A detailed parameter study on the comprehensive characteristics and performance of a parabolic trough solar collector system

Applied Thermal Engineering 63 (2014) 278e289

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

A detailed parameter study on the comprehensive characteristics and performance of a parabolic trough solar collector system Ze-Dong Cheng, Ya-Ling He*, Kun Wang, Bao-Cun Du, F.Q. Cui Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

h i g h l i g h t s
It is to present an analytical method or reference for designing more effective PTC.
Relations between the focal shape and the geometric parameters were analyzed.
Effects of the geometric parameters were studied by the FVM combined with the MCRT.
The comprehensive performance is mainly determined by the defocusing phenomenon.
Optional ranges of geometric parameters with optimized performance can be determined.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 September 2013 Accepted 5 November 2013 Available online 15 November 2013

This paper presents the theoretical analysis results of the relations between the geometric parameters of the reflector of a parabolic trough collector (PTC) system and the focal shape formed by the defocusing phenomenon of the non-parallel solar beam firstly. Then the effects of these designed parameters and the defocusing phenomenon on the comprehensive characteristics and performance of the whole process of the photo-thermal conversion in the PTC system were numerically studied and optimized, using a proposed three-dimensional integrated model combined the Finite Volume Method (FVM) with the Monte Carlo Ray-Trace (MCRT) method. It is revealed that the numerical results can be well explained by the theoretical analysis results, proving that the model and method used in the present study is feasible and reliable. It is also found that the comprehensive characteristics and performance are very different from some critical points determined by the defocusing phenomenon of the non-parallel solar beam. From these critical points, the optional ranges of the geometric parameters of the reflector are determined to collect the entire reflected beam from the reflector, with relative optimized performance. In addition, an improved description for the characteristics of the solar flux density distributions on the absorber tube is further presented. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Geometric parameter Coupled heat transfer Finite Volume Method MCRT method Comprehensive effect Parabolic trough collector

1. Introduction As it is well known, solar energy is the largest and most widely distributed renewable energy resource available on our planet. It has attracted more and more attentions during recent years due to the increasing energy demand and the environmental impact of burning fossil fuel [1e3]. Among numerous technologies for utilizations on solar energy, the parabolic trough collector (PTC) technology of concentrating solar power (CSP) systems is the most proven and lowest cost large-scale solar power technology available today [4]. Moreover, it also has been the most mature and

* Corresponding author. Tel.: þ86 029 82665930; fax: þ86 029 82665445. E-mail addresses: [email protected], [email protected] (Y.-L. He). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.11.011

field-tested solar thermal electric technology up to now [3]. Recently, feasibility analysis also indicates that there is a good condition to utilize the PTC technology in developing countries, especially in China [5,6]. Since it belongs to those so-called solar belt countries for plentiful solar resources and large wasteland areas are widely available in the western and northern part of the country [7,8]. Rapid developments of CSP also occurred in China for the past few years. However, in order to achieve electricity generation costs that are competitive to fossil power plants, further developments of the solar components with regard to cost reduction and increased efficiency are necessary, especially for the solar collector [9]. The PTC system uses mirrored surfaces of a linear parabolic concentrator/reflector to focus direct solar radiation to a tubular absorber tube located along the focal line of the parabola. Then the

Z.-D. Cheng et al. / Applied Thermal Engineering 63 (2014) 278e289

parameters of the reflector will be presented firstly. After that, the effects of these designed parameters and the defocusing phenomenon on the comprehensive characteristics and performance of the whole process of the photo-thermal conversion will be numerically studied and optimized, expecting to present a comprehensive analytical method or a reference for designing a high-performance and more cost effective PTC system later. 2. Model description The present work can be considered as a further study for the PTC system, by combining the FVM and the MCRT method shown in Refs. [10,16,17]. Therefore, the LS2 PTC module tested on the AZTRAK rotating test platform at Sandia National Laboratories (SNL) [11] is also used as the original physical model for simulations in this paper. A schematic diagram of a cross-section of this PTC module without the 2-in diameter solid plug is shown in Fig. 1. From the diagram, it can be seen that the PTC module mainly includes a tubular receiver/absorber and a parabolic trough reflector. Some parameters of the reflector are also presented in details, such as the focal length, the aperture width, the rim angle, and the defocusing phenomenon of the non-parallel reflected beam. To support the receiver located along the focal line of the parabola, a small gap between two halves of the reflector are given to install the flange or support the bracket. It also reduces the cost of the reflector manufacture without losing efficiency, since the receiver always shadows this area. In this paper, the effects of the small gap are also taken into account while other mirror surfaces are assumed continuous. A detailed schematic diagram of the receiver is presented in Fig. 2. It can be seen that the receiver mainly includes an inner absorber tube with a selective cermet coating on its outer surface, an outer glass cover, an annular vacuum space with a high vacuum of 0.0001 torr, and insulations at its ends. An ideal 3D computational fluid dynamics and heat transfer model of the entire receiver of the PTC segment, derived from Ref. [16], should include the following photo-thermal conversion or heat transfer processes: (I) the convection heat transfer between the HTF and the absorber; (II) the conduction heat transfer through the absorber wall; (III) the solar irradiation absorption in the absorber; (IV) the heat transfer from the absorber to the glass cover by molecular conduction and radiation heat transfer; (V) the solar irradiation absorption in the glass cover; (VI) the conduction heat transfer through the glass cover; (VII) the heat transfer from the glass cover to the atmosphere, etc. Particularly, the heat transfer from the glass cover to the atmosphere is caused by both convection and radiation. It could be either forced or natural, depending on whether there is wind around the glass cover [16]. The radiation heat transfer occurs due to the temperature difference between the outer cover surface and

Z a do

2δ

ϕm f

concentrated solar radiation is absorbed and converted into thermal energy by the heat transfer fluid (HTF) flowing through the tube [2,4,5]. It can be seen that the whole process of the photothermal conversion in the PTC system is very complex. It includes the physical process of photon energy concentrating, collecting, converting, and coupling nonuniform heat transfers with nonuniform fluid dynamics. Many significant simulation studies on this complex process of the photo-thermal conversion have been carried out. They generally aimed at the interpretation of the work mechanism, the performance/reliability improvement and the cost reduction. However, most models were assumed that the solar flux and the fluid flow were uniform or constant, or that the correlations in the models were based on an assumption of a uniform or constant temperature [10]. Though good agreements with experimental data were obtained, they may not be used to study the detailed characteristics of the process of the photo-thermal conversion. In reality, because of the nature of nonuniform concentrated solar flux onto the absorber, the inner flow is heated asymmetrically and thus is nonuniform. To predict this type of flow accurately, detailed three-dimensional computational fluid dynamics and coupled heat transfer models need to be further developed, with realistic nonuniform concentrated solar energy flux distributions and working conditions [11]. Because of this, some significant three-dimensional numerical models have been developed in recent years. Eck et al. developed a three-dimensional Finite Element Method (FEM) model with nonuniform solar heat generation distributions derived from heat flux profile from ray tracing simulations [12]. Wang et al. also developed a ray-thermalestructural sequential coupled model to vindicate that the introduced eccentric tube receiver reduces the thermal stresses of the parabolic trough concentrator system [13,14]. Very recently, Wirz et al. studied the three-dimensional heat transfer process in the PTC system by taking into account the incident nonuniform solar radiation distribution and the spectral radiative exchange between receiver surfaces [15]. At the meantime, the authors also developed a three-dimensional computational fluid dynamics and coupled heat transfer model by the Finite Volume Method (FVM), with the nonuniform solar energy flux distribution calculated by the Monte Carlo Ray-Trace (MCRT) method [10,16,17]. Numerical results were compared with experimental data and good agreement was obtained, proving that the model and method used is feasible and reliable. Then some typical working conditions, receiver parameters, HTF types and residual gas conditions were studied only for the receiver while the effects of the geometric parameters of the reflector were not taken into account. However, to the best of our knowledge, few systematic studies have been found in the literature on effects of the geometric parameters of the reflector on the comprehensive (i.e. both optical and thermal) characteristics (including the detailed solar flux and temperature distributions) and performance of the whole process of the photo-thermal conversion in the PTC system [18e29]. Usually, this information may be of great importance to the design and the optimization of the structure of the PTC system. Therefore, they will be further studied here, using the abovementioned integrated 3D FVMeMCRT combined model. In addition, the incident solar beam towards the parabolic trough reflector is always within an optic cone, due to the effect of the finite size of the sun [20e22]. As a result, the corresponding reflected beam from the reflector cannot be concentrated at the deserved focal line of the parabola exactly, but a focal shape is formed (i.e., the divergence of the reflected beam from the reflector or the defocusing phenomenon of the non-parallel incident solar beam named in this paper). It may also have much effect on the comprehensive characteristics and performance. Therefore, the theoretical analysis results of the relations between the focal shape and the geometric

279

Gap

M

Y

O Fig. 1. Schematic of the physical model of a parabolic trough solar collector.

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y or z

Glass tube

Absorber tube VII

V III

IV

VI

I

HTF

x o

II

HTF

Selective coating

Annular space

Fig. 2. Schematic of the photo-thermal computational model of the parabolic trough solar collector.

the sky. To be clearer, values of all parameters or conditions for a basic case to be studied are presented together in Table 1. 3. Methodology description 3.1. Numerical methods Combined model To help the reader understand, a flowchart of the abovementioned 3D FVMeMCRT combined model for the LS2 PTC module is shown in Fig. 3 in details. It can be seen that there are two main loops in the left block of MCRT. The inner loop is that a photon packet propagates in the PTC system, and then it is repeatedly moved and traced until it escapes from the system or is absorbed by the system. The outer loop is that this tracing process is repeated until a desired number of photon packets have been propagated. After necessary checking or validating, the solar energy flux density distribution is counted. Then it is combined with the FVM model as shown in the Coupling and FVM blocks. In the simulations, the HTF inside the absorber tube is Syltherm 800 oil, and the corresponding temperature-dependent thermalephysical properties are taken into account [30]. The Reynolds number is always around the level of 8.8  105, thus the fluid flow is turbulent. In addition, the fluid flow is always in steady state under the typical working conditions. Therefore, all the computational processes can be implemented as the same as mentioned in Refs. [10,16,17]. This includes solving the governing equations for continuity, momentum, and energy, setting the corresponding boundary conditions (especially the nonuniform solar energy flux density calculated by the MCRT method), combing the FVM with the MCRT method, and validating the model (including the MCRT code check, the grid independence test and the FVMeMCRT combined model validation). More information can be found in these references and thus they will not be repeated here for the simplicity of presentation. Method of solution A commercial computational fluid dynamics code (Fluent-code) is used to simulate the fluid flow and heat transfer processes in the Table 1 Parameters and conditions for a computational basic case. Reflector and receiver parameters a f L

rr sg aa

5m 1.84 m 7.8 m 0.93 0.95 0.96

di do

da Di Do

dg

Conditions 0.064 m 0.070 m 0.003 m 0.110 m 0.115 m 0.0025 m

HTF qm Tf Ib Ta Vw

Syltherm 800 5.119 kg s1 650 K 950 W m2 295.15 K 3.2 m s1

receiver. The governing equations are discretized by the FVM [31]. The convective terms in the governing equations for momentum and energy are discretized with the second order upwind scheme. When the natural convection terms caused by the temperaturedependent properties of HTF in the momentum equations are taken into account, the pressure is discretized with the scheme of PRESTO! [32]. The SIMPLEC algorithm is used to ensure the coupling between the velocity and the pressure. The convergence criterions for all the solved variables of the flow and the energy are that the corresponding maximum residual of the cells divided by the maximum residual of the first 5 iterations are less than 105 and 107, respectively. To accelerate the computation convergence of the complicated 3D model, the multi-grid method is adopted [33]. To reduce the computing time enormously, the parallel Fluent-code is run on a large-scale parallel computing platform at the National High Performance Computing Center (Xi’an). 3.2. Parameter definitions Some important parameters are presented here firstly, which will be used in the following section of results and discussion. When the effect of the finite size of the sun is taken into account [20e22], a focal shape will be formed around the focal line by the divergence of reflected beam from the reflector (with d ¼ 160 ). The effect of the focal shape may be significant under some conditions when the absorber cannot collect the entire reflected beam as expected. To absorb as much concentrated solar energy as possible, it can be inferred that the outer diameter of the absorber do should be larger enough than the focal shape width dmin, which can be theoretically expressed as follows [34]:

h i do > dmin ¼ 2 f þ a2 =ð16f Þ sind

(1)

The comprehensive characteristics and performance studied here includes the characteristics of the concentrated solar flux density distributions and the corresponding temperature distributions on the absorber, the optical efficiency, the thermal efficiency, and the collector efficiency of the whole PTC system, etc. To present the characteristics of the concentrated solar flux density distributions and the temperature distributions, the averages and the nonuniformities need to be defined respectively. The averages are defined as the arithmetic means, as follows:

PNc qSR;m ¼

i¼1

qSR;i

Nc

PNc ;

Tm ¼

i¼1

Nc

Ti

(2)

The nonuniformities are defined as the coefficients of mean deviations, which can be expressed as follows:

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MCRT

COUPLING

281

FVM

MCRT for parabolic trough solar collector Initialize photon distributon Y

Continuity equation

Shadowed by glass tube? Momentum equation

General governing equations

N Arrived at reflector surface

Standard k-e equations N

Reflected by reflector?

Absorb photon

Y Reflecting

Energy equation

N

Hit glass tube?

Hit reflector surface?

Thermal radiation heat flux (radiation source term)

Y

Y Arrived at glass tube outer surface

N

N

Transmitted?

Reflecting

Escaped

N Y Absorbed?

Absorb

Y

Thermal radiation between the walls of evacuated space

Count hitting position and new direction

Count hitting position on glass tube surface Absorb Y

Count new direction

Absorbed?

Y

N

Transmitted?

Fluent data

Solar energy flux distribution on absorber tube

N Connective heat transfer on glass tube outer surface

Reflecting Hit absorber tube? Y Absorbed?

N

N

Y Count hitting position on absorber tube surface N

Arrived at glass tube inner surface

Coupling program

Reflecting

Boundary conditions

Thermal radiation heat transfer on glass tube outer surface Other Boundary conditions

Absorb photon

Last photon?

The Stefan-Boltzmann law

Y Count photon distribution

Experimental correlation of heat transfer coefficient

Count heat flux distribution End

Fig. 3. Flowchart of MCRT calculating solar energy flux distribution and coupling with FVM.

sq ¼

 PNc  qSR;i  qSR;m  i¼1

Nc $qSR;m

PNc ;

sT ¼

jTi  Tm j Nc $Tm

i¼1

(3)

Note that the definition of sT is just used for the analysis and comparisons here. It is not a parameter defined to predict the thermal expansion or thermal stress for the absorber tube. However, it could be further studied later. To present the performance of the whole PTC system, the optical efficiency, the thermal efficiency and the collector efficiency based on the net absorbed solar energy (i.e., the value of the statistical absorbed solar energy minus the experimental optical losses and the numerical thermal losses) are specified as follows:



ho ¼ qu þqlost L103

.   qSR;in ; ht ¼ qu = qu þqlost L103 ;

hc ¼ ho ht

ð4Þ

4. Results and discussion In the following, theoretical analysis results of the relations between the geometric parameters of the reflector and the focal shape were presented firstly. Then the results of examining the effects of these geometric parameters and the defocusing phenomenon on the comprehensive characteristics and performance

of the whole process of the photo-thermal conversion will be presented in order. For each examination, only the corresponding checking parameter varies and all the other parameters remain the same as shown in Table 1. 4.1. Studies on parameter relations As we know, an ideal parabolic trough reflector can be mainly determined by a focal length (f) and an aperture width (a). Therefore, to explore the relations of the focal shape and the designed geometric parameters of the reflector, it could be classified into four conditions as follows: varying a with constant f, varying f with constant a, varying both a and f with constant ratios of a /f, and varying both a and f with different ratios of a /f. The practical physical interpretations of examining the effects of the aperture width and the focal length are that examining an existing parabolic trough reflectors intercepted by different planes resulting in different a and examining different parabolic trough reflectors of different f intercepted to keep constant aperture width, respectively. The others could be inferred from them with the corresponding restricted conditions. Fig. 4(a) shows the variations of the width of the focal shape dmin, the geometric concentration ratio GC and the rim angle 4m with a, for different values of f. In the figure, GC (i.e., the ratio of the

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collector aperture area to the tubular absorber surface area) and 4m (as shown in Fig. 1) are specified as the same in Refs. [20e22]. It can be seen that the value of GC is always proportional to a when the outer diameter of the absorber keeps constant of 0.07 m widely available (e.g., SCHOTT PTR70 and UVAC 2010). Whereas, the values of 4m and dmin increase with an increase in a, but they are of different variation trends. The value of the aperture width for dmin ¼ do increases with f increasing when a is greater than 7 m. This means that a larger optional range of the a value can be chosen for a larger focal length to collect the entire reflected beam from the reflector. However, when a is smaller than 7 m the situation is reversed.

a

Fig. 4(b) shows the variations of GC, 4m and dmin with f, for different values of a. It can be noted from this figure that the value of 4m decreases with the increase of f, while an initial rapid decrease in dmin is observed followed by a very slow increase in dmin for higher values of f. There are two values of f for dmin ¼ do under each given value of a, and the range of focal length that can be chosen is determined by these two critical points. It is clear from the figure that the limited range of the focal length becomes smaller for a larger aperture width. Fig. 4(c) shows the variations of GC, 4m and dmin with a, for different values of a /f. In the figure, the symbolic solid lines represent that the values of a /f are larger than 4 (or 4m > 90 ). It can be noted from this figure that the value of dmin increases with an increase in a (or f) while the value of 4m keeps constant for each given ratio of a /f. Moreover, from this figure we can see some features of the conditions of varying both a and f with different ratios of a /f. That is, the rim angle 4m always increases with the increase of a /f. The value of dmin first decreases with a /f increasing, reaching a minimum at the a/f value of 4, and then it starts increasing with a further increase in a /f, for a given value of a. Inversely, the limited span determined by dmin increases firstly, reaching a maximum at the a /f value of 4, subsequently decreases rapidly with a /f increasing. In addition, it should be noted that the data of varying both a and f for different ratios of a /f shown in Fig. 4(c) are only some special cases here, and more studies need to be further carried out later. 4.2. Effects of aperture width

b

c

Fig. 4. Theoretical analysis results of the relations between main parameters/variables of PTC.

The results of the distributions of the solar flux density and the temperature for different values of a are shown in Fig. 5, where only half of the circumferential angles are presented due to the almost symmetrical distributions. Fig. 5(a) and (b) shows the results with the aperture width varied from 3 to 12 m. They also indicate the effects of the geometric concentration ratio GC here, since the outer diameter of the absorber is kept constant of 0.07 m in this study for the most common commercial receivers available. From Fig. 5(a) we can see that all the curves of the solar flux density distributions in this figure could also be divided into 4 parts as guessed and defined in Ref. [17]. Taking the asterisk curve of a ¼ 7.36 m for an example, they are (I) the shadow effect area, (II) the solar flux increasing area, (III) the solar flux reducing area and (IV) the direct radiation area. Note that the shadow effect area does not mean the angle span caused by the shadow of the receiver totally but indicate the area where the shadow affects it at the beginning. The angle spans of part II and III become larger with a increasing, while part I always keeps a constant angle of about 15 here. As the area under each curve could indicate the relative magnitude of solar energy absorbed by the absorber, it can be obviously seen that these curves become higher and wider with a or GC increased. Accordingly, the spaces between two adjacent curves become smaller and smaller with constant increasing intervals. In addition, the solar flux density distributions on the absorber shown in Fig. 5(a) are very heterogeneous in the circumferential direction with large nonuniformities. The variation trends of the temperature distributions shown in Fig. 5(b) are very similar to these. To further evaluate the effects of the aperture width, results of a wider range of a are presented in Fig. 5 (c) and (d), for values of a from 1.5 to 24 m. It can be seen that some of the distribution curves become a little different from our previous guess that all the curves could be divided into 4 parts. First, we can clearly observe that the increasing trend of the solar flux increasing area (part II) decreases when a is smaller than 2 m. A considered reason is that this area cannot receive additional solar beam from a larger reflected light cone under a larger aperture width as before, and the effect of the

Z.-D. Cheng et al. / Applied Thermal Engineering 63 (2014) 278e289

a

b

c

d

283

Fig. 5. Effects of a on solar flux density and temperature distributions.

small gap becomes relative larger when a becomes smaller. Meanwhile, there are no direct radiation areas (part IV) on the curves of the solar flux density distributions when a is larger than 12.9 m. From the corresponding theoretical analysis results shown in Fig. 6, we can see that the width of the focal shape dmin becomes larger than the outer diameter of the absorber do when a is larger than 12.9 m, thus the absorber will be located in relative larger reflected light cones with further increase in a. As a result, the curves of the solar flux density distributions could be considered as interceptions from these reflected light cones. But they are different from each other. This is clearly due to the fact that the size of the reflected light cone from the reflector increases with the increase of the aperture width. Fig. 7 shows the effects of a or GC on the comprehensive characteristics and performance of the whole process of the photothermal conversion in the PTC system. From them following features may be noted. First, qSR, max and qSR,m increase with the increase of a but the increasing trend decreases with further increase in a. The variation trends of Tmax and Tm are very similar to that of qSR, max and qSR,m. However, the variation trends of the nonuniformity of the solar flux density distribution sq and the nonuniformity of the temperature distribution sT are very different, and the values of sq and sT are of different order of magnitude. It can be seen that sq first decreases rapidly with a increasing, reaching a minimum of 30.23%, and then changes slightly with further increase in a. Correspondingly, sT first increases rapidly with the increase of a and then decreases slightly with further increase in a, having a local minimum of 0.77%. Second, qSR, i n, qu and qlost increase with a increasing but they are of different variation trends. Both qu and qlost first increase rapidly when a increases, but eventually the increasing trends decrease, while qSR, i n is always proportional to a. Third, ht always monotonically increases with a increasing, while ho first increases with the increase of a, reaching a maximum of 71.19% at the a value of about 12.9 m, and then it decreases greatly with further increase in a.

From the analysis, different mechanisms of the collector efficiency hc can be revealed as follows. When a is relative small, the variation trend of hc is in accord with that of ht, and it has a maximum of about 68.70% at the a value of about 12.9 m. When a becomes larger than 12.9 m, the variation trend of hc is almost in line with that of ho. The reasons can also be inferred from Fig. 6 clearly. When a increases from a relative small value to about 12.9 m, the absorber can collect the entire reflected beam from the reflector for dmin  do and thus ho changes slightly. At the same time, ht increases rapidly due to larger and larger net incident solar radiation absorbed with a increasing, and it becomes the dominating influential factor of hc. However, ho drops significantly when a is larger than 12.9 m for dmin > do. This is obviously due to the fact that the absorber starts losing reflected beam from this critical point. However, ht changes slightly at this time, due to the intrinsic

Fig. 6. Variations of GC, 4m and dmin with a under the constant f of 1.84 m.

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a

b

c

d

Fig. 7. Effects of a or GC on comprehensive performance of PTC.

limit of heat absorption capability of the HTF. As a result, ho becomes the dominating influential factor of hc finally. From the discussion it is also revealed that all numerical results can be well explained by the theoretical analysis results, proving that the model and method used in the present study is feasible and reliable. 4.3. Effects of focal length Fig. 8(a) and (b) shows the effects of f on the solar flux density and the temperature distributions with f varied from 0.25 to 2.24 m. From Fig. 8(a) we can see that the area under each curve keeps almost constant due to the constant aperture width where the incident solar radiation entering the PTC system. And the curves of the solar flux density distributions could also be divided into 4 parts as defined in Ref. [17] when f is larger than 0.65 m. In addition, the angle spans of the solar flux increasing area (part II) decrease with the increase of f while both the shadow effect area (part I) and the direct radiation area (part IV) increase with f increasing. However, when f becomes smaller than 0.65 m, an additional area with very low solar flux density appears before the previous shadow effect area, and the previous direct radiation area disappears with further decrease in f. Therefore, it also becomes quite different from our previous guess that all the curves could be divided into 4 parts. It can also be seen from the figure that the curves move towards the lower right direction to maintain a constant area under each curve, though the variation trend of each curve becomes different with f decreasing. Meanwhile, the variation trends of the temperature distributions shown in Fig. 8(b) are very similar to that of the solar flux density distributions. But there are no clear boundaries to divide the temperature distributions into 4 parts compared to that of the solar flux density distributions. To further evaluate the effects of the focal length, the results of a wider range of f are shown in Fig. 8(c) and (d) for the value of f from 0.10 to 4.64 m. It can be seen that the above-mentioned additional

area before the previous shadow effect area (part I) appears when f is about 0.75 m. The reason is that the effect of the small gap or the receiver shadow [21] projected to the absorber circumference becomes larger when f becomes smaller. It can be clearly seen from Fig. 9 that the reciprocal values of the shadow projected to the absorber circumference changes from 0.21 (4.66 ) to 0.06 (15.57 ) when f decreases from 0.75 to 0.21 m (where dmin ¼ do). Therefore, this area is also a part of the previous shadow effect area but of lower solar flux density. In addition, from Fig. 9 it can also be inferred that the projected angle of the small gap to the absorber circumference is slightly larger than that of the receiver shadow when f is smaller than 2.69 m. But, the situation is reversed when f is larger than 2.69 m. It may be revealed that a larger gap could be suggested to further reduce the cost of the reflector manufacture for a larger value of f. When the focal length becomes smaller than 0.21 m, the defocusing phenomenon of the non-parallel solar beam becomes the dominating influential factor for dmin > do, thus the absorbed solar energy of the absorber tube drops significantly with further decrease in f. Seen from the other side where f is larger than 2.24 m, the shadow effect area (part I) and the solar flux increasing area (part II) become smaller and smaller until they disappear with f increasing. They are also very different from our previous guess that all the curves could be divided into 4 parts. In addition, the maximum of the solar flux density on the absorber continues to increase when f increases from 2.24 m to about 4.35 m, but a dropping trend appears from f ¼ 4.64 m. We evaluated the reason for this by examining a larger range of f shown in Fig. 8(e) and (f), where f ¼ 7.31 m corresponds to another critical point for dmin ¼ do as shown in Fig. 10. It can be seen that a consistent variation trend of dropping is also revealed when f varies from 4.35 to 7.31 m. When f becomes larger than 7.31 m, the defocusing phenomenon of the non-parallel solar beam becomes the dominating influential factor again, thus the absorber starts losing the reflected beam. As a result, the maximum of the solar flux density and the corresponding

Z.-D. Cheng et al. / Applied Thermal Engineering 63 (2014) 278e289

a

b

c

d

e

285

f

Fig. 8. Effects of f on solar flux density and temperature distributions.

Fig. 9. Variations of angles projected to the absorber circle from the shadow and the small gap with f.

absorbed solar energy drop significantly. From the discussion, it is also revealed that the numerical results can be well explained by the theoretical analysis results. Fig. 11 shows the effects of f on the comprehensive characteristics and performance of the whole process of the photo-thermal conversion in the PTC system. From it, following features may be noted. First, all the comprehensive characteristics of the temperature distributions on the absorber tube are well consistent with that of the solar flux density distributions, which are different from the above-mentioned effects of a. Under the conditions studied, sq and sT have their minimums of 57.27% and 0.36%, respectively. Second, variation trends of ho, ht and hc with f are always very similar to each other. When f is smaller than 0.21 m or larger than 7.31 m, both ho and hc drop significantly due to the effects of the defocusing phenomenon of the non-parallel solar beam for dmin > do shown in Fig. 10. Relative optimal values for ho, ht and hc are 70.67%, 91.17% and 64.43% at the f value of 0.45 m, respectively. Then they decrease slightly towards right or left, as f located between 0.21 m and 7.31 m where the width of the focal shape dmin is smaller than the outer diameter of the absorber tube do.

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Fig. 10. Variations of 4m and dmin with f under the constant a of 5 m.

4.4. Effects of rim angle Similar to the theoretical analysis, examining the effects of the rim angle 4m can also be classified into those similar four conditions. Fig. 12 shows the variations of the distributions of the solar flux density and the temperature with 4m, for constant f ¼ 1.84 m or a ¼ 5 m, respectively. Main comprehensive characteristics and performance are almost the same as those presented in the above two sections. It should be noted here that 4m might not be an independent identifiable parameter to describe the comprehensive characteristics and performance of a PTC system. This is clearly due to the fact that the curves of an equal rim angle shown in these figures are very different. Fig. 13 shows the variations of the distributions of the solar flux density and the temperature with 4m, taking 4m ¼ 90 for an example. We can clearly observe that the comprehensive characteristics are affected by both a and f varying. That is, the curves of these distributions become higher and wider with a increasing,

a

c

and move towards the lower right direction simultaneously with f decreasing. Moreover, the center of the solar flux reducing area (part III) always locates at about 90 when a is smaller than 15 m, though the angle spans of the shadow effect area (part I), the solar flux increasing area (part II) and the direct radiation area (part IV) vary a lot from each other. when a is larger than 15 m, the variation trends of the distributions of the solar flux density and the temperature becomes slightly different. This is because the absorber starts losing the reflected beam and the defocusing phenomenon becomes the predominant influential factor (as shown in Fig. 4c). Similar situations for the center of the solar flux reducing area (part III) also occur in Figs. 12 and 13. Thus, it may be inferred that the center of the solar flux reducing area could indicate the value of the rim angle while the defocusing phenomenon is not predominant. Fig. 14 shows the comparisons between the comprehensive performance of 4m ¼ 90 and that of varying a or f only, respectively. From it, following features may be noted. First, each performance curve of 4m ¼ 90 itself varies, though the rim angle always keeps constant. This also can be observed from Fig. 13. Second, different variation trends compared to that of 4m ¼ 90 can be seen from both sides of the location of a ¼ 7.36 m or f ¼ 1.25 m on these performance curves of varying a or f only. However, the variation trends become very different beyond the abovementioned critical points when the defocusing phenomenon becomes predominant. In addition, further investigations on different ratios of a /f and further examinations or comparation need to be performed, and are currently performed. 5. Conclusions This study presented theoretical and numerical results of the detailed parameter study on the comprehensive characteristics and performance of the whole photo-thermal conversion process of a PTC system. The theoretical analysis results of the relations between the main designed parameters and some important

b

d

Fig. 11. Effects of f on comprehensive performance of PTC.

Z.-D. Cheng et al. / Applied Thermal Engineering 63 (2014) 278e289

a

b

c

d

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Fig. 12. Variations of solar flux density and temperature distributions with 4m under constant f or a.

variables of the reflector were presented firstly. Then the effects of these parameters and the defocusing phenomenon on the comprehensive characteristics and performance were numerically studied and optimized, using an integrated 3D model combined the FVM and the MCRT method. The following conclusions can be made. (1) It is revealed that the numerical results can be well explained by the theoretical analysis results, also proving that the model and method used in the present study is feasible and reliable. Moreover, the comprehensive characteristics and performance are very different from some critical points determined by the defocusing phenomenon. From these critical points, the optional ranges of the geometric parameters of the reflector (e.g., the aperture width and the focal length) can be determined with relatively optimized performance.

a

(2) It is found that the curves of the solar flux density distributions could not be divided into 4 parts under some special conditions as guessed in the previous reference, and that the rim angle may not be an independent identifiable parameter to describe the comprehensive characteristics and performance of a PTC system. This paper presents an improved description of the characteristics of the solar flux density distributions in a wider range of parameters. In addition, a larger gap between two halves of the reflector could be suggested to further reduce the cost of reflector manufacture if the focal length is much larger. Furthermore, investigations on different ratios of a /f, other available PTC prototypes with realistic operating conditions and possible applications based on these comprehensive characteristics and performance obtained need to be further performed, and are currently performed.

b

Fig. 13. Variations of solar flux density and temperature distributions with 4m ¼ 90 under different a and f.

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a

b

c

d

Fig. 14. Comparisons between the comprehensive performance of 4m ¼ 90 and that of constant f or a.

wind speed (m s1)

Acknowledgements

Vw

The study is supported by the National Natural Science Foundation of China (Nos. 51306149, 51176155), and the China National Hi-Tech R&D (863 Plan) Project (No. 2013AA050502). The authors also thank the National High Performance Computing Center (Xi’an) for providing with technology and support of large-scale parallel computing crucial technique.

Greek symbols aa absorptivity of the absorber d finite size of the sun (d ¼ 160 ) da thickness of the absorber tube (m) dg thickness of the glass cover (m) 4, 4’ circle angle ( ) 4g projected angle of small gap ( ) 4m the rim angle ( ) 4s projected angle of receiver shadow ( ) ho the optical efficiency ht the thermal efficiency hc the collector efficiency rr reflectivity of the reflector sq nonuniformity of solar flux density distribution sT nonuniformity of temperature distribution sg transmissivity of the glass cover

Nomenclature a the aperture width (m) Di inner diameter of glass cover (m) Do outer diameter of glass cover (m) di inner diameter of absorber (m) dmin the width of the focal shape (m) do outer diameter of absorber tube (m) f the focal length (m) GC the geometric concentration ratio Ib incident solar radiation (W m2) L the receiver length (m) Nc grid number qlost thermal loss (W m1) qm mean mass flow rate (kg s1) qSR, i solar flux density of the ith grid (W m2) qSR, i n total incident solar radiation (kW) qSR, m average solar energy flux density (W m2) qSR, max maximal solar flux density (W m2) qu net absorbed solar energy (kW) qw local solar energy flux density (W m2) Ta ambient temperature (K) Tf HTF inlet temperature (K) Ti temperature of the ith grid (K) Tm average wall temperature (K) Tw local absorber wall temperature (K) Tmax maximal wall temperature (K)

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