Laminar duct flow Nusselt number

TABLE 5.1 Asymptotic Nusselt Numbers for Laminar Duct Flow (With Fully Developed Velocity Profiles)... 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts... 

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

This chapter has been concerned with the analysis of laminar flows in ducts with various cross-sectional shapes. If the flow is far from the inlet to the duct or from anything else causing a disturbance in the flow, a fully developed state is reached in many situations, the basic characteristics of the flow in this state not changing with distance along the duct. If the diffusion of heat down the duct can be neglected, which is true in most practical situations, it was shown that in such fully developed flows, the Nusselt number based on the difference between the local wall and bulk mean temperatures is constant. Values of the Nusselt number for fully developed flow in ducts of various cross-sectional shape were discussed. 

Flow in Noncircular Ducts The length scale in the Nusselt and Reynolds numbers for noncircular ducts is the hydraulic diameter, D), = 4AJp, where A, is the cross-sectional area for flow and p is the wetted perimeter. Nusselt numbers for fully developed laminar flow in a variety of noncircular ducts are given by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 307). For turbulent flows, correlations for round tubes can be used with D replaced by l. ... 

As a result of the development of the hydrodynamic and thermal boundary layers, four types of laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing). In this chapter, the term fully developed flow refers to fluid flow in which both the velocity profile and temperature profile are fully developed (i.e., hydrodynamically and thermally developed flow). In such cases, the velocity profile and dimensionless temperature profile are constant along the flow direction. The friction factor and Nusselt number are also constant. 

Thermally Developing Flow. Numerous investigators [80, 89-94] have carried out the investigation of turbulent thermally developing flow in a smooth circular duct with uniform wall temperature and uniform wall heat flux boundary conditions. It has been found that the dimensionless temperature and the Nusselt number for thermally developing turbulent flow have the same formats as those for laminar thermally developing flow (i.e., Eqs. 5.34-5.37 and Eqs. 5.50-5.53). The only differences are the eigenvalues and constants in the equations. 

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

Laminar flow and heat transfer in parallel plate ducts are described in this section. The friction factor and Nusselt number are given for practical calculations. 

Uniform Temperature at One Wall and Uniform Heat Flux at the Other. When the two walls of a parallel plate duct are subject to a thermal boundary condition such as uniform temperature at one wall and uniform heat flux at the other, the Nusselt numbers for fully developed laminar flow for qZ = 0 and q"w 0 are determined to be ... 

Uniform and Equal Heat Flux at Both Walls. The local Nusselt number for heat transfer of laminar flow in a parallel plate duct with uniform and equal heat flux at both walls is displayed in Fig. 5.22 for different Prandtl numbers, Pr = 0 [34] and Pr = 0.01, 0.7,1,10, and °° 136]. The thermal entrance lengths obtained from the results presented in this figure are 0.016,0.030,0.017, 0.014,0.012, and 0.0115, for Pr = 0,0.01,0.7,1,10, and °°, respectively. 

In this section, the friction factors and Nusselt numbers for fully developed, hydrodynami-cally developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented. 

TABLE 5.30 Nusselt Number for Fully Developed Laminar Flow in Rectangular Ducts With One Wall or More Walls Heating... 

The fully developed Nusselt numbers NuT and Nuhi for laminar flow in isosceles triangular ducts with one wall or more heated are given in Table 5.40. 

Arbitrary Triangular Ducts. For triangular ducts with arbitrary angles such as that shown in Fig. 5.27/ the fully developed friction factors and Nusselt numbers are presented for fully developed laminar flow in Figs. 5.29 and 5.30 [2]. 

TABLE 5.41 Local and Mean Nusselt Numbers for Thermally and Simultaneously Developing Laminar Flows and Equilateral Triangular Ducts [160]... 

In this section, emphasis will be given to the correlations used for calculating the friction factors and Nusselt numbers for laminar and turbulent flows in curved ducts, helicoidal pipes, and spiral ducts. These will be presented as the ratio of the friction factor in curved ducts to the friction factor in straight ducts fcJfs and the ratio of the Nusselt number in curved ducts to the Nusselt number in straight ducts Nuf/Nus, in most cases. The subscript c represents curved ducts or helicoidal pipes, while the subscript s denotes straight pipes of the same shape. 

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]. 

FIGURE 5.44 Friction factor and Nusselt number for fully developed laminar flow in rectangular ducts with longitudinal thin fins from opposite walls [1]. 

The mean Nusselt number for the laminar flow in isothermal circular ducts with inserted tape can be obtained from the following equation suggested by Manglik and Bergles [272] ... 

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. 

A cardioid duct is shown in Fig. 5.63. Fully developed laminar flow and heat transfer under the boundary condition have been analyzed by Tyagi [294]. The/Re and NuHj values derived from this analysis are 5.675 and 4.208, respectively. The Nusselt number for the thermal boundary condition was found to be 4.097 [1]. 

A careful observation of accurate experimental friction factors for all noncircular smooth ducts reveals that ducts with laminar/Re < 16 have turbulent/factors lower than those for the circular tube, whereas ducts with laminar/Re > 16 have turbulent/factors higher than those for the circular tube [48], Similar trends are observed for the Nusselt numbers. If one is satisfied within 15 percent accuracy, Eqs. 17.87 and 17.88 for/and Nu can be used for noncircular passages with the hydraulic diameter as the characteristic length in / Nu, and Re otherwise, refer to Table 17.16 for more accurate results for turbulent flow. 

Turbulent Flow. The thermal entry length solutions for smooth ducts for several cross-sectional geometries have been summarized [46]. As for laminar flow, the Nusselt numbers in the thermal region are higher than those in the fully developed region. However, unlike laminar flow, Nu,x and NuxHi are very nearly the same for turbulent flow. The local and mean Nusselt numbers for a circular tube with and boundary conditions are [46] ... 

A number of analytical results are available for fully developed Nusselt values for the laminar flow of power law fluids in rectangular channels having aspect ratios ranging from 0 (i.e., plane parallel plates) to 1.0 (i.e., a square duct). Newtonian results (n = 1) are available for the T, HI, and H2 boundary conditions for the complete range of aspect ratios. Another limiting case for which many results are available is the slug or plug flow condition, which corresponds to n = 0. At other values of n, results are available for plane parallel plates and for the square duct. 

The semi-empirical global transport correlation proposed by Hawthorn [15] is the most commonly used for the definition of local Nusselt and Sherwood numbers, applicable to laminar flows in square ducts ... 

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