IN MANY materials processing and manufacturing situations such

Transcription

1 journal J. Am. Ceram. So., 86 [8] (2003) Radiation Charateristis of Glass Containing Gas Bubbles Laurent Pilon* Mehanial and Aerospae Engineering Department, Henri Samueli Shool of Engineering and Applied Siene, University of California, Los Angeles, California Raymond Viskanta Shool of Mehanial Engineering, Purdue University, West Lafayette, Indiana In many materials proessing and manufaturing situations suh as steel, aluminum, eramis, and glass, gas bubbles an form in liquid and solid phases. The presene of suh bubbles affets the thermophysial properties and radiation harateristis of the two-phase system and hene the transport phenomena. This paper presents a general formulation of the radiation harateristis of semitransparent media ontaining large gas bubbles (bubble radius is muh larger than the wavelength of radiation). Sample alulations for the spetral absorption and extintion oeffiients and single sattering albedo of soda lime siliate glass ontaining bubbles are disussed. Partiular attention is paid to the effet of the volumetri void fration and the bubble size distribution. Results learly show that the presene of bubbles strongly affets the radiation harateristis of the semitransparent media ontaining entrapped gas bubbles, partiularly if bubbles, void frations, and the spetral absorption oeffiient of the ontinuous phase are small. I. Introdution IN MANY materials proessing and manufaturing situations suh as steel, aluminum, eramis, and glass, gas bubbles an form in liquid and solid phases. The presene of suh bubbles affets the thermophysial properties and radiation harateristis of the two-phase system and hene the transport phenomena. In glassmelting furnaes, for example, a large number of bubbles are formed by hemial reations during melting of the bath and the thermal deomposition of refining agents. Bubbles that are large enough rise at the surfae of the glass melt while small bubbles are trapped in the flowing molten glass. The quality of glass produts is degraded if gas bubbles and unfused silia grains remain in the molten glass as it is being pulled from the furnae. 2,3 The quality of the glass produed is signifiantly affeted by the flow pattern and the temperature of the glass melt. Both depend strongly on the heat transfer from the ombustion spae to the bath and to the molten glass. The high temperatures and the oxidizing environment in the furnae make experimental measurements very diffiult and unreliable. Therefore, numerial simulations onstitute appropriate alternatives for better design and ontrol of the glass-melting furnaes. General models for prediting the flow and the temperature fields have been reviewed by Viskanta. 14 All the models use the Rosseland diffusion approximation for radiative heat transfer in the molten glass. Reently, Cheong et al. 4 questioned this assumption and showed that the diffusion approximation for radiative transfer is not reommended when the depth of the glass layer is less than 0.5 m. Instead, they suggested the use of the P-1 approximation not only for shallow glass melt layers but also for deep ones. The analyses reported in the literature have negleted the presene of numerous gas bubbles of different sizes in the melt that may affet the radiation heat transfer. The presene of the bubbles will affet the radiation harateristis of the glass melt sine the radiation is sattered by the bubbles. Approximate and rigorous treatements of radiative transfer in glass require spetral radiation harateristis of glass melt whih ontains gas bubbles. Therefore, it is of partiular interest to understand the effet of bubbles on the radiation harateristis of the molten glass in order to better predit radiation heat transfer in glass-melting furnaes, and improve the glass quality and the energy effiieny of the proesses. The objetive of this work is to assess the effet of the bubble radius, the bubble size distribution, and the void fration on the radiation harateristis of glass ontaining spherial gas bubbles with different bubble size distributions and to gain understanding of their importane on radiative transfer in glass. II. Analysis Consider heat transfer within a horizontal layer of ontinuous ondensed phase ontaining bubbles as shown in Fig. 1. In general, onvetion (due to the motion of bubbles), ondution, and radiation heat transfer are present and should be onsidered. We further assume that the ontinuous ondensed phase is a solid or a slowly moving liquid and is essentially isothermal. Then, the first two modes of heat transfer an be safely negleted in omparison to heat transfer by radiation. When gas bubbles are moving with the liquid phase, a time-averaged void fration and bubble size distribution should be used. In addition, we assume the following: (1) all bubbles are spherial, (2) the sattering of a single bubble is not affeted by the presene of its neighbors (independent sattering); and (3) the radiation field within the liquid layer is inoherent (i.e., sattering enters are randomly distributed with zero-phase orrelation). Then, radiative transfer within an absorbing, emitting, and independently sattering medium is governed by the integrodifferential equation expressed in dimensionless optial oordinates: 10,15 di d 1 ŝi b Tŝ I (ŝ,ˆ ) J. R. Hellmann ontributing editor Manusript No Reeived Otober 30, 2001; approved August 20, This work was supported by the U.S. Department of Energy/Glass Industry/ Argonne National Laboratory/University ollaborative researh projet. ŝ I 4 ŝ,ˆ ŝ,ˆ 3 ˆ dˆ (1) ˆ 4 Here, ŝ is the loal spatial oordinate unit vetor, ˆ is a line-of-sight diretion, and I b is Plank s blakbody funtion. The 1313

2 1314 Journal of the Amerian Cerami Soiety Pilon and Viskanta Vol. 86, No. 8 spetral optial thikness and single sattering albedo are defined, respetively, as 0 s ds 0 s ds (2) (3) where,, and ( ) are the absorption, sattering, and extintion oeffiients, respetively. The sattering phase funtion (ŝ,ˆ 3ˆ ) represents the probability of sattering of the radiation from a beam propagating in the diretion ˆ to the diretion ˆ, and it is normalized suh that 1 4ˆ 4 ŝ,ˆ 3ˆ dˆ 1 (4) Equation (1) indiates that the extintion, absorption, and sattering oeffiients together with the sattering phase funtion and the single sattering albedo are major parameters of the radiation transfer. Thus, the following subsetions of the paper are devoted to haraterization of the ontinuous phase layer with dispersed gas bubbles and to the development of detailed models for its effetive radiation harateristis. Let m d n d ik d and m n ik be the spetral omplex indies of refration of the dispersed phase (i.e., gas bubbles), and of the ontinuous phase, respetively. The following setions present the formulations for prediting the spetral radiation harateristis of glass ontaining monodispersed and polydispersed bubbles. (1) Predition of Spetral Radiation Charateristis for Monodispersed Bubbles We assume here that all the bubbles entrapped in the glass melt have a uniform radius a. Then, the effetive extintion oeffiients (due to absorption and sattering) and the sattering phase funtion for the layer an be expressed as 7 Q d abs aa 2 N T Q abs aa 2 N T Q abs a Q d abs aa 2 N T (5) Q d sa aa 2 N T (6) Q abs a Q d ext aa 2 N T (7) where 0 / 0 1/(n ) is the wavenumber of the wave with a frequeny and phase veloity equal to the speed of light in a vauum, 0. (2) Spetral Radiation Charateristis for Polydispersed Bubbles Figure 1 shows a shemati diagram of layer of ontinuous ondensed phase ontaining bubbles of different sizes. In glass melt these bubbles may be generated during the bath fusion and fining reations. 3 Suppose that all bubbles are spherial (in the ase of distorted bubbles one an define an equivalent sphere whih preserves the gas volume fration) and their size (radius a) distribution is given by the so-alled modified gamma funtion (Ref. 10, pp ): na Aa exp(ba ) 0 a (11) The distribution funtion vanishes at a 0 and a 3 and it reahes its maximum at the bubble radius a max (/B) 1/. The four onstants A, B,, and are taken to be positive and real, and they must be determined from measurable quantities suh as the total number of bubbles per unit volume, N T 0 A0 na da a exp(ba )da 1 A (12) B 1/ and the bubble density (i.e., the total volume of bubbles per unit volume or the bubble volume fration) is given by 3a 4 3 na da f v 0 4 4A (13) 3B 4/ Here, (z) is the gamma funtion ( 0 e t t z1 dt), and it is tabulated by Abramowitz and Stegun. 1 Note also that the onstants and are usually hosen to be integers. where Q abs (a), Q sa (a), and Q ext (a) denote the absorption, sattering, and extintion effiieny fators and sattering phase funtion, respetively, for a sphere of radius a, while the supersripts d and refer to the dispersed and the ontinuous phase, respetively. N T is the total number of bubbles per unit volume, whih an be expressed as a funtion of the void fration f v and of the foam and ontinuous densities, f and, respetively: N T 3f v 4a 3 3 4a 31 f (8) Moreover, for independent sattering, the phase funtion in a loud of uniform bubbles () is the same as that for a single partile (a,); 10 it is also the same for a bubble loud, i.e., a, (9) Note that the absorption oeffiient of the ontinuous phase in Eq. (5) an be alulated from the imaginary part (k )ofthe omplex index of refration (m )as 4 0 k (10) Fig. 1. Shemati of the idealized liquid layer ontaining bubbles and the oordinate system.

3 August 2003 Radiation Charateristis of Glass Containing Gas Bubbles 1315 In the present ase, we also assume that all bubbles have the same optial properties throughout the layer. This assumption is valid even if the gas omposition in the bubbles hanges as the bubbles rise and gases diffuse in and out of the bubbles. Indeed, the absorption of radiation by the gas phase is proportional to the bubble pressure and size whereas sattering depends mainly on the interfaial area. The pressure inside large bubbles is relatively small while that for small bubbles is large but the optial path is small. Thus, whether the bubbles are small or large, the gases ontained in the bubbles have little effet on the transfer of radiation. Furthermore, the bubble size distribution funtion n(a) is assumed to be known; future work on prediting the density funtion of bubbles transported by the glassmelt onvetive urrents and growing and shrinking due to gas diffusion will be reported soon. Then, the effetive extintion oeffiients (due to absorption and sattering) and the sattering phase funtion for the medium ontaining bubbles an be expressed as 7 Q abs aa 2 na da Q d abs aa 2 na da 0 0 Q abs a Q d abs aa 2 na da (14) Q d sa aa 2 na da (15) 0 Q abs a Q d ext aa 2 na da (16) Q d sa a a,a 2 na da (17) In alulating the spetral oeffiients, most of the omputer time is used in evaluating the effiieny fators Q abs, Q sa, and Q ext from the Mie theory. In multidimensional and spetral radiative transfer analysis this type of approah beomes impratial. Therefore, it is desirable to have simple approximations for the effiieny fators. The hanges in the sattering pattern due to hanges in the bubble size should be aounted for in the predition of the radiation harateristis of the layer ontaining bubbles. Figure 2 shows the different limiting ases in the domain where simple analytial expressions for Q abs (a), Q sa (a), Q ext (a), and (a,) are available in the literature. For spheres with index of refration lose to 1, the domain an be divided into two limiting ases: 13 (i) The Rayleigh Gans sattering domain orresponds to a near-dieletri sphere with (1) k 0, (2) a refrative index of refration lose to unity i.e., m 1 1, and suh that (3) the phase lag suffered by the entral ray that passes through the sphere along a full diameter is small, i.e., 2m 1 1. Then, refletivity is negligible and the radiation passes through the sphere unattenuated and unrefrated. 13 The Rayleigh Gans sattering domain an itself be divided into two limiting ases namely, 3 0 (Rayleigh sattering) and 3 ( intermediate regime ). (ii) The anomalous-diffration domain is haraterized by 3 and m 3 1 orresponding to a straight transmission and subsequent diffration aording to Huygens priniple. 13 The anomalous-diffration domain an also be divided into two limiting ases, namely, 3 0( intermediate regime ) and 3 ( geometrial optis diffration regime ). Note that the Rayleigh Gans domain and the anomalous diffration domain overlap in the so-alled intermediate regime. The present work is onerned with relatively large bubbles and wavelengths between 0.4 and 10 m for whih the anomalous diffration is valid. In this region of the domain, the large gas bubbles are relatively weak absorbers of radiation and mostly at as the strong radiation satterers. In this ase, the approximate analytial expressions for extintion (absorption and sattering) effiieny fators for a weakly absorbing sphere of arbitrary size an be used as derived by van de Hulst (see Ref. 13, p. 179), os g Q ext, m 2 4 e tan g sin g 4 os g 2 os 2g e tan g os 2g (18) where 2(n 1) 2(n 1)(2a/) and g artan [k/(n 1)] are the van de Hulst s normalized size and absorption parameters, respetively, suh that d tan (g) gives the energy absorbed along the axial ray within the sphere. Beause of the assumptions of the van de Hulst s theory, the expression overestimates the extintion fator for small spheres and underestimates it for larger spheres. To orret this, Deirmendjian 5 proposed to use a orretion fator (1 D i ). The approah was remarkably suessful in improving auray of extintion oeffiient defined by Eq. (18), and the speifi expressions for D i an be found in Ref. 5 (pp ). The absorption and sattering ontributions to the extintion effiieny fator of a single gas bubble are given by the following asymptoti formulas (Ref. 5, p. 35): Fig. 2. Survey of the limiting ases in the domain. tan g e2 Q abs, m 1 tan g e2 tan g 1 2 tan g 2 (19) Q sa, m Q ext, m Q abs, m (20) respetively. Comparison of the absorption and sattering effiienies generated using the approximate expressions given by Eqs. (19) and (20) with numerial results obtained from the exat Mie theory (Ref. 5, pp ) have demonstrated the power of this approah, espeially when the preise diretional sattering pattern and polarizing properties are not desired. Speifially, the results obtained via the orreted van de Hulst s formula (1 D i )Q ext are aurate within about 0.05Q ext for a wide range of sphere sizes and indies of refration. Note that the magnitude of the error inreases with an inrease in a value of the real part (n d )ofthe omplex index of refration for both absorbing and nonabsorbing spheres (Ref. 5, p. 32). Of ourse, if the omplex index of refration of the ontinuous phase (m ) lies within auray limits of the van de Hulst s theory, then Eq. (19) an be suessfully used for alulation of the absorption effiieny fator Q abs (a) ofthe sphere made of the ontinuous phase as well.

4 1316 Journal of the Amerian Cerami Soiety Pilon and Viskanta Vol. 86, No. 8 Fig. 3. Map of the sattering theories and approximations used for determining the extintion effiieny fators of the soda lime siliate ontaining gas bubbles, i.e., Q abs, Q sa, and Q ext in Eqs (5) to (7). III. Results and Disussion The input parameters for the model inlude the spetral index of refration and the spetral absorption oeffiient of both the ontinuous and dispersed phases along with the bubble size distribution and the total number of bubbles per unit volume N T or the void fration f v. Clear soda lime siliate glass (window glass) is used for the sample alulations, and the spetral variation of the real (n ) and imaginary part (k ) of its refrative index (m n ik ) are taken from the literature. 11 The gas mixture ontained inside the gas bubbles is transparent to the inident radiation and its omplex index of refration (m d ) is assumed to be independent of the wavelength and equal to i In this setion we review the different sattering domains for whih simple analytial expressions for the extintion effiieny fators exist. Then, the absorption and extintion oeffiients as well as the single sattering albedo of soda lime siliate glass ontaining monodispersed and polydispersed bubbles with different size distribution and different void fration are alulated in the ase of anomalous diffration (large bubbles). Finally, results giving the apparent refletane, transmittane and absorptane of glass layers of different thikness are disussed. (1) Model Validity for Glass Containing Bubbles (A) Sattering Domains: In defining the limiting ases of the Mie theory for whih simple analytial solutions are known, we used the qualitative riteria 1 for Rayleigh Gans sattering and 1for anomalous diffration. For our partiular appliation the different sattering regimes are delimited arbitrarily as follows: (i) Anomalous sattering approximation is assumed to be valid for 100. This ondition leads to a 100 (21) 2 (ii) The Rayleigh Gans sattering approximation is assumed to be valid for For the gas bubbles, this ondition is equivalent to a (22) 400n 1 2 k 2 Note that unlike the index of refration of the gas ontained in the bubbles, that of the soda lime siliate glass depends on the wavelength and must be aounted for in defining the sattering Fig. 4. Sattering regime map for independent and dependent sattering due to spherial bubbles. domains. The Rayleigh Gans sattering approximation an be made when the ondition expressed by Eq. (22) is valid for both the dispersed and ontinuous phase. (iii) A subdomain of the Rayleigh Gans sattering approximation is the Rayleigh sattering assumed to be valid when 0.01 and For the gas bubbles and glass spheres, these onditions are expressed by Eq. (22) and a (23) 200 Aording to Eqs. (5) to (7) one needs to onsider the absorption effiieny fator for both gas bubbles and the orresponding glass spheres, and the sattering effiieny fator and the sattering phase funtion for the gas bubbles in order to predit the effetive radiation harateristis of the glass slab ontaining bubbles. Figure 3 indiates the theory or limiting ases to be used to predit the extintion effiieny fators for both the glass spheres and the gas bubbles in the wavelength-radius domain. The above representation for the effetive radiation harateristis is valid only if the sattering by the ensemble of bubbles is independent, i.e., sattering by one partile is not affeted by the presene of surrounding partiles. Tien and Drolen 12 presented a sattering regime map whih uses the size parameter ( 2a/) and the volume fration (f v ) as the oordinate axis. They showed that the dependent sattering effets may be ignored as long as f v or / 0.5. Assuming a ubi lattie of bubbles of spaing p, the ondition /0.5 an be expressed in terms of the void fration as f v (32/3)(a/) 16/3. Figure 4 shows the maximum void fration f v for independent sattering as a funtion of the bubble radius a. It suggests that for bubbles larger than 1 m in diameter, independent sattering an be safely assumed. From Figs. 3 and 4, one an onlude that for bubbles with a 0.1 mm, the radiation harateristis of the glass layer an be predited from the anomalous diffration theory. For bubbles less than 1 nm in diameter, the same radiation harateristis of the Table I. Distribution Major Charateristis of the Uniform Bubble Size Distributions a (mm) N T (no./m 3 ) f v 0.2 f v 0.4 f v 0.6 Uniform Uniform Uniform

5 August 2003 Radiation Charateristis of Glass Containing Gas Bubbles 1317 Fig. 6. Effet of void fration on the spetral extintion oeffiient and single sattering albedo for soda lime siliate glass ontaining uniform size bubbles 1 mm in diameter. Therefore, only large bubbles for whih the anomalous diffration theory and the independent sattering assumption are valid will be onsidered further. (B) Ranges of Interest: It an be shown 8 that bubbles are spherial if their radius a is small ompared to the apillary length l (a l ) where the apillary length for gas bubbles surrounded by liquid is defined as l 2 d g (24) Fig. 5. Effet of bubble radius on the spetral absorption, extintion oeffiients, and single sattering albedo for soda lime siliate glass with f v 0.2. glass layer an be estimated from the Rayleigh Gans sattering theory. However, for bubbles having radii between 1 nm and 0.1 mm and/or if the void fration is larger than 0.006, the use of the Mie theory and/or the onsideration of dependent sattering is required for wavelengths between 0.4 and 10 m, making the omputation of the apparent absorptane, refletane, and transmittane of the layer onsiderably more involved. Unfortunately, undertaking the task of solving the Mie theory and/or aounting for dependent sattering is beyond the sope of this work. Here, is the surfae tension (300 mn/m), and (2350 kg/m 3 at around 1400 K 9 ) and d (1.2 kg/m 3 ) are the densities of the molten glass and the air, respetively. For soda lime siliate glass the apillary length is about 4 mm. We assumed that bubbles are spherial for bubble radii up to the apillary length l /4 1 mm. The spetral region where the thermal radiation is the most important is onsidered. The wavelength interval from 0.4 to 10 m is hosen sine it overs nearly 88% of the thermal radiation emitted by a blakbody at the soure temperature of 5800 K and 94.5% at 1200 K. In summary, our study is restrited to the size parameters ( 2a/) ranging from min 0.0 to max and a phase shift, 2m 1, that an take values from 0 (m 1 1 and 1) to infinity (m 1 1 and 3 ). The volume void fration f v defined as the ratio of the volume of gas to the total volume an vary between 0 and 0.74 orresponding to the maximum paking of spheres of uniform size, provided that the assumption of dependent sattering is valid.

6 1318 Journal of the Amerian Cerami Soiety Pilon and Viskanta Vol. 86, No. 8 Table II. Parameters and Major Charateristis of the Bubble Size Distribution Funtions Distribution A B f v 0.2 f v 0.6 a max (mm) N T (no./m 3 ) a max (mm) N T (no./m 3 ) Modified gamma Modified gamma Modified gamma (2) Radiation Charateristis of the Glass Containing Bubbles In this setion we first disuss the effet of the void fration and of the bubble radius on the radiation harateristis of soda-lime siliate glass ontaining bubbles of uniform size. Then, polydispersed bubble louds are onsidered and the results on the effet of their size distribution are presented. (A) Uniform Distribution: Effet of Bubble Radius and Void Fration: As a pratial example, the spetral absorption and extintion oeffiients as well as the single sattering albedo have been predited for lear soda lime siliate glass ontaining monodispersed bubbles for a volumetri void fration of 0.2. Three different radii a were onsidered 0.2, 0.8, and 1.6 mm. Note that in the limiting ase when f v 0, i.e., for dense glass, the sattering oeffiient and the single sattering albedo vanish. Table I summarizes the simulations for monodispersed bubbles and the orresponding number of bubbles ontained per ubi entimeter of glass. From Fig. 5 one an see that the presene of bubbles redues the absorption oeffiient in the spetral region of 0.4 to 4.5 m where the absorption oeffiient of the glass is relatively small. In this same region, the extintion oeffiient is strongly affeted by the presene and the size of the bubbles. The sattering is partiularly important for smaller bubbles and the single sattering albedo is lose to unity. This indiates that the radiative transfer is dominated by sattering rather than by absorption for m. In the spetral region 4.5 to 10 m, however, the absorption oeffiient of the dense glass is large and the presene and the size of the bubbles have little effet of the effetive absorption oeffiient of the glass layer. In other words, the sattering oeffiient is negligible ompared with the absorption oeffiient and the radiative transfer is dominated by absorption. Moreover, the absorption oeffiient dereases signifiantly as the void fration inreases and an be redued by up to one order of magnitude for void frations f v varying from 0.2 to 0.6 (Fig. 6). In ontrast, the extintion oeffiient and the single sattering albedo inrease as the void fration or the number of bubbles Fig. 7. Typial bubble size distributions as summarized in Table I for f v 0.2. Fig. 8. Effet of size distribution on the spetral absorption, extintion oeffiients, and single sattering albedo of soda lime siliate glass with f v 0.2.

7 August 2003 Radiation Charateristis of Glass Containing Gas Bubbles 1319 harateristis of glass ontaining different bubble size distributions but with a onstant void fration. Table II summarizes the onditions simulated while Fig. 7 shows the orresponding bubble density funtions. The effet of the maximum radius a max and of are assessed while the parameter is taken to be unity (gamma funtion). Figure 8 shows the absorption and extintion oeffiients and the single sattering albedo for different bubble size distributions at a void fration of 0.2. Figure 9 illustrates the effet of void fration on the radiation harateristis for the first bubble size distribution. In general, one an see that the bubble size distribution has a strong influene on the extintion oeffiient and single sattering albedo of glass ontaining bubbles but very little on the absorption oeffiient. It is interesting to note that the void fration seems to affet the extintion oeffiient signifiantly for the wavelength range of 0.4 m 4.5 m. Indeed, inreasing in the void fration from 0.2 to 0.6 leads to an inrease of the extintion oeffiient by a fator of 3 while the relative effet of the distribution at f v 0.6 is similar to that at f v 0.2. IV. Conluding Remarks Fig. 9. Effet of void fration on the spetral extintion oeffiient and single sattering albedo for soda lime siliate glass ontaining gas bubbles for modified gamma distribution funtion 1. inrease. This an be explained by the fat that inreasing the void fration inreases the number of satterers, while the absorption by the two-phase mixture dereases. (B) Effet of Bubble Size Distribution: So far, the bubble size distribution was assumed to be uniform. However, in reality bubbles entrapped in the glass melt are of different diameters. Sample alulations were performed for the spetral radiation This paper has presented an analysis of radiative transfer in a semitransparent glass layer ontaining gas bubbles with appliation to glass proessing and manufaturing. The results of sample alulations performed lead to the following onlusions: (1) For gas bubbles smaller than 10 m in diameter and void frations larger than 0.006, the Mie theory should be used and/or onsiderations of dependent sattering are required. (2) For gas bubbles larger than 0.1 mm in radius the analysis developed for glass foams by Fedorov and Viskanta 6,7 an be extended over the entire range of void frations (from 0 to 0.74). (3) Even small void frations affet the total apparent radiation harateristis of the glass layer ontaining large bubbles. The effet of the void fration is even more signifiant for thikner layers and where the bubble size distribution is not uniform. Therefore, in modeling the radiative heat transfer in glass melting furnaes one should onsider the effets of gas bubbles on the radiation harateristis of the glass melt sine bubbles are always present in industrial glass melting furnaes where they exist in large numbers. (4) The model ould be used as a nonintrusive method for measuring void fration and bubble size distribution in two-phase flows by using infrared spetrosopy and inverse methods in spetral regions where the liquid phase is weakly absorbing. Finally, the radiation harateristis of fused quartz ontaining bubbles have been reently determined from experimental measurements of spetral bidiretional transmittane and refletane using an inverse method. 16

8 1320 Journal of the Amerian Cerami Soiety Pilon and Viskanta Vol. 86, No. 8 Nomenlature a Bubble radius 0 Speed of light under vauum D Deirmendjian s orretion fator F Azimuth-averaged sattering phase funtion f v Bubble void volume fration g Speifi gravity or van de Hulst s absorption parameter, artan [k d /(n d 1)] I Radiation intensity I b Blakbody radiation intensity I 0 Inident intensity of ollimated radiation Q Effiieny fator k Imaginary part of the omplex index of refration l Thikness of the slab m Complex index of refration, n ik N T Total number of bubbles per unit volume n Real part of the omplex index of refration r Refletivity of the interfae ŝ Spatial oordinate vetor x Loal depth of the slab (Fig. 1) Greek symbols Size parameter, 2a/ Extintion oeffiient, Eq. (7) Surfae tension Absorption oeffiient, Eq. (5) Sattering oeffiient, Eq. (6) or Stefan Boltzmann onstant Wavelength of the inident radiation Wavenumber of the inident radiation, / 0 Frequeny of the inident radiation ˆ Line-of-sight diretion Single sattering albedo, /( ) Effetive sattering phase funtion, Eq. (9) Sattering phase funtion of the single bubble Azimuth angle Density or phase shift, 2m 1 van de Hulst s normalized size parameter, 2(n 1) Sattering angle Optial depth, x 0 ( )dx Aknowledgments We are indebted to Dr. A. G. Fedorov for helpful disussions and exhange of information. Referenes 1 M. Abramowitz and I. A. Stegun, Handbook of Mathematial Funtions. Dover Publiations, New York, B. Balkanli and A. Ungan, Numerial Simulation of Bubble Behaviour in Glass Melting Tanks. Part 1. Under Ideal Conditions, Glass Tehnol., 37 [1] (1996). 3 R. G. C. Beerkens, The Role of Gases in Glass Melting Proesses, Glasteh. Ber., 71 [12] (1995). 4 K.-B. Cheong, K.-M Moon, and T.-H. Song, Treatment of Radiative Transfer in Glass Melts: Validity of Rosseland and P-1 Approximations, Phys. Chem. Glasses, 40 [1] (1999). 5 D. Deirmendjian, Eletromagneti Sattering on Spherial Polydispersions. Elsevier, New York, A. G. Fedorov and R. Viskanta, Radiative Charateristis of Glass Foams, J. Am. Ceram. So., 83 [11] (2000). 7 A. G. Fedorov and R. Viskanta, Radiative Transfer in a Semitransparent Glass Foam Blanket, Phys. Chem. Glasses, 41 [3] (2000). 8 E. Franses, Interfaial Phenomena: Priniples, Data, and Appliations ; Ch. 2. Class Notes ChE 668, Purdue University, West Lafayette, IN, P. R. Laimbok, Foaming of Glass Melts ; Ph.D. Thesis. Tehnial University of Eindhoven, Eindhoven, Netherlands, M. F. Modest, Radiative Heat Transfer. MGraw-Hill, New York, M. Rubin, Optial Properties of Soda Lime Siliate, Sol. Energy Mater. 12, (1985). 12 C. L. Tien and B. L. Drolen, Thermal Radiation in Partiulate Media with Dependent and Independent Sattering ; pp in Annual Review of Numerial Fluid Mehanis and Heat Transfer, Vol. 1. Edited by T. Chawla. Hemisphere, New York, H. C. van de Hulst, Light Sattering by Small Partiles. Wiley, New York, R. Viskanta, Review of Three-Dimensional Mathematial Modeling of Glass Melting, J. Non-Cryst. Solids, 177, (1994). 15 R. Viskanta and M. P. Mengu, Radiative Transfer in Combustion System, Prog. Energy Combust. Si. 13, (1987). 16 D. Baillis, F. Randrianalisoa, L. Pilon, and R. Viskanta, Identifiation of Radiative Charateristis of Fused Quartz Containing Bubbles Using Disrete Ordinates Method with Fresnel Interfaes ; pp in Computational Thermal Radiation in Partiipating Media (EUROTHERM Seminar 73, Mons, Belgium, April 15 17, 2003). Subsripts Refers to wavelength-dependent quantity abs Refers to absorption Refers to ontinuous phase d Refers to dispersed phase ext Refers to extintion sa Refers to sattering Supersripts d Refers to dispersed phase (gas bubble) Refers to the ontinuous phase