Thermodynamic optimization of free convection film condensation on a horizontal elliptical tube with variable wall temperature

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Abstract

This study focuses on the thermodynamic analysis of saturated vapor flowing slowly onto and condensing on an elliptical tube with variable wall temperature. An entropy generation minimization (EGM), technique is applied as a unique measure to study the thermodynamic losses caused by heat transfer and film-flow friction for the laminar film condensation on a non-isothermal horizontal elliptical tube. The results provide us how the geometric parameter ellipticity and the amplitude of non-isothermal wall temperature variation affect entropy generation during filmwise condensation heat-transfer process. The optimal design can be achieved by analyzing entropy generation in film condensation on a horizontal elliptical tube with further account for the amplitude of non-isothermal wall temperature variation.

Introduction

Filmwise condensation heat transfer of pure vapor flowing onto a body, such as a plate, tube, and sphere has been widely studied by many researchers, like Winkler et al. [1], and Yang and Hsu [2], in view of the practical importance in the design of condensers for power plants, air-conditioning equipments, and many other chemical industrial process equipments. Entropy generation in thermal engineering systems destroys energy available in the system, and reduces its efficiency, such as condenser and heat exchanger. Thus, entropy generation minimization is of great concern in phase-change heat-transfer problems associated with film condensation.

Bejan [3] pioneered the method of entropy generation minimization in heat and mass transfer analysis. Meanwhile, in his book [4], he also conducted the second-law analysis of thermodynamics via the minimization of entropy generation for the single phase convection heat transfer. Jani [5] provided optimization of falling film LiBr solution on a horizontal single tube based on the minimization of entropy generation that irreversibility of non-isothermal heat transfer dominates in comparison with the fluid flow friction and mass transfer. Sahin [6] investigated the effect of temperature-dependent viscosity on the entropy generation rate as well as the ratio of pumping power to heat transfer.

In a study on the forced-convection cooling of an electronic device consisting of a stack of printed circuit boards with heat generation chips, Furukawa [7] employed the entropy generation minimization (EGM) method to determine the optimum board spacing to maximize heat dissipation. The accuracy and reliability of the EGM method were confirmed by a satisfactory agreement between its predicted optimal board spacing and that obtained by the convectional thermal optimization method. Furukawa and Yang used the EGM method to optimize the fin pitch of a plate fin heat sink in free convection [8] and the channel flow in a package of parallel boards with discrete block heat sources [9].

From the above studies, one may see that entropy generation is associated with thermodynamic irreversibility which is common in all types of heat-transfer processes. Film condensation belongs to phase-change heat transfer, but little literature regarding its second-law analysis is investigated. The second-law analysis of the film condensation outside tubes still remains an unsettled question so far.

Adeyinka and Naterer [10] investigated the physical significance of entropy generation in plate film condensation. Their results for optimizing entropy generation and plate size are expressed in terms of a duty parameter. In addition, they observed that entropy generation provides a useful parameter in the optimization of a two-phase system. Lin et al. [11] discussed the second-law analysis on saturated vapor flowing through and condensed in horizontal cooling tubes. They noted that an optimum Reynolds number existed over the parametric range which the entropy generates at a minimum rate. Dung and Yang [12] used the EGM technique to conduct the second-law analysis in a saturated vapor flowing slowly onto and condensing on an isothermal horizontal tube and obtained an optimal diameter that generates a minimum of entropy at a given duty.

As for the enhancement of condensation heat transfer, several researches, such as Yang and Hsu [13] and Yang and Chen [14], Ali and McDonald [15] and Karimi [16] confirmed that tubes, fins, or extended surfaces of elliptical profiles with major axes aligned with gravity are superior to those of circular profiles. In addition to heat-transfer analysis, Li and Yang [17] started to conduct the thermodynamic analysis of saturated vapor flowing slowly onto and condensed on an isothermal elliptical tube. That paper investigated how the geometric parameter-ellipticity affects local entropy generation rate during filmwise condensation heat transfer process. On the other hand, Fujii et al. [18] presented that the wall temperature may often vary significantly over the circumferential length of the tube, even if the condensation on a circular tube with a variable wall temperature (a cosine distribution). For laminar filmwise condensation with vapor flow velocity and inclusion of pressure gradient effect, the mean heat-transfer coefficient is influenced significantly with increasing the wall temperature variation amplitude, as seen in Memory and Rose [19].

The present study will focus on the minimization of total entropy generation number to give an idea of optimal design on free convection film condensation outside an elliptical tube with variable wall temperature. An expression for the entropy generation number accounts for the combined action of the specified irreversibilities. This research on the entropy generation minimization will thus help us achieve the complete thermodynamic analysis, including first and second law, on laminar filmwise condensation outside a non-isothermal elliptical tube.

Section snippets

Thermal analysis

Consider a horizontal elliptical tube with major axis “2a” in the gravitational direction and minor axis “2b”, situated in a quiescent pure vapor which is at its saturated temperature Tsat. Moreover, the wall temperature Tw is considered to be non-uniform and much lower than the vapor saturation temperature Tsat. Thus, condensation occurs on the wall and a continuous film of the liquid runs downward over the tube under the influence of gravity.

The physical model under consideration is shown in

Results and discussion

The variation of dimensionless entropy generation numbers NS with Ra/Ja under the surface tension effects and variable wall temperature for various ellipticities are demonstrated in Fig. 2. Firstly, as in Yang and Chen [14] study, the present result also indicates that the mean heat-transfer coefficients increase with ellipticity. Secondly, the dimensionless entropy generation numbers augment with mean heat-transfer coefficients, i.e., NS is proportional to Ra/Ja. Hence, the dimensionless

Concluding remarks

An analytical study was performed on the entropy generation minimization of free convection film condensation on a non-isothermal horizontal elliptical tube. The result indicated that the optimal entropy generation number is proportional to one-fourth power of group parameter, Ra/Ja and to amplitude of non-isothermal wall temperature variation. The obtained results apply to quiescent vapor condensed outside horizontal elliptical tubes, and to very long elliptical tubes, with negligible

Acknowledgements

Funding for this study is provided by the National Science Council of the Republic of China under Contracts NSC 95-2212-E-151-064.

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