Journal Journal of hemical of hemical Technology and and Metallurgy, 50, 4, 50, 2015, 4, 2015 486-492 MATHEMATIAL MODEL OF THE TEMPERATURE REGIME IN A FURNAE FOR FOAM GLASS GRANULES PRODUTION Georgi E. Georgiev, Luben Lakov, Krasimira Toncheva Institute of Metal Science, Equiment and Technologies Akad. A. Balevski with Hydro-Aerodynamics entre, Bulgarian Academy of Science 67 Shichenski rohod Str. Sofia 1574, Bulgaria, E-mail: g.georgiev@ims.bas.bg Received 19 January 2015 Acceted 20 May 2015 ABSTRAT А mathematical model of the temerature regime of a furnace used for forming foam glass granules is resented. The temerature field in each zone of the furnace is described the software ackage MAGMAsoft. The temerature roblem is solved in a non-stationary and a stationary mode. The heat loss to the environment and heat caacity of the foaming material are taken into account. The calculations rovide to estimate the ower required for the heaters used. Secific solutions are obtained which refer to the aroriate otimum temerature regime for the different stages of the rocess- forming, fixing, temering and cooling. A detailed study of the kinetics of foam glass formation in a single ellet is resented too. A solution concerning the temerature field in the ellet during the rocess of foaming is obtained and discussed. The mathematical interretation takes automatically into account the change of the foam-glass volume and that of any other rocess arameter such as mass density, thermal conductivity, etc. Keywords: heat-mass-transfer, mathematical modelling, foam glass, foam glass formation. Introduction Lately the foam glass granules are widely used in many areas of life and industry. Therefore, the establishment of efficient technologies for the roduction of such materials is becoming very imortant [1, 2]. One of the effective tools for the develoment and otimization of these technologies is the mathematical modelling [3, 4] of the rocesses and the henomena taking lace during the formation of such granules. 3D mathematical models of the temerature field in a horizontal furnace used for the roduction of foam glass granules and of the thermal rocesses in the individual granule are roosed in this aer. Exerimental Mathematical model of the temerature regime in the furnace A simle scheme of the treated device is resented in Fig.1. 486 For the uroses of the mathematical modelling the main arts of the furnace articiating in the heat transfer are built in 3D using the software ackage MAGMA5 [5]. This is illustrated in Fig. 2. The 3-dimensional heat transfer equation used in each material environment is: ( c ρt ) = div( λ grad( T ) (1) t where c is the secific heat caacity of the material, ρ is the mass density of the material, λ is the coefficient of conductivity, while Т(x,y,z,t) is the temerature field. Aroriate boundary conditions have to be added to Eq. (1). They are: On the boundary between the working sections 3&4 (Fig. 2) and the insulation λ Tins Tws ins ws ws ins ( Tws Tins ) n λ = = n α (2) where λ, λ are the coefficients of conductivity of the ins ws
Georgi E. Georgiev, Luben Lakov, Krasimira Toncheva Fig. 1. General View of the device: 1 - Feeder device dosing ellets; 2 - Heating furnace; 3 - Fixing section; 4 - Temering section; 5 - Vents. 6; Transort mesh; 6 - ontinuous steel net; 7 - leaning device; 8 - Receiving module on the formed. Fig. 2. The main thermal arts of the furnace. relevant material environment, αws ins is the heat transfer coefficient of the same boundary, while T, T are the temeratures of surfaces of contact. The heating of the working areas through convection and radiation (the second boundary condition) is described by: q = T T + T T (3) av 4 av 4 i αc( h w ) σ[ h ( w )] ins ws where qi is the heat flux transferred from the heater to the av working environment, This the heater s temerature, T w is the average temerature of the working environment, αc is the heat transfer coefficient of convection, while σ is the coefficient of radiation exchange. The heat transfer from the furnace to the environment is described by: ins q = α ( T T ) = λ T n env wall wall A ins wall (4) where qenv is the heat flux transferred from external walls to the environment, αwall is the heat transfer coefficient of the convection to the environment, Twall is the temerature on the external surface of the walls, while TA is the ambient temerature. The heat loss can be estimated by adding together all the heat flows towards the environment. This gives: 11 W = α [ T( S )][ T( S ) T ] dσ a c i i A i i= 1 Si where S i denotes the surrounding surfaces. (5) 487
Journal of hemical Technology and Metallurgy, 50, 4, 2015 Mathematical model of the temerature field during the foaming The ellets used have a sherical form. In this case Eq.(1) transforms into: 2 ( ρct ( rt, )) Trt (,) 2 Trt (,) = λ[ + ], 2 t r r r t > 0, 0 < r < R 0 (6) where R o is the radius of the ellet. When the temerature in the ellet exceeds 650 o the foaming rocess starts and the ellets volume increases. The change of the radius Ro can be exressed through the following monotonically increasing function: R = F(t), where R = F ). 0 ( t 650 (7). When the temerature of the ellet exceeds 650 o the task is solved as a task with a moving boundary the surface of the shere. The ellet is heated under conditions of gas convection and radiation. Therefore the boundary condition is described by the following equation: T λ = α T T Rt + σ T T Rt r 4 4 S c[ ws (,)] [ h (,)] (8) Fig. 4. omarison between laned and model temerature distribution. where S is the surface of the shere, λ is the coficient of conductivity of the ellet and granule, T is the temerature field of the ellet and granule, αc is the heat transfer coefficient of the gas convection, whilet ws is the average temerature of the air in the heating section. RESULTS and discussion Temerature regime of the furnace After aroriate discretization of the geometry of the furnace using the mesh generator of MAGMAsoft, a solution of the model mentioned in section 2 is obtained with the slower of the same software. The temerature field of the working sections is colour-code resented in Fig. 3. The lanet temerature distribution is comared to that obtained by simulation. This is illustrated in Fig. 4. It can be seen that the constructive solution imlemented a temerature regime close to the desired one. The temerature is resented by control oints 5 (Fig. 2). The heat losses calculated by Eq. (5) are found equal to 10.83 kw. Fig. 5. Volume registration during foaming. Temerature field of the ellet and the granule formed The solutions of the model resented in section 3 are obtained by software develoed by the authors. The temerature deendent thermo-hysical arameters are used taking into account the effect of the volume exansion during the foaming. The following relations are used: 488
Georgi E. Georgiev, Luben Lakov, Krasimira Toncheva For the ellets: = γ ρ = γ ρ λ = γ λ Glass Glass Glass + (1 γ ) + (1 γ ) ρ + (1 γ ) λ (9) For the granules: G G G = χ + (1 χ) ρ = χ ρ + (1 χ) λ = χλ + (1 χ) (10) where γ = 0. 7. The latter value is exerimental obtained. χ ( t) = V / VG ( t) is derived on the ground of the exerimental data ointed in Fig. 5. The evolution of the granule s radius (Eq.(7), section3) is calculated with the alication of R 3 G ( t) = R VG ( t) / V. The data obtained using Eqs. (9) and (10) are shown in Figs. 6-8, resectively. The temerature roblem of two ellets having different radii (R = 5 mm and R = 7.5 mm) is solved. Fig. 8. λ (T ). The main curves obtained (referring to the minimum, the maximum and the average temerature) are shown in Fig. 9 and Fig. 10. It should be noted that the heating of the smaller ellets is faster, while a wider zone is required for the foaming of the larger one. It is ossible to determine the otimum seed of ellets transort on the ground of these considerations. An adjustment aiming to obtain maximum erformance can be done. The second vertical red line in Fig. 9 and Fig. 10 shows the otimal time for ellets movement through the foaming zone. The following two figures (Fig. 11 and Fig. 12) Fig. 6. (T ) Fig. 7. ρ(t ) Fig. 9. R =5 mm. Main temerature curves. 489
Journal of hemical Technology and Metallurgy, 50, 4, 2015 Fig. 12. Radial temerature distribution. R =7.5 mm. Fig. 10. R =7.5mm. Main temerature curves. resent the radial distribution of the temerature within the granules during foaming. The direction of radius increase is shown and the moments of time are marked. The next series of figures shows the temerature field in the cross section of the granules during the foaming. Figs. 13-16 show the results obtained for the ellets of a size referring to R = 5 mm, while Figs. 17-20 illustrate the results obtained for granules of a size referring to R = 7.5 mm. The moments of time and temerature ranges are also shown. The imact of net transort is not reorted. It is not taken into account. Fig. 13. R = 5.5 mm T=648-666 o. Fig. 11. Radial temerature distribution. R =5 mm. Fig. 14. R = 8.2 mm T = 844-852 o. 490
Georgi E. Georgiev, Luben Lakov, Krasimira Toncheva Fig. 15. R = 8.2 mm T=844-852 o. Fig. 18. R = 9.6 mm T=714-750 o. Fig. 16. R = 10 mm, T = 856-859 o. Fig. 19. R = 12.4 mm T = 829-843 o. Fig. 17. R = 7.8 mm T = 657-687 o. Fig. 20. R = 14.85 mm T=855-858 o. 491
Journal of hemical Technology and Metallurgy, 50, 4, 2015 onclusions The mathematical models develoed lead to imortant solutions. The latter are: The tasks are solved at a high level of comlexity of the conditions - the rocess arameters deend on the temerature and the changing geometry; The temerature distribution in the working sections of the furnace is close to that considered rior to the calculations; The heat lost is calculated; The temerature roblem of the ellets is solved under the moving boundary condition; The solution of the temerature roblem during foaming rovides otimization of the rocess erformance. Acknowledgements This work is financially suorted by the National Innovative Fund ometitiveness of Bulgaria through grant BG161Р0003-1.1.06-0094-001. References 1. I.I. Kitajgorodskij, T.N. Keshishian, Penosteklo, Promstrojizdat, 1953, (in Russian). 2. H.I. Karmaus, Schaumglas, Glass Email Keramotechnik, 5, 6, 1957. 3. G.Evt. Georgiev, V. Manolov, L. Lakov, K. Toncheva, I. horbov, Mathematical Model of Heat-Mass- Transfer Processes of the Foam Glass Formation in Vertical Positioned Pilot Installation, The 17th onf. of Glass and eramics, Nessebar, 25-29 Setember 2011. 4. G.Evt. Georgiev, V. Manolov, L. Lakov, K. Toncheva, I. horbov, Mathematical Modelling and Otimization of the Processes in Vertical Positioned Pilot Installation for Glass Foam Formation, J. Mat. Sci. Technology, 19, 2, 2011, 129-147. 5. G.Evt. Georgiev, aabilities of the Software Package Magma5 for Simulation and Otimization of asting Processes, Engineering Sciences, LI, 1, 2014, 5-14. 492