Heat transfer correlations by symbolic regression

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Abstract

We describe a methodology that uses symbolic regression to extract correlations from heat transfer measurements by searching for both the form of the correlation equation and the constants in it that enable the closest fit to experimental data. For this purpose we use genetic programming modified by a penalty procedure to prevent large correlation functions. The advantage of using this technique is that no initial assumption on the form of the correlation is needed. The procedure is tested using two sets of published experimental data, one for a compact heat exchanger and the other for liquid flow in a circular pipe. In both situations, predictive errors from correlations found from symbolic regression are smaller than their published counterparts. A parametric analysis of the penalty function is also carried out.

Introduction

In the design, selection and control of thermal components for industrial and commercial applications, it is necessary to predict their performance under specific conditions of operation. Though in theory this calculation can be carried out from first principles by formulating the governing equations, complexities arising from factors like turbulence, temperature dependence of properties, and the geometry makes it difficult to achieve in practice. As a result, most calculations are based on experimental data. The information is compressed in the form of correlations from which the heat transfer coefficient can be obtained. Most commonly, correlations are developed in terms of dimensionless groups like the Nusselt, Reynolds and Prandtl numbers; sometimes for greater generality geometrical factors are also included. Assuming a functional relationship between the groups with a certain number of free constants, a regression analysis to minimize the error between predicted and experimental values is carried out to determine the appropriate values of the constants.

A disadvantage of this procedure is that predictive errors in the heat transfer rate are normally larger than the experimental uncertainties from which the correlation was generated. Assumptions such as using average transfer coefficients or constant property values [1] and the fact that the error minimization function may have more than one local minimum [2], [3] are among the reasons for this loss of accuracy. Another source of error is the specific form of the correlation function assumed for the regression analysis. The functional form is selected on the basis of simplicity, compactness and common usage [4], but cannot be completely justified from first principles. There is usually not much physics behind the choice of the form. Although power laws are commonly used in heat transfer studies, a variety of other forms have also been used [5], though it is not obvious how the form should be chosen. As an example, for heat exchangers Pacheco-Vega et al. [6] have shown that different functional forms may predict performance with more or less similar accuracy. It would thus be advantageous to have an algorithmic way to determine the best correlation that fits experimental data without the need to assume its functional form.

The genetic algorithm (GA) [7], [8] is an optimization technique based on stochastic, evolutionary principles that is used to find global extrema of a given function. Genetic programming (GP) [9] is a symbolic regression extension that works with a set of possible functions to find the best for a given set of data. Applications of GP to thermal engineering are scarce: the correlations obtained by Lee et al. [10] for critical heat flux for water flow in vertical round pipes and Pacheco-Vega et al. [11] for artificial heat-exchanger data are among the very few.

The aim of the present study is to describe a methodology based on GP to develop heat transfer correlations that can be used to predict the performance of thermal components. Since compact forms of the correlations are to be preferred, the standard procedure will be modified by a penalty function that weights against complicated forms. The procedure is described first. Then, two sets of published experimental data, one corresponding to heat transfer in compact heat exchangers and the other to heating and cooling of liquids in pipes, are used to demonstrate the capability of GP to find accurate correlations. The effect of the parameters of the penalty function on the results is also analyzed.

Section snippets

Description

GP is a soft computing search technique in which computer codes, representing functions as parse trees, evolve as the search proceeds. The objective is to extremize a certain quantity called the fitness function. Developed originally to automatically generate computer programs, it has been used in a variety of applications, e.g., finance [12], electronic design [13], signal processing [14], and system identification [15], among others. GP is discussed in detail in the monograph by Koza [9].

Compact heat exchanger data correlation

The procedure is now applied to data obtained from experimental measurements. Heat exchangers are a common example of thermal components, and empirical correlations have been proposed by Abu Madi et al. [20], Kim et al. [21] and Wang et al. [22] for single-phase flow conditions, and McQuiston [23] and Khartabil [24] for condensing conditions.

We consider experimental data that were obtained and reported by McQuiston [25] from a series of tests on a fin-tube compact heat exchanger. This was a

Effect of penalty parameters

Though different penalty functions may be used to limit the size of the correlations, one of the advantages of the sigmoidal form in Eq. (5) is that, since it is bounded and its denominator is non-zero, it prevents the fitness in Eq. (4) from becoming either unbounded or singular and thus avoids computational problems. However, the choice of a1 and a2 may affect the results. With the other parameters fixed to the values used before, we take the data set of McQuiston [25] and vary a1 and a2 to

Pipe-flow data correlation

Experimental data for heating and cooling of liquids in pipes were reported by Sieder and Tate [26]. These data and the corresponding correlation are frequently used to calculate heat transfer coefficients in laminar flow in pipes during design calculations. Using three distinct oils as working fluids, a total of 67 experimental runs were reported. The experimental results included the Nusselt Nu, Reynolds Re, and Prandtl Pr numbers, as well as the viscosity ratio μ/μw. Here μ and μw are the

Conclusions

Correlations obtained from experimental data are commonly used in the estimations of the heat rate in thermal components. Most often this reduction of experimental data to correlations is based on first choosing a specific functional form of the correlation for which the constants are then determined. Choice of the form determines the least error that can be obtained in the regression process. Power laws are often used, though many other forms appear in the literature. Since digital computers

Acknowledgements

We acknowledge the support of the late Mr. D.K. Dorini and BRDG-TNDR for the activities in the Hydronics Laboratory. A.P.-V. also wishes to thank PROMEP for financial support for a Visiting Professorship under Grant PTC-68.

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