The turbulent flow region

This effect can be demonstrated with a table tennis ball. If this is immersed in water, and released from a position near the surface, it will break through the surface at a high velocity, whereas if it is released several inches below the surface it will barely penetrate it. In the first case the ball s momentum is low but its velocity is high in the second case the momentum is high but the velocity is low. Since momentum is the product of mass times velocity, the mass in the second case must be greater than 

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This general method can be used for geometrically scaled reactors with Newtonian liquids or with Newtonian suspensions in the turbulent flow region (NRe > 1000). 

These dimensionless groups also appear in empirical correlations of the turbulent flow region. Although even in the approximate Eq. (9) of Table 6.7, group He appears to affect the friction factor, empirical correlations such as Figure 6.5(b) and the data analysis of Example 6.10 indicate that the friction factor is determined by the Reynolds number alone, in every case by an equation of the form, / = 16/Rc, but with Re defined differently for each model. Table 6.7 collects several relations for laminar flows of fluids. 

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

Initially, the boundary-layer development is laminar, but at some critical distance from the leading edge, depending on the flow field and fluid properties, small disturbances in the flow begin to become amplified, and a transition process takes place until the flow becomes turbulent. The turbulent-flow region may be pictured as a random churning action with chunks of fluid moving to and fro in all directions. The transition from laminar to turbulent flow occurs when... 

In the turbulent-flow region distinct fluid layers are no longer observed, and we are forced to seek a somewhat different concept for viscous action. A qualitative picture of the turbulent-flow process may be obtained by imagining macroscopic chunks of fluid transporting energy and momentum instead of... 

In the turbulent-flow region, where eA1 > v and eH > a, we define the turbulent Prandtl number as... 

Approximate equations showing the relationship between the friction factor and the Reynolds number in the turbulent-flow region have been developed. Two of these equations follow ... 

The first relation gives the average heat transfer coefficient for die entire plate when the flow is laminar over the etUire plate. The second relation gives the average heal transfer coefficient for the entire plate only when the flow is iiir-bulent over the entire plate, or when the laminar flow region of the plate is too small relative to the turbulent flow region. 

This equation is solved with respect to u under the simplifying assumptions (M31, U5) that is constant in the turbulent flow region, and that bubble gas holdup b is distributed radially according to the relation ... 

Despite that there were two substrates, back-to-back, in the converter, Pb compound was found mainly on the front brick. The front loading phenomena suggests that because the Pb compound is heavy in density, it can therefore settle comparatively rapidly once the linear veloeity of the exhaust gas drops from the turbulent flow region at the endcone of the converter to the laminar flow region inside the honeycomb channel. 

In the turbulent flow region a very efficient mixing takes place, i.e., macroscopic chunks of fluid move across stream-FIGURE 1.7 Property variation with time at some lines and transport energy and mass as well as momentum point in a turbulent boundary layer. vigorously. The most essential feature of a turbulent flow is... 

A major contribution to the study of purely viscous nonnewtonian fluids in the turbulent flow region was made by Dodge and Metzner [72], who proposed the following turbulent pipe flow correlation to predict the friction factor ... 

In the turbulent flow region and at the same pressure drop the tube side often offers better heat transfer conditions than the shell side. This depends, however, on various factors tube length, number of tube passes, tube diameter, physical properties of the fluid, etc. To arrive at the optimum solution in a minimum of time, use of the following calculation method is suggested. The method is best explained by examples. The equations used have been derived from well-known heat transfer principles. 

Draw the equivalent of Fig. 6.13 for a 2-in schedule 40 pipe, on a piece of 2 cycle by 2 cycle log paper. Show (a) the zero-viscosity boundary, (b) the laminar-flow region, (c) the turbulent-flow region, and (d) the transition region. 

The contribution of concentration polarization to overall dialytic mass transfer resistance suggests that dialyzers should be operated with flows in the turbulent flow region to minimize boundary-layer formation. While this is typically the case on the dialysate side of the membrane, dialyzers usually are operated with laminar flow on the feed side. In hemodialysis, this is partly to avoid undue mechanical stress on the blood cells which may result in their destruction. However, a consideration of fluid dynamics dictates that feed-side flow be laminar in nearly all applications. 

In the turbulent flow region, it is not possible to obtain an analytical solution for the friction factor as we do for laminar flow. Most of the data available for evaluating the friction factor in turbulent flow have been derived from experiments. For turbulent flow (Reynolds number above 4(X)0), the friction factor is dependent upon the pipe wall roughness as well as the Reynolds number. For turbulent flow, Colebrook (1939) found an implicit correlation for the friction factor in round pipes. This correlation converges well in a few iterations. 

In general, non-Newtonian fluids behave similarly to Newtonian fluids in the turbulent flow region in that they exhibit relatively constant ap-... 

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