Axial velocity profile

Fig. 59. Relation between the degree of chain extension and the axial velocity profile in...

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

The fomulation of Equation (8.68) gives the fully developed velocity profile, Fz(r), which corresponds to the local values of ix(r) and p(r) without regard to upstream or downstream conditions. Changes in Fz(r) must be gradual enough that the adjustment from one axial velocity profile to another requires only small velocities in the radial direction. We have assumed Vy to be small enough that it does not affect the equation of motion for V. This does not mean that Vr is zero. Instead, it can be calculated from the fluid continuity equation,... 

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. 

At lower Reynolds numbers, the axial velocity profile will not be flat and it might seem that another correction must be added to Equation (9.14). It turns out, however, that Equation (9.14) remains a good model for real turbulent reactors (and even some laminar ones) given suitable values for D. The model lumps the combined effects of fluctuating velocity components, nonflat velocity profiles, and molecular diffusion into the single parameter D. 

The axial velocity profiles, calculated on the basis of Tollmien similarity and experimental measurement in Yang and Kcaims (1980) were integrated across the jet cross-section at different elevations to obtain the total jet flow across the respective jet cross-sections. The total jet flows at different jet cross-sections are compared with the original jet nozzle flow, as shown in Fig. 31. Up to about 50% of the original jet flow can be entrained from the emulsion phase at the lower part of the jet close to the jet nozzle. This distance can extend up to about 4 times the nozzle diameter. The gas is then expelled from the jet along the jet height. 

Fig. 4.8 Illustration of the axial-velocity profile in the entry region of a circular duct.

Unfortunately, these equations cannot be modeled using the simple parallel-flow assumptions. In the entry region the radial velocity v and the pressure gradient will have an important influence on the axial-velocity profile development. Therefore we defer the detailed discussion and solution of this problem to Chapter 7 on boundary-layer approximations. 

Fig. 4.9 Transient nondimensional axial velocity profiles in a long circular duct, responding to a suddenly imposed pressure gradient. The fluid is initially at rest.

Figure 5.14 illustrates the nondimensional velocity profiles (Eq. 5.122) for different values of the cross-stream velocity V. As should be anticipated, for sufficiently low injection velocity V, the parabolic Poiseuille profile is obtained. As the injection velocity increases, the axial velocity profile is skewed toward the upper wall. 

Figure 5.16 shows the product Rej/ / as a function of Rev. For Rev less than approximately 2, the wall-injection has very little effect. In this case the wall friction approaches that of the Hagen-Poiseuille flow (i.e., Reyf = 24). For Rev greater than approximately 2, the V velocity serves to skew the axial velocity profile and thus affect the wall stress. For Rev greater than approximately 20, Re / approaches a linear relationship as... 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

Consider the two axial-velocity profiles in Fig. 17.10 that correspond to the low-strain solution in Fig. 17.9. While at the symmetry plane both solutions must have zero velocity, the inlet-velocity boundary conditions are quite different. In the finite-gap case (here the gap is 3.5 mm), the inlet velocity is specified directly (here as 250 cm/s). In the semiinfinite case, the inlet cannot be specified. Instead, the velocity gradient a = du/dz is specified, with the velocity itself growing linearly away from the surface. In the finite-gap case the strain rate is determined by evaluating the velocity gradient just ahead of the flame, where there is a region in which the velocity gradient is reasonably linear. In the semi-infinite case, the velocity gradient is specified directly, whereas in the finite-gap case it must be evaluated from the solution. 

Kostiuk et al. [40] measured experimentally the flow field of the vertical co-axial turbulent impinging streams with a two-component Laser Doppler velocity meter. The opposing gas streams were ejected from two burner nozzles, which were designed to produce a uniform axial velocity profile at their exits. The turbulence in the flow was generated by a perforated plate located at the end of the contraction section in each nozzle. The air velocity at the exit of the nozzle was varied from 4.1 to 11.4 m s and... 

Figure 1.11 Comparison between calculated non-dimensional axial velocity profile and...

Under the conditions of turbulence, the time-averaged velocity field is symmetric with respect to the free stagnation plane, provided the flow rates from the two nozzles are equal. The mean axial velocity profile has a similar shape to the curve of uju ) vs x. The gradient of the time-averaged axial velocity takes the maximum at the stagnation plane, while it approaches zero near the nozzle. 

Figure 6.5 Down channel, cross channel and axial velocity profiles for various situations that arise in a single screw extruder.

The plots in Figure 10.17 show the axial velocity profiles for various positions along the channel for a Deborah number, De = 1.87. The vertical, dashed lines indicate the locations of each scan of the channel. The initial profile, which is furthest upstream of the cylinder, shows an expected, parabolic shape. As the cylinder is approached, the velocity field becomes distorted, with a larger portion of the flow being diverted toward the lower region where the gap between the cylinder and the boundary of the channel is the largest. 

Figure 10.17 Axial velocity profiles for a Deborah number of De = 1.87. The vertical, dashed lines indicate the axial positions at which the channel was translated.

Fig. 6.14 Cross-channel, down-channel, and axial velocity profiles for various Qp/Qj values, in shallow square-pitched screws [Reproduced by permission from J. M. McKelvey, Polymer Processing, Wiley, New York, 1962.]...

Dead time can result from measurement lag, analysis, and computation time, communication lag or the transport time required for a fluid to flow through a pipe. Figure 2.27 illustrates the response of a control loop to a step change, showing that the response started after a dead time (td) has passed and reaches a new steady state as a function of its time constant (t), defined in Figure 2.23. When material or energy is physically moved in a process plant, there is a dead time associated with that movement. This dead time equals the residence time of the fluid in the pipe. Note that the dead time is inversely proportional to the flow rate. For liquid flow in a pipe, the plug flow assumption is most accurate when the axial velocity profile is flat, a condition that occurs when Newtonian fluids are transported in turbulent flow. 

Figure 9.3. RMS axial velocity profiles. Circles LDV solid line LES.

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective... 

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