The applied simulation parameters are listed in Table 5.1. Two different pump speeds, called “low” and “high”, were applied in the experiments. Measured mass flow rates were used to calculate corresponding average fluid velocities. The velocities vf,max were calculated from these average velocities assuming a Hagen-Poiseuille flow profile. Initially, the particles start with zero velocity at the tube origin.
Table 5.1: Geometrical parameters, process parameters, and material parameters.
Symbol Value Unit
ec 0.94 −
µc 0.325 −
g 9.81 m s−2
l 20.5 m
R 2.5×10−3 m
vf,max,low 0.734 m s−1
vf,max,high 1.122 m s−1
ηf 1002×10−6 kg m−1s−1
ρf 998 kg m−3
ρp 2500 kg m−3
For a particle that reaches the upper limit of typical crystal sizes of 400µm, and for a large relative velocity, which is obtained when a particle is initially at rest in the tube center, Rep in Eq. (5.7) reaches 292 and 447 forvf,max,low andvf,max,high. These values are well inside the validity region of Eq. (5.8).
5.4.1 Stokes Number
The Stokes number St, which is related to the particle velocity, describes how well a particle can follow the fluid when the fluid flow field changes. It is defined as
St= τp τf
whereτf is a time characteristic of the flow field andτpis the relaxation time or momentum response time. The momentum balance in Eq. (5.4) describes the movement of a particle.
Disregarding body forces for particles of a high density compared with the fluid, Crowe (2006) assumes that the Stoke drag force determines the particle motion. Applying only the drag force, which was defined in Eq. (5.5), in the momentum balance in Eq. (5.4), the equation becomes
dvp
dt = 18ηf
dp2ρpfD(vf−vp)
Provided that the correction factor of the Stokes drag force fD is one, the Stokes τp is achieved. The first factor on the right-hand side is the reciprocal of τp
τp = dp2ρp 18ηf In general,
dvp dt = fD
τp (vf −vp) (5.15)
as stated for example by Sommerfeld (2013). Following Eq. (5.15), τp is the time a particle needs to accelerate from rest to 63 % of the fluid velocity (Crowe et al., 2012) after a step-wise change in the relative velocity.
The characteristic time of the fluid τf is the ratio of a characteristic length to the relevant fluid velocity. Following Michaelides et al. (2017), the characteristic length can be the diameter dp of spherical particles, which is often assumed for turbulent flows (Crowe, 2006). According to Kleinstreuer (2017), the tube diameter may be the key length supposing steady laminar particle suspension flow in a pipe, while, for a moving environment, it may be the tube length. Tsai and Pui (1990) apply the tube radius of bends as characteristic dimension and the average velocity as characteristic velocity.
Here, the average fluid velocity and tube diameter are considered as characteristic, and the following correlation is applied
τf = 4R vf,max
For particles of a diameter of 20µm, 50µm, 100µm and 400µm,Stin the Stokes regime is 0.004, 0.03, 0.1 and 1.6 forvf,low, and 0.006, 0.04, 0.2 and 2.5 for vf,high.
For St 1, τp is much smaller than τf, and a particle responds quickly to changes in the fluid velocity. The particle velocity approaches the fluid velocity (Crowe et al., 2012). This is the case for particles of a size below approximately 50µm. For St 1, a particle does not respond or responds only slowly to changes in the fluid velocity. In this case, the particle movement is dominated by the convective flow and gravity following Kleinstreuer (2017). Here, even the largest particles do not reach these values, and they are still affected by the fluid through drag.
5.4.2 Variation of the Initial Particle Position
In the experiments, the glass bead fraction was distributed in the cross section of the tube.
Therefore, in the single particle simulations, five different initial particle positions are compared, as depicted in Figure 5.3c. As visible in Figure 5.3a, an exemplary simulation at vf,max,low reveals that, for a fixed particle size from 100µm to 400µm, and for different initial positions, the RTs deviate less than 1 % from the RT at the tube origin. Comparing initial positions in the upper half of the tube to the origin, as may be expected, the time until the first wall collision is longer, as illustrated in Figure 5.3b. For initial positions at the same vertical height like the origin, the first wall contact happens earlier. The difference is below 1 s, which is negligible compared to the overall RT. Hereafter, only the origin is considered as initial position. Hence, no horizontal particle movement in x-direction can be observed.
5.4.3 Variation of the Wall Collision Coefficients
In this section, the sensitivity of the RT to variations in the wall collision coefficients µc and ec is discussed. The RT is determined by vp,z, which depends on the collision coefficients according to Eq. (5.13c). In Eq. (5.13c), the second term is always negative or zero. After wall collision, the largest reduction in vp,z occurs when ec is zero, in other words inelastic, and µc is large.
Typical values for the coefficients were listed in Table 5.1 and were taken from Groll (2015) for glass particles. The coefficients vary over a large range, not only depending on
a
200 400 600
in s 100
200 300 400
d p in µm
b
0.1 0.2 0.3 0.4 0.5 twall in s
100 200 300 400
d p in µm
c
-R 0 R
x -R
0 R
y
Figure 5.3: (a) RT and (b) time until first wall contact for particles of different size and initial position for vf,max,low and no initial particle velocity; (c) schematic of the cross section of the straight tube with initial particle positions (triangles or x-mark), tube wall (solid line), and position of particle center at wall contact (dotted line).
the material itself, but also on the surface roughness, which may change with material age. A typical value for the friction coefficient for soft glass on soft glass in air is one (Buckley, 1981). Lide (2004) gives a similar maximum value between glass and glass of 0.94 for the static coefficient of friction, and of 0.4 for the dynamic coefficient. For a rubber hemisphere sliding on glass, the sliding friction coefficient for dry conditions is about 2, for wet conditions it is about 0.8, and for elasto-hydrodynamic lubrication it is below 0.2 (Roberts and Richardson, 1981). Between polystyrene and polystyrene, a maximum value of 0.5 is reached for the static coefficient of friction (Lide, 2004). In general, the collision friction coefficient may also exceed one, but for the substances used here, and wet conditions, a limiting value of one is assumed.
Different combinations of values for the collision coefficients ec and µc can be assumed as limiting cases. That is to say,ec = 0 andµc= 1 for perfectly elastic collisions,ec = 0.94 and µc = 0 for no friction collisions,ec = 0 andµc= 1 for inelastic high friction collisions, and ec = 0.94 and µc = 0.325 for typical collisions. For all these combinations of ec and µc, simulations were performed with particles of 100µm to 400µm diameter. In all these cases, the second term in Eq. (5.13c) attributed to less than 1 % of the sum of the equation. The ratio
vp,n
vp,z
was also smaller than 1 %. Therefore, the direct effect of the
collision coefficients on the horizontal velocity in Eq. (5.13c) is negligible.
The elasticity coefficient influences the cross-sectional velocity in Eqs. (5.13a) and (5.13b). It might be expected that perfect elasticity leads to higher bounces after collision and extends the overall RT, but this effect is, also, negligible. The RTs are shortened by less than 1 % from that for the selected coefficients in case of perfectly elastic collisions or no friction conditions. For the opposite limiting case of inelastic collisions and high friction, it may be expected that particles do not reflect from the wall but stay near the wall in regions of small fluid velocities, immediately after first wall contact. In this case, the RT increases by 12 % for 400µm particles and by 5 % for 100µm particles atvf,max,high, and, at vf,max,low, by 22 % for 400µm particles and by 8 % for 400µm particles.