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Solar Energy 83 (2009) 1884–1892

An improved model for natural convection heat lossfrom modified cavity receiver of solar dish concentrator

K.S. Reddy *, N. Sendhil Kumar

Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras,

Chennai, Tamilnadu 600036, India

Received 30 July 2007; received in revised form 25 June 2009; accepted 1 July 2009Available online 29 July 2009

Communicated by: Associate Editor Robert Pitz-Paal

Abstract

A 2-D model has been proposed to investigate the approximate estimation of the natural convection heat loss from modified cavityreceiver of without insulation (WOI) and with insulation (WI) at the bottom of the aperture plane in our previous article. In this paper, a3-D numerical model is presented to investigate the accurate estimation of natural convection heat loss from modified cavity receiver(WOI) of fuzzy focal solar dish concentrator. A comparison of 2-D and 3-D natural convection heat loss from a modified cavity receiveris carried out. A parametric study is carried out to develop separate Nusselt number correlations for 2-D and 3-D geometries of modifiedcavity receiver for estimation of convective heat loss from the receiver. The results show that the 2-D and 3-D are comparable only athigher angle of inclinations (60� 6 b 6 90�) of the receiver. The present 3-D numerical model is compared with other well known cavityreceiver models. The 3-D model can be used for accurate estimation of heat losses from solar dish collector, when compared with otherwell known models.� 2009 Elsevier Ltd. All rights reserved.

Keywords: Natural convection heat loss; Modified cavity receiver; Solar dish concentrator; Nusselt number correlations

1. Introduction

A solar parabolic dish collector is an attractive method toconcentrate direct beam radiation and convert it to thermalenergy in a useful form for electrical power generation. Toretrieve high temperature from the focal plane of solar para-bolic dish concentrator, an open cavity receiver configura-tion is often used. Solar parabolic dish concentratortypically involves several errors such as slope, facet align-ment and tracking errors that causes reflected solar rays todeviate from the focus of the dish. These optical errors arerather random and cause the optical image of the sun tospread at the focus. Therefore, the size of the focal image is

0038-092X/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2009.07.001

* Corresponding author. Tel.: +91 44 22574702; fax: +91 44 22574652/22570509.

E-mail address: ksreddy@iitm.ac.in (K.S. Reddy).

higher than the actual image and casts a fuzzy image at thefocus of the solar dish. The concentrator with fuzzy focusis referred as ‘‘fuzzy focal soar dish”. The focal image imper-fection of the dish with respect to axis of the parabola isdefined as a fuzzy focal dish concentrator. In a fuzzy solarparabolic dish, the reflected cone angle is higher than theincident cone angle. The fuzzy focal solar requires bigger sizereceiver. The size of the receiver greatly influences heat trans-fer surface area, which in turn affect on the performance ofthe receiver. The heat losses from the cavity receiver includethree contributions: (a) conduction heat loss through theinsulation wall, (b) radiation heat loss through the receiveraperture and (c) natural convection heat loss through thereceiver aperture. Among these, convective heat loss is themost difficult to determine and also gives a major contribu-tion to the total energy loss occurring at open cavity receiv-ers. In order to estimate the total heat loss accurately from

Nomenclature

Aw inner surface heat transfer area of the receiver,m2

A1 focal image/aperture area of the receiver, m2

Dcav diameter of the cavity receiver, md aperture diameter of the receiver, mGrD Grashof number based on diameter of the recei-

verhconv inner convective heat transfer coefficient,

W/(m2 K)kf thermal conductivity of the air inside the recei-

ver, W/(m K)Nu inside Nusselt number of the receiverP pressure, Pa

Pr Prandtl numberQConv convective heat loss through receiver, WTw surface temperature of the receiver, KT1 ambient temperature, KV velocity vectors (r, /, w)Xg body force per m3

Symbols

b inclination of the receiver (degree)r, /, w spherical co-ordinatesm kinematic viscosity of the fluid, m2/sr laplace operator

K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892 1885

the receiver, the natural convection heat loss characteristicsneed to be investigated in detail.

In the past, some theoretical and experimental studieshave been reported on natural convection heat loss froman open cavity receiver. An analytical model was presentedfor estimation of convective heat loss of a large cubical cav-ity receiver by Clausing (1981). Subsequently the modelwas refined by including the aperture area and validatedwith experimental data (Clausing, 1983). Koenig and Mar-vin (1981) proposed a model for predicting convective heatloss from an open solar cavity receiver for operating tem-peratures between 550 and 900 �C. LeQuere et al. (1981)presented experimental Nusselt number correlations forisothermal open cubical cavity for different angles of incli-nation. Siebers and Kraabel (1984) suggested simple modelfor the convective heat transfer based on the results ofexperimental studies from cubical solar cavities. Two dis-tinct receivers: semi cavity and modified cavity have beenintroduced for fuzzy focal solar dish collector system(Kaushika, 1993). Stine and McDonald (1989) modifiedmodel given by Siebers and Kraabel (1984) for a cylindricalshaped frustum receiver incorporating aperture size, sur-face temperature and receiver angle. Lage et al. (1992)numerically analyzed natural convection and radiation ina 2-D cavity with open – end and extended simulationsto an ash hopper. Leibfried and Ortjohann (1995) studiedthe dependency of convective heat loss on temperature, tiltangle, aperture-radius, geometry, ribs, and ventilation in anelectrically heated spherical and hemispherical receiver.

Kribus et al. (1999) carried out experiments on a multi-stage solar cavity tubular receiver to minimize the heat lossesby dividing the aperture into separate stages according to theirradiance distribution levels. Kaushika and Reddy (2000)investigated thermal performance characteristics and opti-misation of cavity receivers for a low cost solar parabolicdish and concluded that the conventional cavity receiversare inadequate for fuzzy focal dish concentrator. Khubeizet al. (2002) carried out an experimental analysis of laminarfree convection heat transfer from an isothermal hemispher-

ical cavity. Reynolds et al. (2004) investigated the heat lossfrom the trapezoidal cavity receiver and flow visualizationto study flow pattern within the cavity. Taumoefolau et al.(2004) investigated the natural convection heat loss fromelectrically heated cavity receiver for different inclinationsvarying from �90� (cavity facing up) to +90� (cavity facingdown), with temperature ranging from 450 to 650 �C. Inthe numerical and experimental studies, they found thatthe natural convection heat loss at 90� inclination of thereceiver, which contradict with other heat loss models. Mostof the area of the receiver was filled with stagnant air zone at90� inclination. Only small convection currents were exists atthe bottom plane. The significant amount of convection cur-rents were predicted at the bottom opening of the receiver at90� inclination. In their study, it was observed that the con-vection heat losses are considerable at 90� inclination of thereceiver. This reveals additional information in contrary toearlier reported works (Clausing, 1981; LeQuere et al.,1981; Stine and McDonald, 1989). Their models predictedzero heat loss at 90� inclination of the receiver. Paitoonsuri-karn and Lovegrove (2006) refined the natural convectionNusselt number correlation, for Lovegrove et al. (2003)and Paitoonsurikarn et al. (2004). The new Nusselt numbercorrelation is based on the numerical simulation results ofthree different cavity geometries (model receiver, 20 m2 dishreceiver and 400 m2 dish receiver). A 2-D natural convectionheat loss model was developed for the modified cavity recei-ver (Sendhil Kumar and Reddy, 2007). Because of 2-D nat-ure the model essentially predicts natural convection heatloss from centre plane of the receiver. It was estimated thatthe natural convection heat loss present at 90� of receiverinclination, which is additional information from the earliermodels. A detailed investigation of cavity receivers for fuzzyfocal solar parabolic dish concentrator was carried out bySendhil Kumar (2008). Sendhil Kumar and Reddy (2008)investigated the performance of three receivers: cavity, semi-cavity and modified cavity receivers with solar dish concen-trator. In the present work, 3-D model is presented toinvestigate the natural convection heat loss from an actual

1886 K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892

geometry of the modified cavity receiver of without insula-tion (WOI) at the bottom of the aperture plane. The sche-matic of the modified cavity receiver is shown in Fig. 1.Based on the 3-D numerical studies, a Nusselt number corre-lation is developed for an accurate estimation of natural con-vection heat loss from a modified cavity receiver.

2. Numerical analysis

The numerical investigation of natural convection heatloss in the modified cavity receiver for solar dish concentra-tor has been carried out for ‘‘without insulation (WOI) ataperture plane” of the receiver configuration by varyingthe angle of inclination of the receiver from b = 0 (cavityaperture facing sideways) to b = 90� (cavity aperture facingdown). Once the reflected solar radiation from the solardish concentrator is entered into the receiver, the surfaceof the receiver attains constant or almost constant surfacetemperature. The temperature mentioned in the presentmodel is stagnation temperature. The stagnation tempera-ture of the receiver corresponds to the no flow conditionwhen no useful energy is being collected for constant inso-lation and the system attains peak temperature. The recei-ver is made up of copper tubing with an opening aperturediameter (d) and cavity diameter (D) as shown in Fig. 1.The receiver inner surface is covered with closely woundcopper tubes. The copper tubes are placed very closely totouch each other to form a continuous hemispherical sur-face (not solid). The water is circulated through the tubesto retrieve heat. The temperature distribution of the recei-ver is uniform and ranging between 300 and 700 �C. Themodified cavity receiver is composed of flat section andcavity section. The flat aperture plane is attached at thebase of the receiver. The flat aperture plane is exposed tolow concentration ratio and low fluid temperature. Thecavity region is exposed to higher concentration ratio andhigher fluid temperature. Therefore, the outer surface ofthe cavity region is covered with opaque insulation to min-imized heat losses. The total heat loss of the receiverincludes conductive from the receiver, convective and radi-ative heat losses through the receiver aperture and convec-tive and radiative losses from the external absorbingsurface of the receiver. The natural convection heat lossthrough cavity receiver is most predominant heat lossamong the heat loss contributions. Therefore, an attempt

Fig. 1. Schematic of mod

has been made to investigate convective heat loss frominner surface of the cavity receiver. The geometry and meshwere created using GAMBIT 2.2.30 and duplicating theactual geometry of the modified cavity receiver. For mod-eling a modified cavity receiver, the following two assump-tions have been made: (i) the surfaces of the tubes areassumed to be uniform and smooth, (ii) the temperatureof the fluid (water flowing inside the copper tube) isassumed to be the same as that of the surface temperatureof the tube.

In actual condition of the modified cavity receiver, atemperature difference exists between inner curved surfaceand the aperture plane. The low temperature working fluid(water) enters into the receiver tubes at flat surface of thereceiver. The fluid temperature variation exists betweenthe curved and flat surface of the modified cavity receiver.This is because of the natural convection heat loss from theaperture plane to the ambient. Therefore, in the presentnumerical analysis a temperature difference (DT) of 30 �Chas been considered between curved surface and flat sur-face of the modified cavity receiver. In actual case, themodified cavity receiver is surrounded by an infinite atmo-sphere. To achieve infinite atmosphere in the numericalwork, the receiver was placed centrally in the large spheri-cal enclosure so that the flow domain can establish. Thesize of the spherical enclosure was increased until it hadan insignificant effect on the working fluid and heat istransferred in the vicinity of the receiver. This conditionwas attained, when the diameter of the spherical enclosure(outer domain) is about ten times the diameter of the recei-ver. The grid independence study has been carried out withfine grid for the modified cavity receiver and coarse grid forspherical enclosure. The total number of cells has beenfixed approximately 860725 tetrahedral cells. The grid gen-eration for computational domain is shown in Fig. 2. Thereceiver surfaces were subjected to various boundary con-ditions. The inner surface and bottom of the aperture planeof the modified cavity receiver were taken as constant tem-perature condition, and conditions for isothermal surfacewould be T = Tw. The outer domain of the spherical enclo-sure was assumed as a pressure inlet boundary conditionand corresponding boundary conditions i.e., P = Patm.The top outer surface of the modified cavity receiver is per-fectly insulated to prevent heat losses. The adiabaticboundary condition was imposed on the top outer surface

of the receiver and is given as: @T@/ ¼ 0.

ified cavity receiver.

Fig. 2. Grid generation for computational domain of the modified cavityreceiver at b = 0.

K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892 1887

These boundary conditions have been used to solvemass, momentum and energy equations. A finite volumebased CFD software, FLUENT 6.3 was used for modelingand simulation. Boussinesq approximation is invokedwhile solving the momentum equation. The air is consid-ered as a working fluid. For the present operating rangeof temperature (300–700 �C), it is found that Boussinesqapproximation will lead to reasonable deviations and anon-Boussinesq model paper will follow. The governingequations for the receiver are based on the conservationof continuity, momentum and energy equations, and theseequations are written in vector form (Yuan, 1988) as:

Continuity equation

r � V ¼ 0 ð1Þ

Momentum equation

V � rV ¼ X g �rpqþ mr2V ð2Þ

Energy equation

r � ðkfrT Þ ¼ 0 ð3Þ

The laminar, steady state and 3-D governing equationsare solved in FLUENT 6.3 using an implicit solver. Forpressure velocity coupling, SIMPLEC algorithm has beenused and the pressure term was discretised using body force

Fig. 3. Temperature contours and velocity vectors of the

weighted. The discretization of momentum and energy wascarried out using first order upwind scheme. The minimumconvergence criteria were set at 10�3 for continuity andvelocity and 10�6 for energy. To obtain the heat transferand flow solutions, the solver undertakes the iteration untilthe convergence criterion is satisfied. The properties ofworking fluid are based on the average temperature ofthe surface of the receiver and the ambient air.

3. Validation of the numerical procedure

In order to validate the present numerical procedure ofmodified cavity receiver, a typical case of 3-D laminar nat-ural convective heat transfer from an isothermal horizontalopen hemispherical cavity reported by Khubeiz et al.(2002) has been considered. The results of theoretical anal-ysis, numerical and experimental work were presented inthe Nusselt number and Rayleigh number relations:NuD = 0.296Ra1/4, NuD = 0.340Ra1/4, NuD = 0.316Ra1/4

(1.710 � 105 < Ra < 3.4 � 107 and 376 < Pr < 1140), respec-tively. The present numerical procedure has been validatedby using glycerine as working fluid. The thermophysicalproperties of the glycerine were taken from published data.The isotherms and velocity vectors of the hemisphericalupward open cavity of Rayleigh number of 2.4864 � 106

are shown in Fig. 3. Nusselt number has been predictedusing the present numerical procedure. The comparisonof the present numerical procedure with that of the exper-imental case of Khubeiz et al. (2002) for a hemisphericalupward open cavity in terms of Nusselt number is illus-trated in Table 1. It is observed that present numerical pro-cedure is reasonable agreement with the experimental data.Therefore same numerical procedure has been adopted forextensive studies on natural convection heat loss frommodified cavity receiver.

4. Results and discussions

4.1. 3-D receiver model

The temperature of the modified cavity receiver is variedfrom 300 to 700 �C to estimate the natural convection heatloss from the modified cavity receiver. The temperature

mid plane of the upward open hemispherical cavity.

Table 1Validation of the present numerical procedure using experimental convective Nusselt number.

Rayleighnumber

Nusselt number fromKhubeiz et al. (2002)

Nussel number frompresent numerical procedure

Percentage of deviation fromexperimental value

NuExp NuNum NuAnal

2.4864 � 106 12.55 13.50 11.75 14.03 �11.791.1511 � 107 18.41 19.80 17.24 16.66 9.50

1888 K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892

contours of 3-D modified cavity receiver at 400 �C for dif-ferent inclinations of the receivers are shown in Fig. 4. Themaximum and minimum natural convection heat loss occurat b = 0 and 90�, respectively in both 2-D and 3-D simula-tion. In 3-D receiver model, at b = 0, the receiver apertureplane attains the temperature of only 233 �C. The naturalconvective currents drive cool air into the cavity receiver.As the air rises and becomes hotter and lighter as it absorbsmore and more heat from the receiver surfaces. Because ofthis, hot stagnant air appears only at the top of the cavityreceiver. The receiver at b = 30�, high temperature gradientexists across the aperture plane of receiver as compared toangle of inclination at b = 0. This larger temperature gradi-ent prevents the mixing of ambient air in the receiver. Theconvection zone remains unchanged, only the size of theconvection zone decreases as the inclination increases.The receiver at b = 60�, a clear stagnant zone is visiblefor some distance from the curved portion of the receiver.

Fig. 4. Temperature contours (3-D) at 400 �C for diff

Because of this, temperature across the top and bottomsurfaces of the aperture opening is reduced. At b = 90�,the almost 75% of the cavity is filled with stagnant air zone.Only small convection currents are existing at the apertureplane. A significant convection currents are predicted at thebottom aperture plane of the modified cavity receiver atb = 90�. In the present 3-D numerical simulation results,at 90� inclination of the receiver a small amount the naturalconvection heat loss has been observed. Previous research-ers such as Clausing (1981), LeQuere et al. (1981), Stineand McDonald (1989) assumed that natural convectionheat loss at 90� inclination was zero.

The effect of area ratio (Aw/A1) on the natural convec-tive heat loss has been carried out for the modified cavityreceiver. In the present numerical modeling, Aw is the innerheat transfer area, which includes inner curved and bottomplane area, It is not constant, which depends upon the bot-tom aperture opening area (A1). The size of the aperture

erent inclinations of the modified cavity receiver.

Fig. 5. Temperature contours (3-D) of modified cavity receiver at b = 0 for different area ratio.

0 2 4 6 8 10 12 140

5

10

15

20

25

30

35

40

45

50

Con

vect

ive

Nus

selt

num

ber

Area ratio of the receiver

Fig. 6. Variation of convective Nusselt number with different area ratio atb = 0.

K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892 1889

opening is determined by the flux distribution (focal imagecharacteristics of the dish) of the fuzzy focal solar dish. Theconvection heat loss takes place only through the receiveraperture. Hence, the aperture area has been used to evalu-ate the convection heat loss from the receiver. The temper-ature contours for different area ratio (Aw/A1) at 0inclination of the cavity receiver is shown in Fig. 5. Itclearly indicates that the quantity of air trapped in thereceiver increases as the area ratio increases. As the arearatio increases from 1 to 12, the convection heat loss grad-ually decreases. For lower area ratio, the stagnant air zoneis not visible. It is also observed that the convection heatloss is higher for a smaller area ratio. Thus in a modifiedcavity receiver the maximum convection heat loss occurfor an area ratio (Aw/A1) of 2. The stagnant air zoneappears as area ratio increases from 2 to 12. At the topof the cavity receiver, the air attains highest temperaturedue to stagnant zone. Thus stagnant air zone increases,with the area ratio of the receiver. Due to this, convectiveNusselt number decreases with increase of area ratio ofthe receiver. The variation of convective Nusselt numberwith the area ratio at b = 0 is shown in Fig. 6.

In 3-D modified cavity receiver, significant convectionheat loss is observed at 90� inclination of the modified cavityreceiver. At b = 90�, the temperature contour and the mag-nitude of the convection heat loss are same for both 2-D and3-D model. The variations of the convection heat loss for anarea ratio of 8 with receiver inclinations for different operat-ing temperatures from 300 to 700 �C are shown in Fig. 7.

A 3-D Nusselt number correlation has been developedfor modified cavity receiver based on large set (140) ofnumerical data points. The Nusselt number correlationfor 3-D modified cavity receiver is given as:

NuDðWOIÞ ¼ 0:698Gr0:209D ð1þ cos bÞ0:968

� ðT w=T1Þ�0:317ðd=DÞ0:425 ð4Þ

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Nus

selt

num

ber

(Cor

rela

ted)

Nusselt number (Data)

Fig. 8. Parity plot for convective Nusselt number correlations for themodified cavity receiver.

0 20 40 60 80 1000

50

100

150

200

250

Con

vect

ion

heat

loss

, W

Inclination of the receiver, degree

573 K 673 K 773 K 873 K 973 K

Fig. 7. Variation of 3-D convection heat loss at various receivertemperatures with different inclinations.

1890 K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892

The 2-D convective Nusselt number correlation formodified cavity receiver is given by Sendhil Kumar andReddy (2007).

NuDðWOIÞ ¼ 0:503Gr0:222D ð1þ cos bÞ1:231

� ðT w=T1Þ�0:165ðd=DÞ0:304 ð5Þ

The above correlation is valid for a Prandtl number (Pr)of 0.69 and for a range of parameters of practical interestof solar parabolic dish.

The ranges of parameters include:5.663 � 106

6 GrD 6 7.166 � 107

0.3 6 d/D 6 0.50 6 b 6 90� (in the step of positive angle of 15�)1.91 6 TR 6 3.0, where TR = Tw/T1The goodness of the fit is indicated by a value of corre-

lation coefficient of 0.981 and a standard deviation of0.087. The parity plot for the Nusselt numbers for 3-Dmodified cavity receiver is shown in Fig. 8. The heat trans-fer coefficient is calculated from the Nusselt number as:

Nu ¼ hconvDcav

kf

ð6Þ

The natural convective heat loss through aperture planecan be written as:

Qconv ¼ hconvA1ðT W � T1Þ ð7Þ

4.2. Comparison of 2-D and 3-D Nusselt number correlation

The numerical analysis of 2-D model natural convectionheat loss from modified cavity receiver was reported bySendhil Kumar and Reddy (2007). In the 2-D model ofmodified cavity receiver, at b = 0, most of the convectioncurrents occur at the centre of the receiver. Whereas atb = 0, convection occur at the bottom of the cavity recei-ver. Because of this, small stagnant zone appears at the

top of the receiver. At b = 30�, the top portion the receiveris occupied with stagnant air. But the convection zone isslightly shifted to the bottom of the receiver. At b = 60�,small portion of the bottom of the modified cavity receiveris exposed to cooler air. This reduces the convection zonearea. At b = 90�, the major portion of the receiver is filledwith hot stagnant air. The area of the 2-D convection zone(near the aperture opening) decreases with receiverinclination.

The 3-D and 2-D Nusselt number correlations havebeen developed separately for a modified cavity receiverand are given in Eqs. (4) and (5), respectively. The Grashofnumber directly influences on the natural convection Nus-selt number. The exponent of angle of inclination (b) in the2-D Nusselt number correlation is higher than that of the3-D Nusselt number correlation. This shows that both 2-D and 3-D Nusselt numbers are significantly dependenton the inclinations of the receiver than the other parameter.In evolving the above 2-D and 3-D correlations, (1 + cos b)has been used as the appropriate form for receiver inclina-tion. Because at b = 90�, Nusselt number would be non-zero. The 3-D Nusselt number is partially dependent onthe temperature ratio (Tw/T1). The 2-D exponents arealways higher than the 3-D. Because of this, the magnitudeof 2-D Nusselt number is always higher than the 3-D. Theorder of magnitudes of the exponents are same in 2-D and3-D correlations.

The variation of 2-D and 3-D convection heat loss withinclination of the receiver is shown in the Fig. 9. The com-parison of 3-D and 2-D models for convection heat losswith different temperature of the receiver for b = 0, 45�and 90� are also shown in the Fig. 10. It is evident thatthe 2-D and 3-D model are comparable only at higherangles of inclination. The deviation between 2-D and 3-Dmodel is significant at 45� inclination and marginal athigher angle of inclinations of the receiver(60� 6 b 6 90�). The deviation increases from 26.3% at 0

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

160

180

Con

vect

ion

heat

loss

, W

Inclination of the receiver, degree

2-D Model 3-D Model

Fig. 9. Comparison of 3-D and 2-D models for convection heat loss frommodified cavity receiver of the receiver at 400 � C.

500 600 700 800 900 1000 1100 1200

50

100

150

200

250

300

350

Con

vect

ion

heat

loss

, W

Temperature of the receiver, K

2-D 00

3-D 00

2-D 450

3-D 450

2-D 900

3-D 900

Fig. 10. Variation of 3-D and 2-D convection heat loss with differenttemperature of the receiver.

40

60

80

100

120

onve

ctiv

e he

at lo

ss, W

Le Quere et al. (1981)Stine & Mcdonald (1989)Clausing (1981)Paitoonsurikarn and Lovegrove (2006)Present model

.

K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892 1891

inclinations to 36% at 45� inclination and then decreases to3% at 90� inclination of the receiver. At 45� inclination the2-D model predicts 36% higher than the 3-D model. Forhigher angle of inclinations (60� 6 b 6 90�), the 2-D pre-dictions would deviate about an average deviation of14.5%. Thus for higher inclination of the receiver(60� 6 b 6 90�), 2-D model predicts well. Hence the 2-Dmodel can be effectively used for higher angle of inclinationwith less computational time and grid size. The computa-tional time and grid size involved in 2-D model is 5 and15 times, respectively less than 3-D model.

0

20

0 20 40 60 80 100

Inclination of the receiver, degree

C

Fig. 11. Comparison of modified cavity receiver (WI) of area ratio of one(Aw/A1 = 1) with other heat loss models.

4.3. Comparison of 3-D natural convection heat loss model

with other models

The present 3-D heat loss model has been comparedwith other well known heat loss models such as Clausing

(1981), LeQuere et al. (1981), Stine and McDonald(1989), Paitoonsurikarn and Lovegrove (2006). In orderto compare the present model with other models accurategeometric parameters have been considered in heat lossestimations. The comparison of the present model withother model shown in Fig. 11. The predicted results arecomparable with to other models. All of the other heat lossmodels examined zero natural convective heat loss at 90�inclination. Nevertheless there is a possibility of existingconvection between the inner surface of the cavity andambient. In the present model, heat loss at 90� has beenconsidered for accurate determination of total heat loss.The deviation occurs within these models, which may beprimarily due to geometry of the receiver, and how thecharacteristics length is defined. Moreover, it is alsoobserved that each above described convection heat lossmodel has inherently based on the specific receiver geome-try and operating conditions.

5. Conclusions

The accurate estimation of natural convective heatloss from a modified cavity receiver without insulation(WOI) on the aperture plate has been numerically inves-tigated. The results of 3-D and 2-D numerical analysishave been presented in terms of Nusselt number correla-tions for present receiver configurations. The maximumand minimum convection heat losses occurred at b = 0and b = 90�, respectively for both 2-D and 3-D model.In both the 2-D and 3-D model, significant convectionis noted at b = 90�. The variation difference between 2-D and 3-D Nusselt number increases for lower angleof inclination and decreases for higher angle of inclina-tion of the receiver. The 3-D model can be used for anaccurate estimation of natural convection heat loss atlower inclination of the receiver. The 2-D model canbe used for an accurate estimation of natural convective

1892 K.S. Reddy, N. Sendhil Kumar / Solar Energy 83 (2009) 1884–1892

heat loss for higher angle of inclination of the receiver.The computational time and grid size involved in 3-Dare respectively 5 times and 15 times higher than 2-D.To estimate the natural convective heat loss for higherand lower angle of inclinations, the present 3-D and 2-D model of modified cavity receiver (WOI) may effec-tively be used for a solar dish collector system. To com-pare the heat loss chacteristics of different types ofreceivers, 2-D analysis at higher angle of inclination(60� 6 b 6 90�) of the receiver can effectively be used toreduce the computational time.

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