Майнор / Книги / Chapter-8---Thermal-radiation_2021_Heat-Transfer-Engineering

8.1  Introduction

Any body above a temperature of zero Kelvin emits radiation. This is a fundamental law of nature and is known as Prevost’s law. Radiation occurs due to molecular activity that arises as a consequence of temperature. The molecular activity is due to the translational, rotational, and vibrational energy of matter. Radiation ceases only at absolute zero temperature. Hence, it stands to reason that any body, or anything that exists for that matter, has to and will emit radiation. It is quite intuitive to guess that radiation is a function of temperature and that it shows a monotonic increase with temperature. The challenge now is to characterize this radiation and find ways to quantify it as a mode of heat transfer in engineering problems, when several surfaces at different temperatures and radiation characteristics, whatever they may be, are present.

The theoretical framework for radiation comes from two key game changers in modern physics: (1) electromagnetic theory and (2) quantum theory.

According to electromagnetic theory, the radiation emitted by any surface travels in the form of waves with the velocity of light. In such a case, radiation can be characterized by velocity, frequency, and wavelength.

These are related by

In Eq. (8.1), c is the velocity in m/s, ν is the frequency in Hz, and λ is the wavelength in µm (because meter (m) is typically too big a unit for wavelength in the study of thermal radiation).

Quantum theory, on the other hand, proposed that the radiation emitted from any surface is in the form of discrete packets called quanta.

Here h is the Planck’s constant that has a value 6.627 × 10−34 J s and E is the energy in J.

Table 8.1 shows a very basic classification of radiation based on the wavelength of the emitted radiation.

Radiation that occurs in the wavelength 0 ≤ λ ≤ 100 m is frequently referred to as thermal radiation. In the study of radiation heat transfer, this wavelength range is of primary interest to us. Radiation from a surface or volume can be emitted in all

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234 CHAPTER 8  Thermal radiation

Table 8.1  A very basic classification of radiation.

directions. Hence, we need to understand some basic ideas in geometry to proceed further in our study of radiation.

8.2  Concepts and definitions in radiation

The elemental plane angle (see Fig. 8.1A) dα is defined as

In Eq. (8.3), dα is in radians.

As opposed to this, the solid angle subtended by an elemental area dAn associated with the hemisphere is given by

A schematic representation of solid angle is shown in Fig. 8.1B.

The solid angle subtended by an elemental area dAn about a point on another elemental area dA1 in the spherical coordinate system is given by

where r,θ,φ are the radius, zenith angle, and azimuthal angle, respectively.

FIGURE 8.1

Schematic representation of (A) a plane angle and (B) a solid angle.

236CHAPTER 8  Thermal radiation

8.3.1  Black body

A black body is defined as one that absorbs all the incident radiation, regardless of the direction and the wavelength.

In view of the above definition, it is evident that

1.A black body is an ideal absorber.

2.The black body is a perfect emitter. This logically follows from the definition of a black body and can be proved by a thought experiment involving a black body that is in equilibrium with its surroundings.

3.Black body radiation is independent of position (i.e., homogeneous) and direction (i.e., isotropic).

4.Consequent upon #3, the radiation from a black body is a function of only temperature.

Additionally, if all the incident radiation is absorbed and nothing is reflected, then the body appears black. However, what is visually black need not be radiatively black. Even so, all radiatively black bodies have to be visually black.

8.3.2  Spectral directional intensity

The spectral directional intensity of emission Iλ,e from a black body is given by

From the above mathematical definition, it is clear that the spectral direction intensity of emission Iλ,e is the rate at which the radiant energy emitted by a surface (dQ) in the direction of (θ,φ) per unit area of the surface normal to this direction (dAcosθ) per unit solid angle dw about (θ,φ) per unit wavelength dλ about λ.

For a black body, Ib,λ (λ,θ,φ) ≠ f (θ,φ).

Therefore, the hemispherical spectral black body emissive power of a black body at a temperature T in the wavelength interval dλ about λ is given by

240 CHAPTER 8  Thermal radiation

This is known as the Rayleigh-Jeans distribution; it holds good for longer wavelengths and it fails in the asymptotic limit of λ → 0.

(a)λCT2 = 1

%error = −71.82%

(b)λCT2 = 0.1

%error = −5.17%

(c)λCT2 = 0.01

%error = −0.502%

As the value of C2 decreases, the simplified expression approaches Planck’s

λT

values of Eb,λ. Hence, the Rayleigh-Jeans approximation, works well for longer wavelengths (and so low values of λCT2 )

3. When λCT2 → ∞

C2

eλT 1

C λ−5

Eb,λ = 1C2

eλT

The above equation is known as the Wien’s distribution; while this works well for shorter λs, at higher λs it tends to saturate.

8.3 Black body and laws of black body radiation 241

(a)λCT2 = 1

%error = 36.79%

(b)λCT2 = 10

%error = 0.0045%

(c)λCT2 = 50

%error = 1.93 × 10−20%

1.What are the units of the above quantity?

2.Determine the value of λT at which the value of this function is the maximum.

3.Hence determine the maximum value of this function.

4.What potential use do you see in introducing such a function?

Solution:

Consider,

242 CHAPTER 8  Thermal radiation

Writing this in algorithmic form and solving by the successive substitution method details of which are given in Table 8.2, we have

In the above expression, i refers to the iteration number.

x* = λCT2 = 4.965

Table 8.2  A successive substitution method for example 8.3.