5. THE CONVEYING CHARACTERISTICS OF THE MATERIALS AND
8.6 A Model for Predicting the Limit of Stable Conveying in Suspension
The application of the Kelvin Helmholz instability, to predict the limit of stability in suspension flow, rests on the fact that when the conveying limit is approached, the concentration of powder starts to increase towards the bottom of the pipeline, establishing a free surface. Visual observations of the flow pattern indicate that a pulsating flow occurs before a settled layer of material forms in the bottom of the pipeline. This has also been described previously by several authors [64,65,66]. The model proposed here interprets these pulsations as being a result of the Kelvin Helmholz instability, causing a suspension of particles from the flowing layer at the bottom of the pipeline. When this re-suspension of particles stops, at the marginal stability limit of the KH instability, conditions for blockage are expected to be fulfilled.
For gas-solids flow the density of the fluid should be equal to the suspension density of the fluidized powder. We shall take the simplest approximation to this and equate it to the bulk density of the powder. We can now adapt Equation 8.8 to apply for fluidized two-phase gas-solids flow. First of all, it has to be simplified to contain superficial gas velocity instead of the velocity difference between the powder and the gas. The filling level of the pipeline also has to be inserted into the expression. In accordance with simplifications introduced by Bendiksen and Espedal [62] one can write this as:
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
124 Where the variables are defined as:
vsg Superficial gas velocity.
vg Gas velocity.
vb Bulk material velocity.
K Geometrical factor A/(DAL') equal to π/4 at a filling level of D/2.
D Pipeline inner diameter.
g Gravitational acceleration.
Rg Ratio between cross section area filled with gas and the total area.
ρg Gas density.
ρb Poured bulk density of powder.
Given that vb<<vg and that ρb>>ρg , this can be simplified to:
vsg KDg Rg b
g
2 = ⋅ 3ρ
ρ (8.10)
The question of what value to use for K still remains. The factor AL', which is the rate of change in the cross section area occupied by the "liquid" with filling level, starts out at zero for zero filling level, passes trough a maximum at π/4, and is equal to zero when the pipeline is filled. The corresponding K factor is contained within the interval [∞.. π/4] at a filling level below D/2. At a filling level above D/2 it is contained within the interval [π/4..∞].
The geometry of the pipeline will contribute to stabilise the flow at a filling level below D/2, because a positive perturbation in the filling level will decrease the energy density per surface area which is proportional to the amplitude of the perturbation to a power of two [57]. At a filling level above D/2 the opposite happens and an increase in the the energy density per surface area due to a positive perturbation in the filling level will contribute to destabilize the flow.
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
Considering, as stated initially in this section, that it is assumed that the KH-instability causes re-suspension of particles and thereby prevents blockage, the stabilising effect of the pipeline geometry at a filling below D/2 will cause blockages to occur earlier than without the geometrical effect. In short, a flat bottom pipeline should allow conveying to take place at lower air velocities than a circular pipeline. This has been observed experimentally by Wirth and Molerus [67]. For this reason the K factor selected for this model will be π/4 which corresponds to the highest obtainable filling level without a destabilising effect from the shape of the pipeline cross section.
Before we can use Equation 8.10 for the prediction of stability limits for fluidized powders we must replace the volumetric concentration of gas Rg with an expression containing the solids loading ratio µ. The stability expression incorporates a factor vL/vg that has been asumed to be small. To be able to continue the computation, we also have to assume that vsg/vb≈1. This means that the expression obtained will only be valid for high solids loading ratios. The last assumption may seem contradictory to the assumption that the local air velocity above the moving bed is much larger than the bulk velocity, but it is important to remember that the bulk material has its own voidage, different from Rg= Ag/A, that causes air to flow along inside the bulk of the material. If this fraction of the air flow is large in comparison to the amount of air flowing above the bed, and the slip velocity between the solids and the air flowing inside the bulk of the material is low, the assumption holds.
Defining the bulk voidage of the powder as εb the expression for the superficial air velocity gives:
v v A A v
A v v A v A A v A v
A v R v
sg v
b b b g g
b b b b b g g b
g g b b
g g b
= +
≈ ⇒ ≈ + ⇒ − ≈ ≈
ε ε 1 ε
when Ab≈A which means that the assumptions above are valid when the following condition is satisfied:
( )
v v
b R
g
g b
≈ 1−ε
Which also easily complies with the first assumption when the filling level in the pipeline is
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
126 Using these assumptions we can now write:
( )
Where the variables are defined as:
vsg Superficial gas velocity.
K Geometrical factor A/(DAL') equal to π/4 at a filling level of D/2.
D Pipeline inner diameter.
g Gravitational acceleration.
ρg Gas density.
ρb Poured bulk density of powder.
µ Solids loading ratio.
Equation 8.12 can be used directly to compute the limit of stable conveying in suspension flow. In Figures 8.9 through to 8.15 this model has been compared to the existing models presented in section 6.1.
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
0 2 4 6 8 10 12 14 16 18
5 10 15 20 25 30 35 40
Solids loading ratio
Velocity [m/s]
Velocity minimum Matsumoto Vmin Rose & Duckworth KH
Thomas Pan
Figure 8.9 Comparison between models to predict limit of stable flow in dilute phase for polyethylene pellets, and experimentally obtained velocity minimum.
0 2 4 6 8 10 12 14 16 18 20
0 10 20 30 40 50 60 70 80
Solids loading ratio
Velocity [m/s]
Velocity minimum Matsumoto Vmin Rose & Duckworth KH
Thomas Pan
Figure 8.10 Comparison between models to predict limit of stable flow in dilute phase for rape seed, and experimentally obtained velocity minimum.
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
128
0 2 4 6 8 10 12 14 16 18 20
10 20 30 40 50 60 70
Solids loading ratio
Velocity [m/s]
Velocity minimum Matsumoto Vmin Rose & Duckworth KH
Thomas Pan
Figure 8.11 Comparison between models to predict limit of stable flow in dilute phase for sand, and experimentally obtained velocity minimum.
0 2 4 6 8 10 12
0 5 10 15 20 25 30 35 40
Solids loading ratio
Velocity [m/s]
Velocity minimum Matsumoto Vmin Rose & Duckworth KH
Thomas Pan
Figure 8.12 Comparison between models to predict limit of stable flow in dilute phase for PVC granules, and experimentally obtained velocity minimum.
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
-2 0 2 4 6 8 10 12 14 16 18
0 20 40 60 80 100 120 140 160 180
Solids loading ratio
Velocity [m/s]
Velocity minimum Rose & Duckworth Thomas
Matsumoto Vmin KH
Pan
Figure 8.13 Comparison between models to predict limit of stable flow in dilute phase for alumina, and experimentally obtained velocity minimum.
-2 0 2 4 6 8 10 12 14
50 100 150 200 250 300 350
Solids loading ratio
Velocity [m/s]
Velocity minimum Rizk
Rose & Duckworth Thomas
Matsumoto Vmin KH
Pan
Figure 8.14 Comparison between models to predict limit of stable flow in dilute phase for micronized dolomite, and experimentally obtained velocity minimum.
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
130
-4 -2 0 2 4 6 8 10
400 402 404 406 408 410 412 414 416 418 420
Solids loading ratio
Velocity [m/s]
Velocity minimum Rose & Duckworth Thomas
Matsumoto Vmin KH
Pan
Figure 8.15 Comparison between models to predict limit of stable flow in dilute phase for cement, and experimentally obtained velocity minimum.
As can be seen from Figure 8.9 through to 8.15 the proposed model, with the exception of the cases of high and low solids loading ratio for alumina and high solids loading ratio for PVC, does not underestimate the stability limit. At the points where it does it is just below the error limit of the experimentally obtained value.
In Table 8.1 the results of the comparison above are presented. The worst case prediction errors for all models predicting the limit of stable conveying in suspension flow are shown.
Table 8.1 The worst case error of velocity minimum predictions given in percent of experimentally obtained value.
Model LDPE Rape seed
Sand PVC Alumina Micronized dolomite
Cement Pan -12 8 -31 -29 -57 139 348 Rose &
Duckworth
57 56 -44 -62 -98 -86 -98 Matsumoto Vmin 103 116 71 -8 -80 102 -50 Thomas -48 -46 -66 -77 -107 -123 -228
K-H instability 24 61 133 78 21 140 -95
The two most dominant simplifications included in the model are, the assumption of bulk density for the density of the moving concentrated layer of solids occurring close to
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
blockage or instability, and the assumption of a filling level of D/2. Later improvements to the model should improve on these assumptions and include the effects of the suspension viscosity.
Considering that this model is purely based on theory, and that it has been developed from first principles, without any empirical fitting, it is surprising how well it fits the experimental data.
In Table 8.2 the correlation between model values and experimental data, when varying solids loading ratio µ, is listed for all materials except cement (where we have no observation of blockage). In general, none of the models gives results that correlate well with the experimental data obtained for polyethylene pellets. For the other materials the existing models correlate well with the coarsest materials, while the new model correlates best with the finest materials.
Table 8.2 Correlation with µ for the new and existing models for predicting the limit of stable conveying (negative correlation with µ is shaded).
Model LDPE Rape
seed
Sand PVC Alumina Micronized dolomite Matsumoto 0.10 1.00 0.92 0.99 -0.87 -0.85 Rose & Duckworth 0.15 1.00 0.95 0.98 -0.83 -0.84 Thomas 0.16 1.00 0.94 0.98 -0.93 0.18 Pan 0.04 1.00 0.89 0.99 -0.90 -0.84 KH-instability -0.25 -1.00 -0.98 -0.98 0.83 0.83
It is also possible to evaluate each model's ability to predict the limit of stable conveying for various materials, by computing the correlation coefficient when varying the material type at maximum or minimum solids loading ratio. In Table 8.3 the correlation between model values and experimental data, when varying material type, is listed for all materials.
The high correlation coefficient for Rose and Duckworth's model is artificial, since it has a very limited area of validity. Pan's model gives a correlation coefficient of 0.81 inside its
Ph.D. Thesis S.E.Martinussen Chapter 8, Modelling of Flow Close to the Conveying Limit
132 powders, even though it does not predict the correct solids loading dependency for the coarse materials.
Table 8.3 Correlation with material type for the new and existing models for predicting
the limit of stable conveying.
Model At minimum µ, inside area of validity
At minimum µ, for all materials
At maximum µ, inside area of validity
At maximum µ, for all materials
Matsumoto 0.71 0.19 0.37 0.21
Rose &
Duckworth
0.97 0.14 0.98 0.15
Thomas 0.52 0.22 0.31 0.24
Pan 0.81 -0.16 0.40 -0.15
KH-instability
0.73 0.56 0.64 0.56
8.7 An Empirical Model for Predicting Pressure Minimum based on