Estimation of heat transfer coefficient on the fin of annular-finned tube heat exchangers in natural convection for various fin spacings

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Abstract

The finite difference method in conjunction with the least-squares scheme and experimental temperature data is used to predict the average heat transfer coefficient and fin efficiency on the fin of annular-finned tube heat exchangers in natural convection for various fin spacings. The radiation and convection heat transfer coefficients are simultaneously taken into consideration in the present study. The heat transfer coefficient on this annular circular fin is assumed to be non-uniform. Thus the whole annular circular fin is divided into several sub-fin regions in order to predict the average heat transfer coefficient h¯ and fin efficiency from the knowledge of the ambient temperature, tube temperature and fin temperature recordings at several selected measurement locations. The results show that the h¯ value increases with increasing the fin spacing S, and the fin efficiency decreases with increasing the fin spacing S. However, these two values respectively approach their corresponding asymptotical values obtained from a single fin as S  ∞. The fin temperature departs from the ideal isothermal situation and decreases more rapidly away from the circular center with increasing the fin spacing. In order to validate the accuracy of the present inverse scheme, a comparison of the average heat transfer coefficient on the fin between the present estimates and those obtained from the correlation recommended by current textbooks is made.

Introduction

Annular-finned tube heat exchangers are commonly used in industry. In designing such heat exchangers, it is necessary to note the interactions between the local heat transfer and flow distribution within the fins. The previous works about the effect of the fin spacing of annular-finned tube heat exchangers were limited to the experiments [1]. Thus the present study applies the hybrid inverse scheme in conjunction with experimental temperature data to estimate the heat-transfer characteristics of annular-finned tube heat exchangers. The fin in heat exchangers is always applied to increase the heat flow per unit of basic surface. The analysis of a continuous plate fin pierced by a regularly spaced array of circular tubes in staggered and in-line arrays has many engineering applications [2]. In order to simplify the problem considered, the calculation of the standard fin efficiency usually assumes that the heat transfer coefficient is constant over the plate fin. However, it is well known that there exists a very complex flow pattern within a plate finned-tube heat exchanger due to a plume of the heated air rising above the horizontal circular tube in natural convection. The boundary layer over a heated horizontal tube starts to develop at the bottom of the tube and increases in thickness along the circumference of the tube. The flow forms a low-velocity region above the tube. Thus the heat transfer coefficient is lowest on the top region of the tube. This causes local variations of the heat transfer coefficient on the fin. On the other hand, the heat transfer coefficient on the fin is non-uniform. These phenomena can be found from Refs. [3], [4]. As shown in Ref. [5], the measurements of the local heat transfer coefficient on plain fins under steady-state heat transfer conditions were very difficult to perform, since the local fin temperature and local heat flux were required. Moreover, reliability is an important concept in engineering design, and the use of reliable components enables the designers to utilize more sophisticated techniques to improve the performance [6]. Thus the estimation of a more accurate heat transfer coefficient on the fin is an important task for the device of the high-performance heat exchangers.

It is well known that the physical quantities and surface conditions of the test material can be predicted using the measured temperatures inside this material. Such problems are called the inverse heat conduction problems. These inverse problems have become an interesting subject recently. To date, various inverse methods in conjunction with the measured temperatures inside the test material have been developed for the analysis of the inverse heat conduction problems [7], [8]. However, to the authors’ knowledge, a few investigators performed the prediction of the local heat transfer coefficients on the annular circular fin inside the plate finned-tube heat exchangers.

Lin et al. [9] used the finite difference method in conjunction with the linear least-squares scheme to estimate the space-variable heat transfer coefficient on a heated cylinder normal to the laminar and turbulent air streams. Due to the requirement of the local fin temperature measurements, the estimations of the local heat transfer coefficients on the plate fin under steady-state heat transfer conditions are generally more difficult than those on the boundary surface of a physical geometry, as shown in Ref. [9]. Thus a few researchers predicted the distribution of the local heat transfer coefficients on a plate fin [3], [4], [10], [11], [12]. Saboya and Sparrow [10] and Rosman et al. [11] cast solid naphthalene plates in the form of a plate-fin-and-tube flow passage and used mass transfer techniques to infer the local heat transfer coefficient from the heat-mass transfer analogy. The local mass transfer coefficient was defined by measuring the thickness of naphthalene lost by sublimation during a timed test run. Recently, Ay et al. [12] performed an experimental study with the infrared thermovision to monitor the temperature distribution on a plate-fin surface inside the plate finned-tube heat exchangers and then the local heat transfer coefficient on the test fin can be determined using the obtained experimental temperature measurements. However, the fin efficiency on the fin inside the plate finned-tube heat exchangers was not shown in the work of Ay et al. [12]. Sometimes, it is maybe difficult to measure the temperature distributions on the fin of plate finned-tube heat exchangers using the infrared thermography and the thermocouples for some practical heat transfer problems. Park et al. [13] applied a commercial finite-volume computational fluid dynamics code in conjunction with the sequential linear programming method and weighting method to obtain the optimization of the plate heat exchanger with staggered pin arrays for a fixed volume. In this work [13], the flow and thermal fields are assumed to be periodic fully developed flow and heat transfer with constant wall temperature. Recently, natural convection heat transfer in annular fin-arrays mounted on a horizontal cylinder was experimentally investigated by Yildiz and Yüncü [6]. However, Yildiz and Yüncü [6] only showed the variation of the convection heat transfer rate with the fin spacing and the difference between the base temperature and the ambient temperature. The predicted results of the Nusselt numbers given by them compared with those obtained from Churchill and Chu’s correlation [14] and Morgan’s correlation [15] for natural convection from a horizontal circular cylinder. Chen et al. [3] applied the finite difference method in conjunction with the least-squares scheme and experimental temperature data to predict the fin efficiency and average heat transfer coefficient on the fin inside one-tube plate finned-tube heat exchangers for a single fin and various air speeds. Later, Chen and Chou [4] also applied the same scheme to predict the fin efficiency and heat transfer coefficient on the fin inside one-tube plate finned-tube heat exchangers in natural convection for various fin spacings. Due to the consideration of the radiation and convection heat transfer coefficients, the predicted results of the average heat transfer coefficient on the fin obtained from this scheme [4] are higher than those obtained from the correlation recommended by current textbooks [1], [16]. Thus, in order to validate the accuracy of this scheme further, the present study applies the similar method proposed by Chen and Chou [4] to estimate the fin efficiency and average heat transfer coefficient on the fin of annular-finned tube heat exchangers in natural convection. The present estimated results of the average natural-convection heat transfer coefficient on the annular circular fin also compare with those obtained from the correlation recommended by current textbooks [1], [16].

The inverse analysis of the present study is that the whole fin is divided into several analysis sub-fin regions and then the fin temperatures at these selected measurement locations are measured using T-type thermocouples. Later, the finite difference method in conjunction with these measured temperatures and least-squares method is applied to predict the average heat transfer coefficients on these sub-fin regions. Furthermore, the average heat transfer coefficient on the whole annular circular fin h¯ and fin efficiency can be obtained for various fin spacings under the given conditions of the ambient temperature and tube temperature.

The advantage of the present study is that the governing differential equations for the airflow do not need to be solved. In this study, the effect of the fin spacing on the estimation of the h¯ value and fin efficiency is investigated. The computational procedures for the estimates of the average heat transfer coefficients on each sub-fin region are performed repeatedly until the sum of the squares of the deviations between the calculated and measured temperatures becomes minimum.

Section snippets

Mathematical formulation

The experimental apparatus configuration of the present study is shown in Fig. 1. The schematic diagram of one-tube annular-finned tube heat exchanger in natural convection is shown in Fig. 2. Fig. 3 shows the physical model of the two-dimensional thin annular circular fin inside one-tube annular-finned tube heat exchanger. Ro and Ri denote the outer and inner radii of the annular circular fin, respectively. S and δ respectively denote the fin spacing and fin thickness. The center of the

Numerical analysis

It might be difficult to measure the temperature distributions on the whole annular circular fin using the infrared thermography and thermocouples for some practical heat transfer problems. Relatively, the unknown heat transfer coefficient h(r, θ) on a fin is not easy to be obtained. Under this circumstance, the whole annular circular fin considered can be divided into N sub-fin regions in the present inverse scheme and then the unknown heat transfer coefficient on each sub-fin region can be

Experimental apparatus

The schematic diagram of the experimental apparatus used in the present study for the estimation of the natural-convection heat transfer coefficient on an annular circular fin of one-tube finned-tube heat exchangers is shown in Fig. 1. This experiment is conducted in an open box, as shown in Fig. 1. This box with 550 mm in length, 450 mm in width and 300 mm in height is made of acrylic-plastic sheets. The horizontal circular tube with an outer diameter of 27 mm and 2 mm in thickness and the test

Results and discussion

It can be observed from Ref. [17] that the “insulated tip” assumption is a good approximation when the actual heat rate passed through the tip is negligible relative to the total heat rate drawn from the base wall. For simplicity, the average heat transfer coefficient on the tip surface can be assumed to be the same as that on the lateral surface of the fin. On the other hand, the “insulated tip” assumption will be reasonable provided that the surface area of the fin tip is very smaller than

Conclusions

The present study proposes a numerical inverse scheme involving the finite difference method in conjunction with the least-squares method and experimental fin temperatures at six measurement locations to estimate the average heat transfer coefficient on the whole plate fin h¯ and fin efficiency ηf in natural convection for various To, T and S values. The estimated results show that the fin temperature distributions depart from the ideal isothermal situation and the fin temperature decreases

Acknowledgements

The authors gratefully acknowledge the financial support provided by the National Science Council of the Republic of China under Grant No. NSC 92–2622–E006–146.

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