thermal collector for domestic hot water application

Accepted Manuscript Performance analysis and multi-objective optimization of a hybrid photovoltaic /thermal collector for domestic hot water application

J.F. Chen, L. Zhang, Y.J. Dai PII:

S0360-5442(17)31848-0

DOI:

10.1016/j.energy.2017.10.143

Reference:

EGY 11787

To appear in:

Energy

Received Date:

05 September 2017

Revised Date:

20 October 2017

Accepted Date:

31 October 2017

Please cite this article as: J.F. Chen, L. Zhang, Y.J. Dai, Performance analysis and multi-objective optimization of a hybrid photovoltaic/thermal collector for domestic hot water application, Energy (2017), doi: 10.1016/j.energy.2017.10.143

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ACCEPTED MANUSCRIPT 1.

3-D dynamic thermal-electrical models are developed and validated for PV/T.

2.

2-D temperature distributions of PV/T collector are illustrated and compared.

3.

The dynamic response characteristics of different collectors are analyzed.

4.

The comprehensive performances are studied under various operating conditions.

5.

A multi-objective optimization is carried out for the design of PV/T DHW system.

ACCEPTED MANUSCRIPT

1

Performance analysis and multi-objective optimization of a hybrid

2

photovoltaic/thermal collector for domestic hot water application

3

J.F. Chen, L. Zhang, Y.J. Dai*

4

Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University

5

Engineering Research Centre of Solar Power and Refrigeration, MOE

6

Shanghai, 200240 China

7

ABSTRACT

8

PV/T collector turns out to be a promising alternative for the traditional solar thermal collector for

9

domestic hot water (DHW) application. In this paper, four comprehensive thermal and electrical models

10

including unglazed PV/T, glazed PV/T, PV and flat plate thermal collector are established for the purpose

11

of accurate long-term simulation and optimization. The detailed 2-D temperature distributions of the

12

glazed and unglazed PV/T collector are illustrated and compared for the first time. Besides, the dynamic

13

response characteristics of different collectors are discussed and compared. By running the models under

14

a broad combination of the operating conditions, the comprehensive performances are obtained and

15

found to be significantly influenced by flow rate, temperatures, radiation and wind speed. A multi-

16

objective optimization model coupling TRNSYS and NSGA-II tool is established to study and optimize

17

the PV/T DHW system for a complete year. The Pareto frontier of conflicting objectives (life cycle

18

savings and prime energy saving efficiency) is obtained for the optimal system design. The mass flow

19

rate on the Pareto frontier is between 0.0085kg/s and 0.011kg/s in this study. The optimal value of tank

20

volume on Pareto frontier shows an equally scattering distribution between 99.5 Land 218.6Lfor a 2 m2

21

glazed PV/T collector.

*

Corresponding author: Tel.: +86-21-34204358; fax: +86-21-34206814.

E-mail address: [email protected] (Y.J. Dai). 1

ACCEPTED MANUSCRIPT 22 23

Keywords:

Hybrid

photovoltaic/thermal

(PV/T);

dynamic

simulation;

multi-objective

optimization; co-simulation

24 25

Nomenclature A

area of collector [m2]

Subscripts

C

thermal capacity [J/kg K]

a

ambient

D

diameter [m]

air

air

E

electrical power [W]

abs

absorber

f

friction faction [-]

adh

adhesive

solar radiation intensity

[W/m2]

aux

auxiliary source

h

heat transfer coefficient

[W/m2

c

glass cover/conduction

I

invest [$]

d

discount

k

conductivity [W/m k]

ele

electrical

K

local resistance coefficient [-]

EVA

Ethylene-Vynil-Acetate

m

mass flow rate [kg/s]

f

working fluid

N

life time [year]

fuel

conventional fuel

Nu

Nusselt number [-]

g

PV glass

Pr

Prandtl Number [-]

i

inlet/inner/inflation

Ra

Rayleigh number [-]

ins

insulation

Re

Reynolds number [-]

o

outlet/outer

r

rate

O&M

operation and maintenance

T

temperature [K]

PV

photovoltaic

PWF

present worth factor

pump

circulation pump

v

speed [m/s]

prime

prime energy saving

ted

tedlar

G

k]

Greek symbols α

absorptivity [-]

r

radiation

δ

thickness [m]

ref

refernce

ε

emissivity [-]

sky

sky

η

efficiency [-]

t

tube

θ

[o]

th

thermal

v

convection

angle

[kg/m3]

ρ

density

τ

time constant/ transmittance

26 27 28 29 30 31 32 2

ACCEPTED MANUSCRIPT 33

1. Introduction

34

Solar photovoltaic (PV) modules are now widely used for power generation to lead a sustainable

35

and low-carbon economy. However, the temperature of PV modules is increased by the absorbed solar

36

radiation which is not converted into electricity, causing a decrease in conversion efficiency.

37

Experiments show that every 1 oC rise in operating temperature reduces the electrical conversion

38

efficiency by about 0.5% for the crystalline silicon cells [1]. Thus, a variety of researches have been done

39

to decrease the temperature of PV panels in order to increase their conversion efficiency.

40

During the past few years there have been significant technological advancements concerning all

41

types of PV/T collectors[2-4] and some commercial products start to show up in the market. At present,

42

four types of PV/T collectors that focus on working media: air[5], water[6], refrigerant[7] and heat

43

pipe[8], are available.

44

The sheet and tube PV/T collector working with water is the most common type. Many researches

45

concerning the performance analysis have been reported[9]. Steady state[10] or quasi dynamic

46

models[11, 12] have been developed to evaluate the static performance for PV/T collectors. In these

47

studies, the PV/T collector is assumed to operate under steady conditions. However, the weather

48

conditions as well as boundary conditions vary frequently with time in real application. In this case, a

49

dynamic model is necessary for practical performance analysis.

50

T. T. Chow[13] presented an explicit dynamic model of a glazed PV/T collector. The temperature

51

of each layer is assumed to be uniform. A number of studies based on the 1-D model has been published

52

to evaluate the dynamic performance of PV/T collector[14, 15] and for long-term simulation[16].

53

Although these studies with 1-D model provide an acceptable solution to thermo-electrical model of

54

PV/T collectors, the accuracy can be further improved with a higher dimensional model. On the other 3

ACCEPTED MANUSCRIPT 55

hand, as the temperature of each layer is assumed to be uniform, the temperature distribution in each

56

layer cannot be obtained with the 1-D model.

57

Some studies have been conducted to evaluate the performance of PV/T collector through CFD

58

simulations[17-19]. The accuracy is satisfying, but considering the meshing and calculating time, this

59

type of model is inappropriate for long-term simulation. Ilaria et al[20] presented a finite difference

60

model of glazed PV/T model to predict the dynamic performance as well as the temperature distribution.

61

Therefore, it’s possible to develop a 3-D numerical model for annual performance analysis.

62

As the demand for electricity and thermal energy shows up in the same time, the PV/T system seems

63

to be the most promising renewable energy solution without extending the installation area in residential

64

domestic hot water (DHW) system. Since the PV/T collector produces electricity and heat

65

simultaneously, the design of the DHW system with PV/T will be quite different from the solar thermal

66

systems. The design and operation of the PV/T collector and the system is a trade-off between the

67

operational temperature and the collector efficiencies. For the flat plate PV/T collectors, adding a glazed

68

cover can increase the outlet fluid temperature by reducing the heat loss from the top surface. In this way,

69

a glazed PV/T collector can achieve a higher outlet fluid temperature and thermal efficiency, which is

70

more suitable for a DHW system. However, this results in a lower electrical efficiency when compared

71

with an unglazed PV/T collector under the same condition[9]. On the other side, an increase in fluid flow

72

rate reduces the operational temperature of the PV module. This leads to a higher electrical and thermal

73

efficiency but the outlet fluid temperature will be reduced[21]. However, the increased electrical energy

74

at high flow rates (reducing average PV cell temperature) may not benefit the overall system that requires

75

relatively high temperature due to the drop in the outlet fluid temperature. Therefore, an optimum mass

76

flow rate needs to consider usable outlet fluid temperature and energy extraction[22]. Accordingly, 4

ACCEPTED MANUSCRIPT 77

operation and design of the PV/T collectors and systems must be optimized for the specific purposes of

78

DHW application scenario. The optimal design for PV/T DHW system is rarely reported. Vera et al[23]

79

presented a multi objective optimization model on the PV/T DHW system based on the 1-D thermal-

80

electrical model. The efficiencies and invest are optimized with different size parameters.

81

As a summary, the main contribution of this study is as follows:

82

(1) Four comprehensive thermal and electrical models including unglazed PV/T, glazed PV/T, PV

83

and flat plate thermal collector are established, which are suitable for accurate long-term simulation and

84

optimization. In addition, the flowchart of solving procedures is also presented.

85

(2) The detailed 2-D temperature distributions of the glazed and unglazed PV/T collector are

86

illustrated and compared for the first time under typical operating conditions. Besides, the dynamic

87

response characteristics of different PV/T collectors are discussed and compared with the traditional flat

88

plate solar thermal collectors.

89

(3) The performances of the four collectors are evaluated under a broad combination of the operating

90

conditions. The influences of different operating conditions (flow rate, temperatures, radiation and wind

91

speed) are also discussed. This kind of detailed and comprehensive comparison of solar utilization

92

technologies for residential application has not been reported.

93

(4) A multi-objective optimization model coupling TRNSYS and NSGA-II tool is established to

94

study and optimize the PV/T DHW system for a complete year. The Pareto frontier of conflicting

95

objectives (life cycle savings and prime energy efficiency) is obtained for the optimal system design.

96

This work has not been reported in the previous literatures.

97

2. Configuration of PV/T collector 5

ACCEPTED MANUSCRIPT 98

The configurations of the four different solar systems presented in this study are based on a

99

monocrystalline photovoltaic module MS-M300 and a flat plate solar collector. The configuration of the

100

PV/T collector is illustrated in Fig. 1. As is shown in Fig. 2, the constituent layers of the collectors are

101

as follows:

102

(1). Glass cover (glazing for covered PV/T and flat plate solar collector).

103

(2). Air gap in covered PV/T and solar collector.

104

(3). PV module MS-M300, consisting of the following elements：

105



Protective PV glass.

106



First polymer layer EVA (Ethylene-Vynil-Acetate).

107



PV cell.

108



Second polymer layer EVA.

109



Protective layer of Tedlar.

110

(4). Adhesive layer between the PV module and thermal absorber

111

(5). Thermal absorber.

112

(6). Pipe and heat transfer fluid.

113

(7). Insulation layer.

114

The main parameters are shown in Table 1 and the optical and thermal properties of different layers

115

are listed in Table 2. In a flat plate solar thermal collector, a selective coating on the absorber is necessary

116

to improve the absorptivity of solar radiation. Besides, the selective coating will also reduce the radiative

117

loss of the absorber due to the low emissivity. The emissivity and the absorptivity of absorber in Table 2 6

ACCEPTED MANUSCRIPT 118

are the parameters for flat plate solar thermal collector. However, in a PV/T collector, the PV module is

119

directly encapsulated on the front side of the absorber. In this case, the solar radiation is firstly absorbed

120

and converted into electricity and heat by the PV module. Then, part of the heat is transferred to the

121

absorber by heat conduction through the adhesive layer[24]. As the solar radiation can’t reach the

122

absorber directly, the selective coating in PV/T collector is no longer required. Therefore, the extra

123

material and processing of the selective coating on the absorber is saved.

124 125

Fig. 1 Configuration of studied PV/T collector

126 127

Fig. 2 Detailed layers of the studied collector

128 129 7

ACCEPTED MANUSCRIPT 130

Table 1 Main parameters of PV/T collector Parameters

Value

Collector area (m2)

2 (Length: 2m&Width: 1m)

Nominal efficiency of PV panel (%)

15

Tube spacing(m)

0.1

Tube diameter(m)

0.01

Absorber thickness

0.0003

Number of glass cover

0/1

Glass cover spacing (m)

0.025

Insulation thickness (m)

0.035

Temperature coefficient of PV cell efficiency(1/oC)[25]

-0.0045

131 132

Table 2 Optical and thermal properties of the different layers

Layer

Glass cover/PV glass

EVA

PV cell

Tedlar

Adhesive

Absorber

Parameter

Value

Unit

g

0.01

-

g

0.9

-

g

0.9

-

Cg

790

J/kg K

kg

1.1

W/m k

kEVA

0.35

W/m k

CEVA

2090

J/kg K

 PV

0.9

-

k PV

148

W/m k

CPV

677

J/kg K

kted

0.167

W/m k

Cted

1250

J/kg K

kadh

0.35

W/m k

 abs

0.95

-

 abs

0.05

-

kabs

237

W/m k

Cabs

880

J/kg K

8

ACCEPTED MANUSCRIPT

Insulation

kins

0.025

W/m k

Cins

1670

J/kg K

133 134

3. Mathematical model

135

3.1. Heat transfer model

136

In order to simulate the dynamic thermal performance of PV/T collector, solar thermal collector and

137

PV panel, four dynamic models are established. Considering the thickness of each layer, heat transfer

138

models with different dimensions are chosen. The models are based on the following assumptions:

139

（1） The thermal and optical properties of different layer materials are constant, because the

140

properties (such as conductivity, emissivity and thermal capacity, etc.) barely change within the range of

141

operating conditions[26].

142

（2） The thermal losses from the edge are neglected.

143

（3） The cooling water is distributed equally in the pipes.

144

（4） The boundary conditions of radiation, temperature and wind speed for simulation are uniform.

145

（5） The sky is treated as a blackbody.

146

3. 1.1. Unglazed PV/T component

147

For the dynamic model of PV/T and flat plate collector, as is shown in Fig. 3, the glass cover, PV

148

module and absorber are discretized in x, y direction and the insulation layer is discretized in three

149

dimensions due to the considerable thickness. A group of difference equations based on the discretization

150

is established in order to solve the partial differential energy balance equations numerically in Matlab.

9

ACCEPTED MANUSCRIPT

151 152 153

Fig. 3 Discretization of the heat transfer model for PV/T and flat plate solar collector

The heat transfer and thermal resistance network of the unglazed PV/T collector is shown in Fig.

154

4. The different types of thermal resistances are also marked with corresponding signs. The heat losses

155

of the collector mainly consist of two parts: the radiative loss to the sky and the convective loss to the

156

ambient environment. The solar radiation (G) is firstly absorbed by the PV cell and then converted into

157

heat and electricity (Eele). The heat is transferred through different layers by conduction. The useful

158

heat (Q) is collected by convection of the working fluid in the tubes. The heat resistances in Fig.4 are

159

the reciprocals of the corresponding heat transfer coefficients in the heat transfer models. The detailed

160

calculations of the heat transfer coefficients are listed in section 3.2.

10

ACCEPTED MANUSCRIPT

161 162 163 164 165

Fig. 4 Thermal resistance network of studied PV/T module

With the thermal resistance network presented in Fig.4, the energy balance equations of the unglazed PV/T module are expressed in the following equations[27]:

For the PV glass: Cg  g  g

Tg t

  g G  hv , g  a (Tg  Ta )  hr , g  sky (Tg  Tsky )  hc , g  EVA,1 (Tg  TEVA,1 )  kg g (

166

 2Tg x 2



 2Tg y 2

)

For the first EVA layer:

CEVA,1 EVA,1  EVA,1

TEVA,1 t

 hc , g  EVA,1 (Tg  TEVA,1 )  hc , EVA,1 PV (TEVA,1  TPV )  k EVA,1 EVA,1 (

167

(1)

 2TEVA,1 x 2



 2TEVA,1 y 2

For the PV cell:

11

)

(2)

ACCEPTED MANUSCRIPT CPV  PV  PV

168

TPV   g  PV G  hc , EVA,1 PV (Teva ,1  TPV )  hc , PV  EVA,2 (TPV  TEVA,2 ) t  2T  2T  k PV  PV ( PV  PV )  Eele 2 x y 2

Where Eele is calculated with PV efficiency: 𝐸𝑒𝑙𝑒 = 𝐺𝜂𝑃𝑉 = 𝐺𝜂𝑟𝑒𝑓[1 ‒ 𝛽(𝑇𝑃𝑉 ‒ 𝑇𝑟𝑒𝑓)]

169

(4)

For the second EVA layer:

CEVA,2 EVA,2  EVA,2

170

TEVA,2  hc, PV  EVA,2 (TPV  TEVA,2 )  hc, EVA,2ted (TEVA,2  Tted ) t  2T ,2  2TEVA,2  kEVA,2 EVA,2 ( EVA  ) x 2 y 2

(5)

For the tedlar layer:

Cted  ted ted

171

Tted  hc , EVA,2 ted (TEVA,2  Tted )  hc ,ted  abs (Tted  Tabs ) t  2T  2Tted  kted  ted ( ted  ) x 2 y 2

(6)

For the absorber:

Cabs abs  abs

172

Tabs  hc ,ted  abs (Tted  Tabs )  hc , abs ins (Tabs  Tins )  hc , abs t (Tabs  Tt ) t  2T  2Tabs  kabs abs ( abs  ) x 2 y 2

(7)

For the tube

 ( Do2  Di2 ) 4

173

Cf  f

Tt   hc , abs t Do (Tabs  Tt )  hv ,t  f  Di (Tt  T f )  hc ,t ins (1  ) Do (Tt  Tins ) t 2  ( Do2  Di2 )  2Tt  kt 4 y 2

(8)

For the heat transfer fluid

 Di2 4

174

(3)

Cf  f

T f t

 hv ,t  f  Di (Tt  T f ) 

 Di2 4

vf Cf  f

For the insulation layer:

12

T f dy

(9)

ACCEPTED MANUSCRIPT Tins  2T  2Tins  2Tins  kins ( ins   ) t x 2 y 2 z 2

Cins ins

175

Instant electrical efficiency

Eele,total

ele  176

178

C f m f (T f , o  T f ,i )

For the glass cover: Tc   g G  hv ,c  a (Tc  Ta )  hr ,c  sky (Tc  Tsky )  hv,c  g (Tc  Tg ) t  2T  2T  hr,c  g (Tc  Tg )  kc c ( 2c  2c ) x y

Tg t

  g  g G  hv,c  g (Tc  Tg )  hr,c  g (Tc  Tg )  hc , g  EVA,1 (Tg  TEVA,1 )  kg g (

 2Tg x

2



 2Tg y 2

which is listed in section 3.1.1.

182

3. 1.3. Flat plate thermal collector

)

For the glass cover:

Cc c  c

184

(14)

The heat balance equations for the rest of the layers are the same with the unglazed PV/T module,

181

183

(13)

For the PV glass: Cg  g  g

180

(12)

AG

3. 1.2. Glazed PV/T component

Cc c c

179

(11)

AG

Instant thermal efficiency

th  177

(10)

Tc   g G  hv ,c  a (Tc  Ta )  hr ,c  sky (Tc  Tsky )  hv,c  abs (Tc  Tabs )  hr,c  abs (Tc  Tabs ) t  2T  2T  kc c ( 2c  2c ) x y

For the absorber:

13

(15)

ACCEPTED MANUSCRIPT

Cabs abs  abs

185

Tabs   g  abs G  hv,c  g (Tc  Tabs )  hr,c  abs (Tc  Tabs )  hc , abs  ins (Tabs  Tins )  hc , abs  t (Tabs  Tt ) t  2T  2Tabs  kabs abs ( abs  ) 2 x y 2

(16)

The heat balance equations for the rest of the layers are the same with the unglazed PV/T module,

186

which is listed in section 3.1.1.

187

3.1.4. PV module

188

As the area of the edge is much smaller than the area of aperture of the PV module, the boundary

189

effect can be neglected. In this study，a 1D lumped parameters model is developed to simulate the

190

thermal behavior of the PV module. The structure of the five layers is shown in Fig. 2. Each of the five

191

layers is treated as isothermal and the temperature of the center of layer is assumed to represent the

192

average temperature of the layer.

193

For the PV glass:

Cg  g  g

194

dTg dt

  g G  hv , g  a (Tg  Ta )  hr , g  sky (Tg  Tsky )  hc , g  EVA,1 (Tg  TEVA,1 )

For the first EVA layer:

CEVA,1 EVA,1  EVA,1

195

dTEVA,1 dt

 hc , g  EVA,1 (Tg  TEVA,1 )  hc , EVA,1 PV (TEVA,1  TPV )

dTPV   g  PV G  hc ,eva ,1 PV (Teva ,1  TPV )  hc , PV  EVA,2 (TPV  TEVA,2 )  Eele dt

(19)

The electricity generated by the PV cell is evaluated by the following equation:

Eele  Gref [1   (TPV  Tref )] 197

(18)

For the PV cell:

CPV  PV  PV 196

(17)

(20)

For the second EVA layer:

CEVA,2 EVA,2  EVA,2

dTEVA,2 dt

 hc , PV  EVA,2 (TPV  TEVA,2 )  hc , EVA,2 ted (TEVA,2  Tted ) 14

(21)

ACCEPTED MANUSCRIPT 198

For the tedlar layer:

Cted  ted ted

dTted  hc , EVA,2 ted (TEVA,2  Tted )  hv ,ted  a (Tted  Ta ) dt

(22)

199 200

3.2. Heat transfer coefficients

201

3.2.1. Conductive coefficients

202 203

Because the temperature of each discretized node is assumed to be uniform and is represented by the temperature of the center, the conductive coefficient of the adjacent layer is equal to:

hc ,i  j 

204 205 206

1

(23)

i  j  2 ki 2 k j

3.2.2. Convective coefficients The coefficient of convective heat transfer to the ambient due to wind is estimated by the following equation[26]:

hv ,i  a  2.8  3vwind 207 208

The coefficient of convection heat transfer through the air gap under the glass cover is estimated by the following equation[26]:

hv,c  g 

209 210 211

(24)

  1/3 1.6      kair  1  1.44 1  1708(sin1.8 )  1  1708    Ra cos    1     air  Ra cos       Ra cos    5830  

(25)

This equation is valid for tilt angle ranging from 0o to 75o and the terms with sign (+) mean that only positive values are taken into account.

Convective heat transfer coefficient of fluid inside the tube is given by the following equation:

15

ACCEPTED MANUSCRIPT hv ,t  f  Nu f

212

kf

(26)

Di

Nusselt number for laminar flow is given by the following equation[28]:

Nu f  4.36 213 214 215

(27)

3.2.3. Radiative coefficients The radiative heat transfer coefficient between the glass cover and sky is determined by the following equation[26]: 2 hr , g  sky   g (Tg2  Tsky )(Tg  Tsky )

216 217

(28)

Where  g is the emissivity of the glass and

219

(29)

The radiative heat transfer coefficient between two parallel flat plates is determined by the following equation[26]:

hr ,i  j 

 (Ti 2  T j2 )(Ti  T j ) 1

i 220

is the Stefan Boltzmann constant. The sky

temperature is evaluated by the relation[29]:

Tsky  0.0552Ta1.5 218





1

j

(30)

1

3.3. Numerical solution procedure

221

The dynamic models of the four different collectors are discretized and solved based on an implicit

222

scheme. A flowchart diagram of the numerical solution procedure is presented in Fig. 5. At the beginning

223

of the simulation, the system parameters are read and the initial temperature distribution is set. The time-

224

varying variables (radiation, ambient temperature, fluid inlet temperature and wind speed) are taken as

225

the boundary conditions to solve the model numerically. Since the dynamic models are discretized and

16

ACCEPTED MANUSCRIPT 226

solved in an implicit form, iterations are required in each time step until the maximum residual is less

227

than 10-6. The computations are continued until the end of simulation time.

228 229

Fig. 5 Numerical solution procedure of the dynamic models

230

4. Performance analysis

231

4.1. Model validation

232

In order to validate the proposed model, a PV/T solar collector was fabricated and tested, as is shown

233

in Fig.6. The schematic diagram of the experiment unit is shown in Fig. 7. In order to obtain the efficiency

234

curves, a thermostatic water bath was applied to maintain the inlet temperature of the collector at several

235

controlled levels. The temperatures of the system were measured by a group of PT1000 sensors. The

236

water flow rate was measured by an electromagnetic flow meter. The ambient condition was monitored 17

ACCEPTED MANUSCRIPT 237

by a micro weather station (Davis 6152c) and Kipp & Zone CM22 pyranometer. The power generation

238

of the PV module was monitored by a DC voltage and current sensor. All the data was captured and

239

handled by a Graphtec GL800 acquision system. The accuracies of the sensors are listed in Table 3. By

240

means of equation (11) and (12), the electrical and thermal efficiencies under different working

241

conditions were obtained with the captured data.

242 243

Fig. 6 Picture of the main experiment components

244

18

ACCEPTED MANUSCRIPT

245 246

Fig. 7 Schematic diagram of experiment unit

247 248

Table 3 Components and transducer accuracy Component

Type

Accuracy

Temperature sensor

PT 1000 thermal resistance

Temperature:±0.1oC Ambient temperature:±0.5oC

Davis 6152c/ Kipp & Zone Wind speed:±0.1m/s

Weather station and pyranometer CM22

Radiation: ±5% of the reading KROHNE electromagnetic Flow meter

±0.5% of the reading flow meter

Voltage and current sensor

DC

±5% of the reading

Thermostatic water bath

TZL-1010D

Water temperature:±0.1oC

249 250 19

ACCEPTED MANUSCRIPT 251

The comparison of electrical efficiency of PVT collector is shown in Fig. 8. The simulated result

252

shows a good agreement with the experiment value. The maximum relative error (RE) of the electrical

253

and thermal efficiency is about 8.7% and 9.1%, respectively. Another point worth noting is that the

254

thermal efficiency of PV/T module is significantly lower than the flat plate solar collector. This is due to

255

two main reasons: (1) Part of the solar radiation is converted into electricity, (2) The silicon pellet is not

256

a selective layer for solar radiation so the emissivity is high. The PV/T collector losses partial thermal

257

efficiency but can generate heat and electricity simultaneously, which makes an efficient synthetically

258

utilization of solar energy. Experimental thermal efficiency Linear (Experimental thermal efficiency) 15%

40% 35%

Electrical efficiency

Thermal efficiency

45%

Simulated thermal efficiency Linear (Simulated thermal efficiency)

10%

30% 25% 5%

20% 15% 10%

0% 0

0.005

0.01

0.015

0.02

0.025

0.03

(Tin-Tamb)/I (m2·oC/W) (Tin-Tamb)/I (m2·oC/W)

259 260 261

Fig. 8 Comparison of efficiencies between experiment and simulation（PVT） 4.2. Temperature distribution of PV module

262

The temperature distribution on the PV module has a direct influence on the electrical and thermal

263

performance of the PV/T collector. The temperature distributions of the PV module in unglazed and

264

glazed PV/T collector are shown in Fig. 9. Due to the assumption of the model, the temperature is

265

axisymmetric in the x direction. The maximum temperature appears at middle of the two pipe outlets. 20

ACCEPTED MANUSCRIPT 266

The boundary conditions are as follows: radiation intensity: 1000W/m2, inlet water temperature:25oC,

267

ambient temperature: 25oC, wind velocity: 2m/s and mass flow rate: 0.04kg/s.

268 269

Fig. 9 Temperature distribution of PV module

270

As is shown in Fig. 9, the uniformity of temperature distribution in unglazed PV/T is better than the

271

glazed PV/T collector. The heat loss from the front surface is significantly reduced due to the additional

272

glass cover in glazed PV/T collector. Therefore, a higher average temperature of PV module is achieved

273

which leads to a lower electrical efficiency and a higher thermal efficiency in Table 4. By means of model

274

listed in section 3.1.4, the temperature of standalone PV module is calculated to be 52.7 oC, which is

275

higher than that of the PV/T modules.

276 277 278

21

ACCEPTED MANUSCRIPT 279

Table 4 Performance comparison of different collectors

　

Flow rate (kg/s)

Outlet water

PV module

Thermal

Electrical

temperature

temperature

efficiency

efficiency

(oC)

(oC)

(%)

(%)

Unglazed PV/T

0.04

30.2

36.8

43.9%

14.2%

Glazed PV/T

0.04

31.1

38.7

51.3%

12.7%

0.04

33.6

-

72.6%

-

-

-

52.7

-

13.1%

Flat plate collector PV Operating conditions:

G=1000W/m2,

Tf,i=25oC,

Ta=25oC,

vwind=2m/s

280 281

4.3. Dynamic response characteristics

282

In practical applications of PV/T collectors, the boundary conditions, especially the solar radiation,

283

change frequently with time. The response of the system to a time-varying parameter is an important

284

characteristic for long-time operation purpose. Time constant is usually applied in solar collector tests to

285

show the dynamic response to a change of radiation intensity. Time constant shown in Equation (31)

286

means the elapsed time to reach a ΔT of 63.2%(1-1/e) of the final steady value from the original

287

condition.

c 

T f , o (t )  T f , o (0)

(31)

T f , o ()  T f , o (0)

288

By applying a sudden change of radiation on the collector, the variation process of the outlet

289

temperature is obtained, as is shown in Fig. 10. The radiation changes from 0 to 1000W/m2 at 100s and

290

keeps for 1000s until the end of simulation. The outlet temperature variation of PV/T module is

291

significantly slower than the flat plate solar thermal collector. This can be explained in two aspects: (1)

292

Due to the integration of PV module in the PV/T collector, the overall thermal capacity is increased,

293

compared to the flat plate solar collector. (2) The PV module in the PV/T collector acts as an additional

294

thermal resistance, which reduces the heat conduction speed significantly. Because of the glass cover, 22

ACCEPTED MANUSCRIPT 295

the heat loss coefficient from the top is reduced dramatically which results in a higher overall thermal

296

efficiency in glazed PV/T collector.

297

40

1200

35

900

30

600

25

300

20

Glazed PV/T

Flat plate collector

Solar radiation

Solar radiation (W/m2)

Outlet water temperature (oC)

Unglazed PV/T

0 0

100

200

300

400

500

600

700

800

900

1000

1100

Time (s) Time (s)

298 299

Fig. 10 Outlet temperature response to a step radiation change

300

The value of time constants for the PV/T and flat plate solar collector are shown in Table 5. It’s

301

obvious that when the PV module is integrated in the flat plate collector, the time constant is significantly

302

increased due to the additional thermal resistance and thermal capacitance. Since the silicon pellet is not

303

a selective layer for solar radiation and the emissivity is high, the PV/T collector losses partial thermal

304

energy compared with the flat plate collector. Besides, part of the solar energy is converted into

305

electricity.

306

Table 5 Performance comparison of dynamic response characteristics Time constant

Heat collected

Electricity generated

(s)

(kJ)

(kJ)

Unglazed PV/T

151.4

737.1

286.3

Glazed PV/T

181.5

820.4

256.3

　

23

ACCEPTED MANUSCRIPT Flat plate collector

44.8

1372.5

-

Operating conditions: Tf,i=25oC, Ta=25oC, vwind=2m/s, t=1100s, G changes from 0 to 1000 W/m2 when t=100s.

307 308

4.4. Thermal and electrical efficiency

309

As is shown in Fig.11 the electrical efficiency of both unglazed and glazed PV/T collector increases

310

rapidly when the mass flow rate is lower than 0.01kg/s. In this stage, the mass flow rate is dominating

311

the heat transfer process and temperature distribution. With a higher radiation intensity, a higher flow

312

rate is required to obtain the maximum electrical efficiency. When the mass flow rate reaches 0.04kg/s,

313

the variation of electrical efficiency all becomes relatively flat.

314

The comparison of electrical efficiency of PV/T and PV module is listed in Table 6. Compared with

315

PV module, the electrical efficiency of unglazed PV/T module is increased due to the enhanced heat

316

transfer to the fluid. There are two reasons for the difference of electrical efficiency between unglazed

317

and glazed PV/T modules: (1) The additional glazing reflects part of the solar radiation on the surface.

318

(2) The heat loss coefficient of the front surface with glazing is lower, so the PV temperature is increased,

319

which corresponds to a lower PV efficiency.

320

Table 6 Electrical efficiency comparison of PV/T and PV

　

Flow rate (kg/s)

Electrical efficiency (%) G=200

G=400

G=600

G=800

G=1000

W/m2

W/m2

W/m2

W/m2

W/m2

Unglazed PV/T

0.04

14.9%

14.7%

14.6%

14.4%

14.2%

Glazed PV/T

0.04

13.4%

13.2%

13.0%

12.8%

12.7%

PV module

-

14.8%

14.4%

14.0%

13.6%

13.1%

Operating conditions:

Tf,i=25oC,

Ta=25oC,

vwind=2m/s

321

24

ACCEPTED MANUSCRIPT G=200W/m²

G=400W/m²

G=600W/m²

G=800W/m²

G=1000W/m²

15.5% For unglazed PV/T

Electrical efficiency

14.5% 13.5% 12.5% 11.5%

For glazed PV/T

10.5% 9.5% 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Mass flow rate (kg/s)

322 323

Fig. 11 Variation of electrical efficiency with different mass flow rate and radiation

324

As is shown in Fig.12, the thermal efficiency of both unglazed and glazed PV/T collector increases

325

rapidly when the mass flow rate is lower than 0.04kg/s. The growth rate of thermal efficiency under

326

different radiation intensity is different from the electrical efficiency. When the mass flow rate reaches

327

0.04kg/s, the variation of thermal efficiency also becomes relatively flat. The mass flow rate of 0.04kg/s

328

in this case is consistent with the empirical value of about 0.02kg/s•m2 in solar thermal system design. G=200W/m²

G=400W/m²

G=600W/m²

G=800W/m²

G=1000W/m²

60.0%

Thermal efficiency

50.0% For glazed PV/T 40.0% 30.0% 20.0% 10.0% For unglazed PV/T 0.0% 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

-10.0% Mass flow rate (kg/s)

329 330 331

Fig. 12 Variation of thermal efficiency with different mass flow rate and radiation In order to compare the performance of the four collectors comprehensively, both thermal and 25

ACCEPTED MANUSCRIPT 332

electrical efficiencies are evaluated under a broad combination of operation conditions. As is shown in

333

Table 7, a high and low level of each parameter are chosen and the possible combinations are listed. In

334

order to compare the efficiencies more intuitively in each column, two kinds of bars are added in the

335

table. The yellow bars represent the thermal efficiency and the green bars represent the electrical

336

efficiency. The abnormal values (>100% or <0%) of thermal efficiencies are marked red. The following

337

main conclusions can be draw from Table 7:

338

(1) In most cases, the thermal efficiency of the glazed PV/T collector is higher than the unglazed

339

PV/T collector except when the ambient temperature is high and the fluid inlet temperature is low (e.g.

340

case No. 7 and No. 8, when Tf,i=10oC). The flat plate collector achieves the highest thermal efficiency

341

for all of the cases.

342

(2) The electrical efficiency of the glazed PV/T collector is lower than the unglazed PV/T collector

343

for all of the 8 cases. The difference of electrical efficiency between the unglazed PV/T and PV module

344

depends on the operating conditions. When the working fluid temperature is low enough to cool down

345

the PV module in the unglazed PV/T collector to a lower temperature level than the stand-alone PV

346

module, the electrical efficiency benefits from the configuration of unglazed PV/collector (e.g. case No.

347

7). Otherwise, under the influence of working fluid and the insulation layer, the temperature of the PV

348

module in the unglazed PV/T collector will be higher than the stand-alone PV module. In this way, the

349

electrical efficiency of unglazed PV/T will be lower (e.g. case No. 1 and No. 2), compared with the stand-

350

alone PV module.

351

(3) For case No. 1 and No. 2: The thermal efficiency is negative under such conditions: low

352

radiation, low ambient temperature and high fluid inlet temperature. In this case, the temperature of the

353

absorber or PV module is higher than the ambient temperature, so the heat is extracted from the working 26

ACCEPTED MANUSCRIPT 354

fluid to the ambient environment. In a real DHW system, the controller will stop the working fluid

355

circulation under this kind of conditions.

356

(4) For case No. 3 and No. 4: The thermal efficiency is higher than 100% only under such conditions:

357

low radiation, high ambient temperature and low fluid inlet temperature. In this case, the temperature of

358

the absorber or PV module is lower than the ambient temperature, therefore extra heat is absorbed from

359

the environment. This usually happens at the beginning of the domestic water heating process, when the

360

replenished water temperature is lower than the ambient temperature and the radiation is poor.

361

(5) For case No. 5 No. 6, No. 7 and No. 8: The performance of the unglazed PV/T collector and

362

stand-alone PV module is significantly influenced by the wind speed and ambient temperature, due to

363

high heat loss coefficient from the top surface.

364

27

ACCEPTED MANUSCRIPT 365

Table 7 Performance comparison under various operating conditions

366 367

5. Multi-objective optimization of PV/T DHW system design

368

In practical design of domestic how water (DHW) system, both energetic and economic benefits are

369

the main concerns. The required temperature for DHW is normally higher than 40oC. The thermal

370

efficiency of unglazed and glazed PV/T collector with different water inlet temperature is shown in Fig.

371

13. The boundary conditions are as follows: radiation intensity: 1000W/m2, ambient temperature: 25oC,

372

wind velocity: 2m/s and mass flow rate: 0.04kg/s. It’s obvious that the thermal efficiency of glazed PV/T

373

is significantly higher than the unglazed one’s with a higher water temperature. In this way, the glazed

28

ACCEPTED MANUSCRIPT 374

PV/T collector is more suitable for DHW systems. The energetic and economic optimization of DHW

375

system with PV/T in this section is based on the glazed PV/T collector. Uglazed PV/T

Glazed PV/T

60.0%

THermal efficiency

50.0% 40.0% 30.0% 20.0% 10.0% Boundary conditions:G=1000W/m2 Ta=25oC

0.0% 20

25

30

35

40

45

50

55

Water inlet temperature (oC)

376 377 378

Fig. 13 Variation of thermal efficiency with different water inlet temperature 5.1. Multi-objective optimization model

379

Multi-objective optimization is an effective approach to solve the practical engineering problem

380

with conflicting objectives which need to be considered simultaneously. A set of non-dominating

381

solutions, known as the Pareto set, can be obtained to show the best possible trade-offs between the

382

conflicting objectives.

383

TRNSYS is a powerful tool for energy system analysis, especially in the field of solar application.

384

However, there is no optimization tool provided in TRNSYS to perform a multi-objective optimization.

385

On the other hand, NSGA II optimization tool programmed in Matlab has been proved to be a robust

386

and efficient tool for multi-objective optimization process[30]. As is shown in Fig. 14, the procedures

387

of the NSGA-II algorithm are as follows[31]:

388

(1) A random parent population is generated firstly and the objectives are evaluated.

29

ACCEPTED MANUSCRIPT 389

(2) The individuals are sorted based on their fitness (values of objectives).

390

(3) Then, some individuals are selected based on tournament selection to produce new offspring for

391 392 393 394 395 396

a new population. The new offspring is produced by crossover and mutation operator.

(4) The objectives of the new population are evaluated. After that, the old and new population are merged into a large total population.

(5) The total population is sorted by rank and crowding distance. The individuals with better fitness values are selected by the elitist sorting and form the new parent population.

The procedures from (3) to (5) are repeated until the maximum generation number is reached.

397

30

ACCEPTED MANUSCRIPT

398 399

Fig. 14 Flow chart of NSGA-II algorithm.

400

Thus, a co-simulation system with TRNSYS and Matlab is established in this study. As is shown in

401

Fig. 15, the flowchart contains two main engines: the Matlab engine for optimization and TRNSYS

402

engine for annual energy performance analysis[32]. A DHW system is established in TRNSYS with the

403

component listed in the right of Fig.15. The detailed explanations of the components and parameters

404

are presented in Table 8. The consumption profile of DHW is shown in Fig. 16, which is the typical

405

schedule for a 3-people family in Shanghai[33]. The weather data of typical meteorological year (CN-

406

Shanghai-583670.tm2) is supplied to the model for boundary conditions.

31

ACCEPTED MANUSCRIPT 407

The optimization process is conducted as follows:

408

(1) A group of user-defined initial parameters are transferred to TRNSYS to perform the annual

409 410 411 412 413 414 415

performance analysis first.

(2) The annual performance output data is returned to the Matlab engine to calculate the energetic and economic objectives.

(3) The value of the two objectives are handled by the NSGA II tool, the fitness is evaluated and the optimal value of design variables is chosen and updated.

(4) If the stop criteria are not satisfied, the design data will be updated and transferred to TRNSYS again to perform another annual performance analysis.

416

The optimization is started with 24 individuals and is carried out with 60 generations. The steps

417

from (2) to (4) will be repeated until the last generation. In this way, total 1440 evaluations are evaluated

418

and sorted to find the pareto frontier.

32

ACCEPTED MANUSCRIPT

419 420

Fig. 15 Flowchart of the proposed multi-objective optimization

421 422

Table 8 Main components used in the TRNSYS simulation model Component

Type

Main Parameters 2m2 glazed PV/T collector, model is coded and solved

Glazed PV/T collector

Type 155 in Matlab environment

Storage tank

Type 4e

Volume range: 0.05m3~0.3m3

Circulation pump

Type 3d

Flow rate range: 0.001kg/s~0.1kg/s

Differential controller

Type 2b

temperature difference: 8(on)/4(off)

Auxiliary heater

Type 700

Power: 3kW

Diverter and mixer

Type 11b and 11h

Controlled by the temperature of hot water

Water draw

Type 14b

The consumption profile of DHW is shown in Fig. 16

423 33

ACCEPTED MANUSCRIPT 45

DHW consumption (L/h)

40 35 30 25 20 15 10 5 0 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (h)

424 425 426

Fig. 16 Schedule of DHW consumption[33] 5.2. Objectives

427

5.2.1. Prime energy saving efficiency

428

As is presented in introduction and section 4, the mass flow rate has a complicated effect on both

429

thermal and electrical efficiency. Considering that electricity is a high-grade energy form, prime energy

430

efficiency saving efficiency is proposed as a comprehensive indicator of the PV/T collector’s overall

431

performance[34]. The definition of prime energy saving efficiency is as follows:

 prime  ele /  power  th

(32)

432

Where ηpower is the electrical power generation efficiency of the conventional power plant, which

433

is taken as 38%[34]. The indicator of the primary energy-saving efficiency considers the quality as well

434

as the quantity of the energy that the PV/T system converts solar energy into.

435

5.2.2. Life cycle savings

436

Life cycle savings (LCS) (net present worth) is defined as the difference between the life cycle cost

437

of a conventional fuel-only system and the life cycle cost of the solar plus auxiliary energy system. The 34

ACCEPTED MANUSCRIPT 438

present worth factor is a conversion factor that brings the future cost into the present time. It’s a function

439

of lifetime, discount rate (rd) and inflation rate (ri). The expression is as follows[35]:  1   1  r N  i 1   (1  ri ) j 1     if ri  rd PWF ( N , ri , rd )    r  r 1  r   d i  d    j  j 1 (1  rd )  if ri  rd   N / (1  ri )

(33)

N

440

In this way, the LCS is expressed as follows[35]: LCS  Qload I fuel PWF ( N , ri , rd )   I sys  ( I O & M  I aux ) PWF ( N , ri , rd ) 

441 442

(34)

The main parameters for the economic analysis and capital cost are shown in Table 9 and Table 10. Table 9 Main parameters for the economic analysis Parameter

Description

Value

Evaluation period

Expected useful life

20 years

Annual operating and maintenance cost

1% of original investment

Inflation rate

ri

2.8%

Discount rate

rd

8%

Auxiliary energy prices

Electricity

0.094$/kWh

443 444

Table 10 Capital cost of studied PV/T system Value

　 Module ($/piece) Tank

348.5

($/m3)

1212.1

Inverter ($/piece)

45.5

Pump ($/piece)

50

Temperature difference controller ($/piece)

13.6

Accessories ($/system)

34.1

Assembling cost ($/system)

15.2

445 446

5.3. Results and discussion

447

In multi-objective optimization, when the different objectives are contradictory, an optimal solution

448

is said Pareto optimal when it is not possible to improve an objective without degrading the others. A

449

Pareto optimal solution can then be seen as an optimal trade-off between the objectives. The set of all 35

ACCEPTED MANUSCRIPT 450

Pareto optimal solutions is called the Pareto frontier as it usually graphically forms a distinct front of

451

points. Solutions which do not lay on the Pareto front are called Pareto dominated solutions.

452

The process of multi-objective optimization is adopted to maximize the prime energy efficiency and

453

LCS presented in section 5.2. The results of the two objective indexes are presented in Fig. 17 and the

454

results show that the selected two objectives are conflicting. The Pareto frontier is marked and shown

455

separately in Fig. 18. The LCS is increased by a decrease of prime energy saving efficiency. The slope

456

of the trend is steep when approaching the highest value of LCS. Three representative points are marked

457

in Fig.18, which is the best energetic optimized point A, best economic optimized point B and ideal point

458

C. As is illustrated in Fig. 18, the maximum prime energy saving efficiency is 66.8% with the lowest

459

value of LSC (592.3$) at point A. On the other hand, the highest LCS is 708.7$ while the prime energy

460

saving efficiency is lowest at point B. The ideal design at point C reaches the maximum LCS and prime

461

energy saving efficiency. However, it cannot be achieved in real system design. Feasible solutions

Pareto frontier

Prime energy saving efficiency

70.0%

65.0%

60.0%

55.0%

50.0% 200.0

300.0

400.0

500.0

600.0

LCS ($)

462 463

Fig. 17 Results of the multi-objective optimization

36

700.0

800.0

ACCEPTED MANUSCRIPT Pareto frontier

Prime energy saving efficiency

66.9% 66.6%

C: Ideal design

66.3%

A: Best energetic

66.0% 65.7% 65.4% 65.1% 64.8%

B: Best economic optimiazed design

64.5% 64.2% 580

600

620

640

660

680

700

720

LCS ($)

464 465

Fig. 18 Pareto frontier of the multi-objective optimization

466

The distribution of the variables through the optimization process is presented in Fig. 19 and Fig.

467

20. The optimal designs on the Pareto frontier are also marked. The mass flow rate on the Pareto frontier

468

is between 0.0085kg/s and 0.011kg/s, which is much lower than the empirical value of 0.04kg/s for a 2

469

m2 solar thermal collector. With the analysis in section 4, the increase the mass flow rate will enhance

470

the heat transfer and reduce the average temperature of PV module. In this way, both thermal and

471

electrical efficiency are improved. However, the higher mass flow rate will reduce the time of pump

472

operation, since the system is controlled by a differential controller. In this way, the time of the collector

473

with no water flow will increase, which reduces the thermal and electrical energy collected. Besides, high

474

flow rate may not benefit the overall system that requires relatively high temperature due to the drop in

475

the outlet fluid temperature. Under the comprehensive influence, the optimal value of mass flow rate falls

476

into the range in Fig. 19.

37

ACCEPTED MANUSCRIPT Feasible designs

Pareto frontier

0.1

Mass flow rate (kg/s)

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0

200

400

600

800

1000

1200

1400

1600

Number of designs

477 478

Fig. 19 Distribution of mass flow rate through the optimization process

479

As is presented in Fig. 20, the optimal value of tank volume on Pareto frontier shows an equally

480

scattering distribution between 99.5Land 218.6Lfor a 2 m2 glazed PV/T collector. The tank size is

481

dominating the initial investment of the system at the design stage. During the annual operation, the tank

482

volume has a decisive impact on the temperature level of the system. With a small size tank, the heat

483

storage temperature is easily lifted, which has a negative effect on both thermal and electrical efficiency.

484

On the other hand, when the tank volume is big, the thermal and electrical efficiency will be improved

485

but the initial invest on the system will increase simultaneously. Furthermore, the water temperature in

486

the tank can hardly meet the DHW demand, which will cause a higher auxiliary energy consumption.

487

Under the balance of these factors, optimal value of tank volume falls into the marked range in Fig. 20.

38

ACCEPTED MANUSCRIPT Feasible solutions

Pareto frontier

0.3

Tank volume (m3)

0.25 0.2 0.15 0.1 0.05 0 0

200

400

600

800

1000

1200

1400

1600

Number of design

488 489 490

Fig. 20 Distribution of tank volume through the optimization process

6. Conclusion

491

In this study, 3-D dynamic thermal-electrical models for solar thermal collector, glazed and

492

unglazed PV/T collector are developed. The temperature distributions of PV module under different

493

operation conditions are illustrated and compared.

494

Compared with flat plate collector, the PV/T collectors shows a delay on the dynamic response due

495

to the additional thermal capacity and thermal resistance. The electrical efficiency is improved in

496

unglazed PV/T due to the enhanced heat transfer to fluid.

497

The influence of flow rate on the thermal and electrical efficiency is different with the variation of

498

solar radiation, but both of the thermal and electricity efficiency shows a flat variation when the flow rate

499

is higher than 0.04kg/s. The comprehensive performances of the four collectors are significantly

500

influenced by the operating conditions (flow rate, temperatures, radiation and wind speed).

501

Life cycle savings and prime energy saving efficiency are conflicting objectives in the multi-

502

objective optimization of glazed PV/T DHW, which comprises a Pareto frontier. The mass flow rate on 39

ACCEPTED MANUSCRIPT 503

the Pareto frontier is between 0.0085kg/s and 0.011kg/s, which is much lower than the empirical value

504

of 0.04kg/s for a 2 m2 solar thermal collector. The optimal value of tank volume on Pareto frontier shows

505

an equally scattering distribution between 99.5 Land 218.6Lfor a 2 m2 glazed PV/T collector.

506

Acknowledgement: This study is supported by the National Science Foundation (No.51476099)

507

References

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