Fully developed duct flow

For fully developed duct flows in which it can be assumed that fhe fluid properties are constant, the form of the velocity and temperature profiles do not change with distance along the duct, i.e., considering the variables as defined in Fig. 2.12. if the velocity and temperature profiles are expressed in the form... 

For fully developed laminar flow, the shear stress at the wall of a circular duct is... 

The Rectangular Duct Thermal-Entry Length, with Hydrodynamically Fully Developed Laminar Flow... 

FULLY DEVELOPED LAMINAR FLOW IN A PLANE DUCT... 

Therefore, as was the case with fully developed pipe flow, the velocity profile in fully developed plane duct flow is parabolic. 

Next consider fully developed laminar flow through a plane duct whose wall temperature is kept constant As with pipe flow, for this boundary condition, Eq. (4.69) gives ... 

Water flows in a rectangular 5 mm x 10 mm duct with a mean bulk temperature of 20°C. If the duct wall is kept at a uniform temperature of 40°C and if fully developed laminar flow is assumed to exist, find the heat transfer rate per unit length of the duct. 

Consider constant-property, fully developed laminar flow between two large parallel plates, i.e., in a wide plane duct. One plate is adiabatic and the other is isothermal and the velocity is high enough for viscous dissipation effects to be significant. Determine the temperature distribution in the flow. 

Such flows effectively occur in many practical situations such as flows in heat exchangers. Attention will initially be given to fully developed flow, see References [1] to [12]. This will be followed by a discussion of developing duct flows. Lastly, a brief discussion of the numerical analysis of more complex duct flows will be presented. 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

Shah and London [40] have compiled the heat-transfer and fluid-friction information for fully developed laminar flow in ducts with a variety of flow cross sections as shown in Table 6-1. In this table the following nomenclature applies ... 

Table 6-1 Heat Transfer and Fluid Friction for Fully Developed Laminar Flow In Ducts of Various Cross Sections.

The analysis employed mass transfer coefficients for fully developed laminar flows in square ducts. 

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. ... 

For fully developed turbulent flow, the inner and outer convection coefficients are approximately equal to each other, and the tube annulus can be treated as a noncircular duct with a hydrauLc diameter of - 77, . The Nusselt num-... 

Velocity Profile and Friction Factor. The velocity profile of fully developed laminar flow of a constant-property fluid in a circular duct with an origin at the duct axis is given by the Hagen-Poiseuille parabolic profile, as follows ... 

Heat Transfer on Walls With Uniform Temperature. For this boundary condition, denoted as , temperature distribution in a circular duct for fully developed laminar flow in the absence of flow work, thermal energy sources, and fluid axial conduction has been solved by Bhatti [3] and presented by Shah and Bhatti [2], as follows ... 

Critical Reynolds Number. The Reynolds number, defined as umDhlv, is widely adopted to identify flow status such as laminar, turbulent, and transition flows. A great number of experimental investigations have been performed to ascertain the critical Reynolds number at which laminar flow transits to turbulent flow. It has been found that the transition from laminar flow to fully developed turbulent flow occurs in the range of 2300 Re < 104 for circular ducts [41]. Correspondingly, flow in this region is termed transition flow. More conservatively, the lower end of the critical Reynolds number is set at 2100 in most applications. 

Fully Developed Flow. In this section, the characteristics of the fully developed turbulent flow and heat transfer are presented for both a smooth and a rough circular duct with a diameter of 2a. 

FIGURE 5.7 Power law distribution for fully developed turbulent flow in a smooth circular duct [45]. 

Several friction factor correlations for fully developed turbulent flow in smooth, circular ducts are listed in Table 5.8. According to Bhatti and Shah [45], these formulas were derived from highly accurate experimental data for a certain Reynolds number range. 

The friction factor correlations for fully developed turbulent flow in a rough circular duct are summarized in Table 5.9. The friction factor for turbulent flow in an artificially roughed circular duct can be found in Rao [59]. 

TABLE 5.8 Fully Developed Turbulent Flow Friction Factor in Smooth, Circular Ducts [45]... 

Heat Transfer in Smooth Circular Ducts. For gases and liquids (Pr > 0.5), very little difference exists between the Nusselt number for uniform wall temperature and the Nusselt number for uniform wall heat flux in smooth circular ducts. However, for Pr < 0.1, there is a difference between NuT and NuH- Table 5.11 presents the fully developed turbulent flow Nusselt number in a smooth circular duct for Pr > 0.5. The correlation proposed by Gnielinski [69] is recommended for Pr > 0.5, as are those suggested by Bhatti and Shah [45]. In this table, the / in the equation is calculated using the Prandtl [52]-von Karman [53]-Nikuradse [43] Cole-brook [54] Filonenko [55] or Techo et al. [56] correlations shown in Table 5.8. 

TABLE 5.9 Fully Developed TUrbulent Flow Friction Factor Correlations for a Rough Circular Duct [48] (a = tube radius)... 

TABLE 5.11 Fully Developed Turbulent Flow Nusselt Numbers in a Smooth, Circular Duct for Gases and Liquids (Pr > 0.5) [48]... 

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