Numerical Heat and Mass Transfer

Transcription

1 Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 11-Radiative Heat Transfer Fausto Arpino

2 Nature of Thermal Radiation ü Thermal radiation refers to radiation energy emitted by bodies because of their temperature. All bodies at a temperature above absolute zero emit thermal radiation. ü The energy transport by radiation does not require an intervening medium between the hot and the cold surface. ü The actual mechanism of radiation propagation is not fully understood, but theories are proposed to explain the propagation process. According to Maxwell s electromagnetic theory, radiation is treated as electromagnetic waves, while Max Planck s theory treats radiation as photons, or quanta of energy. ü When radiation is treated as an electromagnetic wave, the radiation from a body at temperature T is considered emitted at all wave length form λ=0 to λ=. ü In most engineering applications the bulk of the thermal energy emitted by a body leis in wavelengths between about 0.1 and 100 μm. This wavelength spectrum is generally referred as thermal radiation. 2

3 Nature of Thermal Radiation ü In the study of radiation transfer, a distinction should be made between bodies which are semitransparent to radiation and those which are opaque. If the material is semitransparent to radiation, then the radiation leaving the body from its outer surfaces results from the emission at all depths within the material. The emission of radiation in such cases is a bulk or volumetric phenomenon. ü If the material is opaque to thermal radiation (metals woods, rocks, etc.) then the radiation emitted by the interior regions cannot reach the surface and the emission is regarded as a surface phenomenon. ü Also n ot e t h a t a ma t e r i al may b eh ave as a semitransparent medium for certain temperature ranges and as opaque for other temperatures. Glass is a typical example of such behavior. 3

4 Blackbody Radiation ü A body at any temperature above absolute zero emits thermal radiation in all wavelengths in all possible directions into space. ü The concept of blackbody is an idealized situation that serves to compare the emission and absorption characteristics or real bodies; a blackbody is considered to absorb all incident radiation from all directions at all wavelengths without reflecting, transmitting, or scattering it. For a given temperature and wavelength, no other bodies can emit more radiation than a blackbody. ü The spectral blackbody radiation intensity I bl (T) into a vacuum was first determined by Planck and given by: The sun emits thermal radiation at an effective surface temperature of about 5760 K and the bulk of energy is in the visible range Planck constant ( J s) ( ) = 2hc 2 I bλ T λ 5 e hc λkt 1 Speed of light in a vacuum Boltzmann constant ( J K) 4

5 Blackbody Radiation ü I bl (T) represents the radiation energy emitted by a blackbody at temperature T, streaming through a unit area perpendicular to the direction of propagation, per unit wavelength about the wavelength λ per unit solid angle about the direction of propagation of the beam. Energy I bλ = ( Area) ( wavelength) solid angle ( ) Solid angle definition Example Determine the solid angles subtended by the surfaces da 1 and da 2, when they are viewed from the point O for the dimensions and the geometric arrangement shown in the figure. ( ) dω 1 = da cos θ 1 1 = = sr 2 r ( ) dω 2 = da cos θ 2 2 = 2 r = sr 5

6 Blackbody Radiation It is of practical interest to know the amount of radiation energy emitted per unit area of a blackbody at an absolute temperature T in all direction into hemispherical space. The spectral radiation emitted by the surface da, streaming through the solid edge dω in any given direction, is given by: d 2 Q bλ = I bλ ( T )dacos( θ)dω = I bλ ( T )dacos( θ)sin( θ)dθdφ d 2 E bλ = d 2 Q bλ da dω = da 1 r 2 ( )( r dφ sin θ) = r dθ r 2 = dθ dφ sin θ 6

7 Blackbody Radiation The spectral blackbody radiation emitted per unit surface area in all directions into the hemispherical space is obtained by integrating the previous equation as follows: E bλ (T) = I bλ (T) 2π π 2 φ=0 θ=0 π 2 cos(θ)sin(θ)dθdφ = 2πI bλ (T) cos(θ)sin(θ)dθ = 2πI bλ (T) sin(θ)dsin(θ) θ=0 = 2πI bλ (T) sin2 (θ) 2 2 = πibλ (T) 0 π π 2 θ=0 I bλ ( T ) = λ 5 (e 2hc 2 hc λkt 1) E bλ T ( ) = C 1 λ 5 e C 2 λt 1 C 1 = 2πhc 2 C 2 = hc k At any given wave length, the emitted radiation increases with increasing temperature, and at any given temperature, the emitted radiation varies with wavelength and shows a peak. The locus of these peaks is given by Wien s displacement law: ( λt ) = C max 3 7

8 Blackbody Radiation: Stefan-Boltzmann Law The radiation energy by a blackbody at an absolute temperature T over all wavelengths per unit time and area is determined by integrating the spectral blackbody radiation equation over the whole wavelengths spectrum: E b ( T ) = λ=0 C 1 C 2 λt 1 Adopting the variable change x=λt, the Stefan-Boltzmann law is obtained, where E b is called the blackbody emissive power: E b λ 5 e ( T ) = σt 4 dλ W σ = Stefan-Boltzmann constant m 2 4 K 8

9 Radiation from real surfaces The spectral radiation intensity emitted by a real surface at a temperature T of wavelength λ is always less than that emitted by a blackbody at the same temperature and wavelength. Furthermore, radiation intensity from a real surface depends on direction, whereas the blackbody radiation intensity is independent of direction. To distinguish these two cases, we use the symbols: Spectral radiation intensity from a real surface: I λ (θ,φ) Radiation intensity: I(θ,φ)= λ=0 I λ (θ,φ)dλ W m 2 µm sr The spectral radiation energy leaving a unit surface area in all direction into hemispherical space is then: q λ = 2π π 2 φ=0 θ=0 q = I λ (θ,φ)cos(θ)sin(θ)dθdφ λ=0 q λ dλ W m 2 9

10 Concept of View Factor ü In engineering applications, problems of practical interest involve radiation exchange between two or more surfaces. ü When the surfaces are separated by a non-participating medium that does not absorb, emit, or scatter radiation, then the radiation exchange among the surfaces in unaffected by the medium. ü For any two or more surfaces, the orientation between them affects the fraction of the radiation energy leaving one surface that strikes the other surface directly. ü To formalize the effects of orientation in the analysis of radiation heat exchange among surfaces, the concept of view factor has been adopted. ü The physical significance of the view factor between two surfaces is that it represents the fraction of the radiative energy leaving one surface that strikes to the other surface directly. 10

11 Concept of View Factor Consider two elemental surfaces da 1 and da 2. Let r be the distance between the two surfaces, θ the polar angle between the normal to the elemental surface the joining segment. The rate of radiative energy leaving da 1 that strikes da 2 is: dq 1 = da 1 I 1 cos(θ 1 )dω 12 dω 12 = da 2 cos(θ 2 ) r 2 dq 1 = da 1 I 1 cos(θ 1 )cos(θ 2 )da 2 r 2 For a diffuse surface (the radiation intensity I is independent of direction), the rate of radiation leaving the surface element da 1 in all direction over the hemispherical space is: Q 1 = da 1 2π π 2 φ=0 θ 1 =0 I 1 cos(θ 1 )sin(θ 1 )dθ 1 dφ π 2 = da 1 I 1 2π sin(θ 1 )dsin θ 1 = da 1 I 1 2π sin2 (θ 1 ) 2 θ 1 =0 π 2 0 Azimuthal angle = da 1 I 1 π 11

12 Concept of View Factor The elemental view factor df da 1-dA2 is by definition the ration of the radiative energy leaving da 1 that strikes da 2 directly to the radiative energy leaving da 1 in all directions in the hemispherical space. Hence: df da1 da 2 = dq 1 Q 1 = da 1 I 1 cos(θ 1 )cos(θ 2 )da 2 r 2 da 1 I 1 π = cos(θ 1 )cos(θ 2 )da 2 πr 2 The elemental view factor df da is now immediately obtained by 2-dA1 Interchanging subscripts: Hence the reciprocity relation: df da2 da 1 = cos(θ 1 )cos(θ 2 )da 1 πr 2 da 2 df da2 da 1 = da 1 df da1 da 2 12

13 View Factor for Finite Surfaces The view factor df is determined by integrating the elemental view factor df da over the area 1-A2 da1-da2 A 2 : F da1 = cos(θ )cos(θ ) 1 2 A 2 da πr 2 2 A 2 The view factor df is determined by integrating the elemental view factor df da over the area 2-A1 da2-da1 A 2 and dividing it by A 2 (the division by A 2 makes the energy striking da 1 a fraction of that emitted by A 2 into the entire hemispherical space): Hence: And: df A2 = 1 da 1 A cos(θ 1 )cos(θ 2 )da 1 2 πr da = da1 2 2 A cos(θ 1 )cos(θ 2 ) 2 πr da 2 2 A 2 F A2 = F A A2 1 da da 1 1 = 1 cos(θ 1 )cos(θ 2 ) A 2 πr da da A1 F A1 = 1 cos(θ 1 )cos(θ 2 ) A 2 A 1 πr da da A 1 A 2 A 1 A 2 A 2 A 1 F A1 A 2 = A 2 F A2 A 1 13

14 View Factor Calculation 14

15 View Factor Calculation 15

16 View Factor Calculation 16