Terminal velocity wall effects

Here Kjj is obtained from Fig. 9.5. Equation (9-27) and the equations of Chapter 5 can be used to determine the decrease in Sh for a rigid sphere with fixed settling on the axis of a cylindrical tube. For example, for a settling sphere with 2 = 0.4 and = 200, Uj/Uj = 0.76 and UJUj = 0.85. Since the Sherwood number is roughly proportional to the square root of Re, the Sherwood number for the settling particle is reduced only 8%, while its terminal velocity is reduced 24%. As in creeping flow, the effect of container walls on mass and heat transfer is much smaller than on terminal velocity. 

All studies of drops and bubbles have been carried out in containers of finite dimensions hence wall effects have always been present to a greater or lesser extent. However, few workers have set out to determine wall effects directly using a series of different columns of varying diameter. Where studies have been carried out, the sole aim has usually been to determine the influenee of X on the terminal velocity. While it is known that the eontaining walls tend to... 

Fig. 9.8 Retarding effect of column walls on the terminal velocity of drops and bubbles of intermediate size.

Above a Reynolds number of the order of magnitude of 1000, bubbles assume a helmet shape, with a flat bottom (Eckenfelder and Barnhart, loc. cit. and Leibson et al., loc. cit.). After bubbles become large enough to depart from Stokes law at their terminal velocity, behavior is generally complicated and erratic, and the reported data scatter considerably. The rise can be slowed, furthermore, by a wall effect if the diameter of the container is not greater than 10 times the diameter of the bubbles, as shown by Uno and Kintner [AlChE 2, 420 (1956) and Collins, J. Fluid Meek, 28(1), 97 (1967)]. Work has... 

It is believed that this can be related to the differences in coefficient of restitution between the conveyed particles and the pipeline walls. On impact with the rubber, the particles will be decelerated, since the rubber will absorb much of the energy of impact. As a consequence, the particles will have to be re-accelerated back to their terminal velocity. The coefficient of restitution of the particles against the steel pipeline wall will be very much lower. This effect is clearly magnified by increase in velocity and explains why there is little difference between the two pipeline materials in low velocity dense phase conveying, but differ by 50% in high velocity dilute phase conveying. The results obtained with the barite were very similar. 

Thus, by measuring the value of rj may be found. The constant velocity u is often called the terminal velocity. The formula holds only if ur is small compared with rjy i.e. for very viscous liquids in the case of spheres of moderate size there are also corrections for the boundary conditions of the walls and base of the cylinder containing the liquid (the formula (1) being deduced for an infinite volume of liquid). If the liquid column is divided into three equal parts, and the centre one is used in timing the fall of the sphere, the correcting factor on the velocity for the wall effect is ... 

Use the problem setup in Problem 10.12 and deduce the wall effect when measuring the terminal velocity by dropping a sphere into a fluid. Compare with Perry and Green (1997, p. 6-54). 

High viscosity of a sonicated liquid lowers the cavitation threshold markedly. Viscous liquids generate bubbles only at high sound pressures. Bubble motion is damped by the dissipative effect of the viscosity and the smaller maximum bubble radii, and the lower inward wall velocities terminate most sonochemical effects. 

When the diameter of of the particle becomes appreciable with respect to the diameter Dw of the container in which the settling is occurring, a retarding effect known as the wall effect is exerted on the particle. The terminal settling velocity is reduced. In the case of settling in the Stokes law regime, the computed terminal velocity can be multiplied by the following to allow for the wall effect (Zl) ioxO JD < 0.05. 

V Terminal settling velocity corrected for wall effects or free-stream velocity... 

There is a crucial difference between the two situations however and this can be seen from the sign of the reflection coefficient. In the closed termination case, the tube ends in a solid wall and the reflection coefficient is -1. From Equation 11.16b we can see that in such a case the volume velocity will be 0. If we find the equivalent pressure expressions for Equation 11.16b, we see that the pressure terms add, and hence the pressure at this point is (pc)/.4i. Intuitively, we can explain this as follows. The solid wall prevents any particle movement and so the volume velocity at this point must be 0. The pressure however is at its maximum, as the wall stopping the motion is in effect an infinite impedance, meaning no matter how much pressure is applied, no movement will occur. In the open end situation, when the reflection coefficient is 1, we see from Equation 11.16b that the volume velocity is at a maximum and now the pressure is 0. Intuitively, this can be explained by the fact that now the impedance is 0, and hence no pressure is needed to move the air particles. Hence the pressure p L, t) is 0, and the volume velocity is at its maximum. 

The walls of the vessel containing the liquid exert an extra retarding effect on the terminal falling velocity of the particle. The upward flow of the displaced liquid, not only influences the relative velocity, but also sets up a velocity profile in the confined geometry of the tube. This effect may be quantified by introducing a wall factor, /, which is defined as the ratio of the terminal falling velocity of a sphere in a tube, V, to that in an imconfined liquid, V, viz.. 

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