K. Toncheva, L. Lakov, J. Dochev, P. Zlatev 47, 4, 2012, 459-464 Journal of the University of Chemical Technology and Metallurgy,
A VARIANT OF A MATHEMATICAL MODEL REFERRING TO SOME AREAS OF AN INDUSTRIAL FURNACE MANUFACTURING FOAMGLASS INSULATION MATERIAL K. Toncheva1, L. Lakov1, J. Dochev2, P. Zlatev2
1
Institute of Metal Science, Equipment and Technologies “Akad. A. Balevski” with Hydro-Aerodynamics Centre, Bulgarian Academy of Science, 67 Shipchenski prohod Str. Sofia 1574, Bulgaria 2 University of Ruse “Angel Kanchev”, 8 Studentska str., Ruse 7017, Bulgaria, E-mail:
[email protected]
Received 05 February 2012 Accepted 12 July 2012
ABSTRACT A new Bulgarian method was developed for continuous shaft furnace operation and manufacture of insulation materials of specific thermo-physical and mechanical properties. It requires the provision of a fixed speed glass block descending with and predetermined values of the heating temperature of the surfaces and the temperature distribution in the cross section of the foamglass block. Variations of a preliminary mathematical model of the temperature field in the foaming section depending on the shape of the heating surfaces and thermo-physical parameters of the forming glass foam are required. It is essential to take into account the changes in the density, specific heat and conductivity in a cross section of the block, as they determine the homogeneity of the structure. A preliminary variant of mathematical model is developed. On the ground of the method of finite elements it outlines the temperature fields in the cross-section of the forming block and hence determines the speed of descent of the block and the constants of tempering and cooling. Keywords: heat transfer, mathematical modeling, foam glass, foam glass formation.
INTRODUCTION The production of foam glass in a shaft furnace has a number of advantages the main of which is the lower cost of getting a product compared to traditional technology used in a tunnel furnace. The processes of foaming, stabilizing, cooling and tempering are done at one facility in the shaft furnace. Several attempts for foam glass production in a shaft furnace are done in Bulgaria [6, 7]. There are other patents concerning the development of a new shaft furnace for the production
of foam glass and products thereof [4, 5]. The disadvantages affecting the structure and properties of foam glass products are avoided there as the process is continuous and the foam glass is obtained as a homogeneous bar of an uniform structure. The development of a new furnace under this patent requires experimental and theoretical studies of a wide range of issues such as the material and heat balance, the temperature fields in the furnace zones, the cooling rates and coefficients of tempering and cooling.
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The purpose of this article is to perform mathematical modeling of the temperature fields in the area of foaming where the processes of heating, sintering and foaming ensure the provision of foam glass of a homogeneous structure. SUMARY The temperature fields modeling is done by zones, taking into account some design details of the new construction. In this case the foaming process occurs in the channel formed by the internal heating unit and the furnace insulated walls rather than in a closed volume as in the previous constructions [1, 2]. A work area of a thickness of 20 mm is obtained. It is heated by electric heating elements placed in a heating unit. Fig. 1 presents the longitudinal section of the shaft furnace, the theoretical temperature regime and the cross sectional views of the channel in the areas of heating sintering and foaming. The modeling of the heat transfer processes in the different zones of the furnace is carried out assuming that:
• the foaming process is divided into three zones, heating, sintering and foaming; • the heat required for foaming of the glass is obtained only by the internal heater ; • the temperature field in the different zones is a two-dimensional one; • the process time in the different zones depends on the descent speed, the constants of tempering and cooling as well as on the geometrical dimensions of the furnace; • in the present case the mathematical model is developed on the ground of a speed of glass descent in the zone of foaming of 1cm/min; • the change of the thermal and physical properties of the foam glass in the different zones is presented as a polynomial one of the type a0 + a1*t + a2*t2 + a3*t3, where a0 to a3 are coefficients and t is the temperature; • the temperature fields modeling is carried out with specialized software by the finite element method; • the geometric dimensions of the oven in the presence of axes of symmetry are assumed to work with 1/4 of the actual size;
1 2
t,°C
t= 20°C
Internal he ater
Zone N1
Zone N1 L=436mm q=0W/m²
3 1
t= 20°C Zone N2 1 2
Zone N2, Heating L=450mm, q= 3888W/m²
3 Zone N3
4 t=100°C
1 2 3 4
Zone N3, Heating L=120mm, q= 3888W/m² Zone N4, Sintering L=120mm, q= 3888W/m²
1
Zone N 5, Foa ming L=220mm, q= 3888W/m²
Zone N4 a nd N5 2 3 4
t=550°C t=650°C
t=860°C Foa m glass
Legend:
Zone N6, L= 170mm Cooling and Fix t=600°C
Zone N7, L=1163mm Temper, Q= 23.4kW
1. Insulation; 2. Foam glass; 3. Interna l heate r ma de from cast iron; 4. Elec tric heater.
t=590°C
Zone N8, L=1163mm Tempe r, Q=18kW
t=580°C
Zone N9, L=1163mm Temper, Q=9kW Zone N10, L= 640mm Cooling
Across sectional view Longitudinal view of of foam glass shaft foam glass shaft furnace zones furnace zones
t=570°C t=100°C
Theoretical temperature regime in different zones of foam glass shaft furnace
Fig. 1 Foam glass shaft furnace zones.
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K. Toncheva, L. Lakov, J. Dochev, P. Zlatev
• a heat flux q = 0 w/m2, (adiabatic boundary) is set along the axes of symmetry for all models; • the heat flow from the inner surface of the internal heating unit for all zones where electric heating elements are present vs accepted as a constant heat flux q=3888,89W/m2, (boundary conditions of the II-order); • the boundary conditions on the outer surface of the insulation covering the furnace channel refer to a convective heat transfer coefficient a=15W/(m2 K) and temperature t=20 oC. The thermo-physical properties of the high temperature insulation and cast iron used in the temperature fields modeling are: • high temperature cast iron - r=7060 kg/m3, Cp=490 J/(kg K), l=53.3 W/(m K); • mineral wool insulation - r=128 kg/m3, Cp=920 J/(kg K), l=0,2 W/(m K); Experimental data for approximation of thermal and physical properties of the foam glass were used reported in ref. [2]. They are presented in a graphical form in Fig. 2. Approximating polynomials are calculated for: heating zone from 0 to 550°C, sintering area from 550 to 650°C and foaming zone from 650 to 860°C.
0 .4 C , J/(kg.K) 0 .35
0 .3
0 .25
0 .2
0 .15 ? , kg /m
0 .1 0
1 00
2 00
3 00
4 00
5 00
6 00
7 00
8 00
3
9 00
Fig. 2. Foam glass thermo-physical properties. The polynomial coefficients, the errors calculated and the regression coefficient obained are presented in a tabular form. It is clear from the theoretical temperature regime of the foaming process, shown in Fig. 1 that it is not necessary to heat the incoming glass because the material enters at 20°C. That is why the modeling of the heat transfer processes and temperature fields refers only to zones 2 and 3. The temperature field of the heating process in zone 2 is pictured in Fig. 3. The duration of this transition process is 2700 s and it is calculated from the de-
Table 1. Curve fit statistics of foam glass properties into the heating zone Coefficients Value Std. Error Value Std. Error
a0 a1 Points rms Bias Error Heat capacity curve fit equation C=849.050904 + 0.675101847*t 8.490509E+02 6.751018E-01 12 3.4242 1.9895E-13 1.901006E+00 5.851559E-03 Conductivity curve fit equation λ=0.116326801 + 0.0000726962108*t 1.163268E-01 1.924879E-05 12 3.4672E-05 2.3130E-18 7.269621E-05 5.925042E-08
2
R ,% 99.92 100
o
Where t is the temperature, C; rms root mean square
Table 2. Curve fit statistics of foam glass properties into the sintering zone. Coefficients Value Std. Error Value Std. Error Value Std. Error
a0 a1 Points rms Bias Error Heat capacity curve fit equation C=852.6852 + 0.6655495*t 8.526852E+02 6.655495E-01 3 9.9587E-02 -7.5791E-14 7.735496E-01 1.278580E-03 Density curve fit equation ρ=-1199.879 + 3.636264*t -1.199879E+03 3.636264E+00 3 1.0483E-01 7.5791E-14 8.142627E-01 1.345873E-03 Conductivity curve fit equation λ=-0.893736264 + 0.00190912088*t -8.937363E-01 1.909121E-03 3 3.1449E-05 3.7007E-17 2.442788E-04 4.037620E-07
2
R ,% 100 100 100
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Journal of the University of Chemical Technology and Metallurgy, 47, 4, 2012
Table 3. Curve fit statistics of foam glass properties into the foaming zone. Coefficients Value Std. Error rms Bias Error
a0 a1 a2 a3 Heat capacity curve fit equation C=852.7836 + 0.6652727*t 8.527836E+02 6.652727E-01 2.460961E-01 3.226953E-04 6.7689E-02 4.1341E-14
Points
R 2, %
11
100
Table 4. Curve fit statistics of foam glass properties into the foaming zone. Coefficients a0 a1 a2 a3 Points R 2, % 2 Density curve fit equation ρ=151143.694 - 561.540337*t + 0.697100818*t - 0.000288728633*t3 Value 1.511437E+05 -5.615403E+02 6.971008E-01 -2.887286E-04 Std. Error 1.283392E+04 5.100971E+01 6.733835E-02 2.952618E-05 11 99.68 rms 1.8566E+01 Bias Error -1.0186E-10 Conductivity curve fit equation λ=14.8263935-0.0517008432*t+0.0000609423078*t 2-2.39413365E-08*t3 Value 1.482639E+01 -5.170084E-02 6.094231E-05 -2.394134E-08 Std. Error 1.151841E+00 4.578108E-03 6.043598E-06 2.649967E-09 11 99.90 rms 1.6663E-03 Bias Error -3.8503E-15
scent speed of the glass and the length of zone 2. The change of foam glass temperature as a function of the thickness of the channels in this zone is shown in Fig. 4. The temperatures obtained at the end of the process of unsteady heat transfer are used as initial for modeling of the temperature field in the next zone. The following initial temperatures are used 300°C for the internal heater, 200°C for the foam glass for zone 3 starting temperature is and 150°C for the mineral wool insulation temperature is. The temperature field of the heat-
ing process in zone 3 is shown in Fig. 5 and the change of foam glass temperature as a function of the the thickness of the channels in this zone is shown in Fig. 6. The following initial temperatures are used for zone 4: 350°C for the internal heater, 270°C for the foam glass starting temperature is and 200°C for the mineral wool insulation temperature. The temperature field of the heating process in zone 4 is shown in the Fig. 7 and the change of foam glass temperature as a funcion of the thickness of the channels in this zone shown on Fig. 8.
Fig. 3 Temperature field in zone 2 at the end of transient process t=2700 s.
Fig. 4 Change of foam glass temperature in thickness of the channel in zone 2.
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K. Toncheva, L. Lakov, J. Dochev, P. Zlatev
Fig. 5 Temperature field in zone 3 at the end of transient process t=720 s.
Fig. 7 Temperature field in zone 4 at the end of transient process t=720 s.
Fig. 6. Change of foam glass temperature in thickness of the channel in zone 3.
Fig. 8. Change of foam glass temperature in thickness of the channel in zone 4.
The following initial temperatures for zone 5 are used: 380°C for the internal heater, 345°C for the foam glass starting temperature and 260°C for the mineral wool insulation temperature. Picture of the temperature field of the heating process in zone 5 is shown on the Fig. 9 and the change of foam glass temperature into the thickness of the channels in this zone is shown on Fig. 10. CONCLUSIONS The resulting patterns of the temperature fields in the different zones of the internal heater unit lead to the following conclusions: • The temperatures in the different zones set out in the theoretical temperature regime of the foaming process can not be achieved on the ground of the internal heater only; • A large difference in the thickness of the foam glass layer is observed at about 150oC which is not desirable; • supplementary outer heaters have to be located to reduce the thickness of foam glass layer.
Fig. 9. Temperature field in zone 5 at the end of transient process t=1320 s.
Fig. 10. Change of foam glass temperature in thickness of the channel in zone 5.
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Journal of the University of Chemical Technology and Metallurgy, 47, 4, 2012
REFERENCES 1. D. Velev, Y. Georgiev, M. Veleva, Temperaturno pole, toplotehnologichen proces i proizvoditelnost na kaskadna pesht za penostyklo. Sb. Nauchni trudove na VIMMESS, t. HH, 1978, (in Bulgarian). 2. I. Kozhuharov, Niakoi izsledvania vurhu toploobmena v shahtova pesht za proizvodstvo na penostyklo. Avtoreferat kandidatska disertaciia, Ruse, 1972.
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3. Y. Georgiev, M. Gospodinov, Edin model na temperaturno pole na kaskadna pesht za penostyklo. Sb. Nauchni trudove na VIMMESS, t. HHII, 1980. 4. Bulgarian Patent No 65718/24.11.2004. 5. Bulgarian Patent No 65745 /26.05.2006. 6. Bulgarian Patent, No 11 26727 S 0311/00, 11 24457S 03S11/00, 25749 MPK G27V17/00. 7. Bulgarian Patent, No 25749 MPK G27V17/00.
