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# Solar thermal: Calculation and validation

nPro helps to generate hourly resolved power profiles for solar thermal collectors. On this page you learn how these are calculated and validated.

## Calculation of solar thermal generation profiles

For the calculation of heat generation profiles nPro uses the Standard ISO 9806. Hereby, nPro supports different calculation approaches based on ISO 9806: ISO 9806:2017 as well as three calculation methods based on ISO 9806:2013: quasi-dynamic, steady-state and unglazed. In the following, the formulas are provided.

With the nPro tool you can generate and download heat generation profiles in hourly resolution for a user-defined solar solar-thermal-calculation collector model.

### Formulas for heat generation based on ISO 9806

• ISO 9806:2017: Newest calculation standard: \begin{gathered} q=\eta_{0, b} K_b\left(\theta_L, \theta_T\right) G_b+\eta_{0 . b} K_d G_d-a_1\left(T_m-T_a\right)-a_2\left(T_m-T_a\right)^2\\ -a_3 u^{\prime}\left(T_m-T_a\right)+a_4\left(E_L-\sigma T_a^4\right)-a_6 u^{\prime} G-a_7 u^{\prime}\left(E_L-\sigma T_a^4\right)\\ -a_8\left(\vartheta_m-T_a\right)^4 \end{gathered}
• ISO 9806:2013: Quasi-Dynamic Approach: This method involves dynamic simulations that provide insights into the system's behavior over time, accounting for fluctuations in solar radiation and other variables. It's particularly useful for capturing real-world dynamics. \begin{gathered} q=\eta_{0, b} \cdot K_{\theta, b}\left(\theta_L, \theta_T\right) \cdot G_b+\eta_{0, b} \cdot K_{\theta, d} \cdot G_d-c_6 \cdot u \cdot G \\ -c_1 \cdot\left(T_m-T_a\right)-c_2 \cdot\left(T_m-T_a\right)^2-c_3 \cdot u \cdot\left(T_m-T_a\right)+c_4 \\ \cdot\left(E_L-\sigma \cdot T_a^4\right) \end{gathered}
• ISO 9806:2013: Steady-State Approach: In contrast to dynamic simulations, the steady-state method simplifies the analysis by assuming constant conditions. This is valuable for quick estimations and comparisons under stable scenarios. \begin{gathered} q=G \cdot \eta_0-a_1 \cdot\left(T_m-T_a\right)-a_2 \cdot\left(T_m-T_a\right)^2 \end{gathered}
• ISO 9806:2013: Unglazed Approach: This method pertains to scenarios where the solar collector lacks a protective cover. It's particularly relevant when studying systems with direct exposure to environmental conditions. \begin{gathered} q=G^{\prime \prime} \cdot \eta_0 \cdot\left(1-b_u \cdot u\right)-\left(b_1+b_2 \cdot u\right) \cdot\left(\vartheta_m-\vartheta_a\right)\\ G^{\prime \prime}=G+\frac{\varepsilon}{\alpha} \cdot\left(E_L-\sigma \cdot T_a^4\right) \end{gathered}

### Pre-defined collector models in nPro

In nPro the following pre-defined collector models are available:

• Flat plate
• Evacuated tube collector
• Air-brine collector

The model parameters of these models are listed in the following:

Table 1: Pre-defined solar thermal collector model for a flat plate collector (model: Vaillant: VFK 155/2 V)
$$\eta_{0,\text{b}}$$ $$K_{\mathrm{d}}$$ $$a_1$$ $$a_2$$ $$a_3$$ $$a_4$$ $$a_5$$ $$a_6$$ $$a_7$$ $$a_8$$
--- --- $$W/(m^2 K)$$ $$W/(m^2 K^2)$$ $$(J)/(m^3 K)$$ --- $$J/(m^2K)$$ $$s/m$$ $$W/(m^2 K^4)$$ $$W/(m^2 K^4)$$
0.784 0.96 3.69 0.012 0 0 7573 0 0 0
Incident angle modifier 10° 20° 30° 40° 50° 60° 70° 80° 90°
Transversal 1.00 0.99 0.98 0.95 0.91 0.84 0.68 0.34 0.00
Longitudinal 1.00 0.99 0.98 0.95 0.91 0.84 0.68 0.34 0.00
Table 2: Pre-defined solar thermal collector model for a evacuated-tube collector (model: Vaillant: VTK 570/2)
$$\eta_{0,\text{b}}$$ $$K_{\mathrm{d}}$$ $$a_1$$ $$a_2$$ $$a_3$$ $$a_4$$ $$a_5$$ $$a_6$$ $$a_7$$ $$a_8$$
--- --- $$W/(m^2 K)$$ $$W/(m^2 K^2)$$ $$(J)/(m^3 K)$$ --- $$J/(m^2K)$$ $$s/m$$ $$W/(m^2 K^4)$$ $$W/(m^2 K^4)$$
0.559 0.96 0.646 0.004 0 0 7914 0 0 0
Incident angle modifier 10° 20° 30° 40° 50° 60° 70° 80° 90°
Transversal 1.01 1.01 1.02 1.02 0.98 1.05 1.14 0.57 0.00
Longitudinal 1.00 1.00 0.99 0.98 0.95 0.89 0.76 0.38 0.00
Table 3: Pre-defined collector model for a air-brine-collectors (see model description: Lott et al., 2022)
$$\eta_{0,\text{b}}$$ $$K_{\mathrm{d}}$$ $$c_1$$ $$c_2$$ $$c_3$$ $$c_4$$ $$c_5$$ $$c_6$$
--- --- $$W/(m^2 K)$$ $$W/(m^2 K^2)$$ $$(J)/(m^3 K)$$ --- $$J/(m^2K)$$ $$s/m$$
0.75 1.01 47.30 0 20.20 0.75 77490 0.09
Incident angle modifier 10° 20° 30° 40° 50° 60° 70° 80° 90°
Transversal 1.04 1.08 1.11 1.15 1.37 1.42 1.74 2.47 0.00
Longitudinal 0.99 0.97 0.94 0.92 0.88 0.8 0.66 0.35 0.00

### User-defined collector models

In addition to pre-defined solar thermal collectors, nPro supports four calculation methods to define your own collector model. These calculation methods are:

• ISO 9806:2017
• ISO 9806:2013, quasi-dynamic
• ISO 9806:2013, unglazed

## How were the heat generation profiles validated?

The profiles generated with nPro have been compared with generation profiles created with the ScenoCalc tool from the SP Technical Research Institute of Sweden for a variety of different locations and orientations. The ScenoCalc tool is an open Excel sheet that is recognized by all institutions for energy yield prediction when certifying collectors according to the Solar Keymark procedure. The tool is limited to the consideration of a single collector. However, it is ideally suited as a basis for almost any type of solar thermal collector. An excerpt of the validation is shown in the tables below. If you want to reproduce the results, please make sure you use the same weather profiles in ScenoCalc and in nPro. For this validation, we used the following weather files:

The weather files can directly be uploaded in nPro. We used ScenoCalc version 4.04 and version 6.1 for the validation. The weather files are already inserted in the ScenoCalc Excel sheets. Please use as "Aperture area" 1 m². In version 4.04, the method "Steady state (EN 12975-2, Chapter 6.1)" refers to the nPro method "ISO 9806:2013: Steady-state" and the method "Quasi Dynamic Testing (EN 12975-2, Chapter 6.3)" refers to the nPro method "ISO9806:2013: Quasi-dynamic". To neglect the IAM, use in the Tab "IAM type" the option "User defined IAM constants", and set all entries to 1. In version 6.1, the method "Quasi Dynamic Testing (ISO 9806:2017, Ch. 23.4.5)" refers to the nPro method "ISO 9806:2017".

### Comparison of nPro and ScenoCalc (flat plate and evacuated-tube)

Table 4: Solar yield for the pre-defined collector model "flat plate" in nPro and ScenoCalc (version 6.1). Collector is faced toward South.
Location Elevation Collector temperature nPro ScenoCalc Deviation
Athens 25° 25 °C 1295.20 kWh/m² 1295.20 kWh/m² 0 %
50 °C 942.58 kWh/m² 942.58 kWh/m² 0 %
75 °C 642.83 kWh/m² 642.83 kWh/m² 0 %
Davos 30° 25 °C 986.77 kWh/m² 986.77 kWh/m² 0 %
50 °C 671.73 kWh/m² 671.73kWh/m² 0 %
75 °C 403.75 kWh/m² 403.75 kWh/m² 0 %
Stockholm 45° 25 °C 644.41 kWh/m² 644.41 kWh/m² 0 %
50 °C 413.77 kWh/m² 413.77 kWh/m² 0 %
75 °C 241.26 kWh/m² 241.26 kWh/m² 0 %
Andrews 25° 25 °C 1102.87 kWh/m² 1102.87 kWh/m² 0 %
50 °C 775.11 kWh/m² 775.11 kWh/m² 0 %
75 °C 498.74 kWh/m² 498.74 kWh/m² 0 %
Table 5: Solar yield for the pre-defined collector model "flat plate" in nPro and ScenoCalc (version 6.1). Location is Athens (Greece) and the mean collector temperature is 25 °C.
Orientation nPro ScenoCalc Deviation
Horizontal 1149.00 kWh/m2 1149.00 kWh/m2 0 %
30° / South 1296.72 kWh/m2 1296.72 kWh/m2 0 %
90° (vertical) / East 574.64 kWh/m2 574.64 kWh/m2 0 %
90° (vertical) / West 557.81 kWh/m2 557.81 kWh/m2 0 %
30° / North 751.96 kWh/m2 751.96 kWh/m2 0 %
Table 6: Solar yield for the pre-defined collector model "evacuated-tube" in nPro and ScenoCalc (version 6.1). Collector is faced toward South.
Location Elevation Collector temperature nPro ScenoCalc Deviation
Athens 25° 25 °C 998.23 kWh/m² 998.23 kWh/m² 0 %
50 °C 918.94 kWh/m² 918.94 kWh/m² 0 %
75 °C 828.07 kWh/m² 828.07 kWh/m² 0 %
Davos 30° 25 °C 873.58 kWh/m² 873.58 kWh/m² 0 %
50 °C 787.55 kWh/m² 787.55 kWh/m² 0 %
75 °C 694.90 kWh/m² 694.90 kWh/m² 0 %
Stockholm 45° 25 °C 563.33 kWh/m² 563.33 kWh/m² 0 %
50 °C 493.48 kWh/m² 493.48 kWh/m² 0 %
75 °C 422.78 kWh/m² 422.78 kWh/m² 0 %
Andrews 25° 25 °C 886.06 kWh/m² 886.06 kWh/m² 0 %
50 °C 805.05 kWh/m² 805.05 kWh/m² 0 %
75 °C 715.96 kWh/m² 715.96 kWh/m² 0 %

### Comparison of nPro and ScenoCalc (version 4.04)

Table 7: User-defined collector model for validation in Tables 8 to 10. IAM was neglected.
$$\eta_{0,\text{b}}$$ $$K_{\mathrm{d}}$$ $$c_1$$ $$c_2$$ $$c_3$$ $$c_4$$ $$c_5$$ $$c_6$$
--- --- $$W/(m^2 K)$$ $$W/(m^2 K^2)$$ $$(J)/(m^3 K)$$ --- $$J/(m^2K)$$ $$s/m$$
0.7 1 3 0.015 0.2 0 0 0
Table 8: Solar yield for a pre-defined collector model as listed in Table 7 in nPro and ScenoCalc (version 4.04). Location is Athens (Greece) and the mean collector temperature is 25 °C.
Orientation nPro ScenoCalc Deviation
Horizontal 1105 kWh/m2 1105 kWh/m2 0 %
25° / South 1217 kWh/m2 1217 kWh/m2 0 %
90° (vertical) / East 563 kWh/m2 563 kWh/m2 0 %
90° (vertical) / West 544 kWh/m2 544 kWh/m2 0 %
Table 9: Monthly solar yield for a pre-defined collector model as listed in Table 7 in nPro and ScenoCalc (version 4.04). Location is Athens (Greece) and the mean collector temperature is 25 °C.
Month nPro ScenoCalc Deviation
January 49.86 kWh/m2 49.86 kWh/m2 0 %
February 55.01 kWh/m2 55.01 kWh/m2 0 %
March 84.50 kWh/m2 84.50 kWh/m2 0 %
April 103.75 kWh/m2 103.75 kWh/m2 0 %
May 128.14 kWh/m2 128.14 kWh/m2 0 %
June 153.74 kWh/m2 153.74 kWh/m2 0 %
July 166.79 kWh/m2 166.79 kWh/m2 0 %
August 158.04 kWh/m2 158.04 kWh/m2 0 %
September 133.31 kWh/m2 133.31 kWh/m2 0 %
October 91.03 kWh/m2 91.03 kWh/m2 0 %
November 50.22 kWh/m2 50.22 kWh/m2 0 %
December 42.65 kWh/m2 42.65 kWh/m2 0 %
Table 10: Solar yield for a pre-defined collector model as listed in Table 7 in nPro and ScenoCalc (version 4.04).
Location Orientation Collector temperature nPro ScenoCalc Deviation
Davos 90° (vertical), West 25 °C 453 kWh/m2 453 kWh/m2 0 %
Davos 90° (vertical), East 25 °C 294 kWh/m2 294 kWh/m2 0 %
Davos 25°, South 50 °C 609 kWh/m2 609 kWh/m2 0 %
Davos 90°, North 25 °C 84 kWh/m2 84 kWh/m2 0 %
Davos 90°, North 50 °C 2 kWh/m2 2 kWh/m2 0 %
Stockholm 25°, South 25 °C 586 kWh/m2 586 kWh/m2 0 %
Stockholm 25°, South 50 °C 362 kWh/m2 362 kWh/m2 0 %
Stockholm 25°, South 75 °C 192 kWh/m2 192 kWh/m2 0 %

### Comparison of nPro and ScenoCalc (version 6.1)

Table 11: User-defined collector model for validation in Table 12.
$$\eta_{0,\text{b}}$$ $$K_{\mathrm{d}}$$ $$a_1$$ $$a_2$$ $$a_3$$ $$a_4$$ $$a_5$$ $$a_6$$ $$a_7$$ $$a_8$$
--- --- $$W/(m^2 K)$$ $$W/(m^2 K^2)$$ $$(J)/(m^3 K)$$ --- $$J/(m^2K)$$ $$s/m$$ $$W/(m^2 K^4)$$ $$W/(m^2 K^4)$$
0.784 0.96 3.69 0.012 0 0 0 0 0 0
Incident angle modifier 10° 20° 30° 40° 50° 60° 70° 80° 90°
Transversal 1.00 0.99 0.98 0.95 0.91 0.84 0.68 0.34 0.00
Longitudinal 1.00 0.99 0.98 0.95 0.91 0.84 0.68 0.34 0.00
Table 12: Solar yield for a pre-defined collector model as listed in Table 11 in nPro and ScenoCalc (version 6.1).
Location Orientation Collector temperature nPro ScenoCalc Deviation
Athens 45°, South 25 °C 1245 kWh/m2 1245 kWh/m2 0 %
Athens 45°, South 50 °C 896 kWh/m2 896 kWh/m2 0 %
Athens 45°, South 75 °C 606 kWh/m2 606 kWh/m2 0 %
Davos 90° (vertical), West 25 °C 462 kWh/m2 462 kWh/m2 0 %
Davos 90° (vertical), East 25 °C 284 kWh/m2 284 kWh/m2 0 %
Davos 25°, South 25 °C 1009 kWh/m2 1009 kWh/m2 0 %
Davos 90° (vertical), North 25 °C 85 kWh/m2 85 kWh/m2 0 %
Davos 90° (vertical), North 50 °C 1.3 kWh/m2 1.3 kWh/m2 0 %
Stockholm 25°, South 25 °C 618 kWh/m2 618 kWh/m2 0 %
Stockholm 25°, South 50 °C 389 kWh/m2 389 kWh/m2 0 %
Stockholm 25°, South 75 °C 221 kWh/m2 221 kWh/m2 0 %

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