. To improve the accuracy of the simulation results, one would resort to more advanced models for the radiative heat transfer, such as that used in [21]. It is observed in this figure that the present numerical calculations are in better agreement with the experimental data upstream of the flame location, while Fu’s calculations show better agreement in close proximity to the flame. Thus, we may not claim superiority of one set of numerical data over the other. To further verify the accuracy of the current model near flash back and blow out conditions, a comparison is made between the present results at such conditions and the corresponding numerical results of [5] and experimental data of [23]. For this purpose, all the dimensions and thermophysical properties of the mixture and solid matrix are set to their corresponding values presented in [5]. The solid matrix consists of two sections with different properties. These properties are given in Table 2. For these conditions, the stable range of flame speed at different equivalence ratios are calculated by the present model and compared with those given in [5,23]. These comparisons are made in Figs. 3 and 4 for the maximum and minimum stable flame speeds, respectively. It is seen in Fig. 3 that the maximum stable flame speed (blow-out limit) is overestimated in both the present study and [5]. However, the flash back limit (minimum stable flame speed) is predicted by the current simulations more accurately than in [5] (except for the case with U = 0.8). The discrepancy between the numerical results presented in Figs. 3 and 4 is believed to be mostly due to the difference between the chemical kinetic models used in these two studies. It is noted that the GRI 1.2 chemical kinetics with 32 species and 177 reactions was used in [5],
Table 2 Porous solid properties used for model validation.
Pore density Pore diameter Porosity Conductivity Density Specific heat
Upstream
Downstream
25.6 PPC 0.00029 m 0.835 0.2 W/m K
3.9 PPC 0.00152 m 0.87 0.1 W/m K 510 kg/m3 824 J/kg K
while a single-step global mechanism is implemented in the present calculations. Considering this fact, one can observe that the overall accuracy of the present data is acceptable when compared with the previous data. This acceptable level of accuracy is due to the fact that the present kinetics is adjusted specifically for this type of flame. 3.3. Baseline simulation As stated previously, baseline parameters are summarized in Table 1. Gas and solid phase temperature profiles are the main thermal characteristics of a porous burner. Transient variations of these profiles, from t = 1 s to steady-state condition, are given in Figs. 5 and 6 for the baseline case. As seen in Fig. 5, the igniter effect disappears as the flame enters the solid matrix. The flame location is approximately fixed at about time t = 30 s; afterwards the temperature of combustion products increases until it reaches a steady value. The gas temperature profile is quite flat at the matrix exit (i.e. region C). A small fracture in the temperature profile can be seen as the combustion products leave the porous region. This is expected as a result of sudden release of gas mixture from the pores of the solid matrix; this effect was also reported in [21]. Fig. 6 shows that the solid temperature profile tends to become more uniform after the reaction zone as steady conditions are approached; the same behavior is also observed in Fig. 5 for the gas temperature. Variation of volumetric heat transfer rate and chemical heat release rate from the combustion reaction along the solid matrix are shown in Fig. 7. It is observed that the chemical energy is released just at the reaction zone, with a peak value of 7.9 105 kW/m2 at the flame location.
2200 DL
FSL
2000
Gas temperature (K)
1800 1600
Computational data [17] Experiment data [20] Present study
1400 1200 1000 800 600 400 200 0.01 0
0.01 5
0.02 0
0.02 5
0.030
x (m) Fig. 2. Comparison of the present numerical results with other studies at FR = 315 kW/m2, and equivalence ratio of 0.9.
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1.4 Present study
1.2 Numerical results [5]
Vmax (m/s)
1
Exprimental results [23]
0.8 0.6 0.4 0.2 0 0.55
0.60
0.65
0.70
0.75
0.80
Equivalence ratio Fig. 3. Comparison of the maximum stable flame speed at different equivalence ratios as predicted by the present study, [5,23].
0.8 Present study
0.7 Numerical results [5]
0.6
Vmin (m/s)
Exprimental results [23]
0.5 0.4 0.3 0.2 0.1 0 0.55
0.60
0.65
0.70
0.85
0.80
0.75
Equivalence ratio Fig. 4. Comparison of the minimum stable flame speed at different equivalence ratios as predicted by the present study, [5,23].
B
A
C
Gas temperature (K)
3100 2600 2100 steady state
1600 t=50(s)
1100
t=30(s) t=10(s) t=1(s)
600 100 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
x (m) Fig. 5. Transient variations of the gas phase temperature from t = 1 s to steady-state condition.
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2200 2000
Solid temperature (K)
1800
steady state
1600 t=50(s)
1400 1200 t=30(s)
1000 800 t=10(s)
600 400 t=1(s)
200 0.0202
0.0252
0.0302
0.0352
0.0402
0.0452
0.0502
0.0552
x (m) Fig. 6. Transient variations of the solid phase temperature from t = 1 s to steady-state condition.
The amount of volumetric heat transfer rate is a function of the solid and gas phase temperatures and the volumetric heat transfer coefficient. At the flame location the local temperature difference between the two phases is a maximum, and the local volumetric heat transfer coefficient also reaches its peak value; hence the volumetric heat transfer rate hits its maximum value at the flame location. A positive value indicates the transfer of heat from the gas mixture to the solid matrix, and vice versa. At the flame location there is a considerable jump in the value of the volumetric heat transfer rate from its minimum value (a negative quantity) to its maximum. This is due to a large gradient in the temperature profiles of both phases. At either side of the porous medium the volumetric heat transfer rate assumes a local peak due to a considerable temperature difference between the two phases.
3.4. Effect of turn–down ratio on the lower flammability limit One of the main advantages of porous burners in comparison with a free flame structure is their extended flammability limits. Several design and operating parameters influence the flammability limits of a porous burner. Inlet firing rate and matrix porosity are two of these parameters. The former is an operating parameter which determines the inlet flow rate, while the latter is a property of the burner material. The simulation results regarding stable performance of the burner under consideration are summarized in Fig. 8, where the flammability limits and turn–down ratio (range of admissible firing rates) are studied. Three distinct regions are marked on this diagram consisting of flash-back, stable and blow-out regions. The burner conditions must be set such that it
900000
60000
B 50000
Convective heat trasfer rate between two phases (kW/m^3)
40000 700000 30000 600000 20000 500000
10000 0 0.015 -10000
400000 0.025
0.035
0.045
0.055 300000
-20000 200000 -30000 100000
-40000
0
-50000
x (m) Fig. 7. Variation of volumetric heat transfer and chemical heat release rates along the solid matrix.
Chemical heat release rate (kW/m^3)
800000
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Flash-back line
Blow-out line
1 0.95 0.9 0.85
Equivalence ratio
Stable region
Flashback region
0.8 0.75 Blow-out region
0.7 0.65 0.6 0.55 0.5 0
200
400
600
800
1000
Firing rate(kW/m^2) Fig. 8. Stability diagram with respect to flammability limits and turn–down ratio.
operates in the stable region, far away from both flash-back and blow-out regions. As shown in the figure, the extent of the stable performance of the burner is increased when the equivalence ratio increases. Generally, with an increase in the equivalence ratio, flash-back and blow-out phenomena occur at higher values of the firing rate. An equivalence ratio of 0.43 is the lower limit at which the flame stabilizes in the matrix at a firing rate of 50 kW/m2. Reduction of the lower limit of flammability in the stable operating regime is one of the main advantages of porous burners that allows economizing the system with respect to fuel consumption. 3.5. Effect of matrix porosity on the lower flammability limit Porosity of the solid matrix is another parameter that controls the burner stability. The matrix porosity influences the location where the flame can form and stabilize. Reticulated ceramics are
FR=400
categorized in the group of high-porosity porous materials [24]. Practically, most of these ceramics have porosities in the range of 0.6–0.9. Shown in Fig. 9 is the stability diagram for the burner operation with respect to the lower flammability limit for different values of matrix porosity in a practical range, and for three different firing rates. The left side of each line identifies the stable region, while the blow-out region is located on the right of the line. It is observed that the blow-out region expands with an increase in the firing rate. Only conditions of the left side of each line must be considered in the burner operation. With fixed values of porosity and firing rate, increasing the equivalence ratio can change the operating regime of the burner from blow-out to stable condition. On the other hand, with fixed values of equivalence ratio and firing rate, an increase in the matrix porosity can cause the flame to blow-out, which is very undesirable. No flash-back line can be observed on this diagram. It means that by varying the porosity in the range of 0.6–0.9, and for the equivalence ratios of
FR=750
FR=1000
1 0.95
Stable region
Equivalence ratio
0.9 0.85 0.8 0.75 0.7 0.65 0.6
Blow-out region
0.55 0.5 0.6
0.65
0.7
0.75
0.8
0.85
0.9
Porosity Fig. 9. Stability diagram with respect to the lower flammability limit and the matrix porosity for three distinct values of firing rate.
M.H. Akbari et al. / Applied Energy 86 (2009) 2635–2643 Table 3 Porosity at blow-out limit for different equivalence rations and firing rates. Equivalence ratio
FR = 400 (kW/m2)
FR = 750 (kW/m2)
FR = 1000 (kW/m2)
0.6 0.7 0.8 0.9 1.0
0.67 ± 0.01 0.85 ± 0.01 / > 0.9 / > 0.9 / > 0.9
/ < 0.6 0.61 ± 0.01 0.75 ± 0.01 0.87 ± 0.01 / > 0.9
/ < 0.6 / < 0.6 0.63 ± 0.01 0.75 ± 0.01 0.85 ± 0.01
Table 4 Stability state of the flame for three firing rates and equivalence ratio of 0.5. FR
Flame situation
400 750 1000
0.6 < / < 0.78 ) extinction / = 0.79 ± 0.01 ) blow-out 0.6 < / < 0.9 ) extinction 0.6 < / < 0.9 ) extinction
more than 0.6, the flame flash-back from the matrix cannot occur. Detailed descriptions of these critical limits for each of the three firing rates are given in Table 3. 3.6. Analysis of the least value of the lower flammability limit As observed in Fig. 8, the least value of the lower flammability limit for all the situations studied is an equivalence ratio of about 0.5. In fact, the more exact value of this limit, not shown in Fig. 8, is an equivalence ratio of 0.43. On the other hand, for the baseline porosity of 0.83 and firing rate of 25 ± 25 kW/m2 the flame is either extinguished or tends to flash-back, while the firing rate of 125 ± 25 kW/m2 causes the flame to blow-out. The near-critical value of U = 0.5 is studied more. Given in Table 4 are the stability state of the flame for three firing rates with U = 0.5. It is seen that for U = 0.5 and firing rates of 400, 750 and 1000 kW/ m2, stabilization of the reaction zone into the porous matrix is impossible. In these conditions the flame extinction or blow-out cannot be avoided even for a low medium porosity of 0.6. 4. Conclusions In this paper a one-dimensional model of a porous burner was employed to numerically investigate the laminar premixed combustion of methane/air mixture using a suitable single-step chemical kinetics. The flame location was not arbitrarily fixed, and the flame was set free to submerge within the solid matrix or blow out or flash back. One of the main advantages of porous burners in comparison with a free flame structure is their extended flammability limit. Simultaneous study of the lower flammability limit and flame stability shows that the inlet firing rate and matrix porosity are two main parameters which influence the lean flammability limit of a porous burner. These quantities must be adjusted such that the burner operating regime is within the stable region. The stable range of the burner is extended with an increase in fuel-lean values of equivalence ratio. When the mixture composition approaches the stoichiometric value, the widest stable range of the flame appears. In other words, by increasing the equivalence ratio, flash-back and blow-out phenomena occur at larger values of
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the firing rate, but the difference in these two limiting values increases considerably. The results indicate that the matrix porosity can influence the flame location. The simulation results show that with fixed values of equivalence ratio and firing rate, an increase in the matrix porosity can lead to flame blow-out. It is also observed that by the variation of porosity in the range of 0.6–.9, and for the equivalence ratios greater than 0.6, flame flash-back cannot occur. An equivalence ratio of 0.43 is the lower flammability limit at which the flame stabilizes in the matrix at a firing rate of 50 kW/m2. It is shown that for an equivalence ratio of 0.5 and at firing rates of 400, 750 and 1000 kW/m2, stabilization of the reaction zone into the porous matrix is impossible. References [1] Abdul Mujeebu M, Abdullah MZ, Abu Bakar MZ, Mohamad AA, Abdullah MK. Applications of porous media combustion technology – a review. Appl Energy 2009;86:1365–75. [2] Barra AJ, Ellzey JL. Heat recirculation and heat transfer in porous burners. Combust Flame 2004;137:230–41. [3] Bouma PH, de Goey LPH. Premixed combustion on ceramic foam burners. Combust Flame 1999;119:133–43. [4] Brenner G, Pickenacker K, Pickenacker O, Trimis D, Wawrzinek K, Weber T. Numerical and experimental investigation of matrix-stabilized methane/air combustion in porous inert media. Combust Flame 2000;123:201–13. [5] Barra AJ, Diepvens G, Ellzey JL, Henneke MR. Numerical study of the effects of material properties on flame stabilization in a porous burner. Combust Flame 2003;134:369–79. [6] Hayashi TC, Malico I, Pereira JCF. Three-dimensional modeling of a two-layer porous burner for household applications. Comput Struct 2004;82:1543–50. [7] Hanamura K, Echigo R. An analysis of flame stabilization mechanism in radiation burners. Wärme-und Stoffübertragung 1991;26:377–83. [8] Hsu PF, Evans WD, Howell JR. Experimental and numerical study of premixed combustion within non-homogeneous porous ceramics. Combust Sci Technol 1993;90:149–72. [9] Mital R, Gore JP, Viskanta R. A study of submerged reaction zone in porous ceramic radiant burners. Combust Flame 1997;111:175–84. [10] Qiu K, Hayden ACS. Increasing the efficiency of radiant burners by using polymer membranes. Appl Energy 2009;86:349–54. [11] Bizzi M, Saracco G, Specchia V. Improving the flashback resistance of catalytic and non-catalytic metal fiber burners. Chem Eng J 2003;95:123–36. [12] Bubnovich V, Henriquez L, Gnesdilov N. Numerical study of the effect of the diameter of alumina balls on flame stabilization in a porous-medium burner. Numer Heat Transfer, Part A 2007;52:275–95. [13] Mohamad AA, Viskanta R, Ramadhyani S. Numerical prediction of combustion and heat transfer in a packed bed with embedded coolant tubes. Combust Sci Technol 1994;96:387–407. [14] Nield DA, Bejan A. Convection in porous media. 3rd ed. New York: Springer Science + Business Media; 2005. [15] Turns SR. An introduction to combustion. 2nd ed. New York: McGraw-Hill; 2000. [16] Fu X, Viskanta R, Gore JP. Measurement and correlation of volumetric heat transfer coefficients of cellular ceramics. Exp Thermal Fluid Sci 1998;17: 285–93. [17] Fu X. Modeling of a submerged flame porous burner/radiant heater. Ph.D. Diss., Purdue University, USA; 1997. [18] Kuo K. Principles of combustion. New York: John Wiley & Sons; 1986. [19] Westbrook CK, Dryer FL. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combust Sci Technol 1981;27:31–4. [20] Mital R. An experimental and a theoretical investigation of combustion and heat transfer characteristics of reticulated ceramic burner. Ph.D. Diss., Purdue University, USA; 1996. [21] Diamantis DJ, Mastorakos E, Goussis A. Simulation of premixed combustion in porous media. Combust Theor Model 2002;6:383–411. [22] Modest MF. Radiative heat transfer. 2nd ed. New Jersey: McGraw-Hill; 2003. [23] Khanna V. Experimental analysis of radiation for methane combustion within a porous medium burner. M.S. Diss., University of Texas, Austin, USA; 1992. [24] Kaviany M. Principles of heat transfer in porous media. 2nd ed. New York: Springer-Verlag; 1995.