Linear correlation of heat transfer and friction in helically-finned tubes using five simple groups of parameters

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Abstract

A linear regression approach was used to correlate experimentally-determined Colburn j-factors and Fanning friction factors for flow of liquid water in helically-finned tubes. Experimental data came from eight enhanced tubes with helix angles between 25° and 48°, number of fin starts between 10 and 45, fin height-to-diameter ratios between 0.0199 and 0.0327, and Reynolds numbers ranging from 12,000 to 60,000. The current study revealed that, in helically-finned tubes, logarithms of both friction and Colburn j-factors can be correlated with linear combinations of the same five simple groups of parameters and a constant. The proposed functional relationship was tested with independent experimental data yielding excellent results.

Introduction

Industrial use of heat transfer enhancement has become widespread. The goal of heat transfer enhancement is to reduce the size and cost of heat exchanger equipment. Webb [1] gives an excellent overview of different enhancement mechanisms available in commercial tubes.

One contemporary enhancement geometry is the helical fin shown in Fig. 1, which is described by several geometric variables. Fig. 1 also provides a pictorial description of these variables, which include: the fin height (e), the fin pitch (p), the helix angle (α), number of starts (Ns), and included angle (β). The fin height is the distance measured from the internal wall of the tube to the top of the fin. The fin pitch is the distance between the centers of two fins measured in the axial direction. The helix angle is the angle the fin forms with the tube axis. The number of starts refers to how many fins one can count around the circumference of the tube. Finally, the angle at which the sides of the fin meet is called the included angle.

An extensive literature survey of research on helically-finned tubes is given in Zdaniuk [2]. Despite a considerable amount of study, the characteristics of flow inside helically-finned tubes are still not very well understood because the physics governing the flow are very complex and experimental data are limited. The current approach to predicting pressure drop and heat transfer in helically-finned tubes is to use algebraic correlations based on least-squares regression. Regression techniques performed on experimental data require mathematical functional form assumptions, which limit their accuracy and generality. To address these limitations, techniques that can effectively overcome the complexity of the problem without ad hoc assumptions are needed. One of these techniques is symbolic regression by means of genetic programming.

Genetic programming (GP) is a method that works with a set of possible operators to obtain optimum functional relationships for a given data set [3]. This method should not be confused with the genetic algorithm (GA) which is an optimization technique based on stochastic, evolutionary principles that is used to find global extrema of a given function [4], [5]. Sen and Yang [6] described the scope of artificial neural networks3 and genetic algorithm techniques in thermal science applications including an exhaustive bibliography. Recently, a methodology for obtaining heat transfer correlations by means of symbolic regression was published by Cai et al. [8]. Otherwise, applications of GP in heat transfer are scarce.

Section snippets

Experimental data

An experimental program devised to measure turbulent pressure drop and heat transfer in helically-finned tubes was conducted at Mississippi State University. The experimental apparatus and procedure are described in detail in Zdaniuk et al. [9]. Eight enhanced tubes and one plain tube were tested inside a double-pipe counter-flow heat exchanger with hot water flowing on the inside and cold water in the annulus side. The tubes were manufactured for condenser applications. The internal geometric

Correlation development

The information presented so far demonstrates that symbolic regression can be a compromise between the complexity of the ANN method and inaccuracy resulting from fitting the data to an assumed functional form (e.g., a power law). The genetic program removes the burden of functional form assumptions and can be controlled to yield uncomplicated equations. The approach taken in this study was to feed the parameters e/D, α, Ns, and ln (Re) into the genetic program in order to correlate them with ln (j

Evaluation of the proposed functional form with independent experimental data

Webb et al. [16] used a double-pipe counter-flow heat exchanger set up with liquid water on the inside and boiling R-12 on the annulus side to measure Fanning friction factors and Colburn j-factors of eight helically-finned tubes at Reynolds numbers between 20,000 and 80,000. Table 2 provides the internal geometry of the helically-finned tubes tested by Webb et al. [16]. Table 2 tubes are numbered W1 through W8 in order to distinguish them from the tubes used in the current study.

Zdaniuk et al.

Conclusions

The principal finding of the current investigation is the fact that in helically-finned tubes both Fanning friction factors and Colburn j-factors can be correlated with exponentials of linear combinations of the same five simple groups of parameters and a constant. The performance of the proposed correlations is much better than that of the power-law correlations and only slightly worse than that of the artificial neural networks. The functional form proposed in Eqs. (8), (9) works very well

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1

Tel.: +44 20 7631 5291; fax: +44 20 7323 4645.

2

Tel.: +1 662 325 3261; fax: +1 662 325 7223.

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