Elliptical ducts

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Elliptical ducts can be thought as a family of ducts, including geometries ranging from lenticular to circular ducts. The major and minor axes lengths are represented by 2a and 2b, respectively, in this section. The origin of the coordinate is the intersection of the major and minor axes. 

Presented in this section are the friction factors and Nusselt numbers for laminar flow in elliptical ducts. 

Fully Developed Flow. The velocity distribution for fully developed laminar flow in elliptic ducts with major axis 2a and minor axis 2b is given by Shah and London [1] as follows ... 

The fully developed incremental pressure drop number K(°°) for elliptic ducts has been found to be independent of the duct aspect ratio a = 2bl2a [187]. The value of (< >) is recommended to be 1.26 for practical calculations [2]. 

FIGURE 5.34 Fully developed Nusselt numbers for elliptical ducts [2]. 

Hydrodynamically Developing Flow. Bhatti [181] has analyzed hydrodynamically developing flow in elliptic ducts. The apparent friction factors and incremental pressure drop numbers can be expressed as ... 

Thermally Developing Flow. The local Nusselt number for thermally developing flow in elliptic ducts with uniform wall temperature NutT was obtained by Dunwoody [183] in terms of a double infinite series. These results are considered the most accurate as x > 0.005. Dun-woody s formula for calculating the mean Nusselt number is as follows ... 

For elliptic ducts subjected to the thermal boundary condition, Someswara et al. [185] have solved thermally developing flow The mean Nusselt number Num Hi can be computed using the following expression ... 

The friction factors for fully developed turbulent flow have been measured by Barrow and Roberts [186] in elliptical ducts with a = 0.316 and 0.415 in the range of 103 < Re < 3.105 and by Cain and Duffy [187] in elliptical ducts with a = 0.5 and 0.667 in the range 2 x 104 < Re < 1.3 x 105. Based on the data presented by Barrow and Roberts [186] and Cain and Duffy [187], Bhatti and Shah [45] have derived the following correlation to calculate the friction factor ... 

Heat transfer in fully developed turbulent flow in elliptical ducts has been determined in several investigations. A comparison of the different results has been presented by Bhatti and Shah [45]. It was concluded that the Gnielinski correlation for circular ducts can confidently be used to calculate the fully developed Nusselt number for elliptical ducts for fluids of Pr 0.5. 

For liquid metals, the fully developed Nusselt numbers can be determined using Eq. 5.236 for elliptical ducts with the boundary condition. The values of Nuslug required in Eq. 5.236 are given in Fig. 5.35. 

FIGURE 5.35 Slug flow Nusselt numbers for elliptical ducts with the boundary condition [169]. 

Dong and Ebadian [234] numerically obtained the friction factor for laminar flow in curved elliptic ducts. The friction factor ratio fjfs is represented by the following expression ... 

In subsequent research [235], thermally developing flow in curved elliptic ducts is analyzed for different a and Prandtl numbers. The local Nusselt numbers along the flow direction are shown in graph form, and the asymptotic values of the Nusselt numbers have been obtained, as is shown in Table 5.46. In a related study, the effects of buoyancy on laminar flow in curved elliptic ducts are discussed by Dong and Ebadian [236]. 

TABLE 5.46 The Asymptotic Values of the Nusselt Number for Curved Elliptic Ducts [235]... 

Elliptical Ducts With Internal Longitudinal Fins... 

FIGURE 5.48 An elliptical duct with internal fins. 

An elliptical duct with four internal longitudinal fins mounted on the major and minor axes, as shown in Fig. 5.48, has been analyzed by Dong and Ebadian [275] for fully developed laminar flow and heat transfer. In this analysis, the fins are considered to have zero thickness. The thermal boundary condition is applied to the duct wall, and / is defined as a ratio of Ha a = Hbib. The friction factors and Nusselt numbers for fully developed laminar flow are given in Table 5.52. 

TABLE 5.52 Friction Factors and Nusselt Numbers for Fully Developed Flow in an Elliptical Duct With Internal Fins [275]... 

A confocal elliptical duct is shown in Fig. 5.64. According to the analysis by Topakoglu and Arnas [295], the friction factor for fully developed laminar flow in confocal elliptical ducts can be computed by... 

TABLE 5.65 The / Re and NuHi for fully Developed Laminar Flow in Confocal Elliptical Ducts [1]... 

Elliptical Ducts With Centered Circular Cores... 

For elliptical ducts with centered circular cores, fully developed laminar flow has been analyzed by Shivakumar [300]. The/Re values are given in Table 5.66, in which a denotes the ratio of the length of the minor axis to the length of the major axis of the ellipse and r is the ratio of the diameter of the circular core to the length of the minor axis. 

TABLE 5.66 Fully Developed Friction Factors for Elliptical Ducts With Centered Circular Cores [300]... 

N. T. Dunwoody, Thermal Results for Forced Convection through Elliptical Ducts, J. Appl. Mech., (29) 165-170,1962. 

S. M. Richardson, Leveque Solution for Flow in an Elliptical Duct, Letters in Heat and Mass Transfer, (7) 353-362,1980. 

H. Barrow, and A. Roberts, Flow and Heat Transfer in Elliptic Ducts, Heat Transfer 1970, paper no. FC 4.1, Versailles, 1970. 

Z. F. Dong, and M. A. Ebadian, Effects of Buoyancy on Laminar Flow in Curved Elliptic Duct, J. Heat Transfer, (114) 936-943,1992. 

For example, from the known solution for laminar flow in an elliptical duct (B4, p. 69) of semiaxes b and c one obtains 2a b + c )l ib c as the appropriate wall-correction factor. 

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