Application of the entropy generation minimization method to a solar heat exchanger: A pseudo-optimization design process based on the analysis of the local entropy generation maps

Abstract

This paper presents an application of the entropy generation minimization method to the pseudo-optimization of the configuration of the heat exchange surfaces in a Solar Rooftile. An initial “standard” commercial configuration is gradually improved by introducing design changes aimed at the reduction of the thermodynamic losses due to heat transfer and fluid friction. Different geometries (pins, fins and others) are analysed with a commercial CFD (Computational Fluid Dynamics) code that also computes the local entropy generation rate. The design improvement process is carried out on the basis of a careful analysis of the local entropy generation maps and the rationale behind each step of the process is discussed in this perspective. The results are compared with other entropy generation minimization techniques available in the recent technical literature. It is found that the geometry with pin-fins has the best performance among the tested ones, and that the optimal pin array shape parameters (pitch and span) can be determined by a critical analysis of the integrated and local entropy maps and of the temperature contours.

Introduction

During the last three decades the EGM (Entropy Generation Minimization) method has become a well-established procedure in thermal science and engineering. Its theoretical foundations are to be found in the well-known Guoy–Stodola theorem which states that for whatever open process whose evolution can be approximated as a succession of quasi-equilibrium states and during which the system is in thermal contact with an ambient at T₀, the lost available power $\left|{\dot{W}}_{\text{rev}}-\dot{W}\right|$, i.e., the difference between the ideally produced power and the one really extracted, is proportional to the global rate of entropy generation ${\dot{S}}_{\text{gen}}:{\dot{W}}_{\text{rev}}-\dot{W}=T_0·{\dot{S}}_{\text{gen}}$ [1]. The lost power $\left({\dot{W}}_{\text{rev}}-\dot{W}\right)$ is always positive, regardless of whether the system is a power source (e.g., an expander) or a power sink (e.g., a compressor). Hence, a reduction in the entropy generation rate leads directly to an increase in the efficiency of the system. The EGM method relies on a simultaneous heat transfer and engineering thermodynamics approach that leads to the construction of realistic models for the systems object of study, be they heat transfer devices, power components or complex plants. Once a model has been established, the strong link between the global entropy generation rate and the efficiency is exploited by the designer who can compare different configurations (“design options”) that either produce the same output with less irreversible losses or use the same amount of resource input to generate a larger output; both cases corresponding of course to a higher resource-to-end use efficiency. Naturally, the minimization of the entropy generation is not an easy task in real design applications; the level of complexity of the model clearly increases when complicated boundary conditions apply and/or when the operating point is varying in time. Nevertheless, several applications of the method can be found in the recent technical literature. Such applications may be divided in two categories, those based on deterministic approach and those based on a heuristic approach. Following the work done in Refs. [1], [2], the overwhelming majority of applications for heat transfer problems belong to the first category. This technique is applied to “classic” problems in Refs. [1], [2], [3], [4], [5]; pin-fins geometries are optimized with this method in Refs. [6], [7], while plate-fins heat sinks are optimized in Refs. [8], [9]. The fundamental characteristic of the deterministic approach is the analytical definition of the global rate of entropy generation (${\dot{S}}_{\text{gen}}$ [W/K]) as a function of critical design parameters, like geometry, dimensions and working conditions. ${\dot{S}}_{\text{gen}}$ is studied numerically and the optimal geometry is defined by the set of parameters which minimize the global rate of entropy generation. Since the function ${\dot{S}}_{\text{gen}}$ is determined using the correlations for average heat transfer rates and fluid friction available in literature, the deterministic approach ultimately results in a lumped-sum parameter model, that is the thermodynamic behaviour of the entire system (or control volume) is described globally by a single mathematical function. While this analytical approach has clear advantages, it is not applicable to complex problems where no reliable and explicit correlations for the mean heat transfer and fluid friction are available. Typical examples are turbomachinery and (convective) heat exchangers design problems. For such kind of problems, the thermodynamic behaviour of the system is more accurately described by a local, distributed-parameter model, and the minimization problem is best solved with a heuristic approach: the initial configuration is successively improved by introducing design changes based on a critical analysis of the local entropy generation maps obtained by means of CFD (Computational Fluid Dynamics) simulations. While this approach has been already adopted in some turbomachinery problems [10], [11], examples for heat exchangers like the one covered in this work appear quite rarely in the archival literature: one example can be found in Ref. [12]. The heuristic approach consists in focussing the attention first on the local entropy generation rates ${\dot{s}}_{\text{T}}$, ${\dot{s}}_{\text{V}}$ and in considering the global one ${\dot{S}}_{\text{gen}}$only after having carefully studied the implications of the local irreversibility on the overall design. The calculation of the local entropy generation rates requires that the velocity and temperature fields are known; therefore a CFD solver, i.e. a distributed-parameter model, needs to be employed. Once a solution is obtained, through the visual analysis of the local entropy maps the practitioner is able to identify the areas where the entropy is produced at the higher rates, that is the areas which demand design modifications. Even if the heuristic approach strongly emphasises the role of the local entropy rates, it must be noted that ${\dot{S}}_{\text{gen}}$ is nevertheless the quantity that reflects the thermodynamic performance of the whole system, that is quantity that allows the designer to discern the better performer from a second-law perspective. The aim of the present work is to show an application of the heuristic approach to an industrial heat transfer problem.

This work is organized as follows: more details about the adopted heuristic approach are given in the second paragraph; the third paragraph describes the heat transfer problem the method has been applied to; the fourth paragraph deals with the numerical simulations; the fifth paragraph presents the results and in the last paragraph conclusions are drawn.