Calculation of long wave radiation to the sky

\[ \text{TempFactSky} = \frac{\left( t_{\text{new}}[1] + 273.15 + th_{\text{abs}} \right)^{3}}{2} \tag{1} \] \[ \text{TempFactOut} = \frac{\left( t_{\text{new}}[1] - t_{\text{out}} + 546.3 \right)^{3}}{2} \tag{2} \]

where

\[ th_{abs} = \left( (1-cc) \cdot 9.36 e^{-6} t_{out,abs}^6 + cc \cdot td_{abs} \right)^{0.25} \tag{3} \]

is the absolute outdoor dew point temperature,

\[ td_{abs} = td + 273.15 \tag{4} \]

is the outdoor dew point temperature in Kelvin,

\[ td_{out,abs} = t_{out} + 273.15 \tag{5} \]

cc is the cloud cover as decimal number.

The heat transfer coefficient for radiation exchange between the exterior surface and the sky is calculated as:

\[ h_{rad,sky} = \left[ \frac{1 + \cos(h)}{2} + (1 - \varepsilon_{ground}) \cdot \frac{1 - \cos(h)}{2} \cdot \varepsilon_{warm} \right] \cdot \sigma \cdot TempFactSky \tag{6} \]

where

ε_warm is the long wave emissivity of the exterior surface,

σ is the Stefan-Boltzmann constant (5.6697e^-8 W/m²K⁴),

ε_ground is the long wave emissivity of the exterior ground (0.90).

In the heat balance of the exterior surface, the following are added to the coefficients of the equation system for the first exterior node point:

\[ b[1] += h_{rad,sky} + h_{rad,out} \tag{7} \]

\[ d[1] += h_{rad,out} \cdot t_{out} + h_{rad,sky} \cdot (th_{abs} + 273.15) \tag{8} \]

Outdoor surface convection coefficient, dependent on wind speed

Next to calculating the long wave radiation effects, the heat transfers coefficient between the outdoor air and the first node point on the exterior side of the construction is calculated as:

\[ h[1] = \frac{1}{outside[1] \cdot m[1] + \frac{1}{h_{conv} + rx[1]}} \tag{9} \]

where outside[1]*m[1] is the thermal resistance between the node point and the surface, if the node point is not exactly at the surface (in tsbi5, m[1] is zero). rx[1] is a possible "extra" surface resistance at the surface—normally zero. h_conv is calculated from the wind speed v (m/s) as:

if v ≤ 5 then

\[ h_{conv} = 5{,}82 + 3{,}96 \cdot v \tag{10} \]

else h_conv = 7.68 \(\cdot\) v^0.75

Both long wave radiation and convection

Normally, both long wave radiation and heat transfer by convection should be calculated in the same calculation, and then the equations from the end of this note's section "Calculation of long wave radiation to the sky" become:

\[ b[1]+ = h[1] + h_{rad,sky} + h_{rad,out} + h[2] \tag{11} \]

\[ d[1]+ = \left( h[1] + h_{rad,out} \right) \cdot t_{out} + h_{rad,sky} \cdot (th_{abs} - 273{,}15) \tag{12} \]

With h[1] calculated as in the section "Outdoor surface convection coefficient dependent on wind speed".

See also

• Calculation of the sky temperature.