# Radiative heat exchange

In building physics, generally only the overall (integral) energy flows due to thermal radiation are relevant; for an ideal emitter (black body), values for radiation from a surface are given according to the Stefan Boltzmann law (Index e for emission):

${\displaystyle \dot{q_{e}}= \sigma T^4 }$

However, real objects do not have 100% emissivity, but for a wavelength range such as that for thermal radiation in the room temperature environment (ca. 0 to 40 °C) they can typically be approximated by means of a 'grey body' with a constant emissivity of ε < 1. A value that is ε-fold of the above mentioned value will then be emitted by these. If two bodies with a different temperature are next to each other, then both will emit heat to and absorb ^{1)} heat from the other (examples: inner and outer panes of double glazing, cloud cover at the ground level or a clear night sky at ground level). In this case one speaks of radiative exchange. As a simple example, let us observe two infinitely extended parallel plane bodies at a certain distance from one another. The net radiant flux $\overset{.}{q} _{rad}$ can be determined as follows from the radiant flux $\overset{.}{q} _{1}$ from 1 to 2 and from $\overset{.}{q} _{2}$ in the other direction

$q_{rad}$= $\overset{.}{q}$ $_{1}$ − $\overset{.}{q}$ $_{2}$ ,

whereby the heat flows consist of the primary thermal radiation (“inherent radiation” emitter) and the reflected radiation:

$\overset{.}{q}$ $_{1}$ =$\overset{.}{q}$ $_{e.1}$−$ϱ_1$$\overset{.}{q}$ $_{2}$ ,

$\overset{.}{q}$ $_{1}$ =$\overset{.}{q}$ $_{e.2}$−$ϱ_2$$\overset{.}{q}$ $_{1}$ ,

The last two equations can be solved based on the directed total heat flows ^{2)}, for the respective generation of radiation $\overset{.}{q}$ $_{e}$ we can then apply the first equation multiplied by the respective emissivity and then determine their differences. In doing so, we can assume that both surfaces are opaque, i.e. $ϱ_j=1−ε_j$. This results in

${ \displaystyle \dot{q}_{rad}= \frac{\sigma}{\frac{1}{\varepsilon_1}+\frac{1}{\varepsilon_2}-1} \left( T_1^4 - T_2^4 \right) ~},$

Thus the net heat flux due to radiation between two window panes (accurately to a great extent) and also the net radiative exchange between the floor and ceiling can be determined in good approximation ^{3)}. The fact that the power/potency Potent $T^4$ must be determined here should not be a problem with the calculators and computers available today. However, it is also permissible (and normal) to use a linear approximation for the dependence on the temperature difference because in the area of building physics the absolute temperatures are all within a quite narrow range (the comfortable range!): $15 °C < \vartheta < 28 °C$. For this we will simply apply the third binomial expansion twice to the difference in the $T^4$- terms:

$T_1^4 - T_2^4 = (T_1^2 + T_2^2) (T_1^2 - T_2^2)= (T_1^2 + T_2^2) (T_1 + T_2)(T_1 - T_2)$

The temperature difference is now especially included in this $(T_1 - T_2)=(\vartheta_1 - \vartheta_2)$. We bring together the two sums of the terms with the net emissivity

${\displaystyle \dot{q}_{rad}=\frac{\sigma (T_1^2 + T_2^2)(T_1 + T_2)}{\frac{1}{\varepsilon_1}+\frac{1}{\varepsilon_2}-1} \left( \vartheta_1 - \vartheta_2 \right) ~,}$

and with the “radiative exchange heat transfer” $h_rad$

${\displaystyle h_{rad}=\frac{\sigma (T_1^2 + T_2^2)(T_1 + T_2)}{\frac{1}{\varepsilon_1}+\frac{1}{\varepsilon_2}-1} }$

which then gives us

${\displaystyle \dot{q}_{rad}= h_{rad} (\vartheta_1 - \vartheta_2) } $

With this, we can write in good approximation the radiative heat exchange as a heat transfer, which however takes place from one surface to the other. An approximation for the factor

${\displaystyle h_{rad}=\varepsilon_n \sigma (T_1^2 + T_2^2)(T_1 + T_2) \approx \varepsilon_n 5,6 }$ W/(m²K)

can also be given for temperatures in the vicinity of 18 °C (average temperatures of the room and interior surfaces of exterior building assemblies). In doing so, we have shortened the net emission factor by $\varepsilon_n=\frac{1}{1/\varepsilon_1 + 1/\varepsilon_2 -1}$ . The radiative heat transfer for emissivities of around 93% that are usual in rooms is thus roughly between 4.8 and 5.4 W/(m²K) and is therefore SIGNIFICANTLY higher than the convective heat transfer.

**Heat exchange in the room thus takes place for the most part via thermal radiation and not via air, as is naively imagined.**

This has far-reaching practical consequences; unfortunately, the naive notion that heat in a room is exchanged via air is still shared by many 'experts'. Radiative heat exchange is much more important here! It also transports the heat directly e.g. from the radiator to the interior surface of the exterior building assembly - or from the heated floor to the interior surface of the roof. This effect contributes at least in part to the frequently observed higher energy consumptions^{4)} of panel heating systems; the effect is not so great that the desired contribution to savings would thereby be cancelled altogether (radiation climate, lower forward flow temperatures and therefore line losses, and especially better COPs of heat pumps). However, if one just wants to raise the temperature of a room quickly for a short period of time (< 1.5 h), then this can be most efficiently achieved by means of an air heater. For one thing, the air will quickly become warm due to the lower thermal capacity; for another, it does not transfer huge quantities of heat to the space enclosing surfaces in the room within 1.5 h, so their temperature will not rise so rapidly. Due to this, the temperature will also fall again quickly after turning the heater off; this is thus a “good trick” for quickly heating up e.g. a small bathroom at reasonable cost.

How good is such an approximation? Within the range of the relevant heat exchange processes inside the building, surprisingly good. If conventional radiators or other hot/heated objects also come into play, it is better to calculate with a higher factor. “Exact” calculation with the $T_4$ expansion can also be carried out, but one should keep in mind that other variables such as the convective heat transfer are far less precisely known, (NZEB = nearly zero energy building). | |

In practice: the house in the middle is NOT unheated, in fact it is comfortably warm. Due to the excellent thermally insulated exterior building assemblies, it hardly loses any heat, as a result of which the exterior surface temperature adjusts to the surroundings, and very little heat radiation is emitted. New York's '1st Passive House' appears “cool” in the thermographic image. This was a retrofit, and these results can therefore also be achieved with the existing building stock. |

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