Presentation on theme: "Momentum Heat Mass Transfer"— Presentation transcript:
1 Momentum Heat Mass Transfer MHMT13Heat transfer-radiationRadiation heat transfer. Boltzmann, Kirchhoff, Lambert’s laws. Radiation exchange between surfaces and in absorbed gases.Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
2 RadiationMHMT13Thermal radiation are electromagnetic waves with wavelengths from 0.1 to about 10 m (these waves are described by Maxwell equations for electric E and magnetic H field intensity). The radiation can be alternatively represented by discrete massless particles, photons, having energy h (proportional to frequency ) and moving in different directions with the speed of light (3.108 m/s) in vacuum.It is not possible to find out precise analogy with the heat and momentum transfer, because photons are massless particles colliding with molecules (atoms) but not between themselves. Thus it is difficult to define viscosity, diffusion coefficients etc.
3 Radiation - fundamentals MHMT13Any material substance emits photons (you can imagine that photons are produced by vibration of small molecular oscillators ). The radiative heat flux emitted from a unit surface is emissive power E [W/m2]. The surface is also capable to absorb incoming photons. Absorbed radiation flux EA is a fraction of the incident flux EI and the flux EI-EA=ER are photons reflected from surface. The ratio EA/EI=A is the surface absorptivity.Emissive power EReflected power ERIncident power EIsome incoming photons are reflected backElmag radiation is generated by motion of particles (molecules) and energy of this motion (internal energy) is determined by temperature T. Therefore EE depends on temperature T and also upon the material property, calledsome photons are absorbed and increase internal energy of material.relative emissivity (emissive power E is directly proportional to emissivity)Surface absorbing all incident photons is characterized by absorptivity A=1.Blackbody surface is an ideal surface emitting maximum power (=1) and at the same time is an ideal absorber A=1. Emissive power of blackbody surface is denoted by Eb and E=Eb (index b-means black body).
4 Radiation – Kirchhoff law MHMT13Emissivity equals absorptivity. =A (ideal emitter of radiation is an ideal absorber). This statement follows from the second law of thermodynamics. Imagine two parallel plates, one, which is a blackbody surface (A=1, b=1) and the other one is gray (A,). Both plates are at the same temperature and therefore there cannot be a non-zero net energy transfer between them.Eb(T)Eb(T)(1-A)Eb(T)reflected radiationradiation emitted by gray plateresulting flux from left to right Eb(T)- (1-A)Eb(T)-Eb(T)=0 A = The Kirchhoff’s law holds not only for the overall emissivity/absorptivity but also for photons of particular energy (or a specific wavelength ), thereforeRemark: a surface can exhibit different emissivity and absorptivity at different wavelenghts (different energies of photons)One and the same material can be a good emitter of high energy photons (small ) and a poor absorber of low energy photons (large )
5 Radiation – Stefan Boltzmann MHMT13Stefan Boltzmann law – total emissive power of blackbody surface is proportional to the 4th power of thermodynamic temperature(s)= [W/(m2K4)] Stefan Boltzmann universal constantStefan Boltzmann law describes power of all photons emitted from a unit surface [W/m2], but doesn’t answer the questions about distributions of energies and directions of the emitted photons.The answer to the second question is easy: distribution of photon directions emitted from the blackbody surface is uniform. This law can be expressed in terms of the radiation intensity as follows: Intensity I( ) =Ib is constant (independent of the direction ). But what is it the intensity of radiation? See next slide
6 Radiation – Intensity MHMT13 Spherical coordinate system and definition of solid angle dTotal power emitted from the small surface dSPower of photons moving inside the solid angle d in the direction and emitted by surface dSs perpendicular to the beam isThe radiation emitted by blackbody surface is isotropic (intensity Ib is independent of the direction). Integrating the radiation power for solid angles covering the whole upper hemisphere results to relationship between the emissive power Eb and the intensity Ib
7 Radiation – Lambert MHMT13 Lambert’s law: If we relate the radiation intensity not to the projected area dSs but to the fixed surface dS, the directional heat flux (sometimes called directional emissive power E’b) will be decreasing with the increasing angle asYour skin absorbs more photons at noon than at evening even if the intensity of radiation is the same (neglecting photon absorption in atmosphere)noonafternoon
8 Radiation – Planck MHMT13 Planck’s law spectral emissive power E as a function of wavelength and temperature TTotal emitted energy at 1000K Is shownas the shaded area (integral of Planck’sequation). This area is proportionalto the 4th power of temperature asdescribed by the Stefan Boltzmann equation.max for T=1000KThis graph corresponds to the blackbody radiation=1Wien’s law (follows directly from the Planck’s law, calculate maximum of E at T=const)temperature x wavelength at max.power = constantCheck validity of this equation using isotherm T=1000 K
9 Radiation - emissivity MHMT13Previus diagram holds only for blackbody surface that reflects no radiation. Real surfaces of plastics, wood, etc are close to the behaviour of blackbody (non reflecting) surfaces having ~0.9, while metals (polished) are characterised by much less ~0.1Surfaces reducing emissivity uniformly ( is independent of wavelength) are gray-bodies.Emissivities of surfaces can be found in different tables (table or table)materialpaper0.93brick0.5 – 0.9wood0.8 – 0.9water0.67Paint white0.9 – 0.95Aluminium0.05Stainless steel (polished)0.22Steel (polished,oxidised)0.08 – 0.8=1 (black body)E=0.9 (gray body)non-gray body
10 Transmitted power (vacuum) MHMT13Heat transfer between two surfaces at different temperatures without participating medium, when photons are generated and absorbed only at surfaces.This is the most frequent case, because e.g. dry air containing only N2 and O2 molecules is almost transparent for photons (photons are emitted and absorbed mainly by heteropolar molecules like CO2, H2O). The situation is simplified as soon as the surfaces are black bodies, because then the intensity of radiation Ib is constant and independent of direction.View factor Fij is the ratio of energy leaving Si and intercepted by Sj related to the total energy leaving SiAssuming constant intensity I along the ray s, the view factor is simplified towhich proves the reciprocity law
11 Transmitted power MHMT13 View factors (available for different geometries of planar, cylindrical or spherical surfaces) enable to calculate resulting radiant energy between black body surfaces asThe case of heat transfer between gray surfaces is more complicated (it is necessary to calculate fluxes reflected from non-ideal surfaces – back and forth in the direction of beams). Simple result is obtained for the case of two bodies where the second one completely surrounds the firstT2,2,S2T1,1,S1Special case: S1=S2 (parallel walls)Special case: S1 << S2 (small object in big space)
12 Radiation between plates MHMT13Try to prove the previous relationship for power transmitted between parallel platesResulting flux transmitted by plate 1 (taking into account reflections from both plates)T1 1T2 2The same result (only with opposite sign) holds for power transmitted by plate 2, giving
13 Radiation in participating medium MHMT13Intensity of radiation I( ) is not a constant in the case that the radiation penetrates through a semitransparent material (for example flue gas).Photons moving through space filled by heteropolar gases (H2O, CO2, CO,…) can be absorbed by molecules of gas (a [1/m] is coefficient of linear absorption) or scattered (s [1/m] is coefficient of linear scattering). On the other hand new photons are emitted by the gaseous molecules having temperature T (this contribution to the intensity of radiation is determined by the Stefan Boltzmann law) and new photons are also coming from outside due to scattering into the direction .H2OIncomming radiationoutgoing radiationemittedabsorbedScattering additionIntensity I is a scalar function of spatial coordinate =(x,y,z) and the direction . I recommend you to read again the precise definition of radiation intensity in previous slides.Variation of intensity I is described by the integro-differential RTE equation that follows from the photon balance in the control volume oriented in the selected direction of beam.
14 Radiation in participating medium MHMT13In the medium which is neither emitting nor scattering the RTE reduces to the Lamberts law (radiation intensity is reduced only by absorption of photons)The unit vector is not a characteristics of radiation (you should not for example imagine that it is a vector of photons velocity), because you can select the direction arbitrary – the RTE just only tells you how quickly is the intensity changing in a selected direction. For a constant linear absorption coefficient the solution is exponentialRemark: Radiation can be sometimes not only dangerous, but confusing. The initial value I0 is not a constant but depends upon a selected direction (beam)!The coefficient of linear absorption a is similar but not the same as the absorptivity A of surface. It should be also distinguished from the attenuation coefficient that sums up not only the absorbed but also the scattered photons. Anyway all these coefficients are proportional to the density of molecules which are capable to absorb photons, and linear absorption coefficient is usually expressed in terms of partial pressures of water and CO2 (prevailing components in flue gases).
15 CFD modelling FLUENT MHMT13 This is example of 2 pages in Fluent’s manual (Fluent is the most frequently used program for Computer Fluid Dynamics modelling)
16 Heat flux in participating medium MHMT13Radiative heat fluxDivergence of the radiative heat flux at constant wavelengthDivergence of the total radiative heat flux (integrated for all wavelengths)(s)= [W/(m2K4)] Stefan Boltzmann universal constantCheck units
17 Fourier Kirchhoff equation MHMT13Additional term in Fourier Kirchhoff equation, respecting radiation can be interpreted as a volumetric source term. The following overall energy balance holds for simplified case af a gray participating medium (where absorption coefficient and incident radiation is independent of wavelength = and G=G)Solution of this equation is very difficult even in CFD (numerically). Note, that this is an integro-differential equation (it is necessary to integrate incoming photons from all directions).
19 What is important (at least for exam) MHMT13Stefan Boltzmann law (relative emissivity and absorptivity, Kirchhoff law)Radiation transfer between the two black body surfaces (view factor)What is it intensity I? Beer Lambert law (absorbing media)