1. Momentum Heat Mass Transfer
MHMT13
Heat transfer-radiation
Radiation 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. Radiation
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Thermal 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
MHMT13
Any 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 E
Reflected power ER
Incident power EI
some incoming photons are reflected back
Elmag 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, called
some 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
MHMT13
Emissivity 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 radiation
radiation emitted by gray plate
resulting 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 ), therefore
Remark: 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
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Stefan Boltzmann law – total emissive power of blackbody surface is proportional to the 4th power of thermodynamic temperature
(s)= [W/(m2K4)] Stefan Boltzmann universal constant
Stefan 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 d
Total power emitted from the small surface dS
Power of photons moving inside the solid angle d in the direction and emitted by surface dSs perpendicular to the beam is
The 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  as
Your skin absorbs more photons at noon than at evening even if the intensity of radiation is the same (neglecting photon absorption in atmosphere)
noon
afternoon

8. Radiation – Planck MHMT13 Planck’s law
spectral emissive power E as a function of wavelength  and temperature T
Total emitted energy at 1000K Is shown
as the shaded area (integral of Planck’s
equation). This area is proportional
to the 4th power of temperature as
described by the Stefan Boltzmann equation.
max for T=1000K
This graph corresponds to the blackbody radiation
=1
Wien’s law (follows directly from the Planck’s law, calculate maximum of E at T=const)
temperature x wavelength at max.power = constant
Check validity of this equation using isotherm T=1000 K

9. Radiation - emissivity
MHMT13
Previus 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 ~0.1
Surfaces reducing emissivity uniformly ( is independent of wavelength) are gray-bodies.
Emissivities of surfaces can be found in different tables (table or table)
material 
paper
0.93
brick
0.5 – 0.9
wood
0.8 – 0.9
water
0.67
Paint white
0.9 – 0.95
Aluminium
0.05
Stainless steel (polished)
0.22
Steel (polished,oxidised)
0.08 – 0.8
=1 (black body) E
=0.9 (gray body)
non-gray body 

10. Transmitted power (vacuum)
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Heat 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 Fij is the ratio of energy leaving Si and intercepted by Sj related to the total energy leaving Si
Assuming constant intensity I along the ray s, the view factor is simplified to
which 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 as
The 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 first
T2,2,S2
T1,1,S1
Special case: S1=S2 (parallel walls)
Special case: S1 << S2 (small object in big space)

12. Radiation between plates
MHMT13
Try to prove the previous relationship for power transmitted between parallel plates
Resulting flux transmitted by plate 1 (taking into account reflections from both plates)
T1 1
T2 2
The same result (only with opposite sign) holds for power transmitted by plate 2, giving

13. Radiation in participating medium
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Intensity 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 .
H2O
Incomming radiation
outgoing radiation
emitted
absorbed
Scattering addition
Intensity 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
MHMT13
In 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 exponential
Remark: 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
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Radiative heat flux
Divergence of the radiative heat flux at constant wavelength
Divergence of the total radiative heat flux (integrated for all wavelengths)
(s)= [W/(m2K4)] Stefan Boltzmann universal constant
Check units

17. Fourier Kirchhoff equation
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Additional 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).

18. EXAM
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Radiation

19. What is important (at least for exam)
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Stefan 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)