Table of Contents
Heat transfer II
Consisting of convection and thermal radiation; temperature dependent
Now that we have been able to clarify the mechanism of radiative heat transfer, and are therefore able to very precisely quantify its values for heat transfer, we shall also examine again in more detail the contribution of convective heat transfer; this proves to be far more complex and for this reason is only possible with lesser accuracy 1). The following explanations thus at first refer only to empirically obtained interrelationships; with modern CFD (Computational Fluid Dynamics), now even more accurate calculations are certainly possible for clearly described individual cases – but translating these into practice generally still fails to work because of the great complexity also of the boundary conditions. On the other hand, the empirical approximations described below are enough for the vast majority of tasks in building physics, particularly for buildings which have a fairly reasonable standard of thermal insulation. The uncertainties still existing here have a much greater influence on heat transfers in the case of tarpaulins or single panes.
Forced flow (convection??): e.g. external 'e' convective 'c' heat transfer $h_{e,c}$
Forced flow may be due to wind for example, or even a fan which lets air flow along a convector 2) . This involves the velocity $v$ as well as its direction $\overrightarrow{v}/v$. However, only extreme documented values are available – which we have provided here.
| Formula for $h_{e,c}$ in W/(m²K) | WIND situation |
|---|---|
| ${\displaystyle 4 + 4 \frac{v}{\text{m/s}} }$ | Parallel incident flow |
| ${\displaystyle 6 + 6 \left( \frac{v} {\text{m/s}} \right)^{0,7} }$ | Windward (facing the wind) |
| ${\displaystyle 6 + 1,4 \left( \frac{v} {\text{m/s}} \right) }$ | Downwind/lee (away from the wind, < 8 m/s) |
Example: in light winds, 4 m/s, parallel incident flow, convective heat transfer
$h_c = (4+4 \cdot 4) \text{W/(m²K)} = 20 \text{W/(m²K)} $.
Free convection (heat transfer at interior surfaces) $h_{i,c}$
Free flow is promoted by the temperature difference $\Delta \vartheta$ (given here in K). With an increasing temperature difference, the difference in density increases so that heat transport then also increases.
| Formula for $h_{i,c}$ in W/(m²K) | Position with regard to heat flow |
|---|---|
| ${\displaystyle 1,31 \Delta \vartheta^{0,33} }$ | Vertical surface |
| ${\displaystyle 1,89 \Delta \vartheta^{0,33} }$ | Upward heat flow |
| ${\displaystyle 0,4 \Delta \vartheta^{0,25} }$ | Downward heat flow |
The mechanism behind this: even if there is no external mechanical impetus, air flow occurs at a surface with a temperature that varies from that of the air 3): as a result of heat transport via heat conduction, the adjacent air is cooled down, which increases its density and the weight force becomes greater than the buoyancy force: the air layer along the surface drops and “free convection” results 4). The exact flow velocities and distributions also depend on the extent and temperatures of other building assemblies; for this reason, the empirical formulas given above are approximations – which consider the typical dimensions of rooms (ca. 2.5 m high) without any additional furniture.
For various reasons, the accuracy of the values that are possible for convective heat transfer is therefore limited, chiefly because the exact boundary conditions are often not even known in individual cases. For simple balance calculations, such as those performed in the PHPP or for single building components, in buildings physics we therefore usually stick to the approximate specification of a total value for convective and radiative heat transfer. For the thermal simulation of a building however, this would make the inaccuracy too high: there must be a clear separation between convection and radiation, since the temperatures of systems in heat exchange can also be quite different. We will deal with some examples below.
Example 1: If the surface temperature of a surface is 0°C 5) in a room with a room air temperature of 20°C, then the value $h_{i,c}$ = 3.52 W/(m²K) results for a vertical surface; with a temperature difference of just 3 K, this value decreases to approximately 1.87 W/(m²K)6).
Example 1a (extended): We will now include heat transfer due to thermal radiation for the surface with 0°C in a room with interior surface temperatures of 20°C 7): the emissivities are applied with 0.85 (for the glass surface) and 0.92 for the “wallpaper” (this results in a radiation factor of Strahlungssichetfaktor von etwas $\phi = 0.79$. With the Stefan-Boltzmann Law, the “radiation heat transfer” results from this as $h_{i,r}$ = 4.08 W/(m²K); even with a very cold single-glazed surface, this value is still more significant than free convective transfer ist dieser Wert immer noch bedeutender freie konvektive Übergang. It must always be kept in mind that radiative exchange takes place with other surfaces in the room, while convective heat transfer takes place with the room air; 'addition' of the two values 8) gives a value of around 7.6 W/(m²K) or an internal heat transfer resistance of 0.13 m²K/W. This is precisely the established 'simple' value if the calculation is greatly simplified.With a temperature difference of only 3 K, radiative heat transfer increases to approximately 4.5 W/(m²K) and the 'simple' total of both heat transfers still amounts to about 6.3 W/(m²K). For well-insulated warm surfaces the predominance of thermal radiation is even more pronounced; the only slightly lower thermal radiation from triple low-e glazing is then no longer subjectively perceived as “clear”.