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This study proposes the concept of the local heat loss coefficient and examines the calculation method for the average heat loss coefficient and the average absorber plate temperature. It also presents an exergy analysis model of flat plate collectors, considering nonuniformity in temperature distribution along the absorber plate. The computation results agree well with experimental data. The effects of ambient temperature, solar irradiance, fluid inlet temperature, and fluid mass flow rate on useful heat rate, useful exergy rate, and exergy loss rate are examined. An optimal fluid inlet temperature exists for obtaining the maximum useful exergy rate. The calculated optimal fluid inlet temperature is 69 °C, and the maximum useful exergy rate is 101.6 W. Exergy rate distribution is analyzed when ambient temperature, solar irradiance, fluid mass flow rate, and fluid inlet temperature are set to 20 °C, 800 W/m^{2}, 0.05 kg/s, and 50 °C, respectively. The exergy efficiency is 5.96%, and the largest exergy loss is caused by the temperature difference between the absorber plate surface and the sun, accounting for 72.86% of the total exergy rate.
Solar energy, a clean and renewable energy source with no harmful environmental effects, has received considerable attention for generating heat and electricity [
Solar collectors have been examined in terms of their exergy, exergy efficiency, and entropy generation [
The present work presents a theoretical model considering nonuniformity in temperature distribution along the absorber plate for the exergy analysis of flat plate solar collectors. The model has also been experimentally verified.
A micro unit
where
According to the integral mean value theorem, the following can be deduced from
where
To simplify the model, the following assumptions are made:
The plate is in a steadystate heat transfer condition.
The temperature gradient in the direction of the fin thickness is negligible.
The fluid mass flow rate inside metal tubes is uniform.
The pressure drop of the fluid inside metal tubes is neglected.
The absorber plate is composed of metal tubes and fins, it is shown in
Establishing coordinate system as
where
The convective heat transfer coefficient between the absorber plate and the glass is calculated from the Nusselt number according to the following
where
The convective heat transferred from the absorber plate to the glass of
The radiative heat transferred from the glass to the sky of
where the temperature of the sky is
The convective heat transfer coefficient between the glass and the environment is as follows [
where
The convective heat transferred from the glass to the environment of
The following equation must hold in steady state:
The top local heat loss coefficient of
The heat loss coefficient consists of the top, back, and edge heat loss coefficients.
The back heat loss coefficient is calculated as follows [
where
The edge heat loss coefficient is estimated as follows [
where
The local heat loss coefficient of
According to
Given that the fin thickness is smaller than the width, the temperature gradient in the direction of the fin thickness is neglected. The temperature distribution of the fin in the
where
In the steady state, the heat obtained by the absorber plate is equal to the heat obtained by the metal tube and fins. The heat obtained by
where
Fin efficiency can be determined as follows [
The heat obtained by the absorber plate is finally transferred to the fluid, thermal contact resistance is neglected, and
The relationship between the metal tube and fluid temperatures can then be deduced by
According to
The average fin temperature of
The average temperature of
From
According to this method, if
The energy equation of
where
Substituting
where
The average heat loss coefficient is:
where
Substituting
The equation of exergy balance takes the following form [
where
At steady conditions,
The inlet exergy rate consists of the inlet exergy carried by fluid flow and the radiation exergy rate from the sun. The rate of inlet exergy carried by fluid flow is as follows [
The radiation exergy rate from the sun on the collector surface can be calculated as follows [
where
The rate of outlet exergy carried by fluid flow is as follows [
where
The exergy loss rate consists of the following four parts:
The first part of the exergy loss rate is caused by heat leakage from the absorber plate to the environment [
The second part of the exergy loss rate is caused by the temperature difference between the absorber plate surface and the Sun [
where (
The third part of the exergy loss rate is due to solar radiation losses from the collector surface to the absorber plate:
The fourth part of the exergy loss rate is caused by the temperature difference between the absorber plate and fluid [
According to
Solar collector exergy efficiency is calculated by dividing the increase in fluid flow exergy by the inlet radiation exergy [
According to
Experiments are conducted to validate the methodology. The flat plate collector is Sangpu PYT/L2.03, the metal tubes are made of copper, the fins are made of aluminum, the insulation material is rock wool board, and the working fluid is water. The relative parameters of the flat plate collector are listed in
The experimental test platform is shown in
The values calculated by the above method and experimental data are compared, and the results are shown in
An experiment is conducted in six time segments of a day. The fluid inlet temperature is controlled by an electrically heated constanttemperature water bath. The fluid inlet temperature of each measurement is different, and the mass flow rate is controlled at 0.05 kg/s by adjusting the valve. When the collector is in steady state (
With the suggested calculation method of this study, the effects of ambient temperature, solar irradiance, fluid mass flow rate, and fluid inlet temperature on the useful heat rate, useful exergy rate, and exergy loss rate are examined.
The useful heat rate and useful exergy rate have conﬂicting behavior with increasing ambient temperature. Although
The variations of the useful heat rate, useful exergy rate, and exergy loss rate
The useful heat rate and useful exergy rate also have conflicting behavior with increasing fluid inlet temperature.
From
When the fluid inlet temperature is lower than the optimized value, the variation trends of
In this study, the first and second laws of thermodynamics are used to study the performance of a flat plate solar collector. A theoretical model for the exergy analysis of flat plate collectors is presented by considering nonuniformity in the temperature distribution of the absorber plate:
The model is experimentally validated, with the results agreeing well with the experimental data.
The useful heat rate increases with increasing ambient temperature, solar irradiance, and fluid mass flow rate and decreases with increasing fluid inlet temperature.
The effects of ambient temperature, solar irradiance, fluid inlet temperature, and fluid mass flow rate on useful exergy rate and exergy loss rate are obtained. The conclusion can be summarized as follows: the useful exergy rate decreases with increasing ambient temperature and increases with increasing solar irradiance. It first increases and then decreases with increasing fluid inlet temperature, and an optimum fluid inlet temperature exists for obtaining the maximum useful exergy rate. When the fluid inlet temperature is lower than the optimized value, the useful exergy rate then decreases with increasing fluid mass flow rate. However, when the fluid inlet temperature is higher than the optimized value, the useful exergy rate increases with increasing fluid mass flow rate, the effect of mass flow rate on performance decreases with increasing mass flow rate, and the mass flow rate should be controlled carefully based on actual demand.
The useful heat rate is greater than the useful exergy rate. The useful heat rate and useful exergy rate have conﬂicting behavior in many cases. Thus, selecting an appropriate evaluation criterion (energy or exergy) for the collector according to specific conditions is recommended.
Solar irradiance considerably affects both the useful heat rate and useful exergy rate. In other words, high performance is based on appropriate solar irradiance.
The optimum fluid inlet temperature varies, and it is mainly affected by heat loss because environmental parameters change during the day. The operating parameters could be adjusted to obtain the maximum useful exergy rate during the day, and exergy performance could be significantly improved by optimizing it.
The exergy loss rate caused by the temperature difference between the absorber plate surface and the sun accounts for the largest exergy loss of the collector. When
Based on this analysis, the appropriate operating conditions for collectors could be determined with the given conditions and are useful for obtaining a higher useful exergy rate and decreasing internal irreversibilities.
Nomenclature
A_{c}  the aperture area  m^{2} 
A_{e}  the side area of the collector  m^{2} 
A_{p}  the area of the absorber plate  m^{2} 
c_{p}  the heat capacity of the fluid  J/(kg·K) 
d  the distance between the absorber plate and glass  m 
D_{i}  the metal tube inner diameter  m 
D_{o}  the metal tube outer diameter  m 
E_{d}  the exergy loss rate  W 
E_{d1}  exergy loss rate caused by temperature difference between the absorber plate surface and the Sun  W 
E_{d2}  exergy loss rate because of solar radiation losses from the collector surface to the absorber plate  W 
E_{d3}  exergy loss rate caused by the temperature difference between the absorber plate and fluid  W 
E_{in}  the inlet exergy rate  W 
E_{in,f}  the inlet exergy rate carried by fluid flow  W 
E_{in,Q}  the radiation exergy rate from the sun on the collector surface  W 
E_{l}  exergy loss rate caused by heat leakage from the absorber plate to the environment  W 
E_{out}  the outlet exergy rate  W 
E_{out,f}  the outlet exergy rate carried by fluid flow  W 
E_{s}  the stored exergy rate  W 
E_{u}  the useful exergy rate  W 

the collector efficiency factor  
g  the gravity  m^{2}/s 
h_{c,p−1}  the convective heat transfer coefficient between the absorber plate and the glass  W/(m^{2}·K) 
h_{f,i}  the convective heat transfer coefficient between the metal tube and fluid  W/(m^{2}·K) 
h_{w}  the convective heat transfer coefficient between the glass and the environment  W/(m^{2}·K) 
I  the solar irradiance  W/m^{2} 
k_{air}  the thermal conductivity of air  W/(m·K) 
k_{l}  the thermal conductivity of the insulation material  W/(m·K) 
L  the metal tube length  m 
L_{b}  the thickness of the insulation material in back  m 
L_{e}  the thickness of the insulation material at the side  m 
m  the fluid mass flow rate  kg/s 
n  the metal tube number of collectors  
Nu  the Nusselt number  
δq_{c,p−1}  the convective heat transfer from the absorber plate to the glass of dy  W 
δq_{c,1−a}  the convective heat transfer from the glass to the environment of dy  W/(m^{2}·K) 
δq_{r,p−1}  the radiative heat transfer from the absorber plate to the glass of dy  W 
δq_{r,1−a}  the radiative heat transfer from the glass to the sky of dy  W 
δ 
the heat obtained by dy  W 
Q_{l}  the heat loss from the solar collector to the environment  W 
Q_{s}  the radiation flux absorbed by the absorber plate  W 
Q_{u}  the useful heat rate gain of the flat plate solar collector  W 
R_{a}  the Rayleigh number  
S  the radiation flux absorbed by a unit area of the absorber plate  W/m^{2} 
T_{a}  the ambient temperature  K 
T_{b}  the metal tube temperature  K 
T_{f}  the fluid temperature  K 
T_{i}  the fluid inlet temperature  K 
T_{n}  the fin temperature  K 
T_{o}  the fluid outlet temperature  K 
T_{p}  the temperature of dA  K 

the average temperature of dy  K 

the integral average of T_{p} over the absorber plate  K 
T_{s}  the apparent solar temperature as exergy source  K 
T_{sky}  the temperature of the sky  K 
T_{1}  the temperature of the glass  K 
U_{b}  the back heat loss coefficient  W/(m^{2}·K) 
U_{e}  the edge heat loss coefficient  W/(m^{2}·K) 
U_{l}  the local heat loss coefficient of dA  W/(m^{2}·K) 

the local heat loss coefficient of dy  W/(m^{2}·K) 

the top local heat loss coefficient of dy  W/(m^{2}·K) 

the integral average of U_{l} over the absorber plate  W/(m^{2}·K) 
U_{L}  the overall heat loss coefficient  W/(m^{2}·K) 
v  the kinematic viscosity  m^{2}/s 
V_{a}  the wind speed  m/s 
W  the width of the metal tube and fins  m 
α  the thermal diffusivity  m^{2}/s 
β  the thermal expansion coefficient  1/K 
δ  the fin thickness  m 
ε_{p}  the emissivity of the absorber plate  
ε_{1}  the emissivity of the glass  
η_{ex}  the exergy efficiency  
η_{n}  the fin efficiency  
θ  the collector tilt relative to the horizontal  ° 
λ  the thermal conductivity of the fins  W/(m·K) 
Σ  the Stefan–Boltzmann constant  W/(m^{2}·K^{4}) 
(τα)_{e}  the effective product transmittance–absorptance 
The study presented in this paper is financially supported by National Natural Science Foundation Programs of China (Grant Nos. 51066002, 51366005, U0937604), Yunnan Provincial Natural Science Foundation Programs (Grant Nos. 2008KA002, 2008CD001), hightech development project by the Development and Reform Commission of Yunnan province (Grant Nos.2008CD001) and Applied Basic Research Projects of Yunnan Province(Grant Nos. KKSY201252024).
Huitao Wang and Zhong Ge contributed to the conception of the study. Huitao Wang and Zhong Ge contributed to the building of model. Hua Wang, Songyuan Zhang and Xin Guan designed the experiment. Zhong Ge, Songyuan Zhang contributed to the data collecting. Zhong Ge performed the data analyses and wrote the manuscript. Hua Wang and Huitao Wang helped in the analysis with constructive discussions.
The authors declare no conflict of interest.
Absorber plate consisting of metal tubes and fins.
Flow chart of simulation program for evaluating
Experimental test platform.
Variations of useful heat rate, useful exergy rate, and exergy loss rate
Variations of useful heat rate, useful exergy rate and exergy loss rate
Variations of useful heat rate, useful exergy rate, and exergy loss rate
Variations of useful heat rate, useful exergy rate, and exergy loss rate
Parameters of the flat plate collector.
Parameter  Value 

Tube center to center distance  0.135 m 
The metal tube length  2 m 
The metal tube number of collectors  7 
The metal tube outer diameter  0.015 m 
Thermal conductivity of the fins  236 W/(m·K) 
Thermal conductivity of the insulation material  0.055 W/(m·K) 
Effective product transmittance–absorptance  0.88 
Emissivity of the absorber plate  0.05 
Emissivity of the glass  0.88 
Size of the flat plate collector  2 m × 1 m ×0.07 m 
Collector tilt relative to the horizontal  45° 
Comparison between calculated values and experimental data.
Time
 

10–11  11–12  12–13  13–14  14–15  15–16  
631  702  713  640  535  374  
25.5  26.4  27.7  27.5  26.8  25.7  
29.4  37.3  44.6  51.9  61.1  67.2  
1.56  3.09  4.10  4.85  4.86  2.44  
1.62  2.96  3.95  4.79  4.90  2.59  
3.70  4.39  3.80  1.25  0.82  5.79 
Exergy rate distribution of collector.
Exergy Rate  Value/W  Percentage/% 

88.860  5.96  
1087.427  72.87  
251.282  16.84  
14.880  1.00  
49.713  3.33 