Piecewise-focusing collectors reduce the mirror area in concentrating solar thermal power plants

Piecewise-focusing (PWF) collectors replace the heliostat ﬁeld and central tower receiver of Gen 2 concentrating solar thermal (CST) power plants. The PWF collector consists of a large base frame supporting a tiered array of reﬂectors that concentrate sunlight into a cavity receiver. The base frame rotates about a central pivot to track the sun in azimuth, while the reﬂectors rotate about individual nearly-horizontal inclined axes to track the sun in elevation. Compressed gas such as CO 2 is used as the heat transfer ﬂuid (HTF) from the receiver to a conventional molten salt thermal reservoir, which supplies heat for a steam turbine power block. Physical and numerical modelling of an approximately paraboloidal PWF collector shows that about 250 separate reﬂectors will be required for adequate focus on the entrance to a cavity receiver at concentration ratio 2000 with little spillage at all sun elevations. With an area 10-20 times larger than the cavity entrance, the absorber functions at peak temperatures that are within range of standard stainless steel tubing, despite using a gaseous HTF. Reﬂectors of PWF collectors have signiﬁcantly better cosine factors than heliostats, and cavity receivers lose much less heat than open cylindrical receivers, so that reﬂector area is reduced to about 62% of heliostat area. Thus the capital cost of a CST plant can be reduced substantially. By continuing to use Gen 2 technology for thermal storage and the power block, PWF-based CST can be developed much sooner than ‘Gen 3’ CST.

of the wagons around their tracks with equal angular velocity is also required, but the cost of doing so is counterbalanced by the savings in single-axis elevation tracking. The orientation of the inclined mounting axis for each reflector depends on its position in the moving field, as does the angle between the reflector and its mounting axis. With this mounting scheme, the aim of the reflector towards the receiver varies slightly with sun elevation and is only perfect at three arbitrarily-chosen elevations, but aiming errors at other elevations are small, less than 1'15" (0.36 mrad) for the conditions analyzed by Baum et al [9]. A 1:50 scale model of part of the proposed design was constructed [9]. To the author's knowledge, no subsequent moving-field design (apart from the present proposal) adopted the single-inclined-axis mounting scheme for reflectors.
Various other designs of moving field collectors were proposed subsequently. For example, Jones [11] designed a platform supported on a large number of rollers to replace the railway wagons, optionally running on an amphitheaterlike embankment. Ruiz et al [12] constructed a version with trolleys that support large conventional heliostats on several concentric tracks at the CTAER in Spain. However, the direct benefit of a moving field is only modest -Ruiz et al [12] found that annual heat collection increased by 8-10% for a certain polar-type heliostat layout -and it is debatable whether this justifies the increased cost and complication if dual-axis reflectors are still used. Similarly, the Jones [11] design was not adopted when compared with standard heliostats in a 2007 cost reduction study [13]. As for a conventional heliostat field, the reflectors in a large moving field at ground level must be widely spaced in order to reduce mutual shading and blocking. elevation azimuth Fig. 1. Concept sketch of a PWF collector [14]. The entire collector tracks the sun in azimuth. Each reflector (mirror unit) tracks sun elevation independently by rotating about a fixed inclined axis.

The piecewise-focusing collector
The piecewise-focusing (PWF) collector was conceived as a large collector divided into independent reflectors, as sketched in Fig. 1, with separation of scales for the rotational axes. A lightweight base-frame provides azimuthal motion and supports a tiered array of a hundred or more slightly curved reflectors, each mounted on its own single, nearly-horizontal inclined axis that allows sun-tracking in elevation. Compared to a moving field at ground level, the tiered PWF collector allows much closer spacing of the reflectors, which can cover a high percentage of the gross aperture area with little shading or blocking. However, the largest practicable size for a PWF collector is likely to be smaller than could be achieved with a ground-level moving field. The concept of using a single inclined axis for each reflector was developed independently by the present author [14], and the method for determining axis inclinations is slightly different from that of Baum et al [9], but the results appear to be equivalent.

Determination of reflector-axis mounting angles
Each reflector (mirror unit) rotates about a single axis fixed to the base frame. For a given reflector, vector a, the direction of its axis relative to the base frame, and the angle α between the axis and the plane tangent to the center of the reflector (Fig.   2), are to be determined. Note that all vectors in the following are of unit length, and therefore only two components of a are independent -there are three unknowns including α.
Coordinate axes are fixed to the base frame at the center of the given reflector. Let r and r be respectively the directions from sun to reflector and from reflector to receiver (Fig. 2); r is fixed, and because of the base frame's azimuthal rotation r depends only on sun elevation. Let n be the inwards normal to the reflector; then for specular reflection it is required that Also, since α is fixed for a given reflector, where C is a constant, or, Select three values of sun elevation, giving three numerically different versions of this equation, and solve simultaneously for the three Cartesian components of (a/C). Finally, The result is a reflector with perfect aim from its center at any three chosen sun elevations, and small aiming errors at other elevations.

The 'segmented dish' as an example PWF collector
In terms of optical performance, the ideal model for a PWF collector is a paraboloidal dish at a fixed angle of tilt, divided into reflectors that cover nearly 100% of the putative dish surface. However, other shapes may be more practical for economical construction, and require only small increases in the spacing between reflectors to avoid shading and blocking.
The following analysis assumes a 'segmented dish' as shown in Fig. 3. It is based on a paraboloidal surface that has a rim angle of 22.5 • (the cone of light reflected to the focus has a vertex angle of 90 • ), and a fixed tilt towards the sun of 30 • . The surface is divided into 80 slightly curved rectangular reflectors, each fixed at a specific angle to its axis of rotation, and that axis is mounted at a specific angle to the base frame (determined as above). Each reflector is controlled by its own independent tilting mechanism (for example, a linear actuator). A planar target, normal to the segmented dish axis and representing the entrance to a cavity receiver, is fixed at the focus. Figure 4 shows a simplified 3D model of the right-hand half of the segmented dish using a smaller number of larger reflectors, with the latter tilted as required at three different sun elevations. The base frame and axis mountings are very stylized in Fig. 4, but the mounting angles and angles of tilt are approximately correct for each reflector. It may be inferred that mutual shading between reflectors is not significant unless the sun is near the horizon.
Numerical models for 32 out of the 80 reflectors ( Fig. 3) were set up [14,15], modelling the reflectors indicated in Fig.   3(b). A physical model of 3.6 m aperture diameter incorporating 10 of those reflectors was constructed [14], and was used  to explore PWF ideas and qualitatively verify numerical results. Since the reflectors are large enough to cause astigmatism of their reflected light as the sun elevation varies, a more accurate, larger-scale model of one reflector was constructed [15] for quantitative comparison with the numerical model, particularly at the corners of the rectangular reflector. Astigmatism results are given in Secs. 4 and 5, but an idea of mirror unit mounting angles can be had from the photographs (Fig. 6) of the larger-scale physical model.

Testing mirror aim
Aiming errors from the centers of several reflectors were examined in the independent numerical model. For a given sun elevation, a specified reflector is rotated about its axis by very small steps in the model until the aiming error of its reflected ray reaches a minimum, which is reported in milliradians. Then the reflected ray is intersected with the target plane and the 2D error on the target is reported in millimetres. Since reflector axes are nearly horizontal, the reflected image on the target 6 Copyright c by ASME  and it was also shown in that paper that small construction errors in reflector mounting would not distort reflector aim unacceptably.

Aiming errors from the corners of a reflector
As noted earlier, the relatively large size of reflectors is potentially a problem for collector focus at different sun elevations. Aims from the corners of a reflector deviate much more than the aim from its center, causing astigmatism and possibly spillage at low and high sun elevations. The PWF segmented dish used for this work (Fig. 3) has only 80 reflectors covering its aperture, whereas the moving field designs mentioned earlier contain much larger numbers of relatively smaller reflectors: 624 [12], 1293 [9], or 2200 [11]. Before the numerical model was used to investigate astigmatism, it was tested against a physical model of the reflector indicated by five dots in Fig. 3 Corner tile aims were set to coincide with the center when the sun was at 45 • elevation. An example of reflections on the target (which has a 50 mm square grid) from the center tile and one corner is given in Fig. 6 (sun elevation = 45 • ). It can be seen that each patch of light spreads considerably because of the finite size of the sun, and therefore it is difficult to shown. An unquantifiable amount of slack was present in the mirror tile aim adjustment screws, which combined with the imprecision in picking the center of each patch of light on the target to add a bit of randomness to the results. Nevertheless, the overall agreement between physical and numerical results is sufficient to give confidence in the numerical modelling that follows.

Images formed on the target plane
The 32 reflectors colored grey in Fig. 3(b) were added to the numerical model, and intersection points on the target plane were calculated for the centers and corners of all 32 reflectors. As for the single reflector above, the PWF collector aperture diameter was 6000 mm, reflector axes were determined using sun elevations of (15 • ,45 • ,75 • ) and the corner aims were set with the sun at 45 • . In Fig. 8 the intersection points are combined into images (focus maps) on the target plane that represents the entrance to a cavity receiver, at a range of sun elevations from 10 • to 80 • above the horizon.  give good results. This analysis applies to the 'segmented dish' of geometry shown in Fig. 3, and it would have to be re-done for any other tiered shape of PWF collector. Also, the practicalities of manufacturing the reflectors must be considered. For example, it may be more cost-effective to approximate the ideal nearly-paraboloidal surface shape with sections of spheres of a small number of different radii, blurring the focus a little further than shown in Fig. 8.
If the ideal size of the PWF 'segmented dish' is an order of magnitude larger than for a conventional paraboloidal dish (see discussion in [14]), e.g. 5000 m 2 , then 250 reflectors would have an area of 20 m 2 each, which is well within the range of sizes for current heliostat designs [16] and likely to be quite economical to produce. At 5000 m 2 aperture area the aperture radius and maximum rim height would both be about 40 m, and average thermal output would be about 4 MW.

Receiver design for PWF collectors
The assessment of collector optical performance in the sections above depended on the size and shape of the receiver to be used, and as noted, a big advantage of the PWF concept is that its converging cone of reflected light is suitable for a cavity receiver. Efficiency of a cavity receiver is likely to be considerably higher than that of an open cylindrical receiver as used for Gen 2 CST (see below). A true cavity receiver has a small opening through which concentrated sunlight passes and then fans out to heat a much larger internal area of absorber surface, of order ten times larger than the opening (or more). If the exterior is well insulated, heat is lost only from the opening, by blackbody radiation and by mixed convection (passive and wind-driven).
As noted by Mehos et al [1], Sedighi et al [17], and others, compressed gases have many advantages as the heat transfer fluid from absorber to heat store. Various common gases are non-toxic, non-flammable, inexpensive, non-condensing and non-freezing, and they allow fast startup after sunrise. Compressed gases have the further advantage over molten salt of simplicity of operation in a distributed network of pipes serving multiple PWF collectors, since trace heating is not required and thermal inertia is low. The big disadvantage of compressed gases is their low forced-convection heat transfer coefficients.
However, heat transfer to a gas can be improved by a combination of (a) large temperature differential from the walls of the absorber, (b) high system pressure to increase gas density, (c) high gas velocity (limited by pumping power losses), and (d) large absorber area. For a steam-Rankine power block with inlet temperature 565 C and heat store (molten salt) at 585 C, the required gas temperature at the receiver outlet will be around 620 C. The absorber surface could be roughly 70 C hotter (inferred from   [20]. Thus the cavity receiver not only facilitates the use of a compressed gas heat transfer fluid, it also gains efficiency quite significantly compared to a Gen 2 molten salt receiver. Several compressed gases were studied as potential heat transfer fluids (HTF), and various gas mixtures also show promise, but the aim in this section is to show that the assumed absorber conditions are feasible with readily available, moderate-cost materials, namely compressed CO 2 and standard sizes of 316 stainless steel tubing [21]. The design concept is that the illuminated wall areas of the cavity are lined with a number of tubes carrying the gas in parallel, with small gaps between them, and following roughly helical paths. Connection to some form of inlet and outlet manifolds will be a crucial design detail, not considered here. Sunlight flux density entering the receiver was taken as 1600 kW/m 2 and thus 80 or 160 kW/m 2 at the absorber. Steel properties used [22] were conductivity 21.4 W/(m·K) and yield stress (at 650 C) 145 MPa; however, rupture stress after 100K hours at 650 C is 94 MPa. Unknown levels of stress caused by thermal gradients and absorber mountings will need to be added to the calculated hoop stresses. The heated area for ∆T calculation was taken conservatively as tube length times tube spacing (instead of half the circumferential area). Inlet and outlet temperatures were 320 C and 620 C, and CO 2 properties were evaluated at 500 C only (these calculations are a first approximation).
The Dittus-Boelter equation [23] was used for Nusselt number, and the Petukhov friction factor equation [23] was used for Some results are given in Table 1 Ideally, temperature differences, circulation power, mass of steel, and complexity (the number of tubes in parallel), would all be kept very low, but in practice there are compromises. Taking the design in column 1 as a base case, increasing the CR (column 2) halves the mass of steel but increases both peak temperature and parasitic power. Increasing the CO 2 pressure (column 3) reduces peak temperature but requires even more steel than in the base case. Combining higher CR and pressure (column 4) gives the best results, subject to further optimization. The design in the last column reduces the mass of steel even further and has smaller temperature differences, at the expense of much greater complexity. In any case, it is clear that a workable high-efficiency cavity receiver can be designed. Also, peak temperature in the tubing can be reduced by using non-uniform sunlight flux density onto the absorber. Another design possibility is to use an absorber based on compact heat exchanger technology [24]. Detailed thermal modelling of cavity receivers will be incorporated into NREL's System Advisor Model (SAM) software [25], which may facilitate the design of the absorber, especially when the PWF collector is not radially symmetrical.

Discussion
The analysis so far has shown the technical feasibility of PWF collectors, so the next step is to consider how their use would affect the cost of a CST power plant. In a Gen 2 power tower plant with a solar multiple of 2.1, the heliostats and receiver made up nearly 60% of the total cost [26], so there is scope for significant cost reduction if they can be replaced effectively. This can be achieved by reducing the area of reflector required through increased efficiency, and by reducing the cost per m 2 of reflector. Some aspects of collector efficiency are set out in Table 2 in terms of their effect on the area of PWF reflectors relative to heliostat area. Other aspects such as mirror reflectance and cleanliness are likely to be very similar for the two technologies. The most important benefit stems from the way the PWF reflectors are comparatively 'square-on' to the sun throughout the day, leading to a better average cosine factor, but the cavity receiver benefits are also important. Note that about two hours elapse after sunrise before a molten salt power tower reaches stable operation [19], whereas the PWF collectors can be brought into operation quite quickly. The product of the five area ratios in the table is 0.62, i.e. the area of reflectors in PWF collectors will be about 62% of the power tower heliostat total area for equal heat collection.
Comparative costing of a PWF collector per m 2 of reflector is not easy at this stage, but some influential factors are noted here. The reflector surface itself is similar to that of a heliostat, but the heliostat pedestal and two-axis precision gear drives can be replaced by simple bearing mounts and one linear actuator. The eliminated gear drives, pedestal and its foundation comprise about 50% of the cost of a large conventional heliostat (Table 3-14 in [13]). On the other hand, the rotating base frame is a necessary new element, but this can be a lightweight steel space-frame anchored to a central pivot point and travelling on a simple horizontal track; it is unlikely to be very expensive per m 2 of reflector. For the cavity receiver, the absorber will be around three times larger in area and require external insulation, but the receiver will not need any of the ancillary reservoirs, drain-back system or trace heating needed for a molten-salt central receiver. Thus the total cost of a PWF collector per m 2 of reflector is likely to be comparable to (but not greater than) that of a central receiver with heliostat surround field. The most significant saving in the cost of a CST power plant arises from from the much smaller area ratio (around 62%).
The fast-start benefit of a compressed-gas receiver was mentioned above, but there are other operational advantages too.
Cylindrical central receivers in a surround field of heliostats are heated differentially throughout the day -hotter on the western side in the morning, and on the eastern side in the afternoon -whereas PWF collectors rotate to follow the sun.
Also, variable cloud cover over part(s) of the heliostat field results in highly uneven heating of the central receiver, making stable operation difficult or impossible [19]. Multiple PWF collectors will be required for a power station of more than a few MW, so those that are not directly affected by a cloud transient will continue normal operation, and the gaseous HTF allows affected collectors to recover quickly. Note also that PWF collectors operating at higher temperatures could be used to charge solid particle thermal storage instead of molten salt, informed by designs for particle-to-sCO 2 heat exchangers [6] but working in reverse (sCO 2 -to-particle heat transfer).
In the ten years since the Gen 2 CST cost reduction study [26] the cost of photovoltaic (PV) panels has dropped most remarkably, and hundreds of GW of PV panels have been deployed in many regions at both rooftop and utility scale. This has led to the so-called 'duck curve' of electricity spot prices, where the price is depressed around the middle of the day but rises rapidly in the late afternoon. Using the Australian National Energy Market as an example, the twelve-month average spot price per MWh for 2019-2020 was A$25 to A$50 for several hours around midday, and rose to about A$180 at 6 pm ( Fig. 1.3 of [27]). CST power plants with storage are well-placed to supply electricity in such situations when demand and prices are high, and in an extreme case the power block would operate for only a few hours per day. The solar multiple would be less than 1.0, reducing the total cost of plant per MW but de-emphasising the proportional cost of the collectors.
However, CST base-load power plants may have a future role as coal-burning power plants are retired, and the power blocks will be designed for continuous rather than intermittent operation. The solar multiple will be nearer 3.0, increasing the value of lower cost, high performance collector fields such as PWF designs. Requirements for island grids and remote sites are similar.
Note: This research did not receive any funding from public, not-for-profit, or commercial agencies.