Annular Ducts

Preliminary experiments showed that it was necessary to oscillate 7% of the total fuel flow to achieve useful attenuation and that the result was insensitive to phase within 30° of the optimum. Control performance was sensitive to the mean equivalence ratio in the pilot stream. Figure 19.4a shows that the attenuation was maximum when the equivalence ratio in the pilot stream was close to unity. Figure 19.46 shows that the attenuation increased to around 12 dB as the equivalence ratio was increased from the lean stability limit of around 0.65 in the annular duct to 0.7. The attenuation declined, however, with further increase in overall equivalence ratio and was negligible for values... 

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,... 

Knaff and Schlunder [9] studied the evaporation of naphthalene and caffeine from a cylindrical surface (a sintered metallic rod impregnated with the solute) to high-pressure carbon dioxide flowing over an annular space around the rod. They studied the diffusion flux within the bar and in the boundary layer. The mass-transfer coefficient owing to forced convection from cylinder to the gas flowing in the annular duct was correlated, using the standard correlation due to Stephan [7]. For caffeine, it does not require a free-convection correction, as the Reynolds dependence is that expected by a transfer by forced convection. This is... 

Oosthuizen. P.H. and Paul, J.T., A Numerical Study of Free Convective Row through a Vertical Annular Duct , ASME Paper 86-WA/HT-81, ASME Winter Annual Meeting, Anaheim. CA, December 7-12, 1986. 

For annular ducts, the accuracy of the Nusselt number given by (5-48) is improved by the following multiplicative factors [Petukhov and Roizen, High Temp., 2,65 (1964)]. 

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. 

Cheng and Hwang [108] analyzed the heat transfer problem in eccentric annular ducts. The Nusselt numbers for fully developed flow in eccentric annular ducts with the and thermal boundary conditions are given in Table 5.26. For eccentric annular ducts with boundary conditions different from the four described in the section entitled Four Fundamental... 

TABLE 5.26 Nusselt Number NuHi and NuH2 for Fully Developed Laminar Flow in Eccentric Annular Ducts [1,108]... 

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. 

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