An analytic evaluation of the flux density due to sunlight reflected from a flat mirror having a polygonal boundary

Computer algorithms for the flux density of reflected sunlight from a heliostat become an essential part of the optical simulation problem for the central receiver system. An exact analytic result is available for heliostats having polygonal boundaries. An analytical method for round heliostats is given in Appendix A, which is extremely complex and requires quartic roots. A useful numerical method is given in Appendix B for heliostats of arbitrary shape. A comparison is made between the analytic method and the Hermite function method, which is much faster but less accurate. The analytic method provides a basis for evaluating all other flux density calculations.     

A numerical approach to the flux density integral for reflected sunlight

A computer model of the central receiver system must evaluate the flux density on the receiver due to sunlight reflected by the heliostats in the collector field. Several approaches are available but each has its limitations. The Monte-Carlo approach represents all of the heliostat behavior but is relatively slow in terms of CPU time and is not suitable for optimization purposes. FLASH is an analytically exact approach for flat polygonal heliostats but is slow and not applicable to dished heliostats or aureole effects. Cone optics programs evaluate the flux density by a direct numerical integration of the double integral, but this method is very slow if accuracy is required. HCOEF is a two dimensional Hermite polynomial method which is relatively fast and can be extended to include canting, focusing, solar limb, and guidance error effects. However, the polynomial approximation breaks down for near heliostats, small guidance errors, and aureole effects. The new image generators based on KGEN overcome this limitation, but running times compare to FLASH and are 3 or 4 slower than HCOEF.The new approach proposed in this study assumes isotropic gaussian guidance errors. Hence, the flux density integral reduces to several iterated single integrals which can be precalculated and stored in a table for interpolation as needed. The LBL solar telescope data are fed into a convolution integral which represents the guidance errors. Aureole effects can be switched on or off at this point. A vector of convoluted solar data is input to another integration which gives the table of normalized flux contributions. The tabular values depend on the position of the flux point with respect to an edge of the heliostat as seen in the image plane. The image map of the heliostat is linear unless ripples or irregularities occur; hence, effects due to canting and dishing can be included by a ray trace of the heliostat vertices.The use of tabular interpolation is not as fast as expected because of the time required to calculate the distance between the flux point and the image of the vertices. The accuracy of this method is limited by interpolation errors, and better results can be obtained with the same CPU time if more core is used for a larger table. It is possible to eliminate the table by introducing a Romberg type of integrator which bisects the interval until sufficient accuracy is achieved; however, this approach is inefficient unless the images are relatively small compared to the receiver.The convolution process in KGEN is fast and can be used to calculate moments for HCOEF and coefficients for FLASH which utilize the LBL data.     

Measurements of solar flux density distribution on a plane receiver due to a flat heliostat

An experimental facility is designed and manufactured to measure the solar flux density distribution on a central flat receiver due to a single flat heliostat. The tracking mechanism of the heliostat is controlled by two stepping motors, one for tilt angle control and the other for azimuth angle control. A x-y traversing mechanism is also designed and mounted on a vertical central receiver plane, where the solar flux density is to be measured. A miniature solar sensor is mounted on the platform of the traversing mechanism, where it is used to measure the solar flux density distribution on the receiver surface. The sensor is connected to a data acquisition card in a host computer. The two stepping motors of the heliostat tracking mechanism and the two stepping motors of the traversing mechanism are all connected to a controller card in the same host computer. A software “TOWER” is prepared to let the heliostat track the sun, move the platform of the traversing mechanism to the points of a preselected grid, and to measure the solar flux density distribution on the receiver plane. Measurements are carried out using rectangular flat mirrors of different dimensions at several distances from the central receiver. Two types of images were identified on the receiver plane—namely, apparent (or visible) and mirror-reflected radiation images. Comparison between measurements and a mathematical model validates the mathematical model.     

An analytical method for reflected flux density calculations

Conception, evaluation and real time control of solar “power tower” systems require the use of fast and accurate computer programs for calculating the flux density distributions on the receiver. Since the classical methods of “cone optics” and “hermite polynomial expansion” have some limitations of speed and accuracy, we have built an analytical model for calculating the convolution of the solar brightness distribution with the principal image of a heliostat (i.e. the fictive image for a “point sun”). We first characterize a principal image of a focusing heliostat by its shape and its geometrical concentration factor. Then this image is projected back onto the central plane (which passes through the center of the mirror), and considered as a flat reflecting surface. And the problem is reduced to density calculation for a flat heliostat. For each point of the receiver, the density of flux reflected by a heliostat is obtained by direct resolution of a convolution integral. The different formulations used to express the density function correspond to the various types of intersections between the image of the solar disk for the considered point and the principal image of the heliostat. Confrontation of this method with a program based on “cone optics” shows a good concordance of results and a strong decrease of computation time. We want to apply this method to the existing “THEMIS” solar plant built in France and to compare our results with real observations. Our density calculation programs will help conceiving fields of focusing heliostats for a new generation of power systems (gaz turbine systems).     

One-point fitting of the flux density produced by a heliostat

Accurate and simple models for the flux density reflected by an isolated heliostat should be one of the basic tools for the design and optimization of solar power tower systems. In this work, the ability and the accuracy of the Universidad de Zaragoza (UNIZAR) and the DLR (HFCAL) flux density models to fit actual energetic spots are checked against heliostat energetic images measured at Plataforma Solar de Almería (PSA). Both the fully analytic models are able to acceptably fit the spot with only one-point fitting, i.e., the measured maximum flux. As a practical validation of this one-point fitting, the intercept percentage of the measured images, i.e., the percentage of the energetic spot sent by the heliostat that gets the receiver surface, is compared with the intercept calculated through the UNIZAR and HFCAL models. As main conclusions, the UNIZAR and the HFCAL models could be quite appropriate tools for the design and optimization, provided the energetic images from the heliostats to be used in the collector field were previously analyzed. Also note that the HFCAL model is much simpler and slightly more accurate than the UNIZAR model.     

Four different views of the heliostat flux density integral

The image due to a single heliostat is represented by its flux density, which can be formulated as an integral over the solid angle of the incoming rays. The initial formulation is transformed into three alternative representations, each having some particular utility. The incoming ray formulation leads to analytic results for flat heliostats with polygonal boundaries. The mirror plane formulation leads to a numerical integration over the mirror plane which can be used to study effects due to distortions of the mirror. The pin-hole view leads to an approximate expression for the flux density integral as a convolution of the image due to a point Sun with respect to the brightness distribution of the real Sun. This formulation allows us to treat the Sun size as though it were a source of guidance errors or alternatively, we can introduce a degraded Sun which includes the guidance errors.     

A system with a tracking concentrating heliostat for lighting underground spaces with beams of sunlight

The results of the introduction of a solar-power installation for lighting and creating light effects in an underground room using mirror-concentrating systems are described.     

Solar flux distribution on central receivers: A projection method from analytic function

This paper presents a methodology to project the flux distribution from the image plane into the panels of any central receiver in Solar Power Tower plants. Since analytic functions derived from the convolution approach are conveniently defined on the image plane, its oblique projection solves the distorted spot found in actual receivers. Because of its accuracy describing the flux distribution due to rectangular focusing heliostats, we make use of the analytic function on the image plane by Collado et al. (1986). Based on the projection method, we have developed a computer code successfully confronted against PSA measurements and SolTrace software, either for flat plate or multi-panel cylindrical receivers. The validated model overcomes the computation time limitation associated to Monte Carlo technique, with a similar accuracy and even higher level of resolution. For each heliostat in a field, the spillage is computed besides the rest of optical losses; parallel projection is used for shading and blocking. The resulting optical performance tool generates the flux map caused by a whole field of heliostats. A multi-aiming strategy is investigated on the basis of the radius of the reflected beams, estimated from error cone angles.     

A solar flux density calculation for a solar tower concentrator using a two-dimensional hermite function expansion

The calculation of flux density on the central receiver due to a large number of flat polygonal reflectors having various orientations is a basic part of the system simulation problem for the tower concept of solar energy collection. A two-dimensional Hermite function expansion is adapted to the simulation problem, and numerical results are contrasted with an analytic integration of the solar flux density at specific nodes on an image plane. Various measures of error in the flux density calculation are monitored vs distance to the image plane and orientation of the reflector. The flux densities predicted by the statistical method compare favorably with those of the analytic model and require approximately one-tenth the computer time.     

An analytic study of boiling heat transfer from a fin

Analytic expressions for the one-dimensional temperature distribution in a pin fin or a straight fin of rectangular profile are derived if various types of boiling occur simultaneously at adjacent locations on such a fin's surface. The heat transfer coefficients for the transition and nucleate boiling are taken as being the power functions of the wall superheat and that for film boiling as being constant. The number of cases analysed is 66. Some of the results obtained are compared with those of experiments carried out elsewhere. A quite reasonable degree of agreement is found between the theory and the experiment carried out in practice.     

An artificial vision-based control system for automatic heliostat positioning offset correction in a central receiver solar power plant

This paper presents the development of a simplified and automatic heliostat positioning offset correction control system using artificial vision techniques and common CCD devices. The heliostats of a solar power plant reflect solar radiation onto a receiver (in this case, a volumetric receiver) placed at the top of a tower in order to provide a desired energy flux distribution correlated with the coolant flow (in this case air mass flow) through the receiver, usually in an open loop control configuration. There exist error sources that increase the complexity of the control system, some of which are systematic ones, mainly due to tolerances, wrong mirror facets alignment (optical errors), errors due to the approximations made when calculating the solar position, etc., that produce errors (offsets) in the heliostat orientation (aiming point). The approximation adopted in this paper is based on the use of a B/W CCD camera to correct these deviations in an automatic way imitating the same procedure followed by the operators. The obtained images are used to estimate the distance between the sunbeam centroid projected by the heliostats and a target placed on the tower, this distance thus is used for low accuracy offset correction purposes. Basic threshold-based image processing techniques are used for automatic correction.     

An analytic study of applying Miller cycle to reduce NOx emission from petrol engine

An analytic investigation of applying Miller cycle to reduce nitrogen oxides (NO_x) emissions from a petrol engine is carried out. The Miller cycle used in the investigation is a late intake valve closing version. A detailed thermodynamic analysis of the cycle is presented. A comparison of the characters of Miller cycle with Otto cycle is presented. From the results of thermodynamic analyses, it can be seen that the application of Miller cycle is able to reduce the compression pressure and temperature in the cylinder at the end of compression stroke. Therefore, it lowers down the combustion temperature and NO_x formation in engine cylinder. These results in a lower exhaust temperature and less NO_x emissions compared with that of Otto cycle. The analytic results also show that Miller cycle ratio is a main factor to influence the combustion temperature, and then the NO_x emissions and the exhaust temperature. The results from the analytic study are used to analyse and to compare with the previous experimental results. An empirical formula from the previous experimental results that showed the relation of NO_x emissions with the exhaust temperature at different engine speed is presented. The results from the study showed that the application of Miller cycle may reduce NO_x emissions from petrol engine.     

An approximate method for calculating the heat flux through a solar collector honeycomb

An approximate calculation of the radiative-conductive flux of heat from the absorber to the cover plate of a flat plate collector equipped with a convection-suppressing honeycomb is developed, using an exponential kernel approximation to the passage transmittance function of the honeycomb cells. Fluxes predicted by this method are found to be in satisfactory agreement with the results of previously reported calculations and experiments. If a selective absorber plate is used in a collector of this type, heat transport by the air in the collector eliminates the radiation slip between the honeycomb and the absorber plate, so that thermal emission by the walls of the honeycomb in close proximity to the hot absorber may largely cancel out the advantage gained by making the absorber plate selective.     

Modelling of the receiver transient flux distribution due to cloud passages on a solar tower thermal power plant

As more and more solar tower thermal power plants are being operated, built or planned, effort is put both on the development and research to bring costs down and increase the plant efficiency. In those plants, the central receiver is one of the key components, accounting for a large investment share. Receivers have to sustain strong thermal stresses caused by irradiation transients, mainly due to cloud passages. To avoid premature failures, increase the receiver cyclic life, and allow longer daily operation periods, an anticipation of the most likely or the worst situations is required. First the calculation of the receiver incident flux distribution is performed, second the cloud and cloud passage characteristics are identified for a given location, third the most likely case is simulated by covering and uncovering the heliostat field, then a worst case configuration is presented, and finally a strategy for the start-up/shut-down of the heliostats is proposed. The value of terms such as the heat flux peak, the maximal flux gradient, the fastest flux transient and total power transients are needed to choose the control strategies regarding heliostat orientation and the receiver operation, as well as the elimination of some bad plant layouts during the design phase.     

Performance of low cost solar reflectors for transferring sunlight to a distant collector

Efficiency of reflection and optical transmission to a distant collector is a critical parameter, along with cost per unit area, in the selection of a heliostat design for the Central Collector Solar Electric Plant. Efficient optical transmission is not easily accomplished because of the large distance to be spanned in a multi-MW facility. Depending on heliostat location, the transmission distance may vary from a few hundred to thousands of feet.Design conditions which influence optical transmission over these long distances are: heliostat pointing accuracy; spreading of the reflected solar beam due to the finite size of the Sun's image; beam spreading due to reflector misalignment or waviness; aberration present if curved heliostat reflectors are used and beam spreading due to microscopic irregularities (characteristic length less than 0.1 mm) in the reflective surface. These factors increase in importance as the transmission distance from heliostat to collector increases. Even the most preliminary heliostat design activity requires a detailed evaluation of beam spreading before the most cost effective heliostat concept, or family or concepts depending on transmission distance, can be defined.Data are presented here which will be of value in assessing one of the factors causing beam spreading. An experimental method has been utilized to determine beam spreading due to microscopic surface irregularities prevalent with “mill finished” materials. The test method provides a nearly independent measure of the effect of surface imperfections.Data are presented for five candidate materials and, as reference, an optical quality first surface mirror.     

Heuristic knowledge-based heliostat field control for the optimization of the temperature distribution in a volumetric receiver

The paper presents the development and implementation of a heuristic knowledge-based heliostat control strategy optimizing the temperature distribution within a volumetric receiver at the Plataforma Solar de Almería (PSA) power tower plant. The experience in operating the plant has been used in the development of an automatic control strategy that provides an appropriate flux distribution within the volumetric receiver in order to obtain a desired temperature profile, and allows for operation without a continuous intervention of the operator, which is one of the main characteristics and drawbacks in the exploitation of these kinds of plants. Experimental results are included and discussed in the paper.     

Estimation of exposure to sunlight of the liner under a tiled roof

One construction for a pitched roof is to use tiles on battens, with a liner attached below the battens. The shape of some types of tiles is such that, at each corner where four tiles overlap, a small gap is formed. At certain positions of the sun solar radiation can penetrate through these gaps and strike the liner, accelerating its degradation. The purpose of this study is to estimate the extent and duration of the exposure. A typical gap is modelled in a ray-tracing program and the size and position of the illuminated area on the liner is calculated for given directions of the light beam relative to the roof. Analytic expressions for the size of the illuminated area are obtained using a thick slit model. The accuracy of the model was assessed by some experimental measurements. The exposure over one year of the roof liner was calculated using the Design Reference Year for Copenhagen, Denmark. Simulations were carried out for a roof tilted at 25°, 35° or 45°, facing SE, S, SW or W. For the particular roof construction and gap studied, the maximum annual exposure of a 25 mm² piece of the liner placed 150 mm below the gap (corresponding to about 100 mm below the base of the tiles) is about 140 kJ m⁻² to UV-B and 4.2 MJ m⁻² to UV-A radiation. Variations in tile size and misalignment of tiles could lead to other gap shapes and sizes, possibly leading to greater exposure. Constructions with the liner placed closer to the tiles would also lead to greater exposures. However the method developed in this study could form the basis of a standard method of assessment.     

An investigation into the use of a wind shield to reduce the convective heat flux to a nocturnal radiative cooling surface

The effect of a wind shield on the convective heat flux from an ambient air stream blowing over a horizontal surface intended for nocturnal radiative cooling has been studied by computational fluid dynamical calculations and by wind tunnel experiments under conditions appropriate for the climate of Thailand. The test unit was a rectangular plate 312 mm×250 mm, with vertical metal strips for the wind shield having heights up to 100 mm along the edges of the plate. It was found that a wind shield of height 25 mm slightly increased the convective heat transfer due to increased turbulence over the surface, but wind shields of height 50 mm and 100 mm reduced the convection due to a separation of the main airflow from the surface. Radiative cooling was reduced by the wind shields. The net cooling of the surface was best with no wind shield at wind velocities less than about 1 m s^–1, and with the wind shield of height 100 mm at wind velocities greater than about 2 m s^–1.     

Calculation of the concentrated flux density distribution in parabolic trough collectors by a semifinite formulation

A first integral of the concentrated radiant flux density for trough concentrators is derived by exploiting the symmetry of these cylindrical systems. The resulting semifinite formulation yields both conceptual advantages and computational efficiency. The formulation is illustrated by application to the parabolic trough concentrator. The elementary example of determining the concentrated flux density at the nominal focal plane is first considered. The more realistic case of a round absorber tube, which can shade the reflector, centered on the focal line is also studied. This more practical geometry, which corresponds to most commercial designs, has previously received little attention in the open literature. These applications illustrate that the semifinite formulation presented herein can be a simple and useful tool for the analysis and design of trough concentrators.     

An analytic approach to the unsteady heat conduction processes in one-dimensional composite media

The transient heat conduction problems in one-dimensional multi-layer solids are usually solved applying conventional techniques based on Vodicka's approach. However, if the thermal diffusivity of each layer is retained on the side of the heat conduction equation modified from the application of the separation-of-variables method where the time-dependent function is collected, then the modified heat conduction equation by itself represents a transparent statement of the physical phenomena involved. This `natural' choice so simplifies unsteady heat conduction analysis of composite media that thermal response computation reduces to a matter of relatively simple mathematics when compared with traditional techniques heretofore employed.