Solar Thermal Collectors Calculation per ISO 9806

From Polysun version 2025.5, solar thermal collectors are calculated according to the current ISO standard 9806.

In this article, you will find out:

• Which parameters the ISO 9806 model uses in Polysun (e.g. Eta0,b, a1, a2, etc.)
• How irradiation and losses are calculated
• What the Incident Angle Modifier (IAM) is and how it is implemented depending on the collector type
• How the calculated temperature of the collector is determined over time (dynamic heat capacity, etc.)

Parameters	Unit	Symbol
Number of collectors	[-]	\(N_{module}\)
Total gross area	\(m^{2}\)	\(A_{gross}\)
Wind speed at the collector array	[-]	\(f_{wind}\)
Orientation	°	\(\gamma\)
Tilt angle	°	\(\beta\)
Rotation	°	\(\phi\)

The catalogue structure depends on the previously selected norm.

Parameters	Unit	Symbol
Eta0,b: Efficiency	[-]	\(\eta_{0,b}\)
a1: Heat loss coefficient	\(\frac{W}{m^{2}K}\)	\(a_{1}\)
a2: Temperature dependence of the heat loss coefficient	\(\frac{W}{m^{2}K^{2}}\)	\(a_{2}\)
a3: Wind speed dependence of the heat loss coefficient	\(\frac{J}{m^{3}K}\)	\(a_{3}\)
a4: Sky temperature dependence of the heat loss coefficient	[-]	\(a_{4}\)
a5: Effective thermal capacity	\(\frac{J}{m^{3}K}\)	\(a_{5}\)
a6: Wind speed dependence of the zero loss efficiency	\(\frac{s}{m}\)	\(a_{6}\)
a7: Wind speed dependence of IR radiation exchange	\(\frac{W}{m^{2}K^{4}}\)	\(a_{7}\)
a8: Radiation loss	\(\frac{W}{m^{2}K^{4}}\)	\(a_{8}\)
Kd: Incidence angle modifier for diffuse solar radiation	[-]	\(K_{d}\)
kt 10 °	°	\(k_{t,10}\)
kt 20 °	°	\(k_{t,20}\)
kt 30 °	°	\(k_{t,30}\)
kt 40 °	°	\(k_{t,40}\)
kt 50 °	°	\(k_{t,50}\)
kt 60 °	°	\(k_{t,60}\)
kt 70 °	°	\(k_{t,70}\)
kt 80 °	°	\(k_{t,80}\)
kt 90 °	°	\(k_{t,90}\)
kl 10 °	°	\(k_{l,10}\)
kl 20 °	°	\(k_{l,20}\)
kl 30 °	°	\(k_{l,30}\)
kl 40 °	°	\(k_{l,40}\)
kl 50 °	°	\(k_{l,50}\)
kl 60 °	°	\(k_{l,60}\)
kl 70 °	°	\(k_{l,70}\)
kl 80 °	°	\(k_{l,80}\)
kl 90 °	°	\(k_{l,90}\)
Collector axis orientation	[-]	\(N_{axis}\)
Volume	\(l\)	\(V\)
Max temperature	° C	\(T_{max}\)

The \(k_{t}\) and \(k_{l}\) parameters modify the angle of incidence for direct solar irradiation for transverse \(k_{t}\) and longitudinal \(k_{l}\) angles of incidence. The relationship is described in more detail in the section explaining the IAM values.

Calculation of \(E_{sol}\) – global solar irradiation

The available irradiation on the collector field is described as \(E_{sol}\).

\(\dot{E}_{sol} = (I_{b}+I_{H}+I_{d})\cdot A_{gross} = I_{sol}\cdot A_{gross}\)

The following definitions apply:

• \(I_{b}\) : direct (beam) tilted solar irradiation with \(G_{b}\) (direct solar radiation) and \(\theta_{I}\) (angle of incidence) 
  
  \(I_{b}=G_{b}\cdot \cos(\theta_{I})\)
  
• \(I_{H}\) : reduced global irradiance taking into account reflection and albedo effects of the surroundings. Here \(G_{H}\) stands for the reduced solar radiation depending on the horizon, \(\alpha\) for the albedo factor and \(F_{p}\) for the reflected view factor 
  
  \(I_{H}=G_{H}\cdot \alpha\cdot F_{\rho}\)
  
• \(I_{d}\) : Diffuse irradiance with \(G_{d}\) (diffuse global irradiance) and \(F_{d}\) (proportion of diffuse irradiance into the collector surface) 
  
  \(I_{d}=G_{d}\cdot F_{d}\)

Calculation of \(Q_{sol}\) – the solar thermal yield

The solar thermal yield is calculated using the following equations:

\(\dot{Q}_{sol}=A_{gross}\cdot (\eta_{0,b}\cdot I_{b}\cdot K_{b}(\theta_{t},\theta_{l})+\eta_{0,b}\cdot I_{d}\cdot K_{d}+\eta_{0,b}\cdot I_{H}-\dot{q}_{loss})\)

The term \(\dot{q}_{loss}\) describes the sum of all losses that result as follows:

\(\dot{q}_{loss} = \dot{q}_{loss,1}+\dot{q}_{loss,2}+\dot{q}_{loss,wind}+\dot{q}_{loss,sky}+\dot{q}_{loss,wind,0}+\dot{q}_{loss,wind,IR}+\dot{q}_{loss,irr}\)

In detail, the loss terms are defined as:

Description	Formula
First order heat loss	\(\dot{q}_{loss,1}=a_{1}\cdot (T_{m}-T_{a})\)
Second order heat loss	\(\dot{q}_{loss,2}=a_{2}\cdot(T_{m}-T_{a})^{2}\)
Wind effect on heat loss	\(\dot{q}_{loss,wind}=a_{3}\cdot f_{wind}\cdot u'(T_{m}-T_{a})\)
Sky temperature on heat loss	\(\dot{q}_{loss,sky}=a_{4}\cdot(\sigma\cdot T_{a}^{4}-I_{L})\)
Wind effect on zero loss efficiency	\(\dot{q}_{loss,wind,0}=a_{6}\cdot f_{wind}\cdot u’\cdot I_{sol}\)
Wind effect on infrared loss	\(\dot{q}_{loss,wind,IR}=a_{7}\cdot f_{wind}\cdot u’\cdot (I_{L}-\sigma\cdot T_{a}^{4})\)
Irradiance loss	\(\dot{q}_{loss,irr}=a_{8}\cdot(T_{m}-T_{a})^{4}\)

With the following variables:

• \(T_{m}\) mean collector temperature
• \(T_{a}\) ambient temperature
• \(u’\) reduced wind speed according to \(u’=u-3\frac{m}{s}\)
• \(I_{L}\) long-wave irradiance \(\sigma\) (Stefan-Boltzmann constant)
  \(I_{L}=G_{long-wave}\cdot \frac{1+\cos(\beta)}{2}+\sigma\cdot T_{a}^{4}\cdot \frac{1-\cos(\beta)}{2}\)

A state of equilibrium is assumed for the second-order heat loss and radiation loss terms if the ambient temperature is greater than the collector temperature:

\(T_{a}>T_{m}\to \dot{q}_{loss,2}=0, \dot{q}_{loss,irr}=0\)

Incident angle modifier (IAM) – \(K_{b}\) calculation

The calculation of the IAM factor \(K_{b}(\theta_{t},\theta_{l})\) depends on the collector type. The following relationship applies for unglazed and flat plate collectors:

\(K_{b}(\theta_{t},\theta_{l})=f_{axis}(\gamma_{l},\theta_{I})\cdot \cos^{2}(\Phi)+f_{axis}(\gamma_{t},\theta_{I})\cdot \sin^{2}(\Phi)\)

For all other collector types: \(K_{b}(\theta_{t},\theta_{l})=f_{axis}(\gamma_{l},\theta_{l})\cdot f_{axis}(\gamma_{t},\theta_{t})\)

Where

\(\theta_{l}=\arctan(\cos(\Phi)\cdot \left| \tan(\theta_{I}) \right|)\)

\(\theta_{t}=\arctan(\sin(\Phi)\cdot \left| \tan(\theta_{I}) \right|)\)

• \(\theta_{l}\) : angle of incidence on the collector surface
• \(\Phi\) : angle as a function of rotation \(\Phi\) (input in user interface) \(\Phi=rad(\phi+90\cdot (1-N_{axis}))+f_{\theta_{I}}\)
• The following applies to \(f_{\theta_{I}}\) :
  • \(\alpha_{s}\) : sun elevation angle
  • \(\beta\) : inclination angle
  • \(\gamma\) : azimuth angle, depending on the tracking system

If \(\cos(\beta)=0\)

\(f_{\theta_{I}}=\arctan(\cos(\alpha_{s})\cdot \frac{\cos(\gamma’)}{\sin(\alpha_{s})})\) , if \(\sin(\alpha_{s})=0\) \(f_{\theta_{I}=0}\)

else if \(\cos(\beta)=\sin(\alpha_{s})\)

\(f_{\theta_{I}}=\arctan(\cos(\alpha_{s})\cdot \frac{\sin(\gamma’)}{\cos(\beta)\cdot (\sin(\beta)-\cos(\alpha_{s})\cdot \cos(\gamma’))})\)

else

\(f_{\theta_{I}}=\arccos(1-\cos(\theta_{I})\cdot \cos(\beta) \cdot \frac{\sin(\beta)-f}{\sin(\theta_{I})\cdot \left[ 1-\sin(\beta)\cdot (\sin(\beta)-f) \right]})\)

where

\(f=\cos(\beta)\cdot \frac{\sin(\beta)-\cos(\gamma’)\cdot \cos(\alpha_{s})}{\cos(\beta)-\sin(\alpha_{s})}\)

The function \(f_{axis}(\gamma,\theta)\) is calculated using the IAM angle values contained in the collector database:

\(f_{axis}(\gamma,\theta)=k_{\gamma,i}+(\theta_{\gamma}-\theta_{i})\cdot \frac{k_{\gamma,i+1}-k_{\gamma,i}}{\theta_{i+1}-\theta_{i}}\)

Calculation of the collector temperature

The temperature in the collector system is determined according to the first law of thermodynamics for open systems. The following diagram illustrates the principle.

As described above, the solar thermal yield is determined according to the following equation:

\(\dot{Q}_{sol}=A_{gross}\cdot (\eta_{0,b}\cdot I_{b}\cdot K_{b}(\theta_{t},\theta_{l})+\eta_{0,b}\cdot I_{d}\cdot K_{d}+\eta_{0,b}\cdot I_{H}-\dot{q}_{loss})=\dot{E}_{sol,net}-\dot{Q}_{loss}\)

Using the energy balance of the first law of thermodynamics, this results in:

\(\frac{dE_{coll}}{dt}=\dot{E}_{sol,net}-\dot{Q}_{loss}+P_{in}-P_{out}\)

Or more specifically as a derivative of the collector temperature over time:

\(m_{coll}\cdot c_{p}\frac{dT_{m}}{dt}=\dot{E}_{sol,net}-\dot{Q}_{loss}+P_{in}-P_{out}\)

From the catalogue entry, the dynamic heat capacity per square metre is given by the value \(a_{5}\). From this, the total heat capacity of the collector field can be reformulated as follows:

\(A_{gross}\cdot a_{5}\frac{dT_{m}}{dt}=\dot{E}_{sol,net}-\dot{Q}_{loss}+P_{in}-P_{out}\)

Rearranging the equation allows the collector temperature in the respective time step to be derived.