Noncircular ducts

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. 

For rectangular ducts Kays and Clark (Stanford Univ, Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for headng and cooling of air in rectangular ducts of various aspect rados. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 X free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. 

Noncircular Ducts Equations (5-50 ) and (5-50/ ) may be employed for noncircular ducts by using the equivalent diameter D = 4 X free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-HiU, New York, 1984) give charts for various noncircular duels encountered in compact heat exchangers. 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

A numerical solution procedure is reasonably flexible in accommodating variations of problems. For example, the Graetz problem could be solved easily for velocity profiles other than the parabolic one. Also variable properties can be incorporated easily. Either of these alternatives could easily frustrate a purely analytical approach. The Graetz problem can also be worked for noncircular duct cross sections, as long as the velocity distribution can be determined as outlined in Section 4.4. 

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. 

When dealing with noncircular ducts it is common to assume that the equations for the heat transfer rate for a circular pipe can be applied to the noncircular duct provided that the correct equivalent diameter is used for the noncircular duct. Now for a circular pipe ... 

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

Irvine, T. R. Noncircular Duct Convective Heat Transfer, in W. lbele (ed.), Modern Developments in Heat Transfer, Academic Press, Inc., New York, 1963. 

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. ... 

Y. Turbulent flow, noncircular ducts Use correlations with j 4 cross-sectional area wetted perimeter Can be suspect for systems with sharp comers. Parallel plates , 2 htv dgn = 4 - 2to + 2 h [141] p. 289... 

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-... 

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. 

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... 

Pressure scaleup factors for noncircular ducts are identical to those for circular ducts provided the shape of the duct (e.g., the width-to-height ratio for a rectangular duct) is not changed. Set S = for a rectangular duct. This conclusion is based on the use of the hydraulic radius, Equation 3.22, but it also applies to other equivalents such as the equivalent diameter of Equation 3.23. The pressure scaleup relationships are the same as those for Sr in circular tubes. 

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. 

R. K. Shah, A Correlation for Laminar Hydrodynamic Entry Length Solutions for Circular and Noncircular Ducts, J. Fluids Eng., (100) 177-179,1978. 

J. P. Hartnet, and T. F. Irvine Jr., Nusselt Values for Estimating Liquid Metal Heat Transfer in Noncircular Ducts, AIChE J., (3) 313-317,1957. 

P. C. Bandopadhayay, and C. M. Ambrose, A Generalized Length Dimension for Noncircular Ducts, Lett. Heat Mass Transfer, (7) 323-328,1980. 

J. P. Zarling, Application of Schwarz-Neumann Technique to Fully Developed Laminar Heat Transfer in Noncircular Ducts, J. Heat Transfer, (99) 332-335,1977. 

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 ... 

Shah and London [19] proposed the following correlations for thermal entrance solutions for circular and noncircular ducts having laminar developed velocity profiles and developing temperature profiles. 

Table 17.18 summarizes the dependence of Ap and h on V for developed and developing laminar and turbulent flows. Although these results are for the circular tube, the general functional relationship should be valid for noncircular ducts as a first approximation. 

Due to the well known corner effect (Nikuradse [12], Gessner [13]) the secondary current occur in noncircular ducts. In this study the values of u component were measured only in (x, X2) plane. The mean values are shown in Fig. 5. 

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. 

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