Two-dimensional thermal analysis of a polygonal fin with two tubes on a square pitch
Introduction
Heat exchangers (evaporators and condensers) are integral parts of vapour-compression refrigeration systems. Many of these heat exchangers employ extended surfaces on the air side comprising spaced thin metal sheets (normally either 0.2 mm or 0.4 mm thick) with tubes passing through each sheet. In these exchangers the evaporation or condensation occurs within the tubes and air flows between the sheets and over the tubes. The film heat transfer coefficient on the air side is normally two orders of magnitude smaller than the condensing or evaporating coefficients and without the extended surface arrangement would dominate the overall coefficient and result in very large units. The metal sheets provide considerable extra area on the air side and hence the combined film coefficient and additional area reduce the air side thermal resistance so that much larger overall heat coefficients are obtained and thus smaller exchangers may be employed. The sheets are mechanically stamped to produce a regular array of circular holes in either square or hexagonal arrangements. The excess metal forms a predetermined rim, which results in the spacing between adjacent sheets in the heat exchangers. The sheets are automatically fed onto the set of tubes to create the heat exchanger block. Finally, the tubes are mechanically expanded to provide intimate contact between the tubes, the rims and the sheet. To complete the heat exchanger, U bends are attached to the ends of adjacent tubes in the block so that various flow paths arrangements for the refrigerant within the tubes can be obtained, see Fig. 1. Each sheet in the final heat exchanger results in the so-called polygonal fin assembly, see Marin et al. [1] and Kraus et al. [2]. Unlike many other types of extended surfaces, the polygonal fin assembly results in direct conductive paths between the tubes passing through the sheet.
Unfortunately, all the sheet area is not fully utilised, because the temperature difference between the sheet and the air falls progressively from the base of the fin (tube outer surface) to the outer regions of the fin geometry. Design methods employ fin performance indicators (fin efficiency [2], [3], [4], fin effectiveness [5] and more recently, fin performance ratio [6]) to accommodate this phenomenon of under-utilisation of the additional area in the evaluation of the thermal resistance on the air side. Fin performance indicators that allow for the two-dimensional temperature distributions within polygonal fins do not exist and designers must resort to approximate techniques. However, these techniques do not attempt to consider the two-dimensionality (radial and angular) of the heat flow and the presence of other tubes in the assembly.
The theory of heat flow along fins attached to a primary surface at one end only has been thoroughly investigated, see e.g. the pioneering work of Harper and Brown [7] and the classical book of Kraus et al. [2]. However, the analysis of fin problems is conventionally based on the assumption that the heat flow is uni-directional because this fact, in general, facilitates an analytical treatment, see e.g. Gardner [3], [4] and Mikk [8]. The early investigations into the applicability of the one-dimensional approximation restricted attention solely to the fin and concluded that two-dimensional effects are negligible provided that the transverse Biot number (Bi = αδf/λf) is much less than unity, see Irey [9], Levitsky [10] and Lau and Tan [11]. However, later investigations of the combined fin and supporting surface, see e.g. Sparrow and Lee [12], Suryanarayana [13] and Heggs and Stones [14], have shown that the presence of fins induces transverse two-dimensional effects within the supporting surface and these may in turn act to produce two-dimensional variations within the fin. However, the two-dimensional transverse effects were found to occur only in long fins, when the coupled wall and fin models were investigated. The previous investigators have only considered the two-dimensional effects in a single fin with the base held at a fixed temperature. This lead to the conclusion that two-dimensional effects occur in short fins, whereas they occur near the base of very long fins and for these fins, the heat flow through the wall will be relatively large.
For polygonal fin assemblies, the relative length of the fins is short and so transverse conduction effects can be ignored. However, it is essential that multi-dimensional analysis in the other two directions of the fin surface is considered when developing performance indicators for the effective design of heat exchangers. Suryanarayana [13] has reported that the difference between heat transfer rates can be as much as 80%. It is therefore essential for the effective design of heat exchangers using polygonal fin assemblies to employ a multi-dimensional analysis.
In this paper, we extend the study of a single isolated polygonal fin on hexagonal and square pitches performed by Marin et al. [1] to the case of two tubes at different temperatures passing through a rectangular fin, see Fig. 2 which shows a photograph of polygonal fin heat exchanger blocks with one and two tubes on a square pitch produced by Elfin Technology Ltd. In the previous study for the single isolated polygonal fin, it was demonstrated that the two-dimensional temperature distribution in square and hexagonal fins, and the resulting heat flow from the fins, could be predicted by a one-dimensional radial rectangular fin with the equivalent surface area of the polygonal fins. Consequently, the conduction/convection problem for square fin assemblies is solved numerically by employing a convergent, stable and consistent two-dimensional boundary element method (BEM) based algorithm. The performance of a particular heat exchanger for which detailed experimental data is available [18] is predicted and compared in the normal energy procedure taking the equivalent radial fin with an adiabatic condition on the outer radius. Moreover, the isotherms corresponding to the geometry under investigation are also presented.
Section snippets
Mathematical formulation
From a geometrical point of view, square fins are characterised by the inner radius (the outer radius of the tube), ri, the radius of the circle which encloses the square fin, , the fin length, , see Fig. 3, and the half-fin thickness, δf. We also consider the corresponding equivalent radial rectangular fin which has the same surface area, inner radius and thickness as the square fin under investigation, the outer radius, , such that , and the fin length
Fin performance indicators
In this section, the performance indicators of the fins are briefly reviewed in the framework of the two-dimensional analysis. In the steady state, the heat flow through the fin, , is obtained from the temperature profile either by considering the heat flow through the base of the fins or by integrating the heat flow from the domain occupied by the fin, see [2], [3], [4], [5], [15], namelyor
The boundary element method
The governing non-dimensionalised partial differential equation (5) can also be formulated in integral form, see e.g. Chen and Zhou [17], as follows:for , where c(X) = 1 for and c(X) = 1/2 for (smooth), and E is the fundamental solution for the modified Helmholtz equation (5), which is given byHere r(X, Y) represents the distance between the load point X and the field point Y.
Numerical results
The numerical analysis presented in this section for the square fin assembly is based on typical geometries and operating conditions used in evaporators and condensers in commercial refrigeration storage units, see e.g. Rizvi [18]. We illustrate the numerical results obtained using the BEM described in the previous section by investigating the convergence, stability and consistency of the numerical method proposed. In addition, the performance of the square fin assembly is studied in the
Investigation of the range of applicability of the use of the one-dimensional analytical performance ratio for a polygonal fin with two tubes on a square pitch
The results reported in Table 1, Table 2 are for two polygonal fins with different thicknesses, δf = 0.1 × 10−3 m and δf = 0.2 × 10−3 m, respectively. The other dimensions and values of the heat transfer coefficient and thermal conductivity are the same in both cases. For the thicker fin, there are conductive paths between the tubes for a wider range of the temperature differences between the two tubes than were obtained for the thinner fin. The parameters for the second case are ri/δf = 35, l/δf = 78.70 and
Conclusions
A polygonal fin with two tubes arranged on a square pitch has been extensively analysed according to the two-dimensional theory. A semi-analytical solution has been obtained by the BEM and this solution is shown to be convergent, stable and consistent. This numerical method has increased accuracy due to the use of Green’s integral identities and, furthermore, the solution function and its normal derivative at the boundary are simultaneously predicted. Only the boundary of the solution domain
Acknowledgement
The authors would like to acknowledge the financial support received from the EPSRC.
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2011, Journal of Heat TransferParameter identification in two-dimensional fins using the boundary element method
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