Convection and Conduction Heat Transfer Part 7 ppt

Convection and Conduction Heat Transfer 170 FORTRAN (Dorn & McCracken, 1972), which has been input in the developed by Microsoft calculation environment of VISUAL FORTRAN PROFESSIONAL (Deliiski 2003b). The software package can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the non-stationary distribution of the temperature fields in the materials containing or not containing ice during their thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after that, taking into account the flexible spatial boundary of the melting ice. The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the non-stationary change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. Based on the calculations it can be determined when the moment of reaching in the entire volume of the heated wood has occurred for the necessary optimal temperatures needed for bending of the parts or for cutting the veneer, as well as the stage of the ennoblement of the wood desired by the clients. 8.1 Non-stationary thermal processing of prismatic wood materials With the help of the 3D model the change in t in the volume of non-frozen beech prisms with 0 0 0tC= and frozen beech prisms with 0 0 10tC=− with d = 0,4 m, b = 0,4 m, L = 0,8 m, b 560 ρ = kg.kg -1 , u = 0,6 kg.kg -1 and 20 fsp 0,31u = kg.kg -1 has been calculated during the time of thermal processing during 20 hours at a prescribed surface temperature 0 m 80 Ct = . The change in m t and t is shown on Fig. 10 in 6 characteristic points of the volume of the prisms with coordinates, which are given in the legend of the graphs. The increase in m t from 0 m0 0Ct = to 0 m 80 Ct = is done exponentially with a time constant equal to 1800 s. -10 0 10 20 30 40 50 60 70 80 0 4 8 121620 Time τ , h Temperature t , 0 C tm d/4, b/4, L/8 d/2, b/2. L/8 d/4, b/4, L/4 d/2, b/2, L/4 d/4, b/4, L/2 d/2, b/2, L/2 0 10 20 30 40 50 60 70 80 0 4 8 121620 Time τ , h Temperature t , 0 C tm d/4, b/4, L/8 d/2, b/2. L/8 d/4, b/4, L/4 d/2, b/2, L/4 d/4, b/4, L/2 d/2, b/2, L/2 Fig. 10. 3D heating at 0 m 80 Ct = of frozen (left) and non-frozen (right) beech prisms with dimensions 0,4 x 0,4 x 0,8 m, b 560 ρ = kg.kg -1 , u = 0,6 kg.kg -1 and 20 fsp 0,31u = kg.kg -1 With the help of the 2D model the change in temperature in 5 characteristic points of cross section of non-frozen oak prisms with 0 0 0Ct = and frozen oak prisms with 0 0 10 Ct =− has been calculated during the time of their thermal processing with prescribed surface temperature 0 m 60 Ct = and during the time of the consequent cooling with surface convection at 0 m 20 Ct = . Transient Heat Conduction in Capillary Porous Bodies 171 The prisms have the following characteristics: d = 0,25 m, b = 0,40 m, L > 1,0 m, b ρ = 670 kg.kg -1 , u = 0,6 kg.kg -1 and 20 fsp 0,29u = kg.kg -1 . The heating of the prisms continues until the reaching of the minimally required for cutting of veneer temperature in their centre, equal to 0 c 50 Ct = . During the time of cooling of the heated prisms a redistribution and equalization of t in their cross section takes place, which is especially appropriate for the obtaining of quality veneer. The change in m t and t is shown on Fig. 11 in 5 characteristic points from the cross section of the prisms with coordinates, which are given in the legend of the graphs. -10 0 10 20 30 40 50 60 0 4 8 1216202428 Time τ , h Temperature t, 0 C tm d=0, b=0 d/8, b/8 d/4, b/4 d/2, b/4 d/2, b/2 0 10 20 30 40 50 60 0 4 8 121620 Time τ , h Temperature t, 0 C tm d=0, b=0 d/8, b/8 d/4, b/4 d/2, b/4 d/2, b/2 Fig. 11. 2D heating at 0 m 60 Ct = and consequent cooling at 0 m 20 Ct = of frozen (left) and non-frozen (right) oak prisms with cross section 0,25 x 0,40 m, b ρ = 670 kg.kg -1 and u = 0,6 kg.kg -1 8.2 Non-stationary thermal processing of cylindrical wood materials With the help of the 2D model the change in the t in the longitudinal section of non-frozen beech prisms with 0 0 0Ct = and frozen beech prisms with 0 0 2Ct =− with D = 0,4 m, L = 0,8 m, b 560 ρ = kg.kg -1 , u = 0,6 kg.kg -1 and 20 fsp 0,31u = kg.kg -1 has been calculated during the time of thermal processing during 20 hours at a prescribed surface temperature 0 m 80 Ct = . The change in m t and t is shown on Fig. 12 in 4 characteristic points of the longitudinal section of the logs with coordinates, which are given in the legend of the graphs. -10 0 10 20 30 40 50 60 70 80 0 4 8 121620 Time τ , h Temperature t , 0 C tm R/2, L/4 R/2, L/2 R, L/4 R, L/2 0 10 20 30 40 50 60 70 80 048121620 Time τ , h Temperature t , 0 C tm R/2, L/4 R/2, L/2 R, L/4 R, L/2 Fig. 12. 2D heating at 0 m 80 Ct = of frozen (left) and non-frozen (right) beech logs with R=0,2 m, L=0,8 m, b ρ = 560 kg.kg -1 , u = 0,6 kg.kg -1 and 20 fsp u = 0,31 kg.kg -1 Convection and Conduction Heat Transfer 172 The change in t in the longitudinal section of non-frozen beech logs with 0 0 0Ct = and in frozen beech logs with 0 0 10 Ct =− has been also calculated with the given above parameters during the time of a 3-stage high temperature thermal processing in autoclave and during the time of the consequent cooling with surface convection at 0 m 20 Ct = (Fig. 13). -20 0 20 40 60 80 100 120 140 0481216 Time τ , h Temperature t , 0 С tm R=0, L=0 R/2, L/4 R/2, L/2 R, L/4 R, L/2 0 20 40 60 80 100 120 140 04812 Time τ , h Temperature t , C tm R=0, L=0 R/2, L/4 R/2, L/2 R, L/4 R, L/2 Fig. 13. 2D high temperature heating in autoclave and consequent cooling of frozen (left) and non-frozen (right) beech logs with R=0,2 m, L=0,8 m, b ρ = 560 kg.kg -1 and u = 0,6 kg.kg -1 Using 3D graphs and 2D diagrams a part of the results is shown on Fig. 14 from the simulation studies on the heat transfer in the radial and longitudinal direction of the frozen beech logs with t 0 = -2°C, whose temperature field is shown on the left side on Fig. 12. The non-stationary temperature distribution during specific time intervals of the thermal processing is clearly observed from the 3D graphs (left columns on Fig. 14). The 2D diagrams which show in more detail the results from the simulations can be used rather for qualitative than quantitative analysis of the thermal processing of the materials (right columns on Fig. 14). On the left parts of Fig. 10, Fig. 11, Fig. 12 and Fig. 13 the characteristic non-linear parts can be seen well, which show a slowing down in the change in t in the range from -2°С to - 1°С, in which the melting of the ice takes place, which was formed in the wood from the freezing of the free water in it. This signifies the good quality and quantity adequacy of the mathematical models towards the real process of heating of ice-containing wood materials. The calculated with the help of the models results correspond with high accuracy to wide experimental data for the non-stationary change in t in the volume of the containing and not containing ice wood logs, which have been derived in the publications by (Schteinhagen, 1986, 1991) and (Khattabi & Steinhagen, 1992, 1993). The results presented on the figures show that the procedures for calculation of non- stationary change in t in prepared software package, realizing the solution of the mathematical models according to the finite-differences method, functions well for the cases of heating and cooling both for frozen and non-frozen materials at various initial and boundary conditions of the heat transfer during the thermal processing of the materials. The good adequacy and precision of the models towards the results from numerous own and foreign experimental studies allows for the carrying out of various calculations with the models, which are connected to the non-stationary distribution of t in prismatic and cylindrical materials from various wood species and also to the heat energy consumption by the wood at random encountered in the practice conditions for thermal processing. Transient Heat Conduction in Capillary Porous Bodies 173 Fig. 14. 3D graphs and 2D contour plots for the temperature distribution with time in ¼ of longitudinal section of beech log with R = 0,2 m, L = 0,8 m, u = 0,6 kg⋅kg −1 and t 0 = -2°C Convection and Conduction Heat Transfer 174 9. Conclusion This paper describes the creation and solution of non-linear mathematical models for the transient heat conduction in anisotropic frozen and non-frozen capillary porous bodies with prismatic and cylindrical shape and at any u ≥ 0 kg.kg -1 . The mechanism of the heat distribution in the entire volume of the bodies is described only by one partial differential equation of heat conduction. For the first time the own specific heat capacity of the bodies and the specific heat capacity of the ice, formed in them from the freezing of the hygroscopically bounded water and of the free water are taken into account in the models. The models take into account the physics of the described processes and allow the 3D, 2D and 1D calculation of the temperature distribution in the volume of subjected to heating and/or cooling anisotropic or isotropic bodies in the cases, when the change in their moisture content during the thermal processing is relatively small. For the solution of the models an explicit form of the finite-difference method is used, which allows for the exclusion of any simplifications in the models. For the usage of the models it is required to have the knowledge and mathematical description of several properties of the subjected to thermal processing frozen and non- frozen capillary porous materials. In this paper the approaches for mathematical description of thermo-physical characteristics of materials from different wood species, which are typical representatives of anisotropic capillary porous bodies, widely subjected to thermal treatment in the practice are shown as examples. For the numerical solution of the models a software package has been prepared in FORTRAN, which has been input in the developed by Microsoft calculation environment of Visual Fortran Professional. The software allows for the computations to be done for heating and cooling of the bodies at prescribed surface temperature, equal to the temperature of the processing medium or during the time of convective thermal processing. The computation of the change in the temperature field in the volume of materials containing ice in the beginning of their thermal processing is interconnected for the periods of the melting of the ice and after that, taking into account the flexible boundary of the melting ice. The computation of the temperature fields is done interconnectedly and for the processes of heating and consequent cooling of the materials, i.e. the calculation of the change in temperatures in the volume of the materials during the time of their cooling begins from the already reached during the time of calculations distribution of temperature in the end of the heating. It is shown how based on the calculations it can be determined when the moment of reaching in the entire volume of the heated and after that cooling body has occurred for the necessary optimal temperatures needed, for example, for bending of wood parts or for cutting the veneer from plasticised wooden prisms or logs. The models can be used for the calculation and colour visualization (either as animation of the whole process or as 3D, 2D, 1D graphs of each desired moment of the process) of the distribution of the temperature fields in the bodies during their thermal processing. The development of the models and algorithms and software for their solution is consistent with the possibility for their usage in automatic systems with a model based (Deliiski 2003a, 2003b, 2009) or model predicting control of different processes for thermal treatment. 10. Acknowledgement This work was supported by the Scientific Research Sector of the University of Forestry, Sofia, Bulgaria. Transient Heat Conduction in Capillary Porous Bodies 175 11. References Axenenko, O. (1995). Structural Failure of Plasterboard Assembles in Fires. In: Australian Mathematical Society Gazette , Available from: http://www.austms.org.au/Gazette/ 1995/Jun95/struct.html Ben Nasrallah, S., Perre, P. (1988). Detailed Study of a Model of Heat and Mass Transfer During Convective Drying of Porous Media, International Journal of Heat and Mass Transfer , Volume 31, № 5, pp. 297-310 Chudinov, B. S. (1966). Theoretical Research of Thermo-physical Properties and Thermal Treatment of Wood, Dissertation for Dr.Sc., SibLTI, Krasnoyarsk, USSR (in Russian) Chudinov, B. S. (1968). Theory of Thermal Treatment of Wood, Publishing Company “Nauka”, Moscow, USSR (in Russian) Chudinov, B. S. (1984). Water in Wood, Publishing Company “Nauka”, Moscow, USSR (in Russian) Deliiski, N. (1977). Berechnung der instationären Temperaturverteilung im Holz bei der Er- wärmung durch Wärmeleitung. Teil I.: Mathematisches Modell für die Erwärmung des Holzes durch Wärmeleitung. Holz Roh- Werkstoff, Volume 35, № 4, pp. 141−145 Deliiski, N. (1979). Mathematical Modeling of the Process of Heating of Cylindrical Wood Materials by Thermal Conductivity. 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Modelling and Automatic Control of Heat Energy Consumption Requi- red for Thermal Treatment of Logs. Drvna Industria, Volume 55, № 4, pp. 181-199 Deliiski, N. (2009). Computation of the 2-dimensional Transient Temperature Distribution and Heat Energy Consumption of Frozen and Non-frozen Logs. Wood Research, Volume 54, № 3, pp. 67−78 Doe, P. D., Oliver, A. R., Booker, J. D. (1994). A Non-linear Strain and Moisture Content Model of Variable Hardwood Drying Schedules. Proceedings of the 4 th IUFRO International Wood Drying Conference, Rotorua, New Zealand, pp. 203-210 Dorn, W. S., McCracken, D. D. (1972). Numerical Methods with FORTRAN IV: Case Studies, John Willej & Sons, Inc., New York Dzurenda, L., Deliiski, N. (2010). Thermal Processes in the Technologies for Wood Processing, TU in Zvolen, ISBN 978-80-228-2169-8, Zvolen, Slovakia (in Slovakian) Ferguson, W. J., Lewis, R. W. (1991). A Comparison of a Fully Non-linear and a Partially Non-linear Heat and Mass Transfer of a Timber Drying Problem, Proceedings of the 7 th Conference on Numerical methods in Thermal Problems, Vol. VII, Part 2, pp. 973-984 Kanter, K. R. (1955). Investigation of the Thermal Properties of Wood. Dissertation, MLTI, Moscow, USSR (in Russian) Convection and Conduction Heat Transfer 176 Khattabi, A., Steinhagen, H. P. (1992). Numerical Solution to Two-dimensional Heating of Logs. Holz Roh-Werkstoff, Volume 50, № 7-8, pp. 308-312 Khattabi, A., Steinhagen, H. P. (1993). Analysis of Transient Non-linear Heat Conduction in Wood Using Finite-difference Solutions. Holz Roh- Werkstoff, Volume 51, № 4, pp. 272-278 Kulasiri, D., Woodhead, I. (2005). On Modelling the Drying of Porous Materials: Analytical Solutions to Coupled Partial Differential Equations Governing Heat and Moisture Transfer. In: Mathematical Problems in Engineering, Volume 3, 2005, pp. 275–291, Available from: http://emis.impa.br/EMIS/journals/HOA/MPE/Volume2005_3 /291.pdf Luikov, А. V. (1966). Heat and Mass Transfer in Capillary Porous Bodies, Pergamon Press Murugesan, K., Suresh, H. N., Seetharamu, K. N., Narayana, P. A. A. & Sundararajan, T. (2001). A Theoretical Model of Brick Drying as a Conjugate Problem. International Journal of Heat and Mass Transfer , Volume 44, № 21, pp. 4075–4086 Sergovski, P. S. (1975). Hydro-thermal Treatment and Conserving of Wood. Publishing Company “Lesnaya Promyshlennost”, Moskow, URSS (in Russian) Siau, J. F. (1984). Transport Processes in Wood, Springer-Verlag, NewYork Shubin, G. S. (1990). Drying and Thermal Treatment of Wood, ISBN 5-7120-0210-8, Publishing Company “Lesnaya Promyshlennost”, Moskow, URSS (in Russian) Steinhagen, H. P. (1986). Computerized Finite-difference Method to Calculate Transient Heat Conduction with Thawing. 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Drying and Thermal Treatment of Wood, University of Forestry in Sofia, ISBN 954-8783-63-0, Sofia (in Bulgarian) Whitaker, S. (1977). Simultaneous Heat, Mass and Momentum Transfer in Porous Media: A Theory of Drying. Advances in Heat Transfer, Volume 13, pp. 119–203 Zhang, Z., Yang, S., Liu, D. (1999). Mechanism and Mathematical Model of Heat and Mass Transfer During Convective Drying of Porous Materials. Heat Transfer – Asian Research, Volume 28 № 5, pp. 337-351 0 Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium Marco T.M.B. de Vilhena, Bardo E.J. Bodmann and Cynthia F. Segatto Universidade Federal do Rio Grande do Sul Brazil 1. Introduction Radiative transfer considers problems that involve the physical phenomenon of energy transfer by radiation in media. These phenomena occur in a variety of realms (Ahmad & Deering, 1992; Tsai & Ozi¸sik, 1989; Wilson & Sen, 1986; Yi et al., 1996) including optics (Liu et al., 2006), astrophysics (Pinte et al., 2009), atmospheric science (Thomas & Stamnes, 2002), remote sensing (Shabanov et al., 2007) and engineering applications like heat transport by radiation (Brewster, 1992) for instance or radiative transfer laser applications (Kim & Guo, 2004). Furthermore, applications to other media such as biological tissue, powders, paints among others may be found in the literature (see ref. (Yang & Kruse, 2004) and references therein). Although radiation in its basic form is understood as a photon flux that requires a stochastic approach taking into account local microscopic interactions of a photon ensemble with some target particles like atoms, molecules, or effective micro-particles such as impurities, this scenario may be conveniently modelled by a radiation field, i.e. a radiation intensity, in a continuous medium where a microscopic structure is hidden in effective model parameters, to be specified later. The propagation of radiation through a homogeneous or heterogeneous medium suffers changes by several isotropic or non-isotropic processes like absorption, emission and scattering, respectively, that enter the mathematical approach in form of a non-linear radiative transfer equation. The non-linearity of the equation originates from a local thermal description using the Stefan-Boltzmann law that is related to heat transport by radiation which in turn is related to the radiation intensity and renders the radiative transfer problem a radiative-conductive one (Ozisik, 1973; Pomraning, 2005). Here, local thermal description means, that the domain where a temperature is attributed to, is sufficiently large in order to allow for the definition of a temperature, i.e. a local radiative equilibrium. The principal quantity of interest is the intensity I, that describes the radiation energy flow through an infinitesimal oriented area d ˆ Σ = ˆ ndΣ with outward normal vector ˆ n into the solid angle d ˆ Ω = ˆ ΩdΩ, where ˆ Ω represents the direction of the flow considered, with angle θ of the normal vector and the flow direction ˆ n · ˆ Ω = cos θ = μ. In the present case we focus on the non-linearity of the radiative-conductive transfer problem and therefore introduce the simplification of an integrated spectral intensity over all wavelengths or equivalently all frequencies that contribute to the radiation flow and further ignore possible effects due to polarization. Also possible effects that need in the formalism properties such as coherence 8 2 Will-be-set-by-IN-TECH and diffraction are not taken into account. In general the Radiative-Conductive Transfer Equation is difficult to solve without introducing some approximations, like linearisation or a reduction to a diffusion like equation, that facilitate the construction of a solution but at the cost of predictive power in comparison to experimental findings, or more sophisticated approaches. The present approach is not different in the sense that approximations shall be introduced, nevertheless the non-linearity that represents the crucial ingredient in the problem is solved without resorting to linearisation or perturbation like procedures and to the best of our knowledge is the first approach of its kind. The solution of the modified or approximate problem can be given in closed analytical form, that permits to calculate numerical results in principle to any desired precision. Moreover, the influence of the non-linearity can be analysed in an analytical fashion directly from the formal solution. Solutions found in the literature are typically linearised and of numerical nature (see for instance (Asllanaj et al., 2001; 2002; Attia, 2000; Krishnapraka et al., 2001; Menguc & Viskanta, 1983; Muresan et al., 2004; Siewert & Thomas, 1991; Spuckler & Siegel, 1996) and references therein). To the best of our knowledge no analytical approach for heterogeneous media and considering the non-linearity exists so far, that are certainly closer to realistic scenarios in natural or technological sciences. A possible reason for considering a simplified problem (homogeneous and linearised) is that such a procedure turns the determination of a solution viable. It is worth mentioning that a general solution from an analytical approach for this type of problems exists only in the discrete ordinate approximation and for homogeneous media as reported in reference (Segatto et al., 2010). Various of the initially mentioned applications allow to segment the medium in plane parallel sheets, where the radiation field is invariant under translation in directions parallel to that sheet. In other words the only spatial coordinate of interest is the one perpendicular to the sheet that indicates the penetration depth of the radiation in the medium. Frequently, it is justified to assume the medium to have an isotropic structure which reduces the angular degrees of freedom of the radiation intensity to the azimuthal angle θ or equivalently to its cosine μ. Further simplifications may be applied which are coherent with measurement procedures. One the one hand measurements are conducted in finite time intervals where the problem may be considered (quasi-)stationary, which implies that explicit time dependence may be neglected in the transfer equation. On the other hand, detectors have a finite dimension (extension) with a specific acceptance angle for measuring radiation and thus set some angular resolution for experimental data. Such an uncertainty justifies to segment the continuous angle into a set of discrete angles (or their cosines), which renders the original equation with angular degrees of freedom a set of equations known as the S N approximation to be introduced in detail in section 3. Our chapter is organised as follows: in the next section we motivate the radiative-conductive transfer problem. Sections 3 and 4 are dedicated to the hierarchical construction procedure of analytical solutions for the heterogeneous radiative-conductive transfer problem from its reduction to the homogeneous case, using two distinct philosophies. In section 4.3 we apply the method to specific cases and present results. Last, we close the chapter with some remarks and conclusions. 2. The radiative conductive transfer problem In problems of radiative transfer in plane parallel media it is convenient to measure linear distances normal to the plane of stratification using the concept of optical thickness τ which is measured from the boundary inward and is related through the density ρ, an attenuation 178 Convection and Conduction Heat Transfer [...]... 0.50164 574 7 671 1904 0.4588 373 32 878 3 077 0.4652 872 682514292 0.46529258462 973 08 0.46529258 379 70406 0.46529258 379 70406 0.46529258 379 70406 0.46529258 379 70406 0.46529258 379 70406 Qr ( ) 1.5158 274 669152312 1.5859091448625833 1. 578 801 177 4222819 1. 578 79602115054 67 1. 578 7960219 574 921 1. 578 7960219 574 921 1. 578 7960219 574 921 1. 578 7960219 574 921 1. 578 7960219 574 921 Q( ) 2.0 174 732145864214 2.04 474 6 477 7408908 2.044088445 673 7112... 0 .77 7583 478 0564881 0 .77 75905224102305 0 .77 75905213060152 0 .77 75905213060152 0 .77 75905213060152 0 .77 75905213060152 Qc ( ) 0.50164 576 99309158 0.4588 373 28586 075 7 0.4652 872 7144 878 63 0.4652925 878 442414 0.4652925 870 115464 0.4652925 870 115464 0.4652925 870 115464 0.4652925 870 115464 Qr ( ) 1.5158 278 540320405 1.5859093685802235 1. 578 80140 975 34921 1. 578 796253 479 04 47 1. 578 7962542859926 1. 578 7962542859926 1. 578 7962542859926... 2.04408860 578 0 277 4 2.04408860 575 453 27 2.04408860 575 453 27 2.04408860 575 453 27 2.04408860 575 453 27 2.04408860 575 453 27 Table 3 The DM LTS350 results for M ranging from 0 to 200, assuming /0 = 0.5 M 0 1 5 10 20 50 150 200 ( ) 0.8 177 175 72 673 2399 0 .76 980833509194 57 0 .77 758332119 076 42 0 .77 75903655034493 0 .77 75903643992450 0 .77 75903643992450 0 .77 75903643992450 0 .77 75903643992450 Qc ( ) 0.50164 573 334 176 28 0.4588 373 3 570 9 278 2... 0.50164 573 334 176 28 0.4588 373 3 570 9 278 2 0.4652 872 662455 577 0.46529258261 278 13 0.46529258 178 00941 0.46529258 178 00941 0.46529258 178 00941 0.46529258 178 00941 Qr ( ) 1.5158 272 165190925 1.585909000 072 7045 1. 578 8010 270 48 973 6 1. 578 795 870 778 96 87 1. 578 795 871 5859125 1. 578 795 871 5859125 1. 578 795 871 5859125 1. 578 795 871 5859125 Q( ) 2.0 174 729498608555 2.04 474 633 578 19826 2.0440882932945312 2.04408845339 174 98 2.0440884533660064... from 0 to 200, and using for N the values 300, 350 and 400, respectively The displayed precision with 16 digits was adopted to show the smooth convergence with increasing M in the three cases for N M 0 1 5 10 20 50 100 150 200 ( ) 0.8 177 176 60285 371 7 0 .76 98083890454525 0 .77 75833829280683 0 .77 75904 272 5686 37 0 .77 75904261526551 0 .77 75904261526551 0 .77 75904261526551 0 .77 75904261526551 0 .77 75904261526551... L and 0 = 1 Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium Non-Linear Radiative-Conductive Gray Plane-Parallel Participating Medium 189 13 s s d d 1 2 1 2 1 2 0 Nc L 0.6 0.4 0.1 0.2 0.3 0.4 1.0 0.5 0.95 1.0 0.05 299 1 2 Table 1 Parameters of case 1 M 0 1 5 10 20 50 100 200 ( ) 0.8 177 177 9550 270 18 0 .76 9808 472 1160902... 1. 578 7962542859926 1. 578 7962542859926 1. 578 7962542859926 Q( ) 2.0 174 736239629563 2.04 474 66 971 662993 2.044088681202 278 5 2.0440888413232861 2.0440888412 975 391 2.0440888412 975 391 2.0440888412 975 391 2.0440888412 975 391 Table 2 The DM LTS300 results for M ranging from 0 to 200, assuming /0 = 0.5 The numerical results for , Qr ( ), Qc ( ) and Q( ) are shown in table 2,3 and 4 The stability and convergence of the method... Journal of Geophysical Research D, Vol 97, No 17 (April 1992), 188 671 8886, 188 67- 18886 Asllanaj, F., Jeandel, G., Roche, J.R (2001) Numerical Solution of Radiative Transfer Equation Coupled with Non-linear Heat Conduction Equation International Journal of Numerical Methods for Heat and Fluid Flow, Vol 11, No 5 (July 2001), 449- 473 Asllanaj, F., Milandri, A., Jeandel, G., Roche, J.R (2002) A Finite Difference... Transfer in Laser Tissue Welding and Soldering Numerical Heat Transfer, Part A, Vol 46, No 1 (January 2004) 23-40 Krishnapraka, C.K., Narayana, K.B., Dutta, P (2001) Combined Conduction and Radiation Heat Transfer in a Gray Anisotropically Scattering Medium with Diffuse-Specular Boundaries International Communications in Heat and Mass Transfer, Vol 28, No 1, (January 2001), 77 -86 Liu, X., Smith, W.L., Zhou,... Solution of Coupled Conductive Radiative Heat Transfer in a Two-layer Slab with Fresnel Interfaces Subject to Diffuse and Obliquely Collimated Irradiation Journal of Quantitative Spectroscopy & Radiative Transfer, Vol 84, No 4, (April 2004), 551-562 Ozisik, M.N (1 973 ) Radiative Transfer and Interaction with Conduction and Convection, John Wiley & Sons Inc., ISBN: 0- 471 -6 572 2-0, New York Pazos, R.P., Vilhena, . kg⋅kg −1 and t 0 = -2°C Convection and Conduction Heat Transfer 174 9. Conclusion This paper describes the creation and solution of non-linear mathematical models for the transient heat conduction. inward and is related through the density ρ, an attenuation 178 Convection and Conduction Heat Transfer Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating. defines the 180 Convection and Conduction Heat Transfer Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium 5 radiative convective transfer problem