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

274 CHAPTER 8  Thermal radiation

Eq. (8.131) requires some caution in its use, as it is not applicable for a black surface.

So, if we know Gi or Ji alone, we can calculate the net radiative heat transfer from any surface.

For a black body, εi = 1, and so Ji is given by

Another surface of interest in the radiation problem is a reradiating surface, which is the radiation equivalent of an insulated surface.

For a reradiating surface, qi = 0.

In the view of the above the key result is,

8.9 The radiosity-irradiation method 275

The key question to be answered now is: How do we engineer a reradiating surface? A very thick wall/surface made of a low thermal conductivity medium and with a vacuum or near vacuum will come close to a re-radiating condition. Else, conjugate

heat transfer has to be considered in a wall, that is, qconduction = qradiation and we need to write out the expressions for qconduction and qradiation and we will obtain an equation in which unknown will be the surface temperature.

Example 8.12: Consider two infinitely long parallel plates that are separated by a distance L, as shown in Fig. 8.26. The plates are very deep in the direction perpendicular to the plane of the paper. The space between the two plates is evacuated. The two plates are maintained at temperatures of 810 K and 490 K, respectively. The two plates have emissivities of 0.95 and 0.45 respectively. Determine the net radiation heat transfer between the two plates.

Solution:

F11 = 0 and F12 = 1

FIGURE 8.26

Two infinitely long parallel plates undergoing surface radiation exchange and under consideration in example 8.12.

8.11 Equation of transfer or radiative transfer equation (RTE) 279

due to high absorption by CO2 and H2O. This figure clearly shows that gas absorption is spectrally dependent and that the atmosphere is truly a participating medium. There are also transparent windows where there is little absorption.

8.11  Equation of transfer or radiative transfer equation (RTE)

Consider a gas volume with a cross sectional area dA and thickness dx as shown in Fig. 8.29. The incoming spectral radiation in the direction x is Iλ,x while Iλ, x+dx is the spectral radiation exiting the gas volume. The area dA is normal to the direction x such that the radiation is traveling in a direction normal to the cross-sectional area. When we neglect scattering, the only two other phenomena involved in this problem are emission and absorption.

The energy absorbed by the gas in the interval dλ is given by

where kλ is the monochromatic or spectral absorption coefficient (m−1). The energy emitted by the gas volume is given by

Eq. (8.150) is known as the RTE, or radiative transfer equation. It is deceptively innocuous as ελ and αλ invariably have strong spectral dependence. Additionally, if “reflection ” or scattering from gas molecules is concerned, the problem gets even more formidable, with scattering being a function of wavelength and angle in many cases.

FIGURE 8.29

Gas volume used in the derivation of the RTE.

280 CHAPTER 8  Thermal radiation

Consider the gas under question in an isothermal cavity at Tg. Let the temperature of the gas be Tg. For this situation we have isotropic radiation; that is, radiation intensity is the same everywhere and so dIλ /dx is 0 everywhere within the isothermal cavity. This can be written as

Let us consider a situation where the absorption is much larger compared to emission. Consider a plane wall at temperature Tw (corresponding to x = 0) surrounded by the gas that is emitting and absorbing. In view of the foregoing assumption, the RTE becomes

The above solution clearly shows that radiation decreases exponentially with x. This is known as Beer-Lambert’s law. A typical example is solar radiation decreasing exponentially from the surface as we go deeper into the ocean waters.

8.11.1  Determination of heat fluxes

The heat flux q is of fundamental importance to engineers. Hence, the intensity I or Iλ, to be more general, must be converted into a flux form.

Now let us consider the full spectrum of electromagnetic radiation, that is, λ = 0 to λ = ∞. The RTE becomes

8.11 Equation of transfer or radiative transfer equation (RTE) 281

(See Balaji, 2014), for a derivation of the above result. In the above equation θ is the angle between a straight path of radiation and a path.) The heat flux at x = 0, going out in the positive direction of x is given by

The integral appearing on the right hand side of Eq. (8.160) is hard to integrate. We can, of course, easily do it on a computer this day and age. Historically, this was solved by an introducing an exponential integral En(t), as defined below. We can introduce the exponential function En(t) here as

The exponential integral of third order is given by

In Eq. (8.162) µ is basically a dummy variable.

E3(x) satisfies the following asymptotic properties.

E3 (0) = 0.5 . The x in Eq. (8.162) corresponds to k.x. The product of k and x is known as the optical depth and is given by τ or τx.

When the optical depth is very small, E3(x) approaches 0. In gas radiation, this is known as the optically thin limit.

qL+ can now be written as