V Eymet

A M Brasil

M El Haÿ

T L Farias
email: fariastl@navier.ist.utl.pt

P J Coelho

Numerical investigation of the e ect of soot aggregation on the radiative properties in the infrared region and radiative heat transfer

Keywords: Soot, Scattering, Fractal aggregates, Rayleigh-Debye-Gans=fractal aggregate (RDG-FA) theory, Integral equation formulation for scattering (IEFS)

The e ect of aggregation on soot radiative properties in the infrared region of the spectrum is numerically investigated using Rayleigh-Debye-Gans theory for fractal aggregates (RDG-FA). In order to use the RDG-FA theory for a wide range of aggregate sizes and wavelengths, the predicted phase functions, scattering and absorption coe cients are compared with a more accurate theory, the integral equation formulation for scattering-IEFS. The importance of scattering when compared with absorption is investigated, as well as the e ect of aggregation on the phase function shape and on the scattering cross section. It is concluded that in the case of small aggregates formed with small primary particles the scattering coe cient is negligible compared with the absorption coe cient, and scattering and aggregation of primary particles can be ignored. Thus, the Rayleigh approximation can be used leading to isotropic scattering. In the case of large aggregates constituted by large primary particles, aggregation becomes important and the scattering cross section is of the same order of magnitude of the absorption cross section. Moreover, the phase function becomes highly peaked in the forward direction. Therefore, the Rayleigh and the equivalent volume Mie sphere approximations are not valid, and the RDG-FA method emerges as a good compromise between accuracy and simplicity of application. However, radiative transfer calculations between two inÿnite, parallel, black walls show that scattering may always be neglected in the calculation of total radiative heat source and heat uxes to the walls. The minor in uence of scattering on the accuracy of the predictions is explained by the shift between the spectral region where scattering is important and the region where the spectral radiative heat source is large.

Introduction

Radiative heat transfer in absorbing, emitting, and scattering media at high temperatures is of particular interest in many engineering technologies, including internal combustion engines, gas turbines, industrial furnaces, ÿre safety, and atmospheric radiation [START_REF] Tien | Flame radiation[END_REF][START_REF] Viskanta | Radiation heat transfer in combustion systems[END_REF]. It is well known that the presence of soot in combustion systems enhances heat transfer rates signiÿcantly due to the continuum radiation in the infrared regions of the wavelength spectrum. Radiation predictions require the knowledge of a wide variety of properties such as soot concentration and temperature distributions in such particulate-laden media. In addition, information about the spectral radiative properties of soot, such as phase function and scattering coe cient is also required. These properties are characterized in terms of the refractive index, size and morphology [START_REF] Siegel | Thermal radiation heat transfer[END_REF].

Earliest literature on soot radiation in ames generally considered soot shape as spherical, so that simple predictions based on the Rayleigh theory for the nanosize primary particles, or the Mie theory with a volume-equivalent diameter were possible. Unfortunately, ultraÿne particles (including soot) produced in combustion systems form aggregates with a wide range of sizes and shapes [START_REF] Megaridis | Morphological description of ame-generated materials[END_REF][START_REF] Faeth | Structure of overÿre soot in buoyant turbulent di usion ames at long residence times[END_REF]. As a result of this complex morphology, Charalampopoulos and Chang [START_REF] Charalampopoulos | E ects of soot agglomeration on radiative transfer[END_REF], and Ku and Shim [START_REF] Ku | Optical diagnostics and radiative properties of simulated soot agglomerates[END_REF] investigated the e ect of soot morphology on ame radiation by considering aggregates of small primary particles having relatively low number of primary particles (¡32). Radiative properties of fractal soot aggregates have recently received considerable attention [START_REF] Farias | Computational evaluation of approximate Rayleigh-Debye-Gans=fractal-aggregate theory for the absorption and scattering properties of soot[END_REF][START_REF] Farias | The range of validity of the Rayleigh-Debye-Gans=fractal-aggregate theory for computing optical properties like fractal aggregates[END_REF][START_REF] Sorensen | Test of static structure factors for describing light scattering from fractal soot aggregates[END_REF][START_REF] Manickavasagam | Scattering matrix elements of fractal-like soot agglomerates[END_REF]. However, these investigations, as well as those reported in [START_REF] Faeth | Optical properties of soot in buoyant turbulent di usion ames at long residence times[END_REF][START_REF] Farias | Radiative heat transfer in soot combustion systems with aggregation[END_REF], were generally limited to fundamental optical cross sections at visible wavelengths because of their speciÿc interest in developing tools for laser diagnostics in particulate-laden media. Farias et al. [START_REF] Farias | Radiative heat transfer in soot combustion systems with aggregation[END_REF] investigated the e ect of aggregation on the soot phase function, scattering albedo and ame emissivity. However, no simple solution concerning the prediction of phase functions and scattering cross sections of soot, such as via Rayleigh-Debye-Gans theory for fractal aggregates (RDG-FA), was proposed which would considerably help researchers to model the radiative heat transfer in combustion enclosures where soot is present. In addition, that study focused on the near infrared and visible wavelengths and for aggregates containing up to 256 primary particles. Furthermore, it is well known that the Rayleigh theory may be used to calculate the absorption coe cient for all but the largest soot primary particles, while scattering may be neglected. However, it is still not clear whether scattering may be neglected or not for the large soot agglomerates found in combustion systems.

Based on the previous considerations, the main objective of the present study is to quantitatively analyze the e ect of soot aggregation on the scattering coe cient and phase function in the infrared region, and to investigate whether scattering may be neglected or not in the calculation of radiative heat transfer in enclosures with high soot concentration and very large soot agglomerates. Moreover, it is our purpose to verify if simple expressions based on the RDG-FA theory can be used to characterize the phase function of aggregated soot particles, in order to easily model radiative heat transfer in a sooting environment. To evaluate the RDG-FA theory in the infrared region, a more accurate theory was used. This theory, referred to as integral equation formulation for scattering (IEFS), accounts for multiple scattering and self-interactions among small spherical particles [START_REF] Farias | The range of validity of the Rayleigh-Debye-Gans=fractal-aggregate theory for computing optical properties like fractal aggregates[END_REF][START_REF] Lou | On the electromagnetic scattering and absorption of agglomerated small spherical particles[END_REF].

The di erent light scattering theories addressed and the numerical method adopted to simulate the aggregates are presented in Section 2. The evaluation of IEFS and RDG-FA theories in the infrared region is presented in Section 3. Then, an investigation of the e ect of aggregation on the scattering properties of soot is presented and the in uence of aggregation is quantiÿed. The in uence of soot aggregation on radiative uxes and radiative heat sources is investigated in Section 5. The main conclusions of the present study are summarized in the last section.

Theoretical methods

Radiative transfer equation

The radiative transfer equation (RTE) is a mathematical statement of the conservation principle applied to a monochromatic pencil (bundle) of radiation traveling along a path through a medium. Radiation traveling along a path is attenuated by absorption and out-scattering, and is enhanced by emission and in-scattered radiation from other directions. All these e ects are accounted for in the RTE [START_REF] Siegel | Thermal radiation heat transfer[END_REF][START_REF] Modest | Radiative heat transfer[END_REF] that can be expressed as

s • ∇I (r; s) = -K I (r; s) + K a I b (r) + K s 4 4 I (r; s ′ ) (s ′ ; s) d ′ ; (1)
where I (r; s) is the spectral radiation intensity at position r along a path with unit vector s, I b is the spectral blackbody radiation intensity, (s ′ ; s) is the scattering phase function, which describes the probability that a ray from direction s ′ is scattered into direction s, and K a , K s and K = K a + K s are the spectral absorption, scattering and extinction coe cients of the medium, respectively. The boundary condition for a di usely emitting and re ecting opaque wall at a point r w on the surface is given by

I (r w ; s) = I b (r w ) + n•s ′ ¡0 I (r w ; s ′ )|n • s ′ | d ′ ; ( 2
)
where n is the local unit outward normal vector, and and are the spectral emissivity and re ectivity of the surface, respectively. Here s is the direction of the outgoing radiation intensity and s ′ is the incoming direction associated with the elementary solid angle d ′ .

Two major di culties may be identiÿed in the analysis of the RTE. First, the RTE is an integro-di erential equation and an exact solution may only be obtained for a very limited set of problems. Hence, a numerical method is generally required to solve the RTE. The second problem deals with the evaluation of the scattering and absorption coe cients, as well as the phase function, which depend on wavelength, media composition, temperature, pressure, type of particles, particle concentration, etc. The present paper addresses the second problem for the particular case of soot particles.

Radiative predictions for aggregated spherical particles

The RDG-FA and the IEFS theories were employed in this study to calculate phase functions, scattering and absorption cross sections of soot aggregates. These two scattering theories are brie y described in the following sections.

The Rayleigh-Debye-Gans theory for fractal aggregates

The RDG-FA scattering theory is based on methods described in [16 -19]. Present considerations, however, will be limited to the extended version due to K oyl u and Faeth [START_REF] Faeth | Optical properties of soot in buoyant turbulent di usion ames at long residence times[END_REF], which allows for the presence of the power-law regime when ÿnding total scattering cross sections, because this regime is important for large aggregates. The major assumptions of this approach with respect to soot aggregate physical properties are as follows: spherical primary particles have constant diameters, primary particles have uniform refractive indices, primary particles just touch one another, and the aggregates are mass-fractal objects. The justiÿcation of these assumptions for soot aggregates is discussed in [START_REF] Faeth | Optical properties of soot in buoyant turbulent di usion ames at long residence times[END_REF][START_REF] Faeth | Radiative properties of ame-generated soot[END_REF].

The mass fractal approximation for aggregates of constant-diameter spherical primary particles implies the following relationship between the number of primary particles in an aggregate, N , and the radius of gyration of the aggregate, R g [START_REF] Julien | Aggregation and fractal aggregates[END_REF]:

N = k g (2R g =d p ) D f ; (3)
where d p is the diameter of a soot primary particle. The fractal dimension, D f , and the pre-factor, k g , in Eq. (3) appear to be relatively universal properties of soot aggregates, as described later on. Thus, Eq. ( 3) provides a critical relationship between N , a quantity that is readily measured, and R g , an important parameter required by the RDG scattering theory.

In addition to the RDG approximation, it is also assumed that the primary particles are small enough to satisfy the Rayleigh scattering approximation as individual particles, and multiple scattering is neglected. In other words, the light scattered in the far ÿeld domain is composed of a vector summation of the scattering light of each primary particle. Assuming that the individual particles act like Rayleigh scatterers is reasonable because primary particle optical size parameters, x p , are generally lower than 0.1 for soot in the infrared wavelength ranges. This implies that the total scattering and absorption cross sections of individual primary particles are within 1% and 5%, respectively, of estimates based on the Rayleigh scattering approximation for typical refractive indices for soot [START_REF] Faeth | Radiative properties of ame-generated soot[END_REF]. This yields the following expressions for the absorption, C p a , and scattering, C p s , cross sections of individual primary particles [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF][START_REF] Kerker | The scattering of light and other electromagnetic radiation[END_REF]:

C p a = 4 x 3 p k 2 Im m 2 -1 m 2 + 2 ; (4a) C p s = 8 x 6 p 3k 2 m 2 -1 m 2 + 2 2 ; (4b)
where m = Á + iÄ is the complex refractive index of soot, k is the wave number and Im denotes the imaginary party of a complex number. According to the RDG approximation, di erential scattering cross sections for aggregates of a given size (after averaging over all orientations of each aggregate within a statistically-signiÿcant monodisperse aggregate population) satisfy the following formulas [START_REF] Kerker | The scattering of light and other electromagnetic radiation[END_REF]:

C a vv ( ) = C a hh ( )=cos 2 ( ) = N 2 C p vv f(qR g ); (5)
where C a and C p denote cross sections of an agglomerate and a primary particle, respectively, the double subscripts identify the direction of polarization of the incident and scattered light (vv: verticalvertical; hh: horizontal-horizontal), respectively, is the scattering angle and q is the modulus of the scattering vector. The form factor, f(qR g ), is expressed as follows in the Guinier and power-law regimes [16 -18]:

f(qR g ) = exp(-q 2 R 2 g =3) Guinier regime (where q 2 R 2 g 6 1:5D f ); (5a)
f(qR g ) = (qR g ) -D f power-law regime (where

q 2 R 2 g ¿ 1:5D f ): (5b)
Here, the boundary between the Guinier and power-law regimes has been chosen according to Dobbins and Megaridis [START_REF] Dobbins | Absorption and scattering of light by polydisperse aggregates[END_REF] to match the value and the derivative of f(qR g ) where the two regimes meet. The total scattering cross section is given by

C a s = N 2 C p s g(k; R g ; D f ); (6)
where the function g(k; R g ; D f ) is given in [START_REF] Faeth | Optical properties of soot in buoyant turbulent di usion ames at long residence times[END_REF]. It is also assumed that absorption is not a ected by aggregation, while the extinction cross section is the sum of the absorption and scattering cross sections by deÿnition, i.e.,

C a a = NC p a ; (7)
C a e = C a a + C a s : (8)
The phase function is given by the following equation:

a ( ) = 4 C a s C a vv ( ) + C a hh ( ) 2 : (9)

The integral equation formulation for scattering (IEFS)

The IEFS theory [START_REF] Farias | The range of validity of the Rayleigh-Debye-Gans=fractal-aggregate theory for computing optical properties like fractal aggregates[END_REF][START_REF] Farias | Radiative heat transfer in soot combustion systems with aggregation[END_REF][START_REF] Lou | On the electromagnetic scattering and absorption of agglomerated small spherical particles[END_REF] divides an aggregate into su ciently small units (the primary particles) so that the internal ÿeld within each particle is assumed to be uniform, and includes multiple scattering e ects, self-interaction contributions, and satisÿes energy conservation. The assumption of uniform electric ÿeld within a primary particle does not hold for touching spheres with large values of |m -1|, regardless of the smallness of x p , and it has been shown that the contact point between the primary particles substantially increases the scattering and absorption properties of an aggregate [START_REF] Mackowski | Electrostatics analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles[END_REF]. However, soot aggregates, like any other type of combustion generated aggregates, are not constituted by single point touching particles. A certain degree of penetration of the neighboring primary particles, generally referred to as partial sintering, is present. This e ect eliminates the single point contact area that is primarily responsible for the substantial increase of scattering and absorption properties. In the IEFS theory, the internal electrical ÿeld of each particle, E j , is obtained from the following system of 3N × 3N linear equations:

E j = 3 m 2 + 2 E inc; j + i m 2 -1 m 2 + 2 x 2 p j 1 (x p ) N l=1;l̸ =j
T jl E l + s j E j ; j = 1; 2; : : : ; N; [START_REF] Sorensen | Test of static structure factors for describing light scattering from fractal soot aggregates[END_REF] where E inc = E o exp(ikz) represents the incident electric ÿeld propagating along the z-axis with a wave number of k = 2 = , j 1 (x p ) is the ÿrst-order spherical Bessel function of the ÿrst kind, s j is the coe cient for self interaction and T is the scattering matrix. Once the internal ÿeld of each spherical particle is known from Eq. ( 10), various optical cross sections can be obtained for an aggregate with N uniform size particles (see, e.g., [1 3]). The di erential scattering cross sections are given by

C a pp (Â; ) = 1 k 2 x 4 p j 2 1 (x p )|m 2 -1| 2 N j=1 exp(-ikr j cos ÿ j )(E j; Â Â + E j; ˆ ) pp 2 : (11)
In this equation the direction of the scattered ÿeld is represented using spherical coordinates, and  and stand for the polar and azimuthal angles (unit vectors denoted by  and ˆ ). The position of each individual particle is deÿned by the coordinates (r j ;  j ; j ). The direction of polarization is denoted by subscript pp, which may stand for vv, vh, hv or hh, and cos ÿ j = cos  j cos  + sin  j sin  cos( j -). The total scattering and absorption cross sections are given by

C a s = 4 3k 2 x 4 p j 2 1 (x p )|m 2 -1| 2 N j=1 N k=1 E j Re( T jk )E * k ; (12a) C a a = 4 k 2 x 2 p j 1 (x p ) Im(m 2 -1) N j=1 |E j | 2 ; (12b)
where the superscript * represents the complex conjugate, and Re and Im denote the real and imaginary parts of a complex number, respectively. The phase function may be determined from Eq. ( 9). This formulation satisÿes the optical theorem, i.e., the sum of the total scattering and absorption cross sections is exactly equal to the extinction cross section of an aggregate [START_REF] Lou | On the electromagnetic scattering and absorption of agglomerated small spherical particles[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. Unfortunately, the IEFS method is computationally intensive for treating relatively large and=or polydisperse soot aggregates, because it requires the knowledge of the spatial position of the primary particles, as well as a large population of aggregates to obtain statistically signiÿcant results.

Numerical simulation of fractal-like aggregates

To compute the radiative properties using the IEFS theory it is necessary to know the primary particle positions within each aggregate. It has been shown that soot aggregates follow a mass fractal-like morphology that can be represented by Eq. ( 3). In this work, aggregates were numerically generated using a practical sequential algorithm that models the statistical fractal relationship given by that equation, rather than using the formal di usion-limited cluster-cluster simulations [START_REF] Farias | Computational evaluation of approximate Rayleigh-Debye-Gans=fractal-aggregate theory for the absorption and scattering properties of soot[END_REF][START_REF] Farias | The range of validity of the Rayleigh-Debye-Gans=fractal-aggregate theory for computing optical properties like fractal aggregates[END_REF][START_REF] Brasil | A recipe for image characterization of fractal-like aggregates[END_REF][START_REF] Brasil | Evaluation of the fractal properties of cluster-cluster aggregates[END_REF]. Brie y, the aggregation process is initiated by randomly attaching individual and pairs of particles to each other. Once the position of every primary particle is known, the radius of gyration of the new aggregate is calculated from its deÿnition, and it is checked if the fractal relationship given by Eq. ( 3) is satisÿed. This procedure continues in order to form progressively larger clusters. Thirty-two di erent realizations of the same aggregate size, each sampled at 16 di erent orientations, were averaged to obtain statistically signiÿcant IEFS predictions with ¡10% numerical uncertainty (95% conÿdence interval).

Soot fractal properties

In spite of the complex appearance, various size and shape soot aggregates have been found to have a universal morphology with D f ∼ = 1:6-1.9. On the contrary, the selection of universal values for the fractal pre-factor for cluster-cluster aggregates has been a controversial issue. Wu and Friedlander [START_REF] Wu | Note on the power-law equation for fractal-like aerosol agglomerates[END_REF], and more recently K oyl u et al. [START_REF] Faeth | Fractal and projected structure properties of soot agglomerates[END_REF], have uniÿed most of experimental and numerical data reported on this property. For example, the results published in the literature for k g in cluster-cluster aggregates indicate values between 1.23 [START_REF] Cai | Analysis of fractal cluster morphology parameters: structural coe cient and density autocorrelation function cuto[END_REF] and 3.47 [START_REF] Samson | Structural analysis of soot aggregates[END_REF]. The range of published results varies by ¿ 200%, clearly showing that this issue deserves further attention. In spite of the range of values published, it was found that the results based on numerically simulated aggregates are systematically lower than the values inferred from experimental data. Oh and Sorensen [START_REF] Oh | The e ect of overlap between monomers on the determination of fractal cluster morphology[END_REF] have investigated the e ect of overlapping between primary particles on the fractal properties and concluded that the fractal dimension, and particularly the fractal pre-factor, may vary considerably with the degree of overlapping. However, no ÿnal conclusion regarding the discrepancies between experimental results and numerical ÿndings was drawn. Brasil et al. [START_REF] Brasil | Evaluation of the fractal properties of cluster-cluster aggregates[END_REF] found that the large discrepancies between the values reported for k g appear to be related to the partial overlapping of primary particles, which leads to an increase of k g while keeping the fractal dimension approximately constant. Therefore, it can be concluded that aggregates without overlapping should be simulated having lower values of k g , while k g will increase as overlapping increases, as systematically observed by several authors.

More recently, Brasil et al. [START_REF] Brasil | Evaluation of the fractal properties of cluster-cluster aggregates[END_REF] combined numerical predictions of light scattering with morphological fractal concepts based on Eq. ( 3) to estimate the fractal dimensions of cluster-cluster simulated aggregates. Their best estimates were k g = 1:3 and D f = 1:8. These results are in very good agreement with the great majority of the values presented by other authors for simulated aggregates (see [START_REF] Brasil | Evaluation of the fractal properties of cluster-cluster aggregates[END_REF] and references cited therein). Based on the previous considerations, aggregates used throughout the present work were simulated in such a manner to have k g ∼ = 1:3 and D f ∼ = 1:8.

Evaluation of the light scattering theories

Preliminary considerations

One of the main goals of the present work is to use the RDG-FA theory to investigate the e ect of aggregation on the radiative properties. However, the RDG-FA theory neglects multiple scattering and, therefore, it is necessary to investigate its accuracy within the range of refractive indices typically exhibited by soot in the infrared region. This issue was addressed by comparing the results of RDG-FA with predictions from the more accurate IEFS theory within a wide range of x p , N and m. The applicability of IEFS within this range was conÿrmed by using a compact aggregate with a spherical shape and comparing IEFS predictions with Mie results for an optically equivalent sphere. Based on these considerations, the applicability of IEFS is investigated ÿrst, and then the RDG-FA theory is evaluated by comparing its predictions with the results obtained from the IEFS theory.

A wide range of primary particle sizes and number of primary particles per aggregate was covered, namely, x p in the range 0.01-0.1 and N between 1 (a single Rayleigh scatterer) and 256. Primary particles in soot aggregates usually have diameters ¡ 60 nm, which combined with wavelengths within the infrared region (1-10 m) indicates that primary particle size parameters fall within the deÿned range. Concerning the refractive index, typical values that characterize soot particles in the infrared range were considered, namely |m -1| = 1 to 4, and the following relationship between the real and imaginary parts of the refractive index was assumed: n = k + 1. This assumption drastically simpliÿes the presentation of the results and closely follows the available data for the complex refractive index of soot in the infrared spectrum [START_REF] Siegel | Thermal radiation heat transfer[END_REF].

Evaluation of IEFS theory

The IEFS theory was checked by computing the scattering properties of sphere-like aggregates, and by comparing the results with Mie scattering predictions for optically equivalent sphere-like aggregates. In order to apply Mie scattering predictions to clusters involving N spherical primary particles with a spherical outer boundary, the following Maxwell-Garnett relationship was used:

(m 2 e -1) (m 2 e + 2) = f f (m 2 -1) (m 2 + 2) : (13)
This equation has been used by several investigators (see, e.g., [START_REF] Ku | A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles[END_REF]) to ÿnd the e ective refractive index of an equivalent sphere, m e , as a function of the ÿlling factor, f f , deÿned as the ratio of the volume of the computational cell ( d 3 p =6 for spherical cells) to the volume of bulk material. The analysis was carried out using a sphere-like aggregate, similar to the arrangement employed in [START_REF] Ku | A comparison of solutions for light scattering and absorption by agglomerated or arbitrarily-shaped particles[END_REF][START_REF] Purcell | Scattering and absorption of light by nonspherical dielectric grains[END_REF], formed by 136 spherical primary particles in a cubical lattice with a spherical outer boundary. The choice of a compact cluster for evaluating the performance of IEFS theory is quite conservative, because it represents the limiting case of a compact aggregate with fractal dimension, D f , approaching three. In such an aggregate, e ects of multiple scattering between primary particles are more intense than for practical fractal-like aggregates, such as soot, that exhibit more open-structure morphologies with D f ¡ 2. Optical cross sections, including absorption, total scattering cross sections and phase function were computed for the sphere-like cluster after averaging the calculations over 12 di erent orientations. This was necessary to reduce the e ect of the staggered shape of the sphere-like cluster, which can a ect the large angle scattering regimes.

The good accuracy of the IEFS theory for the di erent radiative properties can be seen in Figs. 123, which show the contours of the relative error of this theory, taking as exact the Mie equivalent sphere results. In fact, the errors of the absorption and scattering cross sections never exceed 6%, while the errors of the phase function are always ¡ 2:5%, indicating that the IEFS formulation is suitable to estimate the radiative properties.

Evaluation of RDG-FA theory

Fig. 4 shows the percent deviation contours of the RDG-FA predictions from the IEFS results for the absorption cross section of soot aggregates. The percent deviation is deÿned as the ratio of the modulus of the di erence between the values calculated by the two theories to the value computed using the IEFS theory, multiplied by 100. The results show that the RDG-FA approximation is best for small values of |m -1| and x p , as a consequence of the fundamental assumptions involved in its derivation. Similar observations were reported for spheres and cylinders by several investigators, as summarized in [START_REF] Kerker | The scattering of light and other electromagnetic radiation[END_REF]. However, the requirements for the applicability of the RDG-FA theory are not as strict as might be expected, because percent deviations ¡ 35% are found over almost the whole range of parameters under investigation. From a physical point of view, low percent deviations are found whenever the absorption cross sections of an aggregate and the individual particles composing them act similarly, i.e., Eq. ( 7) is satisÿed. The RDG-FA predictions deviate more from the IEFS results as N increases. Nevertheless, the RDG-FA theory seems to be fairly reliable throughout most of the domain considered, except in the regions where large x p and |m -1| are combined.

The percent deviation contours of the RDG-FA predictions from the IEFS results for the total scattering cross sections of aggregates are plotted in Fig. 5. The results exhibit trends similar to those reported for the absorption cross section. They conÿrm that the refractive index of particles does not have to be much smaller than 1, which is one of the classical requirements for the validity of the RDG theory [START_REF] Kerker | The scattering of light and other electromagnetic radiation[END_REF]. In fact, the results suggest that the RDG-FA predictions are in fair agreement with the IEFS predictions for |m -1| ¿ 2 as long as N and x p are small. Similar percent deviation contours for the phase function at scattering angles of 0

• and 90

• are illustrated in Fig. 6. Calculations were also performed for a scattering angle of 180

• , but the results are not shown, since the order of magnitude of the deviation between the two theories is about the same found for = 90

• . Low percent deviations are associated to the range of |m -1| and x p values for which multiple scattering and self-interaction e ects are negligible. It is interesting to note that the RDG-FA theory predicts the shape of the phase function with very good accuracy for all cases considered, as long as the primary particle size of the aggregate is small (i.e., x p ¡ 0:1). In all cases the deviation between IEFS and RDG-FA theories is below 20%.

In summary, the results presented for cluster-cluster fractal-like aggregates are very encouraging, suggesting that the RDG-FA theory is a good approximation for a large range of primary particle sizes and refractive indices. In particular, and apart from a small region in the upper limits of the domain covered, namely x p and |m-1| close to 0.1 and 4, respectively, deviations between RDG-FA and IEFS predictions are always lower than 30%. Therefore, the RDF-FA theory may be used with due caution to investigate the general impact of aggregation on the optical and radiative properties of soot in the infrared region of the spectrum.

E ect of aggregation on the scattering properties of soot

It is well known that scattering is negligible compared with absorption for primary soot particles. This section examines whether this is still true or not for soot agglomerates. Fig. 7 shows the ratio of the total scattering cross section to the absorption cross section for di erent values of the refractive index. It is observed that, for a given complex refractive index, the ratio C a s =C a a increases with N for a ÿxed x p , and increases with x p for a ÿxed N . For an aggregate with given x p , the ratio increases with |m -1|, i.e., with the wavelength. The absorption cross sections are consistently superior to scattering cross sections throughout the range of properties under investigation. However, for large values of m, N and x p , absorption and total scattering cross sections are of the same magnitude, i.e. C a s =C a a ¿ 0:1. For example, scattering may become relevant when compared with absorption for aggregates with N ¿ 200, x p ≈ 0:1 and |m -1| ¿ 2, because of the relatively high scattering cross section combined with a phase function dominated by forward scattering. Eq. ( 6) shows that scattering is directly proportional to N 2 as long as the aggregates are small, i.e., dominated by the small angle (Guinier) regime, and therefore the impact of aggregation grows linearly with N . However, as the aggregate size increases, the power-law regime becomes important, continuously reducing the e ect of aggregation on the total scattering cross section. This trend has been discussed in the past [START_REF] Farias | Computational evaluation of approximate Rayleigh-Debye-Gans=fractal-aggregate theory for the absorption and scattering properties of soot[END_REF], and can be observed in Fig. 8. Apart form a small region where large N and x p are combined, the Guinier regime appears to be dominant throughout the scattering angles. Therefore, it can be concluded that scattering is strongly a ected by aggregation, i.e. the ratio C a s =NC p s is proportional to N . Nevertheless, the region where this e ect becomes less important (large N and x p ) coincides with the one where scattering becomes more important when compared with absorption. This suggests that in most practical situations the impact of aggregation will not be severe.

Fig. 9 shows the phase function of a population of aggregates with N =256; x p =0:1 and |m-1|=1 calculated using the RDG-FA theory. The predictions obtained using the Mie theory for a sphere with the same volume and a single Rayleigh scatterer are also plotted. It is worth to point out that none of these methods (RDG-FA, Mie theory for a sphere with the same volume and single Rayleigh scatterer) re ect reality, and so the results obtained must be regarded with caution. Nevertheless, they have been used here since more accurate theories are not currently available, apart from the IEFS theory, which has been shown to yield results relatively close to the RDG-FA theory for the range of parameters under consideration. Fig. 9 shows that neither Mie equivalent volume spheres nor Rayleigh scatterers predict the highly forward phase functions that fractal-like aggregates such as soot exhibit. In fact, the forward to backward scattering ratio (0

• )= (180 • ) predicted by RDG-FA Fig. 8. E ect of aggregation on the total scattering cross section.

Fig. 9. Phase function of a population of aggregates with N = 256, x p = 0:1 and |m -1| = 1 calculated using the RDG-FA theory, the Mie theory for an equivalent volume sphere and the Rayleigh theory for a single particle. Fig. 10. In uence of aggregation on the shape of the phase function. R is deÿned as [ (0

• )= (180 • )] a =[ (0 • )= (180 • )] p .
is about 10, while the ratio computed using either Mie or Rayleigh theories is of the order of 1. Additional calculations were carried out for values of |m -1| up to 4 using the appropriate wavelengths. It was found that the strong forward peak observed in Fig. 9 for the RDG-FA theory becomes substantial small as m increases, and the di erence between the results predicted by the di erent methods decreases, becoming negligible for |m -1| = 4. The importance of aggregation on the shape of the phase function, namely on the forward scattering peak, is illustrated in Fig. 10, which shows a 3D surface map of R, deÿned as [ (0 180• )] p , the subscripts a and p standing for an aggregate and a primary particle, respectively. The results show that R is large, i.e., aggregation has a strong impact on the shape of the phase function, if both N and x p are large.

• )= (180 • )] a = [ (0 • )= (

In uence of soot aggregation on radiative transfer calculations

The in uence of soot aggregation on radiative transfer calculations is investigated in this section. Radiative transfer between two parallel inÿnite black walls at 300 K is considered. The medium between the walls is at a temperature of 2000 K, and contains a uniform distribution of soot, but no participating gases. The distance between the plates is taken as L = 1 m. It is assumed that soot forms agglomerates with N = 256 and d p = 50 nm. The soot volumetric fraction, f v , varies in the range 10 -6 -10 -5 . The data for the refractive index of acetylene soot as a function of wavelength reported in [START_REF] Dalzell | Optical constants of soot and their application to heat ux calculations[END_REF] is used in the present calculations. The radiative properties of soot are calculated according to the RDG-FA theory.

The problem described above has an exact solution if there is no scattering and the absorption coe cient of the medium is prescribed (see, e.g., [START_REF] Modest | Radiative heat transfer[END_REF]). However, there is no analytical solution if scattering is taken into account. Since our main goal is to investigate the in uence of scattering in the presence of soot aggregates on radiative heat transfer, a radiation model was used both with and without scattering. The calculations were performed using the discrete ordinates method [START_REF] Chandrasekhar | Radiative transfer[END_REF], the step discretization scheme [START_REF] Chai | Evaluation of spatial di erencing practices for the discrete ordinates method[END_REF], a uniform grid with 101 control volumes, and an angular discretization with 10 polar and azimuthal equally spaced divisions per octant, yielding 100 discrete ordinates per octant. The quadrature weights are taken as the areas of the unit sphere that subtend the solid angles resulting from the angular discretization, like in the ÿnite volume method [START_REF] Raithby | A ÿnite-volume method for predicting a radiant heat transfer in enclosures with participating media[END_REF]. This allows a much ÿner angular discretization than if the popular S N quadratures were used, since the order of these quadratures is limited by the appearance of negative weights. It was found that the S 16 quadrature does not accurately integrate the phase functions with a strong forward peak that occur for low values of |m -1|, contrary to the selected angular discretization. The dependence of the radiative properties on the wavelength was accounted for by dividing the spectrum from 0 to 100 m into 40 non-uniform intervals, and integrating the radiative transfer equation over each interval [START_REF] Fiveland | Three-dimensional spectral radiative heat transfer solutions by the discrete-ordinates method[END_REF].

Fig. 11 shows the predicted radiative heat source, ∇•q, where q is the heat ux vector. The proÿle is at in the central region of the domain and increases sharply towards the walls in their vicinity. The radiative source at the center of the domain decreases with the increase of soot volumetric fraction because the optical thickness of the medium becomes so large that the emitted radiation is rapidly absorbed, and radiation loss is only signiÿcant in the neighborhood of the colder walls. However, the heat ux incident on the walls increases with the increase of soot volumetric fraction, as expected.

The radiative source proÿles plotted in Fig. 11 were computed without scattering, but those calculated with scattering are almost indistinguishable from them. This means that scattering may be neglected in heat transfer calculations even for the high soot concentrations and large aggregates under consideration. To give a better insight on the reason why the role of soot scattering is marginal, the predicted spectral radiative heat source, ∇ • q , at the center of the domain (x = L=2) is shown in Fig. 12. Similar conclusions would be drawn if other points were selected, but the in uence of scattering would be even lower. Fig. 12 also shows the relative di erence between calculations of the spectral radiative source neglecting scattering (subscript ns) and those accounting for scattering (subscript s), denoted by :

= |(∇ • q ) ns -(∇ • q ) s |=(∇ • q ) ns : (14)
The calculations performed without scattering were compared with the exact solution [START_REF] Modest | Radiative heat transfer[END_REF], and it was found that the error is of the order of 0.1%. Fig. 12a shows that the spectral radiative source proÿle exhibits a maximum, which is displaced to longer wavelengths if the soot volumetric fraction increases. The relative di erence between the calculations with scattering and those without scattering, , may achieve quite high values for short wavelengths, but decreases sharply with the increase of the wavelength. In fact, it was shown above that the phase function has a strong forward peak for |m -1| = 1, and these values of the refractive index occur for short wavelengths. As the wavelength increases, so does |m -1|, and the phase function approaches that of a Rayleigh scatterer. In this situation, absorption is dominant and scattering is negligible. For example, if f v = 10 -6 , then is ¿ 5% if ¡ 1 m. However, in the region where ¡ 1 m, the contribution of ∇ • q to ∇ • q is very small, and therefore the in uence of scattering is marginal. Although the evaluation of the RDG-FA theory presented above has been restricted to x p up to 0.1, which corresponds approximately to ¿ 1:5 m, previous studies have shown that the theory is also reliable for larger values of x p , i.e., shorter wavelengths, provided that |m -1| does not exceed the unity [START_REF] Farias | Computational evaluation of approximate Rayleigh-Debye-Gans=fractal-aggregate theory for the absorption and scattering properties of soot[END_REF][START_REF] Farias | The range of validity of the Rayleigh-Debye-Gans=fractal-aggregate theory for computing optical properties like fractal aggregates[END_REF]. If f v = 2 × 10 -6 , then is ¿ 5% if ¡ 2 m, and exceeds 10% for ¡ 1:3 m. However, the contribution of these short wavelengths to the total radiative source is small, as illustrated in Fig. 12a, and therefore the in uence of scattering to the total radiative source is again very small. The range of wavelengths where scattering is important increases with the increase of f v , but remains conÿned to a region of the spectrum that has a little contribution to the total radiative source. Even for f v as large as 10 -5 the in uence of scattering on the total radiative source is negligible.

The in uence of the distance between the walls is examined in Fig. 13 for f v = 10 -5 . It can be seen that L has a very important in uence on the value of the radiative heat source, but the in uence of scattering is again negligible. In fact, if the distance between the walls increases, so does the range of wavelengths where scattering is relevant, but the region where the spectral radiative heat source is largest shifts to longer wavelengths.

The radiative heat source at x = L=2 and the incident heat ux on the walls are given in Tables 1 and2, respectively, for calculations with and without scattering, and for the range of f v and L reported above. The radiative heat source does not di er by ¿ 3% between the two sets of calculations, except in the case of f v = 10 -5 and L = 10 m where scattering yields a radiative heat source that is about 6% lower than that calculated without scattering. The incident heat uxes are even closer for the two sets of calculations, di ering by no more than 0.62%.

Additional calculations were performed for a temperature of the medium equal to 1500 K, and the results are very similar to those reported above. Finally, the calculations were repeated for aggregates Fig. 13. Predicted spectral radiative heat source at a point equidistant from two parallel inÿnite black walls as a function of the distance between the walls. (a) Predicted spectral proÿles computed with and without scattering. (b) Relative di erence between the calculations with scattering and the calculations without scattering.

Table 1

Radiative heat source at x = L=2 for N = 256 with N =1024. Although the evaluation of the RDG-FA theory has not been extended to such a high value of N , and the accuracy may be lower, the results obtained show that the incident heat uxes and radiative heat sources are only slightly di erent from those calculated for N = 256. For example, for f v = 10 -5 and L = 1 m, the incident heat ux is 898:6 kW=m 2 and the radiative heat source at x = L=2 is 57:64 kW=m 3 for N = 1024, compared to 899:2 kW=m 2 and 58:24 kW=m 3 , respectively, for N = 256. The maximum di erence between the two sets of calculations does not exceed 0.7% for the incident heat ux and 3.5% for the radiative heat source for the range of parameters under consideration, except for f v = 10 -5 and L = 10 m where the di erence between the radiative heat sources increases to 11.7%. This suggests that scattering remains negligible for such large soot aggregates.

f v L (m) (∇ • q) ns (kW=m 3 ) (∇ • q) s (kW=m

Conclusions

The e ect of aggregation on soot optical properties was investigated emphasizing the infrared region of the spectrum. The numerical study was performed quantitatively using the RDG-FA theory. An evaluation of the accuracy of this theory was performed by means of comparison of predicted phase functions, scattering and absorption coe cients with the results of the more accurate IEFS theory, in order to investigate the applicability of the RDG-FA theory to a wide range of aggregate sizes and wavelengths. Following the successful evaluation, the RDG-FA method was used to investigate the e ects of aggregation on soot optical cross-sections and phase function. The results of this study allow us to distinguish two main situations of interest. If N and x p are both low (small aggregates formed with small primary particles) the results show that the scattering coe cient is negligible compared to the absorption coe cient. Within this range, scattering and aggregation of primary particles can be ignored. Thus, the Rayleigh approximation can be used leading to isotropic scattering. On the other hand, if both N and x p are large, aggregation starts to become important, contributing to an increase in the scattering cross sections, which reach values of the same order of magnitude of the absorption cross sections. In addition, the phase function becomes highly peaked in the forward direction. Therefore, the Rayleigh and equivalent volume Mie sphere approximations appear to be no longer applicable, and the RDG-FA method emerges as a good compromise between accuracy and simplicity of application. However, calculations performed for radiative transfer between two inÿnite, parallel, black walls with a medium constituted by a transparent gas and soot have shown that scattering from soot may be neglected even for large soot aggregates and high soot concentration. In fact, the spectral radiative heat source is small in the region of the spectrum where scattering is signiÿcant, no matter the distance between the walls. Therefore, the spectral region where scattering is important has a marginal contribution to the total radiative source and incident heat uxes.

Fig. 1 .

 1 Fig.1. Contours of the relative error of the absorption cross section calculated using the IEFS theory.

Fig. 2 .

 2 Fig.2. Contours of the relative error of the scattering cross section calculated using the IEFS theory.

Fig. 3 .

 3 Fig. 3. Contours of the relative error of the phase function calculated using the IEFS theory: (a) = 0 • ; (b) = 90 • ; (c) = 180• .

Fig. 4 .

 4 Fig. 4. Percent deviation contours of the RDG-FA predictions from the IEFS results for the absorption cross section of soot aggregates: (a) N = 16; (b) N = 64; (c) N = 256.

Fig. 5 .

 5 Fig. 5. Percent deviation contours of the RDG-FA predictions from the IEFS results for the scattering cross section of soot aggregates: (a) N = 16; (b) N = 64; (c) N = 256.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Percent deviation contours of the RDG-FA predictions from the IEFS results for the phase function of soot aggregates: (a) N = 16; = 0 • ; (b) N = 16; = 90 • ; (c) N = 64; = 0 • ; (d) N = 64; = 90 • ; (e) N = 256; = 0 • ; (f) N = 256; = 90 • .

Fig. 11 .

 11 Fig. 11. Predicted radiative heat source between two parallel inÿnite black walls as a function of soot volumetric fraction.

Fig. 12 .

 12 Fig. 12. Predicted spectral radiative heat source at a point equidistant from two parallel inÿnite black walls as a function of soot volumetric fraction. (a) Predicted spectral proÿles computed with and without scattering. (b) Relative di erence between the calculations with scattering and the calculations without scattering.

Table 2

 2 Incident radiative heat uxes at the wall for N = 256f v L (m) (q w ) ns (kW=m 2 ) (q w ) s (kW=m 2 )

	10 -6	1.00 811.1	807.5
	2 × 10 -6 1.00 870.9 5 × 10 -6 1.00 891.6 10 -5 1.00 904.6	865.6 892.3 899.2
	10 -5	0.01 289.4	289.8
	10 -5	0.10 811.1	807.5
	10 -5	10.00 907.2	904.9

Acknowledgements

The ÿnancial support of ICCTI (Instituto de Coopera cão Cientà ÿca e Tecnolà ogica Internacional) and French embassy in Lisbon within the framework of project 304-B4 is acknowledged. Part of the present research work was funded by FCT (Funda cão para a Ciência e a Tecnologia) through project PRAXIS EME=12035=1998.


L Gaete-Garretã

Y Vargas-Hernandez

A Chamayou

J A Dodds

W Valderama-Reyes

F Montoya-Vitini

Development of an ultrasonic high-pressure roller press

Keywords: Grinding, Ultrasound, Roller press, Mill, Wear, Energy

It is widely known that particle size reduction processes have a low energetic e ciency. In this paper, we present experimental results obtained by a new grinding machine using ultrasound to enhance the performance of a roller mill. The required energy consumption for a comminution operation appears to be signiÿcantly reduced by careful application of an ultrasonic ÿeld in the grinding zone. A lower stress on the shafts and a reduction of the required torque were also observed. The lower mechanical stress should also cause less abrasive wear on the rolls.

Introduction

One of the options to improve the e ciency of comminution is the application of an ultrasonic ÿeld in the stressing zone. There are well-known related applications of ultrasonics in drilling and in the destruction of kidney stones. Since the 1950s, many researchers have been exploring the fragmentation e ect of ultrasonic ÿelds in suspensions and solids under compression. G artner (1959), was probably the ÿrst who intentionally tried the application of ultrasonic waves to the fragmentation of particles in suspension. His results were very poor and have little technical interest. [START_REF] Fleischauer | Forschungsberichte des Landes Nordrhein-Westfalen[END_REF] used ultrasonic waves to fragment coal in suspension for subsequent solvent extraction. The applied power (400 W) is relevant and the active zone design is still interesting. [START_REF] Gra | Ultrasonic comminution[END_REF] arranged many transducers in a multistage stamp mill. Quantitative information about his results is very scarce. [START_REF] Leach | Fragmentation of rocks under ultrasonic loading[END_REF], studied the fragmentation of drill blocks ÿxed to an ultrasonic transducer. They observed preferential fracture at the node and better results with brittle materials. Soft rocks heat up, giving less e cient energy utilisation. Tarpley and Moulder [START_REF] Tarpley | E cient ultrasonic grinding: A new technology for micron sized coal[END_REF] studied the grinding of coal under compression, and developed several devices intended to produce ÿne particles for combustion. Their reported values of energy consumption were as low as 3 kWh=ton, much lower than the 20 kWh=ton usually required in hammer mills to obtain a product 80% under 75 m (200 mesh). For grinding below 10 m, they report 24 kWh=ton, one order of magnitude lower than normal value of 300 kWh=ton. A conical device with an auger spiral was reported to be able to grind 300 lb=h (approx. 135 kg=h) at 5-7 kWh=ton. They designed other devices that ÿnally were not tested. Later, in 1988, other researchers (Link & Killmeyer, 1988) using the same machines were not able to reproduce the previously reported results. Lo and Kientzler (1992, chapter 47) recovered one of the Tarpley and Moulder machines and after redesigning the ultrasonic wave generator, applied the resulting device to grinding of some species. Their results have an energy consumption similar to that of a ball mill, but no better. After identifying some weaknesses in their device, they proposed a new design [START_REF] Lo | Fundamentals and system research in ultrasonic comminution technology[END_REF]. [START_REF] Yerkovic | Exploring the ultrasonic comminution of copper ores[END_REF] made grinding tests comparing standard ore with ultrasonic pre-treated samples in a ball mill. The pre-treated ore exhibited a 32% higher grinding rate.

In most research work, ultrasonic waves have been applied as a means to induce an accelerated fatigue cracking, due to the high frequencies involved. Most devices are composed of a vibrating surface, separated by a gap much smaller than the particle feed size. The amplitude of the vibration is approximately one order of magnitude smaller than the gap size. The process capacities that can be obtained with this design are very limited, in spite of the very low, energy consumption, levels claimed by some authors.

A key point is the in uence exerted by the ultrasonic ÿeld on the material properties. [START_REF] Blaha | [END_REF] found a decrease in yield point of single metal crystals. The softening e ect disappears by switching out the vibration, and comes back when the vibration is restarted. The amount of softening cannot be explained solely by inertial e ects on the load and displacement sensors. An energy dissipation mechanism is proposed at crystalline defects, which could cause local heating and a reduction of the slip threshold. [START_REF] Leach | Fragmentation of rocks under ultrasonic loading[END_REF], mention a 70% reduction of compressive strength of an ore in their experiments. No other data have been found on ultrasonic e ect in mineral or ceramic materials.

E ect of an ultrasonic ÿeld on breakage force of glass balls

An experiment was made on the e ect of ultrasound on the breakage of glass balls. A pneumatic press capable of applying up to 390 kPa was ÿtted with an ultrasonic transducer (Fig. 1). The transducer is of the stepped horn type, resonating at 20:4 kHz. The applied force was measured using a precision pressure gauge.

Three collections, each one formed by 25 glass spheres of 4; 5 mm nominal diameter were tested. The experiments consist of breaking each of the 25 balls individually in di erent conditions. The ÿrst group was broken without ultrasonic activation, the second group was broken with an applied ultrasonic power of 35 W, and the last one with an applied ultrasonic power of 70 W.

Fig. 2 shows the experimental results of single ball breakage tests.

The result of tests under static conditions show a mean breakage force of 11:8×10 6 N=kg with standard deviation of 3:3 × 10 6 (28.2%), whereas the test with ultrasound shows a mean breakage force of 4:8×10 6 N=kg with a standard deviation 1:4 × 10 6 (29.1%). The relative dispersion is the same for both samples, but the breakage force under ultrasonic vibration is only 40% of the static mean breakage force.

These results suggest that an ultrasonic ÿeld reduces the force needed to begin breakage. The main weakness of the high-pressure roll mills is precisely a very high stress imposed on the rolls, which promote rapid wear of their surface and some times even fracture. A vibrating roll could help to reduce this problem by prolonging the useful life of the hard face roll and avoiding, or at least reducing, the risk of fracture. It may also be expected that the additional energy spent on ultrasonic vibration could be compensated by the reduced energy requirements for grinding associated with a lower breakage force.

Design of an ultrasonic grinding mill

In order to be e cient, the design of a grinding machine must follow some general principles. An e cient grinding device would involve the direct ow of particles to the breakage zone, slow compression loading of particles, and immediate removal of the broken fragments [START_REF] Lo | Fundamentals and system research in ultrasonic comminution technology[END_REF]. Because comminution is a high energy consuming process, each stage of the process must be as e cient as possible.

The device that best fulÿls the conditions listed above is a high-pressure roll mill which could be redesigned to work under ultrasonic vibration. One requirement for the correct use of ultrasonics is a good coupling between the generator and the grinding material, which is fulÿlled by the roller stress. For e cient working, it is necessary that the ultrasonic transducer operate in resonant condition. In other words, it is best that the active parts of an ultrasonic mill oscillate in one of its natural frequencies, and best of all in its ÿrst mode which is more e cient for electro-acoustic conversion. It is well known that in this condition the losses in the system decrease and the vibration amplitude is limited only by the internal losses of the system.

To maintain the condition of the free vibration of the roll, that is to say to avoid as far as possible losses other than radiation to the process feed, the system is supported exclusively by its nodal lines. In addition, a special electronic circuit to maintain the frequency in best performance for the whole system was added ( Ramos-Fernà andez, [START_REF] Ramos-Fernã Andez | Automatic system for dynamic control of resonance in high power and high Q ultrasonic transducers[END_REF]. An ultrasonic grinding machine can then be designed in the form of a roller mill constructed over a specially designed ultrasonic transducer, as is shown schematically in Fig. 3.

An active roll, which is itself an ultrasonic transducer, is located in front of a passive roll. The vibration in extensional mode combines compression and shear action of the active roll on the mill feed. The shear of ultrasonic waves are reported to be beneÿcial to liberation [START_REF] Revnivtsev | Selective liberation of minerals[END_REF]. A funnel feeds the material in the gap by gravity. The feed particles are then nipped by the rolls. A spring system furnishes the stress applied to the ore and the stress level can be varied by adjusting the spring tension or by changing the springs. This system also provides a dynamic gap setting, which adjusts automatically with the stress-state of the system. The rotation of the roll is produced by a variable speed electric motor. The ground ore is collected under the rolls in an iron receiver fed by gravity.

Experimental results with the ultrasonic grinding mill

An experimental programme has been performed to assess the real possibilities of the ultrasonic grinding mill. The block diagram of the experimental set-up is shown in Fig. 4. The ultrasonic grinding machine is driven by a 2 HP electric motor coupled with a planetary system that allows the variation of the angular speed. The input power for the drive system was measured with a true power meter. Also the ultrasonic input power was measured at the output of the electronic ampliÿer.

The experimental system allows in-line measurement of the applied torque, the distance between the two rolls, and the angular speed of the rolls. In each experiment, the process variables are measured and recorded. In addition, particle size analysis of feed and products are made by sieving after sample division by the cone and quartering method.

To eliminate consideration of geometrical factors all experiments are made using the same device. For the reference measurements, the machine is operated as a conventional roller mill by switching o the ultrasonic power. Afterwards, the same test is repeated with the ultrasonic system activated at di erent levels.

The behaviour of dynamic gap as a function of the applied ultrasonic power is shown in Fig. 5. Because the very nature of the granular material produces large uctuations in the gap, an average value of di erent gaps must be taken. It is clear in the ÿgure that two well-deÿned gap regimes can be distinguished. For less than 100 W of applied power the dynamic gap is approximately 1 mm. Above this value of applied power, the gap value decreases to approximately half of the original. This suggests that in the ultrasonic fracture process there is a threshold for the activation of some special kind of phenomena. The same behaviour was obtained in other experiments for di erent size feed material.

Although very interesting, this behaviour is out of the scope of this paper and will be considered later in a special research devoted to the basic phenomena of the ultrasonic fracture process.

Fig. 6 shows the curve obtained from a grinding experiment to assess the in uence of ultrasonic energy on the total torque in the system. Ore from Minera Pudahuel, Chile were used in these experiments. The machine was operated as a ÿxed gap machine with a distance between the rolls of 1 mm. The feed is a natural distribution 100%-6# Tyler, that is to say 100% below 3; 350 m. Fig. 6 clearly shows the drop in torque with the applied ultrasonic power. It should be noted that the ow of material is the same for all ultrasonic power levels. The question now is to assess the energy consumed in the process. That is, is the applied ultrasonic power plus the power used to drive the system lower than the power used for the same grinding task without ultrasonic power? In other words, is the total applied ultrasonic power plus the power used for the system drive lower than the power used to grind the same material to the same degree of ÿneness with the same machine without ultrasonic power? For the new technology to be successful, it should show improvements in energy e ciency. These improvements can be appreciated in Fig. 7. Here the total energy used in the same grinding task is plotted against the applied ultrasonic power. It is clear that the total power requirement for the process is reduced with increasing applied acoustic energy.

A special study was conducted of the particle size distribution obtained with the mill operating with and without ultrasonic energy. For this purpose, ore from the Andina Division of Codelco was ground with the machine operating as a conventional high-pressure roller mill and as an ultrasonic grinding machine. These results are given in Fig. 8 and show that for the same feed, the product particle size is almost the same with and without ultrasound. This is a good result because the whole energy spent in the ultrasonic machine is less for the same product.

There is now some certainty on the processing capacity limits of the machine where the linear velocity is approximately 0:3 m=s, that is, one-third of the limiting velocity of the conventional roll mill. However, our experimental set-up does not allow working beyond 120 rpm because the torque of the motor is not high enough to carry out the experiments with the machine without ultrasound. Operating at 120 rpm, the measured ux shows a processing capacity of 500 kg=h. This ÿgure is very interesting since the machine is a small one, the active part of it can be stored in a 35 cm edge length cube.

Wear tests

A special experiment was performed to assess the e ect of the ultrasonic ÿeld on the response of the grinding surfaces to the abrasive action of the grinding material. However, for a machine made with special SAE 2020 heat-treated hard tool steel, the abrasive e ect is negligible over a time scale of 1 h. The experiment, therefore, consists of a new series of grinding tests with the machine ÿtted with a roll made of a low-resistance steel, without any hardening treatment. Two steels were tested, one was an SAE 1020 structural steel and the other was an SAE 4340 steel. In this way, clear wear signs can be observed after a relatively short time of operation and the in uence of the kind of material can be tested. The experiments were planned in such a way that the di erences inherent in destructive tests were minimised. A ÿrst run was made with the machine ultrasonically activated with an initial input power of 100 W. After 1 h of grinding, the test was continued, keeping the same parameters, but without ultrasonic energy applied to the machine. Afterwards, in a second run, the same experiment was done but starting without ultrasonic energy applied to the machine, and activating the ultrasonic energy after 1 h of grinding and for another hour of grinding. In this way, the only in uence that can be noted must be the abrasive e ect of the grinding process. Transient phenomena will be masked because both tests, the non-ultrasonically and the ultrasonically activated test will have the same transient stage. Between each experiment, the grinding roll was weighed and the surfaces were examined with an optical microscope. The results of this test are presented in Fig. 9.

From the data collected, it is clear that another beneÿcial e ect of ultrasound is that, in both the passive and active rolls, abrasive wear is less when the grinding is made under ultrasound. Also some di erences between the behaviour of di erent quality of material are detected. The better is the material, the better is the performance even without any treatment to improve the resistance. It is to be noted that each wear tests requires the construction of a machine made with low-quality material followed by its destruction. For this reason, the wear experiments were done for only two materials. Once the improvement in the wear capabilities of the ultrasonically activated machine were established, this kind of test was abandoned because they are very expensive and time consuming.

Conclusions

An ultrasonically assisted roll mill has been developed, in which one of the contact surfaces vibrates at ultrasonic frequency. Grinding tests performed on copper ore show a reduction in the total power required to operate the process when the ultrasonic energy is applied in the active zone.

As is shown in Fig. 8, the same degree of product ÿneness is obtained with the ultrasonic machine and with a high-pressure roller mill (the machine without ultrasonic activation), but requiring about 6% less energy (see Fig. 7), and with a considerable saving in the grinding medium wear (less steel consumption). This is a very important ÿgure if it is considered that the grinding is the most energy consuming process in a great number of large-scale production processes.

The presence of ultrasonic ÿeld brings about a reduction in the shaft torque required compared with the same operation without ultrasonic energy. In addition, the ultrasonic-assisted operation shows less abrasive wear on the grinding surfaces.

These results show that the use of ultrasonic technology is a very promising approach for improving the e ciency of the breakage in di erent processes. When compared with other approaches in the past, the ultrasonic roll mill does not have the limited process capacity that was characteristic in these developments.

The energy consumption and torque could be even lower than the values we show in this paper. In this research, the power used was that usually employed to evaluate the overall e ciency of a breakage: [START_REF] Austin | Process engineering of size reduction: Ball mill[END_REF] Fig. 10. Copper ore from Andina, CODELCO, Chile. Monosize 1:7 × 1:18 mm sieve feed at the setting gap of 0:2 mm and 120 rpm.

The speciÿc breakage velocity. Curves for this parameter are shown in Fig. 10 which shows that the ultrasonic power, always improves the overall process but that best results are obtained for 20 W of ultrasonic energy. This is the reason why we use this power through all this investigation. Other powers will be considered for di erent processes in further research, but the data provided in this paper are su cient to assess the potentiality of this new technology.

The scale up and the optimisation of the process require more basic research in ultrasonic grinding. Further experiments must be done both from the point of view of machine design and of the fracture phenomena.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Diagram of the uniaxial press used for single ball breakage experiments.

Fig. 3 .

 3 Fig. 3. Schematic view of the ultrasound-assisted roller mill.

Fig. 4 .

 4 Fig. 4. Experimental grinding machine.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Behaviour of the dynamic gap as a function of the applied ultrasonic power.

Fig. 7 .

 7 Fig. 7. Example of in uence of applied acoustic power on the total required power.

Fig. 8 .

 8 Fig. 8. In uence of ultrasound on the particle size distribution.

Fig. 9 .

 9 Fig. 9. Abrasive wear tests performed with ore from "Minera Padahuel" 100%-6# Tyler.

Acknowledgements

This research has been supported by the project FONDEF M2-43. Ore samples were provided by Sociedad Minera Pudahuel and Andina Division of Codelco Chil e are gratefully acknowledged.