Radiasi

Radiation  that leaves a surface can propagate in all possible directions (Figure12.4b), and we are often interested in knowing its directional distribution. Also, radiation incident upon a surface may come from different directions, and the manner in which the surface responds to this radiation depends on the direction. Such directional effects can be of primary importance in determining the net radiative heat transfer rate and may be treated by introducing the concept of  radiation intensity.

12.2.1 mathematical definitions

Due to its nature, mathematical treatment of radiation heat transfer involves the extensive use of the spherical coordinate system. From Figure 1 2.5a, we recall that the differential plane angle dα is defined by a region between the rays of a circle and is measured as the ratio of the arc length dl on the circle to the radius r of the circle. Similarly, from Figure 12.5b, the differential solid angle dω is defined by a region between the rays of a sphere and is measured as the ratio of the area dA_n on the sphere to the sphere’s radius squared. Accordingly,

 

Consider emission in a particular direction from an element of surface area dA₁, as

shown in Figure 12.5c. The direction may be specilied in terms of the zenith and azimuthal angles, θ and ϕ, respectively, of a spherical coordinate system (Figure 12.5d). The area dA_n, through which the radiation passes, subtends a differential solid angle dω when viewed from a point on dA₁. As shown in Figure 12.6, the area dA_n is a rectangle of dimension r dθ x r sinθ d; thus, dA_n,, = r² sinθ dθ dϕ. Accordingly,

When viewed from a point on an opaque surface area element dA₁, radiation may be emitted into any direction defined by a hypothetical hemisphere above the surface. The solid angle associated with the entire hemisphere may be obtained  by integrating Equation 12.3 over the limits ϕ = 0 to ϕ = 2 and θ = 0 to θ = π/12. Hence,

12.2.2 Radiation  Intensity And Its Relation To Emission

Returning to Figure 1 2.5c. we now consider the rate at which emission from dA₁ passes through dA_n. This quantity may be expressed in terms of the spectral intensity

I_λ,e of the emitted radiation. We formally define I_λ,e as the rate at which radiant energy is emmited at the wavelength λ in the (θ, ϕ) direction, per unit area of the emiting surface normal to this direction, per unit solid angle about this direction, and

 

figure 12.7 the projection of dA₁ normal to the direction of radiation.

per unit wavelength interval d λ about λ. Note that the area used to define the intens

ity is the component of dA₁ perpendicular to the direction of the radiation. From Figure 12.7, we see that this projected area is equal to dA₁ cos θ. In effect it is how

dA₁ would appear to an observer situated on dA_n. The spectral intensity, which has

units of W/m² sr μm, is then

where (dq/d λ) dq _λ is the rate at which radiation of wavelength λ leaves dA₁ and

passes through dA_n. Rearranging Equation 12.5, it follows that

where d λ has the units of W/μm. This important expression allows us to compute

the rate at which radiation emitted by a surface propagates into the region of space

defined by the solid angle dw about the (θ, ϕ) direction. However, to compute this rate, the spectral intensity I_λ,e of the emitted radiation must be known. The manner in which this quantity may be determined is discussed later, in Sections 12.3 and 12.4. Expressing Equation 12.6 per unit area of the emitting surface and substituting from Equation 12.3, the spectral radiation flux associated with dA₁ is

If the spectral and directional distributions of I_λ,e are known, that is, I_λ,e (λ,θ, ϕ) is known, the heat flux associated with emission into any finite solid angle or over any finite wavelength interval may be determined by integrating Equation 12.7. For example, we define the spectral, hemispherical e,nissive power E _λ (W/m² μm) as the rate at which radiation of wavelength λ is emitted in all directions from a surface per unit wavelength interval dλ about λ and per unit surface area. Thus. Eλ is the spectral heat flux associated with emission into a hypothetical hemisphere above as shown in Figure 12.8. or

Note that E_λ is a flux based on the actual surface area, where as I_λ,e is based on the

projected area. The cos θ term appearing in the integrand is a consequence of this

difference.

The total, hemispherical emissive power, E (W/m²), is the rate at which radiation is emitted per unit area at all possible wavelengths and in all possible directions. Accordingly,

Or from equation 12.8

Since the term “emissive power” implies emission in all directions, the adjective “hemispherical” is redundant and is often dropped. One then speaks of the spectral

emissive power Eλ, or the total emissive power E.

Although the directional distribution of surface emission varies according to the nature of the surface, there is a special case that provides a reasonable approximation for many surfaces. We speak of adiffuse emitter as a surface for which the intensity of the emitted radiation is independent of direction, in which case I_λ,e (λ,θ, ϕ) = I_λ,e (λ). Removing I_λ,e from the integrand of Equation 12.8 and performing the integration, it follows that

Similarly, from equation 12.10

Where I_e is the total intensity of the emmited radiation. Note that the constant appearing in the above expressions is π,  not 2π, and has the unit steradians.