Geometrical Aspects of the Optics of Linear Fresnel Concentrators: A Review
Abstract
:1. Introduction
2. System Description
2.1. A Semantic Clarification
2.2. Geometrical Description of the Mirror Field
- is the standard Azimuth angle of the local frame x-axis;
- is the inclination of the focal line w.r.t. the horizontal plane, considered positive if the local x-axis points downwards;
- is obtainable from the inclination of the local frame y-axis w.r.t. the horizontal plane (, considered positive if local y-axis points upwards) by the relation .
- all the mirrors have a common aiming line;
- the mirror axes lie on a common plane;
- each mirror is mounted symmetrically w.r.t. its axis and rotates around its central point.
2.3. Incidence Angles and Tracking
- the formula is universal, in the sense that it is correct everywhere is well defined (i.e., when ), and it gives correct angles and signs also when . But if one limits the considered cases to , the simpler formula (without the sign function) can be used:
- formulas for , both in (6) and in (7) versions are not fully explicit, since must be computed before . Fully explicit formulas for from can be obtained substituting with (note that the sign is always correct, since ). Note however, that and are usually computed together, so (6) (or (7)) can be applied.
2.4. Receiver
- A tubular receiver without a secondary reflector: the target is the tube itself, which presents the same section view from every direction. Such a configuration favors the mirror farther from the axis, allowing a higher width ratio for the plant, but the absence of a secondary is a strong disadvantage for tracking.
- A triangular receiver [6], in which the mirrors with aim at an effective target that is flat and inclined to face more favorably this half of the mirror field, and the opposite happens for mirrors with . In fact, from the optical point of view, the two halves of the mirror field can be considered as two independent LFRs, reducing the analysis to the next case;
- A more general case with a flat, but inclined effective target, e.g., for EW-oriented plants: triangular receivers are then just a couple of symmetrical, non-superimposing plants of this kind;
- A flat vertical target, such as the vertical rack of Dewar tubes discussed in [14], can be considered a degenerate triangular receiver.
3. Methodology for Ray-Tracing Simulations
3.1. Receiver Shadow
3.2. Selecting Which Mirror Is Hit, and Computing the Incidence Point
3.3. Reflected Ray
3.4. Is the Reflected Ray Blocked?
3.5. Does the Ray Hit the Receiver?
3.6. Sum of the Contribution of All the Rays
- the reflectivity of the mirror it hits;
- the cosine factor.
- In the plane, choose a horizontal segment of length L above the receiver (), making sure that the radiation coming from this region will “cover” entirely the mirror field;
- From this segment, shoot a sample of rays with a uniform distribution of the starting coordinate y (the starting x coordinate can be set to 0), and with an angular distribution of the rays that reproduces the solar divergence around according to the chosen sun profile. Both the starting points and the directions can be chosen randomly (Monte Carlo ray tracing) or distributed on a uniform grid. The first option is safer (no systematic errors due to the discretization) but significantly slower;
- Shoot each ray of the sample, as illustrated: if it hits the receiver, add the quantity to the “absorbed” energy;
- Normalize the absorbed energy multiplying by the quantity : here is the local Zenith of the sun, that is, the incidence angle of the radiation on the mirror plane, and it accounts for the cosine factor. The following relation holds:The result is the power that enters the receiver per length unit, for a unitary DNI. The procedure actually involves an approximation, since the rays coming from within the sun radiation cone are considered equivalent, neglecting the cosine effect of peripherical rays when they contribute to the DNI; it is clearly a negligible effect (the cosine of the solar angular size differs by about from 1).
4. Quasi-Analytical Analysis
4.1. Collimated Radiation
- are shadowed by the receiver;
- are shadowed by other mirrors;
- reflect radiation that is blocked by adjacent mirrors;
- reflect radiation that does not hit the target.
4.1.1. Receiver Shadow
- or : the mirror n is not shadowed by the receiver.
- and : the mirror region from to is shadowed.
- and : the mirror region from to is shadowed.
- : the mirror region from to is shadowed.
- and : the entire mirror is shadowed (and the analysis for mirror n can terminate here).
4.1.2. Mirror Shadowing
4.1.3. Blocking
- First check: if , the mirror cannot suffer from blocking and the analysis stops here.
- Second check: if the line connecting the rightmost point of the mirror to the rightmost point of the receiver is not blocked by mirror , the mirror cannot suffer from blocking (rays directed to the target cannot be blocked). The line has equation
- Third check: if the rightmost ray reflected from mirror n is not blocked by mirror , the mirror cannot suffer blocking (rays reflected by mirror n cannot be blocked). The local coordinate on mirror of the intersection isIf blocking is not present.
- First check: if , no blocking.
- Second check: if the line connecting the leftmost point of the mirror to the leftmost point of the receiver is not blocked by mirror , the mirror cannot suffer from blocking. The to consider is
- Third check: if the leftmost ray reflected from mirror n is not blocked by mirror , the mirror cannot suffer from blocking. The local coordinate of intersection is
- If none of the above conditions is verified, the inactive part of mirror n can be obtained from the equation
4.1.4. In-Target Reflected Beam
- find all the solutions of both equations , in the interval ;
- if k solutions are found, put the solutions in order on the interval , dividing the interval in segments;
- for each of the segments, compute a probing value with belonging to the segment, in order to check if the segment is active (the ray hits the receiver) or not (the ray does not hit the receiver).
4.1.5. Sum of the Contributions of All Mirrors
4.2. Solar Divergence, Slope and Tracking Errors
4.2.1. Pillbox Sun
- Obtain the tracking angles for each mirror using (8);
- Compute the M integration nodes of (50) and the corresponding values of ;
- For each , compute the function applying the procedure described in this section, i.e., compute the active part for each mirror n, and sum the contribution of all the mirrors applying (47);
4.2.2. Tracking and Slope Errors
- Sampling of the : repeat the computation for a perfect system on a suitable large sample of different sets of , introducing a random deviation of the angles. The set of must be chosen with a standard deviation that is the combined standard deviation of transversal slope errors and random tracking errors. Unfortunately, a quite large sample is required to obtain accurate results.
- Blurred mirrors: perform the computation as for a perfect system, but when computing for the mirror n takes the average on a Gaussian distribution of , with a standard deviation corresponding to the combination of random tracking and transversal slope errors.
Sampling of
Blurred Mirrors
5. Comparison and Possible Integrations between Ray-Tracing and Quasi-Analytical Methods
- The projection of the sun profile on (i.e., the approximation within the solar disk), and the following application of (49) and of the (50) quadrature rule, can also be adopted in ray-tracing, significantly reducing the number of rays that are necessary to reproduce the solar divergence (of course, the analysis of longitudinal slope errors is not possible in this case);
- Relations for the inactive part of a mirror can be used to exclude from ray-tracing analysis the rays that fall within these parts (when computing inactive parts, the extremal rays of the solar cone must be considered).
6. End Effects
- a quick estimation, which is useful to correct the radiation collected by a receiver string by subtracting end effects, but does not supply information on the radiation collected in a given receiver section;
- a method for integrating the end effects analysis in the ray-tracing approach, slightly modifying the procedure described in Section 3;
- a method for integrating the end effects analysis in the quasi-analytical approach presented in Section 4, modifying the computation of for collimated radiation.
6.1. Quick Estimation
- all the rays have the same ;
- for a mirror n, the “representative ray” directed from , to , (from the center of the mirror to the center of the receiver) is chosen to compute the x shift of the radiation;
- mirror defects are neglected.
6.2. End Effects in Ray-Tracing
6.3. End Effects in the Quasi-Analytical Method
- Find the maximum distance , in the section , between a point of mirror n and a point of the receiver (, );
- Find the minimum distance , in the section , between a point of mirror n and a point of the receiver (, );
- Case : if , then mirror n is free from end effects;
- Case : if or , then mirror n is completely inactive due to end effects;
- Case : if , then mirror n is free from end effects;
- Case : if or , then mirror n is completely inactive due to end effects.
- and : the minimum distance is between the right edge of the receiver and the left edge of the mirror;
- and : the minimum distance is between the right edge of the receiver and the left edge of the mirror;
- and : the minimum distance is between the left edge of the receiver and the right edge of the mirror;
- and : the minimum distance is between the left edge of the receiver and the right edge of the mirror.
- Compute the two projections of the two edges of the receiver (at ) on the plane containing the mirror n.
- If lies within the mirror width, compute the distance between the right edge of the receiver () and : this is a possible minimum distance.
- If lies within the mirror width, compute the distance between the left edge of the receiver () and : this is a possible minimum distance.
- Compute the two projections of the two edges of the mirror n on the plane containing the receiver (these will simply be , ).
- If lies within the receiver width, compute the distance between the right edge of the mirror and (that is, ): this is a possible minimum distance.
- If lies within the receiver width, compute the distance between the left edge of the mirror () and (that is, ): this is a possible minimum distance.
- Compute the four distances between the two edges of the receiver and the two edges of the mirror (left-left, left-right, right-left, right-right): these are four possible minimum distances.
- Among the possible minimum distances found so far (of which there will be a number between 4 and 8), choose the shortest one.
7. Examples of Simulations
8. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations and Symbols
CFD | Computational Fluid Dynamics |
CPV | Concentrating Photovoltaic |
DNI | Direct Normal Irradiation |
GAX | Generator-absorber heat exchange |
IAM | Incidence Angle Modifier |
LFR | Linear Fresnel concentrator |
PTC | Parabolic Trough Collector |
PV | Photovoltaic |
PV/T | Photovoltaic/Thermal |
SEV | South-East-Vertical reference frame |
a | Sun angular radius |
Width of the active part of the nth mirror | |
, | Min./max. distance between the receiver and the mirror n, in projection |
Local coordinate on the nth mirror in section | |
of projection of receiver shadow | |
of intersection for blocking | |
of intersection between a ray and a mirror | |
of projection of other mirrors’ shadow | |
Inverse of focal length of the nth mirror | |
H | Height of the LFR receiver (in local frame) |
Incident ray versor | |
Collected power for length unit, for collimated radiation | |
Semi-width of the effective receiver aperture | |
Semi-width of the effective receiver shadowing plane | |
L | Width of the origin of rays for ray-tracing |
M | Number of integration nodes |
N | Number of primary mirrors |
n | Progressive number identifying a mirror () |
Versor orthogonal to the mirror | |
x component of | |
Number of shot rays in ray-tracing | |
Projections of the two edges of the receiver () on the plane containing a mirror | |
Weights associated to integration nodes | |
Projections of the two edges of a mirror on the plane | |
Reflected ray versor | |
x component of | |
y component of | |
z component of | |
Correction factor to a mirror contribution for end effects | |
Global correction factor for end effects, for all the mirror field (i: different approximations) | |
Initial fraction of inactive mirror for end effects | |
Final fraction of inactive mirror for end effects | |
Standard integration nodes | |
Semi-width of the nth mirror | |
Starting x of a ray in ray-tracing | |
x coord. of a section of the receiver (when considering end effects) | |
x coord. at the receiver of a ray reflected from | |
x coord. at the receiver of a ray reflected from | |
, | The mirror field extends in x direction between the two |
, | The receiver extends in x direction between the two |
y coord. of projection of receiver shadow | |
Starting y of a ray in ray-tracing | |
y of intersection for blocking | |
y of intersection with the receiver plane of a ray reflected at | |
y of intersection between a ray and a mirror | |
y coord. of the axis of the nth mirror | |
y of intercept between a ray and the shadowing plane | |
y of intercept between a reflected ray and the receiver | |
Starting z of a ray in ray-tracing | |
z of intersection between a ray and a mirror | |
Transversal incidence angle | |
of the Sun center | |
Integration nodes in | |
Local Azimuth (in frame) | |
Global Azimuth (in SEV frame) | |
Longitudinal incidence angle, or simply incidence angle | |
of the Sun center | |
Random deviation of due to tracking error | |
Longitudinal incidence angle (alternative definition) | |
of the Sun center | |
Optical efficiency of the receiver | |
Auxiliary angle (see Figure 1) | |
Auxiliary angle | |
Mirror reflectivity | |
Standard deviation of slope + tracking errors | |
Local Zenith (in frame) | |
Local Zenith of the Sun center | |
Global Zenith (in SEV frame) | |
Inclination of the nth mirror | |
Incidence angle on the plane of mirror n | |
Azimuth of the focal line in SEV frame | |
Inclination of the focal line w.r.t horizontal plane | |
Third rotation angle | |
Inclination of local y-axis w.r.t. horizontal plane |
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Strategy | = 0 mrad | = 2 mrad | = 5 mrad |
---|---|---|---|
Quasi-analytical (no errors) | 1 | — | — |
Ray-tracing | 4104 | 14,693 | 31,269 |
Quasi-analytical ( sampling) | — | 6144 | 10,614 |
Quasi-analytical (blurred mirrors) | — | 11.1 | 8.5 |
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Grena, R. Geometrical Aspects of the Optics of Linear Fresnel Concentrators: A Review. Energies 2024, 17, 3564. https://doi.org/10.3390/en17143564
Grena R. Geometrical Aspects of the Optics of Linear Fresnel Concentrators: A Review. Energies. 2024; 17(14):3564. https://doi.org/10.3390/en17143564
Chicago/Turabian StyleGrena, Roberto. 2024. "Geometrical Aspects of the Optics of Linear Fresnel Concentrators: A Review" Energies 17, no. 14: 3564. https://doi.org/10.3390/en17143564