2. Heat Transfer in Opaque Materials
2.4 HEATING OF A LOAD IN AN INDIRECTLY-FIRED FURNACE
Indirectly-fired radiant tube furnaces are widely used in metal heat-treating applications where the quality of the final product is a major concern [5]. In this type of furnace, the combustion of fuel and air takes place within the radiant tubes, and the high-temperature products of combustion are kept isolated from the stock being heated. The energy released due to combustion is transferred to the radiant tube wall, and the heating of the stock material in the furnace is accomplished via radiative heat transfer from the heated walls of the radiant tubes and from the refractory surfaces of the furnace enclosure.
Furthermore, the furnace enclosure may be filled with an inert, radiatively
nonpartIcIpating atmosphere, such as nitrogen or argon, to prevent scaling or decarburization of the load during the heating process [5].
Recently, a new type of indirectly-fIred batch reheating furnace has been proposed and its thermal performance has been analyzed [10]. The furnace operates at elevated temperatures; therefore, interreflection of radiation between the walls of the enclosure and the load has been accounted for. This is a generalization of the analysis presented in the preceding subsection. The transport processes occurring in the flat radiant heaters are not treated, but radiation exchange among the walls and load are analyzed using the radiosity method [6 ,7]. Radiant heat transfer calculations are carried out on a spectral- band basis. Details concerning furnace design, operating conditions and analysis cannot be included, but can be found elsewhere [10]. Anunfolded representation of the furnace with a metal load on the hearth of the furnace is illustrated in Fig. 7.
The effect of the load material (including the spectral radiation characteristics) on the load surface temperature variation with time is illustrated in Fig. 8. It should be mentioned that the mass of the load for the four materials was identical, but their thicknesses were different. For purpose of comparison the panel heater temperature is also included in the fIgure. The oxidized iron has a higher temperature than unoxidized iron, because the latter has a higher absorptivity than the latter. The unoxidized iron is heated to a somewhat higher temperature than unpolished aluminum. The reason for this lies in the much higher thermal conductivity of aluminum as heat is conducted more effectively from the surface to the interior of the slab.
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Figure7. Unfolded representation of the low inertia furnace with a one-dimensional (slab)
load model on the hearth [10].
Figure8. Comparison of the load surface temperatures during heating of four different materials using a two-band
radiation model [10].
3. Heat Transfer in Semitransparent Materials
In this section of the paper we discuss heat transfer in semitransparent materials capable of absorbing, emitting and scattering thermal radiation in which radiative transfer is considered to be a volumetric phenomenon. The problems are generally multimodal in nature involving simultaneous conduction and radiation in a solid or conduction, convection and radiation in molten materials such as glass, crystal melts, etc. [4, II, 12].
For the purpose of the present discussion we distinguish two situations: (i) the material is
(9) capable of absorbing external radiation but the material is cold so that the volumetric rate of radiant energy emission is negligible in comparison to absorption, and (ii) the material is at sufficiently high temperature such that the rate of volumetric radiant energy absorption is of the same order of magnitude as emission. The latter situation is much more complex to analyze and several distinct variants can be identified [6, 7, 11, 12].
Here, we discuss a few specific examples which involve the interaction of radiation with semitransparent materials processing and fabrication operations.
Crystals, semiconductors, glass and ceramics are semitransparent to thermal radiation [4, 11, 12]. When such materials are processed at high temperature or manufactured, both volumetric absorption and emission of radiation must be accounted when predicting the temperature distribution in the material. Typically, such materials are homogeneous and scattering of radiation can be neglected in comparison to absorption, and this makes it easier to carry out the analysis of radiative transfer. But the materials are spectrally selective, and this must be considered in predicting the temperature distribution in the material undergoing processing. As an illustration we describe the main features of two models for processing of hot semitransparent materials.
3.1. ANNEALING OF OPTICAL QUALITY GLASS
During the process of annealing of optical quality glass which is used for optical components of imaging systems, the temperature gradients in the glass must be controlled very carefully by imposing a constant cooling rate of the order of 0.01 °C/h, because a true geometric and color image of an object is generated by using the property of refraction. Even small stresses due to the temperature gradients in the annealing range cause permanent strains and inhomogeneities of the refractive index [13], which can be detrimental to optical glasses of high quality. Transient combined conduction-radiation heat transfer in an optical quality glass disk has been analyzed [14], and the analysis is highlighted, and the findings are summarized below.
Transient cooling of a two-dimensional, axisymmetric glass disk (see inset in Fig. 9) cooled by combined convection and radiation has been analyzed by assuming that scattering is negligible in comparison to absorption. The glass is surrounded by isothermal black walls, and the space between the glass and the surrounding walls is occupied by a transparent gas. For wavelengthsA.<5 J..Lm the glass is semitransparent to radiation and for wavelengths 5 < A. < 00 J..Lm it is opaque to radiation. The transient energy equation for the model is [14]
pc aT = ':!"(kr aT) +~(k aT) _ V. F
at rar ar az az
where the spectral radiative flux vector F is found from the solution of the radiative transfer equation [6, 7, 11]. The interface energy balances provide the boundary conditions for the energy Eq. (9). The radiative transfer calculations have been carried out for BK7 optical quality glass (manufactured by Schott Glassworks) as an example using eight (8) spectral bands.
Figure 9 shows the temperature distributions along the center lines in the r- and z- directions. The temperature in the glass disk becomes nonuniform very rapidly. After 10 s, the temperature in the radial direction is nearly uniform within the glass, except in the immediate vicinity of the circumference. However, the temperature in the axial direction has become less uniform. Moreover, when the glass is cooled by free convection and radiation, the temperature in the axial direction becomes non-symmetric. This is clearly
evident from the results given elsewhere [14] which show the isotherms in the glass disk at different times for two different ambient cooling conditions.
h=h(T) - -h'{) /- - - - h=eonsi i
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Q.
E450 {E
350 0 20 40 60 80 100(0) 20 40 50
r(mm) z(mm)
Figure9. Comparison of centerline and midplane temperature distributions during cooling of optical quality glass disk by convection and radiation [14].