Thermal Radiation in Combustion Systems

 

José Bezerra Pessoa-Filho
Divisão de Sistemas Espaciais - ASE
Centro Técnico Aeroespacial – CTA
Instituto de Aeronáutica e Espaço – IAE
12228-904 São José dos Campos, SP Brazil
jbp@iae.cta.br

 

 

Abstract

A numerical procedure for solving the nongray radiative transfer equation (RTE) in two-dimensional cylindrical participating media is presented. Nongray effects are treated by using a narrow-band approach. Radiative emission from CO, CO₂, H₂O, CH₄ and soot is considered. The solution procedure is applied to study radiative heat transfer in a premixed CH₄-O₂, laminar, flame. Temperature, soot and IR-active species molar fraction distributions are allowed to vary in the axial direction of the flame. From the obtained results it is possible to quantify the radiative loss in the flame, as well as the importance of soot radiation as compared to gaseous radiation. Since the solution procedure is developed for a two-dimensional cylindrical geometry, it can be applied to other combustion systems such as furnaces, internal combustion engines, liquid and solid propellant combustion.

Keywords: Radiation, Participating Media, Radiative Transfer Equation, Nongray Gases, Combustion

 

 

Introduction

Due to the high temperatures involved in combustion processes, e.g. 2000 ^oC, radiation heat transfer appears as an important heat transfer mechanism in many combustion devices including, among others, industrial furnaces, internal combustion engines, combustion of liquid and solid propellants, small and large scale flames and exhaust plumes of solid and liquid rockets. Infrared (IR)-active species such as CO₂, CO and H₂O are often present in the products of combustion of hydrocarbon fuels being responsible for the non-luminous radiation. In fuel-rich flames, soot is also formed and, as a consequence, a significant amount of radiation is emitted in the luminous region of the spectrum.

Despite the importance of radiation heat transfer in combustion systems, the inclusion of radiative heat transfer into the analysis of combustion processes is very difficult. Among the difficulties to accomplish such a goal, the following should be mentioned: (i) the solution to the radiative transfer equation is difficult to obtain; (ii) the optical properties of the medium (absorption/emission coefficients) are strongly dependent on wavelength (nongray effects); (iii) the optical properties of the combustion gases are also function of temperature and partial pressures; (iv) a solution to the equation of radiative transfer in multidimensional geometries is often required and it is a formidable mathematical task; (v) the solution of the radiative transfer equation requires the knowledge of the temperature distribution, as well as the concentration of the different species present in the medium. Since the temperature and species concentration distribution come from application of conservation of mass and energy principles, the solution procedure is iterative and computationally intensive.

This work presents part of an investigation that has been conducted with the objective of incorporating radiative heat transfer into an existing computer code of laminar premixed flames (Kee et al., 1985). Flat flame burners, Fig. 1a, have been extensively used over the years to investigate the combustion behavior of premixed gases (Xu et al., 1997). Typically, such flames have a cylindrical shape with 4.0 cm diameter and 3.0 cm length. With the advent of high-speed computers, a significant effort has been devoted towards modeling of the combustion phenomena in premixed laminar flames (Kee et al., 1985). Despite their small size, it has been shown that such flames may experience significant heat losses (Xu et al., 1997). For example, D’Alessio et al. (1973) a temperature drop of 225 ^oC along 1.0 cm of an atmospheric, 2.54 equivalence ratio, CH₄-O₂ flame, Fig. 1.b. In Fig. 1b, the dashed line represents the hypothetical temperature distribution if the flame were adiabatic.

 

 

The main objective of this work is to present a method of solution for solving the nongray radiative transfer equation in two-dimensional cylindrical participating media. Nongray effects are treated by using a narrow-band model. Temperature and species concentration profiles are allowed to vary along the axial direction of the flame. Based on the obtained results it is possible to evaluate the radiative losses in such flames, as well as to verify the relative importance of soot radiation as compared to gaseous radiation.

 

Formulation of the Problem

The radiative transfer equation (RTE) is the equation which describes the conservation of radiative energy within a participating medium. For a nonscattering medium the RTE is given by (Modest, 1993)

, (1)

where I is the intensity of radiation, s is the path length, q and f are the polar and azimuthal angles, respectively, as schematically represented in Fig. 2. K_h is the absorption (emission) coefficient of the medium, I_bh is the Planck function and h is the wavenumber. For a two-dimensional cylindrical geometry, the boundary conditions of Eq. (1) are

 

(2a)

(2b)

(2c)

where T_b=T_sur=298K. Equations (2a), (2b) and (2c) refer to the boundary conditions at the bottom, peripheral and top wall of the cylinder. R and L are the cylinder’s radius and length, respectively. The boundary condition expressed by Eq. (2a) comes from the assumption that the surface of the burner is opaque with unit emissivity, due to roughness and oxidation. Equations (2b) and (2c) result from the assumption that the periphery of the flame is transparent. It should also be mentioned that the assumption of a nonscattering medium is based on the fact that soot particles are small (usually smaller than 0.1 m m; Modest, 1993). In such cases, the size parameter (x=p Dh ), where D is the soot particle diameter, becomes small leading to a negligible scattering efficiency factor (Modest, 1993). Solution of Eq (1) is required in order to obtain the radiative heat transfer terms to be used in the overall energy conservation equation. For example, the radiative heat fluxes along the radial and axial directions of the flame are given by

 

(3)

(4.a, b)

(5a)

(5b)

where

(6)

The subscripts ê_r and ê_z represent the unit vector in the r- and z-directions. The superscripts "+" and "-" represent the positive and negative direction of propagation of the radiative intensity along the z-axis, i.e., 0<q<p/2 and p/2<q<p, respectively. The superscript " represents per unit area.

 

 

Due to the presence of IR-active species in the products of combustion of hydrocarbon fuels, the absorption coefficient, k h (s) of the medium exhibits a strong variation with wavenumber (nongray effects). In other words, the medium absorbs (and emits) significantly in certain regions of the spectrum and is transparent in others. Such a behavior leads to a strong variation of the intensity of radiation within de medium making integration of Eq. (6) quite difficult. Moreover, it requires the RTE to be solved several times. To avoid such difficulties, the gray gas approximation is often used. It consists of establishing a "mean" absorption coefficient in such a way to solve the RTE only once. However, in problems involving gaseous radiation such an assumption may lead to large errors (Viskanta and Mengüç, 1987). Therefore, an adequate treatment of nongray effects is required. There are several ways in which nongray effects can be treated. The two most important are: the narrow-band model and the wide-band model (Edwards, 1976). Considering that the main objective of this investigation is to assess the importance of radiative heat transfer in premixed laminar flames we choose to use the more accurate, and still computationally feasible, narrow-band model to treat nongray effects. In doing so, we use the procedure developed by Grosshandler (1980).

Integration of Eq. (1) along a homogeneous path of length Dt=s-s_o, yields

, (7, 8)

where t_h(s) is the so called spectral transmissivity. The narrow-band model is now introduced by dividing the radiative spectrum (50 cm^-1 < h < 9300 cm^-1) into NB small intervals, D h . The objective of the narrow-band approximation is to establish a spectrally averaged transmissivity, t_h(s) for every mth considered narrow band. Without loss of generality, D h is assumed equal for all NB narrow bands. Integration of Eq. (7) over a narrow band centered around h _m, with band-width D h , yields (Modest, 1993)

(9)

where

(10, 11)

The value of to be used is the one for the medium, i.e., for the mixture of gases and soot which comprise the medium. Let us assume a mixture of CO, CO₂, H₂O, CH₄, O₂, N₂ and soot at uniform temperature, pressure and chemical composition. Since O₂ and N₂ are IR-active only at very high pressures only the transmissivities of CO, CO₂, H₂O, CH₄ and soot need to be considered. In such a situation, the transmissivity of the mixture of gases can be approximated as (Modest, 1993)

(12)

Therefore, before proceeding with the solution of the RTE for the mth narrow-band, it is necessary to evaluate the average spectral transmissivity of each IR-active molecule. Equations (7) and (8) were obtained based on a uniform temperature, pressure and chemical composition along the path length, D s. This is clearly not the case involving radiative heat transfer in combustion systems, in which temperature and chemical composition vary along the path length. In this case, the mostly common used approximation for radiative transfer calculation is the Curtis-Godson approximation (Ludwig et al., 1973). According to the Curtis-Godson approximation, the average spectral transmissivity of a nonuniform gas is replaced by the transmissivity of an equivalent layer of uniform gas

(13)

where the optical depth, X, is dependent on the considered broadening mechanism of the spectral lines. The details about the calculation of X can be found in Ludwig et al. (1973).

 

Application

In this section we intend to illustrate how the nongray concepts developed in the previous section can be applied to solve a problem of practical interest. The two-dimensional, cylindrical, participating medium represents the mixture of gases and soot in a premixed laminar flame. We should start by recalling that the intensity of radiation is a function of five independent variables

(14)

We now consider the evaluation of I_ijklm, Fig. 2. Without loss of generality, we assume that I_ijklm emanates from the bottom wall of the cylinder. To obtain I_ijklm, we integrate the RTE along a path length s’, between s_o, from where the intensity emanates, to s, defined by (r_i,z_j,q _k,f _l), as shown in Fig. 2. Thus, Eq. (1) becomes

(15, 16)

Since the wall at s=s₀ is assumed black at uniform temperature, the boundary condition for the spectral intensity, I_ijklm, is given by the Planck function (Modest, 1993).

To account for the nonuniform temperature and chemical composition distribution along the path length between s_o and s, the path length is divided into ND "homogeneous" elements, Fig. 2, such that each Dd_n element, n=1,2,...,ND, is characterized by an average value of temperature (T_n), soot volume fraction (f_n), and species partial pressures (molar fractions) , H₂O, CH₄, N₂, O₂). For the configuration shown in Fig. 2, ND=j and

(17a, b, c, d)

Therefore, integration of Eq. (15) is divided into a series of integrals, as follows

(18)

where s_ND=s-s_o. Considering that over each Dd_n the temperature is assumed uniform and equal to , we have

(19)

Equation (19) gives the intensity of radiation at a (r_i,z_j) location, (q _k,f _l) direction, and wavenumber h _m. Nonetheless, in heat transfer applications, we are interested in total quantities, i.e., quantities integrated over the entire spectrum. Thus, we have

(20)

The integral in the above equation can be evaluated by applying the narrow band concept described in the previous section. So, the spectrum is divided into NB narrow bands and Eq. (20) becomes

(21)

 

where I_ijklm is given by Eq. (19). It should be recalled that (s_n) represents the transmissivity of the mixture of gases, being given by Eq. (12).

In order to solve the RTE for each wavenumber h_m the method of solution developed by Pessoa-Filho and Thynell (1996a) was used. The technique developed by Pessoa-Filho and Thynell (1996a) is based on the isolation of the discontinuities of the intensity of radiation, prior to the selection of the discrete-ordinates set. The evaluation of Eqs. (19) and (20) was performed by using the computer code RADCAL (Grosshandler, 1980). Besides giving accurate results, RADCAL is structured in such a way that makes its implementation relatively simple within the context of the present work. To account for nongray effects, RADCAL divides the radiative spectrum, 50 cm^-1 < h < 9300 cm^-1, into 432 narrow bands with narrow band widths varying between 5 cm^-1 and 50 cm^-1. H₂O is considered IR-active in all 432 narrow bands, whereas CO, CO₂ and CH₄ are IR-active in, respectively, 32, 254 and 108 of these 432 bands. Since soot emits radiation continuously, its contribution is considered over each one of the 432 narrow bands. To illustrate the effects of such a narrow band distribution on the computational effort, let us consider the situation shown in Fig. 2, in which the path length is divided into ND "homogenous" elements. According to Eq. (19) the evaluation of is required at each one of the ND "homogenous" elements. However, from Eq. (12), results from the contribution of each IR-active species and soot. Therefore, at each one of the ND "homogenous" elements, 1258 evaluations of , is=CO, CO₂, H₂O, CH₄, soot, are necessary. As a consequence, 1258´ ND evaluations of are required for each path length considered. By using the method of solution developed by Pessoa-Filho and Thynell (1996a) and considering that the intensity of radiation within the medium may vary significantly, due to the nonuniform temperature and chemical composition distributions, the RTE is typically solved for 48 directions (path lengths), at each (r_i, z_j) location, requiring, as a consequence, a significant computational effort.

 

Results

The calculation procedure used by RADCAL, i.e. Eqs. (19) and (20), has already been validated by Grosshandler (1980). However, before applying the procedure described in this work to problems of practical interest, it is necessary to validate it by comparing its results against others available in the literature. For example, if the ratio L/R is made sufficiently large, and the two-dimensional solution proposed in this work is accurate, the mean beam length concept can be used and L is replaced by L_mbl (L_mbl= 1.9 R; Edwards, 1976). Thus, we consider a nongray medium formed by a mixture of H₂O-N₂ at uniform temperature and chemical composition. The pressure is uniform and equal to 1.0 atm. To obtain an aspect ratio of at least 10, the cylinder height is made equal to 10 m and R is varied in such a way to obtain p^H₂O X L_mbl varying between 0.01 atm m and 0.2 atm m. In solving the RTE we consider the walls black and at zero temperature, i.e. emission comes only from the medium itself. The r- and z-directions are discretized by using five and eleven nodes, respectively. Since temperature and chemical composition are uniform within the medium, the RTE is solved in 32 directions at each (r_i, z_j) location. Table 1 shows the results obtained for the total hemispherical emissivity at the mid-plane of the cylinder (z=L/2), namely

(22)

where s is the Stefan-Boltzmann constant. Results were obtained for two temperatures: 1000 K and 2000 K. For comparison purposes results obtained by different investigators are also presented in Table 1. An excellent agreement is observed between the results obtained in this work and those presented in the literature. For T=2000 K the agreement with Hottel’s charts is not very good because Hottel’s values were obtained from extrapolation of measurements at lower temperatures.

 

 

To illustrate the application of the present method of solution to a situation of practical interest, we consider the modeling of radiation heat transfer in flat flame burners, Fig. 1a. There is reason to believe that the temperature decrease observed in some experimental measurements, e.g. Fig. 1b., is caused by radiative heat losses across the peripheral wall of the flame. The introduction of radiation heat transfer into the analysis of the combustion problem leads to the addition of the following terms into the overall energy conservation equation: (Pessoa-Filho and Thynell, 1996b). These terms represent, respectively, radiative losses across the peripheral wall of the flame and the transport of radiation along the flame axis. q_êr(R,z) and q_êr(z) are given by Eqs. (3) and (4), respectively.

Figure 3 shows temperature, IR-active species molar fraction and soot volume fraction distributions along an atmospheric, 2.54 equivalence ratio, premixed CH₄-O₂, laminar, flame. The IR-active species and soot volume fraction profiles were obtained experimentally by D’Alessio et al. (1973), whereas the temperature profile was obtained from calculation (Pessoa-Filho and Thynell, 1996b). It is worth mentioning that about 56% of the products of combustion of this flame is comprised of IR-active species. For solving the RTE at each wavenumber, h_m the method of solution developed by Pessoa-Filho and Thynell (1996a) is used. The r- and z-directions were discretized by using nine and thirty-five nodes, respectively. At each (r_i,z_j) location, the RTE was solved along 48 directions. The CPU time required to obtain the results shown in Figs. 4 and 5 was about 5 hours (HP900-700).

 

 

Figure 4 shows the net radiative heat transfer across the peripheral wall of the flame. The temperature of the burner (T_b=298 K), as well as the lower temperature and concentration of the IR-active species in the reaction zone, are responsible for the observed decrease of the peripheral heat flux as z®0 The maximum peripheral heat flux occurs close to the location where soot volume fraction reaches its maximum value (z » 1.2 cm). As z®L there is a decrease in the flame temperature. This fact, together with the presence of the "imaginary" cold wall at z=L, leads to a decrease of the peripheral heat flux as z®L

Figure 5 shows the variation of q_êr(z) along the axial direction of the flame. It is important to recall that q_êr(z) represents the net axial transport of radiative energy at a given z-location, Eq. (4a). A positive value of q_êr(z) indicates a net transport of radiative energy in the positive z-direction. In a similar fashion, a negative value of q_êr(z) indicates a net transport of radiative energy in the negative direction. To help in the understanding of q_êr(z), are also shown in Fig. 5. As we move away from the burner surface there is an increase in the amount of energy propagating along the positive z-direction, As we get close to z=L, there is a slight decrease in Such a behavior is caused by the decrease in the flame’s temperature. The same physical mechanisms as above are responsible for the behavior.

 

 

 

The total heat power lost, through radiative emission, in this flame amounts to 120 W. This value was obtained by adding q_êr(0) and q_êr(L) to the integrated radiative loss across the peripheral wall of the flame. This value is quite large if we consider the small dimensions of the flame (4.0 cm diameter and 3.0 cm length) and its low mass flow rate (about 0.1 g/s). If radiation heat transfer is introduced into the overall energy conservation equation of CHEMKIN (Kee et al., 1985) the resulting temperature profile exhibits a significant decay along the flame, shown by a continuous line in Fig. 1.b. From these results it seems quite reasonable to assume that most of the temperature decay observed by D'Alessio et al. (1973) in their experimental study is due to radiative losses across the flame's peripheral wall.

Based upon the calculation of the radiative terms appearing in the overall energy conservation equation, it is possible to evaluate how much of the total radiation emitted by the flame is due to soot radiation and how much is due to gaseous radiation. Figure 6, shows the net radiative heat transfer across the peripheral wall of the flame. It is observed that about 60 % of the total radiative loss across the peripheral wall of the flame is due to soot emission. Despite being present in high amounts, 56% of the products of combustion, the IR-active species respond for about 40 % of the radiative losses. Since soot particles emit radiation continously over the whole spectrum, they are more efficient emitters than gases. Experimental results obtained by Hamadi et al. (1987), for an atmospheric, 2.9 equivalence ratio, premixed CH₄-O₂ flame, showed that soot radiation comprised about 80% of the radiative emission by the flame. The contribution of each IR-active species to the total gaseous radiation emitted by the flame is not shown but CO₂ and H₂O contributed almost equally with 35% of the gaseous radiation, whereas CO and CH₄ contributed each with 15%.

 

 

Conclusions

As it was pointed out in the introduction, the main objective of the present investigation was to develop a method of solution for solving the nongray RTE in two-dimensional cylindrical media. The motivation behind this study is to incorporate the radiative heat transfer phenomenon modeling into

an existing computer code for simulating premixed, laminar, flat flames (Kee et al., 1985). By doing that, the objective is to investigate how radiative heat transfer affects the temperature distribution along such flames. Considering these facts, this work presented a solution procedure forsolving the RTE in nongray participating media. Based on an accurate representation of nongray and multidimensionality effects, it was possible to obtain a detailed description of the radiative heat transfer phenomenon within the flame. In agreement with the literature, it was shown that most of the radiation emitted by a fuel rich flame comes from soot. The relative importance of each IR-active species on the flame’s emission was investigated, as well. Considering that in practical combustion systems the solution of the RTE is coupled with the solution of the combustion problem, an iterative type of solution is required. The computational cost of such a solution procedure may be high. On the other hand, since the solution procedure has been applied to a two-dimensional cylindrical geometry its application to other combustion systems of practical interest should be relatively simple.

 

Acknowledgements

This work was partly supported by the Brazilian National Research Council (CNPq) through the Research Grant n^o 350885/97-4.

 

References

D’Alessio, A., Di Lorenzo, A., Beretta, F., and Venitozzi, C., 1973, "Optical and Chemical Investigations on Fuel-Rich Methane-Oxygen Premixed Flames at Atmospheric Pressure," Fourteenth Symposium (International) on Combustion, The Combustion Institute, pp. 941-953.        [ Links ]

Edwards, D. K., 1976, "Molecular Gas Band Radiation," Advances in Heat Transfer, Vol. 12, pp. 115-193.        [ Links ]

Grosshandler, W. L., 1980, "Radiative Heat Transfer in Nonhomogeneous Gases: A Simplified Approach," International Journal of Heat and Mass Transfer, Vol. 23, pp. 1447-1459.        [ Links ]

Hamadi, M. B., Vervisch, P., and Copalle, A., 1987, "Radiation Properties of Soot from Premixed Flat Flame," Combustion and Flame, Vol. 68, pp. 57-67.        [ Links ]

Hottel, H. C., 1954, Radiant Heat Transmission, McGraw-Hill, New York, NY.        [ Links ]

Kee, R. J., Grcar, J. F., Smooke, M. D., and Miller, J. A., 1985, "A Fortran Program for Modeling Steady Laminar One-Dimensional Premixed Flames," Sandia National Laboratories, SAND85-8240.        [ Links ]

Ludwig, C. B., Malkmus, W., Reardon, J. E., and Thompson, J. A. L., 1973, "Handbook of Infrared Radiation from Combustion Gases," NASA SP-3080, Scientific and Technical Information Office, Washington, DC.        [ Links ]

Modest, M. F., 1993, "Radiative Heat Transfer," McGraw-Hill, New York.        [ Links ]

Pessoa-Filho, J. B., and Thynell, S. T., 1996a, "Approximate Solution to Radiative Transfer in Two-Dimensional Cylindrical Media," Journal of Thermophysics and Heat Transfer, Vol. 10, No. 3, pp. 452-459.        [ Links ]

Pessoa-Filho, J. B., and Thynell, S. T., 1996b, "Effects of Radiative Heat Transfer on Premixed, Laminar, Flat Flames," AIAA paper, AIAA 96-3960.        [ Links ]

Viskanta, R., and Mengüç, M. P., 1987, "Radiation Heat Transfer in Combustion Systems," Progress Energy Combustion Sciences, Vol. 13, pp. 97-160.        [ Links ]

Xu, F., Sunderland, P. B. and Faeth, G. M., 1997, "Soot Formation in Laminar Premixed Ethylene/Air Flames at Atmospheric Pressure," Combustion and Flame, Vol. 108, pp. 471-493.        [ Links ]

 

 

Nomenclature

d = geometrical distance, [m]

f = soot volume fraction, [ppm]

I = intensity of radiation, [W/m² sr]

I_bh = blackbody intensity of radiation, [W/m² sr]

L = flame's length, [m]

m = considered mth narrow band

ND = number of "homogeneous" elements

p = total pressure, [N/m²]

q = heat flux, [W/m²]

r = radial direction, [m]

R = flame's radius, [m]

s = path length, [m]

T = temperature, [K]

X = optical depth, dimensionless

z = axial distance above the burner, [m]

Greek Symbols:

= variation

e = emissivity, dimensionless

k = absorption coefficient, dimensionless

h = wavenumber, [m^-1]

polar angle, [rad]

s = Stefan-Boltzmann constant [W/m² K⁴]

t = transmissivity, Eq. (8), dimensionless

f = azimuthal angle [rad]

Superscripts:

" = per unit area

+ = positive direction of propagation

negative direction of propagation; average value

~ = boundary

CH₄ = methane

CO = carbon monoxide

CO₂ = carbon dioxide

H₂O = water vapor

N₂ = nitrogen

NB = total number of narrow bands

O₂ = oxygen

is = considered species

mixt. = mixture of gases

soot = soot

Subscripts:

b = blackbody or burner

d = geometrical distance

ê_r, ê_z= unit vectors in the r- and z-direction

i, j = spatial indices

k, l = solid-angle indices

m = narrow band indice

mbl = mean beam length

n = homogeneous element considered

sur = surroundings

h = wavenumber

 

 

Manuscript received: October 1996; revised: October 1998. Technical Editor: Leonardo Goldstein Jr.