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• W2025173894 abstract "Contributed by the Fluids Engineering Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Fluids Engineering Division September 14, 2000; revised manuscript received February 9, 2001. Associate Editor: D. R. Williams. Many researchers have investigated the Fanning factors in circular ducts and proposed many correlating equations to calculate the Fanning factors. As to various noncircular ducts, the frictional pressure drops have rarely been investigated. So it has been common practice in the field of fluid mechanics to use the hydraulic or equivalent diameter in the Reynolds number in predicting turbulent pressure drops along duct lengths having noncircular cross section. But, there is usually large deviation from the circular tube line by using the hydraulic diameter in the Reynolds number. And, therefore, some researchers have proposed various modifying methods to predict friction factors. In the case of rectangular ducts, for example, Jones 1 uses a “laminar equivalent diameter” to form the Reynolds number, which is in turn used in any circular tube correlation for friction factors. And in case of annuli, Brighton and Jones 2 modify the constant C in the Blasius equation on the basis of the experimental data. As to triangular ducts, Nan and Dou 3 use an area equivalent round diameter in the Reynolds number. In the case of regular polygonal ducts, there were no experimental data reported in the literature except equilateral-triangular 4 and square 5 ducts. Therefore, the purpose of the investigation reported here was to obtain friction factors for isothermal, fully developed, laminar, and turbulent flow in smooth equilateral-triangular, square, pentagonal, hexagonal, heptagonal, and octagonal ducts, respectively. Moreover, an area equivalent round diameter is proposed to use in Reynolds number in predicting Fanning factor of turbulent flow in a duct having regular (n-sided) polygonal cross section. The regular polygonal ducts were made of plate glass. Take a hexagonal duct for example. First cut six slabs of plate glass (width 8.10 mm, thickness 3.0 mm). Then precisely work pattern plates of the hexagonal polygon. Put the slabs of glass into the pattern plates. Finally seal the seams between the adjacent slabs with silica gel, as shown in Fig. 1. The ducts made of plate glass, therefore, are hydraulically smooth. The sketch of the experimental procedure used to obtain pressure drop data is illustrated in Fig. 2 and Fig. 3. In the setup used, water was pumped from the water pool into the elevated tank that had an overflow pipe to maintain a fixed water level, then flowed through a duct and past an abrupt entrance and into the hydrodynamic entrance section where it became fully developed. It then entered the test section that consisted of four identical test sections in series. These multiple section served to check on the reproducibility of the measurements and to insure that the flow was fully developed. In the test section, the pressure drop readings were made by means of an inclined manometer. The manometer fluid was benzaldehyde C7H6O with density 1.032 g/cm3 at 20°C. With an inclined manometer and benzaldehyde as the manometer fluid, the probable error is 0.95 percent of the pressure drop readings. After leaving the test section, the water flowed through a duct and back to the water pool. The flow rates were determined by measuring the volume of water during a known time interval. The pertinent dimensions of the duct are listed in Table 1 for the various regular polygonal ducts investigated. In Table 1, the hydraulic diameter de is defined as follows (1)de=4×SPewhere S is the cross-sectional area of fluid flow, and Pe is the wetted perimeter, the length of wall in contact with the flowing fluid at any cross section.The experimental data are presented in the form of Fanning factor against Reynolds number. These two quantities are defined as follows (2)f=de2Lu2ΔPρand (3)Re=deuρμwhere P is pressure, ρ is the density of the fluid, f is Fanning factor, L is the pipe length, de is the hydraulic diameter, u is the average velocity in flow direction, Re is Reynolds number, μ is the dynamic density of the fluid. For each duct, Fanning factors were measured in the laminar, transitional, and turbulent regimes. Figs. 4, 5, 6, 7, 8, and 9 show the relation between Fanning factors and Reynolds numbers for smooth equilateral-triangular, square, pentagonal, hexagonal, heptagonal, and octagonal ducts, respectively. Also shown in the figures, the solid lines are the laminar solutions 6.We suggest an area equivalent round diameter 3 be used in Reynolds number. An area equivalent round diameter may be defined as (4)dSe=4×SPS=4Sπwhere S is the cross-sectional area of fluid flow, and PS is the perimeter of a circle whose area is equal to S, PS=πdSe.For the regular (n-sided) polygonal ducts, as shown in Fig. 10, we can calculate their area equivalent round diameters as follows: According to the geometric relation shown in Fig. 10, the area of the triangle ABC is(5)Sn=a24 tanβ/2where a is the side length of the regular (n-sided) polygon; β is equal to 2π/n.So, the area of the regular (n-sided) polygon is (6)S=nSn=na24 tanπ/nSubstituting Eq. (6) into Eq. (4), we obtain (7)dSe=anπ tanπ/nThe modified Reynolds number may be defined as (8)Re*=dSeuρμThe relation between Re* and Re is (9)Re*=dSeuρμ=dSededeuρμ=k Rewherede=4Sna=atanπ/nk=dSede=n tanπ/nπIf Re* is substituted for Re in the Blasius equation, we have (10)4fRe*0.25=0.3164or(11)4f Re0.25=0.3164k−0.25=CTable 2 shows a comparison about C between Eq. (11) and the experimental results of Schiller, Hartnett et al., and the smooth equilateral-triangular, square, pentagonal, hexagonal, heptagonal, and octagonal ducts. By experimental data or from Figs. 456789, the experimental data are fitted to Eq. (11) when Re>3500. It has been shown in Table 2 that the use of an area equivalent round diameter to calculate a modified Reynolds number yields excellent agreement between Eq. (11) and the experimental data. The maximum deviation from the experimental data is within 1 percent. If the hydraulic diameter is used in the Reynolds number, the deviation from the experimental data, for equilateral-triangular ducts, is 0.3164−0.29420.2942=7.55 percentand, for square ducts, is 0.3164−0.30450.3045=3.91 percentTherefore, the area equivalent round diameter is proposed for use in predicting the Fanning factors of the regular (n-sided) polygonal ducts." @default.
• W2025173894 created "2016-06-24" @default.
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• W2025173894 date "2001-02-09" @default.
• W2025173894 modified "2023-09-23" @default.
• W2025173894 title "Prediction of Fully Developed Pressure Drops in Regular Polygonal Ducts" @default.
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• W2025173894 doi "https://doi.org/10.1115/1.1363699" @default.
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