Correction for the absorber edge effect in analytical models of flat plate solar collectors
Introduction
For economic and technological reasons the vast majority of solar collectors for low temperature applications up to 100 °C are of the flat plate collector type designed for liquid heat carriers. For the same reasons, mostly the tube-and-fin type of absorbers are used. These absorbers are basically made of absorber tubes attached to a plate of sheet-metal which has a selective coating for the absorption of solar radiation. This paper deals with the efficiency of this type of absorbers.
Until the beginning of 1990 most absorbers were made of a number of identical absorber strips. In these absorber strips one single absorber tube is bonded to the middle axis of a strip of coated sheet metal. The bond between the absorber tube and the strip of sheet metal runs continuously over the whole length of the strip. Such absorbers are called ideal, because the heat flow within the strip is one dimensional (i.e., perpendicular to the axis of the absorber tube). Thus, the normalized temperature distribution in the cross-section of a strip is the same for any arbitrary position along the axis of the absorber strip.
In parallel-flow absorbers the strips are connected by header tubes at their inlet and outlet. In meander-type absorbers the strips are connected in series by tube bends. Combinations of these two types were, and still are, commonly implemented. In these combined types, groups of parallel-flow strips are connected in series, which is easily achieved by reducing the cross-section of the header tubes to zero by using a pressing tool at the right place. Both the headers and the tube bends are usually visible beneath the transparent cover of a solar collector and, therefore, are usually painted black. The parallel-flow type and the combined type are pictured in Fig. 1.
By the end of the last century, new selective absorber coatings, based on vacuum-coating processes, were introduced. From the very beginning, the new technology was designed for mass production and, to make the process even more economical, the coil to coil coating of sheet metal of large width. The width of these coils is comparable to that of ordinary flat plate collectors. This and the subsequent development of fully automated welding technologies allowed for the introduction of single plate absorbers which are far more cost effective than absorbers made of a number of strips. But, even in optimized collector designs of today, the material costs still dominate the economics. Therefore, it is essential to maximize the ratio of power to material costs.
The obvious way to do this is to maximize the area of the absorber plate for a given collector aperture area. The result is an absorber of approximately the same length and width as the aperture area with only small amount of space between the edge of the absorber plate and the side walls of the collector. Therefore, headers and tube bends have to be placed beneath the absorber plate, and it is no longer possible to weld straight absorber tubes over the whole length of the absorber plate. In both the parallel flow type and the meander type of absorber, the bonds of the straight absorber tubes end at a certain distance from the edge of the plate. Fig. 2 shows two of many possible designs of single plate absorbers. Continuous lines along the tube axis indicate the thermally conductive bond between tube and absorber sheet, whereas in the region of the dotted lines the tubes are not bonded.
In contrast to ideal absorbers, where the absorber tubes are bonded from edge to edge, these absorbers are considered non-ideal. Because the bonds end a certain distance from the edge, the heat flow within a region along the edge will be two-dimensional. In normal operation of the solar collector, the average temperature in the edge region is higher than the average temperature in the center region. Therefore, the heat loss of such an absorber is higher compared to an ideal absorber, which results in a reduced efficiency of the solar collector. In meander type absorbers it is possible to weld the tube bends if the tubes and the sheet are of the same material. This is done by some manufacturers to reduce heat losses in the edge region. A significant measure for the thermal performance is the fin efficiency, F, which is defined in Section 2. The deviation of the fin efficiency of a non-ideal absorber from the fin efficiency of an ideal absorber is called the edge effect.
Two-dimensional simulations illustrate this edge effect. Fig. 3 shows the temperature distribution in a periodic section of the 300 mm wide edge region of a single plate absorber. The thickness of the plate is 0.12 mm. The material is copper with a heat conductivity of 385 W/m2 K. The distance between the absorber tubes is 132 mm. The solar irradiation is 800 W/m2 and the ambient temperature is 20 °C. The transmission coefficient of the cover plate is τ = 0.91 and the absorption coefficient of the absorber coating is α = 0.95. A heat loss coefficient of UL = 4 W/m2 K was assumed, which is typical for a flat plate collector with single glazing in normal operation. The bond terminates 80 mm before the edge of the absorber plate. The temperature of the center region above the bond is 50 °C. In the ideal case the average temperature of the absorber plate would be 64.4 °C. Due to the incomplete bonding in this case, the average temperature is 74.2 °C, which results in a higher heat loss.
Fig. 4 shows the same section, but with welded tube bends. The distance between the axis of the tube bend and edge of the plate is 84 mm. Here, the average temperature is 70.4 °C. The heat loss will be lower than in the case above, but still higher than in the ideal case.
For ideal tube-and-plate absorbers, Hottel and Woerz (1942) and Hottel and Whillier (1958) developed a theory which is the basis of most of the currently used collector models. These models are widely used for parametric studies and are therefore essential in the development process of new solar collectors. The core of the theory is the analytical solution of the one dimensional energy equation for a straight fin with a rectangular cross-section. Therefore, these models are called analytical models, in contrast to the fully empirical models where the solar collector is characterized by parameters derived from measurements only. The closure relations, however, are empirical correlations which characterize the heat transfer from the center region of the absorber strip to the liquid in the absorber tube, and the heat losses from the absorber strip to the ambient by conduction, radiation, and convection. All models express the efficiency of the collector in the form of dimensionless numbers which are related to the temperature of the heat carrier fluid. Three of these numbers will be introduced in the next section.
Many authors have since contributed to this theory. In the following, only those articles concerning the absorber itself are discussed. Bliss (1959) summarized the previous works and discussed the influence of fin width, sheet metal thickness, and conductivity on the efficiency. Whillier and Saluja (1965) discussed the influence of bond conductance on the efficiency. They concluded that a conductance of per unit length of absorber tube is required for a good collector efficiency, which is fulfilled by the currently used bonding methods like laser- and ultrasonic welding. Florschuetz (1979) adapted the model for photovoltaic–thermal collectors by taking the electrical efficiency of the solar cell into account. Lunde (1981) and Hadorn (1983) derived equivalent representations for the collector efficiency which are related to different fluid temperatures. The representation which relates the efficiency to the arithmetic average of inlet and outlet temperature was found to be the most convenient for the interpretation of experimental results, and was therefore implemented in the international standard ISO 8906-1 as well as in the European Standard EN 12975. However, all of these authors considered the absorber ideal, without considering edge effects due to incomplete bonding.
In the recent years two- and three dimensional simulations were used to study both the temperature distribution of the absorber plate and the flow distributions in the absorber tubes, and its influence on the efficiency of the solar collector (Villar et al., 2009, Fan et al., 2007). Alvarez et al. (2010) presented numerical and experimental results for the efficiency and the temperature distribution of two flat-plate collectors with meander-type absorbers. One absorber was of the ordinary tube-and-sheet type. The other was made of two sheets. The coated, top side sheet was flat while the sheet on the back side was corrugated. The sheets were welded together, forming wide flow channels. These authors, too, considered the analytical model based on ideal absorber geometry as sufficient for their purpose.
Because of the increasing costs of raw materials, especially metals, the design of absorbers was more thoroughly studied in order to optimize the ratio of efficiency and material costs. O’Brien-Bernini and McGowan (1984) derived an analytical model for non-metallic absorbers. Eisenmann et al. (2004) adopted the analytical model to derive useful charts which show the copper content per unit area for various curves of constant collector efficiency. Badescu (2006) extended the analytical model for non-uniform thickness of the absorber plate and discussed cost-optimal fin geometry for both rectangular and triangular fins. Again, all models are based on the assumption that the absorber tube is bonded over the whole width of the plate.
However, lacking an appropriate model, the following questions remain unanswered:
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By how much is the efficiency of a collector reduced if the bonds end a given distance before the edge of the plate?
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To what degree can the welding of the tube bend compensate this edge effect?
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What is the efficiency-equivalent tube length of an absorber with welded tube bends?
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Is it profitable to weld the tube bends?
The goal of this paper is to extend the analytical model for two purposes:
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In order to exploit the full advantage of single plate absorbers, an appropriate analytical model for rigorous efficiency/cost optimization is a necessity.
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By taking the edge effect of non-ideal absorbers into account, the extended analytical model can be used to derive more accurate correlations for the heat loss coefficient from measured collector efficiency.
In order to introduce the necessary numbers and equations, the basic theory of ideal absorber is presented in Section 2 Analytical model for ideal absorbers, 3 Correction of the analytical model to reflect the “edge effect” starts with a description of the two-dimensional modeling and the definition of the range of physical values considered, followed by the derivation of the correction factor. In Section 5, results are presented and discussed.
Section snippets
Analytical model for ideal absorbers
If straight equidistant tubes are used and if these tubes are continuously bonded over the whole width of a single absorber plate, a single plate absorber has practically the same thermal characteristics as an absorber made of an equivalent number of strips.
This single plate absorber can be characterized by a periodic domain of the plate according to Fig. 5, which represents the virtual absorber strip with adiabatic boundaries. The absorber plate is characterized by its thickness, sp, and the
Correction of the analytical model to reflect the “edge effect”
The reduction of collector efficiency due to the non-ideal bonding of the absorber tubes in the edge region of the absorber plate is quite small. It lies in the range of 2–7% for reasonable collector designs. Therefore, it can be captured experimentally only at great expense. It would be necessary to determine the optical and thermal material properties beforehand and with high accuracy to keep the uncertainty at least one order of magnitude smaller than the edge effect. Furthermore, the
Derivation of the correction factor from simulation results
From the simulation results the correction factors for the edge region for all N cases are calculated. These correction factors are to be approximated by a product of n dimensionless functions,
The correlation should be simple and its accuracy, compared to the numerical result, better than 0.5% for all cases,
These dimensionless functions depend on the width of the fin, f, and the distance, e, between the edge of the plate
Results
By introducing the correction factors into the ideal analytical model, the edge effects of non-ideal absorbers are taken into account. The correction factor for the edge effect has the final form
The exponents for absorbers with straight bonds are defined in the upper part of Table 2, the exponents for absorbers with bonded tube bends are defined in the lower part of the same table. The reference value for the heat loss coefficient is .
The
Discussion
For the discussion of the edge effect, the fin efficiency of the edge region is compared to the standard fin efficiency of the ideal absorber.
Fig. 7, Fig. 8, Fig. 9 show the fin efficiency covering the whole range of geometrical parameters. The markers indicate the reference value from the two-dimensional simulations. For clarity, the curves for fin width 44 mm and 64 mm are omitted. For the same edge distance, e, the fin efficiency of absorbers with bonded bends is generally higher. It is even
Conclusion
The edge effect in tube-and-plate absorbers, which is one of the major uncertainties of analytical collector models, was studied using two-dimensional numerical simulations.
From the simulation results, a correlation for a correction factor, Re, was derived which takes this edge effect into account. The standard fin efficiency, F0, for ideal absorbers is multiplied by this correction factor, thereby enabling the modeling of absorbers with relevant geometry.
The improved analytical model allows
Acknowledgements
We would like to express our thanks to the Swiss Federal Department of Energy (BFE) and to the Schwyzer–Winiker foundation for the financing of this research project. We also thank John Kickhofel and Robert Adams for their critical review and fruitful discussions.
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