Flow in Noncircular 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 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)]. 

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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. 

The foregoing arguments may be applied to turbulent flow in noncircular ducts by introducing a dimension equivalent to the diameter of a circular pipe. This is known as the mean hydraulic diameter, which is defined as four times the cross-sectional area divided by the wetted perimeter. The following examples are given ... 

This simple modification does not apply to laminar flow in noncircular ducts. 

Flow in noncircular ducts. In the case of a noncircular duct, the calculation follows that of the circular pipe using the same equations but with the diameter of the circular duct of simply replaced by the hydraulic diameter, Djj. The hydraulic diameter is simply defined as four times the cross-sectional area A divided by the wetted perimeter P . The factor 4 is used to obtain the diameter for a circular pipe. 

This well-known relationship is valid for laminar flow in circular ducts, but it also sets the stage for more general scaling relationships in noncircular cross sections and turbulent flows. 

The critical Reynolds number for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1,900 < Rec < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558-561 [1966]). In triangular ducts, 1,600 < Rec < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., 11, 106-117 [1972] Bandopadhayay and Hinwood, J. Fluid Mech., 59, 775-783 [1973]). 

The mathematical analysis of flow in ducts of noncircular cross section is vastly more complex in laminar flow than for circular pipes and is impossible for turbulent flow. As a result, relatively little theoretical base has been developed for the flow of fluids in noncircular ducts. In order to deal with such flows practically, empirical methods have been developed. 

For circular pipes, Rh = R- The reader is cautioned that some definitions of Rh omit the factor of 2 shown in Equation 3.22 so that the result must be multiplied by 2 for use in equations such as 3.18 and 3.19. The use of Rh is not recommended for laminar flow, but alternatives are available in the literature. Also, the method of false transients applied to PDEs in Chapter 16 can be used to calculate laminar velocity profiles in ducts with noncircular cross sections. For turbulent, low-pressure gas flows in rectangular ducts, the American Society of Heating, Refrigerating and Air Conditioning Engineers recommends use of an equivalent diameter defined as... 

In certain instances, we wish to treat flows in noncircular conduits such as slits, square ducts, and rectangular ducts. 

The problem of two-phase flow through noncircular ducts has received considerable attention in the last decades " and was studied specifically by Ransohoff and Radke. They addressed the problem of the low Reynolds number wetting liquid flow in a noncircular capillary occupied predominantly by a nonwetting gas phase by presenting a solution in terms of a dimensionless flow resistance, fi, depending on corner geometry and fluid parameters. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

This is only a relatively rough guide as very complex flows in the transitional region can arise in some cases involving noncircular ducts. 

As with noncircular ducts the hydraulic mean diameter is employed in formulae that involve diameter. If a channel has a height of a and a width b, the flow area of the channel is ab. In the calculation of the wetted perimeter the free surface is not included so that the wetted perimeter is 2a - - b, and the hydraulic mean diameter... 

For rectangular and other noncircular ducts in turbulent flow, the standard procedure is to replace R with the hydraulic radius... 

For a circular duct, the hydraulic diameter is equal to its physical diameter. For a noncircular duct, it is convenient to use the hydraulic diameter to substitute for the characteristic physical dimension. However, for ducts with very sharp corners (e.g., triangular and cusped ducts), the use of the hydraulic diameter results in unacceptably large errors in the turbulent flow friction and heat transfer coefficients determined from the circular duct correlation. Other dimensions have been proposed as substitutes for hydraulic diameter. These equivalent diameters, provided for specific ducts only, will be presented elsewhere in this chapter. 

It is generally accepted that the hydraulic diameter correlates Nu and /for fully developed turbulent flow in circular and noncircular ducts. This is true for the results accurate to within 15 percent for most noncircular ducts. Exceptions are for those having sharp-angled corners in the flow passage or concentric annuli with inner wall heating. In these cases, Nu and /could be lower than 15 percent compared to the circular tube values. Table 17.16 can be used for more accurate correlations of Nu and /for noncircular ducts. 

Laminar Flow. Based on the solutions for laminar boundary layer development over a flat plate and fully developed flow in circular and some noncircular ducts, /app Re can be correlated by the following equation ... 

Reynolds number for flow outside a cylinder, d G /p, dimensionless Reynolds number for flow past a sphere, d G Jp, dimensionless Reynolds number for flow in a noncircular duct, d G /p, dimensionless Reynolds number computed with x as the length dimension, dimensionless fractional rate of surface-element replacement, 0" cross-sectional area of a duct,... 

A significant heat-transfer enhancement can be obtained when a noncircular tube is used together with a non-Newtonian fluid. This heat-transfer enhancement is attributed to both the secondary flow at the comer of the noncircular tube (23,24) and to the temperature-dependent non-Newtonian viscosity (25). Using an aqueous solution of polyacrylamide the laminar heat transfer can be increased by about 300% in a rectangular duct over the value of... 

A variety of noncircular passage geometries, including the rectangular duct, have been utilized for internal flow applications, for example in compact heat exchangers and solar collectors. The study of the hydrodynamic behavior in a rectangular duct requires a two-dimensional... 

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. 

Previously it was indicated that for noncircular conduits Fig, 6.10 (the friction factor plot) could be used if we replaced the diameter in both the friction factor and the Reynolds number with 4 times the hydraulic radius (HR). The hydraulic radius is the cross-sectional area perpendicular to flow, divided by the wetted perimeter. For a uniform duct this is a constant. For a packed bed it varies from point to point. But if we multiply both the cross-sectional area and the perimeter by the length of the bed, it becomes... 

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