Correlations for natural convective heat exchange in CPC solar collector cavities determined from experimental measurements

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Solar Energy 86 (2012) 2443–2457 www.elsevier.com/locate/solener

Correlations for natural convective heat exchange in CPC solar collector cavities determined from experimental measurements H. Singh a,⇑, P.C. Eames b b

a School of Engineering and Design, Brunel University, Uxbridge, Middlesex UB8 3PH, UK CREST, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK

Received 6 December 2011; received in revised form 11 May 2012; accepted 14 May 2012 Available online 11 June 2012 Communicated by: Associate Editor Brian Norton

Abstract A detailed experimental study was undertaken to analyse the natural convective heat transfer in CPC cavities, a complex function of collector orientation, geometrical aspect ratios and thermal boundary conditions at the enclosure walls. Results are reported for CPC solar collectors with full-, three quarter- and half-height reflectors, CR = 2 and a 100 mm wide flat plate absorber. Experiments were conducted using a purpose built solar simulator under controlled lab environment employing realistic boundary and thermal conditions. The effects of simultaneous tilting of the solar collectors about both transverse and longitudinal axes, truncation of the reflector walls and inlet water (collector heat removal fluid) temperature on the natural convective heat flow characteristics inside the CPC cavity have been determined. It is concluded that the correlations developed for prediction of natural convection characteristics in rectangular, annuli and V-trough enclosures are not appropriate for application to CPC solar collectors with divergence ranging from 150% to 300%. Based on the experimental data a correlation is presented to predict the natural convection heat loss from the absorber plate of solar collectors for a range of water inlet temperatures. Ó 2012 Elsevier Ltd. All rights reserved. Keywords: Natural convection; CPC solar collectors; Heat transfer; Solar simulator; Low concentrating solar collectors; Concentration ratio

1. Introduction Low concentrating (concentration ration (CR) 6 10) Compound Parabolic Concentrating (CPC) solar collectors are of interest due to their suitability for integration into building envelopes either for thermal, photovoltaic or combined photovoltaic-thermal applications (Chemisana, 2011; Norton et al., 2010; Jing et al., 2010; Tchinda, 2008; Alfegi et al., 2007; Othman et al., 2005). Those with CR 6 3 are of particular interest due to their ability to harness a fraction of diffuse solar radiation in addition to direct solar radiation with either minimal or no solar tracking. To achieve high temperatures and good efficiency solar thermal collector design requires low heat loss, convective, radiative and ⇑ Corresponding author. Tel.: +44 1895 265468; fax: +44 1895 256392.

E-mail address: [email protected] (H. Singh). 0038-092X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2012.05.014

conductive, from the hot absorber to the cooler ambient environment. Long wave radiative heat loss from the absorber can be reduced by employing a selective absorber surface characterised by high solar absorptance (>0.9) and low long-wave emittance (<0.1). For a CPC solar collector working at a temperature of 100 °C or higher with a high performance selectively coated absorber, natural convection within the cavity (from the hot absorber to the cold aperture cover and walls) is the dominant mode of heat transfer (Norton, 1991). Previous studies on CPCs report measurements of air velocities, and detail the air flow distributions within the CPC cavities with some information provided on temperature distributions and heat transfer phenomena. For CPC cavities to date only the effect of transverse tilt has been documented in the literature, the effects of longitudinal tilt and combined longitudinal and transverse tilt are not considered. Previous studies are

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H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

Nomenclature surface area of the absorber plate (m2) surface area of the glass cover (m2) surface area of the end walls covered with aluminium reflector (m2) Aswall surface area of the side walls covered with aluminium reflector (m2) CR concentration ratio Cw specific heat of water through the collector tube (J kg1 K1) H height of the collector (m) Iss intensity of radiation received at the absorber plate (W m2) kair thermal conductivity of cavity air (W m1 K1) kins thermal conductivity of the collector insulation (W m1 K1) L length of the solar collector (m) mw flow rate of water flowing through the collector tube (kg s1) Qin amount of radiation incident on the glass cover (W) Qcov amount of radiation absorbed by the glass cover (W) Qswall amount of radiation absorbed by the side walls with aluminium reflector (W) Qewall amount of radiation absorbed by the end walls with aluminium reflector (W) Qabs amount of radiation absorbed by the selectively coated copper absorber plate (W) Qabs-conv natural convective heat loss from the absorber plate (W)

Aabs Acov Aewall

limited to analysis of natural convection within cavities held horizontal (Rabl, 1976; Ozoe et al., 1976; Prapas et al., 1987a; Chew et al., 1988) or inclined longitudinally (Yang et al., 1987; Leong et al., 1998, 1999) or inclined transversely (Hollands and Konicek, 1973; Ozoe et al., 1974a,b, 1975; Abdel-Khalik et al., 1978; Abdel-Khalik and Li, 1978; Meyer et al., 1982a,b; Prapas et al., 1987b; Chew et al., 1989; Eames and Norton, 1993a,b; Yadav et al., 1996). The authors have recently published a detailed review of past studies into natural convection in CPC cavities (Singh and Eames, 2011). In practice a CPC solar collector is installed at an angle of inclination based upon the latitude of the site of location, either in an E–W or in a N–S alignment with respect to the longer axis of the absorber. In both cases, for an absorber without an evacuated envelope, the temperature difference between the warmer absorber and colder cover and cavity walls gives rise to buoyancy forces, which establish and drive natural convective heat transfer within the air filled collector trough. In the case of CPC thermal solar collectors, convection must be suppressed if high absorber temperatures are to be achieved and good efficiency maintained. CPC solar collectors of

Qabs–cov,rad radiative heat exchange from the absorber plate to the cover (W) Qw thermal energy removed by the water from the absorber plate (W) Ra Rayleigh number Texit temperature of water at the collector exit (K) Tin temperature of water at the collector inlet (K) Tabs average absorber plate temperature (K) Tcov average temperature of the collector cover (K) Ucov combined radiative and convective heat transfer coefficient from the glass cover to the lab indoor environment (W m1 K1) W width of solar collector absorber (m) Greek symbols aabs absorptivity of the selective coating surface acov absorptivity of the glass cover db,ins thickness of the collector insulation at back (m) de,ins average thickness of the collector insulation at edges (m) eeff effective emissivity of the absorber surface eabs emissivity of the absorber plate selective surface ecov emissivity of the glass cover h longitudinal tilt angle (°) qcov reflectivity of the glass cover r Stefan–Boltzmann constant (= 5.67  108 Wm 2 K4) / transverse tilt angle (°)

improved design with lower natural convective heat losses will achieve higher efficiencies at high temperatures thus reducing collection area required for a specified load and potentially making systems cheaper. A detailed experimental study was undertaken to analyse the natural convective heat transfer in CPC cavities, a complex function of collector orientation, geometrical aspect ratios and thermal boundary conditions at the enclosure walls. Results are reported here for a CPC solar collector with full-, three quarter- and half- height reflectors, CR = 2 and a 100 mm wide flat plate absorber. 2. Details of the experimental procedure and facility Investigations were performed to experimentally characterise the three-dimensional natural convective heat transfer that occurs in the cavity formed by the absorber, reflectors and aperture cover of tilted CPC line-axis low concentrating (CR 6 2) solar collectors with flat plate absorbers. Heat exchange by all three modes of heat transfer, conduction, convection and radiation were considered. The effects of simultaneous tilting of the solar collectors

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

(a)

End wall

Parabolic side wall

Styrofoam insulation

Collector tube

Absorber Plate

(b) Thermocouple wires CPC solar collector Angular orientation table

Fig. 1. (a) Part-assembled view of the full height CPC solar collector and (b) an instrumented CPC solar collector mounted on the angular orientation table under test.

about both transverse and longitudinal axes, truncation of the reflector walls and inlet water (collector heat removal fluid) temperature on the natural convective heat flow characteristics inside the CPC cavity have been determined. Three CPC solar collectors with full, three-quarter and half height reflectors were manufactured for the experiments based on a flat plate absorber with 100 mm width, the main components of the full height CPC solar collector manufactured are shown in Fig. 1a. For each CPC collector the geometric details and the calculated Rayleigh number range (based on the collector height, the vertical distance between the absorber plate and the collector cover) are presented in Table 1. MiroSilver (Alanod Ltd., 2006) reflective sheet (0.5 mm thick) was glued to a Styrofoam substrate to

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form the parabolic side walls and end walls of the CPC solar collectors. The experimental apparatus consisted of the CPC solar collectors, a solar simulator, an angular orientation table (Fig. 1b) and a water supply system with electric heater. The angular orientation table was employed for simultaneous tilting of the solar collectors mounted on it over 0 6 h 6 10° and 0 6 / 6 40°. Tilt angles were measured to an accuracy of ± 0.15°. The radiation intensity at the CPC aperture cover from the purpose built solar simulator was 350 ± 23 W/m2. The water supply system supplied water at a constant flow rate of 0.011 kg/s per m2 of aperture area and maintained water temperature to within ±0.5 °C of set point temperatures of 10, 20, 30, 40, 50 and 60 °C. Fig. 1b shows an instrumented solar collector mounted on the orientation table under test, Fig. 2a shows a schematic of the test set-up. The temperatures at different locations in the collector structure and within the air filled cavity were identified as the most important parameters to be measured. Temperatures at a number of specific locations on each component of the CPC solar collectors, in the cavity air and in the water supply tank and the ambient temperature were measured using T-type thermocouples with readings recorded by a Keithley2750 data acquisition system (Keithley Instruments Ltd., 2004). The steady state temperatures of the various elements of the solar collectors were used to calculate the thermal energy absorbed by them. The time required to attain steady state conditions varied from 2 h to 3 h. During the experiments the flow rate of water through the solar collectors was kept constant within ±1% of 75 g/min. In the experiments the absorber plates of the CPC solar collectors were heated using radiation from a solar simulator. To observe the effect of inclination on natural convection within the solar collector cavities they were simultaneously tilted both ways; longitudinally between 0 6 h 6 10° in steps of 2° and transversely between 0 6 / 6 40° in steps of 10°. Fig. 2b shows the transverse and longitudinal tilt angles and aspect ratios used for CPC solar collectors.

3. Selection of parametric test conditions A solar simulator was used to supply the energy input to the solar concentrators, the use of either real outdoor sky conditions or a solar simulator (in case of indoor

Table 1 Geometrical specifications of the solar collectors investigated. Collector length L (mm)

Truncation level (final height of CPC solar collectors)

Height, H (mm)

Aperture width, W (mm)

Transverse aspect ratio (Ax)

Longitudinal aspect ratio (Az)

Rayleigh number (Ra)

1000

Full height

259.81

200

0.38

3.85

1000

Three-quarter height

194.86

196.1

0.51

5.13

1000

Half height

129.9

182.44

0.77

7.70

8.2  105– 5.9  106 3.3  106– 2.4  107 8.3  106– 5.7  107

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H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

Overhead planar reflector

Aperture cover

Paraboloidal dish reflector array

Radiation from lamp array

3.5 m

Reflected radiation to CPC collector

Steel structure (tiltable upper part)

CPC solar collector

Steel structure (horizontal lower part)

1.2 m

2.5 m

Fig. 2a. Schematic of the test set-up (dimensions not to scale).

W H

Transverse aspect ratio Ax =

θ Longitudinal tilt angle Transverse tilt angle

L H

Hot Bottom

Co

ld

To p

Longitudinal aspect ratio A z =

Cold Top

Hot Bottom

L

L

θ

H

H

W

W

Fig. 2b. General representation of the tilt angles and aspect ratios used for the CPC solar collectors.

experiments) is recommended in comparison to electrical heating of the absorber for the following reasons: 3.1. Non-uniform solar absorption at collector components In real concentrating solar systems there is absorption and reflection of both solar and long wave radiation at the aperture cover and reflector side walls in addition to absorption at the absorber. Many researchers have neglected the absorption of solar radiation at the aperture cover and the reflector side walls (Abdel-Khalik et al., 1978; Abdel-Khalik and Li, 1978; Hsieh, 1981). The absorption of solar radiation by the components of the solar collector in conjunction with their thermal conductivity and heat losses to ambient

(or heat transfer fluid in the case of the absorber) leads to non-uniform temperatures in these components. It is very difficult (if not impossible) to specify an electrical heating system (Prapas et al., 1987a,b) that can simulate such nonuniform time varying energy inputs and achieve similar component temperatures in a laboratory experimental setup. Electrical heating will not contribute a heat flux directly to all components of the collector, as a real solar energy source will do. For an effective experimental programme boundary conditions should emulate those of real applications, which necessitates experiments to be carried out either using a solar simulator or under the sun in an outdoor environment. In the case of CPC solar collectors, the analytically predicted temperatures of collector components taking into

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

3.2. Controllable applied conditions Solar radiation, wind speed and ambient temperature are transient in nature. Solar radiation reaching any location on the earth varies considerably with atmospheric conditions, time of the day and season of the year. A solar collector therefore always operates in a transient state, making it difficult to characterise natural convective heat transfer in CPC solar collector cavities. A solar simulator provides uniform insolation at the solar collector aperture for the duration of the experiments without sacrificing the realistic energy input to the collector components. 3.3. Radiative heat exchange between elements forming the collector cavity Radiative heat exchange between collector components is an inevitable phenomenon in real systems and was neglected in most previous numerical studies (Chew et al., 1989). Radiative heat transfer influences the convective heat transfer by its effect on the temperature distribution at the reflector side walls. Radiative exchange between different components of a solar collector will increase the Rayleigh number corresponding to the initiation of natural convection in the collector cavity by a factor of two by tending to dampen wall temperature perturbations similar to wall conduction (Edwards and Sun, 1971). 3.4. Thermal conditions at the absorber, aperture cover and reflector side walls The absorber plate and aperture cover of real solar collectors are characterised by temperatures that vary along the longitudinal as well as transverse axis. The reflector side walls of real CPC solar collectors are neither perfectly insulating nor conducting. The studied CPC solar collectors (Prapas et al., 1987a,b; Chew et al., 1989) had thermal boundary conditions with almost linear temperature profiles in the absorber plate and aperture cover. The side walls were internally conducting and externally well insulated, with different temperature profiles observed at the inlet-side half and the exit-side half of the system. These thermal boundary conditions caused the critical Rayleigh number to have values higher than those predicted in past studies that assumed two-dimensional natural convection flow in CPC solar collector cavities. In most of the studies reported in the literature, the temperatures at the heated or cooled surfaces (the absorber and aperture cover respectively for solar collectors) have been assumed to be uniform neglecting the axial variation of the temperature in the heated or cooled surfaces of the enclosure (Hollands and Konicek, 1973; Rabl, 1976; Abdel-Khalik et al.,

1978; Abdel-Khalik and Li, 1978; Hsieh, 1981; Prapas et al., 1987a,b; Chew et al., 1989; Meyer et al., 1982a,b; Symons and Peck, 1984; Yang et al., 1987; Kessler, 1987; Lock and Han, 1990; Leong et al., 1998, 1999; Corcione, 2003). In the present study, the temperature at the absorber was allowed to vary along both the axial directions, longitudinal and transverse, in order to achieve realistic thermal boundary conditions. Realistic thermal boundary conditions at the reflector side walls were achieved by using a conducting reflector sheet (0.5 mm thick) insulated with 100 mm thick insulation applied to its exterior side. 4. Measured absorber plate temperatures Abdel-Khalik and Li (1978) assumed isothermal top and bottom walls and adiabatic side walls leading to unrealistic two-dimensional natural convection in the CPC cavity. In the present study, the mean absorber plate temperatures prevailing at different cross-sections along the longitudinal axis of the CPC solar collectors for water inlet temperature of 10 °C, 20 °C, 30 °C, 40 °C and 50 °C were measured. Fig. 3 shows the rise in absorber plate temperature, above the temperature at the inlet cross-section, along the length of the absorber plate for the full height CPC solar collector. The absorber plate temperature was found to tend to a definite asymptotic value at cross-sections located farther than 900 mm from the inlet. This resulted from a higher temperature of the absorber plate leading to a corresponding increase in heat loss from it. For the CPC solar collectors studied under the range of water inlet temperatures and radiation levels covered here, useful heat gain by the absorber plate at cross-sections located at 900 mm or farther from the inlet cross-section is comparable to the heat loss and thus the mean absorber plate temperature at these locations tends to the asymptotic levels indicated in Fig. 3. Similar trends were seen for the CPC solar collectors

11 ºC

16

20 ºC

30 ºC

40 ºC

50 ºC

600

800

1000

14

Temperature rise (ºC)

account solar radiation absorbed by these components were found to be higher than those predicted when the solar radiation absorbed by these were neglected (Prapas et al., 1987a).

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12 10 8 6 4 2 0 0

200

400

1200

Location on the absorber plate measured from inlet cross-section (mm) Fig. 3. Rise in mean absorber plate temperatures measured at different cross-sections on the absorber plate of full height CPC solar collector held horizontally (h = / = 0°) for a water flow rate of 75 ml/min.

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

with three-quarter and half heights. The mean absorber plate temperatures were not found to vary significantly with varying longitudinal tilt angle with the highest value recorded for the horizontal orientation, a situation with h = / = 0° shown in Fig. 3 for the full height CPC solar collector with a water inlet temperature of 50 °C.

Collector height (mm)

(a)

250

Temperature (°C) 45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

200

150

100

50

-100

-50

0

50

100

5. Measured cavity air temperature The cavity air temperature at three cross-sections, near inlet (100 mm from the inlet), in the central region and near-exit (100 mm before the exit), was found to vary along the transverse, longitudinal and vertical axes of all three

(d)

Temperature (°C) 45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

250

Collector height (mm)

2448

200

150

100

50

-150

150

Temperature (°C)

(a)

45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

200

150

100

50

-100

-50

0

50

100

(e)

100

50

-100

-50

0

50

100

Absorber plate width (mm)

45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

100

-150

-100

-50

0

50

100

Absorber plate width (mm)

45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

150

Temperature (°C)

150

150

Temperature (°C)

200

100

50

150

(f)

Temperature (°C) 45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

250

Collector height (mm)

Collector height (mm)

250

50

200

Absorber plate width (mm)

(c)

0

(d)

250

Collector height (mm)

Collector height (mm)

x (mm)

250

-50

Absorber plate width (mm)

Absorber plate width (mm)

(b)

-100

200

150

100

50

-150

-100

-50

0

50

100

Absorber plate width (mm)

Fig. 4. Experimentally measured cavity air temperatures at the cross-sections (a and d) 100 mm from inlet, (b and e) mid-section and (c and f) 100 mm before exit for the full height CPC solar collector oriented (a–c) horizontally (h = / = 0°) and (d–f) at / = 20° with a water inlet temperature of 40 °C.

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457 250

Temperature (°C) 45 43 41 39 37 35 33 31 29 27 25 23 21 19 17 15

200

Collector height (mm)

CPC solar collectors, whether horizontal or tilted showing a distinctive three-dimensional distribution. This effect was more prominent with higher water inlet temperatures (20– 60 °C) compared to that with an 11 °C water inlet temperature. For example, Fig. 4 shows the cavity air temperature contours, for water inlet temperature of 40 °C and transverse tilt angles of / = 0° and / = 20°, in the cavity of the full height CPC solar collector. Temperature contours clearly indicate the existence of three-dimensional temperature gradients in the CPC cavities. Cavity air temperature decreased with increasing height in the vertical direction at all cross-sections along the absorber length in the cavities of all of the CPC solar collectors with water inlet temperatures of 20–60 °C. In the case of water inlet temperature of 10 °C the vertical temperature gradient at the near-inlet cross-section was positive, due to effects of cold water (10 °C) in the collector tube at the entrance and the mixing of air coming from the central cavity area with the air originally in the entrance area. However, at all cross-sections other than the near-inlet one, the vertical temperature gradient was found to be negative. In the case of water inlet temperatures of 20, 30, 40, 50 and 60 °C the cavity air vertical temperature gradient was found to be negative at all cross-sections along the collector length as shown in Fig. 4 for the full height CPC solar collector. As expected, the temperature of the cavity air was found to be highest adjacent to the absorber plate and lowest adjacent to the collector cover at all cross-sections along the entire length, indicating the existence of negative temperature gradients across the air layers in the cavities, of the full, three-quarter and half height CPC solar collectors tilted between 0 6 h 6 10° and 0 6 / 6 40° with water inlet temperatures of 20–60 °C. For the CPC solar collectors oriented at an angle / > 0°, the temperature of air was higher along the outer tilted side wall than those near the inner side wall showing the effect of tilting the collector. Fig. 4 shows the distribution of cavity air temperatures at three different cross-sections along the collector length in the cavity of the full height CPC solar collector tilted at / = 0° and / = 20° for a water inlet temperature of 40 °C. For the full height CPC tilted at / = 20°, temperatures of air near the right hand side wall of the cavity were found to be greater than those near the left side wall showing the effect of tilting of the collector, measured temperatures in the air spaces of the three-quarter and half height CPC solar collectors also indicated the existence of similar distribution. Fig. 4 signifies the existence of longitudinal (along the collector length) and transverse (along the absorber plate) temperature gradients inside the air space of the full height CPC collector cavities oriented horizontally (Fig. 4a–c) or at a tilt angle of / = 20° (Fig. 4d–f) with a water inlet temperature of 40 °C. The longitudinal temperature gradient existing in the cavity air is expected to cause longitudinal flow (single or multi roll-cell) of the air. The vertical and transverse temperature differentials cause the transverse flow, comprising double roll-cells, as cited previously by Eames and Norton (1993b), for the horizontally

2449

150

100

50

-100

-50

0

50

100

150

Absorber plate width (mm) Fig. 5. Illustration of expected double roll-cell convective flow of air inside the horizontally held (h = / = 0°) cavity of a full height CPC solar collector at the mid-section for water inlet temperature of 40 °C.

held CPC solar collectors. The measured cavity air temperature distribution indicate double roll-cell transverse flow for all CPC solar collectors, investigated in the present study, simultaneously tilted within the range 0 6 h 6 4° and 0 6 / 6 20°. Fig. 5 illustrates the expected double roll-cell convective flow of the cavity air as two dotted ellipses. Temperature distribution inside the cavities of full and three-quarter height cavities for / > 20° was not symmetrical about the transverse axis indicating a shift from double roll-cell to a single roll-cell flow. In the case of half height CPC solar collector at / = 30°, the cavity air temperature distribution indicated the possible existence of a double roll-cell at least in the central cross-section. This phenomena no longer persisted at / = 40°, indicating a shift to a single transverse roll-cell at a tilt angle between / = 30 and 40° for the half height CPC solar collectors. The interaction of the previously mentioned two distinctive flow regimes, consisting of longitudinal and transverse roll-cells, co-existing in the CPC cavities is expected to have resulted in full three-dimensional natural convective flow in the cavity air. The shifting from double roll-cell to single roll-cell flow at / = 20° for full height and three-quarter height CPCs and at / = 30° for the half height CPC indicated the effect of truncation on the natural convective flow. 6. Theoretical analysis The overall coefficient of heat transfer between the absorber plate and water has been determined employing correlations available in the literature (Incropera and Dewitt, 2002; Long, 1999; Duffie and Beckman, 2006). The back loss coefficient has been estimated allowing for enhanced wind velocity due to heating of the ground

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surface as a result of absorption of radiation emanating from the solar simulator (Eames et al., 2001). All three modes of heat exchange to and from the CPC solar components were considered and are shown in Fig. 6. The Nusselt number characterising the natural convective heat transfer occurring from the warm absorber plate to the cold glass cover was calculated for full, three-quarter and half height CPC solar collectors for a range of system parameters described in Section 2 and Table 1. The intensity of radiation incident on the glass cover, Iss, was measured using a pyranometer. Total amount of radiation incident on the collector glass cover, Qin, was calculated using Eq. (1). Qin ¼ Acov I ss

ð1Þ

The energy incident on the glass cover is distributed to the components of the collector, that absorbed by the glass cover being (Qcov), the side walls (Qswall), the end walls (Qewall), the absorber plate (Qabs) and that lost at the glass cover (QcovL), as given in Eq. (2). Qin ¼ Qcov þ Qswall þ Qewall þ Qabs þ QcovL Solar radiation

ð2Þ

1.7

1.1 2.1

3.1

3.2

1.4

1.2 2.3 2.2 1.3 1.5

3.3 3.4

1.6

2.4 Solar radiation 1.1 Reflection at glass cover 1.2 Radiative exchange between glass cover, side & end walls 1.3 Radiative exchange between glass cover, side and end walls 1.4 Reflection from insulation 1.5 Reflection from reflector walls to absorber 1.6 Long wave radiative loss from absorber back insulation to sky 1.7 Short wave reflection at glass cover 2.1 Convective loss from glass cover to ambient air 2.2 Natural convective loss from absorber to cavity elements 2.3 Convective loss from walls' insulation to ambient air 2.4 Convective loss from absorber back insulation to ambient air 3.1 Absorption at glass cover 3.2 Absorption at reflector and end walls 3.3 Absorption at absorber plate 3.4 Useful thermal energy removed

Fig. 6. Heat exchange processes in the CPC solar collectors investigated.

various terms involved in Eq. (2) were calculated employing the following equations. Qcov ¼ acov Acov I ss

ð3Þ

Qswall ¼ aref Aswall I ss ð1  acov Þ

ð4Þ

Qewall ¼ aref Aewall I ss ð1  acov Þ

ð5Þ

Qabs ¼ aabs Aabs I ss ð1  acov Þð1  aref Þ

ð6Þ

QcovL ¼ qcov Acov I ss þ U cov Acov ðT cov  T indoor Þ

ð7Þ

Here Ucov is the overall, radiative and convective, heat transfer coefficient from the glass cover to the indoor environment of the laboratory. Ucov was calculated employing empirical correlations for flat tilted surfaces (Incropera and Dewitt, 2002; Rabl, 1985) and verified using values recommended for flat indoor surfaces provided in CIBSE Handbook B (2006). The experimentally measured mean temperatures of the collector components at steady state conditions were employed to calculate the radiative heat exchange between the elements of the solar collector cavity. The radiative heat flow from the absorber plate to the collector cover (Qabs–cov,rad) has been calculated using Eq. (8) (Rabl, 1985). Qabscov;rad ¼ rAabs ðT 4abs  T 4cov Þeeff

ð8Þ

The average absorber plate temperature employed in Eq. (8) was the arithmetic mean of the temperatures measured at 15 locations on the absorber plate and the collector cover temperature of the temperatures measured at 21 locations. The effective emissivity, eeff, for the collectors was calculated using Eq. (9) (Rabl, 1976).   1 eeff ¼ ð9Þ ð1=eabs Þ þ ð1=ecov Þ  1 The Thermal energy extracted from the absorber plate by the water flowing in the CPC collector tube was calculated using Eq. (10). Qw ¼ mw C w ðT exit  T in Þ

ð10Þ

The physical properties of water (specific heat capacity, thermal conductivity and density) have been evaluated at the arithmetic mean of water temperatures at the collector inlet and exit. The insulation thickness on the reflector sides of the CPC solar collector varied because they were machined out of rectangular cross-section Styrofoam block. The minimum thickness of the insulation was measured to be 100 mm. The temperature at the outer surface of the insulation, in contact with the laboratory air, was measured to have a negligible variation along the collector height. The conductive heat loss from the absorber plate through the back and edge insulation and side walls, Qb, was estimated using Eq. (11) (Rabl, 1985).   k ins 2ðk ins =de;ins ÞðL þ W Þdb;ins Qb ¼ þ Aabs ðT abs  T ins Þ db;ins Aabs ð11Þ

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

The heat balance at the absorber plate has been employed (Eq. (12)) to arrive at the natural convective heat loss from the hot absorber plate to the CPC collector cavities. Qabs ¼ Qw þ Qabscov;rad þ Qb þ Qabsconv

ð12Þ

Finally, the natural convective heat loss from the absorber plate has been characterised in the form of a nondimensional coefficient, Nusselt number, Nu, (Eq. (13)). Nu ¼

ðQabsconv ÞH Aabs ðT abs  T cov Þk air

ð13Þ

7. Natural convective heat transfer in CPC collector cavities The Nusselt numbers characterising the natural convective heat transfer between the absorber plate and the glass cover were calculated for all three CPC solar collectors with the aim of studying the effect of both, longitudinal, transverse and simultaneous tilting of the solar collectors on natural convection. Errors in the Nusselt number values were calculated following a well established method due to Kline and McClintock(1953) and Moffat (1988) and are presented in the form of error bars. Fig. 7 shows the calculated values of the Nusselt number over the complete range of the parameters that applied to the CPC solar collectors during the present experiments. It can be deduced that a progressive reduction in the natural convective heat exchange between the selectively coated hot absorber plate and the glass cover, characterised by the Nusselt number, occured as the height of the CPC solar collector was reduced from the full height (0.259 m) to the three quarter height (0.195 m) and then

50

Half height CPC

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finally to the half height (0.130 m). Errors in the Nusselt number values were found to be within the range of 2.2– 5.8%, 2.2–8.2% and 1.7–6.3% for full height, half height and three quarter height CPC solar collectors respectively. The correlations, cited in ensuing sections, proposed for natural convection occurring for low water inlet temperatures of 10 °C, when applied to those for higher water inlet temperatures (20–60 °C) resulted in variations in the Nusselt number values in excess of 29% from the experimentally determined values. This resulted due to the difference in the way the absorber plate and other cavity elements were heated in the two cases. In the low temperature water inlet (10 °C) case all of the CPC elements were heated by the incident radiation from the solar simulator only. In the case of higher water inlet temperatures of (20–60 °C) the cavity elements received heat from the incident radiation from the solar simulator in addition to that from the hot water entering the absorber tube. The higher water inlet temperatures along with the incident radiation in the second case provided a set of thermal boundary conditions on the cavity elements, which were clearly different from those in which the heat source was the incident solar radiation alone. These two different sets of thermal boundary conditions resulted in the different natural convection phenomena observed in the CPC cavity. No single correlation that can represent both situations with sufficient accuracy could be found, this demanded the development of separate correlations to predict the Nusselt numbers for the two natural convection situations. The spread of the Nusselt number shown in Fig. 7 clearly indicates that the Nusselt number, as expected, is a complex function of Rayleigh number (Ra), water inlet temperature, longitudinal tilt angle (h), transverse tilt angle (/),

Three-quarter height CPC

Full height CPC

45 40

Nusselt number Nu

35 30 25 20 15 10 5 5.0E+05 5.5E+06 1.1E+07 1.6E+07 2.1E+07 2.6E+07 3.1E+07 3.6E+07 4.1E+07 4.6E+07 5.1E+07 5.6E+07

Rayleigh number Ra Fig. 7. Variation of Nusselt number with Rayleigh number for half height, three quarter and full height CPC solar collectors tilted between 0° 6 h 6 10° and 0° 6 / 6 40° with water inlet temperatures of 20, 30, 40, 50 and 60 °C.

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

heights (H), transverse aspect ratio (Ax) and the longitudinal aspect ratio (Az) of CPC solar collector cavities. Previously, Meyer et al. (1982a), Hollands et al. (1976), Kuehn and Goldstein (1976), and Eames and Norton (1993b) reported correlations to predict the Nusselt numbers for natural convective heat transfer in enclosures, which included the effect of one or more of these factors. In view of the correlations presented in these studies, a general form of the correlation (Eq. (14)) to characterise the natural convection in CPC solar collectors is proposed. c2

c3

c6

c7

c8

Nu ¼ c1 H ðT in Þ ðc4 cos h þ c5 cos /Þ ðAx Þ ðAz Þ ðln RaÞ

c9

ð14Þ

where c1, c2, c3, c4, c5, c6, c7, c8 and c9 are empirical constants. Based on the experimental data collected during the current experimental study the coefficients have been determined that best represent the dependence of the Nusselt number on the above listed factors. 8. Correlations proposed covering the complete parametric range studied Two separate correlations are being proposed to specify the natural convective heat transfer within the cavities of CPC solar collectors for lower (10 °C) and higher water inlet temperature (20, 30, 40, 50 and 60 °C) cases. These include the effects of the Rayleigh number, transverse and longitudinal geometrical aspect ratios, transverse and longitudinal tilt angles and the range of water inlet temperature on the Nusselt number for the CPC solar collectors of all three heights. Eq. (15) with a correlation coefficient of 0.90 is proposed to characterise the natural convection situations involving the lower water inlet temperature (10 °C), 8.2  105 6 Ra 6 1.2  107, in the CPC solar collector cavities detailed in Table 1. This correlation has been found to predict the experimentally derived Nusselt number values to within ± 10%. Fig. 8 compares the experimentally determined Nusselt number values with those predicted by Eq. (15) for the full height CPC solar collector tilted at / = 30° with a water inlet temperature of 10 °C. Experimental

Correlation

Nusselt number Nu

18

Experimental-1/2 height CPC Experimental-Full height CPC Correlation-1/2 height CPC

Experimental-3/4 height CPC Correlation-full height CPC Correlation-3/4 height CPC

50 45

Nusselt number Nu

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40 35 30 25 20 15 10 5 5.0E+05

1.1E+07

2.1E+07

3.1E+07

4.1E+07

5.1E+07

6.1E+07

Rayleigh number Ra Fig. 9. Nusselt number calculated from experiments and those using the developed correlation (Eq. (16)) for the three CPC solar collectors held at h = 0° and / = 30° with water inlet temperatures of 20, 30, 40, 50 and 60 °C.

Nu ¼ 1:961HT 0:091 ð0:186 cos h þ 0:003 cos /Þ4:42A0:473 A2:04 in x z ðlnðRaÞÞ

2:631

ð15Þ

This correlation has been derived from experimental data for CPC solar collectors simultaneously tilted over the range 0° 6 h 6 10° and 0° 6 / 6 40°. For the case of raised water inlet temperatures of 20, 30, 40, 50 and 60 °C, Eq. (16) is proposed and is calculated to have a correlation coefficient of 0.92 over the full parametric range. This correlation (Eq. (16)) has been derived for the same parametric range as Eq. (15) except that the former is valid for a higher Rayleigh number range over 1.6  106–5.7  107 and a water inlet temperatures range of 20 °C 6 Tin 6 60 °C. 0:66 Nu ¼ 0:12HT 0:137 ð0:13 cos h  0:127 cos /Þ0:121 A1:17 ðln RaÞ3:175 in x Az

ð16Þ

This correlation was found to predict the experimentally determined Nusselt number values to within ±9%. Fig. 9 shows the curves drawn based on correlation Eq. (16) for the three CPC solar collector heights. Curves shown in this figure have been drawn for the solar collectors held at h = 0°, / = 30° with water inlet temperatures of over 20 °C, 30 °C, 40 °C, 50 °C and 60 °C. The predictions of the Nusselt number for the other parametric ranges covered in the present work were found to have a similar level of fit.

15

8.1. Analysing the performance of the correlations proposed 12 9 6 3 0 9.00E+06

9.50E+06

1.00E+07

1.05E+07

1.10E+07

Rayleigh number Ra Fig. 8. Experimentally derived Nusselt number values and those predicted by correlation, Eq. (15), for the full height CPC solar collector simultaneously tilted at h = 0° and / = 30° with a water inlet temperature of 10 °C.

Figs. 10–16 compare the values of the Nusselt number obtained using the proposed correlations (Eqs. (15) and (16)) with those presented by past researchers for rectangular, cylindrical annuli, CPC and V-trough cavities. It can be seen from Fig. 10 that the Nusselt number values given by the proposed correlation (Eq. (16)) differ significantly from the results for rectangular enclosures. It is worth mentioning that the correlations due to Hollands (1973) and Hollands et al. (1976) were developed to predict the Nusselt number for natural convection in the cavities of flat plate solar collector cavities. A large ratio of length of the enclosure

Hollands, 1973 Ozoe et al., 1975 Hollands, 1976 Present work (3/4 height CPC)

Corcione, 2003 Smart et al., 1980 Present work (half height CPC) Present work (full height CPC)

Nusselt number Nu) (logarithmic scale)

100

10

εw =0.13

Ax=1 Ax=0.5

1 1.E+03

1.E+04

1E+05

1.E+06

1.E+07

Nusselt number Nu (logarithmic scale)

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457 Abdel-Khalik et al., 1978b Iyican et al., 1980 Smart et al., 1980 Present work (3/4 height CPC)

Ax=0.38, εw=0.13 10 Ax=0.38 Ax=0.58 Ax=1.15 1 1.0E+03

Abdel-Khalik et al., 1978b Rabl, 1976 Meyer et al., 1982a Present work (3/4 height CPC)

1.0E+06

1.0E+07

10

Iyican et al., 1980 Eames and Norton, 1993b Present work (half height CPC) Present work (full height CPC)

100

Nusselt number Nu (logarithmic scale)

Nusselt number Nu (logarithmic scale)

1.0E+05

Fig. 12. Nusselt number values predicted using the proposed correlation, Eq. (16), for Tin = 40 °C, h = 0° and / = 10°, and those presented in previous studies for transversely tilted (/ = 10°) CPC cavities.

Abdel-Khalik et al., 1978b Chew et al., 1988 Iyican et al., 1980 Prapas et al., 1987b Present work (3/4 height CPC)

100

10 Ax=0.58 Ax=1.15

Ax=0.58

Ax=0.38 1.E+04

1.0E+04

Rayleigh number Ra (logarithmic scale)

Fig. 10. The Nusselt number values obtained with the proposed correlation (Eq. (16)) drawn for horizontally oriented (h = / = 0°) CPC solar collectors for Tin = 40 °C and those presented in previous studies for horizontally oriented rectangular cavities.

1 1.E+03

Eames and Norton, 1993b Ozoe et al., 1975 Present work (half height CPC) Present work (full height CPC)

100

Rayleigh number Ra (logarithmic scale)

Rabl, 1976 Tatara and Thodos, 1985 Eames and Norton, 1993b Prapas et al., 1987a Present work (half height CPC) Present work (full height CPC)

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1 1.E+03

Ax=1.15

1.E+05

Ax=1.15 1.E+04

Ax=0.38 1.E+05

1.E+06

1.E+07

1.E+08

Rayleigh number Ra (logarithmic scale) 1.E+06

1.E+07

1.E+08

Rayleigh number Ra (logarithmic scale) Fig. 11. Nusselt number values obtained by the proposed correlation (Eq. (16)) for horizontally oriented (h = / = 0°) CPC solar collectors with Tin = 40 °C, and those presented in previous studies for horizontally oriented CPC and V-trough cavities.

to its width, L/W = 44, and the boundary condition employed by Smart et al. (1980) were similar to those adopted by Hollands (1973) and Hollands et al. (1976). The divergence of the present results shown as curves in Fig. 10 from their results highlight the fact that correlations developed to characterise natural convection in flat plate solar collectors are not appropriate for CPC solar collectors. The assumption of two-dimensional natural convection, different aspect ratios and isothermal heated or cooled walls along with either perfectly adiabatic or perfectly conducting or isothermal side walls in these studies are expected to cause this divergence. Corcione (2003) studied natural convection in an air layer enclosed in a two-dimensional rectangular enclosure with the assumptions of Boussinesq behaviour of air and the non-slip boundary condition at the walls.

Fig. 13. Nusselt number values predicted using the proposed correlation, Eq. (16), for Tin = 40 °C, h = 0° and / = 40°, and those presented in previous studies for transversely tilted (/ = 45°) CPC and trapezoidal cavities.

Heated or cooled surfaces were assumed to be isothermal and the side walls adiabatic. It can be seen in Fig. 11 that the curve showing the Nusselt number values predicted by Eq. (16) follow a trend similar to those presented by Chew et al. (1988) and Eames and Norton (1993b), although the absolute values of the Nusselt number differ, apparently due to the different sets of assumptions employed by these researchers. Eames and Norton (1993b) studied two-dimensional natural convective heat transfer in air enclosed in transversely tilted CPC solar collector cavities assuming boundary conditions of constant specified different convective and radiative heat transfer coefficients from tubular absorber, aperture cover and reflector walls. However, Chew et al. (1988) limited their study to two-dimensional natural convective heat transfer in a horizontal CPC solar collector assuming Boussinesq behaviour of air, isothermal

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H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457 Rabl, 1976 Iyican et al., 1980 Abdel-Khalik et al., 1978b Tatara and Thodos, 1985 Present work (3/4 height CPC) Present work (full height CPC)

35

Eames and Norton, 1993b Meyer et al., 1982a Kim and Viskanta, 1984 Chew et al., 1988 Present work (1/2 height CPC)

30

Nusselt number Nu

25

20

15 Ax=1.15 10

Ax=0.58

5 Ax=0.35 Ax=1.15

Ax=0.38 0 0

10

20

30

40

50

60

Transverse tilt angle

70

80

90

(º)

Fig. 14. Nusselt number values obtained using Eq. (16) for Tin = 40 °C, h = 0° and those due to past researchers for transversely tilted cylindrical annuli, rectangular, CPC and trapezoidal cavities at Ra = 1  107. (See above-mentioned references for further information.)

Half height CPC

Three-quarter height CPC

Full height CPC

=0º

30 = 0º

Nusselt number Nu

Transverse tilt angle

Nusselt number Nu

=10º

=20º

=30º

=40º

45

25

20

15

Full height CPC

40 35 30 25 20 15 10

10

5 0

5 0

2

4

6

8

10

12

Longitudinal tilt angle θ (º) Fig. 15. Variation of the Nusselt number with longitudinal tilt angle for the three CPCs held transversely at / = 0° with a water inlet temperature of Tin = 40 °C.

tubular absorber and aperture cover and perfect adiabatic reflector walls. The temperature difference between the absorber and aperture cover adopted by these researchers was similar to that of the present experiments, their Rayleigh number range is much smaller and transverse aspect ratios significantly higher. Results due to Abdel-Khalik and Li (1978) reach a maxima or a seemingly asymptotic level at Ra ffi 5.0  106. This is not the case with the present results shown as curves based on correlation, Eq. (16), and the values presented by other past researchers for similar cavities. Although, the result of Prapas et al. (1987a) follow a trend similar to the present results, shown in Fig. 11, their predicted values are significantly higher than the present

2

4

6

8

10

12

Longitudinal tilt θ (º) Fig. 16. The effect of simultaneous tilting, longitudinal tilting in the range h = 0–10° and transverse tilting in the range / = 0–40°, on the Nusselt number values for the full height CPC solar collector for a Tin = 40°C.

results. The Nusselt number correlation due to Prapas et al. (1987b) has an unrealistically high slope over a narrow Rayleigh number range, 7.7  105 6 Ra 6 2.1  106, for a CPC cavity with Ax = 0.79. Prapas et al. (1987b) employed for their study an unrealistic uninsulated CPC solar collector whose reflector walls and aperture covers were fabricated from a single polished duralumin sheet. Rabl (1976), Tatara and Thodos (1985) and Prapas et al. (1987a,b) have adopted thermal boundary conditions, which are not realistic for a CPC solar collector. Rabl (1976) assumed isothermal collector components and employed correlations based on simple relations involving hot horizontal and vertical plates neglecting the heat exchange between the absorber and the reflector walls. Tatara and Thodos (1985) studied a CPC solar collector with isothermal absorber

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457

and aluminised acrylic filmed aperture cover. The Nusselt numbers for CPC solar collectors tilted over the range of longitudinal tilt angle 0° 6 h 6 10° and the transverse tilt angle 0 6 / 6 40° due to the presently proposed correlation (5.21) have been found to have a good similarity with the results due to Eames and Norton (1993b), as is evident from Figs. 12 and 13. Reasons for divergence of the present results based on correlation (Eq. (16)) from Abdel-Khalik and Li (1978) have been cited previously. Ozoe et al. (1975) studied natural convective heat transfer in air layers enclosed in square cross-section rectangular enclosures with large aspect ratios of 8.4 and 15.5 assuming isothermal heated and cooled surfaces with adiabatic side walls. Fig. 14 compares the Nusselt number values given by the proposed correlation (Eq. (16)), with those presented by past researchers for natural convection in rectangular, veetrough, trapezoidal and CPC solar collector cavities for Ra = 1  107 over the range of transverse tilt angles covered in the present study. Similar results were found for Ra = 1  106. It was found that the Eq. (15) was not able to resolve the significant effects of transverse tilting on Nusselt number. However, correlation given by Eq. (16) was clearly able to resolve the effect of transverse tilting on the variation of Nusselt number as shown in Fig. 14. The results due to Rabl (1976), Abdel-Khalik and Li (1978), Iyican et al. (1980) and Meyer et al. (1982a) fail to show the effect of transverse tilting on the Nusselt number. The present study indicates that the Nusselt number increases initially as the transverse tilt angle is increased from / = 0° till it attains maxima at an angle between / = 40° and / = 60° and then decreases on further tilting, a trend similar to the one reported by Eames and Norton (1993b).

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convective flow in the cavity air. The enhanced natural convection and three-dimensional effects are expected to be the cause of the Nusselt number attaining its maxima at h = 4°. As the tilt angle is increased beyond h = 4° the longitudinal roll-cells merge to form one large roll-cell, which fills the whole cavity. This confines the three-dimensional natural convective flow to the near-wall areas only. In this case, the central portion of the cavity is filled with an essentially two-dimensional flow, resulting in lower Nusselt number values. The longitudinal tilt angle of h = 4° can be termed as the critical tilt angle characterising the transition from multi roll-cell longitudinal flow to single roll-cell longitudinal flow. Fig. 16 shows the Nusselt number values for the full height CPC solar collector simultaneously tilted over the range of h = 0–10° and / = 0–40° respectively for longitudinal and transverse tilting. Longitudinal and transverse tilt angles were increased in steps of 2° and 10° respectively during the experiments. It can be seen that the Nusselt number achieved maxima at a combined tilt of the longitudinal tilt angle of h = 4° and the transverse tilt angle of / = 40°. For the range of longitudinal tilt considered in the present study the maximum Nusselt number values always occurred at transverse tilt angles of / = 40°. Similar results were obtained for other CPC solar collectors studied. 10. Conclusions The results of an experimental investigation into the natural convection heat transfer phenomena that occur in the cavities of full height (CR = 2), three-quarter height and half height CPC solar collectors with flat plate absorbers under normal radiation incidence have been presented. The following conclusions can be drawn:

9. Effect of collector tilt on the Nusselt number Variation of the Nusselt number with longitudinal tilt angle between from 0° and 10° at a fixed transverse tilt angle of / = 0° for all three CPC collectors is shown in Fig. 15. The Nusselt number increased initially as the longitudinal tilt was increased from h = 0° achieving the highest value at h = 4° and then, decreasing on further increase of the longitudinal tilt angle. It can be deduced that as the longitudinal tilt angle is increased from h = 0° to h = 4°, natural convection inside the cavity also intensifies resulting in three- dimensional effects, which become most prominent at h = 4°. In the horizontal position (h = / = 0°), the natural convective flow inside the cavity is expected to consist of transverse roll-cells, characterised by their axes parallel to the longitudinal axis of the solar collector, as depicted in the past by Eames and Norton (1993a,b). On increasing tilt angle from h = 0°, the longitudinal roll-cell(s) with axes perpendicular to the longitudinal axis of the solar collector start appearing. Initially, multiple longitudinal roll-cells occur in the collector cavity, which on further tilting start to merge into each other increasing in size. The interaction of pre-existing transverse roll-cells and newly developing longitudinal roll-cell(s) enhances three-dimensional natural

 The cavity air temperature showed a distinctive threedimensional distribution along the transverse, longitudinal and vertical axes of all three CPC solar collectors, whether horizontal or tilted, with water inlet temperatures of 10, 20, 30, 40, 50 and 60 °C. The longitudinal tilt angle of h = 4° can be viewed to be the critical tilt angle characterising the transition from multi roll-cell longitudinal flow to a single longitudinal roll-cell.  The Nusselt number, characterising the natural convection phenomena taking place within the collector cavity, has been found to increase with the longitudinal tilt angle, when the CPC collectors were oriented at / = 0°, and raised from horizontal position (h = 0°) and attained maximum value at a longitudinal tilt angle of h = 4°. A increase of 24% in the Nusselt number values was calculated as the longitudinal tilt angle increased from 0° to 4° for water inlet temperature of 40 °C for the half height CPC solar collector. It can be concluded that in order to minimise the natural convective loss from the absorber plate of CPC solar collectors studied these are best oriented at an angle between horizontal (h = 0°) and h = 2°. The variation of the

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Nusselt number with transverse tilt angles, when the solar collectors were oriented at h = 0°, has been found to follow a trend similar to that presented previously by Eames and Norton (1993b). The Nusselt number was found to have a maximum increase of 49–56% as the transverse tilt angle was increased from / = 0° to / = 40°.  For the case of the simultaneously orientated CPC solar collectors, the maximum Nusselt number was found to occur at a combined tilt of h = 4° and / = 40° for all three CPC solar collectors. The Nusselt number increase of 63–70% was calculated as the orientation of the CPC solar collectors was changed from horizontal (h = / = 0°) to an orientation of h = 4° and / = 40°.  The correlation given by Eq. (16) is proposed to predict the natural convection heat loss from the absorber plate of solar collectors with geometric and working parameters detailed in Table 1 for water inlet temperatures of 20, 30, 40, 50 and 60 °C. The present study has clearly indicated that the correlations developed for prediction of natural convection characteristics in rectangular, annuli and V-trough enclosures are not appropriate for application to CPC solar collectors with divergence ranging from 150% to 300%. Interestingly, but not surprisingly, the predictions of past numerical and experimental studies diverge from each other and significantly from those of the present study. This is a result of past studies assuming two-dimensional behaviour and boundary conditions at the absorber plate, collector covers and reflecting side walls that are far from the real ones that have been used in the present study. The assumptions that are expected to be the cause of the divergence in the results of past studies have been clearly highlighted. Acknowledgements Valuable support and advice received during the course of this study from Professors Brian Norton and Neil Hewitt, and technicians Alan Carnduff (late) and Phil Dalzell are kindly acknowledged. References Abdel-Khalik, S.I., Li, H.W., Randall, K.R., 1978. Natural convection in compound parabolic concentrators – a final element solution. ASME J. Heat Transfer 100, 199–204. Abdel-Khalik, S.I., Li, H.W., 1978. Natural convection in inclined twodimensional compound parabolic concentrators. In: Proc. ASME Winter Annual Meeting San Francisco, 23–28. ALANOD Ltd., 2006, Property sheet for MIRO-Silver, ALANOD Ltd., Chippenham Drive, Kingston, Milton Keynes MK10 0AN, UK. Alfegi, M.E.A., Sopian, K., Othman, M.Y.H., Yatim, B.B., 2007. Transient mathematical model of both side single pass photovoltaic thermal air collector. ARPN J. Eng. Appl. Sci. 2, 22–26. Chemisana, D., 2011. Building integrated concentrating photovoltaics: A review. Renew. Sust. Energ. Rev. 15 (1), 603–611.

Chew, T.C., Tay, A.O., Wijeysundera, N.E., 1989. A numerical study of the natural convection in CPC solar collector cavities with tubular absorbers. ASME J. Sol. Energy Eng. 111, 16–23. Chew, T.C., Wijeysundera, N.E., Tay, A.O., 1988. An experimental study of free convection in compound parabolic concentrator (CPC) cavities. ASME J. Sol. Energy Eng. 110, 293–297. Corcione, M., 2003. Effects of the thermal boundary conditions at the side walls upon natural convection in rectangular enclosures heated from below and cooled from above. Int. J. Therm. Sci. 42, 199–208. Duffie, J.A., Beckman, W.A., 2006. Solar Engineering of Thermal Processes. John Wiley & Sons, Hoboken, New Jersey. Eames, P.C., Norton, B., 1993a. Validated, unified model for optics and heat transfer in line-axis concentrating solar energy collectors. Sol. Energy 50 (4), 339–355. Eames, P.C., Norton, B., 1993b. Detailed parametric analyses of heat transfer in CPC solar energy collectors. Sol. Energy 50 (4), 321–328. Eames, P.C., Smyth, M., Norton, B., 2001. The experimental validation of a comprehensive unified model for optics and heat transfer in line-axis solar energy systems. Sol. Energy 71 (2), 121–133. Edwards, D.K., Sun, W.M., 1971. Effect of wall radiation on thermal instalbility in a vertical cylinder. Int. J. Heat Mass Transfer 14 (1), 15– 18. Hollands, K.G.T., 1973. Natural convection in horizontal thin-walled honeycomb panels. ASME J. Heat Transfer 95, 439–444. Hollands, K.G.T., Konicek, L., 1973. Experimental study of the stability of differentially heated inclined air layers. Int. J. Heat Mass Transfer 16, 1467–1476. Hollands, K.G.T., Unny, T.E., Raithby, G.D., Konicek, L., 1976. Free convective heat transfer across inclined air layers. ASME J. Heat Transfer 98, 189–193. Hsieh, C.K., 1981. Thermal analysis of CPC collectors. Sol. Energy 27, 19–29. Incropera, F.P., Dewitt, D.P., 2002. Introduction to Heat Transfer, fourth ed. John Wiley & Sons, New York. Iyican, L., Witte, L.C., Bayazitoglu, Y., 1980. An experimental study of natural convection in trapezoidal enclosures. ASME J. Heat Transfer 102, 648–653. Jing, L., Gang, P., Jie, J., 2010. Optimization of low temperature solar thermal electric generation with Organic Rankine Cycle in different areas. Appl. Energy 87 (11), 3355–3365. Keithley Instruments Ltd., 2004, Model 2750 Multimeter/Switch System, Available from Keithley Instruments Ltd., Unit 2 Commerce Park, Brunel Road, Theale, Berksire RG7 4AB, UK. Kessler, R., 1987. Nonlinear transition in three-dimensional convection. J. Fluid Mech. 174, 357–379. Kim, D.M., Viskanta, R., 1984. Study of the effects of wall conductance on natural convection in differently oriented square cavities. J. Fluid Mech. 144, 153–176. Kline, S.J., McClintock, F.A., 1953. Describing uncertainties in singlesample experiments. ASME J. Mech. Eng. 75, 3–8. Kuehn, T.H., Goldstein, R.J., 1976. Correlating equations for natural convection heat transfer between horizontal circular cylinders. Int. J. Heat Mass Transfer 19 (10), 1127–1134. Leong, W.H., Hollands, K.G.T., Brunger, A.P., 1998. On a physicallyrealizable benchmark problem in internal natural convection. Int. J. Heat Mass Transfer 41, 3817–3828. Leong, W.H., Hollands, K.G.T., Brunger, A.P., 1999. Experimental Nusselt numbers for a cubical-cavity benchmark problem in natural convection. Int. J. Heat Mass Transfer 42, 1979–1989. Lock, G.S.H., Han, J.C., 1990. Free convection in a slender, laterally heated cavity: Inclination effects. Math. Comput. Modell. 14, 810–813. Long, C., 1999. Essential Heat Transfer. Longman, Harlow. Meyer, B.A., Mitchell, J.W., El-Wakil, M.M., 1982a. Convective heat transfer in vee-trough linear concentrators. Sol. Energy 28 (1), 33– 40. Meyer, B.A., Mitchell, J.W., El-Wakil, M.M., 1982b. The effect of thermal wall properties on natural convection in inclined rectangular cells. ASME J. Heat Transfer 104, 111–117.

H. Singh, P.C. Eames / Solar Energy 86 (2012) 2443–2457 Moffat, R.J., 1988. Describing the uncertainties in experimental results. Exp. Thermal Fluid Sci. 1, 3–17. Norton, B., 1991. Solar Energy Thermal Technology. Springer-Verlag London Limited, London. Norton, B. et al., 2010. Enhancing the performance of building integrated photovoltaics. Sol. Energy. http://dx.doi.org/10.1016/j.solener.2009. 10.004. Othman, M.Y.H., Yatim, B.B., Sopian, K., Abu Bakar, M.N., 2005. Performance analysis of a double-pass photovoltaic/thermal (PV/T) solar collector with CPC and fins. Renew. Energy 30 (13), 2005–2017. Ozoe, H., Sayama, H., Churchill, S.W., 1974a. Natural convection in an inclined square channel. Int. J. Heat Mass Transfer 17, 401–406. Ozoe, H., Sayama, H., Churchill, S.W., 1975. Natural circulation in an inclined rectangular channel at various aspect ratios and angles – experimental results. Int. J. Heat Mass Transfer 18, 1425–1431. Ozoe, H., Yamamoto, K., Churchill, S.W., Sayama, H., 1976. Threedimensional numerical analysis of Laminar natural convection in a confined fluid heated from below. ASME J. Heat Transfer 98, 202–207. Ozoe, H., Yamamoto, K., Sayama, H., Churchill, S.W., 1974b. Natural circulation in an inclined rectangular channel heated on one side and cooled on the opposite side. Int. J. Heat Mass Transfer 17, 1209–1217. Prapas, D.E., Norton, B., Probert, S.D., 1987a. Thermal design of compound parabolic concentrating solar-energy collectors. J. Sol. Energy Eng. 109, 161–168. Prapas, E., Norton, B., Melidis, P.E., Probert, S.D., 1987b. Convective heat transfer within air spaces of compound parabolic concentrating solar-energy collectors. Appl. Energy 28, 123–135.

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Rabl, A., 1976. Optical and thermal properties of compound parabolic concentrators. Sol. Energy 18, 497–511. Rabl, A., 1985. Active solar collectors and their applications. Oxford University Press, New York. Smart, D.R., Hollands, K.G.T., Raithby, G.D., 1980. Free convection heat transfer across rectangular-celled diathermanous honeycombs. ASME J. Heat Transfer 102, 75–80. Singh, H., Eames, P.C., 2011. A review of natural convective heat transfer correlations in rectangular cross-section cavities and their potential applications to compound parabolic concentrating (CPC) solar collector cavities. Appl. Thermal Eng. 31 (14–15), 2186–2196. Symons, J.G., Peck, M.K., 1984. Natural convection heat transfer through inclined longitudinal slots. ASME J. Heat Transfer 106 (1984), 824– 829. Tatara, R.A., Thodos, T., 1985. Experimental natural convective studies within a compound parabolic concentrator enclosure. In: Proc. ASME Winter Annual Meeting. Miami, Florida, pp. 17–22. Tchinda, R., 2008. Thermal behaviour of solar air heater with compound parabolic concentrator. Energy Convers. Manage. 49 (4), 529–540. Yadav, Y.P., Yadav, A.K., Anwar, N., Eames, P.C., Norton, B., 1996. The fabrication and testing of a line-axis compound parabolic concentrating solar energy collector. Proc. WREC 1996, 572–575. Yang, H.Q., Yang, K.T., Lloyd, J.R., 1987. Laminar natural-convection flow transitions in tilted three-dimensional longitudinal rectangular enclosures. Int. J. Heat Mass Transfer 30 (8), 1637–1644.