Velocity profile distribution

Fluid Kinematics. Water flowing at a steady rate in a constant-diameter pipe has a constant averse velocity. The viscosity of water introduces shear stresses between particles that move at different velocities. The velocity of the particle adjacent to the wall of the pipe is zero. The velocity increases for particles away from the wall, and it reaches its maximum at the center of the pipe for a particular flow rate or pipe discharge. The velocity profile in a pipe has a parabolic shape. Hydraulic engineers use the average velocity of the velocity profile distribution, which is the flow rate over the cross-sectional area of the pipe. 

For this purpose, a simplified scheme [2] presented in Fig. 5.1 is used. The smooth tube 1 with the igniter 2 at its closed end is illustrated in Fig. 5.1a. The primarily smooth combustion front 3 propagates through the tube, where the initial combustible mixture 4 is transformed into combustion products 5. The expanding combustion products push the unburned gas 6 with a velocity profile distributed across the tube cross-section. The flame front 7 starts distorting during propagation and the fresh mixture stream 8 becomes unstable (Fig. 5.1b). After that, the... 

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. 

Flow distribution in a packed bed received attention after Schwartz and Smith (1953) published their paper on the subject. Their main conclusion was that the velocity profile for gases flowing through a packed bed is not flat, but has a maximum value approximately one pellet diameter from the pipe wall. This maximum velocity can be 100 % higher than the velocity at the center. To even out the velocity profile to less than 20 % deviation, more than 30 particles must fit across the pipe diameter. 

The dispersion that takes place in an open tube, as discussed in chapter 8, results from the parabolic velocity profile that occurs under conditions of Newtonian flow (i.e., when the velocity is significantly below that which produces turbulence). Under condition of Newtonian flow, the distribution of fluid velocity across the tube... 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Entrainment ratio is another jet characteristic commonly used in air distribution design practice. Specifically, it is used in analytical multizone models (see Chapter 8) when one needs to evaluate the total airflow rate transported by the jet to some distance from a diffuser face. Airflow rate in the jet, Q,., can be derived by integrating the air velocity profile within the jet boundaries ... 

An analytical solution of the interaction in the case of isothermal main and directing jets, assume that the main stream (Fig. 7.57), supplied with initial velocity (t oi) through a nozzle that has internal diameter (Tqj), is developing within a zone ( -/q, 0) as a free jet. The momentum (/,) of the jet within the zone ( -Iq -F /, 0) remains equal to the initial momentum (/oi)> the velocity distribution in the cross-section of interaction in the plane XY remains the same within the zone (0, X ). The axisymmetric main stream within the zone (0, X j) is substituted by the linear flow with velocity profile that can be described by the formula... 

FIGURE 7.101 Pressure distribution doe to wind effect o) uniform wind velocity profile (6) non-uniform wind velocity profile. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

The advantages of monosized chromatographic supports are as follows a uniform column packing, uniform flow velocity profile, low back pressure, high resolution, and high-speed separation compared with the materials of broad size distribution. Optical micrographs of 20-p,m monosized macroporous particles and a commercial chromatography resin of size 12-28 p,m are shown in Fig. 1.4. There is a clear difference in the size distribution between the monodispersed particles and the traditional column material (87). 

In any liquid flowing down a surface, a velocity profile is established with the velocity increasing from zero at the surface itself to a maximum where it is in contact with the surrounding atmosphere. The velocity distribution may be obtained in a manner similar to that used in connection with pipe flow, but noting that the driving force is that due to gravity rather than a pressure gradient. 

Figure 3.32. (a) Shear stress distribution in pipe (b) Velocity profile for Bingham plastic fluid in pipe... 

The velocity and temperature distributions in a cross-section of a circular microtube are plotted in Figs. 4.22 and 4.23. It is seen that the velocity profile is determined by a single parameter, Z, whereas the temperature profile depends on two dimensionless groups, Z and S. 

Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution. Calculations that assume a parabolic profile will be conservative in the sense that they will predict a lower conversion than would be predicted for the actual profile. The changes in velocity profile due to variations in temperature and composition are normally much more important than the fairly subtle effects due to non-Newtonian behavior. 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature... 

Suppose you are marching down the infamous tube and at step j have determined the temperature and composition at each radial point. A correlation is available to calculate viscosity, and it gives the results tabulated below. Assume constant density and Re = 0.1. Determine the axial velocity profile. Plot your results and compare them with the parabolic distribution. 

Material flowing at a position less than r has a residence time less than t because the velocity will be higher closer to the centerline. Thus, F(r) = F t) gives the fraction of material leaving the reactor with a residence time less that t where Equation (15.31) relates to r to t. F i) satisfies the definition. Equation (15.3), of a cumulative distribution function. Integrate Equation (15.30) to get F r). Then solve Equation (15.31) for r and substitute the result to replace r with t. When the velocity profile is parabolic, the equations become... 

The above derivation assumes straight streamlines and a monotonic velocity profile that depends on only one spatial variable, r. These assumptions substantially ease the derivation but are not necessary. Anal5dical expressions for the residence time distributions have been derived for noncircular ducts,... 

F r) Cumulative distribution function expressed in terms of tube radius for a monotonic velocity profile 15.29... 

Although NMRI is a very well-suited experimental technique for quantifying emulsion properties such as velocity profiles, droplet concentration distributions and microstructural information, several alternative techniques can provide similar or complementary information to that obtained by NMRI. Two such techniques, ultrasonic spectroscopy and diffusing wave spectroscopy, can be employed in the characterization of concentrated emulsions in situ and without dilution [45],... 

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