Ducts of Noncircular Cross Section

Introducing the friction-factor definition / = Tw/(jpU2) produces 

The momentum equation can be solved directly by integration to produce 

It is conventional to take the Re/ product as a positive number, even though the wall stress (and hence the friction factor) are understood to point in the negative z direction. As stated, Eq. 4.68 produces a positive parabolic velocity profile with Re/ assumed to be a positive number. 

At this point in the nondimensional formulation, the factor Re/ is still undetermined. By considering the relationship of the nondimensional velocity profile and the mean velocity, Re/ is determined as an eigenvalue. Equation 4.59, which provides the relationship between U and u(r), may be put in nondimensional form as 

Once the Reynolds number (based on the mean velocity) is known for a given tube and flow situation, the friction factor follows as / = Re//Re. From the friction factor the wall shear stress and pressure gradient are easily determined. 

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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. 

Effective Diameters for Ducts of Noncircular Cross Section... 

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... 

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. 

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. 

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 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 ... 

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... 

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. 

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,... 

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