Concentric annular ducts developing flow

Fully Developed Flow. Velocity distribution, the friction factor, and heat transfer for fully developed laminar flow in concentric annular ducts are described sequentially. 

Velocity Distribution and the Friction Factor. For a concentric annular duct with inner radius r, and outer radius r , the velocity distribution and friction factor for fully developed flow in a concentric annular duct are as follows [1] ... 

Heat Transfer. Fundamental solutions for boundary conditions of the first, second, and third kinds for fully developed flow in concentric annular ducts are given in Table 5.14. The nomenclature used in describing the corresponding solutions can best be explained with reference to the specific heat transfer parameters G) and 0 which are the dimensionless duct wall and fluid bulk mean temperature, respectively. The superscript k denotes the type of the fundamental solution according to the four types of boundary conditions described in the section entitled Four Fundamental Thermal Boundary Conditions. Thus, k = 1,2, 3, or 4. The subscript l in Gj 1 refers to the particular wall at which the temperature is evaluated / = i or o when the temperature is evaluated at the inner or the outer wall. The subscript j in G) 1 refers... 

TABLE 5.14 Fundamental Solutions of the First, Second, and Third Kinds of Boundary Conditions for Fully Developed Flow in Concentric Annular Ducts [1]... 

Thermally Developing Flow. The solutions for thermally developing flow in concentric annular ducts under each of the four fundamental thermal boundary conditions are tabulated in Tables 5.16, 5.17,5.18, and 5.19. These results have been taken from Shah and London [1]. Additional quantities can be determined from the correlations listed at the bottom of each table using the data presented. 

TABLE 5.15 Hydrodynamically Developing Flow Parameters and Constants to Use in Conjunction with Eq. 5.128 for Concentric Annular Ducts [103]... 

TABLE 5.21 Fundamental Solution of the First Kind for Simultaneously Developing Flow in Concentric Annular Ducts for Pr = 0.7 [104]... 

Effects of Eccentricity. In practice, a perfect concentric annular duct cannot be achieved because of manufacturer tolerances, installation, and so forth. Therefore, eccentric annular ducts are frequently encountered. The velocity profile for fully developed flow in an eccentric annulus has been analyzed by Piercy et al. [105]. Based on Piercy s solution, Shah and London [1] have derived the friction factor formula, as follows ... 

Fully Developed Flow. Knudsen and Katz [110] obtained the following velocity distributions for fully developed turbulent flow in a smooth concentric annular duct in terms of wall coordinates u+ and y+ ... 

Dwyer [113] has developed semiempirical equations for liquid metal flow (Pr < 0.03) in a concentric annular duct (0 < r < 1) with one wall subjected to uniform heat flux and the other... 

TABLE 5.27 Nusselt Numbers and Influence Coefficients for Fully Developed Turbulent Flow in a Concentric Annular Duct with Uniform Heat Flux at One Wall and the Other Wall Insulated [111]... 

Hydrodynamically Developing Flow. Hydrodynamically developing turbulent flow in concentric annular ducts has been investigated by Rothfus et al. [114], Olson and Sparrow [115], and Okiishi and Serouy [116]. The measured apparent friction factors at the inner wall of two concentric annuli (r = 0.3367 and r = 0.5618) with a square entrance are shown in Fig. 5.17 (r = 0.5618), where / is the fully developed friction factor at the inner wall. The values of/ equal 0.01,0.008, and 0.0066 for Re = 6000,1.5 x 104, and 3 x 104, respectively [114]. 

Thermally Developing Flow. Kays and Leung [111] present experimental results for thermally developing turbulent flow in four concentric annular ducts, r = 0.192,0.255,0.376, and 0.500, with the boundary condition of one wall at uniform heat flux and the other insulated, that is, the fundamental solution of the second kind. In accordance with this solution, the local Nusselt numbers Nu and Nu at the outer and inner walls are expressed as... 

Simultaneously Developing Flow. Little information is available on simultaneously developing turbulent flow in concentric annular ducts. However, the theoretical and experimental studies by Roberts and Barrow [118] indicate that the Nusselt numbers for simultaneously developing flow are not significantly different from those for thermally developing flow. 

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