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Numerical methods for radiative heat transfer. Centre Tecnol`ogic de Transfer`encia de Calor. Departament de M`aquines i Motors T`ermics.
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Guillem Colomer Rey
(^) Juny 2006
ogic de Transferencia de Caloraquines i Motors TermicsNumerical methods for radiative heat
transfer
Universitat Politecnica de Valencia
Universitat Polit`ecnica de Catalunya
Universidad de Zaragoza
Universitat Polit`ecnica de Catalunya
Universidad de Las Palmas de Gran Canaria
Vull aprofitar aquestes ratlles per donar les gracies a tothom qui ha contribu¨ıt, d’una o altra manera, a que aquest treball s’hagi pogut completar. D’entrada, a l’Assensi Oliva, que em va animar a comenc¸ar aquest projecte, a m´es de proporcionar totes les eines, basicament informatiques, que he emprat tots aquests anys. De ma- nera molt especial, agrair-li la seva comprensi ´o en moments dif´ıcils, permetent-me agafar-me el treball al meu ritme. Molt important ha estat tamb´e la calidesa que han aportat tots els membres del grup, els que hi son i els qui han marxat, i que han fet d’aquest un lloc on hi he treballat realment de gust. I, no podia faltar, el meu agra¨ıment tamb´e al sector in- formatic que mant´e els equipaments en funcionament, aixi com el “cluster”, del que n’he tret profit puntualment. M’agradaria tenir un record per en Miquel Costa, que desafortunadament va en- malaltir al poc de comenc¸ar la meva feina. Vam tenir el temps just de col·laborar en un parell de treballs, per o malauradament, no ha pogut veure aquesta tesi finalit- zada. Aquesta tesi tampoc hauria estat possible sense el suport i el recolzament dels meus pares, i que amb l’Eva i l’Aina han tingut la paciencia d’aguantar-me durant tot aquest temps. I a la Noe, que, en l’elaboraci ´o de la tesi, m’ha hagut de suportar m´es que ning ´u. A vosaltres,doncs , que m’heu donat tantes i tantes mostres d’estimaci ´o i afecte, us la dedico.
The main objective of the present thesis is to study the energy transfer by means of radiation. Therefore, the basic phenomenology of radiative heat transfer has been studied. However, considering the nature of the equation that describes such energy transfer, this work is focussed on the numerical methods which will allow us to take radiation into account, for both transparent and participating media. Being this the first effort within the CTTC (“Centre Tecnol ogic de Transferencia de Calor”) research group on this subject, it is limited to simple cartesian and cylindrical geometries. For this purpose, chapter 1 contains an introduction to radiative energy transfer, and the basic equations that govern radiative transfer are discussed. These are the radiative transfer equation, formulated in terms of the absorption and scattering co- efficients, and the energy equation. It is also given a discussion on when this mode of energy transfer should be considered. In this chapter are also defined all of the magnitudes and concepts used throughout this work. It ends with a brief description of some approximate methods to take radiation into account. The Radiosity Irradiosity Method, RIM, which is suitable only for transparent media, is introduced in chapter 2. In this chapter it is also described a numerical method to calculate the view factors for axial symmetric geometries. The main re- sults obtained in such geometries are also presented. Although a little disconnected from the rest of the present thesis, the algorithm used to handle “de facto” three dimensional geometries with computation time just a little longer than two dimen- sional cases, with no additional memory consumption, is considered worthy enough to be included in this work. In chapter 3, it is detailed one of the most widely used numerical methods to effectively compute radiation energy transfer within either transparent or partici- pating media. This method is called the Discrete Ordinates Method (DOM). The fundamental aspect of this method is the choice of an ordinate set to integrate the radiative transfer equation. The characterization of such valuable ordinate sets is laid out properly. The discretization of the radiative transfer equation, presented in chapter 1, is explained in detail, for both cartesian and cylindrical geometries. The direct solution procedure is also outlined. Finally, illustrative results obtained with the DOM under several conditions are presented. In the moment we wish to solve real problems, we face the fact that the prop- erties on which the radiative transfer equation is based, i. e., the absorption and scattering coefficients, depend strongly on radiation wavelength. Therefore, there are situations where the usually assumed gray behavior of the media involved is far from being true. In the present thesis, special emphasis has been placed on study- ing the radiative properties of real gases in chapter 4. This interest resulted on a
14 Outline
bibliographical research on how the wavenumber dependence of the absorption co- efficient, a fundamental issue regarding radiation heat transfer, is modeled and es- timated. Furthermore, this bibliographical research was focussed also on numerical models capable of handle such wavenumber dependence. Remarks about the differ- ent methods considered can be found in section 4.3. Several methods are discussed, and two of them, namely the Weighted Sum of Gray Gases (WSGG) and the Spec- tral Line Weighted sum of gray gases (SLW), have been implemented to perform non gray calculations. Some significant results are shown. Plenty of tests have been performed to the numerical code that resulted from the elaboration of this thesis. Its results have been compared to other numerical methods, published in several specialized journals, or directly to available analytical solutions. Cartesian and cylindrical geometries are the only ones taken into account. According to the results obtained, the objectives proposed in this thesis have been satisfied. As a demonstration of the usefulness of the implemented code, it has been succesfully integrated to a wider, general purpose, computational fluid dynamics code (DPC), fruit of the effort of many researchers during many years. Results of the above integration lead to the resolution of combined heat transfer problems, that are analyzed in chapters 5 and 6, where radiative heat transfer is coupled to convection heat transfer. The effect of radiation on the total heat transfer is studied in chapter 5, which has been published as International Journal of Heat and Mass Transfer, volume 47 (issue 2), pages 257–269, year 2004. In chapter 6, the influence of some parameters of the SLW model on a combined heat transfer problem is analyzed. This chapter has been submitted for publication at the Journal of Quantitative Spectroscopy and Radiative Transfer. Chapter 7 contains some final general remarks as well as ideas on how the present work could be continued. Finally the appendices include some material which would disturb the normal reading flow of the thesis, such as appendices A and B, and also material that does not fit too well within the title of the thesis in appendix C. All this material follows from the six-year period on which this thesis has been developed.
16 Chapter 1. Radiative heat transfer
putational fluid dynamics to improve its performance. Several works on paralleliza- tion exists [8], with the additional decision on whether it is better to parallelize the spatial or the angular domain. Good performances are obtained for spatial decom- position [9]. Monte Carlo radiation solvers do not face such decision, since they are very well suited to be parallelized [10]. Interpolation schemes are also considered, beyond the typical high-order schemes employed in the convection-diffusion equa- tion. Given the directional nature of radiation, the intensity is not interpolated from neighboring nodes but from its projection along the propagation direction [11]. The situations where radiative heat transfer plays a fundamental role are those where high temperatures are achieved. Notably, when simulating combustion cham- bers, radiation should be taken into account, with a complete solution of the ra- diative transfer equation, as in [12]. However, a survey on combustion oriented literature reveals that this is not exactly the case, since few works on this subject consider radiative heat transfer. And if they do, usually over-simplified models are assumed, models that even do not require the full solution of the radiative transfer equation [13]. Radiative transfer is considered also in turbulent flows. Due to the additional overhead that turbulence implies, earlier works [14] simply inserted a source term due to radiation in the Favre averaged Navier-Stokes equations, and no average was performed at all when solving the radiative transfer equation. Latter works do take into account the interaction between turbulence and radiation, by solving a time averaged radiative transfer equation and modeling the correlated quantities [15, 16]. A detailed review on these correlations can be found in [17]. In fire safety science, where an estimate of the energy output of a fire is desired, radiation transfer is a key aspect to be taken into account [18, 19]. Radiation is also the dominant mode of energy transfer in rocket plumes. These are analyzed in order to improve combustion efficiency and the base protection from radiation heating. The presence of scattering particles in a plume is considered in [20]. Furthermore, the study of industrial furnaces also requires the solution of the radiative transfer equation [21], not to mention any application which involves the use of solar energy.
We refer as radiative energy transfer to the variation of the energy of any system due to absorption or emission of electromagnetic waves. From a physical point of view, such electromagnetic waves can be understood as a group of massless particles that propagate at the speed of light c. Each of these particles carries an ammount of energy, inversely proportional to the wavelength of its associated wave. Such particles are dubbed photons. A single photon represents a plane wave, therefore we are forced to assume that it propagates along a straight path. Because of this, we
§1.2. Mathematical description of radiation 17
can think of a system loosing energy by emitting a number of photons, or gaining energy by absorbing them. Each photon has a well defined polarization state, an internal degree of freedom, which is important in calculating properties such as the fraction of incident energy that is reflected by a medium. However, thermal radiation is not polarized on aver- age, and therefore the polarization of the photons is ignored hereinafter. Radiation heat transfer presents a number of unique characteristics. First of all, the amount of radiative heat transfer does not depend on linear differences of tem- perature, but on the difference of the fourth power of the temperature. This fact implies that, at high temperatures, radiative energy transfer should be taken into account, particularly if large differences in temperature exist. Furthermore, it is the only one form of energy transfer in vacuum. Therefore, in vacuum applications, ra- diation should be taken into account, even an low temperatures. And, of course, it should be taken into account for study of devices which use solar energy. On the other hand, radiation can be neglected if there are no significant temper- ature differences in a given system, and also if highly reflective walls are present. Furthermore, this kind of walls can prevent bodies to radiate energy away, as well as to shield sensible components from damaging radiation. Such shields can be seen in any spacecraft, for instance. In order to fully account for radiation transfer, we are obliged to solve Maxwell equations for the electromagnetic field. Since these equations are linear, any solution can be represented as a superposition of plane waves, i. e., by a certain set of photons. It turns out that, by treating radiative transfer in this way, it won’t be necessary to solve the Maxwell equations. Therefore, all we need is to concentrate on the number of photons present at a single point, and its energy distribution. As the photons propagate along straight paths, we also need to know, at every spatial location, the number of photons propagating in a given solid angle. Under the assumption of straight propagation, we are neglecting the wave properties of radiation. However, these are only relevant for cavities whose size is of the order of magnitude of the radiation wavelength. This is not the case for most of the practical problems, since we can estimate the the longest significant wavelength at room temperature to be less than 20 microns. For high temperature applications, such wavelength is reduced further. Only for cavities of these dimensions wave properties of radiation become relevant. The photon density is defined as the number of photons, with wavelength be- tween λ and λ + dλ, crossing an area dA perpendicular to the photon direction sˆ , within a solid angle dΩ, per unit time dt and per unit wavelength dλ. It is more use- ful to consider the energy density I instead of the photon density. The energy density is simply the number of photons multiplied by the energy of each photon. The en- ergy density is usually named intensity radiation field, and its physical meaning is
§1.3. Radiative properties of materials 19
An important feature of a black body is the emissive power per unit area, which is readily found by integrating equation 1.3 over all wavelengths. The result is the well known Stefan-Boltzmann law, which relates the emissive power to the temperature of the black body:
Ib(T) =
2 π^4 k^4 15 h^3 c^2
σB T^4 π
Using equation 1.3, it is possible to show that the wavelength at which the black body emission intensity is maximum times the temperature is constant. This is known as Wien law. After some calculation, we find that λmax T = 2897.82 μmK.
1.3.2 Surface properties
Each surface will radiate a certain ammount of energy due to its temperature. Such emitted energy, which will depend on the properties of the surface, its temper- ature, the radiation wavelength, the direction of emission... leads to the definition of the emissivity of a surface. As stated before, we can use the black body as a ref- erence to define the surface properties of real bodies. Therefore, if the surface is at temperature T, the emissivity is defined as
ε = Iemitted(T, λ, sˆ ) Ib(T, λ)
where the temperature, wavelength, and incident direction dependence, have been dropped for the sake of clarity. As the black body is defined as the perfect radiation absorber, by the Kirchhoff law, there is no body capable of emitting more radiation energy than a black body for any given temperature. Therefore, the emissivity ε will be between zero and one. On the other hand, when an electromagnetic wave incides on a surface, the in- teraction between this incident wave and the elements of the surface result on a fraction of the energy of the wave being reflected, while the remaining fraction will be absorbed (or transmitted) within the material. These fractions may very well de- pend on the incident angle and the wavelength of the original wave. Specifically, we can define
ρ =
Ireflected Iincoming ; α =
Iabsorbed Iincoming ; τ =
Itransmitted Iincoming
again, the temperature, wavelength, and incident direction dependence, have been dropped. The coefficients ρ, α, and τ, are the reflectivity, absorptivity, and trans- missivity of the surface respectively. As the wave is either reflected, or absorbed, or transmitted, it is clear that we must have ρ + α + τ = 1 if the energy is to be conserved. An important feature of reflection should be pointed out: while the reflected wave has the same angle (with respect to the normal of the surface) of the incident
20 Chapter 1. Radiative heat transfer
wave for perfectly smoot surfaces, if the surface is not polished, the angle of the re- flected wave could have any value, due to multiple reflections. There is a fraction of the incoming intensity that is equally distributed for all possible outgoing direc- tions. Due to this, the reflectivity ρ is usually divided in a diffuse component ρd, and a specular component ρs. Therefore, a fraction ρd of the incoming energy will be reflected equally over all the angles, and a fraction ρs will be reflected in a angle equal to that of the incident wave. We have of course ρ = ρd + ρs. Moreover, a surface that absorbs a fraction α of the incident energy will emit the same amount of energy if it is at thermal equilibrium (the Kirchhoff law again). Therefore, we could assume that α(T, λ, ˆs ) = ε(T, λ, ˆs ). A good approximation is to assume also that the corresponding integrated, wavelength and direction indepen- dent, absorptivity and emissivity are equal. Hence, three parameters, namely ρd, ρs, and ε, will suffice to describe the radiative properties for any surface.
We are assuming that photons propagate along straight lines, hence the most natural way to examine the effect exerted by the medium on the number of photons (or equivalently on the intensity radiation field) is to analyze the intensity radiation field precisely along a straight line. In the most general case, the intensity along the direction defined by the unit vector ˆs will depend both on the distance in this direction, and on time t. Therefore we could write the variation of intensity with respect to and t as
dI = I(+ d, t + dt) − I(`, t) =
d` +
∂t dt. (1.7)
Recalling that photons travel at the speed of light c, we have d` = c dt, and the resulting variation of intensity per unit length along the direction sˆ is
dI d`
c
∂t
The speed of light is very high, resulting on the fact that, for practical purposes, we can think that the intensity radiation field instantly reacts to any changes of the phys- ical conditions that determine it. Therefore, the partial derivative of I with respect to time t in equation 1.8, will be ignored hereinafter. As an electromagnetic wave propagates inside a medium, it looses energy as the charged particles within de medium accelerate in response to the wave. These parti- cles, in turn, release part of its energy in form of electromagnetic waves. In the pho- ton framework, we think of the same process as photons being absorbed and emitted by the medium. However, if the final state of the interacting particle is the same as