Modelling of radiative heat transfer applying computer graphics software

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1 Modelling of radiative heat transfer applying computer graphics software K. Domke Institute of Industrial Electrical Engineering, Poznań University of Technology, Poland Abstract The paper presents theoretical foundations for the modelling of phenomena related to visualisation performed by means of computer graphics software and for the modelling of radiative heat transfer. Since the equations describing both of these processes are very similar, there is a possibility of applying certain computer graphics programmes to resolve problems related to radiative heat transfer. The paper explores all necessary supplements making it possible to perform such calculations. For a system characterised by simple geometry, results of simulation of radiative heat transfer are presented and compared with results of analytical calculations. Keywords: RADIANCE, radiative heat transfer, visualisation, computer graphics software. 1 Introduction Thermokinetics, describing radiative heat transfer; lighting engineering, investigating problems of determination of surface illumination and computergenerated graphics, resolving issues connected with visualisation (it is the creation of seemingly three-dimensional representations of virtual reality on a two-dimensional screen, based on mathematical descriptions) they all examine, to a greater or smaller extent, the same phenomena of emission, transmission and absorption of optical radiation energy. The similarity of phenomena occurring in all of these cases additionally offers the possibility to use similar research tools to investigate them. Only the simplest tasks involving radiative heat transfer or lighting engineering can be solved by Siegel and Howell [1] using analytical methods. Practically, all more demanding problems in these fields are currently é Heat Transfer VIII, B. Sund

2 434 Heat Transfer VIII solved using numerical methods or by means of modelling and simulation: see Jaluria and Torrance [2]. Similarly, when solving computer graphics tasks, different simulation techniques are applied: Ashdown [3] and Ward et al [4]. Their intensive theoretical and practical development (determined chiefly by the needs of visual media), which took place in recent years, has markedly approximated models used in this field to the actual physical phenomena creating the reality described. It thus seems interesting to adapt such sophisticated computer graphics software to solve very complex problems involved in radiative heat transfer. 2 Governing equation Contemporary advanced computer graphics software and interior visualisation applications are based on the visualisation equation proposed by Kajiya [5], given below: L ( x,x ) = g( x,x ) Le( x,x ) + ρ( x,x,x ) L( x,x ) dx (1) Ω where L(x 0,x 1 ) is luminance of point x 0 : the total of luminance of radiation emitted L e (x 0, x 1 ) and reflected (integral value) in the direction of point x 1 ; g(x 0, x 1 ) - factor dependent on the geometry of the system, defining the visibility of point x 1 from x 0 ; ρ(x 0,x 0,x 1 ) specular reflectance of radiation for point x 0, with radiation propagating from the direction of point x 2 and reflected in the direction of x 1. Integration is performed along the whole hemisphere Ω surrounding x 0. This is illustrated by fig. 1. S 1 S 2.X 2.X 1 L(x 2,x 1 ) L(x 0,x 1 ) S 0 L e (x 0,x 1 ).X 0 Figure 1: Illustration of eqn (1). The eqn. (1) was written in the terminology used in lighting engineering and computer graphics, where the concept of luminance L v [lm/m 2 /sr] is used, referring to visible radiation. Thermokinetics, however, uses the concept of

3 Heat Transfer VIII 435 radiance L [W/m 2 /sr] referring to all optical radiation (including thermal radiation). The solution of eqn. (1) for every point of surfaces S 0...S n under consideration consists of determination of luminance of each of these points. This is the basic information, necessary for further construction of visual images of surfaces examined. Unfortunately, the eqn. (1) cannot be solved analytically. Only simulation methods can be applied. A commonly used method is backward ray tracing. The equation describing heat balance of point x 0 in Siegel and Howell [1] has a form that is similar to (1), as given below: p eff ( x + Ω 0, T, θ, φ ) = p ( x, T, θ, φ ), 0 e ρ( x, θ, φ, θ in, φ in )p in (x, T j, θ in, φ in 0 2 ) cos θdω (2) where, in radiative heat transfer terminology: p eff stands for surface density of effective radiant intensity (radiance) of point x 0 in the direction of x 1, defined by angles (θ, φ); T represents temperature and the index in concerns incident radiation. The eqn. (2), when only diffuse radiation is considered, is simplified to a system of linear equations, solved (when the number of points is limited) using exact methods (e.g. matrix methods) or approximate methods. When taking into account both diffuse and specular reflection, the eqn. (2) is solved applying simulation methods, usually radiosity method. However, this method calls for considerable computer resources (memory capacity and the number of calculations), which is a significant limitation in the case of radiative heat transfer systems which are geometrically more complex. 3 Surface illumination and radiative heat transfer Despite the fact that the eqns (1) and (2) describing light radiated from the surface (visible radiation) and heat radiation are practically identical, when considering systems of such surfaces, there are fundamental differences stemming from a range of simplifications which are perfectly legitimate in lighting engineering and computer-generated graphics and which, however, are not acceptable in investigations of radiative heat transfer. Thus when examining surface illumination, the distinction between active surfaces (sources of light) and passive surfaces (reflecting surfaces) is acceptable. It is also possible to disregard visible radiation falling on the source. However, in radiative heat transfer, each surface is at the same time active (since it emits radiation) and passive (since it reflects radiation). No radiation falling on any surface may therefore be ignored. In the case of surface illumination, absorbed radiation is disregarded and only reflected radiation is taken into account. This is permissible because in the case of light effects, the energy state (temperature) of boundary surfaces is not taken into consideration. In radiative heat transfer, the absorbed flux of power cannot be disregarded, as it is an essential element of the power

4 436 Heat Transfer VIII balance calculated for each boundary surface. In illumination investigations, the division of surfaces into active and passive ones, luminance of sources and reflection characteristics of the remaining surfaces are specified as boundary conditions. As a result, luminance distributions on examined surfaces are obtained. In the case of radiative transfer, the boundary conditions are temperature or power density and reflection-absorption characteristics of all surfaces. The final result of calculations is the determination of missing temperature values or power density values and power transfers between surfaces. Still, due to the similarity of eqns (1) and (2), there is a possibility of applying computer graphics software or visualisation based on the eqn. (1) and the method of backward ray tracing in radiative heat transfer simulations. In order to do this, it is necessary: to define an unambiguous relation between thermal values (temperature) on the boundary surface and light values (luminance), to supplement illumination simulation software (which takes into account the division into sources of light and reflecting surfaces) with procedures that include the simultaneous emission and reflection of radiation and also define the power absorbed. 4 Software used for illumination modelling and visualisation There is a very wide range of computer applications that can be used in computer graphics or interior visualisation. At the same time, various methods of determining illumination distributions and rendering are used, with varying accuracy of reproduction of light phenomena. Examples of such software are listed in table 1 below. Only those types of computer graphics or visualisation software are suitable for adaptation for the purpose of solution of problems connected with radiative heat transfer which offer distribution of luminance (or illuminance) on examined surfaces as an intermediate product. An accurate reproduction of the spatial characteristics of emitted and reflected radiation is also desirable. The latter can be achieved both by taking into consideration extreme cases of diffuse and specular reflection and a description of reflecting properties of materials using BRDF. 5 Thermal values and light values In tasks of radiative heat transfer for surfaces that form boundaries of the system under examination, two types of boundary conditions are specified: Dirichlet s boundary condition, i.e. specification of temperature; or Neumann s condition, where the density of power that penetrates the boundary surface from outside is defined. Both cases are illustrated in fig. 2.

5 Heat Transfer VIII 437 Table 1: List of selected computer graphics software which can be used in surface illumination simulations. Name of application Lightscape Visualizat. System - LVS ver Spectr system ver Radiance ver. 2.5 Method of determin. of light distribut. Energybased radiation balance Monte Carlo + bidirect. backward ray tracing Backward ray tracing + Monte Carlo Backward ray tracing Backward ray tracing Backward ray tracing Space distribut. of sources perfectly specular parallel or just any kind Diffuse or IES format possible Method of visualisation (rendering) Characteristics of reflecting materials perfectly specular perfectly specular, BRDF** perfectly specular, BRDF, anisotrop. Intermediate product Distrib. of luminance or illumin.: files* Distrib. of luminance or illumin.: files and drawings Distrib. of luminance or illumin.: files * only in the commercial version ** only for rendering Not considered Spread reflection, sources of parallel radiation Both diffuse and BRDF defined radiation, Selfdivision of surface Secondary sources for some materials set of T Dirichlet s condition radiative heat transfer set L interior illumination set of p Neumann s condition P out interreflection procedure: determines L Figure 2: Radiative heat transfer and an equivalent illumination system. In the interior illumination system, luminance L of the source of light, equivalent to the surface with Dirichlet s condition (set temperature T and emissivity ε) from the radiative heat transfer system is specified by the formula (3) Domke et al [6] given below: σ S ε T L = (3) S π p 4

6 438 Heat Transfer VIII where: S is the plane surface, σ = 5, W/m 2 K 4, and the index p represents a plane perpendicular to the direction of ray propagation. By contrast, in the case of a surface with Neumann s boundary condition (set p zw ), there is no possibility of direct determination of L. The value of equivalent luminance L can be defined in the course of procedure adjusting the current value of L x to external power (P out = p out S) and the value of irradiance E in coming from other surfaces of the system. Also, the balance of power of surface S derived in Domke et al [6] should be considered: Pout + E in S ε L = (4) S π p Therefore, using the formulas given in (3) and (4), it is possible to replace boundary surfaces of the radiative heat transfer system, for which initially either temperature or the density of external power was specified, with energetically equivalent surfaces which are sources of light with known luminance L. An additional procedure is also required for the purpose of taking into consideration the fact described in section 3, i.e. lack of possibility of dividing surfaces into active (sources of light) and passive (reflecting surfaces) in radiative transfer, which is required by computer graphics software. This is achieved by granting a single surface the status of source, while regarding other surfaces as passive ones. Planes of this type experience the process of reflection and absorption of radiation. The status of source is granted successively to each of the surfaces and absorbed radiation is summed up. This procedure, supplementing the RADIANCE system, is described in greater detail in Domke et al [6]. 6 RADIANCE system One of the publicly available [7] software packages setting up a new calculation environment is RADIANCE. Is made up of a set of procedures written in the C language, working in the UNIX environment and designed, according to the intention of its original developers, for interior visualisation in Ward and Shakespeare [4]. In addition to final visualisation images of virtual reality, the RADIANCE package generates the distribution of luminance on set surfaces as an intermediate product (cf. table. 1). These features, following addition of some extra procedures, make it possible to use this package to perform modelling and simulation of radiative heat transfer. The interconnection of the process of visualisation and thermal calculations is illustrated in fig. 3. The difference consists of determining either irradiance E, whose values are necessary for further visualisation process, or absorbed power P a, which is necessary for drawing up thermal balances of surfaces and determining thermal power transferred radiatively between the boundary surfaces of the system.

7 Heat Transfer VIII 439 Data concerning geometry of system and emission properties Simulation of emitted radiation Reflexivity data Examination of ray history for each reflection determination and summation of irradiance absorbed power End of illumination task - luminances known End of energy transfer task - powers and temperatures known Data concerning conditions of vision Visualisation End of visualisation task image of scene known Figure 3: Diagram illustrating the process of simulation applied to visualisation and radiative heat transfer tasks. 7 Example of radiative transfer The case examined involves radiative heat transfer between two opposite rectangles (fig. 4), whose temperatures and emmissivities are known. It is assumed that radiation is diffuse. The configuration coefficient ϕ 1-2 of such system is defined by the following formula in Siegel and Howell [1]: ϕ 1 2= X2Y2 ln + X Y2 arctg πxy X Y X Y2 +, Y + Y X2arctg X arctg ( X) Y arctg ( Y) X2 (5)

8 440 Heat Transfer VIII S 1, T 1, ε 1 p 1-2 S 2, T 2, ε 2 h a b Figure 4: Examined system of radiative heat transfer. Table 2: Values of mean configuration coefficient ϕ 1 2 and density of radiative transfer power p 1-2 determined using the method of simulation and analytical calculations. Configuration coefficient ϕ 1 2 Transfer power densities p 1-2 [W/m 2 ] Height h Analytic calculation RADIANCE calculation Error (%) Analytic calculation RADIANCE calculation Error (%) 0, for: a=3, b=2, T 1 =1000K, ε 1 =0,8, T 2 =800K, ε 2 =0,6 where: X=a/h, X2=1+X 2, Y=b/h, Y2=1+Y 2 power p 1-2 equals: and density of radiative transfer p ) σε 1ε 2ϕ1 2 (T T = (6) 1 (1 ε )(1 ε ) ϕ Table 2 includes the results for configuration coefficient values ϕ 1-2 and densities of radiative transfer power p 1-2 obtained thanks to modelling of radiative heat

9 Heat Transfer VIII 441 transfer based on the computer graphics software package RADIANCE, contrasted with ϕ 1-2 and p 1-2 values resulting from accurate formulas given in eqns (5) and (6). The data included in table 2 regards diffuse radiation. The RADIANCE package also makes it possible to model radiative heat transfer for any spatial nature of emitted and reflected radiation. Modelling and simulation of this type of radiation in such case is the only possible method of calculation, as results of this type cannot be produced through accurate analytical calculations. Examples of results obtained for actual materials with non-diffuse characteristics are given in Domke and Hauser [8]. 8 Conclusion It is possible to apply computer graphics software designed for interior visualisation which makes it possible to define luminance distributions (e.g. RADIANCE) for the purposes of simulation of radiative heat transfer and for the determination of distribution of temperatures and power densities on boundary surfaces. References [1] Siegel R., Howell J.R.: Thermal radiation heat Transfer.: Mc-Graw Hill Book Co.:, New York, [2] Jaluria Y., Torrance K.E.: Computational heat transfer. Hemisphere Pub. Co.: Washington, [3] Ashdown I.: Radiosity A Programmer s Perspective. John Wiley & Sons Inc.: New York, [4] Ward G.L., Shakespeare R.: Rendering with RADIANCE- The Art and Science of Lighting Visualization. Morgan Kaufmann Publ.: San Francisco, [5] Kajiya J. The Rendering Equations. Computer Graphics, 20(4), [6] Domke K, Hauser J., Wandachowicz K.: Calculation of radiation flux transfer using ray tracing method (Chapter X), Computer Applications in electrical engineering. ed. Nawrowski R., Wyd. Inst. Elektrotechniki Przemysłowej Pol. Pozn., Poznań pp , [7] RADIANCE. Home page. polish page: [8] Domke K, Hauser J: Application of RADIANCE procedures for radiative heat transfer modeling (Chapter X), Computer aid design of electroheat devices ed. Hering M., Sajdak Cz., Wciślik M., Wyd. Pol Śląskiej, Gliwice, pp 32-49, 2002.

Global Illumination. CMPT 361 Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller

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