The value of fins as a means of increasing the rate of heat transfer from a surface is widely known. Early fin analyses by Harper and Brown (7) and Gardner considered a constant heat transfer coefficient along the length of the fin. Later studies by Ghai ( 4 ) , Gardner ( 3 ) , Han and Lefkowitz ( 6 ) , and Chen and Zykowski (2) modified their analyses by considering the heat transfer coefficient as a function of position along the fin. All of these studies considered using fins in air.A study by Haley and Westwater ( 5 ) showed that the heat transfer coefficient along a fin during boiling is a function of temperature along the fin, generally expressed as a boiling curve. Using a numerical technique, they solved the general heat transfer equation and developed a design technique for the use of a fin during boiling. The main assumption in this development was that the heat transfer coefficient at any point on the nonisothermal heating surface is the same as would be obtained if the surface surrounding the point were at the same temperature. Their results were presented as graphs of temperature gradient at the fin base versus metal-to-liquid temperature difference at the base of the fin. A further analysis was made to determine the optimum shaped spine (of minimum volume). The optimum spine was found to be turnipshaped. One of the main limitations of the optimum fin was the difficulty of manufacture. This has led to the study of approximate optimum fins by Cash, Klein, and Westwater (1 ) and by Siman-Tov (1 1 ) .In all the previous work with boiling liquids, the interest was centered on a single fin. A discussion of Gardner's ( 3 ) results, in regard to air flowing past transverse fins, brought out the importance of fin spacing. For no interference it was suggested that fins be at least two boundary layers apart. The most effective spacing was found to be 12% larger than one boundary layer. Although some other studies by Jones and Smith (9), Welling and Woodridge Page 1050September, 1971or decrease, and at what spacing does it become noticeable? The spacings in this study refer to the clearance between the fins. EXPERIMENTAL APPARATUS A N D PROCEDUREA schematic diagram of the equipment is shown in
Eddy current (EC) technology is commonly used for detecting flaws, measuring geometric parameters, or determining properties of conducting materials. However, the measurement of a particular parameter can become more challenging if multiple influential parameters vary simultaneously. In particular, eddy current-based measurement of separation (gap) between a pressure tube (PT) and a calandria tube (CT) in the fuel channels of CANDU® reactors is made more difficult by variations in PT wall thickness, resistivity, and probe lift-off. An analytical model of the EC response to changes in PT–CT gap has been developed by approximating the geometry of the PT within the larger diameter CT as a pair of concentric tubes, where gap is varied by changing the CT radius. In this article, this model is used in combination with an error minimization algorithm to construct an inverse algorithm for the extraction of PT–CT gap, PT resistivity (ρ), and PT wall thickness (WT) from measured multi-frequency eddy current signals. Application of a linear regression tool in MATLAB, with fourth-order polynomial fitting of modeled data with varying ρ and WT as a function of PT–CT gap, is used to obtain coefficients that depend on ρ and WT. Output of multidimensional fitting of these coefficients is scaled and rotated to calibration data. Finally, implementation of an error minimization algorithm in MATLAB is used to produce estimates of multiple target parameters from experimental data. Simultaneous extraction of either one, two, or three parameters is examined, using experimental data obtained at frequencies used for in-reactor inspection of 4.2, 8, and 16 kHz, or just two frequencies of 4.2 and 8 kHz. Under full gap variation conditions, the inverse algorithm predicts gap to within 0.1 mm at gaps between 0 and 9 mm and to within 0.4 mm at gaps between 9 and 18 mm. PT resistivity is predicted to within 1 μΩ cm (2% relative error) and PT wall thickness within 0.03 mm (1% relative error) when each is the only extracted parameter. An excellent agreement between actual and predicted values of gap demonstrates the potential of the inverse algorithm for application to in-reactor gap measurement and simultaneous extraction of either PT wall thickness or resistivity when the other parameter is known. The extraction of PT resistivity may be particularly useful, as this parameter cannot otherwise currently be measured in-reactor.
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