Design, simulation and optimization of a compound parabolic collector

Sustainable Energy Technologies and Assessments 16 (2016) 53–63

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Sustainable Energy Technologies and Assessments journal homepage: www.elsevier.com/locate/seta

Original Research Article

Design, simulation and optimization of a compound parabolic collector E. Bellos ⇑, D. Korres, C. Tzivanidis, K.A. Antonopoulos National Technical University of Athens, School of Mechanical Engineering, Thermal Department, Heroon Polytehniou 9, 157 73 Zografou, Athens, Greece

a r t i c l e

i n f o

Article history: Received 12 November 2015 Revised 19 March 2016 Accepted 19 April 2016

Keywords: CPC Solidworks Optical analysis Thermal analysis Collector design

a b s t r a c t In this study, the optical and the thermal performance of a compound parabolic collector (CPC) with evacuated tube are presented. In the first part, the optimization of the reflector geometry is given and in the next part the thermal analysis of the solar collector is presented. The design of the reflector has a great impact on the solar energy exploitation and for this reason is analyzed in detail. In the thermal analysis of the collector, the two most usual thermal fluids, the pressurized water and typical thermal oil, are compared. Pressurized water performs better and it is the most suitable working fluid for transferring the heat because of its properties; something that is analyzed in this study. Moreover, the optical efficiency of the collector for various solar angles (longitude and transverse) is investigated and the heat flux distribution over the absorber is given. In the last part, the temperature distribution over the absorber and inside the fluid are presented and a simple validation of the thermal model is also presented. The model is designed in commercial software Solidworks and simulated in its flow simulation studio. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction The increasing cost of fossil fuels and of the electricity conjugated with the environmental problems caused by the CO2 emissions lead our society to turn its interest in renewable energy sources. Solar energy utilization is a promising way to cover a great part of worldwide energy demand by various ways. The conventional flat plate collectors (FPC) are widely used for domestic hot water production and for low temperature applications (30–90 °C). Concentrating collectors with high concentrating ratios operate in high temperature levels (300–400 °C) [1] by giving suitable heat for electricity production in power plants. Parabolic trough collectors (PTC), Fresnel collectors, central tower receivers and parabolic dish Stirling engines [2,3] are the main solar technologies for electricity production. For the intermediate temperature range from 100 °C to 300 °C lessens number of solar collector types are used while many industrial and residential applications operate in these temperature limits. Applications as desalination, oil extraction, low temperature electricity production, food production, methanol reforming and space cooling with absorption technology [4–13] demand energy sources in the above temperature range. The most suitable solar collector for these conditions is the compound parabolic collector (CPC) with evacuated tube which is able to produce efficiently the processing heat. The use of the evacuated tube is essential in order to overcome the ⇑ Corresponding author. Tel.: +30 210 772 2340. E-mail address: [email protected] (E. Bellos). http://dx.doi.org/10.1016/j.seta.2016.04.005 2213-1388/Ó 2016 Elsevier Ltd. All rights reserved.

limit of 100 °C and the conjugation with a concentrating trough leads to higher levels. CPC belongs to non-imaging concentrators with low concentrating ratio (1–5) [14,15] which exploits mainly the beam radiation and a part of diffuse radiation [8]. The small concentration ratio recuses the tracking demand and many CPC systems are able to operate without tracking which lead to lower cost [16,17]. More specifically, a tracking system with the CPC axis in East–West orientation needs only a small seasonal adjustment in order to perform in a high way [15]. The geometry design of a CPC is related to the application and every manufacturer takes into consideration the operating condition in every case. Important point in the design is the relation between concentration ratio and the acceptance angle which is inversely [14,15]. CPC invented by Winston in 1960 in the U.S.A. and they presented in 1974 [18,19]. The first applications were about hot water supply and many studies have been made for improving their performance. Rabl in 1976 [20] developed a mathematical model for the average number of rays reflections in a CPC, something very important for the optical analysis. Studies for CPC with non-evacuated tubes for thermal performance have been made in order to predict the efficiency in various operating conditions [21,22]. The use of evacuated tubes was first analyzed in Argonne National Laboratory [23] in before 1980. Snail in 1984 [24] analyzed an integrated stationary CPC with evacuated tube. The final results proved an optical efficiency of 65% and a thermal of 50%. Kim et al. [25] compared a stationary and a tracing CPC and proved that the tracking mechanism improves the efficiency at about 15%. Because the tracking system is important

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Nomenclature A C cp D f G H hm k L m Num Pr p Q q Re T UL W

area, m2 concentration ratio specific heat capacity, J/kg K diameter, mm focal length, mm solar radiation, W/m2 parabola length, mm mean convection coefficient, W/m2 K thermal conductivity, W/m K tube length, mm mass flow rate, kg/s mean Nusselt number Prandtl number absorber placement, mm heat flux, W down part placement, mm Reynolds number temperature, °C losses coefficient, W/m2 K aperture length, mm

Greek symbols b peripheral absorber angle, ° c intercept factor e emittance g efficiency h solar incident angle, ° hc half-acceptance angle, ° hL longitude solar angle, ° transverse solar angle, ° hΤ l dynamic viscosity, Pa s q reflectance

parameter, many other researches have been worked in this area. Sallabery [26] analyzed how the tracking error influences on the long-term performance of the system and reported that the yearly energy loss is about 1%. A comparison of N–S and E–W orientated CPCs presented by Kim [5]. Mathematical equations for solar transversal projection angle, longitudinal projection angle are developed by Wang [27] so as the tracking systems to be analyzed more. A lot of research has been conducted worldwide for the CPC performance improvement. The use of asymmetric CPC, which means two different parabolic shapes, has been studied from many researchers. The main idea is the development of a non tracking collector which performs well during the day due to the different shape of the reflector’s parabolas that allows the collector to operate efficiently for a great range of incident angles. Abu-Bakar [28] studied a rotationally asymmetrical compound parabolic concentrator with a PV module, while Souliotis et al. [29] and Kessentini [30,31] analyzed asymmetrical CPC for integrated solar systems with one and two tanks inside the collector respectively. Moreover, Singh et al. [32] made a very interesting review about integrated collector solar water heaters stating the novelties that are able to increase their efficiency. These systems include compound parabolic reflectors, phase change materials and special materials (absorber and cover) in order to achieve high daily performance. The idea of solar cooker examined by Harmin [33] where a booster reflector was located in order to increase the optical efficiency in the stationary mode operation. The use of lens in the trough in order to increase the acceptance angle is an innovative idea which is being examined in recent years. Su et al. [34] made a comparison

(sa) u

transmittance–absorptance product Parabola angle parameter, °

Subscripts and superscripts a aperture abs absorbed am ambient b beam c cover ca cover-air ci inner cover co outer cover d diffuse e exploited fm mean fluid in inlet L Local loss losses m mean max maximum o oil opt optical out outlet r receiver ri inner receiver ro outer receiver s solar th thermal tube receiver tube u useful w water

between lens-walled CPC, a common CPC and a dielectric solid CPC. The results showed that lens-walled CPC has greater acceptance angle, but it has lower optical efficiency in low incidence angles. Guiqiang et al. [14] analyzed also a lens-walled CPC with a PV module and resulted that the lens creates a uniform flux distribution in the PV-module which increases their efficiency. In this study, a CPC with an evacuated tube is designed and simulated, while both an optical and thermal analysis has been conducted. Firstly, an optimization of its geometry is made in order to maximize the optical efficiency. Also, a parametric analysis for different solar angles (transversal and longitudinal) is given to calculate the optical losses for different cases. Moreover, a thermal efficiency comparison between pressurized water and thermal oil as working fluids is presented for different operating conditions. Finally, a deeper analysis for the best working fluid, the pressurized water, is presented. The simulation of the collector has been done with commercial software Solidworks flow simulation. The innovative point of this study is the reflector design. The methodology that is followed lead to an intercept factor close to 1, which is the ideal modeling. Moreover, a comparison between pressurized water and thermal oil is presented in order to determine the most suitable fluid energetically. Examined model A compound parabolic collector with an evacuated tube is examined in this study. The model was designed in Solidworks by a parametrical way in order to optimize its geometry. This optimization gives the opportunity to improve the collector optical

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55

Fig. 1. Geometric parameters of CPC design.

efficiency and to achieve optimum performance. CPC is created by two parabolas and a third part which connects them. In the examined case, this part is created by two identical circular parts which are tangent with the parabolas. This tangency is crucial because leads to a smooth surface and as a consequence to a greater optical efficiency. Fig. 1 shows the CPC scheme and the geometry parameters; the aperture, the height, the acceptance angle, the tube diameters and etcetera. It is important to state that many parameters influence on the optical performance of the collector and the optimization of all them leads to a very complex problem. For this reason, some of these parameters were selected to be constant and the optimization was applied on the rest of them. First of all, the focal distance f and the angle u were selected to be 50 mm and 90° respectively, while the absorber diameter designed at 30 mm. The absorber’s sketch is located in the symmetry axis of the sketch plane and is tangent to the straight line AB which connects the two parabolas focus as well as to the one is defined from C and B. These conditions lead all the rays which are incident vertically on the aperture plane and arrive at the parabolic surfaces to reach to the absorber tube. The parameters were being optimized are the aperture W and the distance q which is related to the down part of the CPC. By knowing these parameters, the height H and the acceptance half-angle hc are easily calculated. The optical efficiency of the collector is the ratio of the absorbed energy from the tube to the total solar energy entering to the aperture. This quantity can be easily calculated by a product of three parameters, as Eq. (1) presents:

Q gopt ðhÞ ¼ abs ¼ q  cðhÞ  ðsaÞ; Qs

ð1Þ

The mirror reflectivity q is the first optical loss and it is affected by the surface quality. The next loss is expressed by the intercept factor c which is depended on the incident angle of the radiation. More specifically, this parameter is calculated as the ratio of the solar irradiation that arrives at the tube to the reflected solar irradiation from the mirror. In addition, the reflector shape determines the intercept factor, thus an optimization of the reflector geometry is presented. Eq. (2) summarizes the above analysis:

c¼

Q tube ; Qs  q

ð2Þ

Qtube symbolizes the reaching radiation to the tube while (sa) is the product of cover transmittance and receiver absorbance. In this point, it is essential to state that the solar radiation, that CPC utilizes, is the beam radiation and a part of the diffuse radiation which is depended on concentration ratio. For this study case, without slope, Eq. (3) describes the exploited solar irradiation:

Ge ¼ Gb þ

Gd ; C

ð3Þ

The incident angle h is a very important parameter which affects on the optical efficiency. However, extra information about the relative position between collector and sun is necessary in order to predict the exact path of the solar rays inside the collector. For this reason, two other angles, the transverse and the longitude, are being introduced to determine the exact position of the sun. These solar angles are used in the optical analysis. Fig. 2 illustrates these solar angles in a 3-D scheme. Eq. (4) gives the incident angle (h) as a function of the transversal and the longitudinal angles:

tan2 ðhÞ ¼ tan2 ðhL Þ þ tan2 ðhT Þ;

ð4Þ

Table 1 includes the basic dimensions of the collector and other important parameters of the simulation. Simulation in Solidworks environment and methodology The examined model was designed in Solidworks and simulated in its flow simulation studio. The first step in the simulation was the determination of the physical problem. The boundary conditions, the materials of the parts and the properties of every material are the main parameters that have been determined. Moreover, the solar radiation parameters are very important for this analysis and emphasis is given to them. The material selection is presented below: – The absorber is selected to be made of cooper.

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Fig. 2. Solar angles and 3-D CPC shape.

– The reflector surface was set as ‘‘symmetry” surface. This choice is necessary for the proper solar rays reflection.

Table 1 Geometry dimensions and simulation parameters. Simulation parameter

Values

Geometry dimensions

Values

er ec q(sa)

0.1 0.88 0.80 1000 W/m2 0.01 kg/s 10 °C 10 W/m2 °C

p L f Dri Dro Dci Dco

17 m 1000 mm 50 mm 30 mm 34 mm 44 mm 48 mm

Ge mw Tam hca

– The cover is transparent to solar energy and is made of glass. – The reflector has a special mirror surface. – Lids have been placed in the inlet and in the outlet of the tube in order to create a closed fluid volume. These lids were selected to be as ‘‘insulators”. By this way, these materials are not taking part in the thermal analysis. The proper boundary conditions of the simulation are the following: – The inlet mass flow rate in the internal face of the inlet lid. – The temperature in the internal face of the inlet lid. – The flow was selected to be fully developed in the inlet of the tube. – The pressure of the outlet was determined in every case and is a necessary boundary condition for the flow simulation inside the tube. – The heat convection coefficient between the outer cover surface and the environment. The next important parameters that have to be taken into consideration are related with the radiation surfaces of the materials: – The absorber outer surface was selected to be selective. – The inner and outer cover surfaces have been selected to have the proper radiation properties (emittance, transparency).

In addition, specific convergence goals were selected in order to take the proper output from Solidworks and to lead the solver to the desirable results. The main outputs of the simulation tool are the following: – The fluid bulk temperature in the outlet. – The mean receiver temperature in its outer surface. – The mean cover temperature in its volume. For the cover, the temperature difference between their surfaces is very low and for this reason the mean volume temperature was selected as output. – The thermal losses of the receiver which are equal to its radiation losses. – The total enthalpy difference between the outlet and the inlet of the tube. The mesh in the computational domain was created by Solidworks and emphasis was given in the fluid domain. For this reason, extra refinement levels were added in fluid and in partial cells. More specifically, in the mesh generation a standard mesh is created by selecting the basic nodes. The next step is a refinement in fluid and partial cells in order to make the mesh better inside the tube. The final mesh contains in every cross cross-section about 50,000 cells and the total mesh is consisted of about 4,000,000 cells. The choice of these values have made after a small sensitivity analysis. Different refinement levels have been tested and the convergence criterion was the fluid outlet temperature. The solar radiation was determined by its intensity and its direction through the general settings of the program. It is essential to state again that the optical analysis is made by Solidworks flow simulation with setting the reflector surface as ‘‘symmetry surface”. The multiple reflections have been taken into account in this model. The model was simulated for various study cases. In every case, some parameters were kept constant and others were varied. The

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temperature of the working fluid in the inlet and the solar radiation direction were the parameters, parametrically investigated. Other parameters as the solar radiation intensity, the ambient temperature and the mass flow rate were kept constant. The energy balances of the thermal model are presented in Appendix A with more details.

1.0 0.9 0.8 0.7

Optical analysis

300 mm

0.6 200

In this paragraph the geometrical parameters which influence on the optical efficiency of the collector are investigated in order to predict the optimum values of the examined geometric parameters. The optimization of the intercept factor leads to the optimization of the optical efficiency which is the goal of this analysis.

250

300

350

400

450

500

W(mm) Fig. 4. Intercept factor for different aperture values.

Table 2 Optimized geometry parameters.

Optimization of the reflector geometry The objective of this section is to present the influence of the geometric parameters (q and W) on the intercept factor. More specifically, the shape of the down part of the CPC as well as the aperture surface value is examined by changing the q and W values respectively. The rest of the parameters were selected to be constant, while the solar rays are vertical to the aperture plane (h = 0°). Fig. 3 depicts the effect that distance q has on the intercept factor, and as a consequence on the optical efficiency. It is obvious that there is a great region of values which lead to an optimum performance of the intercept factor. Especially, when the parameter q takes values in the range of 13–26 mm, all the incident solar rays arrive at the tube, after the reflection on the mirror. This distance was selected at 21 mm which is a value inside the optimum region, in order to take a smooth surface. More specifically, this value for the parameter ‘‘q” makes the two circular parts to be tangent in their contact point, fact that makes them to belong to the same cycle. The next parameter is the aperture length W which is analyzed in Fig. 4. For low aperture values the intercept factor is maximized because all the reflected rays arrive to the receiver. After a critical point, the intercept factor is decreasing something that affects directly on the optical efficiency. This leads us to select the optimum value at the critical point which is the aperture of 300 mm width. It is essential to mention that in the optimum case, the line which connects the focus of each parabola to the opposite upper point is tangent to the receiver (Fig. 1). This is according to the theory which demands this condition in order to determine the optimum aperture. Table 2 gives the final results of the optimization. Heat flux distribution over the receiver The next important issue about the optical analysis is the way that the heat flux is being distributed in the receiver. The following Figs. 5 and 6 show this distribution over the circular geometry of

Geometry parameters

Value

q W H hC Aa C

21 m 300 mm 150 mm 53° 0.3 m2 2.81 0.9921 0.7937

cmax gopt,max

the receiver. The results were taken from Solidworks for the optimum case. In order to present the heat flux distribution with a dimensionless way, we have to calculate the local concentration ratio as Eq. (5) suggests below:

CL ¼

dQ tube

q  Ge  dA

;

ð5Þ

Fig. 5 shows the local concentration ratio in the peripheral line of the absorber. This distribution is very important because there is a great variation from point to point over the absorber. This figure gives a strange profile for the heat flux and the reason is the complex geometry of the reflector. Moreover Fig. 6 depicts the heat flux distribution over the geometry, as it is taken by Solidworks results. From the above figures it is obvious that the maximum solar irradiation is concentrated on the top of the absorber (b = 0°). This can be explained because this part collects beam radiation directly from the sun and simultaneously a part of the reflected radiation from the parabolas. The lower part of the receiver (b = 180°) collects the lessen radiation, because only a small part of the reflector sent rays on this region. Moreover, in the down part of the absorber there are two symmetry regions (b = 145° and b = 215°) where the heat flux takes great values due to the fact that both the parabolas and the down circular part of the mirror reflect the rays to these two specific parts of the absorber. Incident angle impact on intercept factor

1.00 0.95 0.90 0.85 0.80

q=21 mm

0.75 0

5

10

15

20

q (mm)

25

30

35

Fig. 3. Intercept factor for different shapes of CPC down part.

40

The next step in the optical analysis is the determination of optical efficiency for different sun positions. In no concentrating collectors, the incident angle is sufficient to similar calculations, but in the concentrating collectors, a deeper analysis is needed. For this reason, a more analytic determination of the sun position uses two angles, in the longitudinal and in the transversal direction. This means that the position of the sun affects significantly on the optical efficiency. Fig. 7 displays how the intercept factor variation as a function of the longitudinal angle. It is obvious that the optical loss is getting greater as the longitudinal angle increases. It is remarkable that, the loss for small

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6 5

CL

4 3 2 1 0 0

50

100

150

200

250

300

350

400

() Fig. 5. Local concentration ratio distribution in peripheral of the absorber.

Fig. 6. Heat flux distribution over the absorber geometry, with red the maximum values and with blue the lower values. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0

20

40

60 L(

0.0

80

0

o)

10

20

30 T(

Fig. 7. Incident angle as a function of longitude solar angle.

40

50

60

)

Fig. 8. Incident angle as a function of the transverse solar angle.

values of the longitudinal angle is almost inconsiderable. The mathematical approximation of the above curve is presented in Eq. (6):

cðhL Þ ¼ 0:9921  4:1  104  hL  1:92  104  h2L þ 7  107  h3L ; ð6Þ

It is important to state that above the value of 80°, the efficiency is extremely low and the system stops operating. The next analyzed angle is the transversal one which is given in the Fig. 8.

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The next step after determining optical efficiency optimization is the thermal performance investigation of the examined collector. Two different working fluids, pressurized water and thermal oil, are tested in order to make a comparison between them. The water is under a high pressure levels (35 bar) in order to be kept in liquid phase for all study cases. The thermal oil operates in lower pressure level because it is kept in liquid phase up to 300 °C without needing a great pressure increase. The properties of these fluids are taken by Solidworks library. The thermal efficiency of the collector can be calculated as the ratio of the useful energy to the solar energy entering in the collector’s aperture:

ð8Þ

The useful energy from the collector is calculated by the energy balance in the working fluid volume, according to Eq. (9):

_  cp  ðT out  T in Þ; Qu ¼ m

ð9Þ

Thermal oil mass flow rate was determined according to Eq. _  cp Þ to be the same in (10), in order the total heat capacity ðm the two cases. By this assumption, if the two fluids absorb the same amount of heat, then the same temperature increase is being occurred. This will make the absorber to have similar temperature in both cases, something that makes the comparison better.

m_ o ¼ m_w 

cp;w ; cp;o

ð10Þ

In other worlds, the above correction equalizes the different into specific capacitance between the working fluids. For typical values of specific thermal capacities, the mass flow rate of thermal oil is 0.0212 kg/s. More specifically, the specific thermal capacity was selected at 4180 kJ/kg K for water and at 1972 kJ/kg K for thermal oil. Another important parameter is the overall heat loss coefficient which can be calculated by the Eq. (11):

Q Loss UL ¼ ; Ar  ðT r  T am Þ

0.68

Water

0.64

Thermal oil

ð7Þ

Thermal analysis for water and thermal oil as working fluids

Qu ; Qs

0.72

0.60

Moreover, Fig. 8 shows that after 53° no solar rays arrive on the receiver, an expected result, since the half-acceptance angle is also 53°.

gth ¼

0.76

0

0.05

0.1

0.15

0.2

Fig. 9. Efficiency curve for different operating conditions.

It is obvious from Fig. 9 that pressurized water performs better in all cases while the difference on the efficiency is getting greater as the inlet temperature increases. This happens due to the difference in the heat losses and as a consequence in the overall heat loss coefficient between the two fluids, as it is depicted in Fig. 10. This coefficient is calculated according Eq. (11) by using the total thermal losses. Solidworks outputs include these thermal losses and this makes the calculations easier. Fig. 10 shows that the heat loss coefficient (UL) takes greater values in the thermal oil case, fact that leads to greater overall losses and to low thermal efficiency. Furthermore, the difference between the two heat loss coefficients is getting greater as the inlet fluid temperature increases something that explains the divergence in the efficiency curves for the respective operating conditions. The difference in this coefficient is explained by the difference in the receiver temperature, which is depicted in Fig. 11. At this point is essential to state that a warmer absorber lead to higher heat losses and this has to been taken into consideration in the presented analysis. Fig. 11 proves that the absorber temperature is greater when the collector operates with thermal oil. The difference in absorber temperature seems to be constant between the pressurized water and thermal oil case (Fig. 11). However, the difference in the heat loss coefficient (Fig. 10) is getting greater for higher fluid temperature levels. This is a result of the dependence between these parameters. This coefficient is depended on the receiver temperature in the fourth power, because only radiation losses exist inside the evacuated tube, fact that explains the non-linear behavior of Fig. 10. In this point, the reason for the greater receiver temperature in the thermal oil case will be presented. A deeper analysis of the heat transfer between tube and working fluid is given below. Emphasis is given in the heat transfer coefficient inside the tube. Eq. (12) gives the mean Nusselt number for laminar flow [35] and Eq. (13) the heat convection coefficient:

2.1

Thermal oil

1.8

ð11Þ

Thermal comparison of working fluids

0.25

(Tin-Tam)/Ge

Water

1.5

UL (W/m2K)


2 hT hT  25:96 þ 15:37  100 100
3
4
5 hT hT hT  þ 22:5   7:2  ; 100 100 100

cðhT Þ ¼ 0:9921  5:46 

0.80

th

It is obvious that the intercept factor decreases abruptly for low values of the transversal angle something that makes this parameter crucial for the collector performance. A tracking system is used in many applications in order to minimize the transversal angle and to achieve greater optical efficiency. The next equation is the approximation of the Fig. 8 curve:

1.2 0.9 0.6

In this section a comparison between the two working fluids is presented. Fig. 9 illustrates the thermal efficiency of the examined collector for a range of operating conditions. The ambient temperature and the solar irradiation were selected to be 10 °C and 1000 W/m2 respectively. The only parameter that changes is the water inlet temperature which is ranged from 10 °C to 230 °C.

0.3 0 0

0.05

0.1

0.15

0.2

(Tin-Tam)/Ge Fig. 10. Heat loss coefficient for different operating conditions.

0.25

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E. Bellos et al. / Sustainable Energy Technologies and Assessments 16 (2016) 53–63 300

Thermal oil Water

250

Re ¼

Tr (oC)

200

Pr ¼

150 100 50 0 0

0.05

0.1

0.15

0.2

0.25

(Tin-Tam)/Ge Fig. 11. Receivers mean temperature for different operating conditions.

4m

p  Dri  l l  cp k

Fluid

h (W/ m2 K)

Nu

Re

Pr

k (W/ m K)

l (Pa s)

cp (J/ kg K)

Water Thermal oil

150 62

6.7 11.0

1388 73

1.9 144

0.67 0.17

0.00031 0.01240

4180 1972

;

ð14Þ ð15Þ

Table 3 includes the calculated parameters for inlet temperature equal to 90 °C which is an intermediate operating temperature. It is obvious that the heat transfer coefficient is lower for the thermal oil case, something that is explained by the great difference in the dynamic viscosity and the thermal conductivity of thermal oil. Eq. (16) describes the correlation between the heat convection coefficient and the absorber temperature:

T r ¼ T fm þ

Table 3 Mean heat convection coefficient and other properties for working fluid.

;

Qu ; Ari  hm

ð16Þ

This equation suggests that for a constant useful energy amount, a lower heat transfer coefficient leads to a greater absorber temperature. This result makes the heat transfer fluid with higher convection coefficients to be more suitable in thermal solar applications. Additional results for operation with pressurized water

108.3

( C)

108.2

()

108.1

absorber

108.0 107.9 107.8 0

50

100

150

200

250

300

350

400

() Fig. 12. Peripheral temperature distribution in the absorber.

  0:0668  DLri  Re  Pr Num ¼ 3:66 þ  2=3 ; 1 þ 0:04  DLri  Re  Pr

hm ¼

Num  k ; Dri

ð12Þ

ð13Þ

Eqs. (14) and (15) show the definition of Reynolds and Prandtl number respectively:

In this section, additional figures and diagrams are given for operation pressurized water; the optimum working fluid according to the previous analysis. The temperature of the fluid in the inlet is equal to 90 °C for Figs. 12–14. This temperature level is an intermediate and representative value for the collector operation. Firstly, the peripheral temperature of the absorber in the middle of the tube is given in Fig. 12. Fig. 13 shows the temperature distribution (Fig. 13a) and the distribution of the heat losses (Fig. 13b) over the outer tube surface. It is obvious from Fig. 12, that the maximum temperature appears on the top of the tube where the maximum heat flux is observed (Figs. 5 and 6). Moreover, it is essential to state that the variation of the temperature distribution is only 0.4 °C. This result proves that the absorber tube temperature distribution can be approximately assumed as isothermal small rings. Fig. 13a shows that the temperature of the tube increases in the flow direction because the fluid is getting warmer from the inlet to the outlet. In Fig. 13b, the thermal losses have a respective to the temperature distribution due to their dependency, as it was referred above. Fig. 14 depicts the water temperature distribution in the outlet cross-section of the tube. The maximum temperature is observed close to tube walls and especially in the upper and in the lower part of them. In the center core, the temperature is lower because the heat did not manage to reach this part of the flow. The heat

Fig. 13. (a) Absorber temperature distribution, (b) thermal losses distribution over the absorber (the red color illustrates the greater values and the blue the lower values). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E. Bellos et al. / Sustainable Energy Technologies and Assessments 16 (2016) 53–63

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Fig. 14. Water temperature distribution in the outlet cross section.

Fig. 15. Flow chart of the developed numerical model.

conductivity is the factor that determines the temperature distribution on the radial direction of the fluid cross section. Model validation In the final part of this study, a simple validation of the developed model is presented. A simple 1-D numerical model written in the FORTRAN programming language was developed for validating the simulation results. The validation is made for pressurized water because this is the working fluid with the better efficiency. The main developed model is based on the energy balance in the

absorber in order to determine the useful energy and the heat losses [36]. The receiver and the cover are supposed to have uniform temperature levels in every case which is a good assumption; because the examined tube is short and the temperature variation along the tube is low. Fig. 15 illustrates the basic points the developed numerical model. Fig. 16 shows the validation results between model in Solidworks and model in FORTRAN. More specifically, the thermal efficiency, the heat loss coefficient, the receiver and the cover temperature are compared for all the range of operating conditions. It is obvious that the two models give similar results which

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Fig. 16. Validation results (a) thermal efficiency (b) heat loss coefficient (c) receiver temperature (d) cover temperature.

are close to each other. These results validate the developed model in Solidworks.

the peripheral temperature distribution over the absorber has a variation of about 0.4 K; a small variation which leads to the assumption of isothermal tube locally.

Conclusions Acknowledgments In this study a detailed analysis of a CPC collector is presented. First of all, the design of this collector is analyzed in order to optimize its geometry. The goal of the optimization is the maximization of the intercept factor for zero incident angle. The results showed that the optimum aperture width is about 300 mm with a focal distance of 50 mm and a receiver diameter of 34 mm. The shape of the down part has not a significant role in the collector efficiency and for this reason its design is not of great interest. The final reflector geometry is optimum because all the reflected rays are delivered to the receiver which means ideal design. Moreover, the collector tested for various incident angles at the longitude and in the transverse direction, and the results show that the transversal angle variation causes a significant reduction in the optical efficiency. This finding indicates the necessity of tracking this collector in order to minimize the incident angle and mainly the transversal incident angle. The next part of this study is the thermal analysis of the collector in order to predict the efficiency for different operating conditions. Two different working fluids are compared in order to predict the most appropriate as a heat transfer fluid. Pressurized water performed better than thermal oil by giving greater efficiency in the whole operating range. The reason for the water’s better performance is the different thermal properties of it; conductivity and dynamic viscosity. More specifically, the greater values of these properties in pressurized water increase the heat convection coefficient making water the most suitable working fluid. In the last part of this study, the temperature distribution in the absorber and in the fluid are presented for operation with pressurized water. According to the presented figures, the absorber is getting warmer from the inlet to the outlet and the heat losses have a similar distribution. Also, the water temperature in the outlet cross section is characterized by isothermal rings, with the warmer to be closer to the tube. Another important conclusion is that

The first author would like to thank the Onassis Foundation for its financial support. Appendix A The basic equations which describe the general energy balances are given in this appendix. The energy potential of the solar energy is given by Eq. (A.1) and the energy delivered to the absorber by Eq. (A.2):

Q S ¼ Aa  Ge ;

ðA:1Þ

Q abs ¼ Q S  q  c  ðsaÞ;

ðA:2Þ

The useful energy that fluid absorbs is calculated by the energy balance in its volume according Eq. (A.3):

Q u ¼ m  cp  ðT out  T in Þ;

ðA:3Þ

The energy balance in the absorber is given in Eq. (A.4). From this equation is able the direct determination of heat losses:

Q loss ¼ Q abs  Q u ;

ðA:4Þ

Eqs. (A.5)–(A.7) are three different ways to express the heat losses. These equations lead to the receiver temperature, cover temperature and heat loss coefficient:

Q loss ¼ U L  Aro  ðT r  T am Þ; Q loss ¼

r  Aro  ðT 4r  T 4c Þ 1

er

þ 1ecec  AAroci

;

Q loss ¼ Aco  hca  ðT c  T am Þ þ ec  Aco  r  ðT 4c  T 4am Þ;

ðA:5Þ ðA:6Þ ðA:7Þ

E. Bellos et al. / Sustainable Energy Technologies and Assessments 16 (2016) 53–63

The next equation that connects the useful heat and the receiver temperature:

Q u ¼ Ari  hm  ðT r  T fm Þ;

ðA:8Þ

The mean fluid temperature (Tfm) is approximately the average of inlet and outlet temperature. The heat transfer coefficient between absorber and fluid can be calculated by Eq. (A.9) for laminar flow:

hm ¼

" # k 0:0668  Re  Pr  Dri =L :  3:66 þ Dri 1 þ 0:04  ðRe  Pr  Dri =LÞ2=3

ðA:9Þ

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