Calculation of the blocking factor in heliostat fields

Transcription

1 Available online at ScienceDirect Energy Procedia 57 (014 ) ISES Solar World Congress Calculation of the blocking factor in heliostat fields Mustafa E. Elayeb a *, Rajab A. Haan a and Fuad M.F. Siala b a Faculty of engineering,misurata university,misurata P.O.Box 104, Libya b Center for Solar Energy Studies, Tripoli, Libya Abstract This paper presents a ethod for the calculation of the blocking losses of a heliostat, in ters of a blocking factor, arising due to neighboring heliostats in a heliostat field. The proposed ethod is general in the sense that it is valid for any type of heliostat array and orientation. In the ethod the individual blocking eleents are projected on the plane of the heliostat under consideration. The heliostat plane is defined in the heliostat coordinate syste. An analytical expression for the geoetry of the projection is presented, and a nuerical iterative technique is developed for the solution. The solution procedure involves the subdivision of the heliostat surface into a suitable grid. An overlap test is developed to deterine whether a particular sub-area is blocked. In a straightforward anner, all blocked sub-areas are subtracted once fro the overall heliostat area. A progra was written by using MATLAB to test the ethod. In initial runs on lap-top (Intel(R) Core(TM) i3 M GHz), typical rectangular heliostats (9.8x10.7 eters) were subdivided into 5x5 grids. Results for the instantaneous blocking factors for each of 99 such neighboring heliostats, were obtained in an average run tie of less than 3.5 seconds. 014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( 013 The Authors. Published Elsevier Ltd. Selection and/or peer-review under responsibility under responsibility of ISES. of ISES Keywords: CRS; CSP; STE; Heliostat fields; Optical losses; Blocking factor. 1. Introduction The perforance of heliostat field is defined in ters of the optical efficiency. The optical efficiency is defined as the ratio of the net power intercepted by receiver to the direct insolation ties the total irror area. The optical losses include the cosine effect, shading and blocking losses, iperfect irror reflectivity, atospheric attenuation, and receiver spillage losses [1]. Soe authors [1,] think that while both shadowing and blocking increase if the heliostats are closer together, blocking has ore pronounced effect on the layout of heliostat field. Several researchers introduced ethods to deterine the usefulness efficiency of a heliostat surface. Biggs and Vittitoe [3] presented an idea based on projecting the outline of the aligned heliostats that * Corresponding author. Tel.: ; fax: E-ail address: elayeb@yahoo.co The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Selection and/or peer-review under responsibility of ISES. doi: /j.egypro

2 9 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) ight block reflected light onto a unit sphere centered about the ai point. Blocked portions of any heliostat will appear in overlapped regions on this projection. The ain objections to this procedure are that it ay not be accurate in the case of ultiple blocks and for late tie. Also, that it is questionable whether the area ratio in the projection plane is equal to the actual area ratio by which the reflected flux is actually reduced. Sassi [4] derived a cobined analytical/discrete ethod for the calculation of the usefulness efficiency caused by a single heliostat. His idea was based on the projection of a blocking heliostat onto the plane of the heliostat under exaination. He assued that heliostat and projection had the sae geoetry and orientation, which is a siple case selected because it is ore recurrent and because its atheatical developent is siple. By no eans it is a general case. Collado and Turegano [] derived an expression to predict the usefulness efficiency due to the twoshoulder heliostats in a radial aziuth staggered array. This expression is accurate only when the solar aziuth angle, heliostat aziuth angle and heliostat position aziuth angle are the sae. Elsayed et al. [5] developed an analytical expression for the usefulness efficiency of heliostat surfaces when considering all neighboring heliostats in any type of heliostat arrays. Their idea is based on deterining the diensions of the shadow on the plane of the heliostat under testing, and their ain assuption is that both heliostats have approxiately the sae orientation. Again, this assuption is not valid for all cases. Sànchez and Roero [6] calculated the Noralized Blocking using a siplified odel. Aong all the possible tracking positions it is assued that the noral to the heliostat surface points at the aiing point since this represents the worst case. In addition, a grid of neighbors is created to calculate the effect at different positions. Although this approach has been useful for north field configurations, other approaches need to be forulated to generate in a better way surround fields, since the noral of the heliostats located south of the tower would never point at the receiver. The ethod proposed in this paper overcoes the shortcoings of the cited procedures. The blocking losses are due to the interception of part of the reflected sunlight fro heliostat by the backside of another heliostat. The blocking factor defined by Equation (1). F b 1 A A b tr (1) where A b is the blocked area of the heliostat surface and A tr is the total reflecting area of the heliostat. In this paper, a previous ethod to calculate shading factor [7] has been developed to pair with blocking factor calculation. As long as shadow is related to the incidence solar radiation, the blocking loss is related with the reflected sun beas. It ay be possible to calculate a blocking factor by considering that the blocking heliostat acts as if it was a shading heliostat for rays traveling in a reverse direction to the reflected radiation. Starting fro this idea, a vector has been iagined to apply on the reflected sun bea, but on the opposite direction. In other words, it is in the direction fro the receiver to the Heliostat, foring a "virtual shadow" which represents the actual blocked area fro the reflective Heliostat surface by neighboring Heliostat. The effect of blocking is calculated by projecting the outline of the blocking heliostats onto the plane of the blocked heliostat using the heliostat coordinate syste, O. The ain assuptions are: an infinitesial size of the sun, plane surfaces of the heliostats and no tracking errors. Hennet, as cited in

3 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) Collado and Toregano [], established that the axiu relative error caused by these assuptions is less that 5%. Noenclature blocking factor F b A b A tr a blocked area total reflecting area heliostat aziuth angle inclination angle P position vector of point j on heliostat i Q vector joining the center of the blocked heliostat and point j on the blocking heliostat i B cc g vector fro the blocked heliostat to the receiver in the direction of the reflected rays. agnitude of B cc U cc r o 1 N u, N v l,w unit vector, in coordinate syste O, towards the receiver. reflection angle axiu nuber of cells in the u and v directions, respectively length and width of heliostat. Projection of the Blocking Heliostat In Figure 1 let i = 1 be a blocked heliostat and i =, 3 be the blocking heliostats. In our ethod no restrictions are ade on the locations or the orientations of the heliostats. Two coordinate systes, O 1 and O, are utilized. O 1 has its origin at the base of the tower and its X, Y and Z axes are as shown in Figure 1, with unit vectors i, j, and k. O has its origin at the center of the heliostat. Its u-axis is parallel to the horizontal edge of the heliostat and the v-axis is perpendicular to it. The w-axis is parallel to the noral to the heliostat and the unit vectors are u, v and w. In what follows the superscript on the left of any vector indicates the coordinate syste in which it is defined. The transforation of a vector v 1 to v is 1 accoplished through the relation v T v. The atrix T1is defined by [7]: 1 cosa cos si n a si n si n a T1 si n a cos cosa si n cosa () 0 si n a cos

4 94 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) where, a and are the heliostat aziuth and inclination angles respectively. Z heliostat i = 1 v u O r r 1 heliostat i = Q 1 3 P 51 w 5 L Y (north) 4 P 5 1 W P 1 O 1 X (east) Fig. 1. Coordinate systes and vector definition. P is the position vector of point j on heliostat i. The nubering schee is always as shown in Figure 1. The location of any heliostat in the field is defined in ters of its center's position vector: 1 P5 i xii yi j zik (3) L, W, r1 1 In O 1 the vectors and r i i i ay be defined in ters of the length, width and the i aziuth and inclination angles of the heliostat. Then, the position vectors of the corner points of the heliostat are given by the following equations: (4) P1 i P5 i r1i P i P5 i ri P3 i P5 i r1i P4 i P5 i ri (5) (6) (7)

5 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) The vector Q joining the center of the blocked heliostat and point j on the blocking heliostat i can now be defined. Thus, in Figure 1, where i > 1, we have: 1 1 Q P P 1 51 In Figure, a point j' on the plane of the blocked heliostat (i = 1) is the virtual shadow of corner j of heliostat i (> 1). Therefore, B cc is a vector fro the blocked heliostat to the receiver in the direction of the reflected rays, pointing to the receiver center and connecting points j' and j. If g is the agnitude of B cc, then the vector can be expressed as follows: (8) Bcc where g U cc U cc is the unit vector, in coordinate syste O, towards the receiver. (9) In Figure, j'' on the plane of the blocked heliostat is the projection of corner j of heliostat i (> 1). Therefore, (see Figure -b) the following relation is true: g jj Q, cos ro cos r 1 w o1 where, Q, w is the w-coponent of Q and angle) on heliostat 1. (10) r o 1 is the reflection angle (which equal to incidence The virtual shadow of any heliostat on the plane of the blocked heliostat is an area bounded by the four straight lines connecting the points j' belonging to the blocking heliostat. These points are defined by their position vectors, given by the following equation: Q i cc (11) j Q B 3. The Blocking Test A b of the heliostat under consideration is deterined by area of the heliostat intersecting, possibly, with the virtual shadows of all heliostats in the field. Aongst these heliostats, only those located in front of the heliostat under testing ay cast virtual shadow on it, i.e., those for which Q 0. Having deterined the blocking heliostats ; i.e. the active heliostats, A b ay be coputed. This is achieved discretely by generating a rectangular grid on the heliostat under testing. The diensions of the N u x N v cells are given by: 5i, w

6 96 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) (a) (b) Fig.. virtual shadow coordinates on plane of blocked heliostat w u N u and l v N v (1) where N u and N v are the axiu nuber of cells in the u and v directions, respectively. The coordinates of a cell's center are given by C n, k u n, vk for n = 1, N u and k = 1, N v, where, u n 0.5 un u w for n 1 0.5l v and vk 1 u for n 1 vk 1 v for k 1 for k 1 (13) The center of each cell in the grid is tested with respect to the virtual shadows of active heliostats. If a center is found within a virtual shadow, then its associated area, A u v, is added to A b. This center will not be tested again with respect to other virtual shadows, thereby excluding the possibility of virtual shadow overlap. If a cell is not blocked, the process continues, until its center is found to lie within a virtual shadow, or the test is copleted. To accoplish that values are assigned to a variable, p, in the following anner: 0 if Cn, k is on the outline of the virtual shadow p 1 if Cn, k is inside thevirtualshadow (14) if Cn, k is outside thevirtualshadow

7 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) In general, the equation of the (straight line) follows: v v u u j1 j1 v u j j v u j1 j1 j side of the virtual shadow can be written as 1 j (15) The results of substituting the coordinates of the cell's center under testing in Equation (15) can be tabulated in two cases as follows. Case I (sides1 and 3 4 ) Substitute v k for v in Equation (15) applied to each of the two sides in turn and copare the result with u n. In view of Equation (14), the values of p are assigned as shown in Table (1), where: u u u u j j1 j1 vk v j1 v j v j1 Case II (sides 3 and 4 1 ) Here, substitute u n for u in Equation (15) applied to each of the two sides in turn and copare the result with v k. The values of p are assigned as shown in Table (1), where: v v v v j j1 j1 un u j1 u j u j1 If p = for any of the four sides, then the test need not be carried further, because that eans that the cell's center is outside the virtual shadow. Finally, the instantaneous value of the blocking factor for the heliostat under testing ay now be coputed using Equation (1), where A b is found by suing up all the contributing cells. Table 1: p values for case I and case II u n u n State u u u u n P, case I P, case II Side 1 Side 1 State Side 3 Side v k v v k v v v k 4. Perforance To verify the ethod, the heliostats were laid out in a radial staggered arrangeent using procedures forulated by the authors [8] and Collado and Turegano []. These ethods guarantee prevention the blocking loss. The siulation had been applied for a whole year, every single hour a day, which eans

8 98 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) of tie step. The result had confired the nonexistence of any blockage for any heliostat existed in one of the fields. That eans the blocking factor always used to be equal to one, as a proof for the successfulness of the proposed ethod. As an exaple, the proposed ethod was prograed by using MATLAB and applied to two hypothetical heliostat fields located at Misurata (3.4 N, 15.1 E) with 55 tower height. The fields consisted of 9.8x10.7 heliostats were laid out in N-S cornfield and radial stagger arrangeents with separation distance ratio, ds = 0.05, as shown in Figure 3. (a) (b) Fig. 3 the layout of two fields tested; (a) N-S cornfield (b) radial stagger On a Intel(R) Core(TM)i3 M380.53GHz laptop with GB Ra and using a 5x5 grid, the instantaneous results for each heliostats in the field were obtained in a run tie of less than 3.5 seconds for all heliostats, table 3. The distribution of the yearly average blocking factor in the both fields are shown in Figure 4. The figure shows the blocking factor would be in a better state if it was in the situation of radial stagger arrangeents. We can notice decrease in the value of blocking factor as long as we go towards the north away fro the tower. This can be explained by increasing the tilt angle of the heliostat toward the sae direction of the north that goes hand in hand with increasing in the nuber of the heliostats in the first rows which leads to a greater chance of occurrence of blockage loss. Table the results for both fields at solar noon on February 17. Ite cornfield radial stagger No. of heliostats Instantaneous average field blocking factor Run tie (seconds) To test the effect of the grid size results for the annual average of a heliostat blocking factor were obtained for a field containing 99 heliostats in N-S cornfield layout. For grid sizes fro 10x10 to 00x00 the value of the annual average of the heliostat blocking factor, to two significant digits, was 0.83 in all cases except the first case, table 3. We found that only very arginal changes in the value of the blocking factor occur with grids finer than 50x50. Clearly the user has to carefully select the grid size in order to balance the requireents of speed and accuracy.

9 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) (a) (b) Fig. 4. distribution of the yearly average blocking factor in the field; (a) N-S cornfield (b) radial stagger Table 3 the effect of the grid size on run tie and annual average heliostat blocking factor Grid size Run tie (s) Yearly average F b for Heliostat (0,170,5) 10x x x x x Figure 5 shows the relationship between the heliostats nuber and CPU tie required to calculate the annual average blocking factor for all heliostats in the N-S cornfield layout for grid sizes 5x5. Fig. 5 the relationship between the heliostats nuber and CPU tie

10 300 Mustafa E. Elayeb et al. / Energy Procedia 57 ( 014 ) Conclusions We proposed a ethod that allows fast calculation of the blocking factor in heliostat fields. The ethod is not restricted to any particular arrangeent or orientation of heliostats. Although the ethod is nuerical in nature, it sees that accurate results are obtainable with little effort. Further coparisons of the proposed ethod with existing ethods are planned. The proposed ethod relies on tracing the virtual shadows of points on the perieter of the heliostat so that straight lines enclosing the area of the virtual shadow can connect the. The ethod can also be applied to circular heliostats or facets by replacing these by equivalent circuscribing polygons. Obviously, as one increases the nuber of the polygon sides ore accurate results are obtained. References [1] Falcone P.K., A Handbook for Solar Central Receiver Design, Sandia National Laboratories,1986. [] Collado F.J., Turegano J.A., Calculation of the annual theral energy supplied by heliostat field, Solar Energy, 4(1989), p [3] F. Biggs, C. N. Vittitoe, (1979). The Helios Model for The Optical Behavior of Reflecting Solar Concentrators: SANDIA National Laboratories. [4] G. Sassi, Soe notes on shadow and blockage effects, Solar Energy, 31(1983) [5] M.M. Elsayed, M.B. Habeebuallah, O.M. Al-Rabghi, Yearly-Averaged Daily Usefulness Efficiency of Heliostat Surfaces, Solar Energy, 49(199) [6] Marcelino Sànchez, Manuel Roero. Methodology for generation of heliostat field layout in central receiver systes based on yearly noralized energy surfaces, Solar Energy, 80 (006) [7] Siala F.M, Elayeb M.E., Calculation of the shading Factor in Heliostat Fields, EuroSun01, Rijeka, Croatia, 18-0 Septeber 01. [8] Siala F.M, Elayeb M.E., Matheatical forulation of a graphical ethodfor a no-blocking heliostat field layout, Renewable Energy, 3(001),p 77-9.