Study on the effect of container tilt on the falling speed of suspended particles in gravity field

The boycott effect is widely used in wastewater purification, drinking water purification, and mixture treatment in industry and manufacturing. In this paper, we analyze the settling velocity of particles in the clear water and suspended zones, and investigate the reasons for the tilting angle of the vessel to accelerate the settling velocity of particles. This paper uses video tracking software to analyze the settling velocity of particles in the experiment and draws conclusions.


Introduction
In 1920, Arthur Boycott first discovered that the tilting of a container of particulate suspension in a gravitational field affects the setting of red blood cells (RBCs) in a slender vertical tube [1] , and is known as the Boycott effect.The boycott effect is used in the petroleum industry.Under certain inclination angles of the pipe, it is observed that the suspension separation of the oil well is the clearest [2][3][4][5][6] , and inclined deep wells need to be used in the treatment process.[9] .
Particle density, tilt angle, and liquid height all affect the particle settling velocity.In this paper, the effect of container tilt angle on particle settling velocity is studied.It is concluded that in a certain range, increasing the tilt angle will accelerate the particle settling speed.At the same tilt angle, the particle drop rate will be faster due to the particle-to-particle influence, resulting in the drop rate of particles in the suspension area being greater than the drop rate of particles in the clear water area.

Single particle
The descent process when the test tube is vertical is shown in Figure 1 (a): Assume that the particle is at a height of H from the bottom of the container, the liquid viscosity coefficient is μ, the particle radius is r, the mass is m, and the particle is subjected to the combined force of gravity and buoyancy  .
Where  , The descent process when the test tube is tilted is shown in Figure 1 (b): Simplifying the calculation of the tilted container, we assume that the particle collides with the container, and the velocity along the direction of the container wall is reduced to zero.We let the tilt angle of the container wall be , the height of vertical fall be  , and the height of the fall along the wall of the test tube be  .

𝑋 𝑏 𝑣 𝑡
(2) A comparison of the relationship between the sink time of particles sliding down from the highest place and the sink time of the previous kind shows that: a single particle falls vertically faster than the slope (   Ideally) Figure 1.Individual particle drop and container area division [10] .

N-S calculation of a single particle in a particle population
Divide the area inside the container into three sedimentation zones: clear water (I), suspension (II), and sediment (III), as shown in Figure 1 (c).
Clear Water zone: The fluid flow in the clear water zone is laminar, and the settling rate of particles in the vessel is determined by the Reynolds number Re based on the continuum media mechanics approach.
At lower Re numbers, the value ∧ (∧ is the ratio of settlement Grashov number Gr to Reynolds number Re) is higher, and at limit λ →∞, the Clearwater region produces a rate Q (Which can be described by the kinematics Pound Nakamura Kuroda (PNK) formula Where the function   depends on particle size, shape, material, surface charge, and other properties Suspension zone: When the volume concentration of the suspension is not too large (<2~3%), the settling velocity can be calculated using the Stokes formula The simulation only considers the influence of the wall and the inclined container.Due to the more complicated influence between the particles, the particles are not considered for the time being.We set different tilt angles, and specify the number of degrees of tilt for the vertically placed test tube as 0 degrees.The particle velocity after 0.1s release is obtained by releasing the particle cluster from the top of the test tube, as shown in Figure 2(a)-(f).It can be seen from the data shown in the simulation results that the larger the tilt angle is, the smaller the particle drop velocity is.
Particle groups in the finite fluid space of the settlement of individual particles will not only be affected by the direct friction of other particles and collision, but also by other particles through the fluid and the indirect impact.This settlement becomes an interference settlement.From the simulation results can be seen, as the tilt angle increases, the aggregation also increases, and the increase in the aggregation will lead to increased interference settling, increasing the settling resistance of the particles and making the settling end velocity reduced.

Design of experiments
This experiment mainly uses the Control variates to explore the impact of tilt angle on particle drop and uses Tracker video tracking software to track droplets.A protractor is used to divide the angle on the cardboard and specify a vertical tilt angle of 0 degrees.We fix the camera on the control bracket, pour vegetable oil into the test tube to two-thirds of its volume, then add uniform particles, flip the test tube, and let it stand for a period of time.After all the particles settle, we flip the test tube and place it at a certain angle for fixation.The camera records the process of settling.We change the tilt angle (by 10 or 5 degrees in sequence until the tilt angle reaches 60 degrees) as shown in Figure 3.The above operations are repeated.According to the horizontal comparison of (a) and (b), (c), and (d) suspension zones in Figure 4, it is shown that the larger the tilt angle the greater the particle drop velocity.The previous numerical analysis only considers the effects of the wall and inclined container, ignoring the influence between the particles and the particles.The conclusion is that the greater the tilt angle the smaller the particle drop velocity, indicating that the influence between the particles and the particles makes the particle velocity faster.

Conclusion
From the above, we can conclude that increasing the tilt angle will accelerate the particle settling speed within a certain range.And the settlement of the particle population increases with the increase of the tilt angle, and the aggregation degree then increases causing the increase of interference settlement as well as the vortex effect to produce resistance, resulting in the reduction of the settlement end velocity.At the same tilt angle, the particle drop speed becomes faster due to the particle-to-particle influence, causing the drop speed of particles in the suspension area to be greater than the drop speed of particles in the clear water area.At a tilt angle of 10 degrees, the average velocity of particles along the vessel wall increases by 39.91% in the suspension zone compared to the clear water zone.

Figure 2 .
Figure 2. Velocity diagram of particles released at different tilt angles after 0.1s.

Figure 3 .
Figure 3. Set different tilt angles for settling experiments.

Figure 4 .
Figure 4. Particle descent speed at different tilt angles and regions.The experimental data were processed to derive the average velocity of the particles along the vessel wall direction and plotted.From the vertical comparison of the clear water zone with the suspended zone in Figure 4(a) and the suspended area in Figure.4(c), and between the clear water area in Figure 4(b) and the suspended area in Figure.4(d), the velocity of particles in the clear water area is less than that in the suspension area.At an inclination angle of 10 degrees, the average velocity of particles along the wall of the container increases by 39.91% in the suspension zone compared to the clear water zone.According to the horizontal comparison of (a) and (b), (c), and (d) suspension zones in Figure4, it is shown that the larger the tilt angle the greater the particle drop velocity.The previous numerical analysis only considers the effects of the wall and inclined container, ignoring the influence between the particles and the particles.The conclusion is that the greater the tilt angle the smaller the particle drop velocity, indicating that the influence between the particles and the particles makes the particle velocity faster.