Journal of University of Science and Technology Beijing Volume 15, Number 4, August 2008, Page 379
Metallurgy
Numerical simulation of the temperature field of titania-bearing BF slag heated in a microwave oven Liangying Wen, Chenguang Bai, Guibao Qiu, Jian Zhang, Shengfu Zhang, and Zhanjun Long College of Materials Science and Engineering, Chongqing University, Chongqing 400030, China (Received 2007-08-05)
Abstract: Considering the characteristic of selective heating of microwave and the treatment of titania-bearing BF slag, a mathematical model for the heating of a slag specimen is developed. The temperature distribution in the specimen is studied by numerical simulation. The temperature in the center of the cylindrical slag specimen is the highest and the temperature decreases when the radius increases rapidly. In this case, the temperature rising rate decreases with heating time rapidly, and it tends to zero when the heating time is up to 150 s. © 2008 University of Science and Technology Beijing. All rights reserved. Key words: titania-bearing slag; microwave heating; temperature field; numerical simulation
[This study was financially supported by the National Basic Research Program of China (No.2007CB613503).]
At present, titania-bearing BF slag has not been synthetically utilized in Pan Steel Co. As a result, it is necessary to explore it from more aspects and more methods. Microwave has succeeded in mineral treatment [1-5], but it is a creative job and requires a large amount of basic research to deal with the titaniferous-enriched slag. In this article, the temperature field in the titania-bearing BF slag heated by microwave is studied by the method of numerical simulation.
can be taken as an un-state heating problem with a heat source inside the object in the microwave oven. Supposing the heated slag is constant, that is the composition and the physical properties of the slag are uniform and do not change with time, the cylindrical coordinates used for numerical simulation are shown in Fig. 1.
1. Mathematical description of the slag specimen heated by microwave To date, a detection instrument has not been found, which can be used to accurately measure the temperature of an object heated by microwave because of the particularity of microwave heating. Questions such as, how to set up the mathematical model of the temperature field of the slag specimen under microwave, and how to attain the temperature field numerically by computer modeling that can predict the temperature distribution of the specimen, and the variation of temperature, are getting more and more attention [6-8]. 1.1 Governing equation The temperature rising process of the slag specimen Corresponding author: Liangying Wen, E-mail:
[email protected] © 2008 University of Science and Technology Beijing. All rights reserved.
Fig. 1.
Slag sample for heating simulation.
The governing equation for temperature distribution in the sample is Also available online at www.sciencedirect.com
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wt s wW
J. Univ. Sci. Technol. Beijing, Vol.15, No.4, Aug 2008
as[
1 w wt s w 2t s (r ) 2 ] f s (r, z,W ) r wr wr wz
(1)
where t s is the temperature of the slag specimen, qC; W the time, s; a s the thermal diffusivity of the slag specimen; a s k s / U s c s ; k s the heat conduction coefficient of the slag specimen, W/(mqC); U s the density of the slag specimen, kg/m3; c s the specific heat capacity of the slag specimen, J/(kgqC); r , z the radial and axial coordinates, respectively; f s (r , z ,W ) the distributed heat source generated by the electric field of microwave [9], f ( r , z ,W )
2ʌf H 0H ccsH s E 2 /( U s c s )
(2)
where f is the microwave working frequency, f 2450 MHz; H 0 the dielectric constant of free space, H 0 8.854 u 10 12 F/m; H scc the dielectric dissipation factor of the slag specimen. H scc is a function of frequency and temperature, its value increases with temperature rising at a fixed frequency of the microwave. Considering that the dielectric dissipation factor of the slag specimen changes during temperature rising, one can get
H scc H ccsf (H ccsf H ccs 0 )e KW , where H ccs 0 is the dielectric dissipation factor of the slag specimen at the initial heating temperature, H ccs 0 0.08 ; H ccsf the dielectric dissipation factor of the slag specimen at the finial heating temperature; K the coefficient of the slag specimen dielectric dissipation factor as temperature changes, K 0.1 ; E the intensity of the electrical field. 1.2. Initial condition Supposing the slag temperature is uniform before heating, that is t s t 0 (normal temperature) at W z 0~h
0, r
0 ~ r1 ,
(3)
1.3. Boundary conditions As shown in Fig. 1, the slag specimen (2r1h=20 mmu60 mm) is much smaller than the inner volume of a Model WD750 microwave oven (215 mm×350 mm×330 mm). The heat interchange between the slag specimen and gaseous medium (air) in the oven can be taken as a convection loss of the slag specimen into a large space (the volume of the slag specimen is 0.076% of that of the oven). The heat transfer equations on the borders of the slag specimen are: ks ks
wt s wr
r r1
wt s wz
z h
D 1 (t s
r r1
t0 ) , r
r1 , z
0~h
(4)
D 2 (t s
z h
t0 ) , z
h, r
0 ~ r1
(5)
ks
wt s wz
0, z
0,r
0 ~ r1
(6)
z 0
where D 1 is the heat convection coefficient between the side wall of the slag specimen and the air in the oven; D 2 is theheat convection coefficient between the top surface of the slag specimen and the air in the oven. When adiabatic fiber is used to insulate the slag specimen from heat transfer, D 1 0 , D 2 0
2. Heat interchange coefficient of medium for the slag specimen in the oven When the slag specimen rotates at 5 r/min in the oven, the linear speed of the specimen surface is u
r1Z
0.01 u 5 u 2ʌ / 60 0.0052 m/s .
Because the rotation of the slag specimen entrains the surrounding gas, the flow of gas around the specimen is not only on account of the rotation of the slag specimen, but also because of the difference in temperature caused by heating. The relative intensity between natural convection and forced convection can be described by the following dimensionless numbers: Gr / Re 2
(
g E 'tL3
X2
)(
X uL
)2
g E'tL u2
(7)
g E ǻtL3
is the Grashof number, the critiX2 cal Grashof number is 108-109 from laminar to turbu-
where Gr
uL
is the Reynolds number; L is the X characteristic dimension, being the height of the wall when the hot surface is vertical (as Fig. 2), and being 0.9 times the diameter of the specimen (i.e. 0.9 d) when the hot surface is horizontally upwards (Fig. 3); g is the gravity acceleration, g = 9.81 m/s 2 ; E is the
lence flow; Re
1 , 1/qC, 273 t m is the characteristic temperature, which is
thermal expansion coefficient, E
here t m the average of the hot surface temperature and the surrounding medium temperature, that is, t m
1 ˄t s t 0˅; 2
't is the temperature difference between the surface and the surrounding medium, qC; X the kinematic viscosity of the surrounding medium at the characteristic temperature, m2/s; other symbol meanings are the same as before.
At Gr / Re 2 t 10 , the problem can be treated as purely natural convection; at Gr / Re 2 d 1 , the natural convection can be neglected and the problem can be regarded as purely forced convection [10]. In this article, when the temperature on the surface of the slag
L.Y. Wen et al., Numerical simulation of the temperature field of titania-bearing BF slag heated in…
specimen is 250qC higher than that of the atmosphere in the oven, it can be taken as purely natural convection. As the slag sample surface is heated to 670qC, the convection heat transfer around the specimen is calculated. Suppose the temperature of surrounding medium is 30qC, the characteristic temperature is 670 30 tm 350 qC, the heat conductivity of air is 2 k a 0.0491 W/(m2qC), the kinematic viscosity X 55.46 u 10 6 m2/s, and the Prandtl number Prs=0.676. Gr Pr =
3
9.81(670 30)0.06 u 0.676 (273 350)(55.46 u 10 6 ) 2
4.78 u 10 5 % 10 8 ,
which shows that the natural convection is of laminar flow.
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When the hot surface is horizontally upwards and the characteristic dimension is 0.9 times of the specimen diameter, the dimensionless number equation of natural convection heat transfer is as follows [12]: Nu 2
(10)
0.54 ˄Gr Pr˅0.25
where Nusselt number Nu 2˙
D2
Nu 2
D˄ 2 0.9d˅ , therefore, ka
ka W/(m2qC) 0.9d
(11)
According to the earlier discussion, the heat transfer coefficients D1 and D2 can be determined in the oven.
3. Determination of the electric field intensity with microwave From the electromagnetic theory, it is known that the box as microwave heating is multi-module syntony, of which the total power consumption can be divided into three parts, that is, internal cavity storage energy, medium power consumption, and cavity wall energy consumption. Because the cavity wall is made of metal material, the loss through the cavity wall is an extremely small part of the total energy. The microwave energy entering into the cavity is mostly absorbed by the medium and dissipated [13-14]. The unit volume medium absorbing and consuming the microwave power is P/ V
Fig. 2.
Fig. 3.
where P is the effective output power of the microwave oven. Here, the slag sample is heated by a Model WD750 microwave oven with an output power of 750 W and a frequency of 2450 MHz. The microwave effective powers, known as "high power", "middle power", and "low power" are respectively 100%, 81%, and 58% of the output power of the microwave oven, corresponding to 750, 608, and 435 W. V is the sample volume. Other symbol meanings are the same as before.
Vertical hot surface.
Horizontal hot surface.
When the hot surface is a vertical cylindrical wall and the characteristic dimension is the height of the specimen [11], the dimensionless number equation of the natural convective heat transfer [4] is Nu1
0.95 ˄Gr Pr˅0.25
0.95 ˄4.78 u 10 5˅0.25
where the Nusselt number Nu1
D1
Nu1
ka H
24.98 u
0.0 491 0.06
D 1H ka
(12)
2ʌf H 0H ccs E 2
24.98 (8)
, hence
20.44 W/(m2qC) (9)
Considering that the specimen heated is a cylinder, the simulating calculation is a two-dimensional axial-symmetry model. The electric field with axial-symmetry property is distributed as shown in Fig. 4. Its distribution formula is E
E 0 sin(
ʌ ʌ r ) 2r1 2
(13)
where E is the distribution of the microwave electric field; E 0 the maximum of the microwave electric field; r the radial coordinate of the specimen; r1 the radius of the sample, r1
0.01 m
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J. Univ. Sci. Technol. Beijing, Vol.15, No.4, Aug 2008
s, 674.8qC at 150 s, 671qC at 180 s, that is, it tends to stabilize with further heating. With the "middle power" of 608 W, the highest temperature is 561.5qC at 120 s, 564qC at 150 s, 565.1qCat 180 s, and then gets to a constant of 565.4qC till 300 s. With the "low power" of 435 W, the highest temperature is 425.4qC at 120 s, 427.8qC at 150 s, 428.5qC at 180 s, and then gets to a constant of 428.6qC till 300 s.
Fig. 4. Two-dimension axisymmetric cylinder and electrical field distribution.
4. Temperature distribution in the uniform slag specimen Supposing the inside of the slag specimen is isotropic and the composition is uniform, the internal temperature distribution of the specimen is a function of time. Figs. 5-8 show the typical temperature distribution of the specimens heated by "high power (750 W), “middle power (608 W)”, and “low power (435 W)”, as well as the influence of heating time. It is obvious that the center temperature of the cylindrical slag specimen is the highest under the microwave heating condition. The specimen temperature decreases gradually from the centre to the edge and the temperature decreasing rate increases with radius increasing. If the cylinder specimen is vertically put in the oven and is heated without insulation on its top, the temperature of the lower part of the sample is higher than that of its upper part (shown in Fig. 6(a)). The upper border temperature of the specimen is the lowest, owing to the heat exchange between its top surface and the surrounding medium. When insulation that allows the pass of microwave energy is put on top of the sample, the temperature of the upper and lower parts tends to be consistent (shown in Fig. 6(b)), but the temperature of the sample edge is the lowest because of heat loss. However, the highest temperature hardly changes whether or not insulation is put on top of the sample. For example, when the sample is heated with a high power of 750 W, the highest temperature is 675qCwithout top insulation, whereas, the highest temperature is 675.7qCwith top insulation. The inner temperature of the sample changes with time, as shown in Fig. 7. At the initial stage of heating, the inner temperature of the sample increases rapidly because of the microwave energy. After increasing to a certain temperature, the increase rate of temperature evidently decreases and tends to stabilize gradually. As shown in the figure, with the "high power" of 750 W, the highest temperature the sample can reach is about 675qC. The highest temperature is 671qCat 120
Fig. 5. Temperature distribution on cross-section ( z mm, 750 W, heating time 180 s).
30
Fig. 6. Temperature distribution on longitudinal section (750 W, heating time 180 s): (a) without insulation; (b) heat insulation on top.
The temperature rising rate of the sample changes with time as shown in Fig. 8. At the initial stage of heating, the temperature rising rate of the slag specimen ( 'T 'W decreases quickly with heating time.
L.Y. Wen et al., Numerical simulation of the temperature field of titania-bearing BF slag heated in…
When the heating time reaches 150 s, the temperature rising rate is almost zero.
Fig. 7. Variation of the highest temperature in the sample center with time.
Fig. 8. time.
Variation of the heating-up rate of the sample with
With the heat loss by radiation and the thermal effect of the reaction (if it exists) and so on being considered, the temperature rising rate of the sample in the microwave field can be expressed as [4-5] dT dW 1
U sC s
(2ʌf H 0H ccs E 2 D'Tˉ
eaA 4 dF T ¦ n i 'H T0,i i ) V dt
(14) where T is the temperature, W the time, U s the density, C s the thermal capacity, H 0 the dielectric constant in the vacuum, H scc the dielectric dissipation factor, f the frequency of the microwave electric field, E the microwave electric field intensity, D the natural convective heat transfer coefficient, 'T the temperature difference between the slag surface and surrounding atmosphere, e the thermal emissivity of the slag specimen surface, a the Stefan-Boltzman constant, A the surface area of the slag specimen, V the volume of the slag specimen, ni the Moore number of unit i , 'H T0,i the reaction thermal effect, Fi the reaction conversion ratio. It is obvious that the total temperature rising rate is decided by H s" , D , e ,
383
'H T0,i , and dFi dt at a fixed microwave power in-
put. The bigger the dielectric dissipation factor of the components that form the slag specimen, the larger is the total temperature rising rate [15]. Therefore, as the slag specimen is heated by the microwave, if one component with greater permittivity reacts to form another component with smaller permittivity, or if there is an endothermic reaction that consumes plenty of microwave energy, the results will lead to a decrease in the rising temperature rate during the heating process, in the microwave oven. The quicker the reacdFi tion rate is, the more prominent is the decrease dt in the temperature rising rate. If the reaction rate tends to zero or no reaction, the last term on the right of formula (14) can be ignored. When the specimen is heated in the microwave field, the energy dissipation in the specimen and the oven and the surrounding atmosphere heat loss in the oven increases with the rise in temperature. This will lead to a decrease in the slag sample temperature rising rate, and tend to zero when the energy dissipation is equivalent to the energy absorption of the slag sample, and the slag temperature keeps invariant.
5. Conclusions (1) Under the microwave heating condition, the center temperature of the cylinder slag specimen is the highest. The specimen temperature deceases gradually from the centre to the edge and the temperature decreasing rate increases with an increase in radius. (2) When the cylinder slag specimen has no top insulation, the temperature of the lower part is higher than that of its upper part and its top face temperature is the lowest. With top insulation, which allows the microwave energy to pass, the temperature of the lower and upper parts tends to be consistent, but the temperature in the sample edge is the lowest on account of heat loss. However, the highest temperature is hardly changed whether or not insulation is put on top of the sample. (3) At the initial stage of heating, the inner temperature of the sample increases rapidly owing to the action of microwave energy. After increasing to a certain temperature, the temperature rising rate evidently decreases and tends to stabilize gradually. (4) Under this study condition, the slag sample temperature rising rate ( 'T 'W decreases quickly with an increase in heating time. When the heating time reaches 150 s, the temperature rising rate tends to
384
be almost zero.
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