Mathematical Modeling of Molten Slag Granulation Using a Spinning Disk Atomizer (SDA)
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1 Mathematical Modeling of Molten Slag Granulation Uing a Spinning Dik Atomizer (SDA) Hadi Purwanto and Tomohiro Akiyama Center for Advanced Reearch of Energy Converion Material, Hokkaido Univerity Kita 13 Nihi 8, Kita-ku, Sapporo Japan hadi@eng.hokudai.ac.jp Abtract To optimize granulation proce, a mathematical model of a pinning dik atomizer (SDA) wa developed to produce particle from hightemperature molten lag. The model comprie three part: 1) fluid flow model of molten lag on the pinning dik, ) phyical model of ligament formation of lag, and 3) heat tranfer model of lag drop dipered from the ligament. Firt, a -D fluid flow model wa developed to evaluate the film thickne of lag layer and wa calculated uing the calar equation method. The number and diameter of the ligament formed were evaluated uing the phyical model. Finally, the heat tranfer model wa employed to evaluate the quenching rate and temperature ditribution within the drop. The developed model wa verified by comparing the calculated reult with the experimental data of drop diameter, both of which were found to be in reaonable good agreement. The advantage of thi model i that it can alo be ued to predict the cooling rate and temperature ditribution within the particle, which control the lag propertie. KEY WORDS: molten lag, mathematical model, pinning dik atomizer, dry granulation 1. Introduction Spinning dik atomizer (SDA) technology i wellknown and i mainly applied in the chemical, food, and metallurgical indutrie. With regard to the mechanim of lag granulation, everal literature have propoed theoretical formula to evaluate the particle diameter of the drop granulated from the rotary atomizer[1-5]. Fraer wa the firt to propoe an empirical formula to predict the diameter of the drop obtained from the rotary dik experiment, uing oil. Other reearcher have propoed equation for different medium uch a water or liquid with a low vicoity. All thee empirical formula were largely baed on one apect of the operating parameter. In other word, they were derived from a conventional, dimenionle analyi. However, the mechanim of granulation that depend on the device deign and the phyical propertie of the molten lag with very high vicoity i obviouly different from that reported in previou tudie. Depite it engineering ignificance, the phenomena of the lag granulation uing SDA and the heat tranfer behavior in the particle have not yet been theoretically elucidated. Therefore, the objective of thi tudy i to develop a comprehenive model uing SDA through experimental obervation of the lag granulation proce. That i, the mechanim of lag granulation wa firt oberved experimentally uing a high-peed video camera and wa then ued a the bai of thi model, in which the influence of the rotating peed of the dik on the diameter of lag wa mainly examined for validation. In thi model, an unteady tate heat tranfer analyi of a ingle lag particle i alo carried out to predict it cooling rate and the temperature ditribution within the particle during the cooling proce. Thi will provide important information on whether the lag i vitrified or not. The granulated lag, produced by quenching the molten lag, ha propertie of gla and can be ued directly a a raw material in cement production.. Model Overview The model i developed baed on the careful obervation of the granulation phenomena during the experiment. Figure 1 how a chematic diagram of the experimental apparatu. In the experiment, amount of molten lag wa poured onto the pinning dik that wa rotated by a motor at the bottom. The flow of lag followed the gravity force before contact with the dik urface and then prayed out to become granulate. The experiment wa oberved uing a high-peed video camera. The granulated lag wa collected on a particle collecting heet. Figure how the photograph of the molten lag flow on the SDA dik. Thee image reveal that the granulation proce of the molten lag tarted out a a thin film flowing from the dicharged lag on the center of the SDA dik during rotation. Then, the film
2 prayed out of the dik due to centrifugal force to form a number of ligament. Finally, the ligament broke off to become granulate and were cooled while falling. Therefore, a comprehenive granulation model wa contructed according to thee three mode - thin film, ligament, and drop formation. Fluid flow model of a thin film of lag on the rotating dik The phenomena on the dik were aumed a a two dimenional coordinate ytem of an axial ymmetrical cylinder. To obtain the numerical olution, a non-uniform computational meh wa ued in finite differential equation baed on SEM with appropriate initial and boundary condition. A grid wa contructed for the numerical analyi. A fine grid wa made at above the rotating dik, in which the molten lag flow from the center to the dik corner, and it behavior wa analyzed. The partial differential equation ued to decribe the liquid flow are the momentum and continuity equation. The variable to be computed are the volume fraction, lag flow rate, vicoity, rotating peed, and cup diameter. Thi model i developed baed on the kinetic theory, at contant propertie and under tranient condition, a hown by the following equation: u u uv u u P ρ + + = u + t x y + x y x (1) v uv v v v P ρ + + u + t x y = + x y y () Equation 1 and repreent the momentum or motion in x, y direction and S x,y i the momentum ource. The above equation for the conervation of momentum hould be upplemented with the continuity equation for incompreible flow a given below: u v + = 0 x y (3) S x S y
3 Phyical model of ligament formation A imple phyical model baed on a material balance wa ued for evaluating the number and diameter of the ligament formed. That i, we aumed that all the lag reformed from the rotating dik changed into everal ligament with the ame diameter. The number of the ligament around the dik, K, wa etimated uing equation 5 to 7 under the operation condition; additionally, the phyical propertie of the liquid lag and then the film thickne were alo calculated. Accordingly, by ubtituting Eq. 8 in 9 we can calculate the nozzle diameter, D e. Finally, the drop diameter from the liquid pillar wa calculated by equation 8 uing the predicted nozzle diameter. ρω R We = (5) σ µ Z R = (6) ρrσ = 0.31We Z R K (7) ( πr K ) 4h D e = φ (8) π 3µ D p = 1.89De 1 + (9) σ ρde where, D p i the diameter of particle drop. The vicoity of the lag wa evaluated according to the method propoed by Iida et al.[10]. Heat tranfer model of lag drop Heat tranfer analyi of a ingle particle wa conducted according to the tranient heat tranfer method after the molten lag completely granulated [8-9]. Three heat tranfer mechanim are taken into conideration according to the tranport phenomena analyi of convection between the particle urface and air, radiation between wall and particle, and conduction inide the particle. The fundamental equation ued for the heat tranfer analyi of a ingle particle i given a below:
4 T 1 T ρ C p = ( kr ) + Q (10) t r r r The governing equation for the analyi follow the aumption: The lag particle i pherical, and heat conduction occur concentrically. The volume change of the lag i negligible during phae tranformation. The temperature inide the lag i uniform, a an initial condition. Boundary condition of particle: r = R; T k r r = 0; = h ( T p 4 4 T ) + εσ ( T T ) (11) g w Thi model then i olved by uing the calar equation method (SEM) [6-7]. In thi method, denity and vicoity are deduced from the value of variable, a hown in Eq. 4 below: T k r = 0 (1) Φ Φu Φv + + t x y = 0 (4) Initial condition: T = T 0 at t = 0 (13) where, i a conerved calar that ha initial value of zero in one fluid (air) and unity in the other fluid (lag).
5 3. Reult and Dicuion Verifying the model with experimental data To verify the model, the calculation wa demontrated to predict the particle diameter of the granulated lag. The propertie ued for numerical computation are lited in Table 1. Figure 3 how the lag particle diameter calculated baed on the experimental condition, uing the abovementioned method a a function of rotating peed. The olid line repreent the calculated reult from the developed model. The figure clearly indicate that the diameter of the granulated lag decreae with increaing in rotating peed, epecially at an initial over 0 rp. The plot how that the data calculated by the model matche well with the experimental data 11), except when the rotating peed i le than 16.6 rp. Regarding the experimental data, the deviation of the particle diameter at a lower rotating peed may be due to the fact that the hape of all the particle collected wa not pherical and the ditribution of particle ize wa very wide. A very few amount of fine particle are collected, however, large and heavy particle were dominantly produced under thi condition. It may caue the deviation between meaured data and the calculated of drop diameter uing the model. In contrat, the granulated lag drop obtained from experiment at 50 rp are uniformly pherical with a diameter of le than 1 mm. Furthermore, calculation were conducted with difference of lag vicoity. The reult are hown in Fig.4. The plot how that low vicoity of lag (0.1 Pa.) reulted in particle with mall diameter. Decreaing the vicoity of lag lead to produce fine granulated lag. Temperature ditribution of a ingle particle Heat tranfer analye of the granulated lag during cooling proce, were carried out for a ingle particle. In thi analyi, the temperature of the molten lag on the dik wa aumed to be contant. Figure 5 how the change in the calculated temperature of a ingle particle with diameter of 0.7 x 10 - m. The reult how that there i no temperature difference between the center and at the urface of particle. Temperature ditribution in thi particle wa homogeneou. The temperature dropped rapidly in the few econd before reached at contant condition. Experimentally, the time required for the particle to drop onto the collector urface wa approximately 0.5. The time required for cooling the lag wa in the range that for dropping. Therefore, the particle wa cooled before dropping on the ground. It verifie that rapid cooling convert the molten lag into glay and tranparent particle. The high cooling rate reulted in high glay property of granulated lag. In the near future, the model could be ued to optimize the granulation proce uing SDA by conducting numerical experiment, and to control the propertie of the granulate lag by determining the cooling rate baed on particle diameter. 4. Concluion The mathematical model developed in thi work i capable to predict the drop diameter of lag produced by SDA. The model can be ued for optimizing the granulation proce uing SDA from the viewpoint of drop ize, glaification
6 property of the product, and o on. A comparion the calculated reult with the experimental data how the validity of the model for predicting the drop diameter. The agreement i reaonable good. The advantage of thi model i that it can alo be ued to predict the cooling rate and temperature ditribution within the particle, which control the lag propertie. Acknowledgement A part of thi reearch wa upported by the Japan Society for Promotion of Science (JSPS;16P04150). Lit of Symbol and SI Unit C p : Specific heat J kg -1 K -1 D : Diameter m H : Latent heat kj kg -1 h p : Thermal conductivity of ga W m -1 K -1 K : Number of ligament - k : Effective thermal conductivity W m -1 K -1 l : Length m P : Preure Pa R : Radiu of rotating dik m r : Ditance from centre m T : Temperature K t : Time We : Weber number - Z R : Ohneorge number - ω : Rotating peed rp µ : Dynamic vicoity Pa ρ : Denity of liquid Kg m -3 σ : Surface tenion of liquid Dyne m -1 v : Velocity m -1 ε : Emiivity - σ : Boltzmann contant W m - K -4 Subcript: g : Ga : Solid l : Liquid Reference 1. R. P. Frazer and N. Dombrowki, Brit. Chem. Eng. Pr. Tec., 1(1957), S. J. Friedman, F. A. Gluckert and W. R. Marhall, Jr., Chem. Eng. Prog., 48(195), Y. Oyama, M. Eguchi and K. Endo, Kagaku Kogaku Ronbunhu, 17-7(1953), Y. Kitamura, T. Iwamoto and T. Takahahi, Kagaku Kogaku Ronbunhu, (1976), C. Weber and Z. Angew: Math. Mech., 11- (1931), L. Jun and D.B. Spalding, Phyico Chem. Hydrodyn., 10(1988), B. van Leer, J. Compt. Phy., 3(1977), T. Akiyama, Y. Ahizawa and J. Yagi, Tran. JSME, 57B(1991), S.V. Patankar, Numerical Heat Tranfer and Fluid Flow, McGraw-Hill, New York (1980). 10. T. Iida, H. Sakai, Y. Kita and K. Shigeno, ISIJ Int., 40 Suppl.(000), S110-S H. Purwanto, T. Mizuochi, H. Tobo, M. Takagi and T. Akiyama, Mat. Tran., 45(004), H. Purwanto, T. Mizuochi and T. Akiyama, Mat. Tran., 46(005),
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