Regular PaperPrediction of porosity and thermal diffusivity in a porous fin using differential evolution algorithm
Introduction
The porous fins have got several advantages over conventional fins [1], [2], [3], [4], [5], [6], [7], [8], [9] and its investigation is one of the widely investigated topics of research in the area of heat and mass transfer. The heat transfer analysis of porous fins requires the consideration of simultaneous heat and mass transfer of both solid and fluid media [10]. The investigation of literature on porous fins indicates that most studies intend to obtain the performance parameters (temperature, efficiency, etc.) using various schemes using the a priori knowledge of thermo-physical quantities. Such cases fall under the purview of direct (forward) problems [11]. However, the situation becomes different and interesting when the conditions satisfying a given objective are required to be estimated using the inverse method. The procedure of estimating the feasible combination of various unknowns is known as the inverse problem [12], [13]. Inverse problems are generally design-oriented, which can provide possible unknown parameters of an engineering system when subjected to some given requirement.
The situations requiring solution of inverse problems occur quite often in practical engineering problems, where it is easier to measure the temperature in a particular state of the system. Fins form a popular example of such a situation where the inverse problem such as mentioned above is encountered [14], [15]. In this paper, the computation of the temperature for a known set of physical parameters of the fin comprises the forward problem and the focus of the paper is to solve an inverse problem for a porous fin. The problem is solved to compute the unknown physical parameters such as the thermal diffusivity of the fluid medium and the porosity of the fin of the porous fin.
In heat transfer and fluid mechanics, similar to other inverse problems in engineering [16], [17], the inverse problem is difficult to solve due to the ill-posedness and non-linearity of the mathematical model. Although the investigation of inverse problems in porous fins is limited, but inverse problems involving non-porous fins have been studied by many researchers [18], [19], [20], [21], [22], [23]. For inverse problems, it is found that the evolutionary-based optimization methods fare better than conventional gradient-based techniques [24], [25] by performing global searching in the solution space. The differential evolution (DE) [26] is used in the present study for being one of the recently-used evolutionary methods which is receiving considerable attention for inverse problems [27], [28]. The reason of its popularity is due to good convergence properties, simplicity and capability of optimizing objective functions involving difficulties associated with evaluating gradients due to the presence of nonlinear terms (for example due to the involvement of the radiative effects) [29].
In view of the above points, the present work intends to introduce the usage of DE for inverse problems in porous fins. It has been observed from the available literature that DE has not yet been used for studying inverse problem of porous fins involving radiation heat transfer. The work is aimed at implementing DE to a porous fin involving natural convection along with surface radiation for solving an inverse optimization problem. The involvement of radiation mode of heat transfer makes the present problem nonlinear for which the evaluation of gradients does not remain so simple, thereby justifying the usage of an evolutionary method of optimization, such as DE. The performance of DE is also compared with the deterministic Levenberg–Marquardt (LM) algorithm [30] and the meta-heuristic simulated annealing (SA) algorithm [23], [31] along with the genetic algorithm (GA) [15], [15]. It is shown that for the present problem, DE performs better than LM and SA algorithms with enhanced probability of converging to the actual solution and requiring less number of function evaluations. Below the problem and the solution procedure are discussed.
Section snippets
Problem set-up
Let us consider the porous fin geometry as shown in Fig. 1, Fig. 2. It is assumed that the thickness of the fin, t, is small in comparison to the width, i.e., (t/W)≪1, thus, the temperature changes along the longitudinal direction only. Furthermore, the only source of heat is at the west boundary where a base temperature Tb is maintained.
The heat conduction problem through a porous fin is completely described by the fin׳s solid thermal conductivity ks, emissivity ε, permeability K, porosity ϕ,
Inverse problem, optimization variable, and measurement data
The problem becomes interesting when the task is to determine some unknown fin parameters that result into an observed temperature distribution. The parameters of consequence are summarized in Section 2.1. In the present work, the thermal diffusivity, α, of the fluid and the porosity, ϕ, of the fin have been simultaneously estimated for satisfying a given temperature distribution, assuming that the other parameters are known. For convenience, the mathematical vector containing a set of
Numerical simulations and results
In all examples presented below, the dimensions of the fin are given by W=1 m, L=0.1 m, and t=0.03 m. The ambient temperature and the fin׳s base temperature are T∞=30 °C and Tb=330 °C, respectively.
As already mentioned in Section 3.2, the number of grid points used in generating the measurement data , , and is N=61. The same forward solver as presented in Section 2.2 is also used in the inverse approach but the number of grid points is chosen as N=51. This information about the
Conclusion
The inverse problem through a porous fin is considered. The aim of retrieving the fluid׳s diffusivity and the fin׳s porosity using temperature at three points is achieved using DE optimization technique. It is shown that DE can retrieve the value of porosity quite well, but the diffusivity of the fluid is rather difficult to retrieve.
This is explained by the nature of the objective function and the presence of local minima or local minimum valley in the objective function. This indicates that
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