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Renewable Energy 76 (2015) 27e35

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Renewable Energy

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Channel depth, air mass flow rate and air distribution duct diameteroptimization of photovoltaic thermal (PV/T) air collectors linked toresidential buildings

M. Farshchimonfared, J.I. Bilbao, A.B. Sproul*

School of Photovoltaic and Renewable Energy Engineering, University of New South Wales, Sydney, NSW 2052, Australia

a r t i c l e i n f o

Article history:Received 25 August 2014Accepted 16 October 2014Available online

Keywords:Photovoltaic thermal systemsPV/T air collectorOptimizationEffective thermal energy output

* Corresponding author. Tel.: þ61 (0)293854039; faE-mail address: a.sproul@unsw.edu.au (A.B. Sprou

http://dx.doi.org/10.1016/j.renene.2014.10.0440960-1481/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

Photovoltaic thermal (PV/T) air collector design requires an accurate determination of key parameterssuch as the channel depth and the air mass flow rate. This paper focuses on PV/T air collectors linked toan air distribution system with the aim of optimizing the channel depth, the air mass flow rate per unitcollector area and the air distribution duct diameter considering the whole system performance. Aweighted effective thermal energy output which includes electrical and thermal energy generated andfan power usage was utilized to examine the energy performance of the whole system. This study modelsPV/T air collectors with various collector areas (Ac ¼ 10, 15, 25 and 30 m2) and different length to widthratios (L/W ¼ 0.5, 1, 1.5 and 2) linked to an air distribution system of a typical residential building in aclimate with a mild winter (e.g. Sydney). For a constant temperature rise (DT ¼ 10 �C), and maximizingthe rate of effective thermal energy output per unit collector area ( _Qeff ) delivered to the building, theoptimum channel depth (Dopt), the air mass flow per unit collector area ( _m=Ac), and the air distributionduct diameter were optimized. The optimum value of _m=Ac is almost constant and approximately equalto 0.021 kg/s m2. The optimum depth (Dopt) value varies between 0.09 and 0.026 m and the optimum airdistribution duct varies between 0.3 and 0.5 m. The optimum depth increases as the collector L/W ratioand the collector area (Ac) increase.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Photovoltaic thermal (PV/T) systems are capable of generatingheat and electricity simultaneously. A PV/Tcollector typically consistsof a photovoltaic module and a duct or tube carrying a fluid whichextracts the heat from the back of the module and are usually classi-fied in terms of the coolant fluid type (air or water). Furthermore, avariety of PV/T air collector designs including glass covered or un-covered, single pass or double-pass, wall-mounted or roof mountedhave been reported in the literature [1e7]. The heat from a PV/T col-lector can be used, for example, for domestic hot water or spaceheating. The focus of this paper is on a roof mounted, uncovered,single pass PV/T air collector (Fig. 1) connected to an air distributionsystem of a typical residential building to provide space heating.

As photovoltaic thermal systems generate electrical and thermalenergy, a combined energy metric is needed in order to assess thesystem performance. Various metrics have been used to assess theoverall energy yield and efficiency of a PV/T collector. Some

x: þ61 (0)293855412.l).

previous studies have assumed equal value for the electrical andthermal output, some have given greater weight to the electricaloutput and others have utilized exergy to asses PV/T performance.Further a limited number of studies have investigated the optimi-zation of the geometry of solar air heaters and PV/T air collectors.However, previous studies have ignored the energy losses associ-ated with the air delivery system or not treated it thoroughly andhave not optimized all parameters independently (typically only asingle design is optimized).

This paper focuses on the optimization of the channel depth, theair mass flow rate per unit collector area of photovoltaic thermal aircollectors, as well as the building air distribution system in terms ofmaximizing the value of the energy delivered. In this study, theoutput of PV/T air collectors are evaluated considering an electrical/thermal value ratio of 3.22 based on the minimum COP required inAustralia for space heating.

2. Literature review

Previous studies used the energy balance method and consid-ered equal value for the electricity and heat output of a PV/T

Fig. 1. PV/T air collector.

Fig. 2. Air distribution system connected to PVT air collector.

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e3528

collector [8e11]. However, in reality the value of electricity isgreater than the value of heat. As an example the amount of heatgenerated from a heat pump system is significantly higher than itselectrical input. Exergy or availability is the useful work potential ofa system at a specified state and depends on the system state andenvironment [26]. The primary aim of exergy analysis is to quantifyexergy losses which are wasted potential for work. That is, exergyanalysis recognizes the different thermodynamic value of work andheat [26]. A number of studies have evaluated the performance ofthe PV/T collectors based on the exergy method and reported thefollowing equation [12e15]:

εtotal ¼Qth

�1� Tamb

Tf ;out

�þ Qel

G(1)

where εtotal is the total exergy efficiency, Qth is the rate of thermalenergy output of the collector per unit area, Qel is electrical powergenerated per unit area, G is solar irradiance, Tf ;out is the outputfluid temperature from the collector and Tamb is the ambient tem-perature. The exergy method is used when considering the po-tential to convert heat to work. However, this paper argues that theexergy method is not applicable for the analysis of PV/T collectorswhen the aim is to utilize the thermal energy output of the systemfor direct space heating or other low temperature applications.

Other studies have considered different weighting values forelectrical energy and thermal energy and employed a total effi-ciency value for PV/T collectors taking into account efficiency forelectricity generation from a conventional power plant, hpower asbelow [16e18]:

ht ¼hel

hpowerþ hth (2)

where hth and hel are the thermal and electrical efficiency of the PV/T collector respectively. For example, Huang et al. [16] used a valueof 0.34e0.4 for hpower which can be interpreted as an electrical/thermal value ratio of 2.5e2.94. This method is essentially assess-ing the efficiency of converting fuel energy to heat and electricity,which is an approach best suited to assessing the efficiency of co-generation engines where efficient fuel usage is a major concern.

Coventry and Lovegrove [19] developed a value ratio betweenelectricity and heat output of a PV/T system linked to a domestic hotwater system and proposed a ratio of 4.24. This figure was identi-fied employing an analysis of the renewable energy market andlevelised energy costs. They reported that a renewable energymarket is different from an open market due to the various policiesof countries for extending the use of renewable energy (e.g. buyback schemes and capital subsidies). However, it is notable that thisvalue changes due to various applications of the thermal energyoutput from the PV/T collector.

In this paper we propose that PV/T air systems need to beassessed in terms of the value of the energy that can be deliveredwhich will depend on what the energy is being used for. This willchange depending on the actual application or the load that is beingconsidered (e.g. space heating, cooling or hot water). In this studywe investigate PV/T air collectors (Fig. 1) where the thermal outputis used for a direct space heating application. The PV/T air system islinked to an air distribution system (Fig. 2) of a typical residentialbuilding in Sydney. All elements of the system are considered in themodel: that is the optimal design of the PV/T air collector and airdelivery system is determined in terms of 1) the desired outlettemperature 2) air mass flow rate 3) PV/T collector depth, widthand length and 4) air delivery main duct diameter (which is a majorconsideration for minimizing energy loss). It is crucial to considercorrectly the fan energy as well as the electricity and heat producedby the PV/T collector as the value of electricity is higher than thevalue of heat. Therefore, the whole system pressure lossesincluding losses in the PV/T air collector and the air distributionsystem in the specified building are taken into account to identifythe fan energy consumption. Assuming a heat pump system as analternative direct space heating system for the house, theminimumrequirement for the coefficient of performance (COP) of the systembased on the Australian standard is 3.22 (Standards Australia, 2011)[27]. As such this figure is utilized as a weighting value to calculatethe rate of effective thermal energy output. The following equationis an improved equation used in this study to calculate the rate ofeffective thermal energy output per unit collector area, _Qeff :

_Qeff ¼ _Qf þ�

_WPV � _W fan

�� 3:22 (3)

where _Qf , _WPV and _W fan are the rates of thermal energy output, PVpower output and fan electrical power of the PV/T air collector perunit collector area respectively.

When considering the design of a PV/T air collector or solar airheater system, careful determination of key parameters such as theair mass flow rate, the collector geometry, the temperature rise andair handling duct diameter are essential in order to achieve themaximum performance. For systems with constraints on the tem-perature rise and the collector width or length, the remainingvariables are the collector depth, the air mass flow rate and the airhandling duct diameter which need to be optimized. Some authorsconducted research on the optimization of the channel geometry ofthe solar air heaters [20e23].

Hegazy [1] investigated the performances of forced ventilatedPV/T air collectors with various collector models assuming a valueof 2.5�10�3 for the optimum channel depth to length ratio (D/L)optof the collectors based on the results from his previous study on theoptimization of channel geometry for solar air heaters [23]. Theoptimum air mass flow rate per collector areawas determined to bebetween 0.02 and 0.03 kg/s m2 for single pass PV/T air collectors.

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e35 29

However the study by Hegazy [23] did not treat the fan powerconsumption and only considered maximizing the useful thermalenergy of the solar air heater. Both electricity generation and con-sumption as well as heat generation need to be considered whenthe channel depth of a PV/T air collector is optimized.

Brinkworth [28] studied the optimization of the geometry ofnaturally ventilated PV cooling ducts to minimize the loss of effi-ciency of a photovoltaic array. The outcome of this study deter-mined that the optimum ratio of the duct length to its hydraulicdiameter is around 20 (or DH/L ¼ 0.05). Again this value could notbe utilized for a PV/T air collector as the thermal energy output andfan power have not been investigated.

Tiwari et al. [24] have studied the impact of a PVT air collectorlength, depth and air flow rate on the collector overall efficiency.They varied the air velocity through the collector while fixing otherparameters and evaluated the overall efficiency. Then theyrepeated the same approach to investigate the optimum collectorlength and depth. The results of their work indicated that themaximum overall efficiency occurs at the air velocity of 2 m/s andthat an increase of the collector depth leads to an overall efficiencyincrease. In addition they stated that the efficiency does not changeif the collector length value exceeds 3 m. However they did notconsider the fan power consumption in their study. Also there is noair distribution system from the collector back end and they uti-lized the energy balance method which simply assumes the samevalue for the electricity and heat.

Tiwari and Sodha [17] performed a parametric study on a PV/Tair collector with a length of 2.4 m a width of 0.45 m consideringdifferent values for heat and electricity similar to Eq. (2). Theyshowed that an increase in the air flow rate increased the overallefficiency while an increase in the collector length decreased theoverall efficiency. However, again they did not take into account thefan energy required for the system. In addition they assume athermal/electrical value ratio of 0.4 based on the electric generationefficiency from a power plant.

Tonui and Tripanagnostopoulos [2] evaluated the influence ofthe collector length, depth and air mass flow variations on theelectrical and thermal efficiency and fan input power for thevarious PV/T air collector configurations. Initially they studied theeffect of the depth with a fixed air mass flow rate (0.05 kg/s m2)and then they investigated the impact of the collector air massflow rate using a constant value for the depth (0.15 m). Theyshowed that the thermal and electrical efficiency increased as aresult of a decrease in the depth and an increase in the air massflow rate. In addition they reported an increase in fan input powerbecause of an increase in the air mass flow rate and a reduction inthe depth. Finally they stated that an increase in the collectorlength leads to a decrease in electrical efficiency and an increase inthe fan input power and thermal efficiency. Nevertheless, they didnot report anything regarding total energy output and the opti-mum value of the depth and the air mass flow rate for the col-lector. Also they did not investigate the effect of an air distributionsystem from the collector back end on the fan energyconsumption.

Bambrook [25] designed and constructed a PV/T air system inSydney. The system included a PV/T air collector connected to ashort length duct with 500 mm diameter and a 300 mm diameterfan for the air delivery. The author optimized the collector depthand the air mass flow rate to maximize the energy output of thesystem considering thermal and electrical energy output as well asfan power energy. In fact, the total energy output was calculatedusing the weighting value ratio of electricity per heat output pro-posed by Coventry and Lovegrove [19]. The author reported anoptimum channel depth (Dopt) of 0.05 m at the air mass flow rate of0.038 kg/s m2 considering fixed values for the collector length and

width (L ¼ 4 m and W ¼ 1.3 m) which gives a ratio of Dopt/L ¼ 12.5 � 10�3. However, Bambrook's optimization study [25] didnot include consideration of an air distribution system.

In conclusion, there is lack of information for the variation of theoptimum depth and the optimum air mass flow rate of PV/T airsystems with various collector dimensions and typical air distri-bution system components (e.g. duct fittings). Additionally theprevious studies did not focus on the design of the main duct fromthe collector back end to the buildings and a sufficient temperaturerise useful for a city with a mild climate. Also, the previous worksmethod for calculating the rate of the overall energy of PV/T aircollectors/systems needs to be improved.

3. Methodology

This study focuses on a single pass unglazed flat plate PV/T aircollector with crystalline PV array which is installed on a housewith a pitched roof in Sydney. It is also assumed a high level ofinsulation at the back of the collector. An equivalent thermal circuitmodel for the respective PV/T air collector is used in order toinvestigation the heat transfer from the collector to the air.Appendix A gives details of the thermal circuit model evolution andequations to calculate the rates of thermal energy output ( _Qf ), PVpower output ( _WPV) and fan electrical power ( _W fan) as well as thetemperature rise of the outlet air above the ambient temperature(DT). The final equations for above mentioned outputs are asfollows:

_Qf ¼_mCpA

hTeff � Ta þ ða� helÞGRL

i 1� exp

��Ac_mCpRo

�!(4)

_WPV ¼ Gnhref

h1þ bref

�~Tc � TSTC

�io(5)

_W fan ¼_VðDPk þ DPM þ DPmÞ

hfanhmotor(6)

DT ¼hTeff � Ta þ ða� helÞGRL

i�1� exp

� �Ac_mCpRo

��(7)

Please refer to Appendix A and B for the definition of all vari-ables used in the above equations.

The rate of effective thermal energy output ( _Qeff ) is calculatedusing Eq. (3) and inserting equations for _Qf , _WPV and _W fan asbelow:

Q:

eff ¼m:cpAc

"Teff �Taþða�helÞGRL

1�exp

��Ac

m:CpRo

�!#

þ"Ghref

h1þbref

�~Tc�25

�i�V

:

ðDPkþDPMþDPmÞhfanhmotor

#�3:22

(8)

Using Eq. (8) as an objective function, a parametric study isconducted to optimize the channel depth, the air mass flow rateand the air distribution duct diameter of PV/T air systems. For thispaper, a PV/T air system includes a PV/T air collector linked to theair distribution system of a typical residential building in a climatewith a mild winter (e.g. Sydney). In this study the rate of effectivethermal energy output ( _Qeff ) and other parameters were modelledin an Excel spreadsheet. Then using the Excel solver the channeldepth (D), the air mass flow rate per unit collector area ( _m=Ac) and

Fig. 3. Optimum depth as a function of the length to with ratio (L/W) for collectors ofarea 5, 10, 15 and 25 m2.

Fig. 4. Optimum air mass flow rate per unit collector area as a function of the length towidth ratio (L/W) for collectors of area 5, 10, 15 and 25 m2.

Fig. 5. Variation of the air distribution duct diameter as a function of the length towidth ratio (L/W) for collectors of area 5, 10, 15 and 25 m2.

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e3530

the main duct diameter of the air distribution system (∅) wereoptimized based on maximizing _Qeff and considering a fixed valuefor the required temperature rise (DT ¼ 10 �C) and parametersindicated on Table 1. It is notable that various PV/T air collectorswith various collector areas (Ac¼ 10,15, 25 and 30m2) and differentlength to width ratios (L/W ¼ 0.5, 1, 1.5 and 2) are investigated inthis study. Appendix B gives more details for the optimizationmethodology and other assumptions in this paper.

4. Results and discussion

4.1. Optimization of the depth and the air mass flow rate per unitcollector area for PV/T air collectors with various areas anddimensions

PV/T air collectors with various collector areas (Ac ¼ 10, 15, 25and 30 m2) and different values for the length to width ratio (L/W ¼ 0.5, 1.5, 1, 2) were investigated in order to determine the op-timum depth and the air mass flow rate per unit collector area aswell as the optimum air distribution duct diameter that produced amaximum _Qeff for a DT ¼ 10 �C. Fig. 3 shows the results of theoptimization calculations in terms of the optimum depth of thecollector Dopt. It is evident that for a fixed area PV/T air collector, theoptimum depth increases as L/W increases. Also it can be seen thata collector with a larger area requires a greater optimum depth. AsL/W increases, longer collectors suffer greater hydraulic losses so adeeper channel is required. For a fixed L/W, as Ac increases, theabsolute length of the collector increases, hence hydraulic lossesincrease and again a deeper channel is required. Generally speakingthe optimum depth (Dopt) for a collector with the above mentionedareas and L/W values varies between 0.09 and 0.026 m.

Fig. 4 shows the variation of the optimum air mass flow rate perunit collector area based on the different collectors' lengths andwidths. It can be seen that _m=Ac is reduced slightly as a result of anincrease in the collector area and the ratio of L/W. In other words, alonger collector requires slightly less air mass flow rate tominimzsethe hydraulic losses. Taking into account the specified areas anddifferent ranges for the ratio of the L/W from Fig. 5, it is evident thatthe optimum range of _m=Ac is around 0.0213 kg/s m2 ± 1.4%. Notethat the optimum value of _m=Ac is essentially constant as theoptimization has been carried out for a fixed value of DT ¼ 10 �C(which essentially implies a fixed value of _mcpRo=Ac and as Ro alsoremains roughly constant).

Fig. 5 indicates the variation of the optimum duct diameter ofthe air distribution system (∅) within a typical residential buildingbased on the different collectors' lengths and widths. It can beobserved that using a collector with a fixed area and increasing theratio of L/W, the optimum duct diameter is essentially constant dueto slight changes in _m=Ac. However, as the PV/T collector area in-creases the air mass flow rate ( _m) increases and therefore, the mainduct diameter increases. Generally speaking the optimum value ofthe main duct diameter for an air distribution system within atypical residential building connected to a PV/T air collector withthe specified area values varies between 0.3 and 0.5 m.

Fig. 6 illustrates the variation of the optimum depth to length ofPV/T air collectors assuming the specified values for the collectors

Table 1Design data and physical parameters used for this study.

Variable Value

PV absorption coefficient (a) [29] 0.8PV efficiency under standard condition (href) [30] 0.12Temperature coefficient (bref) [30] 0.005/KSpecific heat of the air (cp) [31] 1.005 kJ/kg �C

Fig. 6. Variation of the optimum depth to length ratio of the PV/T air collectors.

Table 2Pressure losses (DP) of optimized air distribution systems and PVT air collectors.

Collectorarea-m2

Length towidth ratio(L/W)

DPads (airdistributionsystemlosses)-Pa

DPPV/T(collectorlosses)-Pa

DPads/DPtotal(percentage)

DPPVT/DPtotal(percentage)

10 0.5 36 96.2 27.2% 72.8%10 1 36.1 98 26.9% 73.1%10 1.5 36.1 99.2 26.7% 73.3%10 2 36.1 100 26.5% 73.5%15 0.5 33.4 97.3 25.5% 74.5%15 1 33.4 99.2 25.2% 74.8%15 1.5 33.4 100.3 25% 75%15 2 33.4 101.2 24.8% 75.2%25 0.5 31.5 98.5 24.3% 75.7%25 1 31.5 100.5 23.9% 76.1%25 1.5 31.6 101.7 23.7% 76.3%25 2 31.6 102.6 23.5% 76.5%30 0.5 31.4 98.9 24.1% 75.9%30 1 31.4 100.9 23.7% 76.3%30 1.5 31.4 102.1 23.5% 76.5%30 2 31.4 103 23.4% 76.6%

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e35 31

area and dimension and this study methodology. It can be seen thatthis figure is not constant and varies between 3.4 � 10�3 to4 � 10�3 depending on the value of L/W and Ac.

Table 2 shows the pressure losses in the air distribution systemand PV/T air collector for various PV/T air systems. It indicates thatfor optimized PV/T air systems, the air duct delivery system ac-counts for 23.4e27.2% of the energy required to operate the fan.This finding has significant implications in the optimization of thewhole system and as stated previously has not been fully treated inprevious optimization studies of PV/T air collectors.

Fig. 7 indicates the variation of the rates of thermal energyoutput, PV power output, fan electrical power and effective thermaloutput of PV/T air systems per unit collector area with variousvalues of the collector area and L/W ratio along with the associatedoptimum collector depth, optimum air distribution duct diameterand optimum air mass flow rates per collector area. It can beobserved that the value of _Qeff , _Qf and _WPV are almost constantbecause DT is being held constant (DT ¼ 10 �C) and this results in

Fig. 7. Variation of the rates of thermal energy output, PV power, fan power and effe

very small variations of the optimum value of _m=Ac. On the otherhand, the value of _W fan slightly decreases as the collector area in-creases. This is because of a slight decrease in the optimumvalue of_m=Ac in larger collectors and the significant impact of the air ve-locity on the fan power. However, as the PVT air systems have beendesigned optimally it can be seen that fan power values are smalland therefore a slight variation on the value of _W fan does notimpact significantly on the system output and finally the value of_Qeff remain almost constant.

4.2. Comparison with previous work

It is useful to compare the results of this study with the previousanalysis presented in the literature. Table 3 compares the results ofthis paper with previous work. The methodology presented in thisstudy is themost suitablemethod for the design of PV/T systems fordirect space heating applications for the following reasons. Firstlythis study takes into account the impact of the sky temperaturewhich is not included in other studies. In addition previous opti-mization studies have not optimized the duct diameter of the airdistribution system and have ignored the energy losses associatedwith the air delivery system or not treated it thoroughly. Further-more it is evident that all previous studies evaluated PV/T collectorswith fixed values for the collector length and width and have notevaluated the effect of the value of L/W and Ac on the value of Dopt

and _m=Ac. Also previous studies have not optimized all parametersindependently and typically only a single design was optimized.

5. Conclusions

The following conclusions can be drawn. When maximizing theperformance of the PV/T collector system for a fixed value of thetemperature rise (DT ¼ 10 �C), and for different values for the PV/Tcollector area and L/W ratio, the optimum channel depth is depen-dent on the collector L/W ratio and area. The optimum channel depthincreases as the L/W ratio and the area of the collector increases.Generally speaking the optimum depth (Dopt) for a collector with thespecified areas and L/W values considered in this paper varies be-tween 0.09 and 0.026 m. Further the associated optimum value of

ctive thermal output of optimally design PVT air systems per unit collector area.

Table 3Results of this study versus previous work outcome for PV/T air collector(s) and their associated air distribution system.

Author(s) Ac (m2) L/W Dopt (mm) ð _m=AcÞopt (kg/s m2) ∅opt (m) Dopt/L � 10�3 DT (�C) Method of analyze

Hegazy [1] 9 9 22.5 0.02e0.03 Not reported 2.5a 17.8 to 24.5 Energy methodTiwari et al. [24] 1.08 5.33 60 0.12 Not reported 50 Not reportedb Energy methodTiwari and Sodha [17] 1.08 5.33 Not reported Not reportedc Not reported Not reported Not reportedb Electricity/heat ¼ 2.5Tonui and Tripanagnostopoulos [2] 0.4 2.5 Not reportedd Not reportedd Not reported Not reported Not reportedd Energy methodBambrook [25] 5.2 3.07 50e 0.038e Not reported 12.5e 5 Electricity/heat ¼ 4.24This study 10, 15, 25 & 30 0.5, 1.5, 1&2 9e26 0.021 0.3e0.5 3.8e4.6 10 Electricity/heat ¼ 3.22

a The author recommended this value for all collector with various dimensions.b The air temperature rise was not reported for the optimum collector.c The authors stated that an increase in the collector air flow rate leads to an increase in the overall efficiency.d In this study the impact of the collector depth and air flow rate variation on the fan power, thermal and electrical efficiency were investigated independently. No optimum

value was reported based on overall efficiency.e The author selected the collector depth and air mass flow rate based on comparing the output results of the system with three different channel depth (D ¼ 0.05, 0.1 and

0.15 m).

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e3532

the air mass flow rate per unit collector area ( _m=Ac) slightly de-creases as a result of an increase in the collector L/W ratio and area.However, it should be noted that this figure is almost constant(0.021 kg/s m2± 1.4%) for the cases investigated due to the constraintof a constant DT. In addition, for a collector with a fixed area as theratio of L/W increases, the optimum air handling duct diameter isessentially constant due to only slight changes in _m=Ac. However, asthe collector area increases, the optimum air handling duct diameterincreases. Generally speaking the optimum value of the main ductdiameter for an air distribution system within a typical residentialbuilding connected to a PV/T air collector with the specified areavalues varies between 0.3 m and 0.5m. Based on the specified valuesfor the PV/T air collector area and dimensions examined in Section 2,the value of the optimum depth to the length ratio (Dopt/L) is notconstant and varies between 3.4�10�3 to 4�10�3 depending on thevalue of L/W and Ac. Additionally, for an optimized PV/T air system,the air duct delivery system accounts for over 23.4% of the energyrequired to operate the fan and hence must be considered whenoptimizing the performance of PV/T systems.

Acknowledgements

The authors would like to acknowledge the financial support ofthe UNSW School of Photovoltaic and Renewable Energy Engi-neering, and one of the authors (MF) particularly acknowledges thefinancial support of the Cooperative Research Centre for Low Car-bon Living under research project RP1001.

Appendix A. Theoretical background and equations for PVTmodel:

Fig. A1 shows the equivalent thermal circuit model for PV/T aircollector presented by Bambrook [25]. In this model, thermal re-sistances, temperatures and thermal energy flow are represented byresistors, voltage and current respectively. The ambient air tempera-ture (Ta) is represented to be a voltage potential. Also, the absorbedfraction of solar irradiance minus the fraction of solar irradianceconverted to electricity, (a � hel)G is equal to the current output.

Fig. A1. Equivalent thermal circuit of PV/T collector [25].

RL,Rf, _qo, _qf , Tc and Tf are the total thermal loss resistance from thetop of the PV module to the ambient, the combined radiative andconvective heat resistance from the PVmodule to the air in the duct,the total rate of heat transferred from the collector to air in duct andambient, the rate of heat transferred from the collector to the air induct, the solar cell temperature and the output fluid temperaturerespectively. Bambrook [25] reported the following equations forthe temperature rise of theoutlet air above the ambient temperature(DT) and the rate of heat output per unit area ( _Qf ):

DT ¼ ða� helÞGRL�1� exp

�� Ac

_mcpRo

��(A1)

_Qf ¼_mcpAc

ða� helÞGRL�1� exp

�� Ac

_mcpRo

��(A2)

where _m is the air mass flow rate hel is the temperature correctedelectrical efficiency, cp is the specific heat of the air, a is the PVabsorption coefficient, G is the solar irradiance, Ro is the total heatresistance Ro ¼ RL þ Rf and Ac, is the collector area (Ac ¼ WL).

Sproul et al. [32] worked on a model similar to Fig. A1 and re-ported the average collector temperature, which for a PV/Tcollectorwould be the average temperature of the solar cell as below:

~Tc ¼ Tin þ DTmh1� F

00i(A3)

where F00is the flow factor which is given by Ref. [33]:

F00 ¼

_mcpRoAc

�1� exp

�� Ac

_mcpRo

��(A4)

Bambrook and Sproul [34] considered model presented byBambrook [25] and reported the maximum temperature rise (DTm)for a PV/T collector assuming as follows:

DTm ¼ ða� helÞGRL (A5)

All of above models did not take into account the impact of thesky temperature. Considering the sky temperature the model re-ported by Bambrook [25] can be modified to the following thermalcircuit model:

Fig. A2. Equivalent thermal circuit of PV/T air collector considering sky temperature.

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e35 33

Rr,o and Rc,o are the radiative and convective thermal resistancefrom the topof the PVmodule to the sky and ambient air respectively.Rr,o is defined using the equation provided by Cengel [31] as below:

Rr;o ¼ 1hr;o

¼ 1

εs�~T2c þ T2sky

��~Tc þ Tsky

� (A6)

where hr,o is the radiation heat transfer coefficient, s is the Ste-faneBoltzmann number (5.67 � 10�8 W/m2 K4), ε is the emissivityof the outer surface, ~Tc is the average solar cell temperature. For thisstudy the average sky temperature at the times of peak solar irra-diance was used taking into account hourly weather data in winterin Sydney from TMY2 data (Tsky ¼ 0 �C) [29].

In addition Rc,o is calculated using the equation reported bySharples and Charlesworth [35] as follow:

Rc;o ¼ 1hc;o

¼ 12:2Vw þ 8:3

(A7)

where hc,o is convective heat transfer coefficient and Vw is the windvelocity.

It is also assumed that the thermal resistance from the collectorto the roof (Rb) is infinite because of high levels of insulation at theback of the collector and the heat loss from the back of the collectoris therefore neglected. A combined radiative and convective heatresistance from the PV module to the air in the duct (Rf) is given bythe following equation:

1Rf

¼ 1Rfc1

þ 1Rrad þ Rfc2

(A8)

where Rfc1 and Rfc2 are the convective thermal resistances from thePV and the back of duct to the air respectively, and Rrad is radiativethermal resistance from the PV array to the air in duct. Rfc1 and Rfc2are calculated using the following expressions given by Cengel [31]:

Rfc1 ¼ Rfc2 ¼DH

kNu(A9)

Nu ¼ 0:023 Re0:8 Prn (A10)

whereNu is Nusselt number, DH is the hydraulic diameter of the PV/T air duct, Re is the Reynolds number, k is the thermal conductivityof air (k ¼ 0.02551 W/m K), and Pr is the Prandtl number(Pr ¼ 0.7296) and n ¼ 0.4 for heating and 0.3 for cooling based onthe data from Cengel [31].

Fig. A3 indicates the evolution of above model to a simplermodel taking into account the combination of ambient air tem-perature (Ta) and the sky temperature (Tsky) as the effective airtemperature (Teff) and combining Rr,o and Rc,o to RL which is totalthermal loss resistance from the top of the PV module to theambient.

Fig. A3. Evolution of equivalent thermal circuit of PV/T collector considering skytemperature.

The effective air temperature (Teff) and total thermal loss resis-tance (RL) are defined by the following equations:

Teff ¼TskyRc;o þ TaRr;o

R þ R(A11)

c;o r;o

RL ¼Rc;oRr;o

Rc;o þ Rr;o(A12)

Eq. (A1), Eq. (A2) and Eq. (A5) are modified by taking into ac-count unequal values for the effective air temperature and ambientair temperature as follows:

DT ¼hTeff � Ta þ ða� helÞGRL

i 1� exp

��Ac_mCpRo

�!(A13)

_Qf ¼_mCpA

hTeff � Ta þ ða� helÞGRL

i 1� exp

��Ac_mCpRo

�!(A14)

DTm ¼ Teff � Ta þ ða� helÞGRL (A15)

Hence the average temperature of the solar cell can be definedas:

~Tc ¼ Ta þhTeff � Ta þ ða� helÞGRL

i�1�

�RLRo

�F

00�

(A16)

Also, the electrical power output per unit area, _WPV, can becalculated as follow:

_WPV ¼ helG (A17)

The correlation between average temperature of the cells andelectrical efficiency is given by Ref. [36]:

hel ¼ href

h1þ bref

�~Tc � TSTC

�i(A18)

Therefor the PV electrical power output can be written as:

_WPV ¼ Gnhref

h1þ bref

�~Tc � TSTC

�io(A19)

where hel is temperature corrected efficiency, href is the PV effi-ciency under standard test conditions, and bref is the temperaturecoefficient, ~Tc is the average temperature of the solar cell and TSTC isthe standard test condition temperature.

This paper focuses on a PV/T air collector linked to an air dis-tribution system utilized for direct space heating in a house inSydney. As a result, the total pressure losses of the system includingthe photovoltaic thermal collector and air distribution systempressure drop is to be computed in order to determine the fanpower for the whole system. The air distribution system is locatedwithin the roof space and consists of flexible duct, fittings, bends, afilter box, an axial fan, a motorized damper, and air grilles.

Cengel and Turner [37] showed that the electrical power of thefan is proportional to hydraulic powerð _WhÞ, fan efficiency andmotor efficiency based on the following equation:

_W fan ¼_Wh

hfanhmotor(A20)

where hfan is, the fan efficiency and hmotor is, the motor efficiency.In this paper combined fan and motor efficiency is assumed to

be equal 0.25 based on the fan manufacturer's data [38].The hydraulic fan power is calculated using the following

expression by Cengel and Turner [37]:

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e3534

_Wh ¼ _VDP ¼ _V�DPk þ DPf

�(A21)

_V ¼_mr

(A22)

where DPk kinetic pressure is drop and DPf is the friction pressuredrop. Cengel and Turner [37] determined these parameters aredetermined as below:

DPk ¼12r�v2e � v2i

�(A23)

DPf ¼ DPM þ DPm (A24)

where r is the average air density and is assumed 1.18 kg/m3 for theaverage air temperature of 25 �C based on the data from Cengel andTurner [37], ve and vi are the exit and input air velocity to thesystem respectively, DPM is the major pressure drop which includesthe pressure loss of collector duct and air distribution ductwork inthe house.

DPM ¼Xni¼1

fiLiDHi

rV2iavg

2(A25)

where L is the length of duct, DH is the hydraulic diameter of duct, fis the friction factor and Vavg is the average velocity of the air induct.

The hydraulic diameter of the duct is given by Cengel andCimbala [39]:

DH ¼ 2DWðDþWÞ (A26)

where D is the depth of duct and W is the width of duct.The friction factor is calculated with the following equation by

Ref. [37].

1ffiffiffif

p ¼ �1:8 log

"6:9Re

þ�ε=DH

3:7

�1:11#

(A27)

where ε is the roughness and Re is the Reynolds number which isgiven by Ref. [39]:

Re ¼ VavgDH

v(A28)

where v is kinematic viscosity of air (1.562 � 10�5 m2/s).DPm account for the minor losses due to the air intake mesh,

bends, fittings, filter box, dampers and air grilles in the house and isgiven by

DPm ¼Xni¼1

Ki

rV2iavg

2(A29)

Finally, the rate of effective thermal energy output ( _Qeff ) iscalculated using Eq. (3) and inserting equations for _Qf , _WPV and_W fan as below:

Q:

eff ¼m:cpAc

"Teff � Ta þ ða� helÞGRL

1� exp

��Ac

m:CpRo

�!#

þ"Ghref

h1þ bref

�~Tc � 25

�i� V

:ðDPk þ DPM þ DPmÞ

hfanhmotor

#

� 3:22

(A30)

Appendix B. Additional details for optimizationmethodology:

This appendix gives further details regarding the optimizationmethodology:

1) The average maximum level of solar irradiance in winter(G ¼ 736 W/m2) was used based on data derived from the Solarenergy laboratory [29] as the peak thermal and electrical outputvalues of the collector are evaluated in this research.

2) As the building was assumed to be in Sydney, the averagemaximum ambient air temperature in winter was used (e.g.17 �C in July), based on the data derived from the AustralianBureau of Meteorology [40]. Assuming a desirable outlet airtemperature of 27 �C for the PV/T air collector based on thesupply air temperature of typical direct air heating systems inSydney, the air temperature rise is considered equal to 10 �C(DT ¼ 10 �C).

3) The minimum wind velocity of 2 m/s was considered based onthe Australian Bureau of Meteorology [40].

4) Although the air distribution system is considered to consist of anumber of fixed components as specified in Appendix A, acareful design is required to minimize the fan input power.Basically an air distribution system with a lower air velocitycauses lower fan energy consumption. The associated ductworkis assumed to be optimally sized based on a fixed air velocity of2.5 m/s for the main ductwork. This velocity is less than theminimum typical main duct air velocity (5 m/s) recommendedby AIRAH DA03 Application Manual [41] for a low velocity sys-tem. This assumption leads to a reduction in the pressure lossesfrom air handling system. In addition, the average velocity of theair through the ductwork is given by:

Vavg ¼_V

Aduct(B1)

Considering the duct as a round duct ðAduct ¼ pð∅=2Þ2Þ andsubstituting Eq. (B1) in above equation the main duct diametercan be calculated from the following equation:

∅ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 _m

prVduct

s(B2)

where Vduct, r, _m are the main duct air average velocity, averageair density (r ¼ 1.204 kg/m3 Cengel and Turner [37]) and air massflow rate respectively. Simply having the value of optimum massflow rate and assuming an average air velocity of 2.5 m/s theoptimum main duct diameter is calculated using Eq. (B2) for PV/Tcollectors with various dimensions. On the other hand the largestavailable size of the domestic flexible duct on the market is500 mm based on the manufacturer data [42]. So if the result ofthe above calculation for a main duct diameter is higher than500 mm, the optimum main duct value needs to be considered tobe equal to 500 mm.

5) Optimization of the depth (D) and the air mass flow rate perunit collector area ( _m=Ac) to produce a maximum rate of effec-tive thermal energy output ( _Qeff ) is achieved by varying thedepth and the air mass flow rate per unit collector area whilstkeeping the value of the air temperature rise constant(DT ¼ 10 �C).

M. Farshchimonfared et al. / Renewable Energy 76 (2015) 27e35 35

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