Fully developed duct flow turbulent

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. 

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. ... 

The Nu and / factors are also dependent upon the duct cross-sectional shape in laminar flow, and are practically independent of the duct shape in turbulent flow. The influence of variable fluid properties on Nu and / for fully developed laminar flow through rectangular ducts has been investigated by Nakamura et al. [57]. They concluded that the velocity profile is strongly affected by the p /p ratio, and the temperature profile is weakly affected by the p /pm ratio. They found that the influence of the aspect ratio on the correction factor (p ,/pm)m for the friction factor is negligible for p /pm < 10. For the heat transfer problem, the Sieder-Tate correlation (n = -0.14) is valid only in the narrow range of 0.4 < p /pm < 4. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

There is also a standardized method based on the estimation of the flow rate on one measurement point only, In this method the velocity probe is placed in the duct so that the measured local velocity is equal to the mean axial velocity. In fully developed turbulent duct flow, this distance from the wall... 

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-... 

In this section, turbulent flow and heat transfer in a circular duct with a diameter of 2a is discussed for fully developed and developing flow. 

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. 

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]... 

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+ ... 

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]. 

Effects of Eccentricity. Jonsson and Sparrow [119] have conducted a careful experimental investigation of fully developed turbulent flow in smooth, eccentric annular ducts. The researchers have provided the velocity measurements graphically in terms of the wall coordinate h+ as well as the velocity-defect representation. From their results, the circumferentially averaged fully developed friction factor is correlated by a power-law relationship of the following type ... 

Fully Developed Flow. Beavers et al. [145] obtained the following friction factor for fully developed turbulent flow in a parallel plate duct for 5000 < Re < 1.2 x 106 from very accurate experimental data ... 

Comparisons of precision using Eqs. 5.220 and 5.221 and Blasius s formula (Table 5.8) in which the diameter of circular duct 2a is replaced by hydraulic diameter 4b, b being the halfspace between two plates, have been conducted by Bhatti and Shah [45]. In the range of 5000 < Re < 3 x 104, Eq. 5.220 is recommended otherwise, Eq. 5.221 should be used to obtain the friction factor for fully developed turbulent flow in a parallel plate duct. However, use of the hydraulic diameter to substitute for the circular duct diameter in the Blasius equation is reasonable for the prediction of the fraction factor [45]. 

Analogous to circular ducts, the fully developed turbulent Nusselt numbers for uniform wall temperature and uniform wall heat flux boundary conditions in parallel plate ducts are nearly identical for Pr > 0.7 and Re > 105. This is also true for the Nusselt number of turbulent thermally developing flow in a parallel plate duct [147]. 

The fully developed friction factor and heat transfer coefficients for turbulent flow in an asymmetrically heated rectangular duct have been reported by Rao [59]. In this investigation, the experimental region of the Reynolds number was from 104 to 5 x 104. 

For fully developed Nusselt numbers for the turbulent flow of liquid metals in rectangular ducts, a simple correlation has been derived for the and boundary conditions [169]. This correlation follows ... 

Isosceles Triangular Ducts. Bhatti and Shah [45] recommended that the friction factor for fully developed turbulent flow in isosceles triangular ducts can be determined using different correlations. For 0 < 2< ) < 60°, the circular duct correlations in Table 5.8 can be used with Dh replaced by Dgt as defined by Bandopadhayay and Ambrose [177] ... 

FIGURE 5.31 Fully developed friction factor for turbulent flow in smooth right-angled and equilateral triangular ducts [45]. 

The friction factors for fully developed turbulent flow have been measured by Barrow and Roberts [186] in elliptical ducts with a = 0.316 and 0.415 in the range of 103 < Re < 3.105 and by Cain and Duffy [187] in elliptical ducts with a = 0.5 and 0.667 in the range 2 x 104 < Re < 1.3 x 105. Based on the data presented by Barrow and Roberts [186] and Cain and Duffy [187], Bhatti and Shah [45] have derived the following correlation to calculate the friction factor ... 

Heat transfer in fully developed turbulent flow in elliptical ducts has been determined in several investigations. A comparison of the different results has been presented by Bhatti and Shah [45]. It was concluded that the Gnielinski correlation for circular ducts can confidently be used to calculate the fully developed Nusselt number for elliptical ducts for fluids of Pr 0.5. 

Fully Developed Turbulent Flow in Curved Rectangular and Square Ducts... 

Note that T 0 = 1 on the fluid side, which does not have fins. For fully developed turbulent flow through constant cross-sectional ducts, the Nusselt number correlation is of the form... 

Transition flow and fully developed turbulent flow Fanning friction factors for a circular duct are given by Bhatti and Shah [46] as... 

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