Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 495—510

Note

Modeling of optical properties for a polydispersion/gas mixture Miguel Caldas, Viriato Semia o* Mechanical Engineering Department, Instituto Superior Te& cnico, Technical University of Lisbon, Av. Rovisco Pais, P-1096 Lisbon lodex, Portugal Received 9 September 1998

Abstract

Adequate modeling of thermal radiation is an essential tool for the design of real-live combustion systems. Predictive methods for solving the radiative heat transfer equation require the values of absorption and scattering coefficients of the participating media. In the present paper, a compromise between accuracy and computational economy is ensured in the evaluation of those coefficients, by using the exponential wide band model for the gaseous components of the mixture, a new curve fitting approach to the Mie theory for intermediate and large particles and power series to represent the Mie coefficients for small particles. Predicted results with those approaches are presented herein to demonstrate the proposed models’ high accuracy and relatively low computational costs.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Radiation modeling; Scattering; Optical properties

Nomenclature A c C (m) D D  g I h k k ¸ ¸? L

ith weighing factor for Gauss—Laguerre integration Speed of light in vacuum curve fitting coefficients (i"1,7) for Eqs. (16a) and (16b) diameter (m) mean quadratic diameter (m) statistical weights for vibrational transition Plank constant Boltzmann constant extinction/scattering coefficient (m\) asymptotic limit of Q when xPR Laguerre polynomial of order n

* Corresponting author. Tel.: #351 1 841 73 78; Fax: #351 1 847 55 45; e-mail: viriato@ navier.ist.ult.pt 0022-4073/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 8 ) 0 0 1 2 0 - 4

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m N(D) N  p Q q S/d w x z G e a o l (*(¹) d I g I Subscripts 0 c e l s u

complex refractive index disperse phase size distribution (m\) number of particles per unit volume (m\) shape parameter of the size distribution extinction/scattering efficiency factor shape parameter of the size distribution mean intensity to line spacing (m kg\) band width parameter (m\) size parameter ith zero of Laguerre polynomial characteristic value of x for curve fitting integrated absorption coefficient (m kg\) density (kg m\) wave number (m\) band intensity temperature variation function change in vibrational quantum number during transition wave numbers of the fundamental bands (m\)

reference value band center or characteristic value extinction lower band limit scattering upper band limit

1. Introduction Radiation is the dominating heat transfer mechanism within the vast majority of industrial combustion systems. The flow field, the temperature and the species concentrations inside combustors are most influenced by the rate of radiant energy exchanged between the flame and the enclosing walls. Hence, the accurate prediction of this heat transfer mechanism is a key issue in the design and operation of industrial combustors. For the achievement of such accurate predictions the values of the absorption and scattering coefficients of the participating media play a determinant role. In addition to radiation, the combustor performance is characterized by a two or a single-phase turbulent combusting flow. If radiation is the determinant heat transfer mechanism, requiring therefore its modeling, the choice of the model has to be effected accordingly to the fluid flow modeling approach. Many radiation models have been developed for emitting, absorbing and scattering media (see, e.g., Ref. [1]), most of them being based on the solution of the radiative heat transfer equation (RHTE). Some of those models are not recommended for coupling with combusting fluid flow modeling, despite their recognized accuracy. These are the cases of the zonal method [2] and the Monte Carlo method [3] that, besides the requirement of too long computing times, make recourse to numerical techniques very different from those used in fluid flow predictions.

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All the other commonly used models make recourse to the flux method, that discretizes the space into a finite number of solid angles, the heat flux being assumed as constant within each solid angle, in order to eliminate the directional dependence of the RHTE. The use of the third order approximation (P3) of the spherical harmonics method (Pn models, n being the method approximation order) has produced accurate results for two-dimensional geometries, both for Cartesian coordinates (see e.g., Ref. [4]) and cylindrical coordinates in axisymmetric geometries [5], as well as for three-dimensional geometries in Cartesian coordinates [6]. However, the method has proved to be mathematically involved to obtain accurate predictions. Two more attractive algorithms for coupling with ordinary fluid flow predictive tools, the discrete ordinates method [7] and the discrete transfer method [8] have produced very accurate results in solving the RHTE (e.g., Refs. [9—12]). The three previous methods require as input the gas and the solid particles absorption coefficients, the particles scattering coefficient and, for anisotropic scattering, the phase function of the participating media. In addition Mengui and Viskanta [5] showed that the radiative heat flux distribution is rather sensitive to the variation of the absorption and scattering coefficients of the participating media. Hence, the accuracy in the calculation of the previous coefficients will determine the precision of the predicted radiation heat fluxes. For the modeling of the spectral absorption coefficient of the gaseous combustion products the exponential wide band model [13] is used in the present work. This model is chosen since it is one of the most effective models ever presented for the calculation of that parameter. Modest [14] refers to it as the model that presents the best compromise between accuracy and computational economy, presenting an average error of approximately 20% when compared to experimental data. The absorption/scattering coefficients of participating particulate could be evaluated by the use of Mie theory [15]. Some authors consider the Mie theory limited for practical applications, such as radiative heat transfer in particle-laden boilers, since Mie scattering describes the far-field scattering of plane waves interacting with isolated, homogeneous, spherical particles. These assumptions mean that particles are considered spherical independent scatterers. Nevertheless, the Mie theory can be applied for practical systems with volume fractions below 0.006 or a clearance-to-wavelength ratio above 0.5 as conservatively stated by Modest [14]. Al-Nimr and Arpaci [16] presented a value of 0.1 for the volume fraction upper limit. As in most particulate combustion systems the volume fraction is below 10\, Mie theory still applies, as widely supported in the literature [1, 17]. As far as the assumption of spherical particles is concerned, Modest [14] states that even for non spherical particles their random orientation in a cloud will produce a similar effect to that of a cloud of spherical particles. Although applicable for the vast majority of practical combustion systems, the use of the Mie theory in numerical predictive tools, even making recourse to efficient algorithms and high-speed computers, is impractical, due to its mathematical involvement and prohibitive computational time requirement. An accurate and economic modeling is, therefore, mandatory. In the present work, models to accurately and economically approximate the results of the Mie theory for the evaluation of the absorption and scattering coefficients for different types of polydisperse solid phase, such as soot, cenospheres, coal particles, char and fly ash, are proposed and assessed. In radiating environments the polydisperse solid phase is the only responsible for the scattering phenomenon, having also a contribution to the radiant energy absorption/emission. The

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corresponding coefficients depend on the solid-phase concentration, on the particles size distribution and on the equivalent coefficients of a single particle. Asymptotic approximations to the Mie theory, both for large (geometrical optics) and small (Rayleigh theory) particles, exist in the literature (e.g., Ref. [15]). However, particles with an intermediate size lack of a simplified approach for a general complex refractive index. For a value close to unity of that parameter, Mengui and Viskanta [17] have tuned an approximation proposed by Van de Hulst [15] for the absorption and scattering coefficients, that is valid for particles of intermediate and large sizes. A different approach to approximate the Mie theory, followed by Blokh [18] and Fiveland et al. [10], is to resort to curve fitting. In the present work the absorption and scattering coefficients are calculated by a combination of the previous approaches. Indeed, a curve fitting approach is used for intermediate and large particles, but, in opposition to the previous works, the values of the absorption and scattering coefficients are herein fixed by the Mie theory at the origin and by the asymptotic limit at infinity. Additionally, for small particles, the above-mentioned coefficients are calculated from power series to represent the Mie coefficients as suggested by Penndorf [19]. As far as the polydisperse solid phase size distribution is concerned, several distributions can be found in the literature. A possible distribution to be used for the purpose of the present work is the normalized form of the Nukiyama—Tanasawa function (see, e.g., Refs. [14, 18]). This normalized distribution is completely defined by three parameters: one related to the size of the particles and the remaining two to the distribution shape. For the case of soot and fly ash, the experimental values of Blokh [18] and Bard and Pagni [20] and the theoretical analysis of Blokh [18] have determined the two shape parameters of the distributions. For the case of cenospheres, consisting of ash and unburnt carbon formed by the liquid-phase pyrolysis in spray combustion, it is reasonable to assume that the distribution exhibits a shape similar to that of the original liquid spray. The theoretical work of Li and Tankin [21] has determined the two shape parameters of spray distributions and Semia o et al. [22] used those parameters for spray calculations attaining good agreement with experimental results. The shape parameters for carbon particles and fly ash were empirically set by Blokh [18]. In the present study, the shape parameters used to define the size distribution of the different types of polydisperse solid phase were retained from the abovementioned works.

2. Gaseous-phase analysis and modeling For the modeling of the spectral absorption coefficient of gaseous combustion products the wide band exponential model (see e.g. Ref. [13]) is used in the present work. In this model the absorption coefficient of each band j for each gas i is assumed to exhibit one of three possible shapes: Symmetric band:


 S d

a " GH exp(!2"l !l"/w ). AGH GH w JGH GH

(1a)

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Band with upper limit head:



 S d

a " GH exp(!(l !l)/w ), SGH GH w JGH GH

S d

l4l , S

(1b)

"0, l'l , S

JGH Band with lower limit head:



 S d

a " GH exp(!(l!l )/w ), l5l , JGH GH J w JGH GH

S d

(1c)

"0, l(l . J JGH In the above equations, S/d is the spectral mean intensity to line spacing, l is the wave number, a is the integrated absorption coefficient given by Eq. (2) and w is the band width parameter given by Eq. (3): (*(¹) , (2) a"a  (*(¹ )  ¹ . (3) w"w  ¹  In Eqs. (2) and (3) ¹ has the value of 100 K, a and w are constants characteristic for each band    and (*(¹) possesses the simplified form given by Eq. (4), provided that all the values of d are I non-negative:








K (*(¹)" 1!exp ! k d I I I where



K (d #g !1)! I (1!exp(!k ))\BI, “ I I (g !1)! I I

hcg I. k" I k¹

(4)

(5)

In the preceding equations, g are statistical weights for vibrational transition, d is the change in I I vibrational quantum number during transition, g are the wave numbers of the fundamental bands, I h is the Plank constant, c is the speed of light in vacuum and k is the Boltzmann constant. For high-temperature thermal radiation (0.5 lm4k410 lm) and for the common combustion products (H O, CO and CO), the bands considered relevant retained from a general table [14]   were the following: H OP71, 6.3, 2.7, 1.87 and 1.38 lm,  CO P15, 4.3 and 2.7 lm  COP4.7 lm.

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The contribution for the total absorption coefficient of the remaining known bands is more than two orders of magnitude inferior to that of the considered ones. Therefore, their contribution to the above-mentioned coefficient is not accounted for in the present work. The value of the constants a , w , l , l or l , g , g and d were taken from Modest [14].   A S J I I I Following experimental results, Edwards [23] concluded that the value of w calculated through Eq. (3) is to be increased by 20% if spectral analysis constitutes the aim of the study. The average absorption coefficient is then calculated through Eq. (6) and the total absorption coefficient due to the gaseous phase is found simply by adding the contributions of the individual bands:




S k "o G d JGH

. (6) JGH In Eq. (6), o is density of the gas i, which can be calculated through the prefect gas equation. G 3. Solid-phase analysis and modeling The monochromatic extinction and scattering coefficients of a polydisperse cloud consisting of spherical particles are given by the following equation:



n  DN(D)Q (x, m) dD, (7) k (N(D), m)"N   4  where k represents the extinction or scattering coefficient, Q is the extinction or scattering   efficiency factor, m is the complex refractive index, x is the size parameter given by x"nD/j, j the wavelength, N(D) is the size distribution and N is the total number of particles per unit  volume. The size distribution N(D) can be adequately represented by a Nukiyama—Tanasawa or modified gamma distribution (see, e.g., Refs. [14, 18]). The following normalized Nukiyama—Tanasawa distribution is used herein: (N>) qC O DN exp(!CDO) (8) N(D)" !( N>) O where p, q and C are distribution parameters and !(x) is the gamma function. Defining a characteristic diameter D as in Eq. (9) it is possible to rewrite the previous A distribution in a more convenient form — Eq. (10):




1 O , D " ! C q N(D)" p#1 D C ! q




(9)




D N exp [!(D/D )O]. ! D !

(10)

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Substituting Eq. (10) into Eq. (7) and performing an adequate change of the integration variable, the extinction and scattering coefficients are given by k "N  

nD  4

1 p#3 ! q










uN>O\ e\u Q (x uO, m) du.  !

(11)

In Eq. (8), x is the size parameter referred to D and D is the quadratic mean diameter obtained A A  from the following definition:





 

p#1#i  DGN(D) dD C q " DG\H. DG\H"  ! GH  DHN(D) dD C p#1#j  q

(12)

In order to permit its numerical integration at lower computational costs, so that the present approach is economically appropriate for radiation predictions, Eq. (11) is modified, resulting in k "N  



1 nD  ¸ (m)#  p#3 4 ! q












uN>O\ e\PF (x uO, m) du ,  !

(13)

where ¸ (m) is the asymptotic limit of the extinction or scattering efficiency factor Q (x, m) for   large values of x and can be found in the work of Van de Hulst [15]. The function F(x, m) appearing in Eq. (13) is defined by Eq. (14a) and has the properties represented by Eq. (14b) and (14c): F (x, m)"Q (x, m)!¸ (m), (14a)    F (0, m)"!¸ (m), (14b)   lim F (x, m)"0. (14c)  V This function as appearing in Eq. (13) poses fewer mathematical difficulties to be integrated than Q (x,m) appearing in Eq. (11). Nevertheless, numerical integration of Eq. (13) still has to be CQ performed. Its form strongly suggests the use of the following Gauss—Laguerre quadrature formula:





 x?e\f (x) dx" A f (z ). (15) G G  G In this formula, z are the zeros of the generalized Laguerre polynomial of order n (¸?(x)) and G L A are the weighting factors (see, e.g., Ref. [24]). G Since the value of (p#3)/q appearing in Eq. (13) is not known beforehand it is necessary to use a quadrature set for a"0 and to include the term uN>O\ in the integrand. This will result in an increase of the number of points required to evaluate the integral with the desired accuracy. At this point it is necessary to calculate the function F (x, m). This could be done using the Mie theory,  however, the computational effort required for these calculations would be prohibitive. An alternative approach is used in this work and consists of finding a new function F(x, m) depending

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on x, the coefficients of which depend on m. Two possible approximations, each using seven coefficients C (m), are described by Eq. (16a) and (16b): G ¸(m)#C (m)#C (m)x! K C (m)    ! (16a) F(x, m)+ 1#C (m)x!K 1#C (m)x! K   C (m) ¸(m)#C (m)   F(x, m)+ ! cos+C (m)[(1#C (m)x)!1],. (16b)    1#C (m)x! K 1#C (m)x! K   Although Eq. (16b) is more accurate to model near-real refractive indexes, where resonance peaks appear, Eq. (16a) is preferred for numerical modeling, because of its smaller computational effort requirements and better accuracy for refractive indexes with larger imaginary part. The coefficients appearing in Eqs. (16a) and (16b) can be calculated for each m, through a least-square regression method that fits these equations to Mie theory, and can be stored as a table for later use. It must be noticed that in some cases, Eqs. (16a) and (16b) become highly inaccurate for small values of x. To overcome this shortcoming it is necessary to resort to a different approach. For small values of the non-dimensional size parameter and of the refractive index the efficiency factors Q (x, m) can be satisfactorily approximated by a power series retaining a few terms. For example,  Penndorf [19] presented a four-term expansion series in the form: Q (x, m)"x(E (m)#E (m)x#E (m)x#E (m)x), (17a)      Q (x, m)"x(S (m)#S (m)x#S (m)x#S (m)x), (17b)      where E (m) and S (m) are complex functions that can be found in his work [19] and are therefore G G not reproduced here. These equations can be inserted into Eq. (11) and integrated analytically, yielding E (m)!(N> )#E (m)!(N>)x# N nD  O  O ! , (18a) k "   x  4!( N>) ! #E (m)!(N>)x#E (m)!(N>)x O O  !  ! O N nD S (m)!(N>)#S (m)!(N>)x#  O  O ! k "   x . (18b) Q 4!( N>) ! #S (m)!(N>)x#S (m)!(N>)x O O  !  ! O The extent to which Eqs. (18a) and (18b) are accurate is strongly dependent on the value of both the real and imaginary parts of the refractive index m. Larger values of m lead to a sooner departure from Mie theory of the values of k and k . It may happen that a range of the non-dimensional size   parameter exists where none of the previously proposed approximations is accurate. In this case, similar expressions to those represented by Eqs. (18a) and (18b) can be used, providing the replacement of the parameters E (m) or S (m) by E*(e, m) or S*(e, m) — Eqs. (19a) and (19b) —   where e is a value of x that makes Eqs. (16a) and (16b) sufficiently accurate:

 





¸(m)#F(e, m) E E E E*(e, m)" ! ! !  , e e e e

(19a)

¸(m)#F(e, m) S S S S*(e, m)" ! ! !  . e e e e

(19b)

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Soot constitutes one important type of polydispersion. Soot particles diameter is usually below 0.2 lm [20, 25]. In the wavelength range of interest this results in a maximum size parameter of 1.26 and, therefore, Eqs. (18a) and (18b) are adequate for the treatment of soot. The shape parameters p and q for the size distribution of soot were determined theoretically by Blokh [18] as p"q"2. These values have been experimentally confirmed [20, 18] and will be used in the present work.

4. Case studies Three different cases are studied in the present work in order to validate the proposed models for different disperse solid phase (soot and char particles) and different size distributions (broad and narrow). The refractive index of soot is calculated through the dispersion equation presented by Lee and Tien [25]. The refractive index of char is considered independent of the wavelength and is made equal to 2.05—1.1275i as suggested by Fiveland et al. [10]. For the simulation of char particles, different size distributions, that is different values of the shape parameters, are considered. In each case, both exact and approximate non-dimensional extinction and scattering coefficient, defined as k* "4k /nN D , are plotted against x , together with the relative error of the     A approximations. For soot, both exact and approximate non-dimensional extinction and scattering coefficients are plotted against j for a size distribution characterized by p"q"2 and D "0.1 lm.  The exact solution is calculated only for comparison purposes and is obtained from the use of Eq. (5), where the efficiency factors are calculated from the Mie theory and the integration is performed using a Gauss—Laguerre quadrature with seventy points, which makes its calculation prohibitively expensive. The approximate solutions to the Mie theory considered herein are the following: Approximation 1 — calculated through Eqs. (13) and (16a), the integration being performed using a Gauss—Laguerre quadrature with 5 points, that gives a compromise between accuracy and economy. Approximation 2 — calculated through Eqs. (13) and (16b), the integration being similarly performed. Approximation 3 — calculated through Eq. (18a) for extinction or Eq. (18b) for scattering. The first case studied in this work (case 1) is characteristic of char particles. The refractive index used (2.05—1.1275i) is that proposed by Fiveland et al. [10] and the distribution shape parameters p"1 and q"1 are those proposed by Blokh [18]. In order to determine the sensitivity of the results to the shape of the size distribution, a similar case as far as the type of particles is concerned was chosen (case 2). The values used for the distribution shape parameters are p"4 and q"4, corresponding to a narrower shape. The values of the constants appearing in Eqs. (16a) and (16b) used for cases 1 and 2 are presented in Tables 1 and 2. The last case studied herein (case 3) refers to the calculation of extinction and scattering coefficients for soot. The refractive index used is given by the dispersion equation of Lee and Tien [25] and the distribution shape parameters (p"2 and q"2) are those proposed by Blokh [18].

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Additionally, for each case study, the total dimensional extinction coefficient (gas plus particles) is plotted against the wavelength for the following mixtures: ¹"2000 K, P"1 atm, H O"15%,  CO "10%, CO"1%, D "0.1 lm and a volume fraction of 10\ for soot or D "50 lm and    a volume fraction of 5;10\ for char particles. Table 1 Curve fitting constants for non-dimensional extinction coefficient (m"2.05—1.1275i)

Eq. (16a) Eq. (16b)

C 

C 

C 

C 

C 

C 

C 

2.4343 1.5778

0.9044 0.3300

0.7053 0.8724

2.3728 5.4386

4.9935 3.5334

!2.5208 0

0.7030 0

Table 2 Curve fitting constants for non-dimensional scattering coefficient (m"2.05—1.1275i)

Eq. (16a) Eq. (16b)

C 

C 

C 

C 

C 

C 

C 

0.1791 0.1818

0.0471 0.0508

0.9061 0.8929

2.3175 2.5362

5.6964 5.4464

!0.0924 0

1.3256 0

Fig. 1. Comparison and error analysis of exact and approximate solutions of non-dimensional extinction coefficient for case 1 — char (m"2.05—1.1275i, p"1, q"1). —— Exact solution; o - Approximation 1; 䊏 - Approximation 2.

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5. Results and discussion Figs. 1 and 2 depict the values of the non-dimensional extinction and scattering coefficients for case 1, respectively, compared against the values obtained from the Mie theory both for Approximations 1 and 2. These figures also display the relative errors of those coefficients for the referred Approximations. It can be observed from both figures, that the agreement is very good — within 5% of the relative error — for all the values of x . A For values of x greater than 2.0, Approximation 1 for extinction is excellent, presenting  a negligible error when compared with the Mie solution. It is clear that the approximated solution containing the cosine function shows poorer agreement, presenting an asymptotic error of 1%. Notice that this is not the case for the scattering coefficient, where both approximations present an asymptotic error of 0.5%. The computational time of approximation 1 is 320 times lower than that requested by the Mie solution even with the use of an efficient algorithm to calculate the Mie coefficients, similar to the one suggested by Crosbie and Davidson [26]. Figs. 3 and 4 show the predicted values of the non-dimensional extinction and scattering coefficients for case 2 and their relative error using Approximations 1 and 2 as well as their comparison against the Mie theory values. As for case 1, it can be observed from these figures that the agreement in case 2 is also very good — within 5% of the relative error, except for the case of

Fig. 2. Comparison and error analysis of exact and approximate solutions of non-dimensional scattering coefficient for case 1 — char (m"2.05—1.1275i, p"1, q"1). —— Exact solution; o - Approximation 1; 䊏 - Approximation 2.

506

Fig. 3.

Fig. 4.

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Fig. 5. Comparison and error analysis of exact and approximate solutions of non-dimensional extinction and scattering coefficient for case 3 — soot (p"2, q"2), with the refractive index depending on j. —— Exact solution, extinction; o Approximation 3, extinction; - - - - - - Exact solution, scattering; 䊏 - Approximation 3, scattering.

the scattering coefficient for very small values of x where the error achieves a magnitude of 12.5%.  Again, for values of x greater than 2.0, Approximation 1 both for extinction and scattering  presents a negligible error when compared with Mie solution, while Approximation 2 presents an asymptotic error of 1% for extinction and a negligible error for scattering. Approximation 1 requires a computational time 66 times lower than that of the exact solution. The reduction in computational economy is due to the effect of the disperse solid-phase size distribution form, which is narrower (p"4, q"4) and, therefore, makes Mie calculations faster by reducing the number of large particles calculations. The previous figures clearly indicate that Approximation 1 is to be used for non-small particles regime, as the precision of the results is higher than that obtained with the use of Approximation 2.

䉳 Fig. 3. Comparison and error analysis of exact and approximate solutions of non-dimensional extinction coefficient for case 2 — char (m"2.05—1.1275i, p"4, q"4). —— Exact solution; o - approximation 1; 䊏 - Approximation 2. Fig. 4. Comparison and error analysis of exact and approximate solutions of non-dimensional scattering coefficient for case 2 — char (m"2.05—1.1275i, p"4, q"4). —— Exact solution; o - Approximation 1; 䊏 - Approximation 2.

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Fig. 6. Extinction coefficient for a mixture of gas and soot (¹"2000 K, P"1 atm), with gaseous species concentrations of (H O: 15%, CO : 10%, CO: 1%) and soot defining characteristics of (D "0.1 lm, fv"10\, p"2, q"2 and    a variable refractive index). —— Gas extinction; o - Soot extinction; 䊉 - Total extinction.

Fig. 7. Extinction coefficient for a mixture of gas and char for two particle size distributions (¹"2000 K, P"1 atm), with gaseous species concentrations of (H O: 15%, CO : 10%, CO: 1%) and char defining characteristics of   (D "50 lm, fv"5 10U, m"2.05—1.1275i). —— Gas extinction; o - Char extinction; 䊉 - Total extinction. 

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Additionally, the number of computational arithmetic operations needed to perform Approximation 1 is smaller than that for Approximation 2. If rigorous calculations in the small particles regime (x ;1) are desired, as in the case of soot  particles, Approximation 3 should be used. Non-dimensional extinction and scattering factors for case 3, given by Approximation 3, are compared against the exact values obtained from the Mie theory in Fig. 5. It can be seen that for wavelengths greater than 1 lm, which covers almost all the spectrum, the agreement is excellent. Figs. 6 and 7 show the extinction coefficient for the gas and particles mixtures described in the previous section. As it can be seen from Fig. 6, soot has a marked influence in the low wavelength region of the spectrum but becomes unimportant at higher wavelengths. Conversely, char particles have a significant influence in all the considered wavelengths, due to its larger dimensions. Those features indicate that, besides an adequate modeling of soot and gas, an adequate modeling of the radiative properties of larger particulate systems is essential for accurate heat transfer predictions.

6. Conclusions In the present work accurate and economic approximations to the Mie theory for the calculation of the extinction and scattering coefficients are proposed. For the sake of accuracy, the x domain is A partitioned into different zones, a different approach being used in each zone. The precision attained through this procedure is, for almost all the x domain, above 95%.  When comparing the two proposed curve-fitting functional forms — Approximations 1 and 2, it is clear that the former is superior in terms of precision and computational time requirements, although the later is also very precise. The high accuracy and the huge economy in computational time requirements achieved, the later depending on the particle size distribution shape and going up to 300-fold when compared with the use of Mie theory, largely justify the incorporation of the present models in predictive tools to be applied to combustion equipment design and optimization.

Acknowledgements This work has been partially performed with the financial support of the European Collaborative Research JOULE Programme, under the contract JOF3-CT95-0010.

References [1] [2] [3] [4] [5]

Viskanta R, Mengui MP. Prog Energy Combust Sci 1987;13:97. Hottel HC, Sarofim AF. Radiative transfer. New York: McGraw-Hill, 1967. Howell JR. Advances in heat transfer, vol. 5, New York: Academic Press, 1968. Ratzel III AC, Howell JR. ASME J Heat Transfer 1983;05:333. Mengui NP, Viskanta R. ASME J Heat Transfer 1986;108:271.

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