The calculation of radiant heat transfer rates, in detail, is beyond the scope of this book and for most food processing operations a simplified treatment is sufficient to estimate radiant heat effects. Radiation can be significant with small temperature differences as, for example, in freeze drying and in cold stores, but it is generally more important where the temperature differences are greater. Under these circumstances, it is often the most significant mode of heat transfer, for example in bakers' ovens and in radiant dryers.
The basic formula for radiant-heat transfer is the Stefan-Boltzmann Law
where T is the absolute temperature (measured from the absolute zero of temperature at -273°C, and indicated in Bold type) in degrees Kelvin (K) in the SI system, and s (sigma) is the Stefan-Boltzmann constant = 5.73 x 10-8 J m-2 s-1K-4 The absolute temperatures are calculated by the formula K = (°C + 273).
This law gives the radiation emitted by a perfect radiator (a black body as this is called though it could be a red-hot wire in actuality). A black body gives the maximum amount of emitted radiation possible at its particular temperature. Real surfaces at a temperature T do not emit as much energy as predicted by eqn. (5.8), but it has been found that many emit a constant fraction of it. For these real bodies, including foods and equipment surfaces, that emit a constant fraction of the radiation from a black body, the equation can be rewritten
where e (epsilon) is called the emissivity of the particular body and is a number between 0 and 1. Bodies obeying this equation are called grey bodies.
vary with the temperature T and with the wavelength of
the radiation emitted. For many purposes, it is sufficient to assume that
as a black body emits radiation, it also absorbs it and according to the
same law, eqn. (5.8).
where 1/C = 1/e1 + 1/e2 - 1, e1 is the emissivity of the surface at temperature T1 and e2 is the emissivity of the surface at temperature T2.
where e is the emissivity of the body, T1 is the absolute temperature of the body and T2 is the absolute temperature of the surroundings.
For many practical purposes in food process engineering, eqn. (5.11) covers the situation; for example for a loaf in an oven receiving radiation from the walls around it, or a meat carcass radiating heat to the walls of a freezing chamber.
In order to be able to compare the various forms of heat transfer, it is necessary to see whether an equation can be written for radiant heat transfer similar to the general heat transfer eqn. (5.3). This means that for radiant heat transfer:
where hr is the radiation heat-transfer coefficient, T1 is the temperature of the body and T2 is the temperature of the surroundings. (The T would normally be the absolute temperature for the radiation, but the absolute temperature difference is equal to the Celsius temperature difference, because 273 is added and subtracted and so (T1 - T2) = (T1 - T2) = DT
Equating eqn. (5.11) and eqn. (5.12)
= es(T1 + T2 ) (T12 + T22)mmmmmm
If Tm = (T1 + T2)/2, we can write T1 + e = Tm and T2 - e = Tm
= 2Tm2 + 2e2
= 2Tm2 + (T1 - T2)2/2
Therefore hr = es(2Tm)[2Tm2 + (T1 - T2)2/2]
Now, if (T1 - T2) « T1 or T2, that is if the difference between the temperatures is small compared with the numerical values of the absolute temperatures, we can write:
q = Aes(T14- T24 )
= 0.0645 x 0.85 x 5.73 x 10-8 (4504- 3734)
By eqn. (5.13)
q = 0.23e (Tm/100)3A DT
= 0.23 x 0.85(411/100)3 x 0.0645 x 77
Notice that even with quite a large temperature difference, eqn. (5.13) gives a close approximation to the result obtained using eqn. (5.11).
Heat-Transfer Theory > CONVECTION-HEAT TRANSFER
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