Radiation and Convection


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 Theory


 
 
 

 

Heat transfer is a common phenomenon encountered in many areas in daily life. Therefore it is an important subject in natural sciences and even more so in engineering and the field of environmental physics. Often one is interested in the various possible ways to cut down on energy use. To economize on home-heating for example one will have to optimize insulation, i.e. minimize heat transfer. Economizing on cooling on the other hand implies maximizing heat transfer to some coolant. Other examples of problems concerning heat transfer in engineering may be found in the application of materials that cannot sustain extreme temperatures and the like. This implies again that heat transport towards this material is to be minimized, whereas the transport away from this material is to be maximized. Generally speaking one is often dealing with minimization or maximization of heat transfer.
To be able to perform calculations like these, one will have to be familiar with the concept of heat transfer and the various mechanisms that result in heat transfer. These mechanisms are:

  • conduction; on a microscopic or atomic scale kinetic energy is transferred through collisions between the atoms so that on a macroscopic scale thermal energy is transferred
  • convection; if the carrier of thermal energy is mobile the thermal energy may be transferred through mass transfer
  • radiation; thermal energy may be transformed into electromagnetic energy, emitted and then absorbed so that it is transformed into thermal energy again

In this particular experiment the second and third mechanism, radiation and convection, are studied in some more detail. Generally conduction always takes place as well, but a set-up will be proposed such that it can be neglected.

 

Radiation

 

Heat transfer through radiation takes place in the form of electromagnetic waves, mainly in the infrared region. The radiation is emitted by some body as a consequence of the thermal agitation of its composing molecules. In a first approach the radiation is described for the case that the emitting body is a so-called 'black' body. A black body is defined as a body that absorbs all radiation that falls on its surface. Actual black bodies don't exist in nature though its characteristics are approximated by the well-known hole in a box filled with highly absorptive material.
The emission spectrum of such a black body was first fully described by Max Planck. The energy emitted per unit time and surface [W m
-2] at some particular wavelength l was found to be given by:

 

 

 

(2.1)

 

Here the constant h is known as Planck's constant with h = 6,626 10-34 [J s], furthermore k is Boltzmann's constant in [J K-1] and c is the speed of light with c = 3,0 108 [m s-1]. Finally the energy emitted also depends on the absolute temperature T of the emitting body.
The total radiation emitted per unit surface is found through integration of the intensity over all wavelengths, leading to the Stefan-Boltzmann law:

 

 

 

(2.2)

 

Here the constant s is known as Stefan-Boltzmann's constant and given by s = 5,67 10-8 [W / m2 K4]. Again the temperature T is the absolute temperature of the radiating surface. This implies that any black body with a temperature other than T = 0 [K] will emit radiation.
If the radiating body is not black the total emission may be described by introducing the emissivity
e(l,T) for some wavelength and temperature. The emissivity or emission coefficient is defined as the ratio of the actual radiation emitted and the radiation that would be emitted if the body at hand were black. A black body therefore has an emissivity equal to one and any other body has an emissivity between zero and one. According to the definition of the emissivity, the spectral intensity of radiation emitted by a non-black body is given by its emissivity multiplied by the emission spectrum of a black body. The intensity of the radiation per unit wavelength per unit surface then is:

 

 

 

(2.3)

 

As before the total radiation is found through integration. For simplicity the emissivity is often assumed to be independent of wavelength and temperature, i.e. e(l,T) = e. Objects that show such a wavelength-independent and temperature independent emissivity are usually called grey. Although grey objects are not found in the physical world, some objects show spectral characteristics that approximate those of a grey body. In practice these 'grey' objects can have emission coefficients up to some 0.8.
For these objects the radiation emitted per unit surface becomes:

 

 

 

(2.4)

 

The net radiation emitted per unit surface by a body at temperature T1, enclosed by some black body at temperature T2 equals to the emission of the body enclosed minus the radiation that is absorbed by the body enclosed. Note that both the emission coefficient and the absorption coefficient refer to the surface of the enclosed body:

 

 

 

(2.5)

 

If the two bodies are in thermodynamic equilibrium no net heat transport takes place and the two bodies are at equal temperature so that T1 T2 . With eq. (2.5), one finds;

 

 

 

(2.6)

 

This is equation is known as Kirchhoff's law of radiation. It implies that for some specific known temperature and wavelength the emission coefficient and the absorption coefficient of some body are equal. Note that since the emissivity of a black body equals one, the absorption of a black body is one as well in accordance with the original definition. Applying Kirchhoff's law to eq. (2.5) one may write:

 

 

 

(2.7)

 

The net power in [W] transferred through radiation from a body with surface A and surface temperature T1 to the enclosure with surface temperature T2 then is:

 

 

 

(2.8)

 

This heat transfer takes place only if the medium between the two bodies is transparent for the relevant spectral region. Furthermore the radiative transfer may be accompanied by other transfer mechanisms. Particularly convection will be of importance and this mechanism is therefore discussed in the following section.

 

Convection

 

Heat transfer through convection arises when a moving fluid absorbs heat from some surface and transports this heat to some other location such that the fluid acts as a carrier.
Two forms of convection are distinguished. In the first place convection may arise naturally. If for example a hot object at temperature T
1 is in contact with a cooler fluid of temperature T2, heat is transported from the object to the boundary layer through conduction. This leads to density changes in the boundary layer and as a result the fluid in the boundary layer will rise and be replaced by cooler fluid that is heated again etc. This phenomenon is called free convection.
The second form of convection arises if the flow is brought about by for example a pump or a fan. In principle this form of convection, known as forced convection, is generally more efficient since the period of contact between the hot object and the fluid is shortened. This effectively comes down to an increase of the temperature difference between the solid and the fluid and with that an increase of the heat transfer to the boundary layer through conduction.

The heat transfer per unit surface through convection was first described by Newton and the relation is known as Newton's law of cooling. According to this law the heat exchange is, under normal atmospheric conditions, proportional to the temperature difference at the interface:

 

 

 

(2.9)

 

Here the heat exchange q'' is measured in [W m-2]. The temperature of the surface and the fluid are indicated by T1 and respectively, where it is implied that the fluid is present in such large quantities that its temperature may be taken to be constant. The constant of proportionality h is called the convection heat transfer coefficient in [W m-2 K-1]. The total power removed from an object with surface A through convection then is:

 

 

 

(2.10)

 

Note that this expression accounts for the heat that is transferred from the object at hand to the 'infinite' body of surrounding fluid.

 

Total Heat Transfer

 

The total power of the heat removed from a hot object at T1, enclosed by a black body under normal atmospheric conditions is given by the sum of the total amounts of heat removed per unit time through both radiation and convection as given in eq. (2.7) and (2.9):

 

 

 

(2.11)

 

Note that the heat loss due to radiation is driven by the temperature difference between the hot object and the black body and the heat loss due to convection is driven by the temperature difference between the hot object and the air.


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