# Solar Heat Transfer Topics

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This Guide is intended to review those aspects of heat transfer that are important in the design and analysis of solar collectors and systems. It begins with a review of radiation heat transfer, which is often given cursory treatment in standard heat transfer.

The role of convection and conduction heat transfer in the performance of solar systems is obvious. Radiation heat transfer plays a role in bringing energy to the earth, but not so obvious is the significant role radiation heat transfer plays in the operation of solar collectors. In usual engineering practice radiation heat transfer is often negligible. In a solar collector the energy flux is often two orders of magnitude smaller than in conventional heat transfer equipment, and thermal radiation is a significant mode of heat transfer.

Contents

## The Electromagnetic Spectrum

Thermal radiation is electromagnetic energy that is propagated through space at the speed of light. For most solar energy applications, only thermal radiation is important. Thermal radiation is emitted by bodies by virtue of their temperature; the atoms, molecules, or electrons are raised to excited states, return spontaneously to lower energy states, and in doing so emit energy in the form of electromagnetic radiation. Because the emission results from changes in electronic, rotational, and vibrational states of atoms and molecules, the emitted radiation is usually distributed over a range of wavelengths.

The spectrum of electromagnetic radiation is divided into wavelength bands. These bands and the wavelengths representing their approximate limits are shown in Figure below. The wavelength limits associated with the various names and the mechanism producing the radiation are not sharply defined. There is no basic distinction between these ranges of radiation other than the wavelength they all travel with the speed of light and have a frequency such that:

where C0 is the speed of light in a vacuum and is the index of refraction.

The wavelengths of importance in solar energy and its applications are in the ultraviolet to near-infrared range, that is, from 0.29 to approximately 25μm. This includes the visible spectrum, light being a particular portion of the electromagnetic spectrum to which the human eye responds. Solar radiation outside the atmosphere has most of its energy in the range of 0.25μm to 3μm while solar energy received at the ground is substantially in the range of 0.29 to 2.5μm.

For some purposes in solar energy applications, the classical electromagnetic wave view of radiation does not explain the observed phenomena. In this connection, it is necessary to consider the energy of a particle or photon, which can be thought of as an “energy unit” with zero mass and zero charge. The energy of the photon is given by:

where is Planck’s constant (6.62607004 × 10-34 m2 kg / s). It follows that as the frequency ν increases (i.e., as the wavelength λ decreases), the photon energy increases. This fact is particularly significant where a minimum photon energy is needed to bring about a required change (e.g., the creation of a hole–electron pair in a photovoltaic device). There is thus an upper limit of wavelength of radiation that can cause the change.

## The Blackbody: Perfect Absorber and Emitter

By definition, a blackbody is a perfect absorber of radiation. No matter what wavelengths or directions describe the radiation incident on a blackbody, all incident radiation will be absorbed. A blackbody is an ideal concept since all real substances will reflect some radiation.

Even though a true blackbody does not exist in nature, some materials approach a blackbody. For example, a thick layer of carbon black can absorb approximately 99% of all incident thermal radiation. This absence of reflected radiation is the reason for the name given to a blackbody. The eye would perceive a blackbody as being black. However, the eye is not a good indicator of the ability of a material to absorb radiation, since the eye is only sensitive to a small portion of the wavelength range of thermal radiation. White paints are good reflectors of visible radiation, but most are good absorbers of infrared radiation.

A blackbody is also a perfect emitter of thermal radiation. In fact, the definition of a blackbody could have been put in terms of a body that emits the maximum possible radiation. A simple thought experiment can be used to show that if a body is a perfect emitter of radiation, then it must also be a perfect absorber of radiation. Suppose a small blackbody and small nonblackbody are placed in a large evacuated enclosure made from a blackbody material. If the enclosure is isolated from the surroundings, then the blackbody, the real body, and the enclosure will in time come to the same equilibrium temperature. The blackbody must, by definition, absorb the entire radiation incident on it, and to maintain a constant temperature, the blackbody must also emit an equal amount of energy. The nonblackbody in the enclosure must absorb less radiation than the blackbody and will consequently emit less radiation than the blackbody. Thus a blackbody both absorbs and emits the maximum amount of radiation.

## Planck’s Law and Wien’s Displacement Law

Radiation in the region of the electromagnetic spectrum from approximately 0.2 to approximately 1000 μm is called thermal radiation and is emitted by all substances by virtue of their temperature. The wavelength distribution of radiation emitted by a blackbody is given by Planck’s law.

where h is Planck’s constant and k is Boltzmann’s constant. The groups and are often called Planck’s first and second radiation constants and given the symbols c1 and c2, respectively.

• C1 : 3.7405 * 108 μm4 /m2
• C2 : 14387.8 μm k

It is also of interest to know the wavelength corresponding to the maximum intensity of blackbody radiation. By differentiating Planck’s distribution and equating to zero, the wavelength corresponding to the maximum of the distribution can be derived. This leads to Wien’s displacement law, which can be written as:

Planck’s law and Wien’s displacement law are illustrated in Figure below, which shows spectral radiation distribution for blackbody radiation from sources at 6000, 1000, and 400 K. The shape of the distribution and the displacement of the wavelength of maximum intensity are clearly shown. Note that 6000 K represents an approximation of the surface temperature of the sun so the distribution shown for that temperature is an approximation of the distribution of solar radiation outside the earth’s atmosphere. The other two temperatures are representative of those encountered in low- and high-temperature solar-heated surfaces.

The same information shown in Figure above has been replotted on a normalized linear scale in Figure below. The ordinate on this figure, which ranges from 0 to 1, is the ratio of the spectral emissive power to the maximum value at the same temperature. This clearly shows the wavelength division between a 6000 K source and lower temperature sources at 1000 and 400 K.

## Stefan-Boltzmann Equation

Planck’s law gives the spectral distribution of radiation from a blackbody, but in engineering calculations the total energy is often of more interest. By integrating Planck’s law over all wavelengths, the total energy emitted per unit area by a blackbody is found to be:

where α is the Stefan-Boltzmann constant and is equal to 5.670367 × 10-8 kg s-3 K-4. This constant appears in essentially all radiation equations.

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