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Figure 7-1 Flow in a slit.

of the conduit as was done for tube flow, a relationship can be determined between flow rate and driving force for laminar flow in a conduit with a noncircular cross section. This can also be done by application of the equivalent integral expressions analogous to Eqs. (6-6) to (6-10). The results for a few examples for Newtonian fluids will be given below. These results are the equivalent of the Hagen-Poiseuille equation for a circular tube and are given in both dimensional and dimensionless form.

### 1. Flow in a Slit

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 Dh = 4A/Wp = 2h, and the solution for a Newtonian fluid in laminar flow is

This can be rearranged into the equivalent dimensionless form fNRe,h = 24 (7-3)

where

p pA

Here, A = Wh, and the Fanning friction factor is, by definition, f =_ef_=_-AO__(7-5)

2 ) \DhJ V 2 ) VDhy because the Bernoulli equation reduces to ef = — AO/p for this system. 2. Flow in a Film

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 6 to the vertical.

For this flow the hydraulic diameter is Dh = 4h (since only one boundary in the cross section is a wetted surface). The laminar flow solution for a Newtonian fluid is

AO h3W pgh3W cos 6

The dimensionless form of this equation is fNRe,h = 24 (7-7)

where the Reynolds number and friction factor are given by Eqs. (7-4) and (7-5), respectively.

### 3. Annular Flow

Axial flow in the annulus between two concentric cylinders, as illustrated in Fig. 7-3, is frequently encountered in heat exchangers. For this geometry the hydraulic diameter is Dh = (Do — Di, and the Newtonian laminar flow solution is

The dimensionless form of this expression is fNRe,h = 16a

where

It can be shown that as Di/Do ! 0, a ! 1 and the flow approaches that for a circular tube. Likewise, as Di/Do ! 1, a ! 1.5 and the flow approaches that for a slit.

The value of fNRe,h for laminar flow varies only by about a factor of 50% or so for a wide variety of geometries. This value has been determined for a Newtonian fluid in various geometries, and the results are summarized in Table 7-1. This table gives the expressions for the cross-sectional area and hydraulic diameter for six different conduit geometries, and the corresponding values of fNRe,h, the dimensionless laminar flow solution. The total range of values for fNReh for all of these geometries is seen to be approximately 12-24. Thus, for any completely arbitrary geometry, the dimension-less expression fNRe£ ^ 18 would provide an approximate solution for fully developed flow, with an error of about 30% or less.

### B. Turbulent Flows

The effect of geometry on the flow field is much less pronounced for turbulent flows than for laminar flows. This is because the majority of the energy dissipation (e.g., flow resistance) occurs within the boundary layer, which, in typical turbulent flows, occupies a relatively small fraction of the total flow field near the boundary. In contrast, in laminar flow the "boundary layer'' occupies the entire flow field. Thus, although the total solid surface contacted by the fluid in turbulent flow influences the flow resistance, the actual shape of the surface is not as important. Consequently, the hydraulic diameter provides an even better characterization of the effect of geometry for noncircular conduits with turbulent flows than with laminar flows. The result is that relationships developed for turbulent flows in circular pipes can be applied directly to conduits of noncircular cross section simply by replacing the tube diameter by the hydraulic diameter in the relevant dimensionless groups. The accuracy of this procedure increases with increasing Reynolds number, because the higher the Reynolds number the greater the turbulence intensity and the thinner the boundary layer; hence the less important the actual shape of the cross section.

Table 7-1 Laminar Flow Factors for Noncircular Conduits

(d + D)(64 - 3c4) c = (D - d)/(D + d) for 0.1 < D/d < 10

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