Assessment of Durability of Solar Absorbers - Performance Criterion

Solar fraction, Fs, defined as the ratio between the delivered energy from a solar domestic hot water (DHW) system and the load (thermal energy necessary to satisfy domestic water heating needs), is widely accepted as performance indicator for this type of systems. Considering solar absorptance αs and thermal emittance εt as the most relevant characteristics of solar absorbers, the relation between the depreciation of these optical properties and depreciation of Fs was used by Hollands et al (1992) to define a performance criterion (PC) for assessment of long-term behavior and service life of selective solar absorbers. The PC was established mainly for solar DHW systems working with solar fractions lower than 50%. In this work, systems working with solar fractions higher than 50%, in climates of south of Europe, are considered and the suitability of solar fraction as performance indicator to develop an adequate PC is studied. As a first step simulations of thermal performance of systems using an in-house software were performed for a reduction of 5% and 10% of Fs. In ISO 22975-3, solar fraction Fs degradation must be lower than 5% to guarantee 25 years of service life for DHW system. The results showed that the parameters obtained to define the PC were incoherent considering solar fractions higher than 50%. In a second step, supplied energy was considered as performance indicator and using similar methodology as Hollands et al (1992), but using as performance indicator energy supplied by the solar system, the PC for systems working with solar fractions higher than 50%, in climates of south of Europe, was establish. The results showed that this is not significantly different from the PC considered in ISO 22975-3.


Introduction
Assessment of long-term behavior and service life of selective solar absorbers used in solar thermal collectors for domestic hot water (DHW) systems can be performed according to ISO 22975-3. The Performance Criterion used in the standard was establish considering that for a service life of 25 years, reduction in solar fraction F s should not be higher than 5%. In the frame of Task X of the IEA Solar Heating and Cooling Programme (Carlsson, B. et al, 1994) and according to Hollands et al (1992) it was possible to establish a performance criterion given by the change in optical properties of solar absorbers, namely, solar absorptance, α s , and thermal emittance,  t : ∆ 0.25∆ 0,05 ( e q . 1 ) where Δα s = α s -α s0 and Δε t = ε t -ε t0, according to ISO 22975-3 (2014).
The selection of this expression was established assuming that this equation is representative of the degradation of solar absorber surfaces used in DHW systems with solar fractions lower than 50% and corresponds to a reduction of 5% of the solar fraction.
Latter, changes introduced by Köhl, M. et al (2004) conduct to the expression presently used in the standard: ∆ 0.50∆ 0,05 ( e q . 2 ) and considered valid for collectors working at higher temperatures.
In this work the procedure for establishment of the performance criterion is revisited considering DHW systems working with higher solar fractions, in climates of the south of Europe. In section 2, the methodology proposed by Hollands et al (1992) was applied considering as performance indicator the Solar Fraction. In section 3, a different performance indicator, supplied energy, is used and deduced. In section 4. conclusions are presented.

Methodology
Considering different combinations of optical properties, through analytical expressions according to (D.E. Roberts and A. Forbes, 2012) it was possible to obtain the thermal performance coefficients η 0 (-), a 1 (Wm -2 K -1 ) and a 2 (Wm -2 K -2 ) for a flat plate collector whose constructive characteristics are known (see Tab. 1). Computer simulations of thermal performance of systems were performed, using an in house software (SolTerm, V5.3, 2017), in order to analyze the solar fraction (F s ) of DHW systems, with a collector area of 4.3 m 2 and storage volumes of 200 L (a), 250 L (b) and 300 L (c), located in Lisbon. These systems have different ratios between store volume and collector area, 46.5 L/m 2 (a), 58.1 L/m 2 (b) and 69.8 L/m 2 (c) respectively. In Tab. 1 are presented the assumed values of input parameters used in the analytical expressions and computer simulations. Symbols in Tab. 1 follow the nomenclature of D.E. Roberts and A. Forbes (2012) and the input parameters signaled with (*) are according with this reference. The methodology proposed by Hollands et al (1992) was also used. Fig. 1 illustrates the solar fraction F s as a function of solar absorptance α s and each curve corresponds to a fixed value of thermal emittance ε t for situation (a). In this figure it is also possible to visualize which combinations of α s and ε t will produce a 5 and 10% loss in solar fraction, F s . In this analysis the initial optical properties considered are α s0 = 0.96 and ε 0 = 0.1, which characterized the initial status of the flat plate collector considered. The simulations considered combinations of 0.025 increments in solar absorptance α s , from 0.1 to 0.96, and 0.1 increments in thermal emittance ε t , from 0.11 to 1.
Two possible states of failure are considered when compared with the initial solar fraction F s0 , i.e., when F s = 0.90 F s0 and F s = 0.95 F s0 , respectively. F s0 represents the solar fraction when the solar DHW system operates with α s = α s0 and ε t = ε t0 . The combination values of α s and ε t that correspond to a reduction of 5% or 10% in the system solar fraction were determined from Fig. 1. The intersection between the horizontal lines and the curves of fixed ε t for F s versus α s gives the combination of α s and ε t that will produce a 5 and 10% loss in solar fraction F s . In Fig. 2, -Δα s is represented as a function of Δε t for the three systems considered, (a), (b) and (c).
From Hollands et al (1992), these results are highly influenced by geographical location. However, it is possible to see that they are not dependent on the variation of system configuration, i.e, the results are very similar when considering storage volumes of 200 L (a), 250 L (b) and 300 L (c) as can be seen in Fig. 2. The relationship between  s and  t is very close to linear, as already presented in Hollands et al (1992), which makes easier further interpretation of these graphs to determine which will be the adequate expression for PC considering solar fractions higher than 50% in climates of South of Europe.
Following the methodology proposed by Hollands et al (1992),  s versus  t can be expressed as: When ∆ is equal to zero, the parameter a is obtained (vertical intercept). When ∆ is equal to zero, the parameter b is obtained (horizontal intercept).
The Eq. 3 can be transformed into Where ( e q . 5 ) ( e q . 6 ) Which gives a general form of the expression of PC presently used in the standard ISO 22975 (2014),

Results
Considering the situation (a) in Fig.2, when and F s = 0.95 F s0 we find a = 0.14 and b = 0.24 and, when F s = 0.90 F s0 we find a = 0.24 and b = 0.48. The parameter b was calculated by determining the linear function that best fits each case, since it is not possible to directly obtain the horizontal intercept from Fig.2 (∆ = 0).
According to Hollands et al (1992), if all solar radiation absorbed by the collector is converted in useful energy, i.e., is transferred to the load, a reduction of the solar fraction F s will be proportional to a reduction on α s , if all other parameters are unchanged, i.e.,  t = 0. This meaning that, in Fig. 2, the vertical intercept (denoted by a) corresponding to  t = 0 would be 0.96 0.05 = 0.048 (for 5% reduction in solar fraction) or 0.96 0.10 = 0.096 (for 5% reduction in solar fraction) since = 0.96.
The difference in a value determined based on the simulations and giving the linear representation of Fig. 2 and equations 3 to 6, can only be explained due to the fact that for higher solar fractions not all energy is transferred to load. There is dumped energy and the proportionality between a decrease in and F s can no longer be considered.
For higher solar fractions we have: Following Hollands et al (1992), the ratio (Eq. 9) versus F s was analyzed taking into account all the simulations performed considering different ratios ∆F s / F s0 for the DHW system studied in this work.  According to the results presented in Tab. 2, the proportionality between ∆α s and ∆F s is clear at low solar fraction (F s < 0.5) since the ratio (a / α s0 ) / (∆F s / F s0 ) shows a tendency to become constant and close to 1.1. Fig. 3 gives the dependence of the ratio (a / α s0 ) / (∆F s / F s0 ) on F s and shows the same tendency for low solar fraction given by Hollands et al (1992).
According to Hollands et al (1992) and the above results, Eq. 10 shows the relationship between k 2 and a, given Eq. 6. ∆ (eq.10) The performance criterion was determined for three different situations in order to meet the main objective of this study, i.e, the adequate expression for performance criterion considering solar fractions higher than 50% in climates of South of Europe. The results are presented in Tab. 3, 4 and 5. The parameter k 1 was calculated considering a and b, according to Eq. 5. The parameter k 2 was calculated considering α s0 = 0.96, the ΔF s / F s0 and the respective ratio, according to Eq.10.
Firstly, the performance criterion was determined for low solar fraction considering an average value for all the solar fractions up to 50%, i.e. the average solar fraction between 20% and 50%, and then, compared with the performance criterion given by Hollands et al (1992). Secondly, the performance criterion was determined for solar fractions lower than 80%, considering an average value of all solar fractions from 20% up to 80%. Thirdly and last, the performance criterion was determined for a solar fraction equal to 80%, considering the ratio (a / α s0 ) / (∆F s / F s0 ) specific for that solar fraction.
For the first situation, shown in Tab. 3, for F s / F s0 = 0.95, a = 0.14 and b = 0.24 are obtained and taking into account α s0 = 0.96 and ratio = 1.1 (low solar fraction), this leads to k 1 = 0.58 and k 2 = 0.05. The approximation k2 ≈ a was not found in this case. This also applies for F s / F s0 = 0.90, where it is obtained a = 0.24 and b = 0.48, for the same α s0 and ratio, leading to k 1 = 0.50 and k 2 = 0.11, where k 2 is clearly different from a. For the second situation, presented in Tab. 4, for F s / F s0 = 0.95, a = 0.14 and b = 0.24 are also obtained and considering α s0 = 0.96 and ratio = 1.68 (solar fraction F s < 0.8), this leads to k 1 = 0.58 and k 2 = 0.08. The approximation k 2 ≈ a was also not found in this case. This also applies for F s / F s0 = 0.90, where it is obtained a = 0.24 and b = 0.48, for the same α s0 and ratio, leading to k 1 = 0.50 and k 2 = 0.16, where k 2 is different from a.  At last, for the third situation given by Tab. 5, for F s / F s0 = 0.95, a = 0.14 and b = 0.24 are also obtained and considering α s0 = 0.96 and ratio = 2.76 (solar fraction F s = 0.8), this leads to k 1 = 0.58 and k 2 = 0.13. The approximation k 2 ≈ a was found for this situation. For F s / F s0 = 0.90, where it is obtained a = 0.24 and b = 0.48, for the same α s0 and ratio, leading to k 1 = 0.50 and k 2 = 0.26, where k 2 is quite close to a. This result is as expected, since the ratio used is obtained for higher solar fraction values. Although for higher solar fractions, values of k 2 and a are similar (Tab. 5), the PC obtained would correspond to a less demanding requirement, i.e., higher depreciation of α s could be considered. Since collectors used in systems working at higher solar fractions are expected to work at higher temperatures, this less demanding requirement is not adequate, i.e., this methodology is unclear regarding to the suitability of the expression for performance criterion considering solar fractions higher than 50%.

Methodology
It was decided to adopt another methodology in order to obtain more enlightening results. The methodology proposed by Hollands et al (1992) was also used but considering, as performance indicator, the supplied energy E of DHW system instead of solar fraction F s . The initial supplied energy E 0 corresponds to the DHW initial state when α s = 0 and ε t = 0.
When replacing the solar fraction F s for supplied energy E, it was considered that supplied energy higher to Load was still useful energy. In this situation the depreciation in supplied energy is proportional do the depreciation in α s which means a reduction of 5 or 10% in α s that will cause a reduction of the supplied energy E.
Still following Hollands et al (1992) methodology, the ratio presented in Eq. 9 is transformed into Eq. 12, where the ratio versus E 0 was analyzed taking into account all the simulations performed considering different ratio E / E 0 for the DHW system studied in this work. Eq. 13 shows the relationship between k 2 and a, given Eq. 6. / ∆ (eq.12) ∝ ∆ (eq.13) Similar to previous analysis, Fig. 4 illustrates the supplied energy E as a function of solar absorptance α s and each curve corresponds to a fixed value of thermal emittance  t for situation (a). From this figure, combinations of α s and  t that will produce a 5 and 10% loss in supplied energy are obtained. Two possible states of failure are represented when compared with the initial solar supplied energy E 0 , i.e., when E = 0.90 E 0 and E = 0.95 E 0 , respectively. E 0 represents the supplied energy when the DHW system operates with α s = α s0 and  t =  t0 . In Fig. 5, -∆α s is represented as a function of ∆ε t .
According to Eq. 12 and the results presented in Tab. 6, the proportionality between a and E 0 is evident for all supplied energies E and not only for a few situations has it happened (see Tab. 2 and Fig. 3) when solar fractions were used in the studied equations. The ratio (a / α s0 ) / (∆E s / E 0 ) shows a tendency to become constant, particularly close to 1.1, for all supplied energies. Fig. 6 shows this dependence of the ratio on supplied energy E.

Results
According to Tab. 6, for E / E 0 = 0.95, a = 0.055 and b = 0.1 were obtained and, considering α s0 = 0.96 and ratio = 1.1, this leads to k 1 = 0.55 and k 2 = 0.055. The approximation k 2 = a is clearly found in this situation. For E / E 0 = 0.90, a = 0.11 and b = 0.2 where obtained and, for the same α s0 and ratio, this leads to k 1 = 0.55 and k 2 = 0.11, where k 2 is equal to a.
Since the ratio (a / α s0 ) / (∆E s / E 0 ) shows to be constant for all supplied energies, there is not a distinction between low or high supplied energy has it happens for solar fraction F s . Then, the performance criterion given in Tab. 7 can be accept as a general equation. These results are consistent with the performance criterion given by ISO 22975-3:2014.

Conclusions
In this work, solar DHW systems working with solar fractions higher than 50%, in climates of south of Europe, were considered. The suitability of solar fraction as performance indicator to develop an adequate PC was studied.
As a first step, simulations of thermal performance of systems using SolTerm software were performed for a reduction of 5% and 10% of F s . According to Hollands et al (1992), if all solar radiation absorbed by the collector is converted in useful energy, i.e., is transferred to the load, a reduction of the solar fraction F s will be proportional to a reduction on α s , if all other parameters are unchanged, i.e.,  t = 0. It was verified that this is only applicable for solar fraction lower than 50% and the results showed that the parameters obtained to define the PC were incoherent. For higher solar fractions, the PC obtained would correspond to a less demanding requirement, i.e., higher depreciation of αs could be considered. Since collectors used in systems working at higher solar fractions are expected to work at higher temperatures, this less demanding requirement is not adequate, i.e., this methodology is unclear regarding to the suitability of the expression for performance criterion considering solar fractions higher than 50%.
In a second step, supplied energy was considered as performance indicator and using similar methodology as Hollands et al (1992), the PC for systems working with solar fractions higher than 50%, in climates of south of Europe, was establish.
When replacing the solar fraction F s by supplied energy E, it was considered that supplied energy higher to load was still useful energy. In this situation the depreciation in supplied energy is proportional to the depreciation in α s which means a reduction of 5 or 10% in α s that will cause a reduction of the supplied energy E. The results showed that the parameters used to define the PC are now not significantly different from the PC considered in ISO 22975-3.
The expression for performance criterion given by ISO 22975-3:2014 is adequate for solar fractions higher than 50% in climate of South of Europe.