Concentric annular ducts

For the prediction of the Nusselt number in ducts of non-circular-cross section (like concentric annular ducts) the same equations can be used for forced convection in the turbulent regime. In this case, the inside diameter should be replaced in evaluating Nu, Re, and (D/L) by the hydraulic diameter defined as,... 

Concentric annular ducts are a common and important geometry for fluid flow and heat transfer devices. The double pipe heat exchanger is a simple example. In this device, one fluid flows through an inside pipe, while the other flows through the concentric annular passages. The friction factor and the heat transfer coefficient are essential for the design of such heat transfer devices. 

As shown in Fig. 5.13, there are two walls in concentric annular ducts. Either or both of them can be involved in heat transfer to a flowing fluid in the annulus. Four fundamental thermal boundary conditions, which follow, can be used to define any other desired boundary condition. Correspondingly, the solutions for these four fundamental boundary conditions can be adopted to obtain the solutions for other boundary conditions using superposition techniques. 

FIGURE 5.13 Fully developed Nusselt numbers for uniform temperatures at both walls in concentric annular ducts [1]. 

In this section, the characteristics of laminar flow and heat transfer in concentric annular ducts are presented, and the effect of eccentricity is discussed. 

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]... 

FIGURE 5.14 Fully developed friction factor and Nusselt numbers for concentric annular ducts [2],... 

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 ... 

It should be noted that when e = 0, the eccentric annular duct is reduced to a concentric annular duct. 

Critical Reynolds Number For concentric annular ducts, the critical Reynolds number at which turbulent flow occurs varies with the radius ratio. Hanks [109] has determined the lower limit of Recrit for concentric annular ducts from a theoretical perspective for the case of a uniform flow at the duct inlet. This is shown in Fig. 5.16. The critical Reynolds number is within 3 percent of the selected measurements for air and water [109]. 

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+ ... 

FIGURE 5.16 Lower limits of the critical Reynolds numbers for concentric annular ducts with uniform velocity at the inlet [109]. 

A critical review of the extensive friction factor data has been made by Jones and Leung [112]. The researchers recommend that the fully developed friction factor formulas for smooth circular ducts given in Table 5.8 be used for calculating the friction factor for concentric annular ducts by replacing 2a with the laminar equivalent diameter Dt for concentric annular ducts. The term D is defined by... 

The fully developed Nusselt numbers Nu and Nu, at the outer and inner walls of a smooth concentric annular duct can be determined from the following relations for uniform wall heat fluxes qo and q" at the outer and inner walls ... 

For r = 1, the concentric annular duct is reduced to a parallel plate duct. The applicable results are given in Table 5.28, the simple Nu being used for the Nusselt number at the heated wall. 

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]... 

For a concentric annular duct with the inner wall heated, the semiempirical equations developed by Dwyer [113] are applicable ... 

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]. 

FIGURE 5.17 Normalized apparent friction factors for turbulent flow in the hydro-dynamic entrance region of a smooth concentric annular duct (r = 0.5168) [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. 

Parallel plate ducts, also referred to as flat ducts or parallel plates, possess the simplest duct geometry. This is also the limiting geometry for the family of rectangular ducts and concentric annular ducts. For most cases, the friction factor and Nusselt number for parallel plate ducts are the maximum values for the friction factor and the Nusselt number for rectangular ducts and concentric annular ducts. 

The local Nusselt number is displayed in Fig. 5.23 for Pr = 0.0,0.01,0.7,10, and °° when one wall of the parallel plate duct is insulated and the other wall is subjected to uniform heat flux heating [140]. Included in Fig. 5.23 are the results for Pr = obtained from the concentric annular duct corresponding to r = 1. The local and mean Nusselt numbers for Pr = 0 were obtained by Bhatti [34]. 

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