Naum Fraidenraich

OPTICAL PROPERTIES OF V-TROUGH CONCENTRATORS

Abstract A new approach to study the optical behavior of V-trough concentrators is developed, based on the use of three characteristic angles defining the appearance, disappearance and return to the outside space of the cavity of a reflection mode. The probability of occurrence of a given number of reflections for beam radiation is determined as a function of these angles and the optical efficiency calculated. It is shown that the optical efficiency can be approximated by a function of two parameters, the angular acceptance function, T, and the mean number of reflections, n, as T·p n T . Deviations between exact and approximate optical efficiency increase as n increases or as p decreases. For troughts with C ≤ 2.5 the maximal error for beam radiation is 3.4% for p ≥ 0.8 (8.3% for p ≥ 0.7). For diffuse radiation the maximal error is less than 2% for configurations whose optical efficiency is above 0.6. A further simplification was introduced to obtain the optical efficiency for diffuse radiation, approximating T by an analytical expression and n by an empirical linear function of the inverse of the vertex angle. Results accurate up to 5% for p = 0.8, were obtained. Increasing the concentration ratio, C, from 1.5 to 2.5 for a vertex angle being one third of the acceptance angle, decreased the optical efficiency from 0.74 to 0.59, for p = 0.8. For a given C, the dependence of the optical efficiency on the vertex angle is rather weak, suggesting that large trough angles might be favoured by cost-benefit analysis.

Analytic solutions for the optical properties of V-trough concentrators

Closed-form analytic solutions that describe the optical behavior of V-trough concentrators are derived. The procedure analyzes the angular and spatial distribution of all the rays that undergo a given number of reflections (0, 1, 2), referred to as reflection modes. Then we obtain (1) the optical efficiency for beam and diffuse radiation, and (2) spatial and angular flux distribution on the absorber, with proper account being taken of reflective losses (in addition to the geometric losses of rejected rays). These are often essential -data for evaluation of V-trough solar energy collectors, in particular for photovoltaic applications where homogeneous flux distributions are desirable.

Performance of solar systems with non-linear behavior calculated by the utilizability method: application to PV solar pumps

Based on the utilizability method we derived, in the first part of the paper, an analytic expression to calculate the time average of physical quantities non-linearly dependent on collected solar radiation. Results are applied to photovoltaic pumping systems (PVPS). Water flow propelled by various types of pumps, centrifugal or progressive cavity displacement, for example, can be conveniently written as a second degree polynomial of the collected solar radiation. In that case, the long-term time average of relevant physical parameters, like water flow or hydraulic power, can be calculated with a very simple expression. The procedure is validated comparing long-term averages of maximum water volume pumped by a PV system, obtained with the utilizability method, with those found by running a 10-year time series. Comparison is made for several climatic regions in Brazil. Results show very good agreement for every month of the year and all locations, with a maximum deviation of 1.7%. The method applied to calculate long-term averages, maximum water volume for example, can be useful for evaluation and design procedures of photovoltaic pumping equipment.

Improved solutions for temperature and thermal power delivery profiles in linear solar collectors

For the conversion of absorbed sunlight into useful thermal power, we demonstrate that the profiles of absorber temperature, fluid temperature and thermal power delivery along linear solar collectors can be solved in closed form even when the collector heat-loss coefficient is far from constant over the collector operating range. This analytic solution eliminates the errors inherent in earlier approximate solutions, and makes the dependence of collector performance on component properties transparent. An example for a realistic solar concentrator illustrates the improvement in prediction accuracy.

Solar energy resource assessment — Brazil

Appropriate information on solar resources is very important for a variety of technological areas, such as: agriculture, meteorology, forestry engineering, water resources and particularly for an innovating technology such as solar energy. In the market entry process of an innovating technology such as solar energy, the enlarged and sustained reproduction of this energy strongly depends on the economy and reliability of the demonstrative solar systems installed (within a restricted or wide scale). The economy and reliability of a system are the consequence of a well-prepared project, resulting from an accurate knowledge of the solar resource available. Therefore, knowing the potential of the solar resource accurately is not only a need, it is also an imperative for a larger diffusion and use of the solar energy.

Exact solutions for multilayer optical structures. Application to PV modules

The analysis of multilayer optical devices is important in solar technology as they can be found in a large variety of equipment, particularly in photovoltaic modules. Absorbed and transmitted light by a set of thick layers are usually obtained by ray tracing procedures. When the optical structure is formed by more than one layer, the calculation of both the absorbed and transmitted light is rather complex due to multiple interactions between the several interfaces of the optical device. This paper describes a general analytic procedure, valid for any number of layers, to calculate the absorbed and transmitted light, e.g. solar radiation, through a set of thick optical layers. Consideration of incoming and outgoing light fluxes at each interface, and the assumption that the last interface acts as a light sink, leads to a closed system of equations that can be solved sequentially. Results are applied to analyze the optical behavior of an encapsulated solar cell and a photovoltaic module.

Simulation model of a CPC collector with temperature-dependent heat loss coefficient

Abstract We describe a mathematical model for the optical and thermal performance of non-evacuated CPC solar collectors with a cylindrical absorber, when the heat loss coefficient is temperature-dependent. Detailed energy balance at the absorber, reflector and cover of the CPC cavity yields heat losses as a function of absorber temperature and solar radiation level. Using a polynomial approximation of those heat losses, we calculate the thermal efficiency of the CPC collector. Numerical results show that the performance of the solar collector ( η vs. Δ T f (0)/ I coll ) is given by a set of curves, one for each radiation level. Based on the solution obtained to express the collector performance, we propose to plot efficiency against the relation of heat transfer coefficients at absorber input and under stagnation conditions. The set of characteristic curves merge, then, into a single curve that is not dependent on the solar radiation level. More conveniently, linearized single plots are obtained by expressing efficiency against the square of the difference between the inlet fluid temperature and the ambient temperature divided by the solar radiation level. The new way of plotting solar thermal collector efficiency, such that measurements for a broad range of solar radiation levels can be unified into a single curve, enables us to represent the performance of a large class of solar collectors, e.g. flat plate, CPC and parabolic troughs, whose heat loss functions are well represented by second degree polynomials.

Analytic solutions for the optical and radiative properties of nonaccepted light radiation of V-trough concentrators.

Knowledge of optical and radiative properties is often essential for the design and evaluation of V-trough solar energy collectors. Using the concept of reflection modes, we derived a set of functions associated with each mode; this allowed us to calculate the optical and radiative properties for rejected light radiation. These expressions, together with those for accepted light radiation published previously, were used to calculate the optical efficiency for beam radiation and the exchange factors (diffuse radiation) between aperture and absorber (accepted light) and between aperture and aperture (rejected light). Numerical results of these factors were obtained for various combinations of concentration ratio and vertex angle. Results are compared between a case in which the reflectivity is constant and one in which the reflectivity varies with incidence angle; the difference does not exceed 1% for a reflectivity of 0.8. Considering the reflectivity as a constant allows us to obtain analytic solutions for the exchange factors, expressed as a sum of trigonometric functions.

Multimode analysis of compound parabolic concentrators with flat absorber

We introduce the concept of the reflection mode in the analysis of data generated by ray tracing for the study of the overall optical behavior of compound parabolic concentrators (CPCs) for beam radiation. The light ray paths in two classical CPC cavities with a flat absorber, full and truncated, were simulated by a system of two recurrent series derived for this purpose. The optical behavior of the cavities is summarized by two sets of functions, P(k)(θ(i)) and S(k)(a, θ(i)), each with an infinite number of terms, that depends on the incident angle θ(i) and on the aperture position a traversed by a light ray. These functions satisfy general properties of symmetry, inclusion, and convergence, either in the angular or in the spatial domains of the aperture and absorber. The use of those functions for calculating the angular acceptance, local and average optical efficiency, and flux density distribution is illustrated. Applications to the design of gaps are also discussed. Although this method of analysis is exemplified for the classical CPCs, the properties and applications of these functions are likely to extend to other types of nonimaging concentrators.

Reverse osmosis desalination: Modeling and experiment

An analytic model for the performance of reverse osmosis desalination systems is derived. Predictions are shown to agree well with extensive measurements conducted on a commercial multistage reverse osmosis desalination unit over a broad range of operating conditions. The model allows a transparent understanding of the dependence of system performance on key design and operating variables. Identifying the characteristic flow rate and length scale of reverse osmosis systems allows a universal description of the variation of the permeate flow rate and recovery factor with the salinity, flow rate, and pressure of the feedwater.