Indo-German Winter Academy

Transcription

1 Indo-German Winter Academy Radiation in Non-Participating and Participating Media Tutor Prof. S. C. Mishra Technology Guwahati Chemical Engineering Technology Guwahati 1

2 Outline Importance of thermal radiation. Fundamental equations in radiative heat transfer. View factor analysis. Basic equations of participating medium radiation. More useful numerical methods for participating medium radiation. Discrete Transfer Method (DTM). Discrete Ordinates Method (DOM). Finite Volume Method (FVM). Chemical Engineering Technology Guwahati 2

3 Importance Radiation becomes more dominant than other two modes of heat transfer in high temperature applications. absence of medium. Evaluation of radiation exchange between surfaces and radiation transport through participating medium is necessary for wide variety of application in Thermal system design Combustion applications. Fire safety. Furnaces. High temperature heat exchangers. High temperature thermal protections. Bio-medical engineering. Boundary layer flow of a participating fluid. Collection of solar energy and greenhouse effect. Inverse problems. Chemical Engineering Technology Guwahati 3

4 Radiation Fundamental Equations Spectral intensity, I λ,e of emitted radiation is the rate at which radiant energy is emitted at the wavelength λ in the (θ,φ) direction, per unit area of the emitting surface normal to this direction,per unit solid angle about this direction, and per unit wavelength interval dλ about λ. Mathematically, I λ,e (λ, θ, φ)= The total hemispherical emissive power, E (W/m 2 ), is the rate at which radiation is emitted per unit area at all possible wavelengths and in all possible directions. Irradiation, G Radiosity, J Chemical Engineering Technology Guwahati 4

5 Fundamentals equations contd. The Planck Distribution The spectral distribution of blackbody emission is given by As blackbody is a diffuse emitter, so it follows emissive power is E(λ,T)= πi(λ,t) Wien s Displacement Law Black-body radiation The blackbody spectral distribution has a maximum and that the corresponding wavelength λ max depends on temperature. λ max T = µm.k Chemical Engineering Technology Guwahati 5

6 Technology Guwahati Fundamentals equations contd. The Stefan-Boltzmann Law The total emissive power of blackbody, E b can be obtained by integrating spectral emissive from limits 0 to. Performing the integration, we get E b = σt 4 This simple yet important result is Stefan-Boltzmann law. Here, σ is Stefan- Boltzmann constant and has numerical value σ = 5.670* 10-8 W/m 2.K 4 Some related terms Emissivity, ε Absorptivity, α Reflectivity, ρ Transmissivity, τ Indian Institute Of

7 Fundamentals equations contd. The interplay of energy exchange is characterize by following equation τ + ρ + α =1 For glass plate, there is mainly ρ and τ and no α. For gaseous medium, there is mainly τ and α and no ρ. For metal plates, there is mainly α and ρ and no τ. Kirchhoff s Law For any surface in enclosure, the emissivity of the surface is numerically equal to its absorptivity. ε λ,θ = α λ,θ Gray surface may be defined as one for which α λ and ε λ are independent of λ over spectral regions of the irradiation and the surface emission. In the absence of a participating medium, radiation is merely a surface phenomenon and its analysis is mainly done using the concept of view factors. Chemical Engineering Technology Guwahati 7

8 Non-participating medium - View Factor Analysis View factor F ij is defined as the fraction of the radiation leaving surface i that is intercepted by the surface j. Mathematically F ij = Some relations A i F ij = A j F ji (reciprocity relation) (summation rule) If the surface is concave, it sees itself and F ii is nonzero. However, for a plane or convex surface, F ii = 0 Chemical Engineering Technology Guwahati 8

9 View factor contd. The net radiative heat flux at a surface in a nonparticipating medium is q i = A i (J i G i ) heat flux Area Radiosity Irradiation Using the definition of radiosity and Kirchhoff's law, we have The term in denominator is surface radiative resistance. The radiation exchange of surface i with the other surfaces j is given by The denominator term in above equation is space or geometrical resistance. Chemical Engineering Technology Guwahati 9

10 View factor contd. Using these relations, any radiation problem in nonparticipating medium can be expressed as an equivalent network representation. The simplest example of an enclosure is one involving two surfaces that exchange radiation only with each other. Chemical Engineering Technology Guwahati 10

11 Radiation with participating medium Basic Difference Generally observed in Gas radiation. In addition to radiation from boundary, medium also takes part. In analyzing transport of radiation, absorption, emission and scattering by medium need to be accounted. So, radiation becomes a surface as well as a volumetric phenomena in the presence of a participating medium. Chemical Engineering Technology Guwahati 11

12 Participating media The need To predict heat transfer in hot furnaces. To study radiation in engine combustion chambers. Radiation transfer is very important in glass and optical fibers manufacturing. Recently, the radiative heat transfer in porous media and packed beds has become a major field of interest. To enhance the predictive capabilities of homogeneous and heterogeneous combustion phenomena. Chemical Engineering Technology Guwahati 12

13 Participating Medium Basic Equations Continuity, momentum, species conservation and energy equations are the governing equations required to solve any fluid flow and heat transfer problems. These equations are coupled and have to be solved simultaneously. The general form of overall energy equation for a single component radiatively participating compressible fluid may be written as (A) The above equation involves all three modes of heat transfer and is solved simultaneously with continuity and the momentum equations. Chemical Engineering Technology Guwahati 13

14 Radiative Transfer Equation (RTE) Solution of Radiative Transfer Equation(RTE) gives the magnitude of radiation intensity at a point in participating medium. Knowing the intensity, we can determine the radiative quantities viz. heat flux, emissive power etc. Important Assumptions Participating medium is in Local Thermodynamic Equilibrium (LTE) Radiative heat transfer has been assumed a steady state process. Medium is assumed to be diffuse and gray. State of polarization has been neglected. Participating medium has been assumed to have constant refractive index. Chemical Engineering Technology Guwahati 14

15 RTE contd. Terms contributing to radiation as it moves from s to s+ds Differential change in radiation intensity through ds Loss by absorption Loss by out-scattering Gain by emission from medium in Ω dir. Radiative energy balance over the control volume Gain by in-scattering radiation 15

16 RTE contd. where κ and σ s are absorption and scattering coefficients of the medium, respectively. i b is Planck s blackbody intensity and the function is called scattering phase function which obeys The change in intensity from location s to s+ds is as follows Chemical Engineering Technology Guwahati 16

17 RTE contd. Above equation can be written in the form where β is the extinction coefficient of the medium and is equal to κ +σ s. In gas-radiation, heat flux, intensity are generally expressed in terms optical thickness expressed as Optical thickness, τ gives measure of extent to which medium participates in the radiative heat transfer process. The above equation can be written in terms of τ as here ω is scattering albedo and is equal to. Chemical Engineering Technology Guwahati 17

18 RTE contd. Previous equation can be written in more compact and convenient form as (B) where the source function writing above equation in terms of θ and φ gives Equation (B) is the differential form of the RTE. As we can see, it is an integropartial differential equation. Chemical Engineering Technology Guwahati 18

19 Complexity of RTE It is an integro-differential equation and except for very idealized conditions, its analytic solution is quite difficult. It is third-order integral equation in intensity i(τ, θ, φ). It deals with seven independent variables. Source function is in general an unknown function of intensity. Usually, temperature and, therefore, blackbody intensity term is not known. Chemical Engineering Technology Guwahati 19

20 Divergence of Radiative Heat Flux RTE gives the conservation of radiative energy along a given a ray direction. Divergence of radiative flux gives a measure of net radiative energy absorbed or emitted in a control volume. The expression for is given by where G is the incident radiation given by Chemical Engineering Technology Guwahati 20

21 Various radiation models in participating medium The complexities in complete solution of RTE pose a challenging task. There has always been a quest for development of efficient and cost-effective methods for predicting radiative heat transfer rates in practical energy conversion systems. A number of numerical methods have been proposed each having there own merits and demerits. Depending upon the problem, one can opt for a particular method after making comparisons against benchmark results. Out of all numerical methods proposed, majority of radiative heat transfer analyses today appears to use one of the five methods: Spherical harmonics method or variation of it. Monte- Carlo method Discrete Transfer Method(DTM) Discrete Ordinate Method(DOM) Finite Volume Method (VOM) 21

22 Discrete Transfer Method(DTM) Considers a finite number of discrete directions and in each direction it finds intensity in a number of discrete steps. Intensity is assumed to originate from the boundary. In one step, intensity travels just between two faces of control volume. Control volume is small enough such that source function can be assumed constant. With this assumption, solution of RTE gives (C) Chemical Engineering Technology Guwahati 22

23 DTM contd. Source term S is calculated at the middle of path length τ. Angular distribution of intensities is known only at grid points. So, source function S can be calculated at four corners. Source value at middle point is found using bilinear interpolation. With S known, eq. (c) is recursively used to compute the intensity at p(i, j). When eq. (c) is used for first time, n lies at boundary and intensity i o is required. For a diffuse-gray boundary with temperature T w and emissivity ε w, i o is given by (D) Chemical Engineering Technology Guwahati 23

24 DTM contd. Once the intensity distributions are known at all the grid-points, the incident radiation G and net flux q R are numerically computed from (E) (F) where M θ and M φ are the number of θ and φ directions considered over the complete span of θ and φ, respectively. For boundary intensity, integration over polar angle θ is performed over and therefore number of directions considered is M θ /2. Calculations become much more simple for azimuthally symmetric radiation. Chemical Engineering Technology Guwahati 24

25 Convergence of DTM With known with DTM, energy eq. (A) is solved using numerical solvers such as FDM or FVM. In first iteration, with known or guessed value of boundary temp. and guessed value of source function, is computed at all grid points. With the solution of energy equation (along with continuity and momentum eq.), a new temp. field is obtained with which is again calculated. The procedure is repeated till the temperature field reaches convergence. Chemical Engineering Technology Guwahati 25

26 Discrete Ordinates Method (DOM) The radiative intensity at any location in the direction Ω about the sub-solid angle dωis given by Modest (1) where κ a is the absorption coefficient, σ s is the scattering coefficient and Φ is the scattering phase function. For a diffuse-gray boundary with emissivity ε, the radiative boundary condition is (2) In the DOM, the radiative transfer equation is written for a finite number of discrete directions. In the Cartesian coordinate system (Fig. 1), for a given direction Ω m having direction cosines μ m,ξ m and η m, Eq. (1) is written as where β is the extinction coefficient and S m is the source term which is approximated as (3) Chemical Engineering Technology, Guwahati 26

27 DOM contd. (4) In Eq. (4), w m is weight corresponding to the direction Ω m. For a discrete direction Ω m, Eq. (2) is written as (5) where is the direction cosine between m direction and the outward normal to the surface of the control volume. For a 3-D control volume, using the approach of the finite volume method, Eq. (3) is written as (6) Chemical Engineering Technology Guwahati 27

28 DOM contd. The intensities at each face of the control volume can be related to the cellcenter intensity I p by a linear relationship α is finite difference weighing factor which in DOM is in general taken as ½. (7) With the help of Eq. (7), the number of intensities unknown at the cell faces in Eq. (6) can be reduced. When all the direction cosines are positive, Eq. (7) is used to eliminate the unknown intensities at the east, north and front cell faces and then the cell-center intensity in terms of known intensities at the south, west and the back cell faces is written as (8) Chemical Engineering Technology Guwahati 28

29 DOM contd. When any of the direction cosines are negative, Eq. (8) is written as (9) In Eq. (9), the subscripts x,y,z on intensity I x,i y and I z refer to the reference faces in x,y and z-directions on which the intensities are known. In Eqs. (8) and (9) the source is given by (10) After all the intensities are known, heat flux at any point is computed from (11) Chemical Engineering Technology Guwahati 29

30 DOM contd. To evaluate the quadratures in Eqs. (5), (10) and (11), once the number M of ordinates/directions is decided, the direction cosines (μ m,ξ m,η m ) of the directions Ω m and their corresponding weights w m are taken from the tabulated values provided in several references. The catch In these tables, for a given order of approximation, the directions are fixed, and thus, one is constrained to select only the prescribed directions. Chemical Engineering Technology Guwahati 30

31 Finite Volume Method (FVM) The radiative transfer equation (RTE) in any direction identified by the solid angle Ω about an elemental solid angle dω is given by (a) where κ a is the absorption coefficient, σ s is the scattering coefficient and Φ is the scattering phase function. Eq. (a) can be written as (b) o where β = κ a + σ s is the extinction coefficient and S is the source term given by (c) Chemical Engineering Technology Guwahati 31

32 FVM contd. Resolving Eq. (b) along the Cartesian coordinate directions and integrating it over the elemental solid angle ΔΩ m, we get (d) If is the outward normal to a surface, then D m is given by (e) o where the direction. (f) Chemical Engineering Technology Guwahati 32

33 FVM contd. When is pointing towards one of the positive coordinate directions,, and are given by (g1) (g2) (g3) For pointing towards the negative coordinate directions, signs of, and are opposite to what are obtained from Eq. (g1), (g2) and (g3). 33

34 FVM contd. In Eq. (d), ΔΩ m is given by (h) Integrating Eq. (d) over the control volume and using the concept of the FVM for the CFD, we get (i) In any discrete direction Ω m, if a linear relationship among the two cell-surface intensities and cell-centre intensity is assumed, then (j) where γ is the finite difference weighting factor and its value is normally considered to be ½. While marching from the first octant of a 3-D enclosure for which, and are all positive, in terms of known cell-surface intensities can be written as Chemical Engineering Technology Guwahati 34

35 FVM contd. where When any one of the, or is negative, marching starts from other corners. In this case, a general expression of in terms of known intensities and source term can be written as (k) (l) (m) where in Eq. (m), x i, y i and z i suffixes over I m are for the intensities entering the control volume through x-, y- and z-faces, respectively and A x, A y and A z are given by (n) Chemical Engineering Technology, Guwahati 35

36 FVM contd. For a linear anisotropic phase function Φ(Ω, Ω ) =1+a cosθ cosθ, the source term S at any location is given by (o) which in terms of the incident radiation G and net radiative heat flux q R is written as (p) In Eq. (p), G and q R are given by and numerically computed from the following (q) Chemical Engineering Technology Guwahati 36

37 FVM Contd. (r) While marching from any of the corners, evaluation of Eq. (m) requires knowledge of the boundary intensity. For a diffuse-gray boundary/wall having temperature T b and emissivity ε b, the boundary intensity I b is computed from (s) In Eq. (s), the first and the second terms represent the emitted and the reflected components of the boundary intensity, respectively. Once the intensity distributions are known, radiative information required for the energy equation is computed from Chemical Engineering Technology Guwahati 37 (t)

38 Conduction-Radiation Problem Radiation combined with conduction has many practical application particularly in case of solid or highly viscous media. Problem formulation Consideration is given to a one-dimensional gray planar medium. South and the north boundaries of the medium are at arbitrary temperatures T S and T N, respectively. The medium is absorbing, emitting and anisotropically scattering. Thermal conductivity of the medium is κ and is assumed constant. Extinction coefficient β of the medium is also assumed constant. In absence of convection and heat generation, equation for conservation of total energy in non-dimensional form is written as (1) Chemical Engineering Technology Guwahati 38

39 Conjugate Conduction-Radiation where θ is the non-dimensional medium temperature. Here for nondimensionalization of temperature, south boundary temperature T S has been taken as reference. Ψ R is the non-dimensional radiative heat flux and N is the conduction radiation parameter defined as For the problem considered, non-dimensional boundary temperatures are given by (2) Right-hand side of Eq. (1) contains divergence of radiative heat flux dψ R /dτ and this can be calculated using any of the three methods discussed before. I am discussing here solution using DTM. In DTM, divergence of radiative heat flux dψ R /dτ appearing in Eq. (1) is given by (3) Chemical Engineering Technology Guwahati 39

40 Conduction-radiation contd. To solve energy equation (1), the divergence of radiative heat flux given by Eq. (3) is substituted in Eq. (1) which yields the desired governing integrodifferential equation to be solved in DTM Solution procedure To solve Eq. (4), it is expressed in finite difference form as (4) (5) Depending upon the values of conduction radiation parameter N, solution of Eq. (5) proceeds in two ways. For N 0.01, first a linear temperature profile is guessed for the right-hand side of Eq. (5). With this guess value of θ, incident radiation G * is calculated. For evaluation of G *, non dimensional effective intensity (EIs) I * are found from Eq. (C). Chemical Engineering Technology Guwahati 40

41 Conduction-radiation contd. The calculation of intensity starts from the bounding wall. To find out intensity values next to the bounding wall where n+1 in Eq. (C) is 1, intensity values at the bounding wall are required. For a gray boundary having emissivity ε w and temperature T w, boundary intensity in DTM is (C) (6) With right-hand side of Eq. (5) known, left-hand side is solved for θ j using the Thomas algorithm. To get the convergence, under-relaxation is used for small values of N. For values of N<0.01, again a linear temperature profile is guessed as in the first case. However, in this case, left-hand side of Eq. (5) and G * are calculated simultaneously. Strong under-relaxation is used to get the convergence. Chemical Engineering Technology Guwahati 41

42 Results Chemical Engineering Technology Guwahati 42

43 Summary Importance Application of radiation transport in industry. Fundamental Equations Planck s Distribution Law Wien s Displacement Law Stefan-Boltzmann Law Kirchhoff's Law Non-participating Medium View factor analysis. Reciprocity relation. Summation rule. Surface radiative resistance. Space resistance. Chemical Engineering Technology Guwahati 43

44 Summary Participating Medium How it differs. Need to study. Basic equations. Radiative Transfer Equation and its derivation. Complexities of RTE. Various radiation models. Discrete Transfer Method. Assumptions. Methodology. Convergence. Discrete Ordinate Method Methodology. Limitations. Finite Volume Method Methodology. Convergence. Chemical Engineering Technology Guwahati 44

45 Summary Conduction-radiation Problem. Problem formulation. Solution Procedure. Results. Chemical Engineering Technology, Guwahati 45

46 References Fundamentals of Heat and Mass Transfer, Frank P. Incropera and David P. DeWitt Radiative Heat transfer- Michael F. Modest PhD Thesis of Mr. Prabal Talukdar. S.C. Mishra, P. Chug, P. Kumar and K. Mitra, Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium, International Journal of Heat and Mass Transfer, 49(11-12), , June P. Talukdar and S.C. Mishra, Analysis of conduction-radiation problem in absorbing, emitting and anisotropically scattering media using the collapsed dimension method, International Journal of Heat and Mass Transfer, 45(10), , May Journal papers given by Prof. S.C.Mishra. World of information- INTERNET 46

47 Acknowledgement I would specially like to thank Professor S.C.Mishra for his enthusiastic support and appreciation during my preparation. Thanks for being patient and for your kind attention. Chemical Engineering Technology Guwahati 47

48 Chemical Engineering Technology Guwahati 48