Velocity profiles in tubes

In turbulent flow, molecular diffusion is augmented by the presence of turbulent eddies and mixing is more intensive though, due to the flatter velocity profiles in tubes, the role of velocity gradient in dispersion diminishes. 

Although, as mentioned above, we are not in a position to determine the interface shape or the distance required for encapsulation, it is at least possible to estimate the conditions when problems are imminent. Following the procedure in Section 7.6.2 it is possible to calculate the velocity profile in tube flow in each layer for various viscosity differences. In Figure 7.55 are shown velocity profiles for two cases. First we consider the case when the viscosity of layer 2, jo2, is less than that of layer 1, r]oi (Note This is for tube flow.) When the flow is stable, the velocity profile of layer 1 is flat, whereas in layer 2 there is a large dependence of the velocity profile on the radius of the capillary. However, when i]02 > rjoi the flow is in an unstable condition, and the velocity profile varies more strongly with r in layer 1 than in layer 2. The variation of y with r is shown in Figure 7.56. When the... 

In the simple Bunsen flame on a tube of circular cross-section, the stabilization depends on the velocity variation in the flow emerging from the tube. For laminar flow (paraboHc velocity profile) in a tube, the velocity at a radius r is given by equation 20 ... 

Fig. 4.3.6 Velocity maps and profiles at differ- mark the NMR foldbacks from the stationary ent heights of the Fano column. The dark ring fluid at the inner surface of the fluid reservoir, surrounding the pipe at z= 1.5 mm (larger In the velocity profiles, the solid curves are the white arrow) is due to a layer of stationary fluid calculated Poiseuille profiles in tube flow, adhering to the pipe exterior following the Velocity images are reprinted from Ref. [20], dipping of the pipe into the reservoir at the with permission from Elsevier, start of the experiment. The small white arrows...

An alternative equation for smooth tubes was derived by Blasius based on observations that the mean velocity profile in the tube could be represented approximately by... 

Momentum Dissipation of a Gas-Solid Two-Phase Jet. Gas velocity profiles in a gas-solid two-phase jet inside a fluidized bed were determined at five different horizontal planes perpendicular to jet direction using a pitot tube (Yang and Keaims, 1980). The velocity profiles were integrated graphically, and gas entrainment into a jet was found to occur primarily at the base of the jet. 

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

Orifice meters, Venturi meters and flow nozzles measure volumetric flow rate Q or mean velocity u. In contrast the Pitot tube shown in a horizontal pipe in Figure 8.7 measures a point velocity v. Thus Pitot tubes can be used to obtain velocity profiles in either open or closed conduits. At point 2 in Figure 8.7 a small amount of fluid is brought to a standstill. Thus the combined head at point 2 is the pressure head P/( pg) plus the velocity head v2/(2g) if the potential head z at the centre of the horizontal pipe is arbitrarily taken to be zero. Since at point 3 fluid is not brought to a standstill, the head at point 3 is the pressure head only if points 2 and 3 are sufficiently close for them to be considered to have the same potential head... 

Figure S-4 Velocity profiles in a tube in plug-flow aid in laminar flow reactors.

Figure 2.10. Flow profiles in tubes and packed columns. (A) Laminar flow. r = tube radius, Vx =stream path velocity at radial position r. V = maximum flow velocity at tube center. (B) Turbulent flow.

Extensive survey of flow and heat transfer in liquid films flowing outside tubes. Measurements of temperature and velocity profiles in films of various liquids are reported, and a heat transfer mechanism is proposed. 

Experimental determination of velocity profiles in films flowing on vertical tube by stereoscopic chronophoto-graphic method. Films included glycerol, aqueous glycerol solutions, liquid paraffin, glycerol + surfactant. In smooth flow, profiles agreed with theoretical semi-... 

Equation (8.11) gave the differential distribution function that corresponds to a parabolic velocity profile in a tube. This specific result is now derived in a more... 

The velocity profile in a tube of length L is Vz(r). The normal case is for Vz(r) to have its maximum value at the centerline and to decline monotonically toward Vz(r) =0 at r = R. The volumetric flow rate is Q. The fraction of that flow rate associated with the region from the centerline to radial position r is found from the following integral ... 

CSTR for most reactions. These conditions are best met for short residence times where velocity profiles in the tubes can be maintained in the turbulent flow regime. In an empty tube this requires high flow rates for packed columns the flow rates need not be as high. Noncatalytic reactions performed in PFRs include high-pressure polymerization of ethylene and naphtha conversion to ethylene. A gas-liquid noncatalytic PFR is used for adipinic nitrile production. A gas-solid PFR is a packed-bed reactor (Section IV). An example of a noncatalytic gas-solid PFR is the convertor for steel production. Catalytic PFRs are used for sulfur dioxide combustion and ammonia synthesis. 

Flfl. 6-3 Influence of heating on velocity profile in laminar tube flow. 

Figure 8.1.1 Schematic representation of laminar velocity profile in a circular tube.

Therefore, the velocity profile in fiiUy developed laminar flow in a tube is parabolic with a maximum at the centerline and minimum (zero) at the tube wall. Also, the axial velocity h is positive for any r, and thus the axial pressure gradient dPIdx must be negative (i.e., pressure must decrease in the flow direction because of viscous effects). 

Because of the relatively flat velocity profile in turbulent flow, the channel geometry has only a small influence on the friction factor (as discussed in the previous section) and the Sherwood and Nusselt numbers. The turbulent Sherwood and Nusselt numbers of rod bundles can therefore be related to those of circular tubes. The few experimental data, as compiled by Ref. 6, suggest that for relative pitches between 1.1 and 2.0 (which... 

The first type of EOF nonuniformity was described in the classical article by Rice and Whitehead [1] written much earlier than the first works on CE. The equation for the EOF velocity profile in the infinitely long tube with radius a was given by... 

The liquid velocity profile in a tube is determined by the formula... 

SOLUTION The model ignores radial gradients and assumes a flat velocity profile in a tube of constant cross section. The a duldz) term in Equation 3.5 is ignored even though density varies with the extent of polymerization. Thns,... 

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