Maximum radial velocity, variation

Figure 1.10 Variation of maximum radial velocity with radial distance.

Fig. 6.4 Streamlines for two axisymmetric Hiemenz stagnation flow situations having different outer velocity gradients, one at a = 1 s 1 and the other at a = 5 s—. Both cases are for air flow at atmospheric pressure and T = 300 K. The streamlines are plotted to an axial height of 3 cm and a radius of 10 cm. However, the solution itself has infinite radial extent in both the axial and radial directions. In both cases the streamlines are separated by 2jt A l = 2.0 x 10-5 kg/s. The shape of the scaled radial velocities V = v/r is plotted on the right of the figures. The maximum value of the scaled radial velocity is Vmax = a/2. Even though streamlines show curvature everywhere, the viscous region is confined to the boundary layer defined by the region of V variation. Outside of this region the flow behaves as though it is inviscid.

A porosity of 23 % was observed at a 75 mm torch-target distance but this was reduced to 11 % at 50 mm, the distance which was found to correspond to the maximum particle velocity on the axis of the jet. The pore morphology was different in each case with much narrower interconnecting porosity in the low porosity coating. The effect of velocity is also shown by the radial variation of porosity in a deposit sprayed onto a fixed target. The particle velocity on the axis of the jet was observed to be 150 m/s greater than that at the periphery (Fig. 58), and the porosity of the deposit was observed to range from 10% on the axis of the jet to 22% at 15 mm radius. 

Figure 5. Schematic of the postulated similarity fiowfield for the steady growth of a liquid-vapor mixture bubble within a superheated liquid. Radial variation of pressure is shown for a bubble radial velocity of 147.1 m/s in water superheated to the spinodal point of 600 K and 2.89 MPa. The bubble velocity corresponds to an evaporation wave velocity of 72.85 m/s, slightly above the CJ velocity of 67.7 m/s but below the maximum velocity of 78.5 m/s.

The maximum values of X are located in the vicinity of the impeller tips. The change of sign of the radial velocity component at the exit of the impeller leads to the existence of two maximum values on each side of the impeller tip. The shear stress is rather important in the volume defined by the impeller. The maximum value is located at the place where there is the greatest variation of the velocity, i.e., at the impeller tip. The comparison of the fields of stresses x and x with the ones obtained in the case of a paddle agitator confirms the considerable shear effect produced by the impeller tip. 

In actual practice, TF reactors deviate from the plug flow model because of velocity variations in the radial direction (see Figure 10.4(b)-(d)). For any of these conditions, the residence time for annular elements of fluid within the reactor will vary from some minimal value at a point where the velocity is a maximum to a maximum value near the wall where the velocity approaches zero. The concentration and temperature profiles, as well as the velocity profile are therefore also not constant across the reactor. The describing equations based on the plug flow assumption are then not applicable. 

Price (1968) made similar measurements, but placed a m.onolith right over the bed. This eliminated the radial components of the flow velocity after the fluid left the bed. Price also operated at an Rep an order of magnitude higher than Schwartz and Smith and his conclusion was that the maximum is at a half-pellet diameter distance from the wall. Vortmeyer and Schuster (1983) investigated the problem by variational calculation and found a steep maximum near the wall inside the bed. This was considerably steeper than those measured experimentally above the bed. 

In terms of Eq. (7.12), the variations of the radial profiles of the dimensionless tangential velocity V with Rew are plotted as shown in Fig. 7.6. It is seen that the dimensionless tangential velocities reach a maximum at r < 0.5 for values of Rew varying from 10 to 30. 

With the measurements subject to fluctuations of 20 or 30%, no accurate description of the profile is possible. All that can be said is that with moderate ratios of tube to particle diameter, the maximum velocity is about twice the minimum, and that when the particles are relatively small, the profile is relatively flat near the axis. It is fairly well established that the ratio of the velocity at a given radial position to the average velocity is independent of the average velocity over a wide range. Another observation that is not so easy to understand is that the velocity reaches a maximum one or two particle diameters from the wall. Since the wall does not contribute any more than the packing to the surface per unit volume in the region within one-half particle diameter from the wall, there is no obvious reason for the velocity to drop off farther than some small fraction of a particle diameter from the wall. In any case, all the variations that affect heat transfer close to the wall can be lumped together and accounted for by an effective heat-transfer coefficient. Material transport close to the wall is not very important, because the diffusion barrier at the wall makes the radial variation of concentration small. 

The water injection pressure, seepage velocity and wetting effect distribute intensively in the borehole under the water injection arrangement of 80 m length hole under the condition of 20 MPa pressure have the same variation characteristics with 50 m length hole. Pressure decay by 76.97% in the scope of the drill hole radial 7.5 m under 20 MPa pressure, maximum seepage speed reach more than 14.52 m/s, and the minimum velocity is 2.9 m/s, the water addition distribute between 13% - 1%, and its minimum addition is 1.2%. 

The measurements of the local properties of two-phase systems during cultivation indicate that radial profiles of ds are fairly uniform. Also, their longitudinal variations are fairly moderate, except in the neighborhood of the aerator (1, 4). The same holds true for the spacial variations of the local relative gas holdups. At low superficial gas velocities the specific interfacial area, a, is fairly uniform also At high superficial gas velocities (turbulent or heterogeneous flow range) the radial profile of a has a shape of an error function, with its maximum in the column center (5). The behavior of these parameters near the aerator depends on the aerator itself and on the medium character. 

The pressure drop due to the change in kinetic energy is of course related to the velocity profile in the capillary which is influenced by the non-Newtonian characteristics of the fluid and radial variations in temperature associated with viscous heating. Calculations Investigating these effects indicate that the differences in kinetic energy due to these modifications of the velocity profile are negligible for fluids which have the characteristics of pol ymer-mod i f i ed lubricants as long as the 10 maximum pressure-drop correction criteria is maintained. 

In laminar flow, the velocity profile across the tube diameter is not flat. If the fluid is Newtonian, and there are no radial variations in temperature or concentration, the velocity profile will be parabolic. In laminar flow, there will be radial concentration gradients at any point along the axis of the tube, since the fluid velocity at the wall approaches zero, whereas the velocity at the center of the tube is at a maximum. The fluid at the wall of the tube spends a long time in the reactor. Therefore, the concentration of reactant is relatively low in this region. The fluid at the centerline of the tube has the highest velocity, so that the reactant concentration is relatively high at this position. 

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