Introduction to modeling of thermal radiation in participating gases

Transcription

1 Project Report 2008 MVK 160 Heat and Mass Transfer May 07, 2008, Lund, Sweden Introduction to modeling of thermal radiation in participating gases Eric Månsson Dept. of Energy Sciences, Faculty of Engineering, Lund University, Box Lund Sweden

2 Abstract Today it is very common in optimization and development work of combustion devices to model the combustion process with computational fluid dynamics CFD. A combustion process mainly consists of a flow field, chemical reactions and heat transfer were heat transfer occurs by conduction, convection and thermal radiation. This small project will briefly concern some basic theory about thermal radiation and modeling of thermal radiation in participating media. Introduction-Blackbody theory The spectrum of electro-magnetic radiation in the atmosphere consist of different types of radiation where thermal radiation is one of them. When a body is exposed of thermal radiation it either reflect, absorb or transmit the electro-magnetic energy, see Figure 1. Nomenclature C Absorption cross section [m 2 /mole] d Line half-width [cm] E emissive, [W/m 2 ] F particle size distribution function [-] I Intensity [W/m 2 sr] k Absorptive index in complex index of refraction [-] m complex index of refraction [-] n Refractive index [-] P Pressure [N/ m 2 ] R Gas constant [J/moleK] r position vector [m] S Line intensity [cm/molecule] ^ s Unite vector into I given direction [-] T Temperature [K] x Size parameter [-] Greek symbols β Extinction coefficient [-] λ Wave number, [m] κ Absorption coefficient [m -1 ] Wave number [cm -1 ] σ Scattering coefficient [-] Ф Phase function [-] Subscripts B blackbody At a given wave number i Direction Figure 1: Reflection, absorption and transmission of a thermal ray [1] A body which absorbs all radiation energy is called a blackbody. According to Kirchoff`s law of thermal equilibrium that state that emissivity of a body is equal its absorptivity, due to that blackbodies emits the maximum amount of radiation as compared with others matters. Prediction of blackbody radiation is done by Max Planck`s law. E Bλ = C 1 e λ 5 c2 / λt 1 (eq 1) Eq 1 can even be written as eq(2) called Wien`s displacement law. EB λ C1 = (eq 2) T C2 / πt ( λt ) { e 1} 16 Were C1 = Wm 2 2 C 2 = mk Figure 2 shows the blackbody radiation for a number of different temperatures after integration of Equation 1.2.[1]

3 Figure 3: Interaction of thermal rays and the medium in a control volume [2] The change of thermal intensity along directions ds can be described in words like; Figure 2: Spectral distribution of blackbody radiation [1] From previous laws radiations between surfaces can be evaluated. This way is not sufficient if concerning for instance a combustion chamber since combustion gas usually contains particles such as unburned hydrocarbons and soot. These particles participate in the heat transfer with emitting, absorbing and even scatter the thermal rays. To cover all parts of thermal radiation in a process with free particles it needs to be model in some way.[2] Radiative transfer equation When concerning thermal radiation processes such as combustion including participation gases a conservation equation for the radiative energy is necessary. Radiation is very much dependent of directions and its spectral properties. The most appropriate scalar for describing the nature of thermal rays is intensity. Intensity is defined as the radiative energy flow per unit solid angle and area normal to the rays. Intensity can either be represented as spectral or total form. The transport equation for thermal intensity so-called radiative transfer equation is derived from a Radiative energy balance along a single line in a specific direction, see Figure 3. Change of radiant energy along ds=-energy emitted out from CV- energy scattered out from CV +absorbed energy by CV+scattered energy in to CV. Equation 2-1 below shows the final transport equation for Radiative transfer, it requires serval mathematical steps to reach this final form. ^ sι ^ b ( r ( r, s) = κ ( r ) Ι (eq 3) ^ σ ) β ( r ) Ι ( r, s) + 4Π s ( r ) ^ ^ ^ Ι r s Φ r si s Π (, ) (,, ) 4 κ is spectral absorption coefficient σ s is spectral scattering coefficient β = κ + σ s The first term in Eq 3 on the right hand side represent the absorption, the second term is combined of emitted energy and scattered energy out from the control volume and the last term take care of inflowing energy due to scattering from the surroundings. Pay attention to that Eq 3 is missing the time dependence, this is due to that the thermal rays are travelling with the speed of light and that it s time dependence is usually neglible.[2,3] dω i

4 Properties of participating gases Modelling of absorption coefficient In engineering assignment were the aim is to solve RTE for a specific case including participating gases, establishing of the different coefficients are necessary. Many models for predictions of absorption coefficient from very simple one to quit complex have been developed. In generally the absorption coefficient depends on the molecular structure regarding electronic energy, vibration and rotational energy. Several models using quantum mechanics such as Line by line calculations, Narrow Band model and wide band model are available. In a Line By Line calculation the change in vibration and rotational energy in a molecule at a specific wave number is studied, that result is called line intensity. In real applications radiation occurs at different numbers of wave number, due to several broadening effects. To completely describe the intensity of the so-called integrated line intensity is introduced in spectral calculations, expression for the spectral absorption coefficient can be read in Eq 4, werec is the spectral absorption or emission cross section of a single line at a given wave number. P a C κ = (eq. 4) RT C = Sf ( 0 ) (eq. 5) Were S is an integrated line intensity and f ( 0) is the line shape function, which is determined from different broadening phenomena. Data for predicting the integrated line intensity and the shape function demands quite a lot of information for the specific modeling case. Such data can be found in several data bases, according to[4]. Band models is another popular and well used model for predicting absorption coefficient. These kinds of models works quite different compares to the previous one, an approximation of the absorption coefficient is done by some average value of some finite band of lines. A number of different models are found even in this category and one example can be seen below (Elsasser model). S sinh 2β κ = (eq 6) d cosh 2β cos( z z 0 ) Were S is spectral emissivity evaluated by using an exponentially probability density function. These types of models are quite computer demanding, empirical and correlations models are available for engineering problems. [2] Modeling of particle scattering Scattering of electromagnetic waves occurs in almost all gases containing free particles. Particle size, shape and material are such variables that very much control the scattering process. The scattering process is done by three different mechanisms: diffraction, reflection and refraction, see Figure 4. It is common to deal with scattering in terms of elastic and inelastic. If electromagnetic waves undergo an elastic scattering process the wave number remains constant. Usually all heat transfer problems are assumed to be elastic processes. If scattering of one particle is not affected by the surrounding particle it is named independent otherwise dependent.

5 particle diameter, as an example of some result from the previous concerned topic. [2, 3, 4] Figure 4: Interaction of electromagnetic waves with a small particle [3] The nature of interaction between particles and thermal radiations is a quite complex and described in The Mie Scattering Theory. Using of Mie Scattering Theory demands that a number of dimensional less parameters are evaluated to classify the specific case. m = n ik Refraction of the particle (eq 7) Πd x = Particle size (eq 8) λ c cr = Clearance to wave number ratio(eq λ 9) Rayleigh scattering: x<<1 and x(m-1)<<1 Goemetric optics: x>>1 and x(m-1)>>1 Rayleigh-Gans scattering: (m-1)<<1 and x(m-1)<<1 Anomalous diffraction: x>>1 and x(m- 1)<<1 Solutions for The Mie Scattering Theory are very complex but a number of simplified cases limited of the previous dimensional less parameters are available, further information see [2]. The results of these solutions may be the phase function (Φ) and the spectral scattering coefficient (σ s). Phase function predics the probability of a thermal ray to be scattered into a specific direction. Figure 5: Spectral scattering coefficient for fly ash [2] Summary of models for solving RTE Exact solutions for the Radiative Transfer Equation are impossible to reach, but with access to gas properties as spectral scattering coefficient and absorption coefficient this equation is possible to be modeled numerically. A number of different models are developed for this assignment. The most common models are: Spherical harmonics method, Discrete Ordinates Method, Finit Volume Method, Discrete Transfer Method, Zonal Method and Monte Carlo Method. When choosing model for RTE there are several important factors that have to be taken into account. It can be factors like solution techniques for other governing equation such as momentum, the computational domain and complexity of geometry. Advantages and disadvantages for a number of different models are presented in Figure 6.[3] Figure 5 shows the spectral scattering coefficient for fly ash as function of

6 [3] Sundén, B, 2008, Course literature folder for Heat and Mass Transfer, Division of Heat Transfer Department of Energy Science, Lund, Sweden [4] Figure 6: Advantages and disadvantages for different models of thermal radiation [3] Conclusion The main topic of this project was to investigate the possibility to model thermal radiation with respect to participating media, like combustion. If you are considering gases with well-known properties like scattering and absorption data, modeling of the thermal radiation should not be an issue for ordinary engineering problem. Many of the commercial CFD codes have several models included for thermal radiation with respect to participating gases. Imagine instead that modeling of a participating gas should be done that has not been examined before. Enormous computer capacity and time will be needed for evaluating the scattering and absorptive data, it is not appropriate for engineering simulations. Final conclusion and advice, use empirical correlations or already done calculations for these parameters. References [1] Sundén, B, 2006, Värmeöverföring, Studentlitteratur, Lund, Sweden [2] Bahador, M, 2007, On Radiative Heat Transfer Modeling with Relevance for Combustion and Biomass Furnaces, Doctoral Thesis, Division of Heat Transfer Department of Energy Science, Lund, Sweden