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Designing solar thermal experiments based on simulation
Applied Thermal Engineering 51 (2013) 1000e1005
Contents lists available at SciVerse ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Designing solar thermal experiments based on simulation Mahmoud Huleihil a, b, *, Gedalya Mazor b a b
The Arab Academic Institute for Education, BeitBerl College, Israel The Sami Shamoon College of Engineering, Israel
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 May 2012 Accepted 14 October 2012 Available online 5 November 2012
In this study three different models to describe the temperature distribution inside a cylindrical solid body subjected to high solar irradiation were examined, beginning with the simpler approach, which is the single dimension lump system (time), progressing through the two-dimensional distributed system approach (time and vertical direction), and ending with the three-dimensional distributed system approach with azimuthally symmetry (time, vertical direction, and radial direction). The three models were introduced and solved analytically and numerically. The importance of the models and their solution was addressed. The simulations based on them might be considered as a powerful tool in designing experiments, as they make it possible to estimate the different effects of the parameters involved in these models. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Designing thermal experiments Concentrated solar systems Lumped system Distributed system Simulation
1. Introduction The study of solar energy [1] as a renewable (pollution free) source of power is a continuously growing field of research. One can find solar systems in various applications, such as solar thermal, phase change, passive buildings, direct and indirect electricity production, and solar production of nano-materials [1e9]. By developing special measurement techniques, the performance characteristics of concentrator solar cells operating in a high irradiance environment could be measured [10]. Hardening steels and other hardening processes are possible with the aid of a solar furnace [11]. Solar furnaces are used to experimentally produce fullerenesdcarbon nanotubesdby direct vaporization of a mixture of powdered carbon and catalysts, such as Co, Ni, Y [12]. A vaporization process that used graphite targets containing different pairs of catalysts was performed using a 2 kW solar furnace. With this small-scale experimental setup the researchers observed the evolution of the structure of the nanotubes that were produced under experimental conditions, for example, where working pressure, the flow rate of the inert gas (argon), and target composition could be manipulated.
* Corresponding author. The Sami Shamoon College of Engineering, Basel/Bialik Sts. P.O. Box 84100, Beer-Sheva, Israel. Tel.: þ972506011900. E-mail address: [email protected] (M. Huleihil). 1359-4311/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2012.10.019
Previous studies found that high pressure favors the production of bundles of single-wall nanotubes (SWNTs), the purity of the produced material depends on the target temperature and the cooling rate of the vapor [13]. In another study, heat and mass transport was obtained in a solar reactor using ‘in situ’ measurements linked to numerical simulation. This facilitated the interpretation of the vaporization process as well as the determination of the cooling regime [14]. For the very high temperature process of carbon vaporization, which is mainly significant at temperatures higher than 3200 K, optimum design of a solar reactor for carbon product processing by vaporization was reported [15]. The method, which accounts for heat transfer and chemical reaction kinetics, was based on the combination of both experimental data obtained at laboratory scale, and numerical simulation. Fullerene mixtures of C60 and C70 were produced in gram quantities in a solar reactor using (50 kW), which was only part of the available power of the 1 MW CNRS solar furnace at Odeillo. Fullerene yield was studied as a function of buffer gas (helium and argon), pressure (in the range 80e500 hPa), and gas flow rate [16]. In a different study, Abanades and Flamant developed a hightemperature solar reactor for co-producing hydrogen-rich gas and high-grade carbon black (CB) from concentrated solar energy and methane. Their approach was based on a single-step thermal decomposition (pyrolysis) of methane without catalysts and without emitting carbon dioxide, because solid carbon is sequestered. In the tested reactor, a graphite nozzle absorbs concentrated solar radiation provided by a solar furnace. The heat is then transferred to the reactive flow [17].
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
Nomenclature A Af Atot B c D H P r T
parameter given by equation (7), K/s front surface area, facing solar radiation, m2 total surface area of the sample, m2 parameter given by equation (6), K/s heat capacity, J/kg K diameter of the cylindrical sample, m height of the cylindrical sample, m solar radiation flux, MW/m2 radial dimension of the cylinder, m temperature distribution of the sample, K
Another design of a solar system is described by Zijffers et al. [18]. Their article describes the design process of the Green Solar Collector (GSC), an area-efficient photobio reactor for efficient outdoor cultivation of microalgae. In recent studies Gordon et al. [19] and Levy et al. [20] used highly concentrated sunlight to generate fullerene-like and nano tubular inorganic nanostructures from Cs2O, SiO2-x, WS2, and MoS2, ranging from single-walled nanotubes and closed-cage structures to their larger multi-walled counterparts. Gordon et al. [21] described a high-irradiance solar furnace that could be used in various high temperature processes, such as testing concentrator photovoltaics and driving high-temperature reactors for the generation of novel nanostructures, with target irradiance up to 12 W/mm2. The opto-mechanical design permits real-sun flash illumination at a millisecond time scale, so that solar cells can be characterized with only insubstantial increases in cell temperature even at irradiance levels of thousands of suns [21]. In a different methodology, solar systems are studied using simulations. Singh and Sulaiman used simulation to look at the designing procedures of a solar thermal cylindrical parabolic trough concentrator (CPTC) [22]. Vidales et al. considered a 1 kW thermochemical solar reactor/receiver fitted with a porous ceramic foam structure, and studied it numerically in order to predict the thermal transfers inside the volumetric solar receiver. The suggested reactor is devoted to the production of hydrogen from twostep thermochemical cycles based on mixed metal oxides, and it features a porous media coated with the reactive ferrite material (MxFe3xO4) that is directly irradiated by concentrated solar energy [23]. Natarajan et al. presented a numerical study of solar cell temperature for concentrating PV, using a concentration ratio of 10. The researchers developed a two-dimensional thermal model to predict the temperature for a PV concentrator system (solar cell and lens) with and without passive cooling arrangements [24]. Min et al. established a metal plate cooling model for 400 single concentrator solar cells [25]. In addition, observations based on simulations could be useful for validating analytic calculations [3], and after model validation based on experimental observations, simulations could be used as a design tool. This type of tool was used to perform many numerical simulations for different design criteria [4]. In this study, a model was developed in order to describe the temperature distribution inside a small piece of solid material under conditions of high solar irradiance and high temperatures. Based on the results, experiments are suggested.
Ta Tf Tb t V z 3
a b k r s
1001
ambient temperature, K front surface temperature, K back surface temperature, K time, s volume of the sample, m3 axial dimension of the cylinder, m hemispherical emissivity, e thermal diffusivity (k/rc), m2/s steady state temperature given by equation (7), K thermal conductivity, W/m K density, kg/m3 StefaneBoltzmann coefficient, W/m2 K4
model is developed following three intuitive steps: lumped system approach (one dimension e time), distributed system approach (two dimensions e time and height) and distributed system approach with azimuthal symmetry (three dimensions e time, height, and radius). The benefits of such a study could be a) educational, b) to increase insight into similar systems, and c) to provide guidance in designing experiments, so that with a simulation and the aid of the computational power of computers, expenses are minimized. 2.1. Lumped system approach Consider a small cylindrical piece of material with diameter D and height H, subject to concentrated solar radiation with constant flux P (MW/m2). The solar flux is assumed constant for simplicity (for photovoltaic applications, uniform flux distributions could be achieved by using a kaleidoscope [28]). The essential properties of the material needed to develop a thermal model are the density r (kg/m3), heat capacity c (J/kg K), thermal conductivity k (W/m K), and hemispherical emissivity 3. In this model the temperature of the material T (K) is assumed to be uniform everywhere. The incoming energy at the front surface is distributed into two parts: energy heating the material and energy lost back to the surrounding environment at temperature Ta (K). It is important to note that in high energy experiments; usually the material is protected in a vacuum, so that convective heat transfer is minute. Although it is an inexact statement; the results of such a lumped system model are of great importance in designing experiments. It will be shown later that these results are confirmed by the other modeling assumptions given in Sections 2.2 and 2.3.
2. Thermal modeling In this section we develop a thermal model to describe the temperature distribution inside a small cylindrical piece of material, subject to a high solar flux on its front side (see Fig. 1). The
Fig. 1. Schematic of the model. A cylindrical piece of diameter D and of height H is subject to concentrated solar flux. Heat is distributed within the sample via heat conduction and losses are introduced via thermal radiation from all sides.
1002
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
dT ¼ A B T 4 Ta4 ; dt
The energy balance of the lumped system is given by
d ðrcVTÞ ¼ PAf 3 sAtot T 4 Ta4 : dt
(1)
The different quantities that appear in eq. (1) are the area of the front Af surface, which is given by
pD2
Af ¼
4
:
V ¼ Af H:
dT ¼ A B T 4 Ta4 : dt
(4)
1 vT v2 T ; ¼ a vt vz2
(5)
Parameter B is related to the initial slope for cooling BðTa4 b Þ and is given by 4
sAtot B ¼ : rcV 3
(6)
Parameter b is the steady state temperature of the heating process, which is also the initial temperature for cooling process, and it is given by
A B
b4 ¼ Ta4 þ :
(7)
Parameter A is the initial slope for the heating process, and is given by
:
(8)
The nonlinear initial value problem (for heating), which is given by eq. (1), could be solved by separation of variables. The solution is given parametrically, with time shown as a function of temperature, by
t ¼
T Ta arctan arctan
2Bb
3
þ
1 3
4Bb
(11)
In the second model, the temperature T(t,z) is assumed to be two-dimensional and is given a function of time t and position along the vertical axis z. The parabolic heat equation is given by
For constant thermal and mechanical properties (which are acceptable assumptions, especially when considering qualitative behavior of the system), the energy equation given by eq. (1) could be rearranged as follows
1
b 1 T arctan arctan Ta Ta 2BTa3 1 T Ta b þ Ta : ln $ b Ta T þ Ta 4BTa3
(3)
and the volume of the cylinder, which is given by
PAf
t ¼
2.2. Distributed system approach e two-dimensional model
Atot ¼ 2Af þ pDH;
rcV
and its solution is given by
(2)
The total surface area of the cylinder Atot, which is given by
A ¼
(10)
ln
b
b þ T b Ta $ : b þ Ta b T
b
(12)
where a ¼ k/rc is the thermal diffusivity. In principle, this equation could be solved by separation of variables, in which case the solution would include an exponential decay in time. After some time the temperature distribution approaches a steady state. In this model the solution is represented by its steady state response. The steady state response is a linear function of the z coordinate. The steady state temperature is given by
z T z ¼ Tf þ Tb Tf : H
(13)
The differential equation (eq. (11)) could be solved for with the initial time and two boundary conditions provided as known factors. At the front surface the boundary condition is given by
vTðt; z ¼ 0Þ P t ¼ k þ 3 s Tf4 Ta4 ; vt
(14)
and the boundary condition at height H is given by
k
vTðt; z ¼ HÞ ¼ 3 s Tb4 Ta4 : vt
(15)
After some algebraic manipulations, in the steady state the thermal fluxes are constant so that equation (14) could be rearranged to represent the front temperature Tf as a function of the back temperature Tb, as given by
Tf ¼ Tb þ
3
sH k
Tb4 Ta4 :
(16)
Finally, equations (14) and (15) are substituted in equation (13) to give a nonlinear equation with one variable, Tb, which is given by
P ¼ 3s
4 3 sH Tb4 Ta4 Tb þ Tb4 Ta4 þ Ta4 : k
(17)
Equation (16) is solved numerically with the aid of Microsoft Excel.
(9)
A similar solution could be given for the cooling process, for which P ¼ 0 and the initial temperature is b. In this case the differential equation (eq. (5)) becomes
2.3. Distributed system approach e three-dimensional model In this section the model described in the previous section is extended to account for changes along the radial direction. The thermal conductivity is assumed to be uniform and independent of direction. The temperature distribution is a function of time t, radial
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
Temperature versus time - Lumped model
k
2500
1003
vTðt; r ¼ D=2; zÞ ¼ 3 s T 4 ðt; r ¼ D=2; zÞ Ta4 : vr
(20)
In order to be able to develop a numerical scheme the partial differential equation (PDE) given by equation (17) should be rearranged for the zero radius and is given by
2000
1 vTðt; r; zÞ v2 Tðt; r; zÞ v2 Tðt; r; zÞ þ : ¼ 2 a vt vr 2 vz2
1500 Density (kg/m 3 ) 5060 2500 1000
1000
(21)
3. Numerical example In this section, a numerical example is given for illustrating the solution for a specific case. The ambient temperature is assumed as 300 K. Based on Ref. [21], solar flux is chosen to be in the range 1e12 MW/m2. The following additional presumptions are made:
500
0 0
10
20
30
40
50
60
70
StefaneBoltzmann constants ¼ 5.667 $ 108 W/m2 K4 Hemispherical emissivity is 0.8 (gray body) The diameter and height of the cylinder are both 4 mm The density of a hypothetical material (with the melting point thus avoided, which could be taken into consideration more properly for a specific material) is r ¼ 5050 kg/m3 The heat capacity is 3977 J/kg K The thermal conductivity k ¼ 0.44 W/m K.
Time (s)
Fig. 2. Predicted temperature based on the lumped system approach. The temperature is plotted vs. time for different values of density. The effect of the response time is highlighted in the figure.
direction r, and vertical direction z. The heat conduction equation in this case is given by
1 vTðt; r; zÞ 1v vTðt; r; zÞ v2 Tðt; r; zÞ r þ : ¼ a vt r vr vr vz2
(18)
The initial temperature distribution is assumed to be uniform at the ambient temperature Ta. The boundary conditions at the front and back surfaces are given by equations (13) and (14) respectively. The additional necessary boundary conditions in the radial direction are at r ¼ 0 and at r ¼ D/2. The boundary condition at zero radius assures physical solution. This condition is equivalent to insulation and it is given by
vTðt; r ¼ 0; zÞ ¼ 0: vr
(19)
The other radial boundary condition that assures energy balance at the outer radius is given by
The properties of MoS2 could be found in the literature [26,27]. In fact, these parameters are lumped in one parameter, the a ¼ k/rc thermal diffusivity. This parameter appears in the PDE and it effects the response time of the system. With these values one could perform calculations and draw some conclusions. In the following paragraph, some calculations are shown next. The temperature of the cylinder based on the lumped system model is plotted against time for different density values (see Fig. 2). An important observation from the plot is that by decreasing the density, a faster time response is achieved. What is important to highlight is the possibility of constructing the sample differently; one might suggest a denser cylindrical sample partially hollow in order to be filled with a less dense sample of the same material, or probably a powder. The predicted steady state temperature is plotted against solar flux for different values of the height of the cylinder, which are depicted by Fig. 3. It is clear from this plot that a specific steady state
Temperature vs. solar flux - Lumped model 3500 Sample's Height (mm) H= 1 2 3 4
Temperature (K)
3000
2500
2000
1500
1000 0
2
4
6
8
10
12
2
Solar flux MW/m
Fig. 3. The steady state temperature as predicted from the lumped system model is plotted vs. solar flux for different values if the height of the cylinder.
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M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
Steady state temperatures of the sample's front and back surfaces vs. solar flux
Temperature at the front surface vs. normalized radius of the sample
4000
4000 3500
Temperature (K)
Steady state temperature (K)
3500
Front surface Sample thickness H (mm) 1 2 3 Back surface 4
3000 2500 2000
3000 2500 2000 1500 1000 500 0
1500
0
0.4
0.6
0.8
1
Normalized radius 2r/D
1000 0
2
4
6
8
10
12 Fig. 6. The temperature distribution at zero height based on the three-dimensional model is plotted vs. radius of the cylinder.
Solar flux MW/m2 Fig. 4. The steady state temperatures of the front side and the back side of the cylinder based on the distributed system approach in the vertical direction are plotted vs. solar flux for different heights of the cylinder.
temperature could be targeted or maybe achieved by choosing a suitable height (as an example of design parameter). Although the lumped system model could not show directional temperature gradients along the cylindrical axis, one could create a design to achieve a gradient by fashioning a proper sample made of two different pieces with different densities. It is important to remember that the lumped system model is insufficient to capture all of the physics (especially the distribution of the temperature along the radial and the axial directions). A step toward determining the real picture of the temperature distribution along the cylindrical axis could be achieved by the second model. Fig. 4 presents the steady state temperatures of the front and back sides of the cylinder against solar flux for different values of the cylinder’s height. By consulting Fig. 4 an obvious conclusion (that the front temperature reaches higher levels than was predicted by the lumped model) can be reached, which is a much more realistic observation. On the other side (the back side), the temperature reaches much lower values, asserting the fact of low thermal conductivity along the vertical direction. This type of result is obtained more accurately through the threedimensional model, as one can see from Figs. 5 and 6. The third
Temperature distribution inside the sample vs. time - 3D model 3500 3100
Temperature (K)
0.2
2700 Location: r=0, z=0 r=D/2, z=0 r, z=H/2 r, H
2300 1900 1500 1100
300 10
20
30
4. Summary and conclusions In this study three different models (see Fig. 1 for the schematic of the models e the same figure fits all the models, the difference is expressed through the modeling assumptions) were studied theoretically. The study progressed from the simpler, onedimensional approach, the lump system (one dimension e time), through a two-dimensional distributed system approach (time and vertical direction) to three-dimensional distributed system approach with azimuthal symmetry (time, vertical direction, and radial direction). The three models are introduced in Section 2 (2.1e2.3) and the appropriate methods of solution were given, analytically whenever it was possible (as in the case of lumped system approach), and numerically in most other cases. The numerical example given in Section 3 enabled the production of some results that are summarized in Figs. 2e6. 5. Discussion
700 0
model, which was presented in Section 2.3, takes into account the heat conduction effect along the radial direction, and heat loss through the sides, through thermal radiation. Fig. 5 shows the plots of the temperatures at different locations along the radial direction, and at different heights. It is clear that the gradients are much higher in the vertical direction. This effect might be explained by uniformity of the flux at the front surface and by the radial symmetry of the cylinder. The effect of the losses from the sides is highlighted by Fig. 6, which presents the temperature distribution at zero height as a function of the radius. It could be seen that toward the end of the radial direction there is a relatively small drop in temperature compared to the drop along the vertical direction (something that can be seen at Fig. 5). The results of the second model and the third model clearly show the vertical gradients in temperature. It is possible to increase such gradients if the sample is replaced by a two part assembly of a denser kind of a container filled with a less condensed piece or powder of the same material.
40
50
Time (s) Fig. 5. The temperature distribution based on the three-dimensional model is plotted at different locations in the sample vs. time.
Although the lumped system model would not show temperature gradients along the vertical direction (appropriate gradients are needed for special materials production [20]), one could suggest that such a gradient could be forced if the uniform sample was replaced by a sample made of two parts, where the outer shell is denser than the inner shell. The other models showed more realistic results and due to the low thermal conductivity of the
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
material, gradients in the vertical direction were much higher than the gradients along the radial direction. This type of behavior could be explained by the uniform input of the solar flux and by the symmetry of the cylindrical sample. After consulting Figs. 2e6 one could observe that a steady state condition at the front surface was reached within seconds. This is a very important observation, especially for designing experiments, because it enables researchers to perform many experiments within a short period of time. This is also most important to researchers in the field of solar energy (when considering the availability of the sun). Lastly, the models analyzed in this study serve as powerful tools in designing experiments, and make it possible to check the different effects of the parameters involved in these models.
References [1] A. Rabl, Active Solar Collectors and their Applications, Oxford University Press, New York, Oxford, 1985. [2] R. Winston, J.C. Minano, P. Benitez, Nonimaging Optics, Elsevier Academic Press, 200 Wheeler Road, 6th Floor, Burlington, MA 01803, USA, 2005, with contributions by Narkis Shatz and John C. Bortz. [3] Ch. Charach, M. Huleihil, Y. Zarmi, Perturbative analysis of planar phase change processes with time dependent temperature at the boundary, J. Appl. Phys. 64 (1988) 4832. [4] D. Faiman, D. Feuermann, M. Huleihil, Numerical model of a rotating prism wall: its validation and predicted device performance for a variety of situations, J. Sol. Energy Eng. 111 (3) (1989) 237. [5] D. Feuermann, J.M. Gordon, M. Huleihil, Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators, Sol. Energy 72 (2002) 195e204. [6] D. Feuermann, J.M. Gordon, M. Huleihil, Solar fiber-optic mini-dish concentrators: first experimental results and field experience, Sol. Energy 72 (2002) 459e472. [7] J.M. Gordon, D. Feuermann, M. Huleihil, Laser surgical effects with concentrated solar radiation, Appl. Phys. Lett. 81 (2002) 2653e2655. [8] J.M. Gordon, D. Feuermann, M. Huleihil, S. Mizrahi, R. Shaco-Levy, Solar surgery, J. Appl. Phys. 93 (2003) 4843e4851. [9] J.M. Gordon, D. Feuermann, M. Huleihil, S. Mizrahi, R. Shaco-Levy, Surgery by sunlight on live animals, Nature 424 (2003) 510. [10] R.D. Nasby, R.W. Sanderson, Performance measurements techniques for concentrator photovoltaic cells, Sol. Cells 6 (1982) 39e47.
1005
[11] Y. Yang, A.A. Torrance, J. Rodriguez, The solar hardening of steels: experiments and predictions, Sol. Energy Mater. Sol. Cells 40 (1996) 103e121. [12] D. Laplaze, P. Bernier, W.K. Maser, G. Flamant, T. Guillard, A. Loiseau, Carbon nanotubes: the solar approach, Carbon 36 (5e6) (1998) 685e688. [13] T. Guillard, G. Flamant, D. Laplaze, Solar production of carbon nanotubes; structure evolution with experimental conditions, J. Mater. Sci. 35 (2000) 419e425. [14] D. Laplaze, L. Alvarez, T. Guillard, J.M. Badie, G. Flamant, Carbon nanotubes: dynamics of synthesis processes, Carbon 40 (2002) 1621e1634. [15] G. Flamant, J.F. Robert, S. Marty, J.M. Gineste, J. Giral, B. Rivoire, D. Laplaze, Solar reactor scaling up: the fullerene synthesis case study, Energy 29 (2004) 801e809. [16] G. Flamant, D. Luxembourg, J.F. Robert, D. Laplaze, Optimizing fullerene synthesis in a 50 kW solar reactor, Sol. Energy 77 (2004) 73e80. [17] S. Abanades, G. Flamant, Production of hydrogen by thermal methane splitting in a nozzle-type laboratory-scale solar reactor, Int. J. Hydrogen Energy 30 (2005) 843e853. [18] J.W. Zijffers, M. Janssen, J. Tramper, R.H. Wijffels, Design process of an areaefficient photobioreactor, Mar. Biotechnol. 10 (2008) 404e415. [19] J.M. Gordon, E.A. Katz, D. Feuermann, A. Albu-Yaron, M. Levy, R. Tenne, Singular MoS2, SiO2 and Si nanostructuresdsynthesis by solar ablation, J. Mater. Chem. 18 (2008) 458e462. [20] M. Levy, A. Albu-Yaron, R. Tenne, D. Feuermann, E.A. Katz, D. Babai, J.M. Gordon, Synthesis of inorganic fullerene-like nanostructures by concentrated solar and artificial light, Isr. J. Chem. 50 (2010) 417e425. [21] J.M. Gordon, D. Babai, D. Feuermann, A high-irradiance solar furnace for photovoltaic characterization and nanomaterial synthesis, Sol. Energy Mater. Sol. Cells 95 (2011) 951e956. [22] B. Singh, F. Sulaiman, Designing a solar thermal cylindrical parabolic trough concentrator by simulation, International Rio3 Congress, World Climate and Energy Event, Rio de Janeiro, 1e5 December 2003. [23] H.I. Villafan-Vidales, S. Abanades, C. Caliot, H.R. Paredes, Heat transfer simulation in a thermochemical solar reactor based on a volumetric porous receiver, Appl. Therm. Eng. 31 (2011) 3377e3386. [24] S.K. Natarajan, T.K. Mallick, M. Katz, S. Weingaertner, Numerical investigations of solar cell temperature for photovoltaic concentrator system with and without passive cooling arrangements, Int. J. Therm. Sci. 50 (2011) 2514e 2521. [25] C. Min, C. Nuo-Fu, D. Jin-Xiag, Thermal modeling and the optimized design of metal plate cooling systems for single concentrator solar cells, Chin. Phys. B 21 (3) (2012) 034216-1e034216-6. [26] R.C. Mclaren, Master thesis in Mechanical Engineering, University of Illinois at Urbana-Champaign, 2009, Last visited on Thursday, April 26, 2012 from: http://www.ideals.illinois.edu/handle/2142/14668. [27] http://en.wikipedia.org/wiki/Molybdenum_disulfide. [28] J. Sun, T. Israeli, T.A. Reddy, K. Scoles, J.M. Gordon, D. Feuermann, Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators, Trans. ASME 127 (February 2005) 138e145.
Contents lists available at SciVerse ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Designing solar thermal experiments based on simulation Mahmoud Huleihil a, b, *, Gedalya Mazor b a b
The Arab Academic Institute for Education, BeitBerl College, Israel The Sami Shamoon College of Engineering, Israel
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 May 2012 Accepted 14 October 2012 Available online 5 November 2012
In this study three different models to describe the temperature distribution inside a cylindrical solid body subjected to high solar irradiation were examined, beginning with the simpler approach, which is the single dimension lump system (time), progressing through the two-dimensional distributed system approach (time and vertical direction), and ending with the three-dimensional distributed system approach with azimuthally symmetry (time, vertical direction, and radial direction). The three models were introduced and solved analytically and numerically. The importance of the models and their solution was addressed. The simulations based on them might be considered as a powerful tool in designing experiments, as they make it possible to estimate the different effects of the parameters involved in these models. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Designing thermal experiments Concentrated solar systems Lumped system Distributed system Simulation
1. Introduction The study of solar energy [1] as a renewable (pollution free) source of power is a continuously growing field of research. One can find solar systems in various applications, such as solar thermal, phase change, passive buildings, direct and indirect electricity production, and solar production of nano-materials [1e9]. By developing special measurement techniques, the performance characteristics of concentrator solar cells operating in a high irradiance environment could be measured [10]. Hardening steels and other hardening processes are possible with the aid of a solar furnace [11]. Solar furnaces are used to experimentally produce fullerenesdcarbon nanotubesdby direct vaporization of a mixture of powdered carbon and catalysts, such as Co, Ni, Y [12]. A vaporization process that used graphite targets containing different pairs of catalysts was performed using a 2 kW solar furnace. With this small-scale experimental setup the researchers observed the evolution of the structure of the nanotubes that were produced under experimental conditions, for example, where working pressure, the flow rate of the inert gas (argon), and target composition could be manipulated.
* Corresponding author. The Sami Shamoon College of Engineering, Basel/Bialik Sts. P.O. Box 84100, Beer-Sheva, Israel. Tel.: þ972506011900. E-mail address: [email protected] (M. Huleihil). 1359-4311/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2012.10.019
Previous studies found that high pressure favors the production of bundles of single-wall nanotubes (SWNTs), the purity of the produced material depends on the target temperature and the cooling rate of the vapor [13]. In another study, heat and mass transport was obtained in a solar reactor using ‘in situ’ measurements linked to numerical simulation. This facilitated the interpretation of the vaporization process as well as the determination of the cooling regime [14]. For the very high temperature process of carbon vaporization, which is mainly significant at temperatures higher than 3200 K, optimum design of a solar reactor for carbon product processing by vaporization was reported [15]. The method, which accounts for heat transfer and chemical reaction kinetics, was based on the combination of both experimental data obtained at laboratory scale, and numerical simulation. Fullerene mixtures of C60 and C70 were produced in gram quantities in a solar reactor using (50 kW), which was only part of the available power of the 1 MW CNRS solar furnace at Odeillo. Fullerene yield was studied as a function of buffer gas (helium and argon), pressure (in the range 80e500 hPa), and gas flow rate [16]. In a different study, Abanades and Flamant developed a hightemperature solar reactor for co-producing hydrogen-rich gas and high-grade carbon black (CB) from concentrated solar energy and methane. Their approach was based on a single-step thermal decomposition (pyrolysis) of methane without catalysts and without emitting carbon dioxide, because solid carbon is sequestered. In the tested reactor, a graphite nozzle absorbs concentrated solar radiation provided by a solar furnace. The heat is then transferred to the reactive flow [17].
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
Nomenclature A Af Atot B c D H P r T
parameter given by equation (7), K/s front surface area, facing solar radiation, m2 total surface area of the sample, m2 parameter given by equation (6), K/s heat capacity, J/kg K diameter of the cylindrical sample, m height of the cylindrical sample, m solar radiation flux, MW/m2 radial dimension of the cylinder, m temperature distribution of the sample, K
Another design of a solar system is described by Zijffers et al. [18]. Their article describes the design process of the Green Solar Collector (GSC), an area-efficient photobio reactor for efficient outdoor cultivation of microalgae. In recent studies Gordon et al. [19] and Levy et al. [20] used highly concentrated sunlight to generate fullerene-like and nano tubular inorganic nanostructures from Cs2O, SiO2-x, WS2, and MoS2, ranging from single-walled nanotubes and closed-cage structures to their larger multi-walled counterparts. Gordon et al. [21] described a high-irradiance solar furnace that could be used in various high temperature processes, such as testing concentrator photovoltaics and driving high-temperature reactors for the generation of novel nanostructures, with target irradiance up to 12 W/mm2. The opto-mechanical design permits real-sun flash illumination at a millisecond time scale, so that solar cells can be characterized with only insubstantial increases in cell temperature even at irradiance levels of thousands of suns [21]. In a different methodology, solar systems are studied using simulations. Singh and Sulaiman used simulation to look at the designing procedures of a solar thermal cylindrical parabolic trough concentrator (CPTC) [22]. Vidales et al. considered a 1 kW thermochemical solar reactor/receiver fitted with a porous ceramic foam structure, and studied it numerically in order to predict the thermal transfers inside the volumetric solar receiver. The suggested reactor is devoted to the production of hydrogen from twostep thermochemical cycles based on mixed metal oxides, and it features a porous media coated with the reactive ferrite material (MxFe3xO4) that is directly irradiated by concentrated solar energy [23]. Natarajan et al. presented a numerical study of solar cell temperature for concentrating PV, using a concentration ratio of 10. The researchers developed a two-dimensional thermal model to predict the temperature for a PV concentrator system (solar cell and lens) with and without passive cooling arrangements [24]. Min et al. established a metal plate cooling model for 400 single concentrator solar cells [25]. In addition, observations based on simulations could be useful for validating analytic calculations [3], and after model validation based on experimental observations, simulations could be used as a design tool. This type of tool was used to perform many numerical simulations for different design criteria [4]. In this study, a model was developed in order to describe the temperature distribution inside a small piece of solid material under conditions of high solar irradiance and high temperatures. Based on the results, experiments are suggested.
Ta Tf Tb t V z 3
a b k r s
1001
ambient temperature, K front surface temperature, K back surface temperature, K time, s volume of the sample, m3 axial dimension of the cylinder, m hemispherical emissivity, e thermal diffusivity (k/rc), m2/s steady state temperature given by equation (7), K thermal conductivity, W/m K density, kg/m3 StefaneBoltzmann coefficient, W/m2 K4
model is developed following three intuitive steps: lumped system approach (one dimension e time), distributed system approach (two dimensions e time and height) and distributed system approach with azimuthal symmetry (three dimensions e time, height, and radius). The benefits of such a study could be a) educational, b) to increase insight into similar systems, and c) to provide guidance in designing experiments, so that with a simulation and the aid of the computational power of computers, expenses are minimized. 2.1. Lumped system approach Consider a small cylindrical piece of material with diameter D and height H, subject to concentrated solar radiation with constant flux P (MW/m2). The solar flux is assumed constant for simplicity (for photovoltaic applications, uniform flux distributions could be achieved by using a kaleidoscope [28]). The essential properties of the material needed to develop a thermal model are the density r (kg/m3), heat capacity c (J/kg K), thermal conductivity k (W/m K), and hemispherical emissivity 3. In this model the temperature of the material T (K) is assumed to be uniform everywhere. The incoming energy at the front surface is distributed into two parts: energy heating the material and energy lost back to the surrounding environment at temperature Ta (K). It is important to note that in high energy experiments; usually the material is protected in a vacuum, so that convective heat transfer is minute. Although it is an inexact statement; the results of such a lumped system model are of great importance in designing experiments. It will be shown later that these results are confirmed by the other modeling assumptions given in Sections 2.2 and 2.3.
2. Thermal modeling In this section we develop a thermal model to describe the temperature distribution inside a small cylindrical piece of material, subject to a high solar flux on its front side (see Fig. 1). The
Fig. 1. Schematic of the model. A cylindrical piece of diameter D and of height H is subject to concentrated solar flux. Heat is distributed within the sample via heat conduction and losses are introduced via thermal radiation from all sides.
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M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
dT ¼ A B T 4 Ta4 ; dt
The energy balance of the lumped system is given by
d ðrcVTÞ ¼ PAf 3 sAtot T 4 Ta4 : dt
(1)
The different quantities that appear in eq. (1) are the area of the front Af surface, which is given by
pD2
Af ¼
4
:
V ¼ Af H:
dT ¼ A B T 4 Ta4 : dt
(4)
1 vT v2 T ; ¼ a vt vz2
(5)
Parameter B is related to the initial slope for cooling BðTa4 b Þ and is given by 4
sAtot B ¼ : rcV 3
(6)
Parameter b is the steady state temperature of the heating process, which is also the initial temperature for cooling process, and it is given by
A B
b4 ¼ Ta4 þ :
(7)
Parameter A is the initial slope for the heating process, and is given by
:
(8)
The nonlinear initial value problem (for heating), which is given by eq. (1), could be solved by separation of variables. The solution is given parametrically, with time shown as a function of temperature, by
t ¼
T Ta arctan arctan
2Bb
3
þ
1 3
4Bb
(11)
In the second model, the temperature T(t,z) is assumed to be two-dimensional and is given a function of time t and position along the vertical axis z. The parabolic heat equation is given by
For constant thermal and mechanical properties (which are acceptable assumptions, especially when considering qualitative behavior of the system), the energy equation given by eq. (1) could be rearranged as follows
1
b 1 T arctan arctan Ta Ta 2BTa3 1 T Ta b þ Ta : ln $ b Ta T þ Ta 4BTa3
(3)
and the volume of the cylinder, which is given by
PAf
t ¼
2.2. Distributed system approach e two-dimensional model
Atot ¼ 2Af þ pDH;
rcV
and its solution is given by
(2)
The total surface area of the cylinder Atot, which is given by
A ¼
(10)
ln
b
b þ T b Ta $ : b þ Ta b T
b
(12)
where a ¼ k/rc is the thermal diffusivity. In principle, this equation could be solved by separation of variables, in which case the solution would include an exponential decay in time. After some time the temperature distribution approaches a steady state. In this model the solution is represented by its steady state response. The steady state response is a linear function of the z coordinate. The steady state temperature is given by
z T z ¼ Tf þ Tb Tf : H
(13)
The differential equation (eq. (11)) could be solved for with the initial time and two boundary conditions provided as known factors. At the front surface the boundary condition is given by
vTðt; z ¼ 0Þ P t ¼ k þ 3 s Tf4 Ta4 ; vt
(14)
and the boundary condition at height H is given by
k
vTðt; z ¼ HÞ ¼ 3 s Tb4 Ta4 : vt
(15)
After some algebraic manipulations, in the steady state the thermal fluxes are constant so that equation (14) could be rearranged to represent the front temperature Tf as a function of the back temperature Tb, as given by
Tf ¼ Tb þ
3
sH k
Tb4 Ta4 :
(16)
Finally, equations (14) and (15) are substituted in equation (13) to give a nonlinear equation with one variable, Tb, which is given by
P ¼ 3s
4 3 sH Tb4 Ta4 Tb þ Tb4 Ta4 þ Ta4 : k
(17)
Equation (16) is solved numerically with the aid of Microsoft Excel.
(9)
A similar solution could be given for the cooling process, for which P ¼ 0 and the initial temperature is b. In this case the differential equation (eq. (5)) becomes
2.3. Distributed system approach e three-dimensional model In this section the model described in the previous section is extended to account for changes along the radial direction. The thermal conductivity is assumed to be uniform and independent of direction. The temperature distribution is a function of time t, radial
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
Temperature versus time - Lumped model
k
2500
1003
vTðt; r ¼ D=2; zÞ ¼ 3 s T 4 ðt; r ¼ D=2; zÞ Ta4 : vr
(20)
In order to be able to develop a numerical scheme the partial differential equation (PDE) given by equation (17) should be rearranged for the zero radius and is given by
2000
1 vTðt; r; zÞ v2 Tðt; r; zÞ v2 Tðt; r; zÞ þ : ¼ 2 a vt vr 2 vz2
1500 Density (kg/m 3 ) 5060 2500 1000
1000
(21)
3. Numerical example In this section, a numerical example is given for illustrating the solution for a specific case. The ambient temperature is assumed as 300 K. Based on Ref. [21], solar flux is chosen to be in the range 1e12 MW/m2. The following additional presumptions are made:
500
0 0
10
20
30
40
50
60
70
StefaneBoltzmann constants ¼ 5.667 $ 108 W/m2 K4 Hemispherical emissivity is 0.8 (gray body) The diameter and height of the cylinder are both 4 mm The density of a hypothetical material (with the melting point thus avoided, which could be taken into consideration more properly for a specific material) is r ¼ 5050 kg/m3 The heat capacity is 3977 J/kg K The thermal conductivity k ¼ 0.44 W/m K.
Time (s)
Fig. 2. Predicted temperature based on the lumped system approach. The temperature is plotted vs. time for different values of density. The effect of the response time is highlighted in the figure.
direction r, and vertical direction z. The heat conduction equation in this case is given by
1 vTðt; r; zÞ 1v vTðt; r; zÞ v2 Tðt; r; zÞ r þ : ¼ a vt r vr vr vz2
(18)
The initial temperature distribution is assumed to be uniform at the ambient temperature Ta. The boundary conditions at the front and back surfaces are given by equations (13) and (14) respectively. The additional necessary boundary conditions in the radial direction are at r ¼ 0 and at r ¼ D/2. The boundary condition at zero radius assures physical solution. This condition is equivalent to insulation and it is given by
vTðt; r ¼ 0; zÞ ¼ 0: vr
(19)
The other radial boundary condition that assures energy balance at the outer radius is given by
The properties of MoS2 could be found in the literature [26,27]. In fact, these parameters are lumped in one parameter, the a ¼ k/rc thermal diffusivity. This parameter appears in the PDE and it effects the response time of the system. With these values one could perform calculations and draw some conclusions. In the following paragraph, some calculations are shown next. The temperature of the cylinder based on the lumped system model is plotted against time for different density values (see Fig. 2). An important observation from the plot is that by decreasing the density, a faster time response is achieved. What is important to highlight is the possibility of constructing the sample differently; one might suggest a denser cylindrical sample partially hollow in order to be filled with a less dense sample of the same material, or probably a powder. The predicted steady state temperature is plotted against solar flux for different values of the height of the cylinder, which are depicted by Fig. 3. It is clear from this plot that a specific steady state
Temperature vs. solar flux - Lumped model 3500 Sample's Height (mm) H= 1 2 3 4
Temperature (K)
3000
2500
2000
1500
1000 0
2
4
6
8
10
12
2
Solar flux MW/m
Fig. 3. The steady state temperature as predicted from the lumped system model is plotted vs. solar flux for different values if the height of the cylinder.
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Steady state temperatures of the sample's front and back surfaces vs. solar flux
Temperature at the front surface vs. normalized radius of the sample
4000
4000 3500
Temperature (K)
Steady state temperature (K)
3500
Front surface Sample thickness H (mm) 1 2 3 Back surface 4
3000 2500 2000
3000 2500 2000 1500 1000 500 0
1500
0
0.4
0.6
0.8
1
Normalized radius 2r/D
1000 0
2
4
6
8
10
12 Fig. 6. The temperature distribution at zero height based on the three-dimensional model is plotted vs. radius of the cylinder.
Solar flux MW/m2 Fig. 4. The steady state temperatures of the front side and the back side of the cylinder based on the distributed system approach in the vertical direction are plotted vs. solar flux for different heights of the cylinder.
temperature could be targeted or maybe achieved by choosing a suitable height (as an example of design parameter). Although the lumped system model could not show directional temperature gradients along the cylindrical axis, one could create a design to achieve a gradient by fashioning a proper sample made of two different pieces with different densities. It is important to remember that the lumped system model is insufficient to capture all of the physics (especially the distribution of the temperature along the radial and the axial directions). A step toward determining the real picture of the temperature distribution along the cylindrical axis could be achieved by the second model. Fig. 4 presents the steady state temperatures of the front and back sides of the cylinder against solar flux for different values of the cylinder’s height. By consulting Fig. 4 an obvious conclusion (that the front temperature reaches higher levels than was predicted by the lumped model) can be reached, which is a much more realistic observation. On the other side (the back side), the temperature reaches much lower values, asserting the fact of low thermal conductivity along the vertical direction. This type of result is obtained more accurately through the threedimensional model, as one can see from Figs. 5 and 6. The third
Temperature distribution inside the sample vs. time - 3D model 3500 3100
Temperature (K)
0.2
2700 Location: r=0, z=0 r=D/2, z=0 r, z=H/2 r, H
2300 1900 1500 1100
300 10
20
30
4. Summary and conclusions In this study three different models (see Fig. 1 for the schematic of the models e the same figure fits all the models, the difference is expressed through the modeling assumptions) were studied theoretically. The study progressed from the simpler, onedimensional approach, the lump system (one dimension e time), through a two-dimensional distributed system approach (time and vertical direction) to three-dimensional distributed system approach with azimuthal symmetry (time, vertical direction, and radial direction). The three models are introduced in Section 2 (2.1e2.3) and the appropriate methods of solution were given, analytically whenever it was possible (as in the case of lumped system approach), and numerically in most other cases. The numerical example given in Section 3 enabled the production of some results that are summarized in Figs. 2e6. 5. Discussion
700 0
model, which was presented in Section 2.3, takes into account the heat conduction effect along the radial direction, and heat loss through the sides, through thermal radiation. Fig. 5 shows the plots of the temperatures at different locations along the radial direction, and at different heights. It is clear that the gradients are much higher in the vertical direction. This effect might be explained by uniformity of the flux at the front surface and by the radial symmetry of the cylinder. The effect of the losses from the sides is highlighted by Fig. 6, which presents the temperature distribution at zero height as a function of the radius. It could be seen that toward the end of the radial direction there is a relatively small drop in temperature compared to the drop along the vertical direction (something that can be seen at Fig. 5). The results of the second model and the third model clearly show the vertical gradients in temperature. It is possible to increase such gradients if the sample is replaced by a two part assembly of a denser kind of a container filled with a less condensed piece or powder of the same material.
40
50
Time (s) Fig. 5. The temperature distribution based on the three-dimensional model is plotted at different locations in the sample vs. time.
Although the lumped system model would not show temperature gradients along the vertical direction (appropriate gradients are needed for special materials production [20]), one could suggest that such a gradient could be forced if the uniform sample was replaced by a sample made of two parts, where the outer shell is denser than the inner shell. The other models showed more realistic results and due to the low thermal conductivity of the
M. Huleihil, G. Mazor / Applied Thermal Engineering 51 (2013) 1000e1005
material, gradients in the vertical direction were much higher than the gradients along the radial direction. This type of behavior could be explained by the uniform input of the solar flux and by the symmetry of the cylindrical sample. After consulting Figs. 2e6 one could observe that a steady state condition at the front surface was reached within seconds. This is a very important observation, especially for designing experiments, because it enables researchers to perform many experiments within a short period of time. This is also most important to researchers in the field of solar energy (when considering the availability of the sun). Lastly, the models analyzed in this study serve as powerful tools in designing experiments, and make it possible to check the different effects of the parameters involved in these models.
References [1] A. Rabl, Active Solar Collectors and their Applications, Oxford University Press, New York, Oxford, 1985. [2] R. Winston, J.C. Minano, P. Benitez, Nonimaging Optics, Elsevier Academic Press, 200 Wheeler Road, 6th Floor, Burlington, MA 01803, USA, 2005, with contributions by Narkis Shatz and John C. Bortz. [3] Ch. Charach, M. Huleihil, Y. Zarmi, Perturbative analysis of planar phase change processes with time dependent temperature at the boundary, J. Appl. Phys. 64 (1988) 4832. [4] D. Faiman, D. Feuermann, M. Huleihil, Numerical model of a rotating prism wall: its validation and predicted device performance for a variety of situations, J. Sol. Energy Eng. 111 (3) (1989) 237. [5] D. Feuermann, J.M. Gordon, M. Huleihil, Light leakage in optical fibers: experimental results, modeling and the consequences for solar concentrators, Sol. Energy 72 (2002) 195e204. [6] D. Feuermann, J.M. Gordon, M. Huleihil, Solar fiber-optic mini-dish concentrators: first experimental results and field experience, Sol. Energy 72 (2002) 459e472. [7] J.M. Gordon, D. Feuermann, M. Huleihil, Laser surgical effects with concentrated solar radiation, Appl. Phys. Lett. 81 (2002) 2653e2655. [8] J.M. Gordon, D. Feuermann, M. Huleihil, S. Mizrahi, R. Shaco-Levy, Solar surgery, J. Appl. Phys. 93 (2003) 4843e4851. [9] J.M. Gordon, D. Feuermann, M. Huleihil, S. Mizrahi, R. Shaco-Levy, Surgery by sunlight on live animals, Nature 424 (2003) 510. [10] R.D. Nasby, R.W. Sanderson, Performance measurements techniques for concentrator photovoltaic cells, Sol. Cells 6 (1982) 39e47.
1005
[11] Y. Yang, A.A. Torrance, J. Rodriguez, The solar hardening of steels: experiments and predictions, Sol. Energy Mater. Sol. Cells 40 (1996) 103e121. [12] D. Laplaze, P. Bernier, W.K. Maser, G. Flamant, T. Guillard, A. Loiseau, Carbon nanotubes: the solar approach, Carbon 36 (5e6) (1998) 685e688. [13] T. Guillard, G. Flamant, D. Laplaze, Solar production of carbon nanotubes; structure evolution with experimental conditions, J. Mater. Sci. 35 (2000) 419e425. [14] D. Laplaze, L. Alvarez, T. Guillard, J.M. Badie, G. Flamant, Carbon nanotubes: dynamics of synthesis processes, Carbon 40 (2002) 1621e1634. [15] G. Flamant, J.F. Robert, S. Marty, J.M. Gineste, J. Giral, B. Rivoire, D. Laplaze, Solar reactor scaling up: the fullerene synthesis case study, Energy 29 (2004) 801e809. [16] G. Flamant, D. Luxembourg, J.F. Robert, D. Laplaze, Optimizing fullerene synthesis in a 50 kW solar reactor, Sol. Energy 77 (2004) 73e80. [17] S. Abanades, G. Flamant, Production of hydrogen by thermal methane splitting in a nozzle-type laboratory-scale solar reactor, Int. J. Hydrogen Energy 30 (2005) 843e853. [18] J.W. Zijffers, M. Janssen, J. Tramper, R.H. Wijffels, Design process of an areaefficient photobioreactor, Mar. Biotechnol. 10 (2008) 404e415. [19] J.M. Gordon, E.A. Katz, D. Feuermann, A. Albu-Yaron, M. Levy, R. Tenne, Singular MoS2, SiO2 and Si nanostructuresdsynthesis by solar ablation, J. Mater. Chem. 18 (2008) 458e462. [20] M. Levy, A. Albu-Yaron, R. Tenne, D. Feuermann, E.A. Katz, D. Babai, J.M. Gordon, Synthesis of inorganic fullerene-like nanostructures by concentrated solar and artificial light, Isr. J. Chem. 50 (2010) 417e425. [21] J.M. Gordon, D. Babai, D. Feuermann, A high-irradiance solar furnace for photovoltaic characterization and nanomaterial synthesis, Sol. Energy Mater. Sol. Cells 95 (2011) 951e956. [22] B. Singh, F. Sulaiman, Designing a solar thermal cylindrical parabolic trough concentrator by simulation, International Rio3 Congress, World Climate and Energy Event, Rio de Janeiro, 1e5 December 2003. [23] H.I. Villafan-Vidales, S. Abanades, C. Caliot, H.R. Paredes, Heat transfer simulation in a thermochemical solar reactor based on a volumetric porous receiver, Appl. Therm. Eng. 31 (2011) 3377e3386. [24] S.K. Natarajan, T.K. Mallick, M. Katz, S. Weingaertner, Numerical investigations of solar cell temperature for photovoltaic concentrator system with and without passive cooling arrangements, Int. J. Therm. Sci. 50 (2011) 2514e 2521. [25] C. Min, C. Nuo-Fu, D. Jin-Xiag, Thermal modeling and the optimized design of metal plate cooling systems for single concentrator solar cells, Chin. Phys. B 21 (3) (2012) 034216-1e034216-6. [26] R.C. Mclaren, Master thesis in Mechanical Engineering, University of Illinois at Urbana-Champaign, 2009, Last visited on Thursday, April 26, 2012 from: http://www.ideals.illinois.edu/handle/2142/14668. [27] http://en.wikipedia.org/wiki/Molybdenum_disulfide. [28] J. Sun, T. Israeli, T.A. Reddy, K. Scoles, J.M. Gordon, D. Feuermann, Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators, Trans. ASME 127 (February 2005) 138e145.