10. Radiative heat transfer

Content ‰

10.1 The problem of radiative exchange

‰

10.2 Kirchhoff’s law

‰

10.3 Radiant heat exchange between two finite black bodies

10. Radiative heat transfer

College of Energy and Power Engineering

JHH

1

10. Radiative heat transfer

College of Energy and Power Engineering

JHH

10. Radiative heat transfer

2

10.1 The problem of radiative exchange

The electromagnetic spectrum

10.1 The problem of radiative exchange

College of Energy and Power Engineering

JHH

10. Radiative heat transfer

3

10.1 The problem of radiative exchange

College of Energy and Power Engineering

10. Radiative heat transfer

Black bodies ‰

JHH

4

10.1 The problem of radiative exchange

Absorptance, reflectance and transmittance

Perfect thermal radiator

‰

ƒ Absorbs all energy that reaches it (include visible light and other radiation)

translucent slab ƒ ƒ ƒ ƒ

Incident energy flux Absorbed energy q a Reflected energy q r Transmitted energy

q = αq , absorptance α = qa / q = ρq , reflectance ρ = qρ / q qt = τq , transmittance τ = qτ / q

α + ρ +τ = 1 cross section of a spherical hohlraum. ‰

ƒ For single wavelength

‰The

hole has the attributes of a nearly perfect thermal black body

αλ + ρλ + τ λ = 1

ƒ For black body

α b = α λb = 1 ρ b = τ λb = 0

ƒ For opaque solid

τ =0 College of Energy and Power Engineering

JHH

5

α + ρ =1 College of Energy and Power Engineering

JHH

6

10. Radiative heat transfer

10.1 The problem of radiative exchange

10. Radiative heat transfer

Emissive power ‰

Diffuse and specular emittance and reflection

Total emissive power

‰

dQ

dQ e(T ) = W / m2 dA

ƒ The flux of energy radiating from a body per unit time and per unit area ‰

10.1 The problem of radiative exchange

Energy emitted and reflected by a non-black surface may leave the body diffusely or specularly ƒ Specular and diffuse reflection of radiation.

dA

2 Monochromatic emissive power eλ (λ , T ) W / m ⋅ μ m

ƒ Energy emitted per unit area and per unit time within a unit small wavelength interval centered around the wavelength λ

eλ ( λ , T ) =

de(λ , T ) or e(λ , T ) = dλ

e(T ) ≡ e(∞, T ) =

∫

∞

0

∫

λ

0

e λ ( λ , T ) dλ

eλ (λ , T )dλ

ƒ Black bodies are diffusive

College of Energy and Power Engineering

JHH

10. Radiative heat transfer

7

10.1 The problem of radiative exchange

College of Energy and Power Engineering

10. Radiative heat transfer

Solid angle ‰

‰

Definition

ƒ Directional intensity of radiation

ƒ For full sphere dA

4π r 2 ω = 2 = 4π r

i=

dAa (rdθ ) × (r sin θ dφ ) = = sin θ dθ dφ r2 r2

Radiation that leaves dA within dω stays within dω as it travels to dAa College of Energy and Power Engineering

9

10.1 The problem of radiative exchange

‰

For diffuse surface i =const

‰

The energy emitted per unit time in any direction

College of Energy and Power Engineering

Area seen by dAa

JHH

10. Radiative heat transfer

10

10.1 The problem of radiative exchange

Relation between e and i ‰

Radiation that leaves dA pass through the entire hemisphere qoutgoing = e = ∫

2π

φ =0

dQ(θ , φ ) = e(θ , φ )dAdω = idA cosθ dω

π /2

∫θ

=0

i cos θ (sin θ dθ dφ ) = π i

ƒ For black body

dAa

ib =

e(θ , φ ) = i cosθ = e(θ ) ƒ For θ=0, e(0)=i dA

iλ =

Lambert’s law, or Cosine law JHH

eb

π

ƒ Monochromatic intensity

e(θ ) = e(0) cosθ

College of Energy and Power Engineering

θ

dQλ (W/m 2 ⋅ μ m ⋅ steradian) cos θ dAdω d λ

Diffusive radiation and Lambert’s Law

ƒ Directional emissive power

(W/m 2 ⋅ steradian)

cos θ dAdω

iλ =

JHH

10. Radiative heat transfer

dQoutgoing

ƒ Monochromatic, directional intensity of radiation

ƒ For a element area dAa

‰

10.1 The problem of radiative exchange

⎧radiant energy from dA dQoutgoing = (idω )(cos θ dA) = ⎨ ⎩ that is intercepted by dAa

dA steradian r2

dω =

8

Intensity of radiation

Definition

dω =

JHH

11

eλ

π

College of Energy and Power Engineering

JHH

12

10. Radiative heat transfer

10.1 The problem of radiative exchange

10. Radiative heat transfer

Radiation from a black body ‰

Radiation from a black body

Wien’s law and Planck’s law

‰

ƒ Heating a poker in the fire ¾ first dull red ¾ White-hot

The Stefan-Boltzmann law ƒ Stefan established experimentally in 1879 ƒ Boltzmann explained on the basis of thermodynamics in 1884

at long wavelength at short wavelength

ƒ Wien’s Law (in 1893)

eb (T ) = σT 4

(λT ) eλ = max = 2898 μm ⋅ K

ƒ Stefan-Boltzmann constant

ƒ Max Planck’s law (in 1901)

eλb =

10.1 The problem of radiative exchange

2π hc λ [exp(hc0 / k BTλ ) − 1]

σ = 5.670400 ×10 −8 W/m 2 ⋅ K 4

2 0

5

ƒ T is absolute temperature

¾ Light speed

c0 = 2.99792458 × 108 m / s

¾ Planck’s constant

k B = 1.3806503 × 10 −23 J / K

¾ Boltzmann’s constant

h = 6.62606876 × 10 −34 J ⋅ s

Monochromatic emissive power of a black body– predicted and observed by Lummer and Pringsheim (1899)

College of Energy and Power Engineering

JHH

13

10. Radiative heat transfer

10.1 The problem of radiative exchange

Radiation from a non-black surface ‰

ƒ The stefan-Boltzmann law can be derived by integrating Planck’s law

College of Energy and Power Engineering

JHH

10. Radiative heat transfer

10.1 The problem of radiative exchange

Radiation from a non-black surface

Emittance

‰

ƒ Monochromatic emittance

ελ =

eλ (λ , T ) eλb (λ , T )

ƒ Total emittance

ε≡

e(T ) = eb (T )

ƒ Real bodies: ƒ Black body:

∫

∞ 0

eλ (λ , T )d λ

σT

4

=

∫

∞ 0

‰

ε λ eλ (λ , T )d λ

Total emittances for a variety of surfaces

ε is low for metals

b

σT 4

0 < ε