- Email: [email protected]
T) water collector based on exergy concept
Renewable Energy 68 (2014) 356e365
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimization of a solar photovoltaic thermal (PV/T) water collector based on exergy concept F. Sobhnamayan a, F. Sarhaddi a, *, M.A. Alavi b, S. Farahat a, J. Yazdanpanahi a a b
Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2013 Accepted 30 January 2014 Available online
In this paper, the optimization of a solar photovoltaic thermal (PV/T) water collector which is based on exergy concept is carried out. Considering energy balance for different components of PV/T collector, we can obtain analytical expressions for thermal parameters (i.e. solar cells temperature, outlet water temperature, useful absorbed heat rate, average water temperature, thermal efficiency, etc.). Thermal analysis of PV/T collector depends on electrical analysis of it; therefore, five-parameter currentevoltage (I eV) model is used to obtain electrical parameters (i.e. open-circuit voltage, short-circuit current, voltage and current at the point which has maximum electrical power, electrical efficiency, etc.). In order to obtain exergy efficiency of PV/T collector we need exergy analysis as well as energy analysis. Considering exergy balance for different components of PV/T collector, we obtain the expressions which show the exergy of the different parts of PV/T collector. Some corrections have been done on the above expressions in order to obtain a modified equation for the exergy efficiency of PV/T water collector. A computer simulation program has been developed in order to obtain the amount of thermal and electrical parameters. The simulation results are in good agreement with the experimental data of previous literature. Genetic algorithm (GA) has been used to optimize the exergy efficiency of PV/T water collector. Optimum inlet water velocity and pipe diameter are 0.09 m s1, 4.8 mm, respectively. Maximum exergy efficiency is 11.36%. Finally, some parametric studies have been done in order to find the effect of climatic parameters on exergy efficiency. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Solar photovoltaic thermal (PV/T) water collector Exergy analysis Optimization
1. Introduction Nowadays using renewable energies is considered more because of fossil fuels high price and the environmental problems which are caused by them. One of an abundant renewable energy sources is solar energy. Most conventional systems which are used in order to convert solar energy in to other kinds of energies are as follows 1. Photovoltaic (PV) systems: systems which receive solar energy in order to give us electricity. 2. Solar collector systems: systems which receive solar energy in order to give heat to the agent fluid In a PV system, when the solar cells temperature increases, electrical voltage decreases. The electrical efficiency of PV system * Corresponding author. Tel.: þ98 541 2426206; fax: þ98 541 2447092. E-mail address: [email protected] (F. Sarhaddi). http://dx.doi.org/10.1016/j.renene.2014.01.048 0960-1481/Ó 2014 Elsevier Ltd. All rights reserved.
decreases as a result of decrease in electrical voltage. In a solar collector system, an external electrical source is needed to circulate the agent fluid through the system. Therefore, if we combine a PV module and a solar collector system, the electrical source can be provided for solar collector system. On the other hand, the additional heat which is absorbed from PV module causes solar cells temperature to decrease and as a result electrical efficiency increases. Combining these two systems, we can also use optimum installation space. Using a PV/T collector, we can convert solar energy in to electricity and heat. As we know, the quality of these two kinds of energies is not the same. Using exergy analysis, we can compare these two kinds of energies, which are different in their quality. Using exergy optimization, we can determine optimum operating mode of PV/T water collector. It has been shown that PV/ T collector energy payback time (EPBT) can be decreased if PV/T collector works in its optimum operating mode [1]. Experimental and theoretical research which is carried out in the field of exergy analysis of PV and PV/T systems, can be expressed briefly as follows
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
357
Thermodynamic (exergy) limitations of photovoltaic phenomenon and its conversion efficiency have been investigated in Refs. [2e7]. Exergy analysis of PV systems based on net output exergy or exergy losses have been investigated in Refs. [8e11]. Exergy analysis of PV/T air collectors has been widely investigated in Refs. [12e17]. Exergy analysis of PV/T water collectors has been carried out in Refs. [18e26]. The deficient points of the previous studies are considered as follows The exergy loss due to pressure drop in flow pipes is neglected The exergy efficiency of PV/T water collector which is used in the previous literature [18e26], has a significant error at low solar radiation intensity. In this condition, the exergy efficiency tends to the nominal electrical efficiency (hex z hel,ref) at the reference conditions (Tcell,ref ¼ 25 C, G ¼ 1000 W m2). There has not been a comprehensive optimization of PV/T water collector which is based on exergy concept regarding to design and operating parameters before the one carried out in this paper. In this paper, the following items are carried out respectively in order to optimize a solar PV/T water collector which is based on exergy concept. Energy analysis is done in order to obtain thermal and electrical parameters Exergy analysis is done in order to introduce exergy efficiency Exergetic optimization is carried out in order to determine optimum operating mode of PV/T water collector Finally, some parametric studies are done in order to find the effect of climatic parameters on exergy efficiency. Fig. 1. (a) The cross-sectional view of a PV/T water collector and (b) its thermal resistance circuit diagram on a flow duct.
2. Thermal analysis The proof of governing equations on the thermal analysis of PV/T water collector is not mentioned to have a brief note. More details of the derivation of governing equations are found in Refs. [1,24]. Fig. 1 shows the cross-sectional view of a PV/T water collector and its equivalent thermal resistant circuit diagram on a flow duct. Analytical expressions for thermal parameters and thermal efficiency of PV/T water collector are obtained by writing energy balance equation for each component of the PV/T water collector.
electrical efficiency of PV module, respectively. An expression for the solar cells temperature can be obtained from Eq. (1) as follows
i. h Tcell ¼ ðasÞeff G þ Ut Tamb þ UT Tbs ðUt þ UT Þ
(2)
where (as)eff is the product of effective absorptivity and transmissivity and it is defined as follows:
ðasÞeff ¼ sg ½ac bc þ aT ð1 bc Þ bc hel
(3)
2.2. Energy balance for the back surface of tedlar
2.1. Energy balance for glassetedlar PV module
sg ½ac bc G þ aT ð1 bc ÞGD dx ¼ ½Ut ðTcell Tamb Þ þ UT ðTcell Tbs Þ D dx þ sg bc hel GD dx (1) where Tcell, Tamb, Tbs, G, D, dx, ac, aT, bc, sg and hel are solar cells temperature, ambient temperature, back surface temperature, solar radiation intensity, absorber plate width on a flow duct, elemental length of flow duct, absorptivity of solar cell, absorptivity of tedlar, packing factor of solar cell, transmittivity of glass cover and
UT ðTcell Tbs ÞD dx ¼ hT Tbs Tf D dx
(4)
Back surface temperature of tedlar can be obtained from Eq. (4) as follows
i. h Tbs ¼ hp1 ðasÞeff G þ UtT Tamb þ hT Tf ðUtT þ hT Þ
(5)
An expression for the heat transfer rate, transferred to the back surface of tedlar, is obtained by substituting Eqs. (2) and (5) into Eq. (1).
358
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
UT ðTcell Tbs Þ ¼ hT Tbs Tf ¼ hp1 hp2 ðasÞeff G Utw Tf Tamb
Tf ¼
(6)
" ¼
2.3. Energy balance for absorber plate
ZL Tf ðxÞdx x¼0
0 3 #2 UL L 1 exp bF hp1 hp2 ðasÞeff G _ mC 5 41 0 p Tamb þ bF UL L UL
Energy balance for an elemental length of absorber plate and the flow duct under it, gives the following equation
h i ð1=NÞ dQ_ u =dx ¼ WF 0 hp1 hp2 ðasÞeff G UL Tf Tamb
1 L
þ Tf ;in
0 UL L 1 exp bF _ mC 0 p
_ p mC
bF UL L _ p mC
(7)
where UL, W, N and F 0 are overall heat transfer coefficient from PV/T collector to surrounding, distance between two adjacent flow duct, number of flow ducts and collector efficiency factor, respectively.
2.5. Thermal efficiency of PV/T water collector
2.4. Energy balance for the element of agent fluid in flow duct (Fig. 2)
_ p Tf ;out Tf ;in Q_ u ¼ mC i h ¼ FR bL hp1 hp2 ðasÞeff G UL Tf ;in Tamb
_ _ 1 m m Cp Tf Cp Tf dQ_ u dx ¼ 0 N N N xþdx x 8 < dTf þ WF 0 UL Tf Tamb ¼ _ =N mC dx :
p
WF 0 hp1 hp2 ðasÞeff G _ p =N mC
(8)
(9)
x ¼ 0/Tf ¼ Tf ;in
An expression for agent fluid temperature is obtained by solving Eq. (9) as follows
Tf ðxÞ ¼
hp1 hp2 ðasÞeff G UL bF 0 UL x exp _ p mC
!
Tamb þ þ Tf;in
bF 0 UL x 1 exp _ p mC (10)
The outlet temperature of agent fluid (Tf,out) is obtained from the above equation by considering boundary condition as Tf ¼ Tf,out, at x¼L
Tf;out
hp1 hp2 ðasÞeff G ¼ Tamb þ UL bF 0 UL L þ Tf;in exp _ p mC
!
bF 0 UL L 1 exp _ p mC (11)
where L is flow duct length and b ¼ NW is PV/T collector width. The average temperature of agent fluid along the length of a flow duct is obtained as follows
The rate of useful thermal energy of PV/T water collector is obtained as follows
FR ¼
_ p mC bF 0 UL L 1 exp _ p UL bL mC
(13)
(14)
_ and Cp are heat removal factor, mass flow rate and heat where FR, m capacity of agent fluid, respectively. Thermal efficiency of PV/T water collector is defined as follows
hth
3 2 UL Tf;in Tamb Q_ u 5 ¼ FR 4hp1 hp2 ðasÞeff ¼ G bLG
(15)
Because of the presence of the electrical efficiency of PV module (hel) in Eq. (3), thermal parameters of PV/T water collector and its electrical parameters are dependent to each other. 3. Electrical analysis In order to calculate electrical parameters and electrical efficiency of PV/T water collector five-parameter currentevoltage (Ie V) model (Fig. 3) is used as follows [10,11]
V þ IRs ðV þ IRs Þ 1 I ¼ IL Io exp Rsh a
(16)
where I and V represent current and voltage at load, a, IL, Io, Rs and Rsh are ideality factor, light current, diode reverse saturation current, series resistance and shunt resistance, respectively. In order to calculate five reference parameters (aref, IL,ref, Io,ref, Rs,ref and Rsh,ref), five pieces of information are needed at reference conditions. These five pieces of information are defined as follows [10,11]: At short-circuit current: I ¼ Isc,ref, V ¼ 0 At open-circuit voltage: I ¼ 0, V ¼ Voc,ref
Fig. 2. The element of agent fluid in flow duct.
(12)
Fig. 3. Equivalent electrical circuit in fiveeparameter photovoltaic model.
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
At the maximum power point: I ¼ Imp,ref, V ¼ Vmp,ref At the maximum power point: ½dðIV Þ=dVmp ¼ ½VdI=dV + Imp ¼ 0 At short circuit: ½dI=dVsc ¼ - 1=Rsh;ref The subscript ‘ref’ indicates the value of parameters at the reference conditions. Reference conditions or standard rating conditions (SRC) are defined as follows [10,11] Solar cells temperature at reference conditions: Tcell,ref ¼ 25 C Solar radiation intensity at reference conditions: Gref ¼ 1000 W m2. Substituting the five mentioned pieces of information into Eq. (16), following equations are obtained
"
Isc;ref Rs;ref aref
F1 ¼ Isc;ref þ IL;ref Io;ref exp
!
# 1
Isc;ref Rs;ref ¼0 Rsh;ref
IL ¼
! i G h IL;ref þ a Tcell Tcell;ref Gref
F2 ¼ IL;ref Io;ref
Voc;ref exp aref
!
"
Vmp;ref þ Imp;ref Rs;ref aref
F3 ¼ Imp;ref þ IL;ref Io;ref exp
!
#
Isc ¼ Isc;ref þ DI
(28)
Voc ¼ Voc;ref þ DV
(29)
2
Vmp;ref þImp;ref Rs;ref
Io;ref exp
6 6 F4 ¼ Vmp;ref 6 4 Io;ref Rs;ref 1þ
aref
exp
3 1 Rsh;ref
aref
Vmp;ref þImp;ref Rs;ref
R
aref
7 7 7 þ Imp;ref ¼ 0 5
s;ref þ Rsh;ref
aref
(20) 2 6 6 F5 ¼ 6 4
Io;ref exp
Isc;ref Rs;ref aref
aref
1þ
Io;ref Rs;ref exp
3 1
Rsh;ref
Isc;ref Rs;ref aref
aref
þ
Rs;ref Rsh;ref
7 1 7 ¼ 0 7þ 5 Rsh;ref
Io;ref
Tcell Tcell;ref
exp
3Nc aref
mp þImp Rs a
aV
Io;ref exp
3
mp þImp Rs a a
R1
Vmp;ref þImp;ref Rs;ref aref
aref Io;ref Rs;ref exp
sh
7 5
þRRs
sh R
Vmp;ref þImp;ref Rs;ref aref
3 1
sh;ref
aref
R
1
Tcell;ref Tcell
(30)
7 7 7 5
þR s;ref
sh;ref
(31)
where 3, Nc, a and b are semiconductor band gap energy, solar cells number in series, current temperature coefficient and voltage temperature coefficient, respectively. PV module manufacturers usually give temperature coefficients. Vmp and Imp is a point on IeV characteristic curve. The area of rectangle under the IeV characteristic curve with the length of Vmp and width of Imp should be maximum. The consumed electrical power by pump to circulate working fluid in PV/T water collector is calculated from [27]
m1:83 D0:5 1:2465 104 APV=T h3:5 f h k2:33 Cp1:17 r2 hpump
(32)
(21)
(22) !3
V
Io Rs exp
1þ
Dh ¼
a Tcell ¼ aref Tcell;ref
¼
1þ
G ¼ 2 Gref 6 6 6 4
Ppump ¼
where Voc, Vmp, Isc and Imp are open-circuit voltage, maximum power point voltage, short-circuit current and maximum power point current, respectively. Functions F1, F2, F3, F4 and F5 form a set of five nonlinear equations in five unknown variables x1 ¼ aref, x2 ¼ IL,ref, x3 ¼ Io,ref, x4 ¼ Rs,ref and x5 ¼ Rsh,ref. Since the Eqs. (17)e (21) are implicit and nonlinear, NewtoneRaphson method is used to solve them. Solving the equations set with NewtoneRaphson method gives the value of five parameters (aref, IL,ref, Io,ref, Rs,ref and Rsh,ref), at the reference conditions (Tcell,ref ¼ 25 C, Gref ¼ 1000 W m2). In order to calculate the model parameters at new climatic and operating conditions (G,Tcell), a set of translation equations is used such as follows [10,11]
Io
Io;ref exp
Rsh zRsh;ref (19)
2
(26)
(27)
1
Vmp;ref þ Imp;ref Rs;ref ¼ 0 Rsh;ref
! G 1 Isc;ref Gref
DV ¼ bDT Rs DI
6 4 (18)
(25)
! G DI ¼ a DT þ Gref
#
Voc;ref ¼ 0 1 Rsh;ref
(24)
DT ¼ Tcell Tcell;ref
(17) "
359
! (23)
4 Cross sectional area of duct 4dD ¼ Wetted perimeter 2ðd þ DÞ
(33)
The electrical efficiency of PV/T water collector is defined as the ratio of actual electrical output power to input solar energy incident rate on PV/T collector surface as follows:
hel ¼
Vmp Imp Ppump GAPV=T
(34)
In the previous equations, Dh, APV/T, d, hf, m, k, r and hpump are hydraulic diameter of flow duct, PV/T surface area, duct depth, viscosity, conductivity, density of agent fluid and pump efficiency, respectively.
4. Exergy analysis Exergy is defined as the maximum amount of work that can be produced by a system or a flow of mass or energy as it comes to equilibrium with a reference environment [28,29]. The general form of exergy balance equation for a control volume is written as [9]
360
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
X
_ ðmexÞ in
X
_ ðmexÞ out þ
X
_ _ _ Ex Q ExW ¼ I
(35)
P P P_ _ W and I_ are inlet exergy _ _ ExQ , Ex where ðmexÞ ðmexÞ in , out , rate related to inlet water flow, outlet exergy rate related to outlet water flow, inlet or outlet heat exergy rate, inlet or outlet work exergy rate and irreversibility rate in control volume, respectively.
These exergy rates are given by [28,29]
X
_ _ ðmexÞ in ¼ mCp Tf;in Tamb Tamb
Exergy efficiency of PV/T water collector is defined as the ratio of net output (desired) exergy rate to net input exergy rate [9]
P
hex ¼
P _ _ _ ðmexÞ I_ ðmexÞ out in þ ExW ¼ 1 _Ex _Ex Q ;sun
_ DPf;in Tf ;in m ln þ r Tamb
_ p Tf;out Tf;in Tamb ln TTf;out m_ Dr Pf þ Vmp Imp Ppump mC f;in hex ¼ 4 þ 13 TTamb GAPV=T 1 43 TTamb sun sun
(36) X
_ ðmexÞ out
_ DPf;out T m _ p Tf ;out Tamb Tamb ln f;out þ ¼ mC r Tamb (37)
hex
where DPf,in and DPf,out are the pressure difference between water flow and surroundings at entrance and exit of PV/T collector. 4.2. Heat exergy rate Heat exergy rate includes solar radiation intensity exergy rate _ ðEx Q ;sun Þ. According to the Petela theorem, it is given by [30,31]
(38)
where Tsun is sun temperature in Kelvin scale. 4.3. Work exergy rate Work exergy rate includes the difference between outlet electrical power of PV module and consumed electrical power by pump:
_ W ¼ Vmp Imp Ppump Ex
(39)
4.4. Irreversibility rate Irreversibility rate is the summation of exergy losses rate from control volume and exergy destructions rate in control volume
I_ ¼
(42) where DPf is pressure drop in flow pipes. In the previous studies [18e26], exergy efficiency of PV/T water collector has been calculated from following equation
n h i o amb þ hel;ref 1 bref Tcell Tamb;ref GAPV=T Ppump Q_ u 1 TTf;out ¼ 4 1 Tamb þ GAPV=T 1 43 TTamb Tsun 3 sun
" # 4 Tamb 1 Tamb 4 _Ex þ Q ;sun ¼ GAPV=T 1 3 Tsun 3 Tsun
X
_ Ex loss þ
X
_ Ex des
(41)
Q ;sun
Substituting Eqs. (36)e(39) into Eq. (41), exergy efficiency of PV/ T water collector is obtained as follows:
4.1. Inlet and outlet exergy rate related to water flow
4.5. Exergy efficiency of PV/T water collector
(40)
The amount of exergy goes out of control volume and cannot be used, is called exergy loss. On the other hand, the amount of exergy produced by for example flow friction or mixture of two kinds of liquids, etc. and does not go out of control volume is named exergy destruction [9].
(43)
This equation has some deficiencies; first, it does not include the exergy loss due to pressure drop in flow pipes. Second, it has a significant error at low solar radiation intensity. At low solar radiation intensity, PV/T exergy efficiency defined by Eq. (43) tends to the electrical efficiency of the reference conditions (hex z hel,ref). The equivalence of solar cells temperature and ambient temperature and negligible amount of Q_ u are the reasons of this fact. 5. Formulation of optimization problem Optimization of PV/T water collector is necessary to increase its efficiency and reduce its EPBT. A computer simulation program has been developed based on thermal and electrical models which are presented in the previous sections. The formulation of optimization problem, considering the quantities Tamb, Tamb,ref, Tcell,ref, Tf,in z Tamb, Tsun, G, Gref, hel,ref, APV/T, L, W, etc. as constant parameters is given by
8 > Maximize hex ¼ Eq: ð42Þ; > > > > subject to > > > > Eqs: ð1Þ ð34Þ > > > > and < 0:01 Vin 2 m s1 ; > > 0:001 Dh 0:01 m; > > > > _ Cp ; Io ; IL ; Rs ; Rsh ; a; Isc ; Voc ; Imp ; Vmp ; > Tcell ; Tf ;out ; Tbs ; T f ; m; > > > _ > Q ; Q ; Ut ; UL ; UT ; UtT ; DPf ; r; ðasÞeff ; hp1 ; hp2 ; hf ; hth ; hel 0; > > : u loss other nonlinear constraints: where Vin and Dh are independent parameters and Tcell, Tf,out, Tbs, T f , _ Cp, Io, IL, Rs, Rsh, a, Isc, Voc, Imp, Vmp, Q_ u , Q_ loss , Ut, UL, UT, UtT, DPf, r, m, (as)eff, hp1, hp2, hf, hth and hel are dependent parameters in the optimization procedure. The objective function and its constraint equations are nonlinear. Therefore, a real coded genetic algorithm
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
program has been developed to optimize the objective function [32]. 6. Experimental validation The simulation results of our computer program have been validated by the experimental results of Huang et al. [18] for a sample PV/T water collector. The features of sample PV/T water collector come as follows. The PV/T water collector made in their study consists of a polycrystalline PV module (Solarex MSX60) and a heat-collecting plate. The heat-collecting plate adheres directly to the back of the PV module. Thermal grease is used between the plate and the PV module for better contact. Below the heat-collecting plate, a thermal insulation layer is attached using a fixing frame. Fig. 4 shows schematic diagram of the experimental setup of Huang et al. [18]. The experimental data of Huang et al. [18] include solar radiation intensity, ambient temperature, solar cells temperature, inlet and outlet water temperature and average value of electrical efficiency. The simulated values of solar cells temperature, outlet water temperature, thermal efficiency, electrical efficiency and exergy efficiency have been validated by their corresponding experimental values in Ref. [18]. It should be mentioned that zero-phase digital filtering is used to obtain smooth curves from Fig. 4 in Huang et al.’s work. Then curve fitting methods are used to convert the smooth curves in to numerical values. The design parameters of PV/T water collector are described in Table 1. More details about experiments and measurement instruments can be found in Ref. [18]. In order to compare the simulated results with the experimental result, a root mean square percentage deviation (RMS) has been used by following equation
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 P 100 Xsim;i Xexp;i Xexp;i RMS ¼ n
(44)
where n is the number of the experiments carried out. Variations of solar radiation intensity and the simulated and experimental values of solar cells temperature, outlet water temperature, inlet water temperature and ambient temperature during the test day are shown in Fig. 5. Since, PV/T water collector of Huang et al.’s setup has worked in a closed loop therefore the inlet water temperature is higher than the ambient temperature. According to this figure, it is observed that there are good agreements between the experimental and simulated values of solar cells temperature and outlet water temperature. Furthermore, the root mean square percentage deviations of these parameters are 6.13%, and 4.52%, respectively.
Fig. 4. Schematic diagram of Huang et al.’s setup [18].
361
Table 1 Design parameters of the PV/T water collector. Parameter
Value
PV Module Lb W Isc,ref Voc,ref Imp,ref Vmp,ref
Solarex MSX60, Poly-crystalline silicon 60 W 1.108 m 0.502 m 0.037 m 3.8 A 21.1 V 3.5 A 17.1 V 2.06 mA C1 0.077 V C1 0.9 0.0045 C1 1 m s1
a b bc bref Vw
Experimental and simulated values of thermal and electrical efficiency during the test day are shown in Fig. 6. A comparison between the experimental and simulated values of thermal efficiency is carried out in this figure. Since in Ref. [18], only the average value of electrical efficiency (hel,exp,avg ¼ 9%) during the course of experiments has been reported, the RMS of the average values of experimental and simulated electrical efficiency has been calculated. The root mean square percentage deviations of these parameters are 9.68% and 5.55%, respectively. It is observed that there is a fair agreement between the experimental and simulated values of these efficiencies. Simulated and experimental values of the exergy efficiency of present work and the one given by previous studies during the test day are shown in Fig. 7. The values of root mean square percent deviation are 6.36% and 6.85%, respectively. It is observed that there is a good agreement between the simulated and experimental values of these efficiencies. The relative values of the exergy efficiency given by previous studies are more than the exergy efficiency values of present work. The exergy efficiency given by previous studies does not include the exergy loss due to pressure drop in flow pipes. Therefore, its values are more than the ones given by present study.
Fig. 5. Variations of solar radiation intensity and the simulated and experimental values of various temperatures of PV/T water collector during the test day.
362
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Fig. 8. Behavior of exergy efficiency as a function of inlet water velocity (Vin) and pipes diameter (Dh). Fig. 6. Experimental and simulated values of thermal and electrical efficiency during the test day.
7.2. Parametric studies
Optimization process is carried out for the selected values of the environmental and design conditions according to Table 1. The variations of objective function (hex) with respect to the variation of inlet water velocity (Vin) and pipes diameter (Dh) are drawn in Fig. 8. Fig. 8 is presented a range of operational and design conditions where the exergy efficiency takes a global maximum value. It is observed that there is one global maximum point. The coordinate of this point shows the values of optimized parameters. The calculated values of global maximum point are Vin,opt ¼ 0.09 m s1, Dh,opt ¼ 4.8 mm, hex,max ¼ 11.36%.
In order to plot the next figures some parameters are assumed which are mentioned above each figure. The rest of parameters needed to plot the following figures are used from Table 1. The variations of the exergy efficiency of present work and the exergy efficiency of previous studies with respect to inlet water velocity are plotted in Fig. 9. It is observed that by increasing inlet water velocity from 0.01 to 2 m s1, initially the exergy efficiency of present study increases from w9% to w11.36% and after attaining the inlet water velocity of about Vin ¼ 0.09 m s1, it decreases to 0%. This indicates the optimum value of inlet water velocity for given climatic and design parameters (Table 1). The increase of inlet water velocity increases pressure drop in flow pipes, therefore the consumed electrical power by pump increases and it causes a significant drop in the exergy efficiency of PV/T water collector. On the other hand, the exergy efficiency of previous studies has the same behavior with
Fig. 7. Simulated and experimental values of the exergy efficiency of present work and the one given by previous studies during the test day.
Fig. 9. Variations of the exergy efficiency of present work and the exergy efficiency of previous studies with respect to inlet water velocity.
7. Results and discussion 7.1. Optimization results
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Fig. 10. Variations of the exergy efficiency of present work and the one given by previous studies with respect to pipes diameter.
respect to the variation of inlet water velocity. Since it does not include the exergy loss of pressure drop, its relative values are more than the exergy efficiency values of present work. Fig. 10 shows the variations of the exergy efficiency of present work and the one given by previous studies with respect to pipes diameter. According to this figure, the exergy efficiency of present study increases from w8.5% to its optimum value and then it tends to a constant value (w11.3%). The coordinate of optimum point is Dh ¼ 4.8 mm. It is impossible to determine an optimum value for mass flow rate regarding to the optimum value of inlet water velocity and pipes diameter because there are some values of inlet water velocity and pipes diameter which are not optimum but give us the same mass flow rate as the one obtained for optimum inlet water velocity and pipes diameter. Fig. 11 shows the effect of solar radiation intensity on the exergy efficiency of present work and the one given by previous studies.
363
Fig. 12. Variations of the exergy efficiency of present work and the one given by previous studies with respect to wind speed.
It is observed that by increasing solar radiation intensity from 0 to 1000 W m2, the exergy efficiency of the present work increases from w0% to w11.5%, exponentially but the exergy efficiency of previous studies increases from w11.4 to w13%, quasilinearly. According to the figure, it is observed that the exergy efficiency given by previous studies has a significant error at low solar radiation intensity. At low solar radiation intensity, the exergy efficiency given by previous studies gives the value of PV/T exergy efficiency almost near the value of electrical efficiency of the reference conditions (hex z hel,ref). The equivalence of the solar cells and ambient temperature and the negligible amount of Q_ u are the reasons of this fact. Fig. 12 shows the variations of the exergy efficiency of present work and the one given by previous studies with respect to wind speed. According to this figure, it is observed that the two exergy efficiencies have a slight change with respect to wind speed variations. 8. Conclusion Based on present study, the following conclusions have been drawn:
Fig. 11. Effect of solar radiation intensity on the exergy efficiency of present work and the one given by previous studies.
the numerical simulation results are in good agreement with the experimental measurements of Huang et al. [18]; since the exergy efficiency given by previous studies does not include the exergy loss due to pressure drop in flow pipes, its relative values are more than the ones given in the present study; the exergy efficiency of the previous studies has a significant error at low solar radiation intensity. This deficiency has been corrected in the modified exergy efficiency of present study; we know that energy efficiency changes without having extremum points with respect to operating and design parameters variation. Therefore, energy optimization gives optimum points in start or end of optimization parameters ranges. However, the exergy efficiency has extremum points and shows a point of global maximum; exergy efficiency behavior of the present work is the same as the one given by previous studies with respect to the changes of inlet water velocity, pipes diameter and wind speed.
364
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Nomenclature a ideality factor (eV) b width of PV/T water collector (m) A area (m2) Cp specific heat capacity of water (J kg1 K1) D width of flow duct, hydraulic diameter of flow duct (m) ex specific exergy flow (J kg1) _Ex exergy rate (W) F function FR heat removal factor F0 collector efficiency factor G solar radiation intensity (W m2) h heat transfer coefficient (W m2 K1) hp1 penalty factor due to the presence of solar cell material, glass and EVA hp2 penalty factor due to the presence of interface between tedlar and working fluid I circuit current (A) I_ irreversibility rate (W) IeV currentevoltage L dimensions of solar module, the length of PV/T water collector (m) _ m mass flow rate of water (kg s1) n number of the experiment carried out N number of flow duct Nc number of cells in PV module P power (W), pressure (Pa) Q_ heat transfer rate (W) R resistance (U) RMS root mean square percentage deviation (%) T temperature (K) T average temperature (K) UL an overall heat loss coefficient from the PV/T collector to the environment (W m2 K1) Ut an overall heat transfer coefficient from solar cell to ambient through glass cover (W m2 K1) UT a conductive heat transfer coefficient from solar cell to absorber plate through tedlar (W m2 K1) UtT an overall heat transfer coefficient from glass to tedlar through solar cell (W m2 K1) Utw an overall heat transfer coefficient from glass to agent fluid through solar cell (W m2 K1) V circuit voltage (V), wind speed (m s1) W distance between two adjacent flow duct (m) X experimental or simulated parameter Greek symbols a absorptivity, current temperature coefficient (mA C1) (as)eff product of effective absorptivity and transmittivity b packing factor, voltage temperature coefficient (V C1) the temperature coefficient of electrical efficiency ( C1) d duct depth (m) D difference in current, pressure, temperature, voltage 3 semiconductor band gap energy (eV) h efficiency (%) r density (kg m3) m viscosity (kg m1 s1) s transmittivity Subscripts amb ambient avg average bs back surface of tedlar cell cell, module des destroyed
el ex exp f g h i in inlet loss L max mp o oc opt out pump Q PV/T ref s sc sh sim sun t T th u w
electrical exergy experimental fluid flow glass hydraulic i-th parameter inlet inlet loss light current maximum maximum power point reverse saturation open circuit optimum outlet pump heat transfer PV/T reference series short-circuit shunt simulated sun top tedlar thermal useful wind, work
References [1] Tiwari A, Sodha MS. Performance evaluation of solar PV/T system: an experimental validation. Sol Energy 2006;80:751e9. [2] Ross RT, Hsiao TL. Limits on the yield of photochemical solar energy conversion. J Appl Phys 1977;48(11):4783e5. [3] Landsberg PT, Markvart T. The carnot factor in solar cell theory. Solid-State Electron 1998;42(4):657e9. [4] Markvart T, Landsberg PT. Thermodynamics and reciprocity of solar energy conversion. Physica E 2002;14:71e7. [5] Würfel P. Thermodynamic limitations to solar energy conversion. Physica E 2002;14:18e26. [6] Smestad GP. Conversion of heat and light simultaneously using a vacuum photodiode and the thermionic and photoelectric effects. Sol Energy Mater Sol Cells 2004;82:227e40. [7] Bisquert J, Cahen D, Hodes G, Ruhle S, Zaban A. Physical chemical principles of photovoltaic conversion with nanoparticulate, mesoporous dye-sensitized solar cells. J Phys Chem B 2004;108(24):8106e18. [8] Sahin AD, Dincer I, Rosen MA. Thermodynamic analysis of solar photovoltaic cell systems. Sol Energy Mater Sol Cells 2007;91:153e9. [9] Joshi AS, Dincer I, Reddy BV. Thermodynamic assessment of photovoltaic systems. Sol Energy 2009;83(8):1139e49. [10] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergetic performance evaluation of a solar photovoltaic (PV) array. Aust J Basic Appl Sci 2010;4(3):502e 19. [11] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergy efficiency of a solar photovoltaic array based on exergy destructions. Proc Inst Mech Eng Part J Power Energy 2010;224(6):813e25. [12] Polivoda FA. Analysis of the exergy efficiency and the energy efficiency of photothermal-and-voltaic systems. Therm Eng 1998;45(7):609e12. [13] Saitoh H, Hamada Y, Kubota H, Nakamura M, Ochifuji K, Yokoyama S, et al. Field experiments and analyses on a hybrid solar collector. Appl Therm Eng 2003;23:2089e105. [14] Joshi AS, Tiwari A. Energy and exergy efficiencies of a hybrid photovoltaice thermal (PV/T) air collector. Renew Energy 2007;32(13):2223e41. [15] Dubey S, Solanki SC, Tiwari A. Energy and exergy analysis of PV/T air collectors connected in series. Energy Build 2009;41(8):863e70. [16] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergetic performance assessment of a solar photovoltaic thermal (PV/T) air collector. Energy Build 2010;42:2184e99. [17] Agrawal S, Tiwari GN. Energy and exergy analysis of hybrid micro-channel photovoltaic thermal module. Sol Energy 2011;85(2):356e70.
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365 [18] Huang BJ, Lin TH, Hung WC, Sun FS. Performance evaluation of solar photovoltaic/thermal systems. Sol Energy 2001;70(5):443e8. [19] Zondag HA, De Vries DW, Van Helden WGJ, Van Zolengen RJC, Van Steenhoven AA. The thermal and electrical yield of a PVethermal collector. Sol Energy 2002;72(2):113e28. [20] Zondag HA, De Vries DW, Van Helden WGJ, Van Zolingen RJC, Van Steenhoven AA. The yield of different combined PVethermal collector designs. Sol Energy 2003;74:253e69. [21] Dubey S, Tiwari GN. Thermal modeling of a combined system of photovoltaic thermal (PV/T) solar water heater. Sol Energy 2008;82:602e12. [22] Dubey S, Tiwari GN. Analysis of PV/T flat plate water collectors connected in series. Sol Energy 2009;83:1485e98. [23] Bosanac M, Sorensen B, Ivan K, Sorensen H, Bruno N, Jamal B. Photovoltaic/ thermal solar collectors and their potential in Denmark. Final report. EFP Project 2003, 1713/00-0014, www.solenergi.dk/rapporter/pvtpotentialindenmark.pdf; 2003. [24] Tiwari A, Dubey S, Sandhu GS, Sodha MS, Anwar SI. Exergy analysis of integrated photovoltaic thermal solar water heater under constant flow rate and constant collection temperature modes. Appl Energy 2009;86(12):2592e7.
365
[25] Chow TT, Pei G, Fong KF, Lin Z, Chan ALS, Ji J. Energy and exergy analysis of photovoltaicethermal collector with and without glass cover. Appl Energy 2009;86:310e6. [26] Mishra RK, Tiwari GN. Energy and exergy analysis of hybrid photovoltaic thermal water collector for constant collection temperature mode. Sol Energy 2013;90:58e67. [27] Kakaç S, Liu H. Heat exchangers: selection, rating and thermal design. 2nd ed. Boca Raton, Florida 33431, USA: CRC Press LLC, 2000 N.W. Corporate Blvd.; 2003. [28] Kotas TJ. The exergy method of thermal plant analysis. Malabar, FL: Krieger Publish Company; 1995. UK. [29] Wong KFV. Thermodynamics for engineers. USA: University of Miami/CRC Press LLC, Taylor & Francis; 2000. [30] Petela R. Exergy of undiluted thermal radiation. Sol Energy 2003;74:469e88. [31] Petela R. An approach to the exergy analysis of photosynthesis. Sol Energy 2008;82:311e28. [32] Osyczka A. Evolutionary algorithms for single and multicriteria design optimization. 1st ed. Verlag Heidelberg; 2001.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimization of a solar photovoltaic thermal (PV/T) water collector based on exergy concept F. Sobhnamayan a, F. Sarhaddi a, *, M.A. Alavi b, S. Farahat a, J. Yazdanpanahi a a b
Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 April 2013 Accepted 30 January 2014 Available online
In this paper, the optimization of a solar photovoltaic thermal (PV/T) water collector which is based on exergy concept is carried out. Considering energy balance for different components of PV/T collector, we can obtain analytical expressions for thermal parameters (i.e. solar cells temperature, outlet water temperature, useful absorbed heat rate, average water temperature, thermal efficiency, etc.). Thermal analysis of PV/T collector depends on electrical analysis of it; therefore, five-parameter currentevoltage (I eV) model is used to obtain electrical parameters (i.e. open-circuit voltage, short-circuit current, voltage and current at the point which has maximum electrical power, electrical efficiency, etc.). In order to obtain exergy efficiency of PV/T collector we need exergy analysis as well as energy analysis. Considering exergy balance for different components of PV/T collector, we obtain the expressions which show the exergy of the different parts of PV/T collector. Some corrections have been done on the above expressions in order to obtain a modified equation for the exergy efficiency of PV/T water collector. A computer simulation program has been developed in order to obtain the amount of thermal and electrical parameters. The simulation results are in good agreement with the experimental data of previous literature. Genetic algorithm (GA) has been used to optimize the exergy efficiency of PV/T water collector. Optimum inlet water velocity and pipe diameter are 0.09 m s1, 4.8 mm, respectively. Maximum exergy efficiency is 11.36%. Finally, some parametric studies have been done in order to find the effect of climatic parameters on exergy efficiency. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Solar photovoltaic thermal (PV/T) water collector Exergy analysis Optimization
1. Introduction Nowadays using renewable energies is considered more because of fossil fuels high price and the environmental problems which are caused by them. One of an abundant renewable energy sources is solar energy. Most conventional systems which are used in order to convert solar energy in to other kinds of energies are as follows 1. Photovoltaic (PV) systems: systems which receive solar energy in order to give us electricity. 2. Solar collector systems: systems which receive solar energy in order to give heat to the agent fluid In a PV system, when the solar cells temperature increases, electrical voltage decreases. The electrical efficiency of PV system * Corresponding author. Tel.: þ98 541 2426206; fax: þ98 541 2447092. E-mail address: [email protected] (F. Sarhaddi). http://dx.doi.org/10.1016/j.renene.2014.01.048 0960-1481/Ó 2014 Elsevier Ltd. All rights reserved.
decreases as a result of decrease in electrical voltage. In a solar collector system, an external electrical source is needed to circulate the agent fluid through the system. Therefore, if we combine a PV module and a solar collector system, the electrical source can be provided for solar collector system. On the other hand, the additional heat which is absorbed from PV module causes solar cells temperature to decrease and as a result electrical efficiency increases. Combining these two systems, we can also use optimum installation space. Using a PV/T collector, we can convert solar energy in to electricity and heat. As we know, the quality of these two kinds of energies is not the same. Using exergy analysis, we can compare these two kinds of energies, which are different in their quality. Using exergy optimization, we can determine optimum operating mode of PV/T water collector. It has been shown that PV/ T collector energy payback time (EPBT) can be decreased if PV/T collector works in its optimum operating mode [1]. Experimental and theoretical research which is carried out in the field of exergy analysis of PV and PV/T systems, can be expressed briefly as follows
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
357
Thermodynamic (exergy) limitations of photovoltaic phenomenon and its conversion efficiency have been investigated in Refs. [2e7]. Exergy analysis of PV systems based on net output exergy or exergy losses have been investigated in Refs. [8e11]. Exergy analysis of PV/T air collectors has been widely investigated in Refs. [12e17]. Exergy analysis of PV/T water collectors has been carried out in Refs. [18e26]. The deficient points of the previous studies are considered as follows The exergy loss due to pressure drop in flow pipes is neglected The exergy efficiency of PV/T water collector which is used in the previous literature [18e26], has a significant error at low solar radiation intensity. In this condition, the exergy efficiency tends to the nominal electrical efficiency (hex z hel,ref) at the reference conditions (Tcell,ref ¼ 25 C, G ¼ 1000 W m2). There has not been a comprehensive optimization of PV/T water collector which is based on exergy concept regarding to design and operating parameters before the one carried out in this paper. In this paper, the following items are carried out respectively in order to optimize a solar PV/T water collector which is based on exergy concept. Energy analysis is done in order to obtain thermal and electrical parameters Exergy analysis is done in order to introduce exergy efficiency Exergetic optimization is carried out in order to determine optimum operating mode of PV/T water collector Finally, some parametric studies are done in order to find the effect of climatic parameters on exergy efficiency. Fig. 1. (a) The cross-sectional view of a PV/T water collector and (b) its thermal resistance circuit diagram on a flow duct.
2. Thermal analysis The proof of governing equations on the thermal analysis of PV/T water collector is not mentioned to have a brief note. More details of the derivation of governing equations are found in Refs. [1,24]. Fig. 1 shows the cross-sectional view of a PV/T water collector and its equivalent thermal resistant circuit diagram on a flow duct. Analytical expressions for thermal parameters and thermal efficiency of PV/T water collector are obtained by writing energy balance equation for each component of the PV/T water collector.
electrical efficiency of PV module, respectively. An expression for the solar cells temperature can be obtained from Eq. (1) as follows
i. h Tcell ¼ ðasÞeff G þ Ut Tamb þ UT Tbs ðUt þ UT Þ
(2)
where (as)eff is the product of effective absorptivity and transmissivity and it is defined as follows:
ðasÞeff ¼ sg ½ac bc þ aT ð1 bc Þ bc hel
(3)
2.2. Energy balance for the back surface of tedlar
2.1. Energy balance for glassetedlar PV module
sg ½ac bc G þ aT ð1 bc ÞGD dx ¼ ½Ut ðTcell Tamb Þ þ UT ðTcell Tbs Þ D dx þ sg bc hel GD dx (1) where Tcell, Tamb, Tbs, G, D, dx, ac, aT, bc, sg and hel are solar cells temperature, ambient temperature, back surface temperature, solar radiation intensity, absorber plate width on a flow duct, elemental length of flow duct, absorptivity of solar cell, absorptivity of tedlar, packing factor of solar cell, transmittivity of glass cover and
UT ðTcell Tbs ÞD dx ¼ hT Tbs Tf D dx
(4)
Back surface temperature of tedlar can be obtained from Eq. (4) as follows
i. h Tbs ¼ hp1 ðasÞeff G þ UtT Tamb þ hT Tf ðUtT þ hT Þ
(5)
An expression for the heat transfer rate, transferred to the back surface of tedlar, is obtained by substituting Eqs. (2) and (5) into Eq. (1).
358
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
UT ðTcell Tbs Þ ¼ hT Tbs Tf ¼ hp1 hp2 ðasÞeff G Utw Tf Tamb
Tf ¼
(6)
" ¼
2.3. Energy balance for absorber plate
ZL Tf ðxÞdx x¼0
0 3 #2 UL L 1 exp bF hp1 hp2 ðasÞeff G _ mC 5 41 0 p Tamb þ bF UL L UL
Energy balance for an elemental length of absorber plate and the flow duct under it, gives the following equation
h i ð1=NÞ dQ_ u =dx ¼ WF 0 hp1 hp2 ðasÞeff G UL Tf Tamb
1 L
þ Tf ;in
0 UL L 1 exp bF _ mC 0 p
_ p mC
bF UL L _ p mC
(7)
where UL, W, N and F 0 are overall heat transfer coefficient from PV/T collector to surrounding, distance between two adjacent flow duct, number of flow ducts and collector efficiency factor, respectively.
2.5. Thermal efficiency of PV/T water collector
2.4. Energy balance for the element of agent fluid in flow duct (Fig. 2)
_ p Tf ;out Tf ;in Q_ u ¼ mC i h ¼ FR bL hp1 hp2 ðasÞeff G UL Tf ;in Tamb
_ _ 1 m m Cp Tf Cp Tf dQ_ u dx ¼ 0 N N N xþdx x 8 < dTf þ WF 0 UL Tf Tamb ¼ _ =N mC dx :
p
WF 0 hp1 hp2 ðasÞeff G _ p =N mC
(8)
(9)
x ¼ 0/Tf ¼ Tf ;in
An expression for agent fluid temperature is obtained by solving Eq. (9) as follows
Tf ðxÞ ¼
hp1 hp2 ðasÞeff G UL bF 0 UL x exp _ p mC
!
Tamb þ þ Tf;in
bF 0 UL x 1 exp _ p mC (10)
The outlet temperature of agent fluid (Tf,out) is obtained from the above equation by considering boundary condition as Tf ¼ Tf,out, at x¼L
Tf;out
hp1 hp2 ðasÞeff G ¼ Tamb þ UL bF 0 UL L þ Tf;in exp _ p mC
!
bF 0 UL L 1 exp _ p mC (11)
where L is flow duct length and b ¼ NW is PV/T collector width. The average temperature of agent fluid along the length of a flow duct is obtained as follows
The rate of useful thermal energy of PV/T water collector is obtained as follows
FR ¼
_ p mC bF 0 UL L 1 exp _ p UL bL mC
(13)
(14)
_ and Cp are heat removal factor, mass flow rate and heat where FR, m capacity of agent fluid, respectively. Thermal efficiency of PV/T water collector is defined as follows
hth
3 2 UL Tf;in Tamb Q_ u 5 ¼ FR 4hp1 hp2 ðasÞeff ¼ G bLG
(15)
Because of the presence of the electrical efficiency of PV module (hel) in Eq. (3), thermal parameters of PV/T water collector and its electrical parameters are dependent to each other. 3. Electrical analysis In order to calculate electrical parameters and electrical efficiency of PV/T water collector five-parameter currentevoltage (Ie V) model (Fig. 3) is used as follows [10,11]
V þ IRs ðV þ IRs Þ 1 I ¼ IL Io exp Rsh a
(16)
where I and V represent current and voltage at load, a, IL, Io, Rs and Rsh are ideality factor, light current, diode reverse saturation current, series resistance and shunt resistance, respectively. In order to calculate five reference parameters (aref, IL,ref, Io,ref, Rs,ref and Rsh,ref), five pieces of information are needed at reference conditions. These five pieces of information are defined as follows [10,11]: At short-circuit current: I ¼ Isc,ref, V ¼ 0 At open-circuit voltage: I ¼ 0, V ¼ Voc,ref
Fig. 2. The element of agent fluid in flow duct.
(12)
Fig. 3. Equivalent electrical circuit in fiveeparameter photovoltaic model.
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
At the maximum power point: I ¼ Imp,ref, V ¼ Vmp,ref At the maximum power point: ½dðIV Þ=dVmp ¼ ½VdI=dV + Imp ¼ 0 At short circuit: ½dI=dVsc ¼ - 1=Rsh;ref The subscript ‘ref’ indicates the value of parameters at the reference conditions. Reference conditions or standard rating conditions (SRC) are defined as follows [10,11] Solar cells temperature at reference conditions: Tcell,ref ¼ 25 C Solar radiation intensity at reference conditions: Gref ¼ 1000 W m2. Substituting the five mentioned pieces of information into Eq. (16), following equations are obtained
"
Isc;ref Rs;ref aref
F1 ¼ Isc;ref þ IL;ref Io;ref exp
!
# 1
Isc;ref Rs;ref ¼0 Rsh;ref
IL ¼
! i G h IL;ref þ a Tcell Tcell;ref Gref
F2 ¼ IL;ref Io;ref
Voc;ref exp aref
!
"
Vmp;ref þ Imp;ref Rs;ref aref
F3 ¼ Imp;ref þ IL;ref Io;ref exp
!
#
Isc ¼ Isc;ref þ DI
(28)
Voc ¼ Voc;ref þ DV
(29)
2
Vmp;ref þImp;ref Rs;ref
Io;ref exp
6 6 F4 ¼ Vmp;ref 6 4 Io;ref Rs;ref 1þ
aref
exp
3 1 Rsh;ref
aref
Vmp;ref þImp;ref Rs;ref
R
aref
7 7 7 þ Imp;ref ¼ 0 5
s;ref þ Rsh;ref
aref
(20) 2 6 6 F5 ¼ 6 4
Io;ref exp
Isc;ref Rs;ref aref
aref
1þ
Io;ref Rs;ref exp
3 1
Rsh;ref
Isc;ref Rs;ref aref
aref
þ
Rs;ref Rsh;ref
7 1 7 ¼ 0 7þ 5 Rsh;ref
Io;ref
Tcell Tcell;ref
exp
3Nc aref
mp þImp Rs a
aV
Io;ref exp
3
mp þImp Rs a a
R1
Vmp;ref þImp;ref Rs;ref aref
aref Io;ref Rs;ref exp
sh
7 5
þRRs
sh R
Vmp;ref þImp;ref Rs;ref aref
3 1
sh;ref
aref
R
1
Tcell;ref Tcell
(30)
7 7 7 5
þR s;ref
sh;ref
(31)
where 3, Nc, a and b are semiconductor band gap energy, solar cells number in series, current temperature coefficient and voltage temperature coefficient, respectively. PV module manufacturers usually give temperature coefficients. Vmp and Imp is a point on IeV characteristic curve. The area of rectangle under the IeV characteristic curve with the length of Vmp and width of Imp should be maximum. The consumed electrical power by pump to circulate working fluid in PV/T water collector is calculated from [27]
m1:83 D0:5 1:2465 104 APV=T h3:5 f h k2:33 Cp1:17 r2 hpump
(32)
(21)
(22) !3
V
Io Rs exp
1þ
Dh ¼
a Tcell ¼ aref Tcell;ref
¼
1þ
G ¼ 2 Gref 6 6 6 4
Ppump ¼
where Voc, Vmp, Isc and Imp are open-circuit voltage, maximum power point voltage, short-circuit current and maximum power point current, respectively. Functions F1, F2, F3, F4 and F5 form a set of five nonlinear equations in five unknown variables x1 ¼ aref, x2 ¼ IL,ref, x3 ¼ Io,ref, x4 ¼ Rs,ref and x5 ¼ Rsh,ref. Since the Eqs. (17)e (21) are implicit and nonlinear, NewtoneRaphson method is used to solve them. Solving the equations set with NewtoneRaphson method gives the value of five parameters (aref, IL,ref, Io,ref, Rs,ref and Rsh,ref), at the reference conditions (Tcell,ref ¼ 25 C, Gref ¼ 1000 W m2). In order to calculate the model parameters at new climatic and operating conditions (G,Tcell), a set of translation equations is used such as follows [10,11]
Io
Io;ref exp
Rsh zRsh;ref (19)
2
(26)
(27)
1
Vmp;ref þ Imp;ref Rs;ref ¼ 0 Rsh;ref
! G 1 Isc;ref Gref
DV ¼ bDT Rs DI
6 4 (18)
(25)
! G DI ¼ a DT þ Gref
#
Voc;ref ¼ 0 1 Rsh;ref
(24)
DT ¼ Tcell Tcell;ref
(17) "
359
! (23)
4 Cross sectional area of duct 4dD ¼ Wetted perimeter 2ðd þ DÞ
(33)
The electrical efficiency of PV/T water collector is defined as the ratio of actual electrical output power to input solar energy incident rate on PV/T collector surface as follows:
hel ¼
Vmp Imp Ppump GAPV=T
(34)
In the previous equations, Dh, APV/T, d, hf, m, k, r and hpump are hydraulic diameter of flow duct, PV/T surface area, duct depth, viscosity, conductivity, density of agent fluid and pump efficiency, respectively.
4. Exergy analysis Exergy is defined as the maximum amount of work that can be produced by a system or a flow of mass or energy as it comes to equilibrium with a reference environment [28,29]. The general form of exergy balance equation for a control volume is written as [9]
360
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
X
_ ðmexÞ in
X
_ ðmexÞ out þ
X
_ _ _ Ex Q ExW ¼ I
(35)
P P P_ _ W and I_ are inlet exergy _ _ ExQ , Ex where ðmexÞ ðmexÞ in , out , rate related to inlet water flow, outlet exergy rate related to outlet water flow, inlet or outlet heat exergy rate, inlet or outlet work exergy rate and irreversibility rate in control volume, respectively.
These exergy rates are given by [28,29]
X
_ _ ðmexÞ in ¼ mCp Tf;in Tamb Tamb
Exergy efficiency of PV/T water collector is defined as the ratio of net output (desired) exergy rate to net input exergy rate [9]
P
hex ¼
P _ _ _ ðmexÞ I_ ðmexÞ out in þ ExW ¼ 1 _Ex _Ex Q ;sun
_ DPf;in Tf ;in m ln þ r Tamb
_ p Tf;out Tf;in Tamb ln TTf;out m_ Dr Pf þ Vmp Imp Ppump mC f;in hex ¼ 4 þ 13 TTamb GAPV=T 1 43 TTamb sun sun
(36) X
_ ðmexÞ out
_ DPf;out T m _ p Tf ;out Tamb Tamb ln f;out þ ¼ mC r Tamb (37)
hex
where DPf,in and DPf,out are the pressure difference between water flow and surroundings at entrance and exit of PV/T collector. 4.2. Heat exergy rate Heat exergy rate includes solar radiation intensity exergy rate _ ðEx Q ;sun Þ. According to the Petela theorem, it is given by [30,31]
(38)
where Tsun is sun temperature in Kelvin scale. 4.3. Work exergy rate Work exergy rate includes the difference between outlet electrical power of PV module and consumed electrical power by pump:
_ W ¼ Vmp Imp Ppump Ex
(39)
4.4. Irreversibility rate Irreversibility rate is the summation of exergy losses rate from control volume and exergy destructions rate in control volume
I_ ¼
(42) where DPf is pressure drop in flow pipes. In the previous studies [18e26], exergy efficiency of PV/T water collector has been calculated from following equation
n h i o amb þ hel;ref 1 bref Tcell Tamb;ref GAPV=T Ppump Q_ u 1 TTf;out ¼ 4 1 Tamb þ GAPV=T 1 43 TTamb Tsun 3 sun
" # 4 Tamb 1 Tamb 4 _Ex þ Q ;sun ¼ GAPV=T 1 3 Tsun 3 Tsun
X
_ Ex loss þ
X
_ Ex des
(41)
Q ;sun
Substituting Eqs. (36)e(39) into Eq. (41), exergy efficiency of PV/ T water collector is obtained as follows:
4.1. Inlet and outlet exergy rate related to water flow
4.5. Exergy efficiency of PV/T water collector
(40)
The amount of exergy goes out of control volume and cannot be used, is called exergy loss. On the other hand, the amount of exergy produced by for example flow friction or mixture of two kinds of liquids, etc. and does not go out of control volume is named exergy destruction [9].
(43)
This equation has some deficiencies; first, it does not include the exergy loss due to pressure drop in flow pipes. Second, it has a significant error at low solar radiation intensity. At low solar radiation intensity, PV/T exergy efficiency defined by Eq. (43) tends to the electrical efficiency of the reference conditions (hex z hel,ref). The equivalence of solar cells temperature and ambient temperature and negligible amount of Q_ u are the reasons of this fact. 5. Formulation of optimization problem Optimization of PV/T water collector is necessary to increase its efficiency and reduce its EPBT. A computer simulation program has been developed based on thermal and electrical models which are presented in the previous sections. The formulation of optimization problem, considering the quantities Tamb, Tamb,ref, Tcell,ref, Tf,in z Tamb, Tsun, G, Gref, hel,ref, APV/T, L, W, etc. as constant parameters is given by
8 > Maximize hex ¼ Eq: ð42Þ; > > > > subject to > > > > Eqs: ð1Þ ð34Þ > > > > and < 0:01 Vin 2 m s1 ; > > 0:001 Dh 0:01 m; > > > > _ Cp ; Io ; IL ; Rs ; Rsh ; a; Isc ; Voc ; Imp ; Vmp ; > Tcell ; Tf ;out ; Tbs ; T f ; m; > > > _ > Q ; Q ; Ut ; UL ; UT ; UtT ; DPf ; r; ðasÞeff ; hp1 ; hp2 ; hf ; hth ; hel 0; > > : u loss other nonlinear constraints: where Vin and Dh are independent parameters and Tcell, Tf,out, Tbs, T f , _ Cp, Io, IL, Rs, Rsh, a, Isc, Voc, Imp, Vmp, Q_ u , Q_ loss , Ut, UL, UT, UtT, DPf, r, m, (as)eff, hp1, hp2, hf, hth and hel are dependent parameters in the optimization procedure. The objective function and its constraint equations are nonlinear. Therefore, a real coded genetic algorithm
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
program has been developed to optimize the objective function [32]. 6. Experimental validation The simulation results of our computer program have been validated by the experimental results of Huang et al. [18] for a sample PV/T water collector. The features of sample PV/T water collector come as follows. The PV/T water collector made in their study consists of a polycrystalline PV module (Solarex MSX60) and a heat-collecting plate. The heat-collecting plate adheres directly to the back of the PV module. Thermal grease is used between the plate and the PV module for better contact. Below the heat-collecting plate, a thermal insulation layer is attached using a fixing frame. Fig. 4 shows schematic diagram of the experimental setup of Huang et al. [18]. The experimental data of Huang et al. [18] include solar radiation intensity, ambient temperature, solar cells temperature, inlet and outlet water temperature and average value of electrical efficiency. The simulated values of solar cells temperature, outlet water temperature, thermal efficiency, electrical efficiency and exergy efficiency have been validated by their corresponding experimental values in Ref. [18]. It should be mentioned that zero-phase digital filtering is used to obtain smooth curves from Fig. 4 in Huang et al.’s work. Then curve fitting methods are used to convert the smooth curves in to numerical values. The design parameters of PV/T water collector are described in Table 1. More details about experiments and measurement instruments can be found in Ref. [18]. In order to compare the simulated results with the experimental result, a root mean square percentage deviation (RMS) has been used by following equation
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 P 100 Xsim;i Xexp;i Xexp;i RMS ¼ n
(44)
where n is the number of the experiments carried out. Variations of solar radiation intensity and the simulated and experimental values of solar cells temperature, outlet water temperature, inlet water temperature and ambient temperature during the test day are shown in Fig. 5. Since, PV/T water collector of Huang et al.’s setup has worked in a closed loop therefore the inlet water temperature is higher than the ambient temperature. According to this figure, it is observed that there are good agreements between the experimental and simulated values of solar cells temperature and outlet water temperature. Furthermore, the root mean square percentage deviations of these parameters are 6.13%, and 4.52%, respectively.
Fig. 4. Schematic diagram of Huang et al.’s setup [18].
361
Table 1 Design parameters of the PV/T water collector. Parameter
Value
PV Module Lb W Isc,ref Voc,ref Imp,ref Vmp,ref
Solarex MSX60, Poly-crystalline silicon 60 W 1.108 m 0.502 m 0.037 m 3.8 A 21.1 V 3.5 A 17.1 V 2.06 mA C1 0.077 V C1 0.9 0.0045 C1 1 m s1
a b bc bref Vw
Experimental and simulated values of thermal and electrical efficiency during the test day are shown in Fig. 6. A comparison between the experimental and simulated values of thermal efficiency is carried out in this figure. Since in Ref. [18], only the average value of electrical efficiency (hel,exp,avg ¼ 9%) during the course of experiments has been reported, the RMS of the average values of experimental and simulated electrical efficiency has been calculated. The root mean square percentage deviations of these parameters are 9.68% and 5.55%, respectively. It is observed that there is a fair agreement between the experimental and simulated values of these efficiencies. Simulated and experimental values of the exergy efficiency of present work and the one given by previous studies during the test day are shown in Fig. 7. The values of root mean square percent deviation are 6.36% and 6.85%, respectively. It is observed that there is a good agreement between the simulated and experimental values of these efficiencies. The relative values of the exergy efficiency given by previous studies are more than the exergy efficiency values of present work. The exergy efficiency given by previous studies does not include the exergy loss due to pressure drop in flow pipes. Therefore, its values are more than the ones given by present study.
Fig. 5. Variations of solar radiation intensity and the simulated and experimental values of various temperatures of PV/T water collector during the test day.
362
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Fig. 8. Behavior of exergy efficiency as a function of inlet water velocity (Vin) and pipes diameter (Dh). Fig. 6. Experimental and simulated values of thermal and electrical efficiency during the test day.
7.2. Parametric studies
Optimization process is carried out for the selected values of the environmental and design conditions according to Table 1. The variations of objective function (hex) with respect to the variation of inlet water velocity (Vin) and pipes diameter (Dh) are drawn in Fig. 8. Fig. 8 is presented a range of operational and design conditions where the exergy efficiency takes a global maximum value. It is observed that there is one global maximum point. The coordinate of this point shows the values of optimized parameters. The calculated values of global maximum point are Vin,opt ¼ 0.09 m s1, Dh,opt ¼ 4.8 mm, hex,max ¼ 11.36%.
In order to plot the next figures some parameters are assumed which are mentioned above each figure. The rest of parameters needed to plot the following figures are used from Table 1. The variations of the exergy efficiency of present work and the exergy efficiency of previous studies with respect to inlet water velocity are plotted in Fig. 9. It is observed that by increasing inlet water velocity from 0.01 to 2 m s1, initially the exergy efficiency of present study increases from w9% to w11.36% and after attaining the inlet water velocity of about Vin ¼ 0.09 m s1, it decreases to 0%. This indicates the optimum value of inlet water velocity for given climatic and design parameters (Table 1). The increase of inlet water velocity increases pressure drop in flow pipes, therefore the consumed electrical power by pump increases and it causes a significant drop in the exergy efficiency of PV/T water collector. On the other hand, the exergy efficiency of previous studies has the same behavior with
Fig. 7. Simulated and experimental values of the exergy efficiency of present work and the one given by previous studies during the test day.
Fig. 9. Variations of the exergy efficiency of present work and the exergy efficiency of previous studies with respect to inlet water velocity.
7. Results and discussion 7.1. Optimization results
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Fig. 10. Variations of the exergy efficiency of present work and the one given by previous studies with respect to pipes diameter.
respect to the variation of inlet water velocity. Since it does not include the exergy loss of pressure drop, its relative values are more than the exergy efficiency values of present work. Fig. 10 shows the variations of the exergy efficiency of present work and the one given by previous studies with respect to pipes diameter. According to this figure, the exergy efficiency of present study increases from w8.5% to its optimum value and then it tends to a constant value (w11.3%). The coordinate of optimum point is Dh ¼ 4.8 mm. It is impossible to determine an optimum value for mass flow rate regarding to the optimum value of inlet water velocity and pipes diameter because there are some values of inlet water velocity and pipes diameter which are not optimum but give us the same mass flow rate as the one obtained for optimum inlet water velocity and pipes diameter. Fig. 11 shows the effect of solar radiation intensity on the exergy efficiency of present work and the one given by previous studies.
363
Fig. 12. Variations of the exergy efficiency of present work and the one given by previous studies with respect to wind speed.
It is observed that by increasing solar radiation intensity from 0 to 1000 W m2, the exergy efficiency of the present work increases from w0% to w11.5%, exponentially but the exergy efficiency of previous studies increases from w11.4 to w13%, quasilinearly. According to the figure, it is observed that the exergy efficiency given by previous studies has a significant error at low solar radiation intensity. At low solar radiation intensity, the exergy efficiency given by previous studies gives the value of PV/T exergy efficiency almost near the value of electrical efficiency of the reference conditions (hex z hel,ref). The equivalence of the solar cells and ambient temperature and the negligible amount of Q_ u are the reasons of this fact. Fig. 12 shows the variations of the exergy efficiency of present work and the one given by previous studies with respect to wind speed. According to this figure, it is observed that the two exergy efficiencies have a slight change with respect to wind speed variations. 8. Conclusion Based on present study, the following conclusions have been drawn:
Fig. 11. Effect of solar radiation intensity on the exergy efficiency of present work and the one given by previous studies.
the numerical simulation results are in good agreement with the experimental measurements of Huang et al. [18]; since the exergy efficiency given by previous studies does not include the exergy loss due to pressure drop in flow pipes, its relative values are more than the ones given in the present study; the exergy efficiency of the previous studies has a significant error at low solar radiation intensity. This deficiency has been corrected in the modified exergy efficiency of present study; we know that energy efficiency changes without having extremum points with respect to operating and design parameters variation. Therefore, energy optimization gives optimum points in start or end of optimization parameters ranges. However, the exergy efficiency has extremum points and shows a point of global maximum; exergy efficiency behavior of the present work is the same as the one given by previous studies with respect to the changes of inlet water velocity, pipes diameter and wind speed.
364
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365
Nomenclature a ideality factor (eV) b width of PV/T water collector (m) A area (m2) Cp specific heat capacity of water (J kg1 K1) D width of flow duct, hydraulic diameter of flow duct (m) ex specific exergy flow (J kg1) _Ex exergy rate (W) F function FR heat removal factor F0 collector efficiency factor G solar radiation intensity (W m2) h heat transfer coefficient (W m2 K1) hp1 penalty factor due to the presence of solar cell material, glass and EVA hp2 penalty factor due to the presence of interface between tedlar and working fluid I circuit current (A) I_ irreversibility rate (W) IeV currentevoltage L dimensions of solar module, the length of PV/T water collector (m) _ m mass flow rate of water (kg s1) n number of the experiment carried out N number of flow duct Nc number of cells in PV module P power (W), pressure (Pa) Q_ heat transfer rate (W) R resistance (U) RMS root mean square percentage deviation (%) T temperature (K) T average temperature (K) UL an overall heat loss coefficient from the PV/T collector to the environment (W m2 K1) Ut an overall heat transfer coefficient from solar cell to ambient through glass cover (W m2 K1) UT a conductive heat transfer coefficient from solar cell to absorber plate through tedlar (W m2 K1) UtT an overall heat transfer coefficient from glass to tedlar through solar cell (W m2 K1) Utw an overall heat transfer coefficient from glass to agent fluid through solar cell (W m2 K1) V circuit voltage (V), wind speed (m s1) W distance between two adjacent flow duct (m) X experimental or simulated parameter Greek symbols a absorptivity, current temperature coefficient (mA C1) (as)eff product of effective absorptivity and transmittivity b packing factor, voltage temperature coefficient (V C1) the temperature coefficient of electrical efficiency ( C1) d duct depth (m) D difference in current, pressure, temperature, voltage 3 semiconductor band gap energy (eV) h efficiency (%) r density (kg m3) m viscosity (kg m1 s1) s transmittivity Subscripts amb ambient avg average bs back surface of tedlar cell cell, module des destroyed
el ex exp f g h i in inlet loss L max mp o oc opt out pump Q PV/T ref s sc sh sim sun t T th u w
electrical exergy experimental fluid flow glass hydraulic i-th parameter inlet inlet loss light current maximum maximum power point reverse saturation open circuit optimum outlet pump heat transfer PV/T reference series short-circuit shunt simulated sun top tedlar thermal useful wind, work
References [1] Tiwari A, Sodha MS. Performance evaluation of solar PV/T system: an experimental validation. Sol Energy 2006;80:751e9. [2] Ross RT, Hsiao TL. Limits on the yield of photochemical solar energy conversion. J Appl Phys 1977;48(11):4783e5. [3] Landsberg PT, Markvart T. The carnot factor in solar cell theory. Solid-State Electron 1998;42(4):657e9. [4] Markvart T, Landsberg PT. Thermodynamics and reciprocity of solar energy conversion. Physica E 2002;14:71e7. [5] Würfel P. Thermodynamic limitations to solar energy conversion. Physica E 2002;14:18e26. [6] Smestad GP. Conversion of heat and light simultaneously using a vacuum photodiode and the thermionic and photoelectric effects. Sol Energy Mater Sol Cells 2004;82:227e40. [7] Bisquert J, Cahen D, Hodes G, Ruhle S, Zaban A. Physical chemical principles of photovoltaic conversion with nanoparticulate, mesoporous dye-sensitized solar cells. J Phys Chem B 2004;108(24):8106e18. [8] Sahin AD, Dincer I, Rosen MA. Thermodynamic analysis of solar photovoltaic cell systems. Sol Energy Mater Sol Cells 2007;91:153e9. [9] Joshi AS, Dincer I, Reddy BV. Thermodynamic assessment of photovoltaic systems. Sol Energy 2009;83(8):1139e49. [10] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergetic performance evaluation of a solar photovoltaic (PV) array. Aust J Basic Appl Sci 2010;4(3):502e 19. [11] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergy efficiency of a solar photovoltaic array based on exergy destructions. Proc Inst Mech Eng Part J Power Energy 2010;224(6):813e25. [12] Polivoda FA. Analysis of the exergy efficiency and the energy efficiency of photothermal-and-voltaic systems. Therm Eng 1998;45(7):609e12. [13] Saitoh H, Hamada Y, Kubota H, Nakamura M, Ochifuji K, Yokoyama S, et al. Field experiments and analyses on a hybrid solar collector. Appl Therm Eng 2003;23:2089e105. [14] Joshi AS, Tiwari A. Energy and exergy efficiencies of a hybrid photovoltaice thermal (PV/T) air collector. Renew Energy 2007;32(13):2223e41. [15] Dubey S, Solanki SC, Tiwari A. Energy and exergy analysis of PV/T air collectors connected in series. Energy Build 2009;41(8):863e70. [16] Sarhaddi F, Farahat S, Ajam H, Behzadmehr A. Exergetic performance assessment of a solar photovoltaic thermal (PV/T) air collector. Energy Build 2010;42:2184e99. [17] Agrawal S, Tiwari GN. Energy and exergy analysis of hybrid micro-channel photovoltaic thermal module. Sol Energy 2011;85(2):356e70.
F. Sobhnamayan et al. / Renewable Energy 68 (2014) 356e365 [18] Huang BJ, Lin TH, Hung WC, Sun FS. Performance evaluation of solar photovoltaic/thermal systems. Sol Energy 2001;70(5):443e8. [19] Zondag HA, De Vries DW, Van Helden WGJ, Van Zolengen RJC, Van Steenhoven AA. The thermal and electrical yield of a PVethermal collector. Sol Energy 2002;72(2):113e28. [20] Zondag HA, De Vries DW, Van Helden WGJ, Van Zolingen RJC, Van Steenhoven AA. The yield of different combined PVethermal collector designs. Sol Energy 2003;74:253e69. [21] Dubey S, Tiwari GN. Thermal modeling of a combined system of photovoltaic thermal (PV/T) solar water heater. Sol Energy 2008;82:602e12. [22] Dubey S, Tiwari GN. Analysis of PV/T flat plate water collectors connected in series. Sol Energy 2009;83:1485e98. [23] Bosanac M, Sorensen B, Ivan K, Sorensen H, Bruno N, Jamal B. Photovoltaic/ thermal solar collectors and their potential in Denmark. Final report. EFP Project 2003, 1713/00-0014, www.solenergi.dk/rapporter/pvtpotentialindenmark.pdf; 2003. [24] Tiwari A, Dubey S, Sandhu GS, Sodha MS, Anwar SI. Exergy analysis of integrated photovoltaic thermal solar water heater under constant flow rate and constant collection temperature modes. Appl Energy 2009;86(12):2592e7.
365
[25] Chow TT, Pei G, Fong KF, Lin Z, Chan ALS, Ji J. Energy and exergy analysis of photovoltaicethermal collector with and without glass cover. Appl Energy 2009;86:310e6. [26] Mishra RK, Tiwari GN. Energy and exergy analysis of hybrid photovoltaic thermal water collector for constant collection temperature mode. Sol Energy 2013;90:58e67. [27] Kakaç S, Liu H. Heat exchangers: selection, rating and thermal design. 2nd ed. Boca Raton, Florida 33431, USA: CRC Press LLC, 2000 N.W. Corporate Blvd.; 2003. [28] Kotas TJ. The exergy method of thermal plant analysis. Malabar, FL: Krieger Publish Company; 1995. UK. [29] Wong KFV. Thermodynamics for engineers. USA: University of Miami/CRC Press LLC, Taylor & Francis; 2000. [30] Petela R. Exergy of undiluted thermal radiation. Sol Energy 2003;74:469e88. [31] Petela R. An approach to the exergy analysis of photosynthesis. Sol Energy 2008;82:311e28. [32] Osyczka A. Evolutionary algorithms for single and multicriteria design optimization. 1st ed. Verlag Heidelberg; 2001.