Velocity profiles

Fig. IV-3. Velocity profiles for particles suspended in water with elapsed time, due to spreading of oleic acid. Time after onset of spreading O, 5 sec, , 5 sec. (From Ref. 31.)...

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

C3.3.4.2 VELOCITY PROFILES FOR TRANSLATIONAL-ROTATIONAL EXCITATION OF THE BATH... 

This result describes a parabolic velocity profile, as sketched in Fig. 9.5b. 

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Most flow meters are designed and caHbrated for use on turbulent flow, by far the more common fluid condition. Measurements of laminar flow rates may be seriously in error unless the meter selected is insensitive to velocity profile or is specifically caHbrated for the condition of use. 

Enough space must be available to properly service the flow meter and to install any straight lengths of upstream and downstream pipe recommended by the manufacturer for use with the meter. Close-coupled fittings such as elbows or reducers tend to distort the velocity profile and can cause errors in a manner similar to those introduced by laminar flow. The amount of straight pipe required depends on the flow meter type. For the typical case of an orifice plate, piping requirements are normally Hsted in terms of the P or orifice/pipe bore ratio as shown in Table 1 (1) (see Piping systems). 

This equation is appHcable for gases at velocities under 50 m/s. Above this velocity, gas compressibiUty must be considered. The pitot flow coefficient, C, for some designs in gas service, is close to 1.0 for Hquids the flow coefficient is dependent on the velocity profile and Reynolds number at the probe tip. The coefficient drops appreciably below 1.0 at Reynolds numbers (based on the tube diameter) below 500. 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

When the fluid is incmpressible and the velocity profiles are identical, = P2, (3 = /52, the velocity terms disappear and T is independent of. Equation 52 reduces to ... 

This integrates direcdy to yield a parabohc velocity profile ... 

The equiHbrium approach should not be used for species that are highly sensitive to variations in residence time, oxidant concentration, or temperature, or for species which clearly do not reach equiHbrium. There are at least three classes of compounds that cannot be estimated weU by assuming equiHbrium CO, products of incomplete combustion (PlCs), and NO. Under most incineration conditions, chemical equiHbrium results in virtually no CO or PlCs, as required by regulations. Thus success depends on achieving a nearly complete approach to equiHbrium. Calculations depend on detailed knowledge of the reaction network, its kinetics, the mixing patterns, and the temperature, oxidant, and velocity profiles. 

Fig. 8. Vertical velocity profile near impeller blade tip where the shear rate = AV/AV.

An alternative to traditional mass flow bin design is to use a patented BINSERT, which consists of a hopper-within-a-hopper below which is a single-hopper section (Fig. 15). The velocity pattern in such a unit is controded by the position of the bottom hopper. A completely uniform velocity profile can be achieved which results in an absolute minimum level of segregation. Alternatively, by changing the geometry at the bottom of the hopper, a velocity profile can be developed in which the center section moves faster than the outside, thus providing in-bin blending of the materials (7). 

Figure 5 shows the variation of the droplet mean axial velocity at the same axial location. The primary feature of this velocity profile is that the maximum velocity peaks at the centerline. The velocity magnitude and direction in the center region tend to be related to the hquid swid strength and axial distance. A reverse (recirculation) flow with negative velocity is possible if the swid is intense. Under such conditions, the maximum velocity tends to shift away from the centerline. 

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 ... 

By feeding the mixture through a converging nozzle, the velocity profile may be made nearly flat or uniform. A Bunsen flame in such a flow has a smaller range of stabiUty but the mechanism is essentially the same and the flame very closely approximates a cone. If the apex angle of the flame is , then S can be obtained from equation 21... 

The flow pattern efficiency depends solely upon the shape of the velocity profile in the circulating gas. In terms of the integrals appearing in the gradient equation, the flow pattern efficiency is given by equation 86. 

To evaluate the flow pattern efficiency, a knowledge of the actual hydrodynamic behavior of the process gas circulating in the centrifuge is necessary. Primarily because of the lack of such knowledge, the flow pattern efficiency has been evaluated for a number of different assumed isothermal centrifuge velocity profiles. 

The Optimum Velocity Profile. The optimum velocity profile (41), that is the velocity profile that yields the maximum value for the flow pattern efficiency, is one in which the mass velocitypv is constant over the radius of the centrifuge except for a discontinuity at the wall of the centrifuge (r = rP). This optimum velocity profile is shown in Figure 14a. For this case the following values for the separation parameters of the centrifuge are obtained... 

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