Noncircular conduits

All the relationships presented in Chapter 6 apply directly to circular pipe. However, many of these results can also, with appropriate modification, be applied to conduits with noncircular cross sections. It should be recalled that the derivation of the momentum equation for uniform flow in a tube [e.g., Eq. (5-44)] involved no assumption about the shape of the tube cross section. The result is that the friction loss is a function of a geometric parameter called the hydraulic diameter  

By either integrating the microscopic momentum equations (see Example 5-9) or applying a momentum balance to a slug of fluid in the center 

Flow between two flat parallel plates that are closely spaced (h W) is shown in Fig. 7-1. The hydraulic diameter for this geometry is Db = 4A/Wp = 2h, and the solution for a Newtonian fluid in laminar flow is 

This can be rearranged into the equivalent dimensionless form. AW = 24 where 

The flow of a thin film down an inclined plane is illustrated in Fig. 7-2. The film thickness is h W, and the plate is inclined at an angle 0 to the vertical. 

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It is also informative to express these results in dimensionless form, i.e., in terms of appropriate dimensionless groups. Because this is a noncircular conduit, the appropriate flow length parameter is the hydraulic diameter defined by Eq. (5.48) ... 

Table 7-1 Laminar Flow Factors for Noncircular Conduits...

The circular tube expressions for/ and 7VRe can also be transformed into the equivalent expressions for a noncircular conduit by the substitution... 

The cross section of the fluid in the partially full pipe will not be circular (see Fig. 7-6), so the methods used for flow in a noncircular conduit are applicable, i.e., the hydraulic diameter applies. Thus, Eq. (7-61) becomes... 

This result can be used to apply the previous equations for circular pipes to conduits of any other shape, by replacing D in the appropriate equation with for the noncircular conduit. This gives excellent results for turbulent flows, for which the boundary layer is generally thin relative to the flow area dimensions, since the wall resistance (i.e., friction loss) is confined to a region very near the wall and, consequently, is not very sensitive to the shape of the cross section. 

For laminar flows, the laminar sublayer fills the entire cross section. Thus, the effect of wall drag is influenced to a greater extent by the shape of the cross section. For many noncircular conduits with laminar flows, either theoretical analyses (similar to the Hagen-Poiseuille equation) or numerical analyses for the pressure-flow relation have been conducted. The results can be expressed in dimensionless form as... 

For turbulent flows in noncircular conduits, the Moody diagram or ChurchiU equation can be used with good results, if the relative roughness is taken as e/Hj. 

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow... 

Laminar flowlin a few kinds of noncircular conduits can be analyzed by the same technique as used for circular pipes. 

Turbulent flow in many kinds of noncircular conduits can be estimated by substituting 4j times the hydraulic radius for the diameter in the friction factor plot and the friction factor equation. 

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... 

For noncircular conduits the channel aspect ratio or aspect factor (see Eqs. 2 and 3) has a profound influence on the friction factor Cf = /Re = ApD / piuL, which has a constant value of 16 for circular conduits. For example. Fig. 2c provides a comparison of the simulation results with an empirical correlation of Wu and Cheng [12], i.e.,... 

ISOTHERMAL FLOWS OF MOLTEN AND THERMALLY SOFTENED POLYMERS IN NONCIRCULAR CONDUITS... 

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

Studies on noncircular conduit cases of heat transfer to and from polymer systems have been quite limited. What little work that has been done has concentrated on the parallel-plate case. 

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