Research ArticleEffects of Nonuniform Incident Illumination on the ThermalPerformance of a Concentrating Triple Junction Solar Cell

Fahad Al-Amri1 and Tapas Kumar Mallick2

1 College of Technology PO Box 7650 Dammam 31472 Saudi Arabia2 Environment amp Sustainability Institute University of Exeter Penryn Campus Cornwall TR10 9EZ UK

Correspondence should be addressed to Fahad Al-Amri alamrifahadhotmailcom

Received 24 February 2014 Revised 20 May 2014 Accepted 21 May 2014 Published 16 June 2014

Academic Editor Mark van Der Auweraer

Copyright copy 2014 F Al-Amri and T K MallickThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A numerical heat transfer model was developed to investigate the temperature of a triple junction solar cell and the thermalcharacteristics of the airflow in a channel behind the solar cell assembly using nonuniform incident illumination The effectsof nonuniformity parameters emissivity of the two channel walls and Reynolds number were studied The maximum solar celltemperature sharply increased in the presence of nonuniform light profiles causing a drastic reduction in overall efficiency Thisresulted in two possible solutions for solar cells to operate in optimum efficiency level (i) adding new receiver plate with highersurface area or (ii) using forced cooling techniques to reduce the solar cell temperature Thus surface radiation exchanges insidethe duct and Re significantly reduced the maximum solar cell temperature but a conventional plain channel cooling system wasinefficient for cooling the solar cell at medium concentrations when the system was subjected to a nonuniform light distributionNonuniformity of the incident light and surface radiation in the duct had negligible effects on the collected thermal energy

1 Introduction

Nonuniformity of the incident illumination in concentratedphotovoltaic (CPV) systems occurs as a result of usingconcentrating optics that reflect or refract light flux on thesolar cell surface The optimal optical system design wouldconcentrate radiation on the solar cell surface in a uniformdistribution However existing concentrators condense themajority of the incident flux onto a limited area of the cellsurface and most areas of the cell receive the remainingsmall amount of radiation This situation produces a highlynonuniform irradiance distribution on the photovoltaic cellsThis nonuniformity has two main effects on the solar cell(i) electrical impacts such as high ohmic drops and theflow of an internal current and (ii) thermal impacts suchas increased temperatures in some regions of the cell thatcause hot spots and drastically reduce the overall efficiencyand power outputThis reduction of power limits the viabilityof photovoltaic cells as an alternative energy source

Previous studies reported thermal behavior of the CPVsolar cells such as Min et al [1] and Cotal and Frost [2]

who developed a theoretical thermal model for predictingthe temperature of a multijunction solar cell based on passivecooling systems Vincenzi et al [3] used a silicon waferwith microchannels circulating water directly beneath thecells as an active cooling system Moshfegh and Sandberg[4] performed both a numerical study and an experimentalstudy of the flow and heat transfer characteristics of naturalair convection behind solar cells The results demonstratedthat surface radiation significantly affects the temperatureand conversion efficiency of the solar panel Bhargava etal [5] Garg and Adhikari [6] and Hegazy [7] analyzedthe performance of hybrid photovoltaicthermal (PVT) airheating collectors and reported the effects of different designand controlling parameters on system performance Al-Amriand Mallick [8 9] recently developed a numerical heattransfermodel for predicting themaximum cell temperaturesof multijunction concentrating solar cell systems that wereactively cooled by water-forced and air-forced convectionThe maximum cell temperature was strongly dependent onthe inlet velocity and the channel width Teo et al [10]experimentally investigated an active cooling system for

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2014 Article ID 642819 12 pageshttpdxdoiorg1011552014642819

2 International Journal of Photoenergy

photovoltaic modules A parallel array of ducts with aninletoutlet manifold was attached to the back of the PVpanel When an air-based active cooling mechanism wasemployed the temperature dropped effectively leading toa 12 to 14 increase in the efficiency of the solar cellsKumar et al [11] presented a numerical heat transfer modelfor a novel concentrating photovoltaic design for the activesolar panel initiative system (ASPIS) The incorporation ofmicrochannels is an innovative way to remove a large amountof heat from a small cell surface that is associated with a highconcentration [12 13] Tuckerman and Pease [14] pioneeredthe use of the microchannel heat sink for cooling planarintegrated circuits Several additional studies [15ndash17] wereperformed to analyze the heat transfer performance and thepressure drop of flows in microchannels Qu and Mudawar[18] investigated the pressure drop and heat transfer perfor-mance of a laminar flow in a microchannel both experimen-tally andnumericallyThemicrochannelwas 213120583mwide and713 120583m deep The results of this study indicated that the flowin this system is still governed by conventional equations forthe conservation of mass momentum and energy Micheliet al [19] recently presented an overview of micro- andnanotechnologies that are applicable to passive CPV coolingand associated manufacturing technologies (eg monolithicapplications) The authors reported that carbon nanotubesand high-conductive coating are the technologies that offerthe best CPV cooling performanceThis technological reviewwas also critically assessed

The problem of nonuniform illumination patterns hasbeen studied extensively However most of these studiesfocused on the effects of nonuniformity on electrical perfor-mance [12] Few studies investigated the thermal impacts ofnonuniformity in radiation profiles Franklin and Conventry[20] demonstrated both numerically and experimentally thata silicon solar cell under nonuniform focused illuminationexperiences a drop in both open-circuit voltage and effi-ciency compared to a cell under uniform illumination Inaddition the results indicated that the temperature profileclosely resembles a Gaussian distribution Coventry et al[21] measured the thermal and electrical output from aconcentrating PVthermal collector at 25ndash30 suns Whilethe overall concentration ratio was approximately 30x theillumination flux intensity across the width of the solarcell was far from even Domenech-Garret [22] investigatedthe effects of a combination of nonuniform illuminationand nonuniform temperature on a silicon solar cell Heused Gaussian and inverse Gaussian profiles to describe thetemperature in the cell which simulated a general cell coolingdevice Chemisana and Rosell [23] studied the effects ofGaussian and anti-Gaussian temperature profiles on the cellrsquoselectrical parameters under different types of concentratedillumination both numerically and experimentally The elec-tric conversion efficiency increases under a Gaussian temper-ature profile and decreases when the cell is subjected to aninverse Gaussian temperature curve The authors concludedthat the temperature profile could be tailored to maximizeefficiency for determined radiation conditions

The studies presented above failed to adequately addressthe effects of nonuniformity of the incident flux on

the temperature profiles of CPV systems Thus furtherstudies of heat transfer characteristics under nonuniformfluxare necessary to understand the effect of this nonuniformityon the overall efficiency of the solar cell The objective of thisinvestigation was to extend the work of Al-Amri and Mallick[8] by assessing the effect of nonuniform incident light onthe temperature distribution of a concentrated triple junctionsolar cell

A heat transfer model was developed to simulate thetemperature distribution on the concentrated solar cellsunder a nonuniform incident light The results of this studyare important for determining the overall efficiency of aCPV system and for promoting the development of a coolingsystem and new concentrators

2 Mathematical Formulations

The geometry of the physical problem and the coordinatesystem considered in this study are presented in Figure 1Thecomponents consist of a triple junctionGalnPGaAsGe solarcell a Cu-Ag-Hg front contact to the solar cell a 2mm glasscover a 15mm thick aluminum back plate and a 05mmadhesive material used to attach the solar cell to the coverglass and the rear aluminum plateThe cooling air is forced toflow within the ducts behind the back plate whose externalwall is assumed to be adiabatic The two walls of the duct areassumed to be gray opaque and diffuse surfaces The inletand exit channel areas are assumed to be black at the ambienttemperature (119879

infin= 27

∘C) and the exit bulk temperature(119879119890) respectively The efficiency of the triple junction solar

cell (120578) is 038 and the reflectivity is assumed to be zero forall solar cell components except for the front contact metalwhich is assumed to be 085 Table 1 shows the dimensionsand the thermophysical properties of the components usedin the model

The illumination distribution incident on the cell surfacein a concentrator system consisting of reflective troughs ishighly non-uniform and takes the shape of a Gaussian profile[12] Thus the illumination intensity formula adopted in thecurrent model is

119868 (119911) =2119862

104SDradic2120587119890minus(2(119911minus1199110)

2)SD2

(1)

where 1199110is the position of the maximum illumination on the

cell surface and SD is the standard deviation whichmeasureshow far incident illumination is spread out

The intensity of incident light 119868(119911) can be written in thenondimensional form as follows

119868 (119885) =119868 (119911)

119868avg (2)

where 119868avg is the average incident illumination across theentire cell surface

The physical problem depicted in Figure 1 is governedby equations of energy conservation in both the multiwalls(solar cell assembly) and the fluid inside the duct and byequations of continuity momentum in the fluid region andradiation constraint along each of the two duct surfaces as

International Journal of Photoenergy 3

y = b

Glass

Airflow

z

y

Cu-Ag-Hg

Adhesive material

Ga In P

Ga As

Germanium

Adhesive material

Aluminum back plate

ZcLc

Incident radiation

s8

s7

s6 s5

s4

s3

s2

s1y = ys2-3

Figure 1 Schematic of the geometry of the employed system

Table 1 Thermophysical properties assumed in the simulation

Component Material Length (mm) Thickness (mm) Thermal conductivityWmsdotK Absorptivity

Solar cellGuInp 10 01 73 004GuAs 10 02 65 040Ge 10 02 60 050

Front contact Cu + Ag + Hg 2 0025 300 015Glass cover Low-iron glass 10 2 12 004Adhesive 10 05 50 002Back plate Aluminium 10 15 238

a result of surface radiation exchange between the two wallsof the channel Using typical boundary layer assumptions andneglecting the axial conduction of heat in both the solid andthe fluid regions the corresponding equations governing theconjugate laminar mixed flow and the heat transfer in theentry region of the two parallel plates can be expressed in thefollowing dimensionless forms

Continuity Equation Consider

120597119881

120597119884+120597119880

120597119885= 0 (3)

Momentum Equation Consider

119881120597119880

120597119884+ 119880

120597119880

120597119885= minus

119889119875

119889119885+119866lowast

119903

Re(120579 minus 120579

infin) +

1205972

119880

1205971198842 (4)

Energy Equation for the Fluid Consider

119881120597120579119891

120597119884+ 119880

120597120579119891

120597119885=

1

Pr1205972

120579119891

1205971198842 (5)

Energy Equation for Each Solid Material in Solar Cell Assem-bly Consider

1205972

120579119904

1205971198842= 0 (6)

The integral formof the continuity equation (1) can bewrittenin the following form [24]

119865 = int

1

0

119880119889119884 = 1 (7)

The radiation constraint equations can be derived by applyingthe first law of thermodynamic per unit area of the surface on

4 International Journal of Photoenergy

No incidentlight in thisportion ofsolar cell

Airflow

z

y

I(z)

P = Ps5+ Ps4 + Ps3

qs7

qs8

qs5

qs4

qs3

Ps4

Ps5

Ps5

Ps3

(1 minus 120572s6) I(z)

qs8 = 120572s8 I(z)

qs7 = 120572s7 I(z)(1 minus 120572s8)

qs6 = 120572s6 I(z)(1 minus 120572s8 minus 120572s7)qs6

qs5 = 120572s5 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs4 = 120572s4 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs3 = 120572s3 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

Ps5 = 120572s5 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps4 = 120572s4 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps3 = 120572s3 120578I(z)(1 minus 120572s8 minus 120572s7)

s8

s7

s6s5

s4

s3

s2

s1

Figure 2 Energy fluxes in solar cell assembly

the channel plate element [25] which yields the following twoequations

Surface 1 (119884 = 0) Consider

119873rad1205794

infin[1

2minus (

119885Re2

) (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)Re2

) (1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205762

1205762

(120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+) + 119873rad120579

4

1199082(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205761

1205761

[minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+]

+ 119873rad1205794

1199081

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+

Where 119873rad = 1205901199023

avg1198874

1198964119891

(8)

Surface 2 (119884 = 1) Consider

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1++ 119873rad120579

4

infin[1

2minus (

119885Re2

)

times (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)

2Re)

times(1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205761

1205761

(minus120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=0+) + 119873rad120579

4

1199081(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205762

1205762

[120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+]

+ 119873rad1205794

1199082

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(9)

When surface radiation is absent (ie 120576 = 0) (7) and (8)are reduced to

120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0

= 0

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+=120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(10)

The conjugate convection field equations are subject to thefollowing dimensionless boundary conditions

In the Fluid Region For 119885 = 0 and 0 lt 119884 lt 1

119880 = 1 119881 = 119875 = 0 120579 = 120579infin (11a)

International Journal of Photoenergy 5

Uniform flux

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

S = 00005

S = 0001

S = 0002

(a)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

zm = 2 zm = 5 zm = 8

(b)

Figure 3 Incident illumination profiles for 119862 = 100 at (a) different values of SD and (b) different values of 1199110

0

100

200

300

400

500

600

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

SD= 0002

SD= 0001

SD= 00005

Figure 4 Variations in solar cell temperature obtained usingdifferent values of SD when 119911

0= 05 Re = 500 120576 = 095 and

119862 = 100

For 119885 gt 0 and 119884 = 0

119880 = 119881 = 0 (11b)

For 119885 gt 0 and 119884 = 1

119880 = 119881 = 0 (11c)

In the Solid Region (solar cell assembly) In the present modelwe assumed that some of the incident light is absorbed byeach surface of solar cell assembly layers based on the valueof absorptivity and converted into thermal andor electricalenergyThen heat is diffused to the next layer by conductionEach layer of the solar cell is in the order of 100 to 200microns therefore the skin depth of each layer is assumedto be thin and the radiation effects are confined to the surface

050

100150200250300350400450500

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

z0 = 2mmz0 = 5mm

z0 = 8mm

Figure 5 Variations in solar cell temperature obtained usingdifferent values of 119911

0 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

layerThe boundary conditions in this region can be deducedby applying the continuity of temperature and heat flux atthe solid-solid interface Taking a surface energy balance ateach solid-solid interface (see Figure 2) leads to the followingboundary condition equations

For 119885 gt 0 and 119884 = 119884119904(119894)minus(119894+1)

119894 = 1 and 2

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= KR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11d)

For 119884 = 119884119904(119894)minus(119894+1)

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

119894 = 3 and 4120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

+(1 minus 120578) 120572

119904119894(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR119904119894minus119891

(11e)

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

2 International Journal of Photoenergy

photovoltaic modules A parallel array of ducts with aninletoutlet manifold was attached to the back of the PVpanel When an air-based active cooling mechanism wasemployed the temperature dropped effectively leading toa 12 to 14 increase in the efficiency of the solar cellsKumar et al [11] presented a numerical heat transfer modelfor a novel concentrating photovoltaic design for the activesolar panel initiative system (ASPIS) The incorporation ofmicrochannels is an innovative way to remove a large amountof heat from a small cell surface that is associated with a highconcentration [12 13] Tuckerman and Pease [14] pioneeredthe use of the microchannel heat sink for cooling planarintegrated circuits Several additional studies [15ndash17] wereperformed to analyze the heat transfer performance and thepressure drop of flows in microchannels Qu and Mudawar[18] investigated the pressure drop and heat transfer perfor-mance of a laminar flow in a microchannel both experimen-tally andnumericallyThemicrochannelwas 213120583mwide and713 120583m deep The results of this study indicated that the flowin this system is still governed by conventional equations forthe conservation of mass momentum and energy Micheliet al [19] recently presented an overview of micro- andnanotechnologies that are applicable to passive CPV coolingand associated manufacturing technologies (eg monolithicapplications) The authors reported that carbon nanotubesand high-conductive coating are the technologies that offerthe best CPV cooling performanceThis technological reviewwas also critically assessed

The problem of nonuniform illumination patterns hasbeen studied extensively However most of these studiesfocused on the effects of nonuniformity on electrical perfor-mance [12] Few studies investigated the thermal impacts ofnonuniformity in radiation profiles Franklin and Conventry[20] demonstrated both numerically and experimentally thata silicon solar cell under nonuniform focused illuminationexperiences a drop in both open-circuit voltage and effi-ciency compared to a cell under uniform illumination Inaddition the results indicated that the temperature profileclosely resembles a Gaussian distribution Coventry et al[21] measured the thermal and electrical output from aconcentrating PVthermal collector at 25ndash30 suns Whilethe overall concentration ratio was approximately 30x theillumination flux intensity across the width of the solarcell was far from even Domenech-Garret [22] investigatedthe effects of a combination of nonuniform illuminationand nonuniform temperature on a silicon solar cell Heused Gaussian and inverse Gaussian profiles to describe thetemperature in the cell which simulated a general cell coolingdevice Chemisana and Rosell [23] studied the effects ofGaussian and anti-Gaussian temperature profiles on the cellrsquoselectrical parameters under different types of concentratedillumination both numerically and experimentally The elec-tric conversion efficiency increases under a Gaussian temper-ature profile and decreases when the cell is subjected to aninverse Gaussian temperature curve The authors concludedthat the temperature profile could be tailored to maximizeefficiency for determined radiation conditions

The studies presented above failed to adequately addressthe effects of nonuniformity of the incident flux on

the temperature profiles of CPV systems Thus furtherstudies of heat transfer characteristics under nonuniformfluxare necessary to understand the effect of this nonuniformityon the overall efficiency of the solar cell The objective of thisinvestigation was to extend the work of Al-Amri and Mallick[8] by assessing the effect of nonuniform incident light onthe temperature distribution of a concentrated triple junctionsolar cell

A heat transfer model was developed to simulate thetemperature distribution on the concentrated solar cellsunder a nonuniform incident light The results of this studyare important for determining the overall efficiency of aCPV system and for promoting the development of a coolingsystem and new concentrators

2 Mathematical Formulations

The geometry of the physical problem and the coordinatesystem considered in this study are presented in Figure 1Thecomponents consist of a triple junctionGalnPGaAsGe solarcell a Cu-Ag-Hg front contact to the solar cell a 2mm glasscover a 15mm thick aluminum back plate and a 05mmadhesive material used to attach the solar cell to the coverglass and the rear aluminum plateThe cooling air is forced toflow within the ducts behind the back plate whose externalwall is assumed to be adiabatic The two walls of the duct areassumed to be gray opaque and diffuse surfaces The inletand exit channel areas are assumed to be black at the ambienttemperature (119879

infin= 27

∘C) and the exit bulk temperature(119879119890) respectively The efficiency of the triple junction solar

cell (120578) is 038 and the reflectivity is assumed to be zero forall solar cell components except for the front contact metalwhich is assumed to be 085 Table 1 shows the dimensionsand the thermophysical properties of the components usedin the model

The illumination distribution incident on the cell surfacein a concentrator system consisting of reflective troughs ishighly non-uniform and takes the shape of a Gaussian profile[12] Thus the illumination intensity formula adopted in thecurrent model is

119868 (119911) =2119862

104SDradic2120587119890minus(2(119911minus1199110)

2)SD2

(1)

where 1199110is the position of the maximum illumination on the

cell surface and SD is the standard deviation whichmeasureshow far incident illumination is spread out

The intensity of incident light 119868(119911) can be written in thenondimensional form as follows

119868 (119885) =119868 (119911)

119868avg (2)

where 119868avg is the average incident illumination across theentire cell surface

The physical problem depicted in Figure 1 is governedby equations of energy conservation in both the multiwalls(solar cell assembly) and the fluid inside the duct and byequations of continuity momentum in the fluid region andradiation constraint along each of the two duct surfaces as

International Journal of Photoenergy 3

y = b

Glass

Airflow

z

y

Cu-Ag-Hg

Adhesive material

Ga In P

Ga As

Germanium

Adhesive material

Aluminum back plate

ZcLc

Incident radiation

s8

s7

s6 s5

s4

s3

s2

s1y = ys2-3

Figure 1 Schematic of the geometry of the employed system

Table 1 Thermophysical properties assumed in the simulation

Component Material Length (mm) Thickness (mm) Thermal conductivityWmsdotK Absorptivity

Solar cellGuInp 10 01 73 004GuAs 10 02 65 040Ge 10 02 60 050

Front contact Cu + Ag + Hg 2 0025 300 015Glass cover Low-iron glass 10 2 12 004Adhesive 10 05 50 002Back plate Aluminium 10 15 238

a result of surface radiation exchange between the two wallsof the channel Using typical boundary layer assumptions andneglecting the axial conduction of heat in both the solid andthe fluid regions the corresponding equations governing theconjugate laminar mixed flow and the heat transfer in theentry region of the two parallel plates can be expressed in thefollowing dimensionless forms

Continuity Equation Consider

120597119881

120597119884+120597119880

120597119885= 0 (3)

Momentum Equation Consider

119881120597119880

120597119884+ 119880

120597119880

120597119885= minus

119889119875

119889119885+119866lowast

119903

Re(120579 minus 120579

infin) +

1205972

119880

1205971198842 (4)

Energy Equation for the Fluid Consider

119881120597120579119891

120597119884+ 119880

120597120579119891

120597119885=

1

Pr1205972

120579119891

1205971198842 (5)

Energy Equation for Each Solid Material in Solar Cell Assem-bly Consider

1205972

120579119904

1205971198842= 0 (6)

The integral formof the continuity equation (1) can bewrittenin the following form [24]

119865 = int

1

0

119880119889119884 = 1 (7)

The radiation constraint equations can be derived by applyingthe first law of thermodynamic per unit area of the surface on

4 International Journal of Photoenergy

No incidentlight in thisportion ofsolar cell

Airflow

z

y

I(z)

P = Ps5+ Ps4 + Ps3

qs7

qs8

qs5

qs4

qs3

Ps4

Ps5

Ps5

Ps3

(1 minus 120572s6) I(z)

qs8 = 120572s8 I(z)

qs7 = 120572s7 I(z)(1 minus 120572s8)

qs6 = 120572s6 I(z)(1 minus 120572s8 minus 120572s7)qs6

qs5 = 120572s5 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs4 = 120572s4 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs3 = 120572s3 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

Ps5 = 120572s5 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps4 = 120572s4 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps3 = 120572s3 120578I(z)(1 minus 120572s8 minus 120572s7)

s8

s7

s6s5

s4

s3

s2

s1

Figure 2 Energy fluxes in solar cell assembly

the channel plate element [25] which yields the following twoequations

Surface 1 (119884 = 0) Consider

119873rad1205794

infin[1

2minus (

119885Re2

) (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)Re2

) (1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205762

1205762

(120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+) + 119873rad120579

4

1199082(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205761

1205761

[minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+]

+ 119873rad1205794

1199081

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+

Where 119873rad = 1205901199023

avg1198874

1198964119891

(8)

Surface 2 (119884 = 1) Consider

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1++ 119873rad120579

4

infin[1

2minus (

119885Re2

)

times (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)

2Re)

times(1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205761

1205761

(minus120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=0+) + 119873rad120579

4

1199081(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205762

1205762

[120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+]

+ 119873rad1205794

1199082

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(9)

When surface radiation is absent (ie 120576 = 0) (7) and (8)are reduced to

120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0

= 0

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+=120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(10)

The conjugate convection field equations are subject to thefollowing dimensionless boundary conditions

In the Fluid Region For 119885 = 0 and 0 lt 119884 lt 1

119880 = 1 119881 = 119875 = 0 120579 = 120579infin (11a)

International Journal of Photoenergy 5

Uniform flux

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

S = 00005

S = 0001

S = 0002

(a)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

zm = 2 zm = 5 zm = 8

(b)

Figure 3 Incident illumination profiles for 119862 = 100 at (a) different values of SD and (b) different values of 1199110

0

100

200

300

400

500

600

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

SD= 0002

SD= 0001

SD= 00005

Figure 4 Variations in solar cell temperature obtained usingdifferent values of SD when 119911

0= 05 Re = 500 120576 = 095 and

119862 = 100

For 119885 gt 0 and 119884 = 0

119880 = 119881 = 0 (11b)

For 119885 gt 0 and 119884 = 1

119880 = 119881 = 0 (11c)

In the Solid Region (solar cell assembly) In the present modelwe assumed that some of the incident light is absorbed byeach surface of solar cell assembly layers based on the valueof absorptivity and converted into thermal andor electricalenergyThen heat is diffused to the next layer by conductionEach layer of the solar cell is in the order of 100 to 200microns therefore the skin depth of each layer is assumedto be thin and the radiation effects are confined to the surface

050

100150200250300350400450500

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

z0 = 2mmz0 = 5mm

z0 = 8mm

Figure 5 Variations in solar cell temperature obtained usingdifferent values of 119911

0 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

layerThe boundary conditions in this region can be deducedby applying the continuity of temperature and heat flux atthe solid-solid interface Taking a surface energy balance ateach solid-solid interface (see Figure 2) leads to the followingboundary condition equations

For 119885 gt 0 and 119884 = 119884119904(119894)minus(119894+1)

119894 = 1 and 2

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= KR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11d)

For 119884 = 119884119904(119894)minus(119894+1)

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

119894 = 3 and 4120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

+(1 minus 120578) 120572

119904119894(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR119904119894minus119891

(11e)

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal ofPhotoenergy

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International Journal of

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ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Photoenergy 3

y = b

Glass

Airflow

z

y

Cu-Ag-Hg

Adhesive material

Ga In P

Ga As

Germanium

Adhesive material

Aluminum back plate

ZcLc

Incident radiation

s8

s7

s6 s5

s4

s3

s2

s1y = ys2-3

Figure 1 Schematic of the geometry of the employed system

Table 1 Thermophysical properties assumed in the simulation

Component Material Length (mm) Thickness (mm) Thermal conductivityWmsdotK Absorptivity

Solar cellGuInp 10 01 73 004GuAs 10 02 65 040Ge 10 02 60 050

Front contact Cu + Ag + Hg 2 0025 300 015Glass cover Low-iron glass 10 2 12 004Adhesive 10 05 50 002Back plate Aluminium 10 15 238

a result of surface radiation exchange between the two wallsof the channel Using typical boundary layer assumptions andneglecting the axial conduction of heat in both the solid andthe fluid regions the corresponding equations governing theconjugate laminar mixed flow and the heat transfer in theentry region of the two parallel plates can be expressed in thefollowing dimensionless forms

Continuity Equation Consider

120597119881

120597119884+120597119880

120597119885= 0 (3)

Momentum Equation Consider

119881120597119880

120597119884+ 119880

120597119880

120597119885= minus

119889119875

119889119885+119866lowast

119903

Re(120579 minus 120579

infin) +

1205972

119880

1205971198842 (4)

Energy Equation for the Fluid Consider

119881120597120579119891

120597119884+ 119880

120597120579119891

120597119885=

1

Pr1205972

120579119891

1205971198842 (5)

Energy Equation for Each Solid Material in Solar Cell Assem-bly Consider

1205972

120579119904

1205971198842= 0 (6)

The integral formof the continuity equation (1) can bewrittenin the following form [24]

119865 = int

1

0

119880119889119884 = 1 (7)

The radiation constraint equations can be derived by applyingthe first law of thermodynamic per unit area of the surface on

4 International Journal of Photoenergy

No incidentlight in thisportion ofsolar cell

Airflow

z

y

I(z)

P = Ps5+ Ps4 + Ps3

qs7

qs8

qs5

qs4

qs3

Ps4

Ps5

Ps5

Ps3

(1 minus 120572s6) I(z)

qs8 = 120572s8 I(z)

qs7 = 120572s7 I(z)(1 minus 120572s8)

qs6 = 120572s6 I(z)(1 minus 120572s8 minus 120572s7)qs6

qs5 = 120572s5 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs4 = 120572s4 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs3 = 120572s3 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

Ps5 = 120572s5 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps4 = 120572s4 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps3 = 120572s3 120578I(z)(1 minus 120572s8 minus 120572s7)

s8

s7

s6s5

s4

s3

s2

s1

Figure 2 Energy fluxes in solar cell assembly

the channel plate element [25] which yields the following twoequations

Surface 1 (119884 = 0) Consider

119873rad1205794

infin[1

2minus (

119885Re2

) (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)Re2

) (1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205762

1205762

(120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+) + 119873rad120579

4

1199082(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205761

1205761

[minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+]

+ 119873rad1205794

1199081

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+

Where 119873rad = 1205901199023

avg1198874

1198964119891

(8)

Surface 2 (119884 = 1) Consider

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1++ 119873rad120579

4

infin[1

2minus (

119885Re2

)

times (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)

2Re)

times(1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205761

1205761

(minus120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=0+) + 119873rad120579

4

1199081(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205762

1205762

[120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+]

+ 119873rad1205794

1199082

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(9)

When surface radiation is absent (ie 120576 = 0) (7) and (8)are reduced to

120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0

= 0

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+=120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(10)

The conjugate convection field equations are subject to thefollowing dimensionless boundary conditions

In the Fluid Region For 119885 = 0 and 0 lt 119884 lt 1

119880 = 1 119881 = 119875 = 0 120579 = 120579infin (11a)

International Journal of Photoenergy 5

Uniform flux

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

S = 00005

S = 0001

S = 0002

(a)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

zm = 2 zm = 5 zm = 8

(b)

Figure 3 Incident illumination profiles for 119862 = 100 at (a) different values of SD and (b) different values of 1199110

0

100

200

300

400

500

600

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

SD= 0002

SD= 0001

SD= 00005

Figure 4 Variations in solar cell temperature obtained usingdifferent values of SD when 119911

0= 05 Re = 500 120576 = 095 and

119862 = 100

For 119885 gt 0 and 119884 = 0

119880 = 119881 = 0 (11b)

For 119885 gt 0 and 119884 = 1

119880 = 119881 = 0 (11c)

In the Solid Region (solar cell assembly) In the present modelwe assumed that some of the incident light is absorbed byeach surface of solar cell assembly layers based on the valueof absorptivity and converted into thermal andor electricalenergyThen heat is diffused to the next layer by conductionEach layer of the solar cell is in the order of 100 to 200microns therefore the skin depth of each layer is assumedto be thin and the radiation effects are confined to the surface

050

100150200250300350400450500

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

z0 = 2mmz0 = 5mm

z0 = 8mm

Figure 5 Variations in solar cell temperature obtained usingdifferent values of 119911

0 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

layerThe boundary conditions in this region can be deducedby applying the continuity of temperature and heat flux atthe solid-solid interface Taking a surface energy balance ateach solid-solid interface (see Figure 2) leads to the followingboundary condition equations

For 119885 gt 0 and 119884 = 119884119904(119894)minus(119894+1)

119894 = 1 and 2

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= KR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11d)

For 119884 = 119884119904(119894)minus(119894+1)

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

119894 = 3 and 4120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

+(1 minus 120578) 120572

119904119894(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR119904119894minus119891

(11e)

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

4 International Journal of Photoenergy

No incidentlight in thisportion ofsolar cell

Airflow

z

y

I(z)

P = Ps5+ Ps4 + Ps3

qs7

qs8

qs5

qs4

qs3

Ps4

Ps5

Ps5

Ps3

(1 minus 120572s6) I(z)

qs8 = 120572s8 I(z)

qs7 = 120572s7 I(z)(1 minus 120572s8)

qs6 = 120572s6 I(z)(1 minus 120572s8 minus 120572s7)qs6

qs5 = 120572s5 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs4 = 120572s4 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

qs3 = 120572s3 (1 minus 120578)I(z)(1 minus 120572s8 minus 120572s7)

Ps5 = 120572s5 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps4 = 120572s4 120578I(z)(1 minus 120572s8 minus 120572s7)

Ps3 = 120572s3 120578I(z)(1 minus 120572s8 minus 120572s7)

s8

s7

s6s5

s4

s3

s2

s1

Figure 2 Energy fluxes in solar cell assembly

the channel plate element [25] which yields the following twoequations

Surface 1 (119884 = 0) Consider

119873rad1205794

infin[1

2minus (

119885Re2

) (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)Re2

) (1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205762

1205762

(120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+) + 119873rad120579

4

1199082(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205761

1205761

[minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+]

+ 119873rad1205794

1199081

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0+

Where 119873rad = 1205901199023

avg1198874

1198964119891

(8)

Surface 2 (119884 = 1) Consider

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1++ 119873rad120579

4

infin[1

2minus (

119885Re2

)

times (1 + 1198852Re2)

minus12

]

+ 119873rad1205794

119890[1

2minus (

(119871 minus 119885)

2Re)

times(1 + (119871 minus 119885)2Re2)

minus12

]

+ int

119871

0

1 minus 1205761

1205761

(minus120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=0+) + 119873rad120579

4

1199081(119885)

lowast1

2[1 + Re2(119885 minus 119885

1015840

)2

]

(minus32)

Re 1198891198851015840

=1 minus 1205762

1205762

[120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minusminus120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+]

+ 119873rad1205794

1199082

(119885) minus120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(9)

When surface radiation is absent (ie 120576 = 0) (7) and (8)are reduced to

120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=0

= 0

KR1199041minus119891

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=1+=120597120579119891

120597119884

100381610038161003816100381610038161003816100381610038161003816119884=1minus

(10)

The conjugate convection field equations are subject to thefollowing dimensionless boundary conditions

In the Fluid Region For 119885 = 0 and 0 lt 119884 lt 1

119880 = 1 119881 = 119875 = 0 120579 = 120579infin (11a)

International Journal of Photoenergy 5

Uniform flux

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

S = 00005

S = 0001

S = 0002

(a)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

zm = 2 zm = 5 zm = 8

(b)

Figure 3 Incident illumination profiles for 119862 = 100 at (a) different values of SD and (b) different values of 1199110

0

100

200

300

400

500

600

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

SD= 0002

SD= 0001

SD= 00005

Figure 4 Variations in solar cell temperature obtained usingdifferent values of SD when 119911

0= 05 Re = 500 120576 = 095 and

119862 = 100

For 119885 gt 0 and 119884 = 0

119880 = 119881 = 0 (11b)

For 119885 gt 0 and 119884 = 1

119880 = 119881 = 0 (11c)

In the Solid Region (solar cell assembly) In the present modelwe assumed that some of the incident light is absorbed byeach surface of solar cell assembly layers based on the valueof absorptivity and converted into thermal andor electricalenergyThen heat is diffused to the next layer by conductionEach layer of the solar cell is in the order of 100 to 200microns therefore the skin depth of each layer is assumedto be thin and the radiation effects are confined to the surface

050

100150200250300350400450500

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

z0 = 2mmz0 = 5mm

z0 = 8mm

Figure 5 Variations in solar cell temperature obtained usingdifferent values of 119911

0 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

layerThe boundary conditions in this region can be deducedby applying the continuity of temperature and heat flux atthe solid-solid interface Taking a surface energy balance ateach solid-solid interface (see Figure 2) leads to the followingboundary condition equations

For 119885 gt 0 and 119884 = 119884119904(119894)minus(119894+1)

119894 = 1 and 2

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= KR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11d)

For 119884 = 119884119904(119894)minus(119894+1)

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

119894 = 3 and 4120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

+(1 minus 120578) 120572

119904119894(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR119904119894minus119891

(11e)

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Photoenergy 5

Uniform flux

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

S = 00005

S = 0001

S = 0002

(a)

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10Distance along solar cell (mm)

Inci

dent

illu

min

atio

n (W

m2)

times105

zm = 2 zm = 5 zm = 8

(b)

Figure 3 Incident illumination profiles for 119862 = 100 at (a) different values of SD and (b) different values of 1199110

0

100

200

300

400

500

600

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

SD= 0002

SD= 0001

SD= 00005

Figure 4 Variations in solar cell temperature obtained usingdifferent values of SD when 119911

0= 05 Re = 500 120576 = 095 and

119862 = 100

For 119885 gt 0 and 119884 = 0

119880 = 119881 = 0 (11b)

For 119885 gt 0 and 119884 = 1

119880 = 119881 = 0 (11c)

In the Solid Region (solar cell assembly) In the present modelwe assumed that some of the incident light is absorbed byeach surface of solar cell assembly layers based on the valueof absorptivity and converted into thermal andor electricalenergyThen heat is diffused to the next layer by conductionEach layer of the solar cell is in the order of 100 to 200microns therefore the skin depth of each layer is assumedto be thin and the radiation effects are confined to the surface

050

100150200250300350400450500

Uniform flux

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

z0 = 2mmz0 = 5mm

z0 = 8mm

Figure 5 Variations in solar cell temperature obtained usingdifferent values of 119911

0 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

layerThe boundary conditions in this region can be deducedby applying the continuity of temperature and heat flux atthe solid-solid interface Taking a surface energy balance ateach solid-solid interface (see Figure 2) leads to the followingboundary condition equations

For 119885 gt 0 and 119884 = 119884119904(119894)minus(119894+1)

119894 = 1 and 2

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= KR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11d)

For 119884 = 119884119904(119894)minus(119894+1)

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

119894 = 3 and 4120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

+(1 minus 120578) 120572

119904119894(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR119904119894minus119891

(11e)

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

6 International Journal of Photoenergy

0

100

200

300

400

500

600

700

0 2 4 6 8 10Distance along solar cell (mm)

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(a) 1199110= 2mm

0 2 4 6 8 10Distance along solar cell (mm)

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

120576 = 0

120576 = 05

120576 = 095

(b) 1199110= 5mm

0

100

200

300

400

500

600

700

800

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(c) 1199110= 8mm

0

50

100

150

200

250

300

350

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

120576 = 0

120576 = 05

120576 = 095

(d) Uniform flux

Figure 6 Variations in solar cell temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

For 119884 = 1198841199045minus7

and 0 lt 119885 lt 119885119888or (119885119888+ 119871119888) lt 119885 lt 119871

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199045minus7

= kR1199047minus5

120597120579119904

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199045minus7

+(1 minus 120578) 120572

1199045(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

KR1199045minus119891

(11f)

For 119884 = 119884119904(119894)minus(119894+1)

and 119885119888le 119885 le (119885

119888+ 119871119888) 119894 = 3 4 and 5

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus119904(119894)minus(119894+1)

= kR119904(119894+1)minus(119894)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+119904(119894)minus(119894+1)

(11g)

For 119884 = 1198841199046-7

and 119885119888le 119885 le (119885

119888+ 119871119888)

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199046-7

= kR1199047-6

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199046-7

+1205721199046(1 minus 120572

1199048minus 1205721199047) 119868 (119885)

kR1199046minus119891

(11h)

For 119885 gt 0 and 119884 = 1198841199047-8

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884minus1199047-8

= kR1199048-7

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=119884+1199047-8

+1205721199047(1 minus 120572

1199048) 119868 (119885)

kR1199047minus119891

(11i)

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Photoenergy 7

0

100

200

300

400

500So

lar c

ell t

empe

ratu

re (∘

C)

0 2 4 6 8 10Distance along solar cell (mm)

(a) 1199110= 2mm

0

100

200

300

400

500

600

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

(b) 1199110= 5mm

0

100

200

300

400

500

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(c) 1199110= 8mm

0

50

100

150

200

250

300

Sola

r cel

l tem

pera

ture

(∘C)

0 2 4 6 8 10Distance along solar cell (mm)

Re = 500

Re = 1000

Re = 2000

(d) Uniform flux

Figure 7 Variations in solar cell temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

For 119885 gt 0 and 119884 = 1198841199048

120597120579

120597119884

10038161003816100381610038161003816100381610038161003816119884=1198841199048

=1205721199048119868 (119885)

kR1199048minus119891

(11j)

where kR119904(119894+1)minus119894

= 119896119904119894+1119896119904119894and kR

119904119894minus119891= 119896119904119894119896119891

3 Method of Solution

The conjugate convection field equations (ie continuitymomentum and energy) the two radiation constraint equa-tions and the corresponding boundary conditions are trans-formed into finite difference equations The conjugate con-vection field equations are initially solved using a numerical

marching technique The two radiation constraint equationsare then solved iteratively to update the wall temperaturesusing the Gauss-Seidel technique These updated wall tem-peratures are used to resolve the conjugate convection fieldequations The process is then repeated until convergenceis achieved The details of the solution procedure and thevalidation of the numerical code can be found inAl-Amri andMallick [9]

4 Results and Discussion

Figures 3(a) and 3(b) show the incident illumination profilesobtained using (1) for the concentration ratio 119862 = 100 atthree selected standard deviation values (SD = 00005 0001

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 International Journal of Photoenergy

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Distance along channel (mm)

Uniform flux

Bulk

tem

pera

ture

(∘C)

S = 00005

S = 0001

S = 0002

Figure 8 Variations in fluid bulk temperature obtained usingdifferent values of SD when 119911

119898= 05 Re = 500 120576 = 095 and

119862 = 100

z0 = 2mmz0 = 5mmz0 = 8mm

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

0 21 3 5 7 94 6 8 10Distance along channel (mm)

Uniform flux

Figure 9 Variations in fluid bulk temperature obtained usingdifferent values of 119911

119898 when SD = 0001 Re = 500 120576 = 095 and

119862 = 100

and 0002) and three selected values for the position of themaximum illumination on the cell surface (119911

0= 2 5 8mm)

respectivelyThese profiles were used as input data in the codeto study the effects of these parameters on the temperaturedistribution of the solar cell and the fluid bulk temperature Inaddition the effect of the emissivity of the two duct walls andthe Reynolds number in the presence of nonuniform incidentlight was investigated

41 Effect of a Nonuniform Light Profile on the Solar CellTemperature Figure 4 shows the variations of solar celltemperature with 119911

0= 5mm and with the axial distance

from the channel entrance for both uniform and nonuniformincident light at SD = 00005 0001 and 0002 For auniform distribution the results demonstrate that the solar

cell temperature first increases sharply near the entrance andthen gradually increases until reaching its maximum valueat the channel exit In addition there is a sudden increasein cell temperature at 119911 = 4mm and a sudden decreaseat 119911 = 6mm These changes occur due to the fact that nocurrent is generated in this portion of the solar cell and allof the light is converted to heat due to the presence of theCu-Ag-Hg front contact However this increment is small(asymp7∘C) and has minor effects on the overall efficiency For anonuniform distribution the temperature of the first quarterof the solar cell remains constant at ambient temperaturethen rapidly increases and reaches its maximum value atthe center of the cell The temperature then drops sharplyand gradually in the fourth quarter of the solar cell wherethe temperatures are higher than those observed in the firstquarter The profile of the temperature becomes thinner andthe maximum temperature increases as the SD decreases asshown in Figure 4 In addition the maximum temperature ofthe solar cell is much higher when using a nonuniform lightdistribution than when using a uniform light distributionIn this example the maximum temperature obtained using auniform light distributionwas 240∘Cwhich indicates that thesolar cell can be operated with a reasonable efficiency as themodern multijunction solar cells can adequately be operatedat this level of temperature [25 26] In contrast the resultsobtained using a nonuniform light distribution indicate thatthe cell efficiency will be drastically reduced due to the hightemperature of some portions of the cell surface

The effects of the position of maximum incident light onthe solar cell temperature distribution are shown in Figure 5A hot spot formed on the portion of the cell surface that wasexposed to a large amount of light This hot spot is slightlyhotter when 119911

0= 5mm (ie at the center of the solar cell)

due to the presence of the front contact metal in this regionFigures 6(a)ndash6(d) show the effect of the emissivity of the twoduct walls on the temperature profile of the solar cell for bothuniform and nonuniform incident light when 119911

0= 2 5 and

8mm and when 120576 = 0 (ie pure forced convection) 05 and095 The results presented in these figures demonstrate thatthe maximum solar cell temperature decreases significantlyas surface emissivity increases However the presence ofsurface radiation inside the channel does not change theshape of the temperature distribution for the nonuniformlight profile In this example as emissivity increased from120576 = 0 to 095 the maximum solar cell temperature wasreduced by approximately 35 when using nonuniformillumination and 20 when using uniform illuminationThelarge contribution of surface radiation to the nonuniformlight profile is due to the increase in the effective temperaturedifferential for surface radiation between the two duct wallsNevertheless the maximum temperature of the solar cellwhen using nonuniform light was still higher than thatobtained using a uniform light distribution For the uniformlight profile surface radiation exchange between the two ductwalls and between the heated duct wall and the inlet and exitareas of the channel leads to a reduction of maximum celltemperature from305∘C to 240∘Cwhich enables the solar cellto operate with a reasonable efficiency

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Photoenergy 9

120576 = 0

120576 = 05

120576 = 095

0

10

20

30

40

50

60

70

0 2 4 6 8 10Distance along channel (mm)

Bulk

tem

pera

ture

(∘C)

(a) 1199110= 2mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

120576 = 0

120576 = 05

120576 = 095

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Figure 10 Variations in fluid bulk temperature obtained using different values of 120576 when SD = 0001 Re = 500 and 119862 = 100

The results presented in Figures 7(a)ndash7(c) illustrate thevariations in solar cell temperature that occur with changesin axial distance from the channel entrance when usinguniform and nonuniform light distributions three selectedvalues of Reynolds number (500 1000 and 2000) and agiven value of 120576 = 095 The cell temperature decreases withincreasing Re but the profile maintains its general shapeThepercent reductions of maximum solar cell temperature thatoccurred as Reynolds number increased by 300 from 500to 2000 were 207 134 and 137 for 119911

0= 2 5 and

8mm respectivelyThis value was approximately 225whena uniform light distribution was employed These resultsindicate that for nonuniform light profiles increasing flowrate is more useful when the maximum incident light islocated near the channel entrance because the differencebetween the wall temperature and the fluid temperature ishigh in this region which enhances convective heat transfer

42 Effect of a Nonuniform Light Profile on the Fluid BulkTemperature Figures 8 and 9 show the variation of fluidbulk temperature with increasing axial distance from thechannel entrance under different incident light profiles Theresults presented in these two figures demonstrate that theincident light distribution strongly influences the fluid bulktemperature profile inside the channel However the effectof the incident light profile on the fluid bulk temperatureat the channel exit is insignificant Figures 10(a) 10(b) and10(c) show the effect of the emissivity of the two duct wallson the fluid bulk temperature when 119911

0= 2 5 and 8mm

respectively Surface radiation increased the temperature ofthe fluid closer to the entrance than pure convection (ie120576 = 0) due to radiative heat exchanges between the heatedand the adiabatic walls which redistribute the temperatureprofiles of the walls and hence the convective heat transferredto the flowing fluid The effect of emissivity on the fluid bulk

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

10 International Journal of Photoenergy

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70Bu

lk te

mpe

ratu

re (∘

C)

(a) 1199110= 2mm

Re = 500Re = 500Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

80

Bulk

tem

pera

ture

(∘C)

(b) 1199110= 5mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(c) 1199110= 8mm

Re = 500Re = 2000Re = 1000

0 2 4 6 8 10Distance along channel (mm)

0

10

20

30

40

50

60

70

Bulk

tem

pera

ture

(∘C)

(d) Uniform flux

Figure 11 Variations in fluid bulk temperature obtained using different values of Re when SD = 0001 120576 = 095 and 119862 = 100

temperature at the channel exit is insignificant as shown inFigures 10(a)ndash10(c)

The effects of Re on the fluid bulk temperature are shownin Figures 11(a)ndash11(d) for uniform and nonuniform lightflux The results demonstrate that the fluid bulk temperaturedecreases with increasing Re In this example the fluid bulktemperature decreases by approximately 43 as Re increasesby 300 for given values of controlling parameters Howeverthe collected thermal energy increases with increasing Re asa result of increasing flow rate

5 Conclusions

The effects of nonuniformity of the incident light on thetemperature and fluid bulk temperature of a triple junctionsolar cell were studied Nonuniformity was simulated using aGaussian distributionThe results indicated that nonuniform

illumination significantly alters the temperature distributionof the solar cell causing hot spots in some regions of thesolar cell Although increasing the emissivity of the two ductwalls and increasing Re can reduce the maximum solar celltemperature by as much as 35 and 20 respectively solarcell temperatures in the region exposed to the most incidentlight are high enough to cause degradation in solar cellefficiency at medium concentrationsThus secondary opticalconcentrators are required to redistribute the incident lighton the cell surface into a uniform shape Alternatively aninnovative cooling system such as the introduction of a sec-ondary jet flow into a channel should be used to redistributesolar cell temperatures and reduce themaximum temperatureto an acceptable value Surface emissivity and nonuniformillumination affect the fluid bulk temperature distributioninside the channel However the effect of these parameterson fluid bulk temperature at the channel exit is insignificant

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Photoenergy 11

Nomenclature

119887 Channel width m119862 Concentration ratioGrlowast Modified Grashof number

119892120573119902avg1198874

1205922

119870119891

119868 Intensity of incident lightWm2

119896 Thermal conductivityWm sdot K

ℓ Channel length m119871 Dimensionless plate length

= ℓ(119887 Re)119901 Pressure of fluid at any cross

section Nm21199011015840 Pressure defect at any cross

section 119901 minus 119901119904 Nm2

119901119904 Hydrostatic pressure minus120588

infin119892119911

Nm2119901infin Pressure of fluid at the

channel entrance Nm2119875 Dimensionless pressure at any

cross section= (1199011015840

minus 119901infin)(120588infin1199062

infin)

119873rad Radiation number 1205901199023avg1198874

1198964

119891

119902avg Average input heat flux= (1 minus 120578)119868avg Wm2

Re Reynolds number = 119906119900119887120592

119905 Solar cell assembly thicknessm

119879 Temperature at any point K119879infin Inlet temperature K

SD Standard deviation119906infin Entrance axial velocity ms

119906 Longitudinal velocitycomponent at any point ms

119880 Dimensionless longitudinalvelocity 119880 = 119906119906

infin

V Transverse velocitycomponent at any point ms

119881 Dimensionless transversevelocity 119881 = 119887 lowast V120592

119910 Horizontal coordinate m119884 Dimensionless horizontal

coordinate 119910119887119911 Vertical coordinate m1199110 Position of the maximum

incident light m119885 Dimensionless vertical

coordinate 119911(119887 lowast Re)

Greek Symbols

120592 Kinematic fluid viscosity120588 Fluid density kgm3120583 Dynamic fluid viscosity

kg119898 sdot 119904

120572 Absorptivity120578 Efficiency of the solar cell

120579 Dimensionless temperature atany point [=119896

119891119879119902avg119887]

120579infin Dimensionless inlettemperature at any point[=119896119891119879infin119902avg119887]

120576 Wall emissivity120590 Stefan Boltzmann constant =

567 lowast 10minus8

Wm2k4

Subscripts

119888 A Cu-Ag-Hg front contact119890 Exit119891 Fluid119904 Solid119908 Wall of the channelinfin Ambient or inlet1 Duct wall at 119884 = 0

2 Duct wall at 119884 = 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by King Abdulaziz City for Scienceand Technology in Saudi Arabia

References

[1] CMin C Nuofu Y XiaoliW Yu B Yiming and Z XingwangldquoThermal analysis and test for single concentrator solar cellsrdquoJournal of Semiconductors vol 30 no 4Article ID044011 2009

[2] H Cotal and J Frost ldquoHeat transfer modeling of concentratormultijunction solar cell assemblies using finite difference tech-niquesrdquo in Proceedings of the 35th IEEE Photovoltaic SpecialistsConference (PVSC 10) pp 213ndash218 Honolulu Hawaii USAJune 2010

[3] D Vincenzi F Bizzi M Stefancich et al ldquoMicromachined sili-con heat exchanger for water cooling of concentrator solar cellrdquoin Proceedings of the PV in Europe Conference and ExhibitionmdashFrom PV technology to Energy Solutions Conference Record pp7ndash11 Rome Italy 2002

[4] B Moshfegh and M Sandberg ldquoFlow and heat transfer in theair gap behind photovoltaic panelsrdquo Renewable and SustainableEnergy Reviews vol 2 no 3 pp 287ndash301 1998

[5] AK BhargavaH PGarg andRKAgarwal ldquoStudy of a hybridsolar system-solar air heater combined with solar cellsrdquo EnergyConversion and Management vol 31 no 5 pp 471ndash479 1991

[6] H P Garg and R S Adhikari ldquoConventional hybrid pho-tovoltaicthermal (PVT) air heating collectors steady-statesimulationrdquo Renewable Energy vol 11 no 3 pp 363ndash385 1997

[7] A A Hegazy ldquoComparative study of the performances of fourphotovoltaicthermal solar air collectorsrdquo Energy Conversionand Management vol 41 no 8 pp 861ndash881 2000

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

12 International Journal of Photoenergy

[8] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof high concentration solar cell by active coolingrdquo WorldRenewable Energy Forum ASES Denver Colo USA 2012

[9] FAl-Amri andTKMallick ldquoAlleviating operating temperatureof concentration solar cell by air active cooling and surfaceradiationrdquo Applied Thermal Engineering vol 59 no 1-2 pp348ndash354 2013

[10] H G Teo P S Lee and M N A Hawlader ldquoAn active coolingsystem for photovoltaic modulesrdquo Applied Energy vol 90 no 1pp 309ndash315 2012

[11] N S Kumar K Matty E Rita et al ldquoExperimental validationof a heat transfer model for concentrating photovoltaic systemrdquoAppliedThermal Engineering vol 33-34 no 1 pp 175ndash182 2012

[12] H Baig K C Heasman and T K Mallick ldquoNon-uniformillumination in concentrating solar cellsrdquo Renewable and Sus-tainable Energy Reviews vol 16 no 8 pp 5890ndash5909 2012

[13] A Royne C J Dey and D R Mills ldquoCooling of photovoltaiccells under concentrated illumination a critical reviewrdquo SolarEnergy Materials and Solar Cells vol 86 no 4 pp 451ndash4832005

[14] D B Tuckerman and R F W Pease ldquoHigh-performance heatsinking for VLSIrdquo Electron Device Letters vol 2 no 5 pp 126ndash129 1981

[15] TM HarmsM J Kazmierczak and FM Gerner ldquoDevelopingconvective heat transfer in deep rectangular microchannelsrdquoInternational Journal of Heat and Fluid Flow vol 20 no 2 pp149ndash157 1999

[16] W Owhaib and B Palm ldquoExperimental investigation of single-phase convective heat transfer in circular microchannelsrdquoExperimental Thermal and Fluid Science vol 28 no 2-3 pp105ndash110 2004

[17] J Y Min S P Jang and S J Kim ldquoEffect of tip clearance on thecooling performance of amicrochannel heat sinkrdquo InternationalJournal of Heat and Mass Transfer vol 47 no 5 pp 1099ndash11032004

[18] W Qu and I Mudawar ldquoExperimental and numerical study ofpressure drop and heat transfer in a single-phasemicro-channelheat sinkrdquo International Journal of Heat and Mass Transfer vol45 no 12 pp 2549ndash2565 2002

[19] L Micheli N Sarmah X Luo K S Reddy and T KMallick ldquoOpportunities and challenges in micro- and nano-technologies for concentrating photovoltaic cooling a reviewrdquoRenewable and Sustainable Energy Reviews vol 20 pp 595ndash6102013

[20] E Franklin and J Conventry ldquoEffect of highly non-uniformillumination distribution on electrical performance of solarcellrdquo in Proceedings of Solar Australian and New Zealand SolarEnergy Society 2003

[21] J Coventry E Franklin and A Blakers ldquoThermal and electricalperformance of a concentrating PVthermal collectors resultfrom ANU CHAPS collectorrdquo

[22] J-L Domenech-Garret ldquoCell behaviour under different non-uniform temperature and radiation combined profiles using atwo dimensional finite elementmodelrdquo Solar Energy vol 85 no2 pp 256ndash264 2011

[23] D Chemisana and J I Rosell ldquoElectrical performance increaseof concentrator solar cells underGaussian temperature profilesrdquoProgress in Photovoltaics Research and Applications vol 21 no4 pp 444ndash455 2013

[24] F G Al-Amri and M A I El-Shaarawi ldquoCombined forcedconvection and surface radiation between two parallel platesrdquo

International Journal of Numerical Methods for Heat and FluidFlow vol 20 no 2 pp 218ndash239 2010

[25] F G Al-Amri andM A I El-Shaarawi ldquoMixed convection withsurface radiation between two asymmetrically heated verticalparallel platesrdquo International Journal of Thermal Sciences vol58 pp 70ndash78 2012

[26] D Meneses-Rodrıguez P P Horley J Gonzalez-HernandezY V Vorobiev and P N Gorley ldquoPhotovoltaic solar cellsperformance at elevated temperaturesrdquo Solar Energy vol 78 no2 pp 243ndash250 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of