Thermal performance analysis of a tilted single glazed flat plate collector considering the effect of clear sky black body temperature under different ambient conditions encountered in Delhi.


 The present study concerns with the thermal analysis of a flat plate collector covered with a single glass having a tilted surface facing south, located at Delhi (28064’ N, 77012’ E), India under different operating and climate conditions. Solar radiations falling on the tilted surface were noted throughout the year and based on the availability of these solar radiations optimum tilt angle was estimated by using empirical relation for every month of a year. The lowest optimum tilt angle was estimated for June (5.260), and the highest was estimated for December (51.950) with an overall optimum tilt angle of 31.10 for a year. The maximum variation in estimated values of UL is 11% and the collector heat efficiency factor has a maximum variation of 1.24% to 2.52%. Further, the use of the Swinbank equation provides the lower value of heat loss which in turn provides the highest value of useful heat gain and thus the efficiency is 4.596% to 9.94% higher than the efficiency calculated by using Angstrom relation which gives the lowest value of efficiency.


Introduction
Increase in global population has increased demand in energy consumption at building sectors for applications such as space heating, lighting, etc., and production of this excess amount of energy has become a matter of concern as in recent time availability of fossil fuels such as natural gas, coal, etc has been reduced due to which supply of this excess amount of electrical energy to building sectors looks like a challenging task to ful ll the customer's demand. Moreover, the consumption of fossil fuels to produce electricity also harms the environment and causes global warming. That's why nding an alternate source of energy that is less hazardous to the environment has become necessary. Solar energy, a renewable kind of energy, available free at every place on earth, is the source of energy that causes no environmental hazards and also easy to be convertible into electrical energy. Flat plate collectors are the most popular devices to collect solar radiations for producing heating effects to heat water for building sectors. FPC is the kind of heat exchanger in which heat is transfer to working uid by collecting solar radiations. These are simple to construct as well as install and have a low maintenance cost. Besides these advantages, the main drawback of FPC is its poor thermal e ciency because of low density and larger exposed surface area for heat loss to the surrounding.
While designing any kind of energy storage device, engineers must reduce heat loss for better performance. Many kinds of research have been done in reducing the major heat loss, in turn, to increase the thermal performances of FPC. The e ciency of FPC can be increased by coating the collector with several selected coaters. FPC with selective coaters has improved thermal performance as compared to black-coated FPC due to an increase in average temperature obtained by chrome-coated surface as compared to the black-coated surface (Shrestha et al. 2021). Use of carbon-coated absorber plate can give up to 35.04% higher e ciency as compared to black-chrome coated and up to 28.4% higher e ciency as compared black-painted absorber plate due to reduction in heat removing parameters  Further, the implementation of a corrugated riser with reduced pitch length and with little increment in roughness height can also increase the thermal performances of FPC (Sayed Ali  (2019) simulated the serpentine ow conduit, both experimentally and theoretically, and predicts that an increase in inclination angle results in less area for convective heat loss inside the tube's gap and e ciency increases. Besides these advantages, the use of multiple riser tubes in FPC cause increased exposed surface area for heat loss, which is a major drawback with its complex design. To reduce the complexity of the circuit, multiple numbers of riser tubes were replaced by a single spiral-shaped tube collector (

Mathematical Modeling
A liquid at collector consists of an absorber plate that absorbs all short wave solar radiations and transfers this amount of heat energy to liquid circulating through tubes or other kinds of ow channels installed inside the absorber plate. This heat energy transferred to liquid is known as the useful heat gain of a at plate collector. In the present study, a single glazed at plate collector, as shown in Fig. 1, is considered for the estimation of thermal performances considering the effect of black body sky temperature estimated by the equations proposed by different authors as discussed in the literature.
There are three kinds of heat loss to surrounding for a at plate collector viz; heat transfer from the bottom surface, heat transfer from the top surface, and heat transfer from the side surface. Heat loss from bottom and side surfaces can be reduced by providing insulation effectively, which is not possible for top surface heat loss reduction, which, contributes as a major area of energy loss.

Calculations of Solar Radiations:-
In the present study, a at plate collector facing south is taken for analysis at Delhi (28 0 64' N, 77 0 12' E).
Solar radiations falling on at plate collector were analyzed using empirical relation given by Liu and Jordan (1963), in which total radiations falling on collector are considered as a sum of diffused and beam radiations and given as: Where the value of ρ g (Diffused ground re ectance factor) is taken as 0.2 for simpli cation and R b , known as tilt factor, is the ratio of Cosine of "Incidence angle (θ)" to the Cosine of "Zenith angle (θz)" and is given as: Whereas the incidence angle made by incident solar radiations with normal to the plane surface of collector is given as: As the at plate collector is facing south so the "Surface Azimuth angle, 'γ'," equals zero and the above equation become: Here angle 'δ' is the declination angle and calculated by the relation given by Cooper  And φ is the latitude angle having a value equals to 28.64 0 for Delhi. Hour angle 'ω' is given by the relation: The Zenith Angle (θz) can be calculated by putting β = 0 in Eq. (4) and can be written as:

Estimation of Optimum Tilt Angle:-
For analyzing the maximum amount of solar radiation falling on the at plate collector, it is necessary to nd the value of the optimum tilt angle. In the present study, the variation of clear sky temperature is being considered which value changes seasonally in India, thus the optimum value of tilt angle 'β' also changes signi cantly and given by the relation (Sukhatme and Nayak 2013) Radiative heat transfer coe cient between glass cover and plate is given as: [(1/ϵ p + 1/ϵ g ) -1] (10) And h rg−a is given as: Where K i , and t b are known as the thermal conductivity of insulation and thickness of insulation respectively.

Calculation of Collector E ciency:-
The most important measuring tool for designing a energy storage or energy conversion device it its thermal e ciency. In case of a at plat collector, the e ciency of the collector reveals the ability of the collector to convert maximum amount of the incident solar radiations in useful heat gain. Thus the collector e ciency is given as:η = F R Q u / A C I T (21) Where, F R, known as the collector heat removal factor, an important design parameter the thermal resistance, is calculated as:- In the present study, the e ciency of the collector is taken as a function of top heat loss coe cient, or in terms, effective clear sky black body temperature, and heat removal factor.

Estimation of total solar radiations and optimum tilt angle:-
In the present study, a single glazed at plate collector with a tilted surface situated at Delhi (India) has been considered for studying the effect of environmental conditions encountered throughout the year. India is a multi-season country in which the environmental condition changes very rapidly throughout the year. Due to changes in climate conditions, there is a signi cant variation in total solar radiations, and hence there is a signi cant change in total available solar energy for a at plate collector. In the present study, global, beam, and diffused solar radiations falling on the tilted surface of a at plate collector located in Delhi with south-facing has been estimated [31] and plotted in Fig. 2. From Fig. 2, it is seen that there are huge variations in the availability of solar radiation throughout the year every month. The optimum tilt angle varies with variation in the availability of solar radiations, thus for simplicity of the research calculations, the optimum tilt angle is estimated for a complete month by using Eq. 8. From

Estimation of overall heat loss coe cient:-
Due to change in climate conditions like vapor pressure, relative humidity, dew point temperature, etc clear sky emissivity also changes which affect the calculation of effective black body sky temperature, thus there is remarkable variation in the thermal performances of a at plate collector. It is cleared in the literature that the sky temperature mostly depends upon clear sky emissivity. Figure 5 reveals that with increase in ambient sky temperature, there is signi cant increase in plate temperature and the total amount of energy loss from the collector increases which means that the overall energy loss, in terms of heat, from the solar plate is directly affected by ambient temperature, or in turns, the clear sky emissivity (Fig. 5). Also, the radiative heat transfer coe cient is also affected by the clear sky emissivity and it is noted that the clear sky emissivity has reverse impact on radiative heat loss coe cient. It is also noted that the use of Angstrom relation for sky temperature results in the highest value of radiative heat transfer coe cient and use of Swinbank ] relation for T s results in the lowest value of radiative heat transfer coe cient between the glass cover and ambient when the ambient temperature is 318 K (Fig. 6).
The effect of clear sky emissivity on the total energy loss encountered for the collector was estimated using the relation provided by different authors and from the Fig. 5, it is clearly seen that the Angstrom relation estimate the maximum heat loss, whereas, Swinbank relation provide the lower value of heat transfer coe cient. As there was approx 11% variation in the values of heat loss factor estimated by the relation provided by different authors, so, while designing a solar panel for heat utilization direct from the Sun; a designer should estimate the heat loss occurring in at plate collector using Angstrom relation for better accuracy.
3.3 Calculation of collector heat removal factor and collector e ciency:-For calculating collector e ciency factor, collector heat removal factor, and also collector e ciency, a at plate collector having inner and outer tube diameter equal to 12.5mm and 13.7mm respectively with tube pitch equals to 113mm has been considered for the study. The mass ow rate of the liquid (water) owing through the tube passage is taken as 70Kg/h and the uid heat transfer coe cient is taken as 500W/m 2 -K for ease of simpli cation. The collector e ciency factor is a function of the overall heat loss coe cient, which is varying with sky temperature, that's why the collector e ciency factor also varies with clear sky ambient temperature as shown in Fig. 7. From Fig. 7, it is clear that as the sky temperature, or in turn, plate temperature increases, the collector e ciency factors values decreases. The collector e ciency factor has the maximum values when the ambient temperature is being calculated by using Swinbank relation while the use of Angstrom relation gives the lower values of collector e ciency factor. Figure 8 shows the variation of collector heat removal factor with plate temperature considering the effect of clear sky temperature calculated by using the equations provided by different authors as mentioned in the literature. From gure 8, it is noticed that the use of Angstrom relation gives the lowest value while the use of the Swinbank equation gives the highest values of collector heat removal factor. There is approx 1.24% to 2.52% variation in the respective lowest and highest values of collector e ciency factor calculated by using Angstrom and Swinbank relation which is a signi cant variation to be considered. Similarly, the overall collector e ciency calculated by Swinbank relation gives the highest values as compared to others. Use of Swinbank equation for calculating overall heat loss coe cient provides the lower value which in turns provides the highest value of useful heat gain and thus the e ciency is 4.596% to 9.94% higher than the e ciency calculated by using Angstrom relation which gives the lowest value of e ciency (Figure 9).

Conclusion
A tilted single glazed south-facing at plate collector located in Delhi, India has been considered for studying the effect of clear sky black body temperature on the thermal performance of a at plate solar collector. The sky temperature has been estimated using correlations provided by different authors as mentioned in the literature. From the literature, it is cleared that radiative heat transfer coe cient and top heat loss coe cient values vary with clear sky temperature and thus the overall heat loss coe cient. The solar radiations falling on the surface of at plate collector has been estimated on monthly basis and it was found that there is huge variation in the availability of solar radiation in different months as Delhi is a multi-seasonal city and there is huge variation in climate conditions in every season due to which there is a variation in the availability of solar radiations. Thus due to these variations in climate conditions, different tilt angle values are estimated for different months and also there is variation in thermal performances of the collector due to change in climate conditions. The main results of the study are:i) Due to variation in available solar radiation, the optimum tilt angle has been estimated on monthly basis, as well as, seasonal basis. It was found that the optimum tilt angle has different values for each month. The lowest value of the optimum tilt angle was found to be 5.26 0 for June whereas, the highest value of the optimum tilt angle was found to be 51.95 0 for December month. The average optimum tilt angle for the whole year was estimated and found to be 31.1 0 .
ii) The seasonal variation in the values of optimum tilt angle has been also calculated and it was found that the average value of optimum tilt angle for winter, spring, summer, and monsoon season is 50 0 ,31 0 , 7.34 0 , and 26.34 0 respectively.
iii) For a selected range of working parameters, the overall heat loss coe cient and heat loss due to radiation between glass and ambient has been estimated, and it was found that there are approx 14% variations in the values of total energy loss and also the maximum and minimum absolute values of radiative heat transfer between absorber plate and glass cover was found to vary up to 121% when estimated by using correlations provide by different authors.
iv) With an increase in clear sky temperature, or in turns, plate temperature of at plate collector, the values of collector heat removal factor and collector e ciency factor decrease. The collector e ciency factor has the maximum values when the ambient temperature is being calculated by using Swinbank relation while the use of Angstrom relation gives the lower values of collector e ciency factor. There is approx 1.24-2.52% variation in the respective lowest and highest values of collector e ciency factor calculated by using Angstrom and Swinbank relation which is a signi cant variation to be considered.
v) Use of Swinbank equation for calculating overall heat loss coe cient provides the lower value which in turn provides the highest value of useful heat gain and thus the e ciency is 4.596-9.94% higher than the e ciency calculated by using Angstrom relation which gives the lowest value of e ciency.   Figure 1 Different kinds of heat transfer and thermal losses occur in at plate collector.  The optimum tilt angle of south-facing plat plate collector located at Delhi under climate conditions encountered in different seasons.

Figure 5
Variations in values of the overall heat loss coe cient considering the effect of clear sky emissivity.

Figure 6
Variations in values of the radiative heat loss coe cient considering the effect of clear sky emissivity.

Figure 7
Variation of collector e ciency factor with plate temperature considering the effect of effective black body sky temperature under ambient conditions encountered at Delhi.

Figure 8
Variation of collector e ciency factor with ambient temperature considering the effect of effective black body sky temperature under ambient conditions encountered at Delhi.

Figure 9
Variation of collector e ciency factor with ambient temperature considering the effect of effective black body sky temperature under ambient conditions encountered at Delhi.