Heat Removal Factor in Flat Plate Solar Collectors: Indoor Test Method

: This paper presents a couple of methods to evaluate the heat removal factor F R of ﬂat plate solar collectors, as well as a parametric study of the F R against the tilt angle β , and ( T i − T a )/ G , and its effects on the a 0 -factor ( F R τα ) and the a 1 -factor ( F R U Lmin ). The proposed methods were based on indoor ﬂow calorimetry. The ﬁrst method considers the ratio of the actual useful heat to the maximum useful heat. The second takes into account the slopes of the family of efﬁciency curves ( F R U Lmin ) according to ANSI/ASHRAE 93-2010, and the minimum overall heat loss coefﬁcient, U Lmin . In both methods, a feedback temperature control at collector inclinations from horizontal to vertical allows the inlet temperature and the emulating of the solar radiation to be established by electrical heating. The performance of the methods was determined in terms of the uncertainty of the F R . Method 1 allowed a three-fold improved precision compared to Method 2; however, this implied a more detailed experimental setup. According to the ﬁrst method, the effects of the tilt angle β , and the ( Ti − Ta )/ G , on the a 0 -factor were considerable, since F R is directly proportional to the a 0 -factor. The changes in ( Ti − Ta )/ G caused an average change in F R of 32% The F R shows almost linear behavior for inclinations from horizontal to vertical with a 14.5% change. The effects of β on the a 1 -factor were not considerable, due to the compensation between the increase in F R and the decrease in U Lmin as β increased.


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The effects of tilt angle and (Ti − Ta)/G on the F R , a 0 -factor and a 1 -factor, were investigated using indoor flow calorimetry.

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The method based on the ratio of the actual useful heat to the maximum useful heat shows considerably improved behavior in terms of uncertainty.

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The F R shows linear behavior for inclinations from horizontal to vertical with a change of 14.5%.

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The changes in (Ti − Ta)/G caused an average change in F R of 32%. • The effect of the variation of the tilt angle and the (Ti − Ta)/G-value on the a 0 -factor is considerable since the F R is directly proportional to the a 0 -factor. • The changes in inclination do not considerably affect the a 1 -factor due to the compensation between the increase in F R and the decrease in U Lmin when β increases.

Introduction
In 2014, flat plate collector heating represented 22.4% (83.9 GW th ) of the total worldwide solar heating, and this percentage continues to grow [1]. The characterization and simulation have improved commercialization in many types of process, first for low temperature (<80 • C) and more recently in the medium temperature range (80-250 • ). The characterization reduces estimating uncertainty and allows better economic scenarios, which are welcome for the solar heating industry.
Flat plate solar collector characterizations are now used to determine the a o , a 1 , and the IAM as indicated by [2][3][4][5][6], among other standards and publications. The F R refers to the thermal effectiveness of solar collectors viewed as heat exchangers, while the a 0 (F R τα) is the fraction of the solar radiation GA a that gains the collector, and the a1 (F R U Lmin ) is the factor of heat losses of the solar collector due to the ambient effect. Most of the solar collector's studies consider the U Lmin as a constant; it is the case in which the efficiency against (Ti − Ta)/G has almost linear behavior. The U Lmin has linear behavior when the efficiency curve is fitted as the second order polynomial approach; however, in this case, the U Lmin is considered a function of (Ti − Ta)/G only. In the same way, the IAM mainly concerns the optical effects of the incidence angle, but does not concern the effect of the confined fluid flow and these optical effects simultaneously, as is the case when U Lmin is considered constant.
The convective flow pattern of the confined fluid between the absorber and the collector glazing is a function of the hot and cold wall temperature changes [7]. In a tilted differentially heated cavity, it has been found that different classes of natural convection flows appear due to changes in tilt angle and the differences in temperature [8][9][10]. The overall heat transfer coefficient considerably reduces a tall cavity turns from horizontal to vertical [11]. The above-mentioned implies disregarding the fact that the figures of merit F R , a 0 , and a 1 are a strong function of the U Lmin , which is in turn dependent on the collector inclination due to changes in the difference in outlet-inlet temperature and collector tilt angle. Meanwhile, the influence of collector inclinations mainly takes into account the incidence angle (optical effects: τα) according to ANSI/ASHRAE 93-2010, 2014 [2]. Awasarmol and Pise [12] studied the natural convection heat transfer from a fin array and angles of inclination. They found a decrease in heat transfer coefficient with the increase in angle of inclination and an optimal angle at 45 • . Montoya-Marquez and Flores-Prieto [13] experimentally shows that the changes in collector tilt considerably affects the U L and the calculations of the efficiency, due to variations in the flow pattern into the air cavity between the absorber plate and glazing.
The F R is also understood as the ratio of the actual useful heat to the maximum useful heat [4], which is the thermal effectiveness of the solar collector. Thus, one way to show a more detailed picture of the performance or effectiveness of the solar collector, like most heat exchangers, is through the heat removal factor. Currently, the heat removal factor F R is calculated as the ratio of actual useful energy gains to the useful energy gains, when the whole collector surface is at the fluid inlet temperature. However, this last experimental condition is difficult to achieve because inlet fluid increases its temperature as it flows through the absorber. Additionally, most studies determining F R have been theoretical, using a combination of thermal parameters, such as the collector fin efficiency factor F [4]. In addition, the efficiency curve slope at outdoor conditions (a 1 -factor), which is F R U Lmin , can be determined by a standard test [2]. To achieve F R , the U Lmin must be determined separately.
In line with this, the a 0 -factor and a 1 -factor strongly depend on F R , but it is usually calculated by a method where U Lmin is considered as a constant or with linear behavior [14][15][16]. For its part, Malvi et al. [17] reported the performance of a solar flat plate collector for various flow configurations; experiments at indoor conditions were conducted to determine the F R . The results indicate that, for the same conditions, parallel flow receives a double F R value compared with the serpentine flow. Experimental methods to evaluate F R , the a 0 -factor and the a 1 -factor separately, and its uncertainty have an incipient development. On the other hand, the relationships between F R and β and between F R and (Ti − Ta)/G, which affect the a 0 and a 1 , have become briefly studied, and characterized collectors at a specific latitude have different performance predictions at other latitudes, although these run at the same solar incidence angle. Therefore, the effects of tilt angle β, in a range of (Ti − Ta)/G, due to changes in the convective flow patterns, on F R , which affects the a 0 -factor and a 1 -factor, were experimentally studied by two proposed methods. The tilt collector was studied under inclinations from horizontal to vertical, and a T i from 60 to 90 • C. The proposed methods were based on indoor heat flow calorimetry and ANSI/ASHRAE 93-2010 [2]. The indoor condition consideration fixes the solar radiation, ambient temperature, wind velocity, and background radiation, in order to improve the experimental uncertainty.

Sampling
The study involves the manufacture and instrumentation of the solar collector to work in indoor conditions. The sample is shown in Figure 1; this is a glazing collector with 2.00 m 2 of gross area Aa and an aspect ratio AR of 40. The 10-90% water-glycol is the working fluid, in which heat capacity is considered variable [18]. The absorber is comprised of a couple of header tubes d 1 , and five raised finned tubes d 2 , all of which of copper and joined by tin-lead solder. The absorber solar absorbance α, is 0.94 and the glazing solar transmittance τ is 0.86 [19]. The absorbance and transmittance were obtained by normalizing the measured spectral [20]. A Shimadzu UV-3100 (Shimadzu Corporation, Kyoto, Japan) is used from 300 to 2500 nm, every 2.0 nm with ±0.1% of photometric uncertainty and 1.0% of wavelength uncertainty. Table 1 shows the manufacture characteristics of the sample.
Energies 2018, 11, x FOR PEER REVIEW 3 of 12 solar radiation, ambient temperature, wind velocity, and background radiation, in order to improve the experimental uncertainty.

Sampling
The study involves the manufacture and instrumentation of the solar collector to work in indoor conditions. The sample is shown in Figure 1; this is a glazing collector with 2.00 m 2 of gross area Aa and an aspect ratio AR of 40. The 10-90% water-glycol is the working fluid, in which heat capacity is considered variable [18]. The absorber is comprised of a couple of header tubes d1, and five raised finned tubes d2, all of which of copper and joined by tin-lead solder. The absorber solar absorbance α, is 0.94 and the glazing solar transmittance τ is 0.86 [19]. The absorbance and transmittance were obtained by normalizing the measured spectral [20]. A Shimadzu UV-3100 (Shimadzu Corporation, Kyoto, Japan) is used from 300 to 2500 nm, every 2.0 nm with ±0.1% of photometric uncertainty and 1.0% of wavelength uncertainty. Table 1 shows the manufacture characteristics of the sample.     Figure 2 shows that the glazing collector heats up the absorber plate, which in turns heats the working fluid that flows through the raising tubes. At this time, part of the supplied energy GAτα heats the working fluid and the rest is transferred to the ambient as heat losses Q L , which is dependent from the β and (Ti − Ta)/G. The incoming heat flux GAτα is considered independently of β, and it is the sum of the useful energy Q u (β, (Ti − Ta)/G) plus the heat loss flux Q L (β, (Ti − Ta)/G).

Experimental Desing
Energies 2018, 11, x FOR PEER REVIEW 4 of 12 Figure 2 shows that the glazing collector heats up the absorber plate, which in turns heats the working fluid that flows through the raising tubes. At this time, part of the supplied energy GAτα heats the working fluid and the rest is transferred to the ambient as heat losses QL, which is dependent from the β and (Ti − Ta)/G. The incoming heat flux GAτα is considered independently of β, and it is the sum of the useful energy Qu(β, (Ti − Ta)/G) plus the heat loss flux QL(β, (Ti − Ta)/G). The experimental design allows for the determination of the FR and FRULmin as a function of β and (Ti − Ta)/G at indoor conditions by a couple of methods, under inclinations from horizontal to vertical. The a0 is equal to FRτα, and the a1 is equal to FRULmin. The indoor condition consideration fixes the solar radiation, ambient temperature, wind velocity, and background radiation, in order to improve the experimental uncertainty. In the first proposed method, the FR was determined by the ratio of Qu_a to the maximum useful heat Qu_max, by indoor flow calorimetry. In the second proposed method, the FRUL_min was determined according to ANSI/ASHRAE 93-2010 [2] by achieving the slopes of the families of the efficiency curves, and the UL_min by flow calorimetry at indoor conditions too. The indoor condition is the main difference of the second method with the standard technique to obtain the factor FRUL_min. In both proposed methods, a feedback temperature control works at a set of collector inclinations from horizontal to vertical. At indoor conditions, the solar heating was emulated by the Joule effect and the PID control, using an electrical heater, making it possible to replace (Ti − Ta)/G by (Ti − Ta)/(VI/Aτα), to achieve better experimental uncertainty. Thus, the solar heating is given by Equation (1):

Experimental Desing
where τ and α are the glazing solar transmittance and the solar absorbance of the absorber respectively, and V and I are the electrical voltage and current, respectively. The performance of both methods was evaluated in terms of uncertainty, which was determined by the propagation error method and by the RMSE and R2, respectively. The experimental design allows for the determination of the F R and F R U Lmin as a function of β and (Ti − Ta)/G at indoor conditions by a couple of methods, under inclinations from horizontal to vertical. The a 0 is equal to F R τα, and the a 1 is equal to F R U Lmin . The indoor condition consideration fixes the solar radiation, ambient temperature, wind velocity, and background radiation, in order to improve the experimental uncertainty. In the first proposed method, the F R was determined by the ratio of Q u_a to the maximum useful heat Q u_max, by indoor flow calorimetry. In the second proposed method, the F R U L_min was determined according to ANSI/ASHRAE 93-2010 [2] by achieving the slopes of the families of the efficiency curves, and the U L_min by flow calorimetry at indoor conditions too. The indoor condition is the main difference of the second method with the standard technique to obtain the factor F R U L_min . In both proposed methods, a feedback temperature control works at a set of collector inclinations from horizontal to vertical. At indoor conditions, the solar heating was emulated by the Joule effect and the PID control, using an electrical heater, making it possible to replace (Ti − Ta)/G by (Ti − Ta)/(VI/Aτα), to achieve better experimental uncertainty. Thus, the solar heating is given by Equation (1): where τ and α are the glazing solar transmittance and the solar absorbance of the absorber respectively, and V and I are the electrical voltage and current, respectively. The performance of both methods was evaluated in terms of uncertainty, which was determined by the propagation error method and by the RMSE and R 2 , respectively.
Method 1: F R as a Function of the Ratio Q u_a /Q u_max Equation (2) shows that the F R is the ratio of the actual useful heat to the maximum useful heat, according to Duffie and Beckman [4]. The Q u_max occurs when T p is equal to T i , or T o is equal to T i , because Q L tends to be minimal. In the case, that T p or T o is greater than T i , the Q L is greater than zero, and the Q u_a is then greater than zero too.
The Q u_a (β, (Ti − Ta)/G) can be correlated by the change in enthalpy of the working fluid between outlet and inlet, at constant pressure. The Q u_max occurs when the whole collector is at the inlet fluid temperature, minimizing heat losses, Q Lmin . For this, T p = T i , or T o = T i , as is shown in Equation (3).
The F R value is determined by two parallel tests. The first determines the Q u_a at a fixed value of GAτα. In the second, the Q Lmin , and the Q u_max , are determined considering T p = T i and adjusting the GAτα. In this case, the F R is a function of β and (Ti − Ta)/G, as well as Q u_a , Q Lmin or (VI) 2 , Q u_max , and T p , as is shown in Figure 3. Equation (2) shows that the FR is the ratio of the actual useful heat to the maximum useful heat, according to Duffie and Beckman [4]. The Qu_max occurs when Tp is equal to Ti, or To is equal to Ti, because QL tends to be minimal. In the case, that Tp or To is greater than Ti, the QL is greater than zero, and the Qu_a is then greater than zero too.
The Qu_a(β, (Ti − Ta)/G) can be correlated by the change in enthalpy of the working fluid between outlet and inlet, at constant pressure. The Qu_max occurs when the whole collector is at the inlet fluid temperature, minimizing heat losses, QLmin. For this, Tp = Ti, or To = Ti, as is shown in Equation (3).
The FR value is determined by two parallel tests. The first determines the Qu_a at a fixed value of GAτα. In the second, the QLmin, and the Qu_max, are determined considering Tp = Ti and adjusting the GAτα. In this case, the FR is a function of β and (Ti − Ta)/G, as well as Qu_a, QLmin or (VI)2, Qu_max, and Tp, as is shown in Figure 3. The FRULmin was determined based on ANSI/ASHRAE 93-2010 [2], the ULmin factor was obtained separately, using heat flow calorimetry as per Beikircher et al. [3] and Montoya-Marquez and Flores-Prieto [13]. A set of tests is used to determine the family of efficiency curves against (Ti − Ta)/G, at inclinations from horizontal to vertical, which in turn allows for a set of FRULmin, each of which represents the slope of linear regression of each efficiency curve. The ULmin is determined to achieve the value of the FR once the FRULmin is achieved by Method 1 (Test 2), setting the GAτα as the compensation heat flux (VI)2 and setting the difference in temperature (Ti − Ta) according to Equation (4): The F R U Lmin was determined based on ANSI/ASHRAE 93-2010 [2], the U Lmin factor was obtained separately, using heat flow calorimetry as per Beikircher et al. [3] and Montoya-Marquez and Flores-Prieto [13]. A set of tests is used to determine the family of efficiency curves against (Ti − Ta)/G, at inclinations from horizontal to vertical, which in turn allows for a set of F R U Lmin , each of which represents the slope of linear regression of each efficiency curve. The U Lmin is determined to achieve the value of the F R once the F R U Lmin is achieved by Method 1 (Test 2), setting the GAτα as the compensation heat flux (VI) 2 and setting the difference in temperature (T i − T a ) according to Equation (4): Equation (5) gives the collector efficiency (ANSI/ASHRAE 93-2010, 2014) [2]: (5)

Experimental Setup
The experimental setup entails that the sample is mounted with a variable angle β at 0-90 • , with an uncertainty of ±0.1 • . The absorber heating (VI) is homogeneously distributed by means of the electrical heater and remains almost constant over each test. The electrical heater is supplied with a maximum of 2000 W, with an uncertainty of ±5 W. The temperature differences (T o − T i ) and (Ti − Ta) were measured using a thermopile type T thermocouple and 32 gauge wires, with an uncertainty of ±0.1 • C. A thermal bath supplied with a 10-90% water-glycol mixture was used as working fluid, with an uncertainty of ±0.01 • C. The mass flow rate was 0.016 kg/s [21,22]; it was monitored with a turbine flowmeter, with an uncertainty of 3%. It was also verified by weighing the water-glycol mixture, at specified time steps during the experiments. The experimental setup is shown in Figure 4.

Experimental Setup
The experimental setup entails that the sample is mounted with a variable angle β at 0-90°, with an uncertainty of ±0.1°. The absorber heating (VI) is homogeneously distributed by means of the electrical heater and remains almost constant over each test. The electrical heater is supplied with a maximum of 2000 W, with an uncertainty of ±5 W. The temperature differences (To − Ti) and (Ti − Ta) were measured using a thermopile type T thermocouple and 32 gauge wires, with an uncertainty of ±0.1 °C. A thermal bath supplied with a 10-90% water-glycol mixture was used as working fluid, with an uncertainty of ±0.01 °C. The mass flow rate was 0.016 kg/s [21,22]; it was monitored with a turbine flowmeter, with an uncertainty of 3%. It was also verified by weighing the water-glycol mixture, at specified time steps during the experiments. The experimental setup is shown in Figure 4. The experimental indoor conditions allow uniform surrounding temperature and surrounding emissivity, and the experiments can be run with non-considerable changes in solar heating, ambient temperature, and wind velocity. The working fluid was a 10-90% water-glycol mixture to minimize adverse boiling effects. A programmable FPGA (NI-cRIO9022, 32 bit data acquisition, Lab-VIEW software) (National Instruments, Austin, TX, USA) was used to monitor, record, and calculate the experimental variables at time steps of 1 s. The steady state was verified by monitoring experimental data without considerable changes in the experimental variables over 30 min. Each reported data point corresponds to an average of over 1800 measurements, taken over a 30 min period. The tests were run at a specified range and steps of the tilt angle, and the remaining variables involved in the experiment were considered without significant variations. Each test was carried out by in triplicate for comparison. The experimental indoor conditions allow uniform surrounding temperature and surrounding emissivity, and the experiments can be run with non-considerable changes in solar heating, ambient temperature, and wind velocity. The working fluid was a 10-90% water-glycol mixture to minimize adverse boiling effects. A programmable FPGA (NI-cRIO9022, 32 bit data acquisition, Lab-VIEW software) (National Instruments, Austin, TX, USA) was used to monitor, record, and calculate the experimental variables at time steps of 1 s. The steady state was verified by monitoring experimental data without considerable changes in the experimental variables over 30 min. Each reported data point corresponds to an average of over 1800 measurements, taken over a 30 min period. The tests were run at a specified range and steps of the tilt angle, and the remaining variables involved in the experiment were considered without significant variations. Each test was carried out by in triplicate for comparison.
The experimental conditions T p = T i and T p = f (T i , V I, U L ) was verified by infrared imaging on the absorber plate. As seen in Figure 5, the field temperature of the absorber plate was as expected, thanks to the PID control losses compensation. Figure 5a shows the T p = T i condition, where the standard deviation was only 0.14 • C. Figure 5b shows the case in which T p = f (T i , V I, U L ), where we can find a conventional temperature profile of the solar collector absorber. The experimental conditions Tp = Ti and = ( , , ) was verified by infrared imaging on the absorber plate. As seen in Figure 5, the field temperature of the absorber plate was as expected, thanks to the PID control losses compensation. Figure 5a shows the Tp = Ti condition, where the standard deviation was only 0.14 °C. Figure 5b shows the case in which = ( , , ), where we can find a conventional temperature profile of the solar collector absorber.

Results
The experiments were conducted highlighting the behavior of the FR and FRULmin as a function of the β and (Ti − Ta)/G. As noted above, the a0-factor is direct proportional to FR and the a1-factor is equal to FRULmin. The experimental campaign comprises five sets of four experiments, each one performed in triplicate. The parametric study was conducted for β as follows: 0°, 30°, 45°, 60°, and 90°. The (Ti − Ta)/G was 0.044, 0.056, 0.069, and 0.083.

Comparative Performance of both Methods
The uncertainty, in terms of RMSE and R2, of each method disclose its performance. Figure 6 shows a similar shape of FR against β for both methods. The FR grew were 0.14 and 0.27 for Methods 1 and 2, respectively. The experimental uncertainties of Methods 2 and 1 were ±0.049 and ±0.016 respectively; this is 306% times greater with Method 2 compared with Method 1. As seen in Figure 6, both shadow gaps of the experimental uncertainty are cross practical. The linearity values are (0.053, 0.9630) and (0.104, 0.9708) for Methods 1 and 2, respectively. The RMSE was considerably higher in Method 2. Method 1 allows for achieving an FR value for every studied value of (Ti − Ta)/G, unlike Method 2, where the FR is just an average value determined from the slope of each efficiency curve. Therefore, the remainder of this paper focuses on Method 1.

Results
The experiments were conducted highlighting the behavior of the F R and F R U Lmin as a function of the β and (Ti − Ta)/G. As noted above, the a 0 -factor is direct proportional to F R and the a 1 -factor is equal to F R U Lmin . The experimental campaign comprises five sets of four experiments, each one performed in triplicate. The parametric study was conducted for β as follows: 0 • , 30 • , 45 • , 60 • , and 90 • . The (Ti − Ta)/G was 0.044, 0.056, 0.069, and 0.083.

Comparative Performance of both Methods
The uncertainty, in terms of RMSE and R 2 , of each method disclose its performance. Figure 6 shows a similar shape of F R against β for both methods. The F R grew were 0.14 and 0.27 for Methods 1 and 2, respectively. The experimental uncertainties of Methods 2 and 1 were ±0.049 and ±0.016 respectively; this is 306% times greater with Method 2 compared with Method 1. As seen in Figure 6, both shadow gaps of the experimental uncertainty are cross practical. The linearity values are (0.053, 0.9630) and (0.104, 0.9708) for Methods 1 and 2, respectively. The RMSE was considerably higher in Method 2. Method 1 allows for achieving an F R value for every studied value of (Ti − Ta)/G, unlike Method 2, where the F R is just an average value determined from the slope of each efficiency curve. Therefore, the remainder of this paper focuses on Method 1.         Figure 8 shows F R against β for the studied range of (Ti − Ta)/G. The change in F R was 14.5% on average over the set of β, as mentioned above. On the other hand, the variation of (Ti − Ta)/G means average changes in F R of 32% over the set of (Ti − Ta)/G, and the a 0 is affected considerably due to changes in (Ti − Ta)/G, only if τα remains almost constant, because the flow pattern between the absorber plate and glassing cover is modified considerably. The latter is in the same line of the experimental work of Montoya-Marquez and Flores-Prieto [13].  Figure 8 shows FR against β for the studied range of (Ti − Ta)/G. The change in FR was 14.5% on average over the set of β, as mentioned above. On the other hand, the variation of (Ti − Ta)/G means average changes in FR of 32% over the set of (Ti − Ta)/G, and the a0 is affected considerably due to changes in (Ti − Ta)/G, only if τα remains almost constant, because the flow pattern between the absorber plate and glassing cover is modified considerably. The latter is in the same line of the experimental work of Montoya-Marquez and Flores-Prieto [13].  The a1-factor shows variations over the set of β that are not considerable. However, the a1-factor can change by 2.3% over the studied range of (Ti − Ta)/G. Over the range of (Ti − Ta)/G, the changes in a1-factor as a function of β is not considerable due to the compensation between the increase in FR and the decrease in ULmin when β increases.   The a 1 -factor shows variations over the set of β that are not considerable. However, the a 1 -factor can change by 2.3% over the studied range of (Ti − Ta)/G. Over the range of (Ti − Ta)/G, the changes in a 1 -factor as a function of β is not considerable due to the compensation between the increase in F R and the decrease in U Lmin when β increases.  Figure 8 shows FR against β for the studied range of (Ti − Ta)/G. The change in FR was 14.5% on average over the set of β, as mentioned above. On the other hand, the variation of (Ti − Ta)/G means average changes in FR of 32% over the set of (Ti − Ta)/G, and the a0 is affected considerably due to changes in (Ti − Ta)/G, only if τα remains almost constant, because the flow pattern between the absorber plate and glassing cover is modified considerably. The latter is in the same line of the experimental work of Montoya-Marquez and Flores-Prieto [13].  The a1-factor shows variations over the set of β that are not considerable. However, the a1-factor can change by 2.3% over the studied range of (Ti − Ta)/G. Over the range of (Ti − Ta)/G, the changes in a1-factor as a function of β is not considerable due to the compensation between the increase in FR and the decrease in ULmin when β increases.

Conclusions
A couple of experimental indoor test methods for determining F R and F R U Lmin (a 1 -factor) were conducted. Method 1 determined the ratio of Q u_a to Q u_max ; Method 2, by achieving the slopes, determined the F R U Lmin of the family of the efficiency curves, (ANSI/ASHRAE 93-2010, 2014) [2] and the U Lmin by indoor flow calorimetry. The effects of tilt angle and (Ti − Ta)/G on the F R and F R U Lmin factors were investigated, considering that the a 0 is directly proportional to F R and a 1 is equal to F R U Lmin . Both methods determine the behavior of F R against tilt angle and (Ti − Ta)/G. However, Method 1 shows considerably improved behavior in terms of uncertainty of F R -a three-fold lower uncertainty. Thus, a method was conducted to obtain the F R with the use of a PID temperature control at fixed indoors conditions, with which it is possible to obtain the F R with lower uncertainty.
In addition, Method 1 shows some advantages over Method 2-more data points, less uncertainty, and a complete view of the collector's efficiency-but a more detailed experimental setup is needed.
The changes in inclination from horizontal to vertical caused an almost linear increase in F R , (14.5%), which represents a change of 45% due to (Ti − Ta)/G, which caused an average change in F R of 32%. Thus, the effects of changes in tilt angle and (Ti − Ta)/G-value on the a 0 -factor is considerable, since a 0 is directly proportional to F R . The inclination changes do not considerably affect the a 1 -factor due to the compensation between the increase in F R and the decrease in U Lmin when β increases. Funding: This research was funded by Consejo Nacional de Ciencia y Tecnolgía México, CONACyT and the Tecnológico Nacional de México (Project. 5685.16), and the APC was funded by the Tecnológico Nacional de México.

Acknowledgments:
The authors are grateful to Consejo Nacional de Ciencia y Tecnolgía, CONACyT and Tecnológico Nacional de México, TecNM, whose financial support made this work possible.

Conflicts of Interest:
The authors declare that they have no conflict of interest.