A liquid flat 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 flow channels installed inside the absorber plate. This heat energy transferred to liquid is known as the useful heat gain of a flat plate collector. In the present study, a single glazed flat 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 flat 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.

## 2.1 Calculations of Solar Radiations:-

In the present study, a flat plate collector facing south is taken for analysis at Delhi (28064’ N, 77012’ E). Solar radiations falling on flat 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:

IT = Ib×Rb + Id× (1 + Cosβ)/2 + (Ib+Id) ×ρg× (1-cosβ)/2 (1)

Where the value of ρg (Diffused ground reflectance factor) is taken as 0.2 for simplification and Rb, known as tilt factor, is the ratio of Cosine of “Incidence angle (θ)” to the Cosine of “Zenith angle (θz)” and is given as:

Rb = (Cosθ/Cosθz). (2)

Whereas the incidence angle made by incident solar radiations with normal to the plane surface of collector is given as:

Cosθ = Sinφ (SinδCosβ + CosδCosγCosωSinβ) + Cosφ (CosγCosωCosβ – SinδCosγSinβ)

+Cosγ SinγSinωSinβ (3)

As the flat plate collector is facing south so the “Surface Azimuth angle, ‘γ’,” equals zero and the above equation become:

Cosθ = SinδSin (φ-β) + Cosδ CosωCos (φ-β) (4)

Here angle ‘δ’ is the declination angle and calculated by the relation given by Cooper [30] as

δ0 = 23.45 Sin ([{360 × (284 + n)}/365]) (5)

And φ is the latitude angle having a value equals to 28.640 for Delhi. Hour angle ‘ω’ is given by the relation:

ω = [{(1200h – LAT) × 15}/60]. (6)

The Zenith Angle (θz) can be calculated by putting β = 0 in Eq. (4) and can be written as:

Cosθz = SinδSinφ + Cosδ CosωCosφ (7)

## 2.2. Estimation of Optimum Tilt Angle:-

For analyzing the maximum amount of solar radiation falling on the flat plate collector, it is necessary to find 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 significantly and given by the relation (Sukhatme and Nayak 2013)

βopt = Tan− 1[{ ∑Ib×Tan(φ-δ)}/∑Ib] (8)

## 2.3 **Estimation of top heat loss coefficient:-**

Different authors have proposed relations for estimation of top heat loss (S. A. Klein 1975; Agarwal and Larson 1981; Malhotra and Garg 1981; Mullick and Samdarshi 1988). In the present work, the top heat loss is estimated considering the effect of clear sky temperature and is given as (Mullick and Samdarshi 1988):-

(Ut)-1 = (hrg-a + hw)-1 + (hrp-g + hcp-g)-1 + (tg )/ Kg (9)

Radiative heat transfer coefficient between glass cover and plate is given as:

(hrp−g) = σ[(Tp )2+(Tg)2]×[ (Tp ) + (Tg)]

[(1/ϵp + 1/ϵg) – 1] (10)

And hrg−a is given as:

(hrg−a) = ϵg σ× [(Tg )4– (Ts)4]

[Tg – Ta] (11)

Many empirical relations have been proposed by different authors to calculate sky temperature. In 1915, Angstrom (1915) estimated the effect of ambient conditions on clear sky emissivity as:

Es= [0.82-(0.25× (10) -0.168×Pv)] (12)

In 1932, Brunt (1932), relates the clear sky emissivity with vapour pressure considering the effect of various environmental conditions and proposed the following relation:

Es= [0.52+ (0.065× (Pv) 0.5)] (13)

According to Swinbank (1963) the ambient sky temperature is calculated as:

Ts = 0.0552*(Ta^1.5) (14)

Berger (1984) computed spectral sky radiations and the study reveals that the clear sky emissivity is affected by weather conditions as suggested in the following relation:

Es= [0.77+ (0.0038×Tdp)] (15)

In the present work, the effect of clear sky emissivity and thus, ambient sky temperature on energy loss, in term of heat, from the solar panel will be estimated. Further, convection energy loss for plate with glass cover is estimated by following equation:

hcp−g = [5.78(Tp - Tg)0.27] /[(Tmp−g)0.31 (Lp−g)0.21] (16)

Where,

Tmp−g = (Tp+Tg)/2 (17)

And,

Tg=Ta+hw−0.42[0.6336ϵp+(Tp/346)–0.6547–1.16exp{-0.072(Tp-Ta)}]×(Tp-Ta)] (18)

## 2.4 Calculation of Overall Heat Loss Coefficient:-

The overall heat loss coefficient represents the total amount of energy loss occurred in a collector and given as (Sukhatme and Nayak 2013):-

UL = Ut + Ub + Us (19)

And Ub = Ki / tb (20)

Where Ki, and tb are known as the thermal conductivity of insulation and thickness of insulation respectively.

## 2.5 Calculation of Collector Efficiency:-

The most important measuring tool for designing a energy storage or energy conversion device it its thermal efficiency. In case of a flat plat collector, the efficiency 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 efficiency is given as:-

η = FRQu / ACIT (21)

Where, FR, known as the collector heat removal factor, an important design parameter the thermal resistance, is calculated as:-

FR = ṁ×CP [{1- exp (-(F′×UL ×AP)/ ṁ×CP))}]

UL×AP (22)

Where F′ is the collector efficiency factor and is calculated as:-

F′ = (1/UL) / W [1 / UL {D+ (W-D) ×ε} + (1 / ∏×D×hf)] (23)

In the present study, the efficiency of the collector is taken as a function of top heat loss coefficient, or in terms, effective clear sky black body temperature, and heat removal factor.

Thus η = f (FR, Ut, Ts) (24)