Elsevier

Applied Energy

Volume 87, Issue 7, July 2010, Pages 2243-2255
Applied Energy

Analytic method for thermal performance and optimization of an absorber plate fin having variable thermal conductivity and overall loss coefficient

https://doi.org/10.1016/j.apenergy.2010.01.008Get rights and content

Abstract

The absorber of a collector receives solar energy which is delivered to the transport medium to be carried away as useful energy. During this process, temperature of the absorber plate increases and therefore, thermophysical parameters engaged to determine the thermal performance of an absorber plate varies with temperature of the plate. The present study demonstrates analytically to determine the performance of an absorber plate fin with temperature dependent both thermal conductivity and overall heat loss coefficient. The decomposition method is proposed for the solution methodology. An optimum design analysis has also been carried out. A comparative study has been executed among the present results and that of existed in the published work, and a notable difference in results has been found. Finally, unlike published work, dependency parameters on the performances and optimum design have been highlighted.

Introduction

The transformation of solar radiant energy into internal energy of the transport medium is taken place in a special kind of heat exchanger known as solar collector. Among the various types of collectors, flat-plate collectors are the common choices for this conversion probably due to its simple design and ease of fabrication. The flat plate collector has the same area of intercepting and for absorbing solar radiation. When solar insolation passes through the glazing cover and impinges on the blackened absorber plate of high absorptivity, the large portion of energy is absorbed by the plate and then transferred to the transport medium in the fluid tubes to be carried away as utilizable energy. The underside of the absorber plate and the side casing are well insulated to reduce conduction losses. The transport medium may be liquid or gases, usually water or air. The flat plate collector is very much useful in supplying thermal energy at moderate temperature up to the normal boiling point of water. The major applications of flat-plate collectors are in domestic hot water and space heating, and to a lesser degree in industrial processes. Nowadays, they are widely employed to supply hot water in absorption refrigeration systems for space cooling. The water flows through the tubes. The tubes can be welded to the plate or they can be an integral part of the plate. The water tubes are connected at both ends by large headers. From the heat conduction through the plate, a heat transfer module of the plate is repeated. For the thermal analysis of the whole absorber plate, the only analysis of symmetric module can be considered. The performance of solar collector is mainly dependent upon the performance of the absorber plate on which the solar energy is absorbed. Nowadays, flat-plate collectors are widely employed in vapor absorption LiBr/H2O refrigeration systems to heat the generator for running an absorption cycle.

A through research activities have been presented first by Duffie and Beckman [1] for thermal analysis of an absorber plate. During the analysis of an absorber plate, a simplified one-dimensional model of constant thickness plate with constant thermal conductivity and overall loss coefficient has been considered. The pioneering works by Hottel and Whiller [2] and Bliss [3] that developed a model to calculate the conversion of solar radiation into useful thermal energy. The collector heat loss coefficient is considered as a constant along the absorber. A differential equation had been derived by Cooper and Dunkle [4] to take into account the variation of the heat loss coefficient. The drawback of their work is that the temperature dependence of collector heat loss is expressed in terms of fluid, rather than absorber temperature. Kalosgirou [5] presented a survey of the various types of solar thermal collectors and applications.

Thermal performance of the solar air collectors is dependent upon the material, shape, dimension and layout of the collector. Performance improvement can be achieved using diverse materials, various shapes and different dimensions and layout. The collector plate absorbs as much of the irradiation as possible through the glazing, while loosing as little heat as possible upward to the atmosphere and downward through the back of the casing. The collector plate transfers the retained heat to the transport fluid. The absorptance of the collector surface for shortwave solar radiation depends on the nature and color of the coating and on the incident angle. Usually black color is employed. The modifications to improve the heat transfer coefficient between the absorber plate and air include the use of an absorber with fins attached, corrugated absorber, matrix type absorber, with packed bed, with baffles and different configurations were mentioned in the literature [6], [7], [8].

The calculation of heat loss from the collector to its surroundings is required for the design or simulation of the performance of solar collectors. The top heat loss coefficient is evaluated by considering convection and radiation losses from the absorber plate in the upward direction. There are two approaches for calculating top heat loss coefficient, namely numerical solution/iterative procedure and approximate, non-iterative method. The approximate method is also very useful and convenient for the designers of solar thermal systems. Hottel and Woertz [9] were the first to propose a simple empirical equation to estimate top heat loss coefficient of flat-plate collectors. With the development of selective coatings for the absorber plate, a need was felt for better correlation at the lower values of plate emittance. The basic equation of Hottel and Woertz [9] was modified by Klein [10] to provide a better correlation with emittance of absorber plate.

As the solar energy absorbed on the plate and then conducted in the plate, it may be required a variable cross section in the direction of energy transferred for better utilization of plate material. An extensive works [11], [12], [13], [14] have already been devoted for the thermal analysis of variable thickness absorber plate. However, for all these analyses, thermophysical parameters were considered which is independent with temperature. For increase in better utilization of material as well as ease of fabrication, the analysis of absorber plate with step profile [11], [12], [13] and rectotrapezoidal profile [14] have been demonstrated.

It is well known fact that the energy transferred through the absorber plate to the collector fluid is dependent upon the size of the absorber plate and thermophysical parameters involved with this process. Generally in a design application, thermophysical parameters are taken constant for the simplicity of the design. The optimization of an absorber plate is commonly made on the basis of the two approaches. In the first approach, the optimum shape (parabolic, circular, etc.) [15], [16] of the symmetric module of an absorber is determined either maximizing energy transfer through the plate for a given volume or minimizing plate volume for a specified energy transferred according to the requirement of a design. In this standpoint, it can be mentioned that the optimization followed by this approach is rarely used in practices due to inherent difficulties in manufacturing and fabrication for the complex shape of the absorber plate obtained. To avoid this method, a profile shape (rectangular, trapezoidal, triangular, etc.) is chosen according to the ease of fabrication and then optimum dimensions for this profile shape are determined satisfying either maximization of heat transfer rate for a given volume constraint or minimization of fin volume for a constrain energy transfer rate. The second approach of optimization is very much attractive from the practical point of view and thus lot of research activities [11], [13], [14] have already been engaged to establish an optimum condition by using this approach.

The investigations on the relations between collector efficiency factor and material content are important both for efficiency and costs. Several papers in the literature have contributed to this field [3], [17], [18]. All the mentioned papers take into account the complete costs for collector production, dividing them up into three parts: the fixed costs per absorber area (for frame, glazing, insulation, supports, etc.), the costs per volume for the absorber plate, and the costs for the tubing (per unit length). In Bliss’ investigation the quantity to be optimized is collector efficiency factor over total costs. In the other mentioned papers the ratio of thermal collector power over total costs has been maximized. Unfortunately the collector modeling used in the cited literature is not quite satisfactory, as for example the strong influence of the heat transfer from the fin base to the fluid on the collector efficiency is not taken into account.

In recent publications, different types of collectors have been analyzed by many authors [19], [20], [21], [22], [23], [24]. Dubey et al. [19] have developed an analytical expression for electrical efficiency of photovoltaic and thermal (PV/T) hybrid air collectors with and without flow as a function of dimatic and design parameters. Kostic et al. [20] investigated the influence of reflectance from flat plate solar radiation concentrators made of aluminum sheet and aluminum foil on energy efficiency of PV/T collector. Chow [21] has carried out a review on PV/T hybrid technology. To improve the heat transfer efficiency, Ho et al. [22] have suggested a new device for inserting an absorber plate to divide a flat plate channel into two parts with fins attached by baffles and external recycling at the ends. The modeling of stratification inside an aluminum solar collector-storage system and its validity by comparing results with experimental data has been explored by Garnier et al. [23]. Singh et al. [24] have studied the thermal performance of linear Fresnel reflecting solar concentrator with trapezoidal cavity absorber and shown from the result that the higher thermal efficiency of solar device with round pipe absorber as compared to that of the rectangular pipe absorber.

From the literature survey summarized above, it can be highlighted that an obtaining temperature distribution in absorber plates is required the overall loss coefficient which was taken a constant by many researchers for getting analytical solutions. In their analyses, it was also assumed that the thermal conductivity of absorber plate is constant. However, in practical situations, both the overall loss coefficient and thermal conductivity of the plate material depend upon the temperature. With considering this actual condition, the energy equation of the absorber plate becomes nonlinear and thus for its solution, it may be quite difficult. All the published works establishes an optimization technique with adopting constant thermophysical parameters. On the other hand, thermophysical parameters are not a constant which depend upon the temperature as a result the optimization study makes complex also. For all these considerations, the present problem has been assigned.

In this study, an effort has been devoted to analyze the performance and optimization of an absorber plate analytically with the consideration of variable thermophysical parameters. The overall loss coefficient is expressed as a quadratic function with the temperature. In addition, the thermal conductivity of the absorber plate material is expressed by a linear function with its temperature. With the imposition of the above fact, the governing equation of an absorber plate is derived which is a nonlinear in nature and thus it cannot be solved by conventional analytical techniques. However, it may be solved by Adomian decomposition method. Thermal performances are evaluated for a wide range of thermogeometric parameters. As the present analysis is analytical it is easily extendable to the optimization analysis. The optimization study is carried out in a generalized analysis such that either maximization of heat transfer for given volume or minimization of plate volume for given energy transfer rate is satisfied. The different design parameters such as ambient temperature, base temperature and absorbed solar flux on the optimum conditions have been studied. Finally a comparative study of performances and optimization between the present and published results has been executed and adequate differences in results have been noticed.

Section snippets

Statement of physical problem and mathematical formulations

The main components of the collector are absorber plate and fluid carrying tubes. The arrangement of absorber plate and fluid carrying tubes is shown in Fig. 1. The solar energy is absorbed on the plate and that energy is carried out by collector fluid in the form internal energy. The collector fluid flows through the tubes. As the conduction heat transfer takes place in the absorber plate, a symmetric heat transfer module repeats between two fluid carrying tubes. The symmetric length of the

Optimization analysis

The volume of a symmetric heat transfer module of an absorber plate can be written in dimensionless form asU=Ul02Vka2=Bi2ψ

The optimization of a known geometric absorber plate can be made either by maximization of energy transfer rate for a given plate volume or by minimization of plate volume for a desired energy transfer rate. However outcome from both the approaches give the same value. For a design requirement, a generalized optimization scheme is provided in which the aforementioned

Results and discussion

The main objective of the present paper is to establish an analytical expression for determination of thermal performance and optimization of an absorber plate with a variation of thermal conductivity of plate material (α = 1 × 10−4 K−1) and overall heat loss coefficient with the temperature. The Adomian decomposition method is used as an analytical tool. Before furnishing outcomes obtained from the present model, it is required to validate with the any other results published in the literature. For

Conclusions

The present analysis is paying attention to establish an analytical model for performance and optimization of an absorber plate fin for the variation of thermal conductivity of the plate material and overall heat loss coefficient with the temperature. By regression analysis, the overall heat loss coefficient is expressed in a second order polynomial with the temperature. Adomian decomposition method is suggested for this analytical establishment. For the validation of the proposed method, the

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