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Virginia Tech | tested with steam. After verifying that the steam is operating correctly, the first superheated steam
drying tests began.
The preliminary superheated steam drying tests presented the first troubleshooting issues
for the system. When steam was injected into the cylinder to dry a nitrogen formed particle cake,
the cylinder became flooded with condensate in under 10 seconds. Testing was continued with
higher steam pressure, and higher superheat temperatures. While this was able to slightly reduce
the condensate formed in the cylinder, it still flooded and could not be reevaporated by the
prolonged injection of superheated steam. At this point testing was stopped and it was necessary
to develop a solution to the condensate issue.
2.3.3 Experimental Apparatus Troubleshooting and Modifications
During preliminary testing it had become apparent that the amount of condensate generated
by the steam passing through the system and entering the cylinder was significant. There were
three separate system components contributing to the formation of condensate. The piping
immediately downstream of the boiler and continuing until the heat exchanger, the piping and
flexible hose downstream of the heat exchanger, and the cylinder itself. Figure 13 shows a coal
cake that was removed from the cylinder after an attempted steam drying experiment. The filter
became flooded with condensate and began to leak water from the discharge. The cake itself could
not begin to dry because it was saturated with water and the system overflowed before condensate
could be re-evaporated. In order to successfully dry the coal cake using this system, the three
problematic sources of condensate needed to be remedied.
30 |
Virginia Tech | Figure 13: A wet coal filter cake after being removed from the condensate flooded filter. The water on the
wood surface around the cake was from steam condensing in the filter cylinder.
By adding an electrical heating tape to the piping and the flexible hose, the formation of
condensate in these areas could be eliminated. The insulation was removed from the piping/hose,
the heating tape was installed, and then the insulation was reapplied. With this addition to the
system, the piping/hose heating tapes could be set at the saturation temperature of the steam used
for drying.
Condensation forming in the cylinder itself was a more difficult problem to solve. The
cylinder itself could not be wrapped in a heating tape and set to a high temperature for multiple
reasons. If the cylinder temperature were raised before filtration, it would be dangerous to pour the
cool solvents into the cylinder. If the cylinder temperature were raised after filtration, but before
the initiation of steam drying, the heating tape itself would begin to dry out the cake and the
experimental data generated would not be an accurate portrayal of the drying effect of the steam
itself. The most feasible solution appeared to be adding an insulation layer to the inside of the
cylinder to separate the steam from the stainless-steel cylinder wall. The diameter of the cylinder
itself was only 2.5”, so the insulation layer would have to be thin to still allow formation of a cake
with reasonable diameter.
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Virginia Tech | The first option considered was an aerogel type of insulation. The cost to insulate the inside
of the cylinder with aerogel was relatively cheap and it was easy to acquire. A square foot sheet of
¼” thick aerogel was less than $100. However, it presented challenges in that it would be difficult
to affix it to the cylinder wall, and it was very porous and would become saturated with the steam
drying medium. Different sealant or waterproofing methods of the aerogel insulation were
researched, but most of these materials did not have the high temperature tolerance required for
superheated steam.
A second option to consider was using a rigid insulation, such as a high temperature
resistance, low thermal conductance plastic. PEEK (polyether ether ketone) met all these
requirements. PEEK material is readily available from many suppliers and is machinable. The first
step in acquiring the PEEK insulation tube was to perform calculations and a simulation of its
insulating properties with the superheated steam used for the drying experiments. These
calculations were performed using a finite element package COMSOL. The goal of this calculation
was to determine the amount of condensate produced by the existing cylinder baseline model, and
then to create a model with a selected PEEK insulating tube to verify that it will reduce condensate
formation significantly.
The COMSOL Multiphysics software is a finite element solver used for many different
applications including heat transfer. The software allows users to construct a physical model of a
test system, develop a mesh, and then the heat conduction equation can be solved to simulate
conduction through the system walls. First, a representative model of the steel pressure cylinder
was constructed including the coal filtration cake. The dimensions of the cylinder itself as well as
the cake were input into the software. Next, the heat capacity, density, and thermal conductivity
of the steel and the coal cake were defined.
32 |
Virginia Tech | Figure 15: Temperature profile of the steel cylinder with PEEK insulation tube after injecting 15 psig
steam for 60 seconds. The cylinder dimension scale is in inches, and the temperature color scale is in K.
The saturation temperature of the 15 psig steam is 394K.
In Figure 14, after 60 seconds, nearly the entire cylinder except for the bottom cap portion
below the coal cake has reached the saturation temperature of the steam. It is obvious that a
significant amount of energy is wasted heating the cylinder steel, and forming condensate, rather
than drying the coal cake. After the addition of the PEEK tube in Figure 15, the heat transfer
through the cylinder side walls is substantially reduced. It is observed that only about half of the
½” thick PEEK tube experiences any temperature change at all. Most of the heat loss is through
the cap at the top of the cylinder that has no PEEK insulation. The large majority of the steel
remains near the 273K initial system temperature. While it is visually apparent that a 0.5” thick
35 |
Virginia Tech | PEEK tube will significantly reduce condensate formation, a more precise estimation can be
calculated by integrating the temperature (and thus internal energy) change over the volume of the
cylinder to obtain the heat heat transfer through the inner surface of the cylinder. Dividing this
value by the volumetric latent heat of vaporization yields the volume of condensate formed for
both models. Condensate formed vs time for both scenarios is plotted in Figure 16.
Figure 16: Condensation as a function of time for the steel cylinder with no PEEK insulation tube in black,
and with the PEEK insulation tube installed in blue. The PEEK insulation reduces the volume of condensate
formed inside the cylinder over the course of 60s from over 120ml to approximately 18ml.
Observing Figure 16 it is seen that the PEEK tube reduces condensate by nearly a factor of
7 at 60s. Additionally, because the condensate volume in non-insulated cylinder rises sharply, and
that in PEEK-induced cylinder rises much shallower, condensate is reduced by close to a factor of
10 at the 15-20 second mark and thus is even more beneficial for experiments less than ~40
seconds.
Having determined that 0.5” is an acceptable PEEK wall thickness, the interior of the
cylinder was measured with a micrometer to provide the sales company with a highly accurate
36 |
Virginia Tech | measurement so that the PEEK tube would fit into the cylinder properly. The supplier took a solid
cylinder of the PEEK material, and machined it to the required inside and outside diameters. When
the tube arrived at the lab, mining department machinists drilled the additional holes required for
the steam and nitrogen supply, and the thermocouples. The installed tube is shown in Figure 17.
Figure 17: The pressure filtration cylinder with the PEEK insulation tube installed.
With the PEEK tube installed, and the additional heating tapes added to the system, preliminary
testing was able to continue. Tests were run with 5, 10, and 15 psig steam superheated to 150
degrees C. The reduction in condensate was significant. There was no longer any flooding of the
drying cylinder. The coal cake was much dryer upon inspection and crumbled when being removed
from the cylinder. It formed a fine, dry powder when inspected visually. Moisture content was
tested in the lab analyzer and was found to be below five percent. Figure 18 is a photo of a coal
cake removed from the filter after steam drying with the newly installed PEEK tube.
37 |
Virginia Tech | Figure 18: A crumbled coal cake removed from the cylinder after steam drying. There was no condensate
visually present on the cake when it was removed. The PEEK insulation tube appeared to solve the steam
condensate problem.
There was also no visible condensate buildup on the particle cake itself, or on the walls of the
cylinder. Whatever small amount of condensate did form during the drying experiment appears to
have been reevaporated by the superheated steam. Unfortunately, before final testing could begin,
another issue arose with the heat exchanger solder connections. The solder material had not been
durable enough to withstand the electrical heating tape, and the high temperature super-heated
steam. A small leak had formed at the heat exchanger and was preventing the system from
maintaining pressure. Again, the system had to be disassembled and the solder was redone with a
higher temperature resistant solder material. After this small fix, the system was running smoothly,
and official testing and sample collection could begin.
Given the requirement for the filtration/drying system to perform several different
functions, a valve control diagram was developed for five different gas flows. The system can
utilize unheated nitrogen for filtration or drying, heated nitrogen for drying, steam for drying,
38 |
Virginia Tech | Figure 20: The final fabrication design of modifications to the filtration equipment to accommodate
nitrogen filtration, nitrogen drying, heated nitrogen drying, and superheated steam drying. Electrical heating
tape has been added to piping to prevent condensation, and high temp solders have been added to reinforce
connections to the heat exchanger plate. The PEEK tube has been added to the pressure filter.
Before drying, filtration was completed by bypassing the filtration gas around any heated
parts of the system. The filtration cake was left in the cylinder. To begin drying, all system
temperatures were verified to be at the desired setting. Valves are then set to the desired flow
setting, and the drying gas was injected into the cylinder until the desired time interval has been
reached. During the drying process, the gas temperature is monitored and recorded to ensure
consistent results. At the end of the drying process, the drying gas supply valve was closed, and
the coal cake was removed from the cylinder. Samples of the coal cake were collected and stored
for chromatography testing. The final experimental procedure is described below.
2.3.4 Experimental Procedure
1. Perform pre-start system checks;
40 |
Virginia Tech | Table 5: Experimental Parameters for the vaporization stage of the two-step solvent removal process.
Test Drying Solvent Drying gas Drying gas Drying time Cake
number gas type pressure temperature intervals (s) formation
(psig) (°C) method
1 Nitrogen Pentane 20 20 30, 45, 60, pressure
75, 90, 105, filtration using
120 20psig N₂
2 Nitrogen Hexane 20 20 60, 120, 180, pressure
240, 300 filtration using
20psig N₂
3 Nitrogen Hexane 20 100 60, 120, 180, pressure
240 filtration using
20psig N₂
4 Nitrogen Hexane 20 150 60, 120, 180 pressure
filtration using
20psig N₂
5 Nitrogen Hexane 30 150 60, 120, 180, pressure
240 filtration using
20psig N₂
6 Steam Hexane 5 150 20, 35, 65 pressure
filtration using
20psig N₂
7 Steam Hexane 10 150 15, 30, 60 pressure
filtration using
20psig N₂
8 Steam Hexane 15 150 5, 10, 15 pressure
filtration using
20psig N₂
9 Steam Hexane 5 180 20, 35, 65 pressure
filtration using
20psig N₂
10 Steam Hexane 15 150 3, 6, 9, 15 Vacuum
filtration
2.4 Chromatography Analysis
In typical drying experiments, drying is monitored by continuously measuring the mass of the
sample being dried. This is because the sample is often isolated from the housing of the drying
vessel, and low operating pressure was involved. In the process studied here, continuous
monitoring of the drying is challenging to implement. The amount of solvent in the filtration cake
to be vaporized and removed is extremely small (typically less than a gram), and the final residual
43 |
Virginia Tech | solvent must be reduced to tens of milligrams. The drying vessel (the pressure cylinder), however,
weighs more than 5 kg. Therefore, a highly sensitive measurement of the solvent content of the
cake must be conducted, and building a continuous solvent content measurement system is beyond
the resources available to us. Therefore, many tests were performed with different drying times.
Samples of filtration cakes obtained at the end of these tests were analyzed using gas
chromatography to determine the solvent concentration in them. This reduced the time resolution
of the drying curve, but accurate drying data are available at pre-selected drying time, and the
overall drying trends are still apparent.
After each drying experiment, the coal cake was removed from the cylinder and broken apart.
3 to 4 grams of the coal sample were placed into a sterile vial and refrigerated. Later, the samples
were delivered to the chemistry chromatography lab for testing. Solvent concentration results are
returned as parts per million (ppm) value.
All organic solvents (Hexane and pentane) and SPME fibers were obtained from Sigma-
Aldrich (St. Louis, MO) and used as received. All standards were prepared by direct transfer of
0.6, 1, 2, 3 and 4 µL of hexane or pentane to a 5 x 40 mL of screw-top, septum-sealed headspace
vial containing a magnetic stir bar. Next, each vial was heated using a water jacket at 56°C for 10
minutes and stirring to obtain equilibrium. Then, an SPME fiber
(Divinylbenzene/Carboxen/Polydimethylsiloxane, 50/30µm DVB/CAR/PDMS, 24 ga, 1cm length
for manual injection) was placed inside the 40 mL vial and 5 minutes sampling was
obtained. Finally, SPME fiber was injected onto the GC/FID and quantitative results were
obtained.
For quantitative analysis, each sample was prepared by direct weighing a known mass of
sample in a 40 mL of screw-top, septum-sealed headspace vial containing a magnetic stir bar. Each
44 |
Virginia Tech | Chapter 3. Results
This section presents the results of in-situ solvent recovery that combines filtration and solvent
vaporization. Results on liquid-solid separation through filtration are presented first, followed by
solvent removal from the filtration cake through vaporization and convection by a carrier gas. As
mentioned earlier, to be viable for integration with the HHS-based coal dewatering technology,
the filtration and drying steps must be completed in 60 seconds and 10 seconds, respectively. This
is to ensure that the process can effectively be scaled up to the pilot or commercial stage using
readily available filtration and drying equipment. In addition, the mass loading of residual solvents
in the filtration cake at the end of the drying step should be below 1,400 ppm.
3.1. Liquid-solid separation through pressure filtration
The first experiments conducted were filtration kinetics tests using various coal slurries. These
tests allowed an appropriate percent solid and filtration pressure to be chosen for the subsequent
drying experiments. Figure 21a and b present the filtration curves for pentane-coal slurries with
10% and 15% solid mass loading, respectively. The filtration curves show several common
features. First, the liquid mass collected has an initial value above zero at 𝑡 = 0 (see, e.g., the
curve at 60psig in Figure 21a). Next, though not obvious in all cases, the filtrate mass increases
nonlinearly with time before reaching a peak value. Finally, the mass collected decreases very
slightly at large time.
46 |
Virginia Tech | Figure 21: Mass of filtrate collected vs. time during filtration of pentane-coal slurries with a solid mass
loading of 10% (a) and 15% (b) under a filtration pressure drop of 20,40, and 60 psig.
The initial mass collected is due to a small portion of solvents passing through the filter via
gravity after the slurry was poured into the vessel, but before the filtration pressure was applied.
The nonlinear filtrate mass curve is caused by the three stages of a filtration process as elucidated
by Huang et al (2018) [10]. Specifically, a filtration process involves three stages. Taking the
filtration at 60 psig and 10% solid mass as an example (see Figure 21a), in the first stage, which
ends at 𝑡~ 10s, the filtration rate decreases as the filter cloth/paper beneath the slurry (referred to
as filter medium hereafter) becomes increasingly clogged. In the second stage, second region, from
10 to 25 s approximately, the filtration rate decreases further. During this stage, more and more
coal particles collect on the filter paper and the filtration cake forms and becomes thicker. Liquid
solvents flow through the capillaries formed between coal particles. Since the length of the
capillaries, and thus the flow resistance, increases with time, the filtration rate decreases with time.
In the first and second stages, the filtration is governed by the liquid-solid two-phase flow in
the filtration cake. In the final stage, which begins at ~26 s where a sharp upwards jump of filtrate
mass is seen, the filtration is dominated by gas-liquid-solid three-phase flows: the beginning of the
jump in filtrate mass is when the filtration gas begins to penetrate the filtration cake, and marks
47 |
Virginia Tech | the transition from two- and three-phase flow in the filtration cake. The abrupt end of the jump is
when the filtration gas exits the filtration cake. The jump in filtrate mass is also caused partly by
the impact of the last remaining film of solvent liquids being forced out of the filtration cake on to
the filtrate collection vial that sits on the mass balance.
The time at which the abrupt end of the jump in filtrate mass is taken as the filtration time 𝑡 .
(cid:3033)
At 𝑡 = 𝑡 , the third stage of the filtration process finishes, and in-situ evaporation and removal of
(cid:3033)
solvent vapor by filtration gas (here, N ) start. Clearly, the filtration time is dominated by the first
2
and second filtration stages. Figure 21a shows that the filtrate mass decreases slightly from 27 s to
50 s. This decrease is due to the vaporization of the highly volatile pentane collected in the filtrate
collection vial.
From the data shown in Figure 21, filtration times for different slurries and filtration pressure
are obtained. Figure 22 shows that, at the same filtration pressure, the filtration times for slurries
with a 15% solid loading is about 10-15% shorter than with a 10% solid loading. This is caused
by the smaller amount of solvent in the latter slurries. Figure 22 also shows that, as the filtration
pressure increases, filtration time decreases nonlinearly, e.g., the reduction of filtration time as
filtration pressure increases from 40 to 60 psig is ~40% of that which occurs when the filtration
pressure increases from 20 to 40 psig. The nonlinear dependence of filtration time on filtration
pressure is attributed to the different filtration cake structure under different filtration pressure.
48 |
Virginia Tech | Figure 22: Dependence of the filtration time on filtration pressure (Δp) for pentane-coal slurries with 10%
and 15% solid loading.
As noted earlier, the filtration time is controlled primarily by the duration of the first and
second stages of a filtration process. As filtration pressure increases, the filter medium becomes
more highly clogged at an earlier time, which tends to prolong the first filtration stage.
Furthermore, as filtration pressure increases, the filtration cake formed during the second filtration
stage becomes more compact, resulting in decreased radii of capillaries between coal particles and
thus increased flow resistance through the filtration cake [10]. Therefore, even though higher
filtration pressure provides a larger driving force for solvents to flow through the filter medium
and develop a filtration cake, it also elevates the resistance for solvent flow through them.
Consequently, as filtration pressure increases, filtration time does not decrease linearly. In practice,
the filtration pressure may be optimized by balancing energy consumption and throughput, which
is out of the scope of the present work.
Next, filtration tests were conducted for hexane-coal slurries. Figure 23 shows the filtration
kinetics for slurries with 10% and 15% mass loading of coal at three different filtration pressures.
The shape of the filtration curves is qualitatively similar to those shown in Figure 21. A notable
difference is that, after the third filtration stage when the N gas breaks through the filtration cake,
2
49 |
Virginia Tech | the filtrate mass collected shows no noticeable decrease. This is expected because hexane is much
less volatile than pentane and mass loss due to evaporation is suppressed for hexane. Figure 24
presents the filtration time for hexane-coal slurries with 10% and 15% solid loadings at pressure
differences of 20, 40, and 60 psig. A nonlinear dependence of filtration time on filtration pressure
is again observed. Compared with the filtration time for pentane-coal slurries (see Figure 22), for
the same filtration pressure, the filtration time for hexane-coal slurries is considerably longer. The
longer filtration time is consistent with the fact that hexane has a viscosity 15% higher than
pentane.
Figure 23: Mass of filtrate collected vs. time during filtration of hexane-coal slurries with a solid mass
loading of 10% (a) and 15% (b) under a filtration pressure drop of 20,40, and 60psig.
Figure 22 and Figure 24 show that filtration can be completed within 60 s for pentane-coal and
hexane-coal slurries with solid loading of 10% and 15%, even at the lowest filtration pressure
studied (20 psig). Although the filtration time is slightly shorter for pentane, the more volatile
pentane is more dangerous to handle than hexane and thus hexane is preferred in practical
applications.
50 |
Virginia Tech | Figure 24: Dependence of the filtration time on filtration pressure (Δp) for hexane-coal slurries with 10%
and 15% solid loading.
Having completed filtration testing, it is useful to characterize the filtration cake, whose
structure can greatly affect the subsequent drying step. Such characterization, however, is very
challenging. Because of the small size of coal particles, the size of pores within the filtration cake
are small. Quantifying the size of these pores and their connectivity is difficult, not only because
the filtration cakes are relatively delicate, but also because they are partially saturated by relatively
volatile solvents. To overcome these difficulties, the approach developed by Huang et al (2018) is
utilized [10]. In this approach, the filtration kinetics curves (e.g., those shown in Figure 21) are
fitted to a model to extract the effective radius of the capillaries in the filtration cake. The model
was derived from first-principles using the Navier-Stokes equation under the assumption that the
filter cake consisted of a bundle of well-defined capillaries of identical radius. The model
considered the two- and three-phase flows through a filter cake during the cake formation. Eq. (2)
shows that the model for two-phase flow during cake formation:
𝑑𝑉 𝐴(cid:2870)∆𝑃
𝑄 = =
𝑑𝑡 8𝑋 𝑉
𝜇(cid:3428) (cid:3020) +𝐴𝑅 (cid:3432)
𝑘𝑅(cid:2870)𝜌 (1−𝑘) (cid:3040) (2)
(cid:3046)
51 |
Virginia Tech | This model relates the filtration rate 𝑄 (i.e., volume of filtrate removed per unit time, dV/dt) to
the filtration conditions including pressure difference (ΔP), solids content in the feed slurry (X ),
s
cross sectional area of the filter cake (A), particle density (ρ ), solvent viscosity (μ), cake surface
s
porosity (k), pore (capillary) radius (R), and resistance of filter medium (R ).
m
An integral of Eq. (2) gives the volume of filtrate (V) for a given filtration time (t). Note here
that the filtrate volumes measured as a function of time (V vs. t) as well as ΔP, X , A, ρ , μ, and k
s s
can be readily obtained from each filtration test. Therefore, one can determine the other model
parameters, such as pore radius (R), and filter medium resistance (R ) by fitting V vs. t curves
m
calculated by integrating Eq. (2) to those obtained from filtration tests.
In the case of the pentane-coal slurry filtration shown above, (ΔP) varied between 20,40, and
60psig. (X ), the % solids, was either 10% or 15%. The cake area (A) was 3.17x10⁻³m². (ρ ) for the
s s
coal sample used in all experiments was 1,400 kg/m³. The viscosity (μ) of pentane at 20°C is 0.250
mpa·s. Cake porosity (k), a function of particle size, was estimated as 0.35. Finally, capillary (pore)
radius (R) and medium resistance (R ) are determined from fitting the model to the experimental
m
curve.
Figure 25 displays the results of the curve fitting for the 10% solids in the 20 psig pentane
filtration experiment, and quality of the fitting is good considering the complexity of the filtration
process and the simplicity of the model. The curve fitting leads to a capillary radius of 1.28 µm in
the filtration cake. This small radius is consistent with the small size of coal and will be used later
in this paper to provide more insight into the drying behavior of the fine coal particle cake.
52 |
Virginia Tech | Figure 25: Experimental filtration results for the 10% solids pentane slurry at 20 psig (every third data
point denoted by black circle) are fitted to the curve produced by the Huang et al (2018) model (red curve).
The fitting parameters (R) and (Rₘ) resulted in a capillary radius of 1.28 µm, and a medium resistance of
0.27x10¹²m⁻¹.
Finally, it is useful to identify preferred filtration conditions from results in this section. The
solid loading in slurries in the HHS process can vary between 10% and 15%. Given that filtration
is slower with a 10% solid loading than 15% solid loading, it is beneficial to focus on slurries with
an initial 10% solid loading so that the worst-case scenario is studied. Further, because the 20 psig
tests were completed in less than 60 seconds, there is no need to use higher filtration pressures
during drying testing.
3.2 Drying through solvent vaporization and convection
3.2.1 Room-temperature N₂ drying
Drying tests were conducted with room-temperature nitrogen as the carrier gas to obtain a
baseline drying performance. In addition to drying filtration cakes obtained by filtration of hexane-
coal slurries, drying of filtration cakes with residual pentane was also tested to serve as a reference
for comparison. Figure 26 presents the drying curves for filtration cakes with pentane and hexane
as a function of time. The pressure difference driving the nitrogen flow through the cake is 20 psig.
53 |
Virginia Tech | Figure 26: The drying curve of filtration cakes with pentane and hexane as residual solvents. 20 psig
nitrogen at 20°C is used as the carrier gas. The target concentration of 1,400 ppm is denoted by a horizontal
dashed line. The initial hexane loading is 24,500 ± 250 ppm.
Figure 26 shows that pentane is removed more quickly from filtration cakes than hexane. The
residual pentane in filtration cakes reaches the target concentration of 1,400 ppm at ~105 s, while
it takes ~230 s for hexane. To understand these results, it is noted that the drying of filtration cakes
involves physical processes including phase charge, heat transfer, and mass transfer by diffusion
and convection. The last two processes greatly affect the drying rate, and they depend strongly on
the distribution of liquid solvents in filtration cakes. Establishing such distribution in a filtration
cake by imaging is difficult because of the small pore size and low liquid saturation in the cake.
The drying data in Figure 26, however, can help provide insights into the liquid distribution in
filtration cakes. As mentioned earlier, there exist two limiting scenarios for the distribution of
residual liquid solvent in a filtration cake. In the first scenario (see Figure 1a), at the end of the
liquid-solid filtration step, liquid solvents form a continuous film transversing the filtration cake
and are in close contact with the pathways of the carrier gas flow. In the second scenario (see
Figure 1b), liquid solvents are trapped in tiny pores sparsely distributed in the filtration cake. Here,
each major pathway of the carrier gas flow is in contact with a few liquid solvent clusters. To
54 |
Virginia Tech | examine which of these scenarios is closer to the actual scenario, analysis is performed for the
drying time based on the first scenario. Without losing generality, our analysis focuses on the
drying of filtration cake with pentane as residual solvents.
Figure 27a shows a pore-scale model of the drying process based on the first scenario, where
a representative pore inside the filtration cake is initially covered by a thin film of residual solvents.
Since our measurements indicated that the temperature of a typical filtration cake decreases less
than 6C during the entire drying process, the drying is approximated as an isothermal process.
Furthermore, the slow drying shown in Figure 26 is not limited by the kinetics of evaporation at
liquid-vapor interfaces. Instead, drying is governed by the diffusion of solvent vapor from the
evaporation site (pore walls) to the interior of the pore and the carrier gas flow along the pore.
Because of the pore's small radius R and its large length-to-radius ratio (L/R ~ 104), the gas flow
in the pore is laminar and fully developed. Hence, the solvent removal rate from the pore can be
estimated using the classical model for fully developed flows and mass convection in round pipes
[27].
Figure 27: (a) A schematic of a pore-scale model for the drying (solvent vaporization and mass convection)
of a filtration cake based on the assumption that the residual solvents form a continuous film on the pore
walls. (b) Variation of the solvent vapor density along a representative pore (radius R = 1.28 µm; length L
= 0.01 m).
Specifically, the average gas velocity in the pore is given by the Hagen-Poiseuille equation:
55 |
Virginia Tech | where 𝑅(cid:3364) is the universal gas constant, 𝑇 is temperature, 𝑝 is the saturation pressure of solvent, 𝑀
(cid:3046) (cid:3046)
is the molecular mass of solvent, 𝑚 is the initial liquid solvent mass inside the filtration cake,
(cid:3046),(cid:3030)
𝑅 is filtration cake's radius, and 𝑘 is filtration cake's porosity. Using the properties of filtration
(cid:3030)
cake (𝑅 = 0.032 m, L=0.01 m, R = 1.28 µm, k = 0.35, 𝑚 = 0.563 g) and pentane (𝑀 = 72.15
(cid:3030) (cid:3046),(cid:3030) (cid:3046)
g/mol and 𝑝 =57.3 kPa), 𝑡 is found to be 1.88 s.
(cid:3046) (cid:3031)(cid:3045)(cid:3052)
The drying time estimated above is almost 100 times smaller than that observed experimentally
(120 s, see Figure 26). Such a dramatic discrepancy suggests that the basic assumption underlying
the above estimation, i.e., the residual liquid solvents form continuous films lining the walls of the
pathway of carrier gas flowing through the filtration cake, is inaccurate. Therefore, the second
scenario for the distribution of residual solvent in the filtration cake is more likely, i.e., solvents
exist as isolated liquid clusters sparsely distributed in filtration cakes, is more reasonable. The
carrier gas is thus only in contact with "patches" of liquid solvents whose surface area is far smaller
than the surface area in contact with the carrier gas flow. The remaining solvents are trapped in
microcapillaries perpendicular to the main carrier gas pathway (see Figure 1b). Under this
scenario, the carrier gas flow will not be saturated with solvent vapor when it leaves the filtration
cake and thus the solvent removal is mainly limited by the transport of vaporized solvent from the
evaporation site to the carrier gas flow pathway rather than by the convection of solvent vapor by
the carrier gas flow.
The above analysis helps understand the drying curves in Figure 26. For example, the drying
curves show that the drying rate decreases as solvents are removed from filtration cakes. Similar
falling drying rate has been reported in many studies of the drying process in porous media, where
the carrier gas usually flows over a porous media's surface [13, 28-30]. In those works, drying rate
falls as the liquid saturation in a porous media becomes low. Under that condition, the capillary
57 |
Virginia Tech | flow of water and diffusion of water vapor to the porous media's surface become difficult, which
slows down transport of water to the surface of porous media and hence the drying rate.
In the present study, although the carrier gas flows through filtration cakes, the distribution of
the liquid solvent inferred from our above analysis is qualitatively similar to those revealed in
previous drying studies. As such, similar falling drying rate can be expected. Figure 26 also shows
that the drying of hexane is slower than pentane. This can be understood as follows. Here, the
drying process is limited by the transport of solvent toward the pathway of the carrier gas flow.
Because hexane has a lower vapor pressure than pentane, the flux of hexane from the evaporation
site (solvent patch in contact with the carrier gas flow or solvents that have receded into micro
capillaries connecting to the gas flow pathways) to the carrier gas flow pathway tends to be smaller
than that of pentane. Consequently, the drying of filtration cakes with hexane is slower than that
of pentane-loaded filtration cakes.
Overall, Figure 26 shows that the hexane and pentane content in a filtration cake can be reduced
to the target level (1400 ppm) using room-temperature nitrogen flow. However, the cutoff drying
times, defined as the time when the solvent content is reduced to the target level, are ~10-20 times
longer than the 10 s interval needed by commercial deployment of HHS-based dewatering
technologies. To reduce the cutoff drying time, one possible strategy is to increase the temperature
of the carrier gas flow. This will increase the vapor pressure of solvent and facilitates the transport
of solvent vapor to the carrier gas pathway within filtration cakes, thereby accelerating drying.
3.2.2 Drying using heated N₂
Having shown that room-temperature nitrogen cannot dry filtration cakes to the desired
level within the required time, heated nitrogen with a pressure of 20 psig is used to dry cakes
58 |
Virginia Tech | obtained from pressure filtration of hexane-coal slurries with 10% solid loading. Figure 28 displays
the drying curves when the nitrogen temperature is 20°C, 100°C, and 150°C. The results in Figure
28 exhibit several interesting features. First, there was a significant reduction in drying time when
N temperature is increased from 20°C to 100°C. With 100°C N , the hexane concentration in
2 2
filtration cakes reaches the target concentration at ~150 s, 60-70 s shorter than when 20°C N is
2
used. Second, as N temperature increases to 150°C, the improvement of drying performance is
2
negligible. Finally, at elevated temperatures, the drying rate also decreases with time.
Figure 28: Hexane drying curves using 20 psig nitrogen at a temperature of 20°C, 100°C, and 150°C as the
carrier gas. The target solvent concentration of 1,400 ppm is denoted by a dashed horizontal line. The initial
hexane loading is 24,500 ± 250 ppm.
The reduction in drying time as N₂ temperature is increased from 20°C to 100°C is attributed
to the fact that when the heated N enters the filtration cake, it raises the temperature of the filtration
2
cake and the liquid solvents in the cake. The latter increases the solvent vapor density in the cake,
thereby facilitating more rapid solvent removal through mass convection. When the temperature
of N is elevated to 150°C, the cutoff drying time is shortened only marginally. It is interesting to
2
note that hexane's boiling points are 69.0°C and 98.6°C at 0 and 20 psig, respectively. It thus
appears that raising the carrier gas temperature to 100°C likely causes rapid vaporization of liquid
59 |
Virginia Tech | solvent that is absent when the temperature is 20°C. Further temperature rise above 100°C had less
of an impact on the phase change of the liquid hexane. In the future, it would be interesting to
perform experiments to examine how drying behavior varies as temperature increases from 69.0°C
to 100°C.
The fact that the drying rate still falls after the solvent concentration inside the filtration cake
falls to a low value (here, around 2000 ppm) even when the carrier gas temperature is increased
above hexane's boiling point suggests that mass transfer from the evaporation site to carrier gas
pathways remains a limiting factor in drying. The latter implies that violent phase change such as
boiling, which should perturb coal particles in filtration cake to release trapped solvents and
diminish the resistance for solvent transport to carrier gas flow through filtration cakes, does not
occur at the temperature investigated here. The absence of boiling is likely caused by the fact that,
for liquids trapped in sub-micrometer cavities, boiling is suppressed; instead, liquids become
superheated, and phase change occurs only at their interfaces with solvent vapor or carrier gas.
Having seen little benefit to drying by raising N temperature to 150C, the next set of
2
experiments will determine what effect is produced by increasing the pressure difference driving
N flow through the filtration cake. Figure 29 compares the drying curves when 150°C N flows
2 2
through the filtration cake at 20 and 30 psig pressure differences. It is observed that increasing the
pressure of the carrier gas entering the filtration cake from 20 to 30psig resulted in a reduction in
the cutoff drying time of ~30 seconds.
60 |
Virginia Tech | Figure 29: Hexane drying curves using 20 psig nitrogen at 150°C and 30 psig nitrogen at 150°C. The target
solvent concentration of 1,400 ppm is denoted by a dashed horizontal line. The initial hexane loading is
24,500 ± 250 ppm.
The enhanced drying accompanying the increase of carrier gas pressure difference may be due
to several factors. First, the higher N pressure increases the density (and thus mass flowrate) of
2
N₂ carrier gas entering the filtration cake. Therefore, the heat supplied by N gas to the filtration
2
cake and its residual solvent increases, which facilitates the evaporation of solvents in the filtration
cake and thus drying. Second, the work from Huang and colleagues showed that increasing fluid
pressure could change filtration cakes' structure, e.g., a cake can be compacted [10]. Such structure
change, especially the relative movement of coal particles during the drying process, may help
release liquid solvent initially trapped inside the cake's microcapillaries, thus enhancing drying.
Overall, the above results show that nitrogen with temperature up to 150°C and pressure up to
30 psig can reduce the cutoff drying time by ~50% compared to 20°C nitrogen tests. Despite the
noticeable improvements, the drying time is still ~10 times longer than the 10 s required by the
HHS technologies for coal dewatering.
3.2.3 Drying using superheated steam
61 |
Virginia Tech | Unable to produce satisfactory drying results using heated nitrogen, drying experiments using
superheated steam as the drying medium and carrier gas were conducted. Figure 30 displays the
drying curves obtained using 150°C superheated steam with a pressure of 5, 10, and 15 psig at the
entrance of the drying vessel. Two key observations are that (1) superheated steam offers a
significant improvement of drying performance over the heated N₂ with the same temperature and
(2) increasing steam pressure from 5 to 15 psig greatly enhances the drying performance.
Figure 30: 150°C superheated steam drying curves at 5 psig, 10 psig, and 15 psig. The target solvent
concentration of 1,400 ppm is denoted by a dashed horizontal line. The initial hexane loading is 24,500 ±
250 ppm.
Superheated steam offers much better drying performance over the heated N₂ with the same
temperature. For 10 psig superheated steam, the target solvent concentration is reached at
approximately 30 s. This drying result is in contrast to the fact that N₂ flow driven by twice the
pressure difference (20 psig) did not reach the target concentration until approximately 130 s (see
Figure 28). Furthermore, by reducing the solvent concentration to below 1400 ppm within
approximately 6 s, the 15 psig superheated steam meets the requirement for HHS-based drying
technology. It should be possible to optimize the steam pressure further so that the target solvent
62 |
Virginia Tech | concentration is reached at 10 s. However, this optimization is not pursued here since 15 psig steam
provides a reasonable safety margin for drying time, which will be helpful in practical scale-up.
The much better drying performance of superheated steam over N of the same temperature is
2
attributed to the different drying mechanisms of these two carrier gases. Unlike N , when steam
2
enters the filtration cakes at room temperature, it condenses rapidly. The heat thus released is
significantly greater than that afforded by heated N . For example, when 150°C steams at 5, 10,
2
and 15 psig condense into saturated liquids at the same pressure, their specific enthalpies decrease
by 2320.8, 2289.1, and 2261.9 kJ/kg, respectively. On the other hand, even if 150°C N is cooled
2
to 20°C, its specific enthalpy decreases only by about 136.5 kJ/kg. As such, solvents are vaporized
at a higher rate when steam is used, which facilitates the solvent removal from filtration cakes and
results in faster drying. Conceivably, the condensation of water in the filtration cake can block
some pores in the cake, trap liquid solvents, and slow down drying. However, the superior drying
performance shown in Figure 30 suggests that these processes are not very significant.
For superheated steam, the drying rate increases significantly as its pressure increases from 5
to 15 psig. This can likely be attributed to the increased steam flow into the filtration cake as steam
pressure increases, whose condensation facilitates the vaporization of solvents in the cake.
During the early stage of drying, the steam condenses on the inner surface of filtration cakes.
Driven by the heat transfer from the superheated steam, the liquid condensates are later vaporized
and removed from the filtration cake by convection. Because the ultimate goal of HHS technology
and the present work is coal dewatering, it is necessary to determine the residual water content in
the filtration cake at the end of drying. Tests were conducted on all three cake samples, and the
water content was all below 5%. In the industrial handling of coal, the moisture content of coal is
an important parameter to monitor because if coal becomes too dry, it can be a safety hazard. If it
63 |
Virginia Tech | is too wet, the heating value is reduced and can cause problems during transportation. Typical
market specifications require coal moisture to be below 8% [2]. Therefore, the water content in
coal dried here is acceptable. Further study of controlling moisture content by varying superheated
steam temperatures and pressures can be conducted if more precise moisture control is required.
As shown in Figure 30, 10 psig and 5 psig steams with a temperature of 150°C are not able to
reduce the solvent content in filtration cake to the target concentration within the desired 10 s time
period. It is worthwhile to determine if higher superheat temperatures may address this limitation.
Therefore, drying experiments were conducted with 5 psig steam at 180°C, the highest temperature
that can be tolerated by our experimental apparatus. Figure 31 compares the 150°C 5 psig steam
drying curve with that of the 5 psig 180°C steam to determine the benefit of raising superheat
temperature while maintaining steam pressure. It is observed that, as the superheat increases by
30°C, the beginning part of the drying curve changes little. Drying with 180°C steam becomes
faster than that with 150°C steam after the solvent concentration drops below ~5000 ppm, and the
target solvent concentration is nearly reached at about 65 s. This, however, represents only a minor
improvement compared to the drying with 150°C steam. Therefore, increasing the steam pressure
rather than temperature is a more effective way to enhance drying, at least within the range of
temperature and pressure explored in this work. Overall, 15 psig steam at 150°C is the best carrier
gas studied in this study.
64 |
Virginia Tech | Figure 31: 5 psig superheated steam drying curves at 150°C and 180°C. The target solvent concentration
of 1,400 ppm is denoted by a dashed horizontal line. The initial hexane loading is 24,500 ± 250 ppm.
All the above drying tests were conducted using filtration cakes prepared by pressure filtration
of solvent-coal slurries (pressure difference: 20 psig). In commercial applications, filtration
equipment is available in either a vacuum-type filter model or a high-pressure model. The latter
can at times be significantly more expensive, and thus vacuum filtration is often preferred.
However, in vacuum filtration, the available pressure difference for driving the formation of
filtration cake is considerably lower than the atmosphere pressure (14.5 psia) and thus the 20 psig
used in our pressure filtration work. As discussed earlier, Huang and colleagues showed that an
increase in pressure while filtering a fine clay led to the cake being more compacted and having
smaller pore radii [10]. Therefore, if a vacuum filter were used to form a fine coal cake, it would
be expected to be less compact and have greater pore radii than a cake formed by 20 psig nitrogen.
The effects that the cake formation method have upon cake structure may increase or decrease
drying time. Therefore, it is desirable to investigate how drying would be affected if a vacuum
filter forms the filtration cake.
Figure 32 compares the drying curves of filtration cakes formed by vacuum filtration and 20
psig nitrogen but dried using the same 15 psig and 150°C superheated steam. It is observed that,
65 |
Virginia Tech | with the filtration cake formed by vacuum filtration, the solvent content is reduced to the target
concentration of 1400 ppm in approximately 9 s. While this represents a nearly 50% increase in
drying time from the filtration cakes formed by N₂ pressure filtration, it is still below the target
drying time of 10 s. The slower drying of vacuum-formed coal cakes suggests that a more tightly
formed filtration cake promotes more efficient drying. This is likely caused by the fact that, in
more tightly packed filtration cakes, the carrier gas flows more uniformly through the cake and
thus, the transport of vaporized solvent to carrier gas streams is more facile. On the contrary, in
loosely packed filtration cakes, the carrier gas may flow preferentially through a limited number
of wide pores across the filtration cake. As a result, the vapor from liquid solvents trapped in tiny
pores must travel a significant distance to reach the main carrier gas streams, hindering drying.
Figure 32: 15 psig, 150°C steam drying curves for vacuum-formed filtration cakes and 20 psig nitrogen-
formed filtration cakes. The target solvent concentration of 1,400 ppm is denoted by a dashed horizontal
line. The initial hexane loading is 24,500 ± 250 ppm.
66 |
Virginia Tech | Chapter 4. Conclusions
A recently developed technology called hydrophobic-hydrophilic separation (HHS) has been
used to simultaneously clean and dewater fine coals. The HHS operations produce fine coal
dispersed in hydrophobic solvents. To achieve economic viability, it is necessary to reduce the
residual solvent concentration in produced coal to below 1400 ppm. In this work, an in-situ solvent
recovery scheme is used to achieve this goal. The scheme includes a liquid-solid filtration stage to
recover most solvents and a drying stage to reduce the solvent concentration to the target
concentration. The drying stage immediately follows the filtration stage and involves the
vaporization of solvents in the filtration cake and the subsequent removal through convection by
a carrier gas. A series of experiments were carried out to identify filtration and drying operations
that allow the solvent concentration to be reduced to the target concentration within the time
required by the commercial application of the HHS technologies.
Our experiments showed that pressure filtration with 20 psig nitrogen could finish the filtration
step in 60 s. Analysis aided by the classical model of mass convection in round pipes revealed that
the residual solvents at the end of filtration exist in the form of isolated clusters trapped in small
cavities that are sparsely distributed in the filtration cake. Room-temperature N nor N heated to
2 2
150°C, even at an elevated pressure of 30 psig, cannot reduce the solvent concentration in filtration
cakes to the target concentration within 10 s. It is found that drying using 15 psig steam superheated
to 150°C can reduce solvent concentration below 1400 pm in less than 10 s, and the water content
in the final filtration cakes does not exceed the limit set for commercial clean coal. Increasing the
temperature and pressure of superheated steam using a carrier gas does not markedly improve the
drying performance. Even if filtration cakes are formed by vacuum filtration, which involves less
67 |
Virginia Tech | equipment cost but produces less compact cakes, drying can still be accomplished effectively with
15 psig steam superheated to 150°C.
The results and methods of this work can be further optimized and studied. First, it possible to
explore using different superheated steam temperatures and pressures to control the final water
content in the dried coal cake. This is desirable because the final preferred moisture content can
vary depending on the future use of the coal. Second, it was discovered that a coal cake formed by
a vacuum pump results in a slightly different drying time than that of a cake formed by 20 psig
nitrogen. It would be worthwhile to study the effects of cake formation method and filtration
pressure on drying time for a wider set of parameters.
Third, 15 psig 150°C steam was determined to be the optimal drying setting out of the options
tested. This can be further refined by testing in between 10 psig and 15 psig, and by altering
superheat temperature to determine a more precise drying setting arriving at an exact ten second
drying time. Fourth, the simple model developed in this thesis was for isothermal room
temperature drying. It would be helpful to develop a computational method for predicting
superheated steam drying depending on temperature and pressure.
Fifth, the filtration and drying experimental apparatus used for this work prevented testing
different cake depths. It would be useful to study filtration and drying on a number of thicker
cakes. Additionally, this could include using different coal samples with different material
properties, or even other minerals. Last, from rough calculations it is expected that using
superheated steam as a drying medium will provide cost savings compared to currently utilized
methods. Constructing an economic analysis of the HHS drying process will provide insight into
the most sensitive cost parameters, and how the process can be optimized to further reduce energy
use.
68 |
Virginia Tech | ABSTRACT
TURBULENCE CHARACTERISTICS IN STIRRING VESSELS:
A NUMERICAL INVESTIGATION
Vasileios N. Vlachakis
Understanding the flow in stirred vessels can be useful for a wide number of industrial applications, like
in mining, chemical and pharmaceutical processes. Remodeling and redesigning these processes may have a
significant impact on the overall design characteristics, affecting directly product quality and maintenance
costs. In most cases the flow around the rotating impeller blades interacting with stationary baffles can cause
rapid changes of the flow characteristics, which lead to high levels of turbulence and higher shear rates. The
flow is anisotropic and inhomogeneous over the entire volume. A better understanding and a detailed
documentation of the turbulent flow field is needed in order to design stirred tanks that can meet the
required operation conditions. This thesis describes efforts for accurate estimation of the velocity
distribution and the turbulent characteristics (vorticity, turbulent kinetic energy, dissipation rate) in a
cylindrical vessel agitated by a Rushton turbine (a disk with six flat blades) and in a tank typical of flotation
cells.
Results from simulations using FLUENT (a commercial CFD package) are compared with Time
Resolved Digital Particle Image Velocimetry (DPIV) for baseline configurations in order to validate and
verify the fidelity of the computations. Different turbulence models are used in this study in order to
determine the most appropriate for the prediction of turbulent properties. Subsequently a parametric analysis
of the flow characteristics as a function of the clearance height of the impeller from the vessel floor is
performed for the Rushton tank as well as the flotation cell. Results are presented for both configurations
along planes normal or parallel to the impeller axis, displaying velocity vector fields and contour plots of
vorticity turbulent dissipation and others. Special attention is focused in the neighborhood of the impeller
region and the radial jet generated there. This flow in this neighborhood involves even larger gradients and
dissipation levels in tanks equipped with stators. The present results present useful information for the
design of the stirring tanks and flotation cells, and provide some guidance on the use of the present tool in
generating numerical solutions for such complex flow fields.
Keywords: Stirring tank; Turbulence; DPIV; FLUENT; Rushton turbine; |
Virginia Tech | ACKNOWLEDGMENTS
The completion of this thesis would not have been possible without the help of many
individuals. First I would like to thank my advisor, Dr. Pavlo Vlacho, for helping me in my first
difficult steps here in US and for giving me the opportunity to work on this project. Your patience
and your comments helped me to make this work better. I would also like to thank Professor
Telionis for providing me with useful directions and discussions in my quest for solutions as well
as for showing the scientific path in order to enrich my knowledge. I will always be grateful to
Professor Yoon for having me involved to his flotation group from which I gained a lot. Last but
not least I am thankful to Professor Tafti for having his “door” open every time a question arises.
The collaboration with him helped me to develop my “computational thought and intuition”.
I will never forget the guys in my office and in the fluids lab who helped me a lot and kept
me company all this time. A big thank you goes to Mike Brady who was as a second advisor in my
way to explore the world of the experiments and to Yihong Yang who helped me a lot the last
month. I will never forget all my Greek, Cypriot and International friends that I made here at
Virginia Tech when I started my graduate studies. You gave me freely your company and I am
honored to be your friend. I also am thankful to my friends in Greece who did not forget about me
when I decided to continue my studies on the other side of the world.
Finally, I am grateful to my parents for believing in me in all the steps of my life. My
gratitude is the least reward for their contribution. Thanks for your love and for being there to
support me in everything, even from so far away.
Above all, I would like to thank God for getting me to the position I am today and being
the source of my internal strength.
"If we knew what it was we were doing, it would not be called research, would it?"
Albert Einstein
Dedicated to the memory of my grandmother Elisavet Vlachaki
iii |
Virginia Tech | INTRODUCTION
This thesis represents part of the work of a team of researchers from mining engineering,
mechanical engineering and engineering mechanics on flotation processes. The present work is
computational modeling of stirring tanks and flotation cells, using a CFD code (Computational
Fluid Dynamics) by Fluent.
The thesis is prepared in a format recently approved by the Dean of the VA Tech Graduate
School. The material is presented in very concise manner, in the form of a journal publication.
The idea is that, a paper can be submitted for publication without any further modifications.
Results that were not included in the main body of the thesis due to lack of space can be found
in the Appendix.
In the first chapter we explore the flow in a typical stirring tank. This is a cylindrical container
equipped with an agitating impeller. This is a classical problem that has been extensively
investigated before. The aim of the present work is essentially to use this case as a benchmark, in
order to validate the computational method. The results are compared with earlier numerical and
experimental data, as well as the experimental results obtained by members of the VA Tech team.
The present work was extended to consider two configurations not yet explored by other
investigators. This is the effect on the hydrodynamics of placing the impeller closer to the bottom
of the tank. This configuration is closer to the needs of flotation technology and the mining
industry. Another issue that was explored with greater detail than earlier reported is the effect of
the Reynolds number.
In the second chapter we examine a different configuration, typical of a flotation cell. This
chapter is designed so that it can be presented as another independent paper. As a result there is
some repetition of the fundamental material found in the first chapter. Actually this chapter
includes a larger number of results, because it will also serve as a final report to the sponsor. It
will therefore require some shortening before submission to an archival journal.
xi ii |
Virginia Tech | C H A P T E R 1
1.1. Background and Introduction
In many industrial and biotechnological processes, stirring is achieved by rotating an
impeller in a vessel containing a fluid (stirred tank). The vessel is usually a cylindrical tank
equipped with an axial or radial impeller. In most cases, baffles are mounted on the tank wall
along the periphery. Their purpose is to prevent the flow from performing a solid body rotation
(destroy the circular flow pattern) [1], to inhibit the free surface vortex formation which is
present in unbaffled tanks [2] and to improve mixing. However, their presence makes the
simulations more difficult and demanding as they remain stationary while the impeller rotates.
There are two types of stirring, laminar and turbulent. Although laminar stirring has its
difficulties and has been studied in the past [1] by many authors, in most industrial applications
where large scale stirring vessels are used, turbulence is predominant. Turbulent flows are far
more complicated and a challenging task to predict due to their chaotic nature [4], [5]. In the
case of stirred tanks, not only is the flow fully turbulent, but it is also strongly inhomogeneous
and anisotropic due to the energetic agitation induced by the impeller. In addition, the flow is
periodic, because of the interaction between the blades and the baffles. This leads to periodic
velocity fluctuations, which are often referred to as pseudo-turbulence [6]. Energy is
transported from the large to the small eddies and then dissipated into the smallest ones
according to the Kolmogorov’s energy cascade. The size of these smallest eddies can be
calculated from the following equation:
η= D
Re−3
4 (1.1)
I
However in stirring tanks we have additional energy coming from the rotation of the impeller
that is smaller than the energy from the large eddies but larger than that is dissipated into the
smallest scales. Thus this energy is located in the middle of the energy spectrum [7]. There are
many parameters such as the type and size of the impeller, its location in the tank (clearance),
and the presence of baffles that affect the nature of the generated flow field. All these
1 |
Virginia Tech | geometrical parameters and many others (e.g. rotational speed of the impeller) make the
optimum design of a stirring tank a difficult and time consuming task. [8]
Accurate estimation of the dissipation rate (ε) distribution and its maximum value in
stirred tanks, especially in the vicinity of the impeller is of great importance. This is because of
a plethora of industrial processes such as particle, bubble breakup, coalescence of drops in
liquid-liquid dispersions and agglomeration in crystallizers require calculation of the eddy sizes
which are related directly to the turbulent kinetic energy and the dissipation rate [1], [9], [10]
In the last two decades significant experimental work has been published contributing to
the better understanding of the flow field and shedding light on the complex phenomena that
are present in stirred tanks. In most of these studies accurate estimation of the turbulent
characteristics and the dissipation rate were the other important aspects [2], [4], [8]
There is a wealth of numerical simulations of stirring vessels. In most of these studies
the Rushton turbine [10], [11], [12], [13], [14], [15] was used while in others, the pitched blade
impeller (the blades have an angle of 45 degrees) [9] or combination of the above two [8], [15]
was considered. Only a few of these were carried out using Large Eddy Simulations (LES) in
unbaffled [2] and baffled stirred tanks [1], [5], [16]. A variety of different elevations of the
impellers and Reynolds numbers were considered.
Today, continuing increase of computer power, advances in numerical algorithms and
development of commercial Computational Fluid Dynamics (CFD) packages create a great
potential for more accurate and efficient three dimensional simulations.
In this study we employ a CFD code to analyze the flow in a baffled tank agitated by a
Rushton impeller. The selection of the impeller and the stirring tank geometrical parameters
were made to facilitate comparisons with available experimental data. The primary objective of
this study is to produce numerically a complete parametric study of the time-averaged results
based on the Reynolds number, the turbulence models and the clearance of the impeller. The
CFD results are compared with those obtained by the present team via a Digital Particle Image
Velocimetry. These data were generated with sufficient temporal resolution capable to resolve
the global evolution of the flow. The commercial package FLUENT [17] was used for the
simulation and the package GAMBIT [18] as a grid generator. MIXSIM [19] is another
2 |
Virginia Tech | commercial package specialized in stirring having a library with a variety of industrial
impellers.
A number of different turbulence models are available in FLUENT as in most of the
CFD commercial packages. The choice of an appropriate turbulence model is of great
significance for getting an accurate solution. In this work we used mostly the k-ε two-equation
models. These models have been extensively tested in many applications. The k-ε models are
based on the assumption of homogenous isotropic turbulent viscosity, which is not strictly
consistent with our case. And yet they predict quite well the velocity distribution and the
dissipation rate. We also used the RSM model. Although this model is more general than the
two-equation models, it is computationally time consuming because it consists of seven
equations for the turbulence modeling, allowing the development of anisotropy, or re-
orientation of the eddies in the flow. It should therefore provide good accuracy in predicting
flows with swirl, rotation and high strain rates.
1.2. Stirring Tank Model
Three dimensional simulations were carried out in a baffled cylindrical vessel with
diameter D =0.1524m(Figure 1a). Four equally spaced baffles with widthw =0.1D and
T bf T
thickness th = D /40 were mounted on the tank wall. The tank was agitated by a Rushton
bf I
turbine (disk with six perpendicular blades) with diameterD = D /3, disk
I T
diameterD =0.75D , blade widthw = D /4, blade height h =0.2D blade
D I bl I bl I
thicknessth =0.01D . (Figure 1b, 1c) The working fluid was water and its height was equal to
bl I
the height of the tank. This model is identical to the model employed in the experimental work
carried out by the present team.
3 |
Virginia Tech | Although body forces include gravitational, buoyancy, porous media and other user defined
forces in our case includes only the gravitational force. In the case of the steady state
simulations the first term on the left hand side in both the continuity and the momentum
equation is zero.
Velocity (u) and pressure (p) can be decomposed into the sum of their mean (u , p) and the
fluctuation components (u', p'):
u =u+u' and p= p+ p' (1.6)
Noting the rules for averaging:
u =u and u' =0 (1.7)
The above two rules state that the average of the average velocity is equal again to the
average and that the average of the fluctuating component is equal to zero.
By applying (1.6) and (1.6) into (1.1) and (1.2) in which already we had got rid of the first
terms we obtain the time–averaged governing equations:
∇⋅u =0 (1.8)
( ) 2 ( ) ( )
ρ∇ uu =−∇p+2μ∇⋅S− μ∇ ∇⋅u +ρg−ρ∇ u'u' (1.9)
3
The last term is the divergence of the Reynolds stress tensor. It comes from the convective
derivative. So strictly, the Reynolds stresses are not stresses; they are the averaged effect of
turbulence convection.
Modeling is usually achieved using the Boussinesq’s hypothesis. In this case the
turbulent viscosity depends only the turbulent structure and not on the fluid properties [16].
2 ⎛∂u ∂u ⎞
u'u' = kδ −ν⎜ i + j ⎟ (1.10)
i j 3 ij t⎜∂x ∂x ⎟
⎝ j i ⎠
1.4. Stirring Tank Simulations
Three-dimensional simulations were carried out in three sets of calculations, using three
different clearance heights of the impeller. In the first the clearance was set to
5 |
Virginia Tech | C/D =1/2(impeller in the middle of the tank) and five Reynolds numbers (Re) in the range
T
20000 to 45000 were chosen. The Reynolds number was based on the impeller diameter,
Re=ND2/ν. For every Re number three turbulence models, namely Standardk−ε, RNG k−ε
I
and Reynolds Stress model were tested in order to investigate their predictive accuracy in this
kind of flow. In the second set of calculations the clearance of the impeller was set to
C/D =1/15(impeller almost at the bottom of the tank) and simulations were performed again
T
but this time using only one of the above three turbulence models that had the best overall
performance. In the last configuration the proximity to the ground was set to C/T=1/3 and two
turbulence models were tested, the standard k-ε and the Reynolds Stress model. The latter had
been chosen due to the availability of experimental data from the literature about the Reynolds
stresses. More information about the test cases is listed in Table 1.1. In all sets of calculations,
the origin of the coordinate system was fixed in the center of the impeller.
The domain of integration was meshed with the aid of the commercial grid generator
package GAMBIT creating a hybrid three dimensional grid. The hybrid grid is actually an
unstructured grid that contains different types of elements. In our study 480.000 quadrilaterals
and triangle elements were used to construct the mesh. The choice of having an unstructured
grid versus a structured one was made due to the fact that in a complex flow like the present,
details of the flow field everywhere in the tank and especially in the discharge area of the
impeller and behind the baffles must be captured. Then the model was launched to FLUENT
for the simulation part. Although FLUENT does not use a cylindrical coordinate system, all the
notation in this study was converted to cylindrical making the following changes in
notation:x=r, y=ϕ and z≡z
The simulations were accomplished using the steady-state Multi Reference Frame (MRF)
approach that is available in FLUENT. In this approach the grid is divided in two or more
reference frames to account for the stationary and the rotating parts. In the present case the
mesh consists of two frames, one for the tank away from the impeller and one including the
impeller. The latter rotates with the rotational speed of the impeller but the impeller itself
remains stationary. The unsteady continuity and the momentum equations are solved inside the
rotating frame while in the outside stationary frame the same equations are solved in a steady
6 |
Virginia Tech | form. At the interface between the two frames a steady transfer of information is made by
FLUENT. One drawback of the MRF approach is that the interaction between the impeller and
the baffles is weak.
Table 1.1: Simulation Test Matrix
103⋅Re 20 25 35 40 45
Standard k-e 1a 2a 3a 4a 5a
C/D =1/2
RNG k-e 1b 2b 3b 4b 5b
T
Reynolds 1c 2c 3c 4c 5c
Stresses
103⋅Re 20 25 35 40 45
C/D =1/15
Standard k-e 6 7 8 9 10
T
C/D =1/3 Standard k-e 11 - 12 13 -
T
1.5. Results-Discussion
Figure 1.2 shows the distribution of the radial velocity across the centerline of the impeller
for all the turbulence models and configurations tested. We observe that all the models predict
the radial velocity quite well. This plot can also verify the low speed radial jet produced by the
impeller for low clearance configuration (C/T=1/15). Figure 1.3 shows the contour plots of the
spatial distribution of the radial velocity superimposed with streamlines in the baffle plane
(r/D =0) for the three configurations. In the case of the low clearance, only one large
T
recirculation area on each side of the tank is observed, while in the other cases, two distinct
toroidal zones above and below the impeller divide the flow field in two parts (in half for the
case of C/T=1/2 and in one third in the case of C/T=1/3). According to [1] these large-scale
ring vortices [21] act as a barrier to stirring by increasing the blend time. It can also be noticed
that in the latter cases the radial jet is more energetic than it is in the first.
Figure 1.4 presents the contours of the dissipation rate in the impeller e plane (r/D =0)
T
for all the tested configurations. The maximum dissipation was calculated in a rectangular
region along the tip of the blade. Although in this study we present the maximum dissipation
7 |
Virginia Tech | found in this “box”, the result is very sensitive to its definition, especially as we go closer to the
blade, because the value of the maximum dissipation changes drastically. Despite the fact that
the RNG k-e model overpredicted the results more for the radial velocity in the centerline of
the impeller than the other two models, it had superior behavior among the studied turbulence
models in predicting the Turbulent Dissipation Rate (TDR) as illustrated in Figure 1.5. A
parametric study for all the turbulence models, elevations and Re numbers for the maximum
dissipation is presented in Figure 1.6. As the Re number increases the maximum TDR for the
C/D =1/2configuration decreases. This is in agreement with our experimental data and
T
those of Baldi’s et Al [24]. Unfortunately, no experimental data are available for the
configuration where the impeller is almost at the bottom of the Tank. For this case the line of
the maximum dissipation levels off.
Figure 1.7 demonstrates the normalized Turbulent Kinetic Energy (TKE) in a plane that
passes through the middle plane of the impeller (z/D =0) [15]. It can be observed that the
T
TKE has smaller values in the case of the low configuration. In Figure 1.8 a difference between
the experimental and the computational results can be observed. This apparent discrepancy is
due to the periodicity that characterizes the flow, since with every passage of a blade strong
radial jet is created. By removing (filtering) this periodicity matching of the experimental and
the computational results can be achieved.
Figure 1.9 shows the Zvorticity in the middle plane of the impeller (z/D =0). For all the
T
configurations, vortices behind the baffles are formed [24]. For the cases of clearance
C/D =1/2 trailing vortices that form behind the impeller blades can also be observed. Figure
T
1.10 demonstrates the nondomensionalized Xvorticity in the baffle plane (r/D =0) in which
T
the trailing vortices from the rotating blades can be seen. For the low clearance case only one
large ring vortex forms.
In Figure 1.11 radial velocity profiles obtained by simulations, using three different
turbulence models, and measurements are compared at two locations, namely: r/T=0.19 (very
close to the blades) and r/T=0.315 (further downstream) for Re=35000.All the turbulence
models predict quite well the profile of the radial jet in the first location while in the second
start to diverge from the measurements. The velocity is normalized with the tip velocity of the
8 |
Virginia Tech | blade and the axial distance with the blade width. Figure 1.12 shows the dissipation rate
profiles again for Re=35000 at r/T=0.19 and r/T=0.315. Although the dissipation profiles
obtained by the Standard K-e and the Reynolds Stress model still overpredict the experimental
data, the RNG K-e model seems to be promising. By observing the shape of the velocity
profiles of the jet we see that as we go further downstream they open up as we expect from the
jet theory.
1.6. Conclusions
In the simulation of the flow in a stirred vessel equipped with a Rushton turbine, three
different turbulence models and three different configurations of the impeller have been
simulated for five Re numbers. The turbulence models tested were the Standard k-e the RNG
k-e and the Reynolds stress model and the elevations of the impeller were C/T=1/2, C/T=1/3
and C/T=1/15 Calculations were carried out for Reynolds numbers of 25000-45000. All the
simulations were for time-averaged, three-dimensional, single-phase flow, and were carried out
using the commercial CFD package FLUENT.
One of the main aims of this study was to characterize the complex hydrodynamic field
and to calculate the flow velocities and the turbulence quantities accurately in the entire vessel.
Emphasis was given in the discharge area of the impeller. In addition, the effect of the
clearance of the Rushton turbine on the turbulence characteristics and especially on the
maximum dissipation rate was of great interest.
A grid study of three different meshes showed that the results are sensitive to the
number of elements that the grid contains in order to resolve better the sharp gradients that
occur next to the impeller’s blade region. The finest grid used in this study consists of about
five hundred thousand elements. All the results presented are based on the finest grid. All the
grids were unstructured and were generated in the commercial package GAMBIT. The coarse,
fine and the finest grid had 220,000, 350,000 and 480,000 elements respectfully. Therefore
there is an increase of 59% from the coarse to the fine grid and a 37% from the fine to the
finest one.
9 |
Virginia Tech | Good agreement between the computational and experimental data was found for the
radial velolcity in the case where the impeller was in the middle of the tank (C/T=1/2). More
specifically, both the Standard and the RNG k-e turbulence models slightly overpredicted the
experimental data by 1.4% and 4.2% respectfully, while the Reynolds stress model
underpredicted them by 2.8%. For the other configurations no in-house experimental results
were available. From the comparison of the turbulence models used here, it is concluded that
all under-predicted the turbulence kinetic energy even when the periodicity was removed from
the experimental data. The percents of underprediction for the Standard k-e, RNG k-e and for
the RS were 304.2%, 450% and 378.9 respectfully. As far as the dissipation rate is concerned
both the Standard and the Reynolds stress models overpredicted the dissipation rate by 58.9%
and 23% respectfully while the RNG k-e overpredicted them by 39% . In some cases the
standard k-e model had the best behavior but in others it gave poor results if compared to the
superior behavior of the RNG k-e. This happened because of the anisotropy that characterizes
the flow in the area next to the impeller region. We expected that the Reynolds stress model
should have given the best results due to the fact that it accounts for anisotropy but this did not
prove to be the case. The problem may lie in the grid size; maybe for the Reynolds stress
model should be larger due to the fact that it consists of seven equations instead of two and it
needs better resolution because it accounts for anisotropy.
The flow patterns for the cases where the impeller clearance was C/T=1/2 and C/T=1/3
were almost the same. Two recirculation regions above and below the impeller were observed
with the one below the impeller to be smaller in the case where the impeller clearance was set
to C/T=1/3. In the case where the impeller is very close to the bottom, only one recirculation
area was observed. In this case the radial velocity was very low (low speed jet and not so
energetic) compared to the other two configurations. On the other hand, the axial velocity was
larger in that case compared to the other two. Secondary circulation loops form behind the
baffles in all the cases.
Good agreement of the maximum dissipation rate with experimental results from other
researchers [24] in the case where the clearance was set to C/T=1/3 was achieved. The
maximum dissipation rate curves show that the higher the Reynolds number, the lower the
maximum dissipation. But for the case of the lowest position of the impeller, the dissipation
1 0 |
Virginia Tech | C H A P T E R 2
2.1 Introduction
In the last few years computational fluid dynamics (CFD) has extensively been used to
explore the flow in stirred vessels. Although most of the researchers have carried out
simulations in stirring tanks with the standard Rushton turbine, only a few have reported results
in flotation cells equipped with more complex rotor-stator geometries. Simulations can provide
accurate and useful information about the hydrodynamics in a flotation cells. More specifically,
in the flotation process the interest is concentrated in predicting the bubble size, the collisions
between particles and bubbles as well as the probability of attachment and detachment of the
floated particles. In order for all the above to be calculated, there is a need of information on
the turbulent quantities, and most importantly, the maximum value of dissipation rate. This is
because all models of bubble/particle collision and attachment depend on the rate of turbulent
dissipation. Therefore the primary objective of this study is to estimate accurately the velocity
field and the turbulent quantities in a flotation cell. The Dorr-Oliver flotation cell is the
configuration chosen for this study.
Computational methods have been employed extensively for the calculation of the flow in
stirring vessels [1], [2], [5], [9], [11] but only in the past three years authors attempted to
compute the flow field in flotation cells. The new and challenging element in this problem is
the proximity of the impeller to the ground and the effect of stators. Koh et al. (2003) show
predicted velocity vectors in a Metso Minerals and Outokumpu cell. Titinen et al. (2005)
compare velocity vectors from LDV experiments and CFD simulation in an Outokumpu
flotation cell. They also show contour plots of gas and liquid volume fractions. Koh et al.
(2005) have carried out three-phase simulations in a stirring tank equipped with a Rushton
turbine and in a CSIRO flotation cell. For the first one they present results like attachment rates
for particles and bubbles, the fraction of particles that remain in the cell, but for the CSIRO
flotation cell they present only the net attachment rate for particles after some time of the
flotation process. These authors presented multi-phase flow simulations, but they do not show
the detailed hydrodynamic field for single phase flow as well as the effect of the stator on the
2 4 |
Virginia Tech | flow field and on the turbulence quantities. The present study will shed more light on the
hydrodynamics of a flotation cell by varying some of the parameters that control the process.
Simulations were carried out first with the impeller at two different elevations above the
floor of the cell. (Table 2.1) Finally a Dorr-Oliver stator was added to the model in order to
investigate its effect to the flow field and to the turbulent characteristics.
2.2 The Flotation process
The main objective of this work is to study the detailed hydrodynamics of flotation
cells using Computational Fluid Dynamics (CFD). Froth Flotation [26], [27], [28] is an
important process which is widely used in mining, coal, chemical, pharmaceutical and lately in
the environmental industries in order to separate mixtures. The technique relies on the surface
properties of different particles which need to be separated. There are two types of particles:
hydrophobic and hydrophilic. The first are the ones that we want to separate from the latter and
need to be floated. Mostly flotation is carried out using mechanically agitated vessels. These
vessels are usually huge cylindrical tanks where a rotating impeller is stirring the flow. The
rotation is transferred to the impeller from a motor through a shaft. The shaft is hollow and air
is passing through it in order to generate bubbles which will be the “vehicle” for the
hydrophobic particles to attach and float. Particles accumulate as froth on the top of the tank
and are collected while some others that couldn’t attach to the surface of the rising bubbles fall
to the bottom of the tank because of gravity. In most of the applications that flotation cells are
involved the flow is highly turbulent (Re∼106 −108 in industrial cells) and turbulence is an
important ‘key’ to better understand the flotation process. Three effects of turbulence are
important in flotation [26]:
a. the turbulent transport phenomena (particle suspension)
b. the turbulent dispersion of air
c. the turbulent particle-bubble collisions
2 5 |
Virginia Tech | The first is controlled by the macro turbulence while the last two are controlled by the
micro- turbulence. Studies in the past have shown that the collision rate between particles and
bubbles depend on the maximum value of the dissipation rate which can be calculated from the
turbulence models through out the simulations.
2.3 Configuration and parts of the Dorr-Oliver flotation cell
The Dorr-Oliver stirring tank is a cylindrical tank with a conical lower part (Figure 2.1)
which is agitated by a six-blade (Figure 2.2) impeller. Unlike stirring tanks employed in
chemical industry, the impeller is positioned at the bottom of the tank, with very little
clearance. A stator (Figure 2.3) is mounted also at the bottom of the tank. Simulations were
carried out for two different elevations of the impeller with and without the stator in order to
predict the effects in the flow field and in the turbulence characteristics. The parameters of the
cases chosen are given in Table 2.1
Table 2.1: Simulation Matrix for the Dorr-Oliver Tank
Distance of the Without With Stator
impeller from the Stator
Re=35000 bottom of the tank
3cm 1 -
1cm 2 3
2 6 |
Virginia Tech | Figure 2.3: Dorr-Oliver Stator
Figure 2.4 shows the contour plots of the normalized velocity magnitudes superimposed
with streamlines for the three different cases. The first two cases correspond to two different
elevations of the impeller from the ground. In the third case, we added the stator shown in
Fig 2.3. In the first two cases we observe that two recirculation regions form while in the last
case with the stator we have only one. In addition, in the case where the impeller is closer to
the bottom of the tank we can see that the stagnation points have moved from the inclined wall
closer to the bottom of the tank. This may be a desirable feature for flotation processes,
because more turbulence is generated with higher dissipation rate which is directed towards the
floor of the tank, and thus can lift particles that may have settled there. The collision rate of
bubbles and particles depends on the maximum dissipation (usually occurring in the discharge
zone of the impeller) which means the higher the dissipation, the higher the collision rate.
Therefore more particles will attach to the surface of the rising bubbles and will not settle in the
bottom of the tank. Thus, the configuration with the impeller closer to the bottom of the tank
will scrape the floor better and as a result the flotation process will be more efficient. On the
other hand, in the case where the stator is present, the flow field is completely different. First of
all no stagnation points exist and the streamlines indicate that the flow instead of following the
tank wall while ascending, and returning back along the center, they do the opposite. This
means that the flow is going mostly upwards along the center of the tank, thus reinforcing
flotation.
2 8 |
Virginia Tech | Figure 2.5: Contour Plots of the radial velocity using the Standard k-ε model for the three
configurations (impeller blade plane)
We observe that the closer the impeller is to the floor of the tank, more turbulence is
generated. We attribute this to the blockage of the return flow to the impeller. With a flat disk
blocking the return of the flow on the top of the impeller and the ground blocking the return
form the bottom, the edges of the impeller produce slower flow but much higher levels of
turbulence. In the case where the stator is present even more turbulence is generated and not
only inside the stator but outside too. (Orange zones at the end of the stator indicate a high
TKE). Probably this is happening because vortices form around the stator blades, and these
break down to turbulence.
Similar behavior is observed in the dissipation rate shown in Figure 2.7. The only
difference is that even if high values of dissipation can be seen around the stator, the maximum
is observed inside the stator blades. In those regions there is a high probability of collisions
between particles and bubbles which could lead to more efficient flotation.
3 2 |
Virginia Tech | Figure 2.9: Contour Plots of the normalized Z-vorticity using the Standard k-ε model for the
three configurations in a plane between the rotor blades (x/Dves =0)
Figures 2.10, 2.11 and 2.12 show the streamlines of all the configurations along three
different horizontal cut-planes of the impeller, namely aty/H =−1/4, y/H =−1/2 and
imp imp
aty/H =−3/4. The minus (-) sign indicate that these planes are below the impeller disk
imp
which has been defined as the zero (0) elevation. In Figure 2.10 in the first case the turbine
pushes all the fluid out. In the second case in which the impeller has been moved closer to the
bottom of the tank we notice again that most of the fluid is directed out but a distinct circle can
be seen. This is the envelope of the streamlines separating those that spiral inward from those
that spiral outward. Lastly, in the case with the stator, the fluid in its effort to pass through the
stator blades form small vortices between them, while some of the fluid is returning back as it
was indicated in Figure 2.4.
As we proceed to next slice which is located in the middle of the impeller (Figure 2.11) in
the first two cases we can observe that some of the flow is returning back from the two
recirculation regions next to the impeller while some of it is still going out (on the opposite
3 9 |
Virginia Tech | direction in the big recirculation region). In the case of the low clearance (1cm from the bottom
of the tank) the inside circle is smaller. This is due to the fact that the vortices do not extend as
much as in the other case. On the other hand, in the case with the stator the streamlines do not
merge as before to create a circle. Therefore most of the flow is returning back from the huge
recirculation region (outside from the stator) while some of it inside the stator still tries to pass
through.
Finally in Figure 2.12 (one forth from the bottom of the tank) the circle for the first two
cases grows because more flow is returning back to the impeller. The same behavior is
observed in the case in which we have the stator. This can be explained because in the lower
part of the tank, instead of sixteen stator blades there are only four and therefore they do not
block as much the flow as at the higher elevations
4 0 |
Virginia Tech | In Figure 2.13 in the first case we can notice that vorticity is high around the impeller blade
where an extended vortex is formed as we saw in Figure 2.8. The second vortex (the big one)
starts further up and cannot be captured very well in this slice. As far as the second case is
concerned, the small vortices are very close to impeller blades and the second one starts closer.
This is why high values of vorticity are extended further away (orange-yellow zones). The
outside recirculation regions are not as spinning as violently as the inner, so no red values (the
highest values based on the color zones) are observed. In the case of the stator/rotor
configuration high values of vorticity can be observed everywhere due to the vortices that form
around the stator blades.
Figure 2.14 indicates that high values of vorticity still exist but they do not extend too
much further out. In the case with the stator, higher values are observed next to rotor blades
than in the stator blades. This can be attributed to the fact that now more flow is moving inward
than outward, and therefore the impeller ‘makes’ a bigger effort to push the flow outside.
Lastly in Figure 2.15 although we observe the same behavior for the first two cases with
even less high values of vorticity as we move further away from the impeller in the case with
the impeller no higher vorticity values can be seen. This is due to the lower number of the
stator blades as we mentioned before. Only four stator blades are present, and therefore the
flow is not so much interacting with them.
4 5 |
Virginia Tech | Figure 2.15: Normalized Y-vorticity using the Standard k-ε model for all three configuration
in a horizontal slice that passes through the first one forth of the impeller (y/H =−3/4)
imp
Figures 2.16, 2.17 and 2.18 illustrate the normalized Turbulent Kinetic Energy (TKE) for
all the studied cases in three different elevation slices.
Figure 2.16 demonstrates that the configuration which has the impeller further up from the
bottom of the tank has the lowest TKE. The flow is only directed outwards in that case. In the
second case where the impeller is closer to the bottom generates more turbulence and therefore
more TKE. Finally, when the stator is added the flow finds more resistant, creates vortices
around it and higher values of TKE can be observed inside and outside the stator blades.
Figure 2.17 shows that the TKE for the first case increased while in the second the opposite
happened. In the case with the stator it decreased in the boundary of it but increased in the area
close to the rotor blades. As we mentioned before more flow is coming in than going out which
results to happen the above phenomenon.
Lastly, Figure 2.19 indicates that low values of TKE are dominated in the first two
configurations while in the one with the stator higher values of TKE than before appear next to
the impeller blades.
5 0 |
Virginia Tech | CFD simulations were carried out for three different configurations with the Standard k-ε
turbulence model and for Re=35000 in order to investigate the effect of the clearance and the
stator on the velocity field. In the first two configurations chosen, the impeller was placed at
two distances from the floor of the flotation cell, namely at 1cm and 3cm but the stator was
removed. In the third case a full stator configuration was added for a clearance of the impeller
of 1cm from the bottom of the tank. The flow field without the stator looks similar for both
clearance cases; two recirculation regions can be observed. In case of the low clearance
configuration, namely, at 1cm from the bottom of the tank, the impeller jet attaches on the floor
of the tank. Since the impeller jet convecting turbulent structures, we conjecture that
attachment on the floor involves scraping and therefore agitating and lifting particles that tend
to settle by gravity. In the case with the stator the flow field is completely different. No
stagnation points exist and the flow is mostly directed upwards in the middle of the tank,
reinforcing flotation. Two small vortices form between the edge of the disk that forms a barrier
on top of the impeller and the stator ring. This is because of the jet that is generated from every
passage of the impeller which impinges on the stator ring.
Our simulations indicate that the presence of the stator creates more turbulence. High
values of turbulent kinetic energy and dissipation rate can be seen in the immediate
neighborhood of the stator. This is due to the vortices that form between the stator blades. In
order to analyze in greater detail the changes that occur in the neighborhood of the impeller
region, we cut three horizontal slices at three elevations of the impeller, namely, one third, one
half and three forths (y/H =−3/4) of impeller diameters below the cover of the impeller.
imp
From these contour plots we can draw the conclusion that the higher values of the turbulent
kinetic energy and the dissipation rate are observed in the first plane (y/H =−1/4) because
imp
more flow is directed outwards and at a higher speed, than inwards and it finds more resistance
from the stator blades. In the lowest plane (y/H =−3/4) due to the fact that all the flow is
imp
entering the impeller domain the impeller ‘makes’ greater effort to push the flow outside. The
point is that a rotating blade tends to move fluid radically outward. But in the present situation,
the top of the impeller is blocked by the circular roof plate of the impeller and the bottom is
blocked by the floor of the tank. Therefore, with a strong centrifugal action on the top where
6 2 |
Virginia Tech | the impeller blades have a larger diameter, the flow exits the impeller domain thee and thus it is
forced into the impeller domain lower. In this region therefore, the blades tend to move the
flow outward conservation of mass forces it inward. To conclude, the presence of the stator
plays an important role in the flotation process because more turbulence is generated which
means that the values of the maximum dissipation rate are larger and are located inside the
stator and very close to the impeller outflow. Therefore the collision rates between air bubbles
and particles is larger (more particles will attach to the air bubbles and will float) and as a result
the flotation process will be more efficient. Moreover, the bubble/particle aggregates will find a
mild upward flow motion in the middle of the tank, which will improve the flotation process.
In all other cases studied the recirculation patterns involve downward motions in the middle of
the tank, which would tend to oppose the flotation process.
6 3 |
Virginia Tech | APPENDIX I
RUSHTON TURBINE
A. Outline of turbulence models for the stirring tank simulations
i. The Two- Equation k-ε Models
Among the existing turbulence models the family of the two equation k-ε models is the
most simplest to use. The purpose of these models is to predict an eddy viscosity. A k-ε model
consists of two equations; one for the Turbulent Kinetic Energy (TKE) and one for the
dissipation rate (ε). At high Reynolds numbers the rate of dissipation and the production (P) are
of similar order of magnitude. Therefore, we estimateε≈ P.This correlation implies that once
Turbulent Kinetic Energy is generated at the low wave number end of the spectrum (large
eddies) it is immediately dissipated at the same location at the high wave number end (small
eddies). [11]
a. The Standard k-ε Model
The Standard k-ε model was initially developed by [25] and was the first two-equation
model used in applied computational fluid dynamics. This model is based on the Boussinesq
hypothesis (isotropic stresses) that the Reynolds stress is proportional to the mean velocity
gradient, with the constant of proportionality being the turbulent or eddy viscosity. The
equation for the turbulent viscosity (ν ) is given by:
turb
k2
ν =c ⋅ (I.1)
t μ ε
Where k is the kinetic energy, ε is the dissipation rate and c is a parameter that depends on the
μ
k-ε turbulence model. In this case is equal toc =0.09.
μ
6 4 |
Virginia Tech | The Turbulent Kinetic Energy (TKE) for three dimensional flows is given by:
1
k = ⋅( u'2 +v'2 +w'2) (I.2)
2
This model became a “workhorse” for many engineering application because it combines
reasonable accuracy, time economy and robustness for a wide range of turbulent flows. The
governing transport equations for this model can be written as:
∂(ρk) ∂(ρuk) ∂(ρuk)⎛μ ∂k ⎞
k-equation: + i = i ⎜ t ⋅ ⎟+ρ⋅(P−ε) (I.3)
∂t ∂x ∂x σ ∂x
⎝ ⎠
i i k i
∂(ρε) ∂(ρuε) ∂(ρuk)⎛μ ∂ε⎞ 1
ε- equation: + i = i ⎜ t ⋅ ⎟+ρ⋅ ⋅( c P−c ε) (I.4)
∂t ∂x ∂x σ ∂x τ 1,ε 2,ε
⎝ ⎠
i i ε i d
In the above equations τ is the dissipation rate time scale that characterizes the dynamic
d
process in the energy spectrum and P is the production for turbulence. The equations of these
terms are given by:
k
τ = (I.5)
d ε
⎛∂u ∂u ⎞∂u
P=v ⎜ i + j ⎟ j (I.6)
t⎜∂x ∂x ⎟ ∂x
⎝ j i ⎠ i
The empirical constantsc ,c , c ,c , c ,are given in Table A.1
μ k ε 1,ε 2,ε
Table I.1: Parameters for the Standard k-ε model
Model Parameters
c =0.09 σ =1 σ =1.314 c =1.44 c =1.92
μ k ε 1,ε 2,ε
b. The RNG k-ε Model
The RNG k-ε model is one of the most popular modifications. It has become known due to
some weakness of the Standard k-ε. It was derived from the Standard k-ε using the
6 5 |
Virginia Tech | renormalization group theory (a statistical method) in order to be more accurate in rapidly
strained and swirling, flows. This theory is first used to resolve the smallest eddies in the
inertial range. After resolving these small eddies a small band of the smallest eddies is
eliminated by representing them in terms of the next smallest eddies. This process continues
until a modified set of the Navier Stokes equations is obtained which can be solved using even
a coarse grid. Changes have been made deriving a formula for the effective viscosity that
accounts for low Re number effects as well. The governing equations for the RNG k-ε model
are:
∂(ρk) ∂(ρuk) ∂(ρuk)⎛ ∂k ⎞
k-equation: + i = i ⎜a μ ⎟+ρ⋅(P−ε) (I.7)
∂t ∂x ∂x k eff ∂x
⎝ ⎠
i i i
∂(ρε) ∂(ρuε) ∂(ρuk)⎛ ∂ε⎞ 1
ε-equation: + i = i ⎜a μ ⎟+ρ⋅ ⋅( c* P−c ε) (I.8)
∂t ∂x ∂x ε eff ∂x τ 1,ε 2,ε
⎝ ⎠
i i i d
μ =μ+μ (I.9)
eff t
⎛ η⎞
η⋅⎜1− ⎟
η
⎝ ⎠
c* =c − 0 (I.10)
1,ε 1,ε 1+β⋅η3
S⋅k
η= (I.11)
ε
S = 2⋅S ⋅S (I.12)
ij ij
The main difference between the Standard k-ε model and the RNG k-ε is the additional sink
term in the ε equation (second term in equation (I.8))
The empirical constants for this modelc ,σ ,σ , η,β,c , c are given in Table I.2
μ k ε 0 1,ε 2,ε
Table I.2: Parameters for the RNG k-ε model
Model Parameters
c =0.0845 α =1.39 α =1.39 η =4.38 β=0.012 c =1.42 c =1.68
μ k ε 0 1,ε 2,ε
6 6 |
Virginia Tech | c. The Realizable k-ε Model
This model is relatively new and differs from the Standard k-ε model because it contains a
new form for the ν and a new transport equation for the ε derived from an exact equation for
turb
the transport of the mean square vorticity fluctuations. The term ‘realizable’ means that it
consistent with the physics of turbulence (mathematical constraints on the Reynolds stresses).
One disadvantage of this model is that when a multi reference frame (MRF) or a sliding mesh
(SM) are used it produces non-physical turbulent viscosities due to the fact that includes the
mean rotation effects in the ν equation. Thus, the use in the above frames should be taken
turb
with some caution.
In this case c is no longer a constant but is given from the following equation:
μ
1
c = (I.13)
μ k⋅U*
A + A ⋅
o s ε
which in its turn consist of some other parameters as follows:
A = 6⋅cosϕ (I.14)
s
1
ϕ= ⋅cos−1( 6⋅W) (I.15)
3
S S S
W = ij ij ki (I.16)
S3
S = S S (I.17)
ij ij
U* = S S +Ω(cid:106) Ω(cid:106) (I.18)
ij ij ij ij
(cid:106)
Ω =Ω −2e ω =Ω −e ω (I.19)
ij ij ijk k ij ijk k
Ω is the mean rate of the rotation tensor presented in a rotation frame with angular velocityω
ij k
and S is given by
ij
1 ⎛∂u ∂u ⎞
S = ⋅⎜ i + j ⎟ (I.19a)
ij 2 ⎜∂x ∂x ⎟
⎝ j i ⎠
6 7 |
Virginia Tech | p⎛ ∂u' ∂u' ⎞
Pressure-rate of strain Tensor:R = ⎜ i + j ⎟ (I.27)
ij ρ⎜∂u ∂u ⎟
⎝ j i ⎠
⎛∂u' ∂u' ⎞
Dissipation Tensor:ε =2ν⎜ i j ⎟ (I.28)
ij ⎜∂x ∂x ⎟
⎝ k k ⎠
The convection, the molecular diffusion and the production terms are in closed forms due to
the fact that they contain only the dependent variable u'u' and mean flow gradients so they
i j
don’t need to be modeled. However, modeling of the turbulent diffusion, the pressure-strain
and the dissipation rate (calculated from ε equation) are required. Although as we said it is
more general model than the two equation models it is computationally time consuming
because it consists of seven equations for the turbulence modeling. This model allows the
development of anisotropy, or orientation of the eddies in the flow. Therefore, has good
accuracy in predicting flows with swirl, rotation and high strain rates.
B. Grid Study
In this section results from a grid study of three different grids and three turbulence models
are presented. All the grids were unstructured and were generated in the commercial package
GAMBIT. The coarse, fine and the finest grid had 220,000, 350,000 and 480,000 elements
respectfully. With the Reynolds stress model only the finest grid was tested, due to the fact that
it consists of seven equations instead of two and it needs better resolution because it accounts
for anisotropy. As it can be seen from the Figure I.1 as the grid size becomes finer results tend
to be closer to the experimental data. In the case of the turbulent kinetic energy (Figure I.2) all
the models with all the grids underpredicted the experimental results. We also observed that
even the Reynolds stress model gave poor results. As far as the dissipation is concerned
(Figure I.3), only the finest grid of the RNG k-e gave good results, even in the area next to the
blade were the steepest gradients are present and the Reynolds stress models should have
superior behavior, again because it accounts for anisotropy.
6 9 |
Virginia Tech | 30
25
20
15
10
5
0.2 0.25 0.3 0.35 0.4
r/Dtank
7 1
)2pmiD3N(/ε
Finest grid using the K-e model
Fine grid using the K-e model
Coarse grid using the K-e model
Finest Grid using the RNG K-e model
Fine grid RNG K-e model
Coarse grid RNG K-e model
Finest grid using the Reynolds Stresses
Experimental results
Figure I.3: Normalized Dissipation rate at the centerline of the impeller for all the grids and
turbulence models
C. CFD Simulations of a Rushton tank and a flotation cell
CFD simulations were carried out for the geometry of a stirring tank adopted for a
parallel experimental study. This consisted of a small-scale cylindrical stirring vessel made of
Plexiglas with diameter D =0.1504m (shown in Figure I.4). Four equally-spaced axial baffles
T
with width (w ) one tenth of the tank diameter were mounted along the wall. The tank was
b
closed at the top, and was filled with deionized water as the working fluid. A Rushton impeller
used to agitate the fluid was located at a height 0.0762m from the bottom. (in the middle of the
tank). The ratio between the vessel and the impeller diameter was D D =1/3.The ratio of
imp/ T
blade width (w) to the impeller diameter (D ) was 0.25. The height of the water was
i imp
maintained at 0.1504 m, which is equal to the tank diameter. The cylindrical tank was housed
in an outer rectangular tank filled with water to eliminate the optical distortion of light beams
passing through a curved boundary of media with different indices of refraction. |
Virginia Tech | The region most interesting in the above configuration is the region immediately
adjacent to the impeller. We chose this region because of the extreme increase in velocities and
turbulent statistics from the rest of the tank. For this reason CFD calculations for Re number
equal to 35000 are performed to check how the results with different turbulence models are
compared with those from the Experiments.
z
r
(a) (b)
Figure I.4: Stirring Tank configuration: (a) Three dimensional and (b) Cross sectional view of
the Tank
D. The Multi-Reference System
This report discusses the application of different turbulence models in a stirring tank
equipped with a Rushton impeller. Although that better numerical techniques and increasing
computational power permits unsteady simulations, there is a wide range of applications that
require steady-state calculations such as biotechnological and chemical reaction where other
physical phenomena are involved. Therefore, a Multi Reference System (MRF) is applied for
calculating time averaged velocities and other turbulence quantities. As mentioned earlier this
model allows the flow simulation in baffled stirring tanks with rotating or stationary internal
parts. It uses at least two frames, one containing the frame where moving or rotating parts
7 2 |
Virginia Tech | (impellers, turbines) are present and another one for the stationary regions. The first is rotating
with the impeller and thus in this frame of reference the impeller is at rest. The second one is
stationary. We mentioned that MRF could involve at least two frames, but it can have multiple
frames in case where stirring tanks with multiple turbines have to be simulated. With this
approach several rotating frames can be modeled while the remaining space can be modeled as
stationary. The momentum equations inside the rotational frame are solved in the unsteady
form while those outside the rotating frame are solved in a steady form. In Figure I.5 a plot of
the 3D stirring tank grid with the two frames of reference are presented. The green area is the
rotating frame of reference that encloses the Rushton turbine while the white one is the
stationary frame. The rotational frame of reference has Diameter 1.2 times the Diameter of the
impeller.
Figure I.5: Computational Grid for simulations with the MRF model
7 3 |
Virginia Tech | E. Non Dimensional numbers
A number of dimensionless parameters that are important for the flow in stirring vessels are
presented.
a. The Reynolds number
One of the most common but important dimensionless parameter is the Reynolds
number (Re). It controls the scale of the model and the corresponding flow velocities. It
characterizes the region of the flow as laminar, transitional and turbulent. It is defined as the
ratio of the inertial forces to viscous forces and is given by the following equation:
ρuD uD
Re= = (I.29)
μ ν
where u is the characteristic velocity and D is the characteristic diameter.
In the case of stirring tanks D is the diameter of the impeller and u is the tip velocity of the
blade. The tip velocity is given by:
D
u =ωr =2πN =πDN (I.30)
tip 2
where N is the rotational speed of the impeller. Usually in the literature before substituting
(I.30) into (I.29) investigators drop the π in the u . Therefore, the Re number is finally given
tip
by:
ND2
Re= (I.31)
ν
Depending on the problem the region where the flow is laminar, transitional or turbulent is
different. For example for pipe flow the flow is laminar when Re<2000, transitional for
2000<Re<4000 and turbulent for numbers larger than 4000. In stirring tanks the flow is laminar
for Re<50 transitional 50<Re< 5000 and fully turbulent for Re>5.000. But this still depends on
the power number of the impeller.
7 4 |
Virginia Tech | b. Power Number
The power number parameter is defined by the ratio of the drag forces applied to the
impeller to the inertial forces and is given by:
P
Po= (I.32)
ρN3D5
The Power delivered to the fluid can be calculated straight out from the CFD calculations or
from the following equation:
P=2πNF (I.33)
torque
The torque can be calculated by integrating the Pressure on the impeller blade.
F =ω⋅∫r×(τ⋅dA) (I.34)
torque
A
There is a relation in the CFD packages from which the Power number can be obtained:
b
⎛5w ⎞⎛ N ⎞
Po=a b b (I.35)
⎜ ⎟⎜ ⎟
⎝ D ⎠⎝ 6 ⎠
a,bare constants (5, 0.8), D is the impeller diameter w is the width of the blade and N is the
b b
number of blades. Our Rushton turbine has:N =6 withw =0.0127m. By substituting them
b b
into (I.35) we obtain:Po=6.25
The above number is correct assuming unbaffled tank. In case where baffles are present the
power number is significantly affected.
w D
In our case:N b =0.4 and =0.33thusPo=1.05.
b T T
To conclude, we can say that the power number for a single impeller depends on the impeller
type, the size of the impeller comparing to the size of the tank, the Re number, the elevation of
the impeller, the number of baffles and their width if any are present.
7 5 |
Virginia Tech | c. Froude Number
The Froude number is defined as the ratio of inertial forces to gravitational forces and it
is given by:
u2 N2D2 N2D
Fr = = = (I.36)
Dg Dg g
The Froude number is an important parameter for studying the motion of vortices in unbaffled
tanks. Although the effect of the Froude number is negligible for baffled tanks or unbaffled
with Re<300, for unbaffled with Re>300 we can define a new function (I.37) that contains both
the Po and the Fr non-dimensional numbers
b⋅Po
f = (I.37)
(a−logRe)Fr
By plotting f as a function of Re number we get a line similar to the one that relates the Po
number with the Re number.
d. Flow number
The Flow number (Fl) is a parameter that characterizes the pumping capacity of an impeller
and it can be defined as:
Q
Fl = (I.38)
ND3
Q can be calculated by integrating the outflow in the discharge area of the impeller. For axial
impellers the area is circular while for the radial ones it is a section of a cylinder. The Flow
number for a radial disk impeller can also be calculated theoretically from the following
equation:
b c
⎛ N ⎞ ⎛w ⎞⎛ D⎞
Fl =a b blade
⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ 6 ⎠ ⎝ D ⎠⎝T ⎠
a=6 (I.39)
b=0.7
c=0.3
7 6 |
Virginia Tech | F. More results concerning the Rushton turbine
In the main body of the dissertation we present the most basic results that will be needed
for an archival publication. In this section additional results for the case of the Rushton tank
can be found. More specifically non-dimensional line plots of radial and tangential velocity,
velocity magnitude, dissipation rate, turbulent kinetic energy and vorticity at the centerline of
the impeller are presented for three Re numbers (20000, 40000 and 45000).
Figure I.8 illustrates the predicted profiles of the radial velocity and the velocity
magnitude at the impeller radial center line for all the tested turbulence models, as well as the
in-house experimental data measured via Digital Particle Image Velocimetry (DPIV). It can be
seen that the Standard k-e model predicts quite well the maximum values. As we move
radically further from the impeller until the pointr/D =0.27, it underpredicts them, and
tank
beyond that point the Standard k-e model results agree relatively well with the experimental
data. The RSM model in this case gives the poorest results from all the turbulence models. The
RNG K-e overpredicts the experimental results near the impeller but further out into the filed it
agrees very well with the experiment. In-house experimental data are not available for the other
two cases of impeller clearance. Especially for the low clearance configuration experimental
results do not exist in the literature at all. We observe that the radial component of the velocity
for this case is very small compared to the other two configurations. In the bottom frame when
we present the velocity magnitude we have calculated only the root of the sum of the squares
of two (radial and axial) of the three components of the velocity in order to compare it with the
experimental data. Measurements have been taken in a perpendicular plane that passes through
the impeller blades and therefore no out of plane component (circumferential) is available.
7 7 |
Virginia Tech | Comparisons for the dissipation rate and the turbulent kinetic energy are shown in Figure
I.9. All the turbulence models overpredict the dissipation rate close to the impeller blades. This
happens because in that area we have the steepest gradients which make the calculations more
complex. A little further from the impeller tip the values of the dissipation drop sharply. The
model that captures most details in that case is the RNG K-e. As far as the turbulent kinetic
energy is concerned all the turbulence models underpredict its value, mostly at the impeller
discharge zone. This can be explained by the fact that the experimental results contain
contributions from the periodic motion of the impeller blades, since every time a blade passes
by the point of interest; a strong radial jet is created. In the literature it is stated [14] that by
removing (filtering) that periodicity, the values of turbulent kinetic energy drop and in most
cases would have matched computational results. Figure I.9 confirms the idea that
experimental values result following this procedure, but there is sill some discrepancy with the
calculated results. Although the Reynolds stress model is the one that is closer to the
experimental results without the periodicity, it still underpredicts them.
7 9 |
Virginia Tech | The top frame of Figure I.10 displays the distribution of the vorticity along the centerline of
the impeller. Clearly none of the turbulence models can capture what is happening. Every time
that a blade is passing a strong radial jet is created followed by the trailing vortices that
propagate downstream as indicated from the experimental results. The nature of the vorticity
seems to have this oscillating behavior in the experiments due to the periodicity and due to the
fact that trailing vortices is present. It may also be possible that the method of calculating
vorticity from the measured velocity components introduces errors. The bottom frame of the
same figure compares the tangential velocities at the impeller centerline. What is shown in this
Figure agrees with the theory in the sense that the tangential component of the velocity has
high absolute values close to the impeller, and as we move radically out takes smaller values.
Unfortunately in-house experimental results are not available, and it is difficult to compare
them with results from the literature because different configurations and different Reynolds
number are available. Figures I.8 to I.10 correspond to Re=20000.
Figures I.11-I.13, I.14-I.15 show similar profiles for all the studied variables (radial
velocity, velocity magnitude, tangential velocity, turbulent kinetic energy, vorticity and
dissipation rate) for different Reynolds numbers, namely 40000 and 35000. The results are
qualitatively similar to those obtained at Re=20000.
8 1 |
Virginia Tech | In Figure I.18 (top frame) radial velocity profiles obtained by simulations, using three
different turbulence models, and measurements are compared at r/T=0.19 (very close to the
blades) and for Re=35000.All the turbulence models predict quite well the profile of the radial
jet. The velocity is normalized with the tip velocity of the blade and the axial distance with the
blade width. The second frame in this Figure shows the dissipation rate profiles again at
r/T=0.19 and for Re=35000. Either the Standard k-e model or the Reynolds stress models are
able to capture the rapid changes of the dissipation rate while the RNG K-e underpredicts it.
Figure I.19 and I.20 illustrate the same variables for the same Re number but in two stations
further downstream from the impeller blades (r/T=0.256 and r/T=0.315). In the first case
(r/T=0.256) although the velocity profiles are still well captured by all the turbulence models,
the dissipation rate is overestimated by the Standard K-e and the Reynolds stress model and
only a little by the RNG K-e. As we continue downstream (r/T=0.315), we observe that the
profiles of the velocity obtained from the simulations start to diverge from the measurements.
Although the dissipation profiles obtained by the Standard K-e and the Reynolds Stress model
still overpredict the experimental data, the RNG K-e model seems to be promising. By
observing the shape of the velocity profiles of the jet we see that as we go further downstream
they open up as we expect from the jet theory.
9 0 |
Virginia Tech | Chapter 1
Introduction
Since the inception of the wheel as a viable means of ground transportation, man has
been on a never-ending quest to optimize its use for the transport of people and cargo.
Vehicles of all shapes, sizes, and weights have been built to accomplish one task or
another. Although vastly different in design and intended application, we could classify
most ground vehicle in terms of a single design feature; the number of wheels. This
classification does not predicate advantages of one vehicle over another. However, it does
provide a metric against which the designer may estimate of a vehicle’s potential
performance characteristics and general capabilities. Therefore, it stands to reason that
the historical record should demonstrate mankind’s quest to classify the dynamic
characteristics and performance advantages of vehicles with every conceivable number of
wheels. This is in fact the case. Simply by examining the design and use of ground
transportation throughout history, we can see both experimentation and refinement in the
design of everything from vehicles having no wheels (tracks or legs) to those containing
hundreds of wheels (trains). Figure 1.1 presents the best known single-wheel vehicle, the
unicycle. Although this would have been the only possible configuration at the moment
of the wheel’s inception, the design has never proven itself as an effective means in the
1
Introduction |
Virginia Tech | transportation of people and cargo. However, it remains in mainstream society as a
source of entertainment and amusement.
Figure 1.1: One-Wheel Vehicle Figure 1.2: Standard Two-Wheel Vehicle
Likewise, we see in figure 1.2 the common perception of the two-wheel vehicle, the
bicycle. This design, though inherently unstable, has found widespread use and
acceptance throughout the world. Although the standard bicycle has met with great
success in both human and engine-powered transportation its overall utility as a
workhorse remains a point of debate. Millions of people all over the world rely on the
standard bicycle as their primary mode of transportation. However, cargo capacity is
meager at best.
At this point, we could make a strong argument for the correlation between how many
wheels are on a vehicle and its relative usefulness to society. Indeed, we could continue
this pattern by examining some of the more successful three-wheel designs. Though not
as prevalent in number as bicycles and motorcycles, this design shows up in everything
from toy tricycles to commercially successful off and on-road vehicles. Figure 1.3
presents a very successful three-wheel car marketed by the Morgan motor company
during the late 1920’s. Even though the design lost favor compared to vehicles with more
wheels, these types of vehicles are still highly acclaimed and sought after by both
collectors and driving enthusiasts. Naturally, they also tend to be much more stable than
bicycles and motorcycles, but problems still exist. In fact, it was the high-speed
2
Introduction |
Virginia Tech | moved over the world’s roadways every year. Compared to the success of the four-wheel
vehicle class, the popular two-wheelers and nearly forgotten three-wheelers are primitive
in their capabilities. However, even with the incredible success of the four-wheelers,
increasing utility does not end there. Larger trucks designed specifically for cargo
handling can have anywhere from 10 to 22 wheels. These examples effectively support
the thesis that more wheels inherently lead to more utility when considering the
transportation of people and cargo.
Finally, if we take the utility to number of wheels correlation toward the limit, we find
one of the most influential vehicle types since the development of the wheel itself, the
train (Figure 1.5). Largely responsible for United States expansion in the West, the train
represents to limit of the wheel-utility correlation. Most of a train’s volume is dedicated
to cargo. Its efficiency in ground transport is therefore undeniable. Even today when
most Americans do not travel by train, it remains at the forefront of industrial
transportation.
Figure 1.5: Multiple-Wheel ground vehicle: The Train
We have made an argument supporting the idea that more wheels are better. In light of
this apparent correlation, one would assume that investigation of the two-wheel concept
would prove fruitless. However, what must be considered here is that the historical
development of ground vehicles has focussed on efficiency in business, commerce, and
personal transportation. Further, designers of ground vehicles have in general worked
4
Introduction |
Virginia Tech | under the assumption that vehicle control would ultimately fall into the hands of a human
pilot. If another metric of utility is employed, we see much different results.
Consider the case in which the motivating force for vehicular design is that of movement
through harsh and discontinuous terrain as would be expected in cases such as warfare.
Vehicles with multiple wheels are used for troop and cargo over prepared road surfaces
but tracked vehicles have by far been the design class of choice for traversing off-road
terrain. Further, tracked vehicles have proven effective in other conditions where the
terrain is not groomed or conditioned for use with wheels. Planetary exploration,
traversal of snow, and any application requiring a zero turn radius have been particularly
attractive for this design concept. Clearly, some applications warrant a considerably
different design approach than the one by which ground transportation has traditional
been motivated.
Vehicular design for transport has evolved from focussing on the refinement of
mechanics and suspension to focussing more on the integration of electronics and control
systems. Therefore, most work in vehicular mechanical design is being done for non-
traditional applications such as planetary exploration, traversal of discontinuous terrain,
stair climbing, and mine clearance. One could argue that the engineering community has
entrenched itself with mindsets developed over years of manned transport system design.
With new attention being given to autonomous robotics and their use in unconditioned
environments, vehicle designers must rethink the old views of ground traversal and
release the assumptions inherent to traditional human-driven transport methods. This
thesis rethinks one of the earlier wheeled vehicle approaches and helps develop a new
class of vehicle that should be considered for autonomous applications, namely, the
biplanar bicycle.
To understand the concept, consider first a pair of uniform, balanced wheels set abreast of
one another and coupled by an axle pivoted at the center of each wheel. Further, assume
that a mass has been added at the rim of each wheel at a point closest to the ground.
5
Introduction |
Virginia Tech | Reaction Masses in the Driving Position
All that remains to create a functional biplanar bicycle is to create a means for actuating
the reaction masses. This could be a motorized carriage that moves along the wheel rim
on a track. An equally effective but simpler mechanical arrangement uses a motor at the
center of the wheel driving an eccentric mass that moves relative to the wheel. Figure 1.8
shows an early prototype vehicle, constructed using two cordless drills and radio-
controlled vehicle electronics.
Figure1.8: Early Prototype of the Biplanar Bicycle
The use of independent drive motors and masses on each wheel allows independent
control of the motion of each wheel. Alternatively, we can use a single, centrally located
reaction mass and drive each wheel relative to the mass. In either case, steering is
provided via differential drive. The prototype shown in Fig. 1.8 has been used to
successfully demonstrate both modes of operation. The single reaction mass is
mechanically simpler and has become the preferred embodiment in our work to date.
Either of these configurations results in an extremely nimble vehicle that can follow any
path, even those that include zero radius turns.
It seems intuitive that the biplanar bicycle would be inherently poor at traversing terrain
that involved steep grades or obstacles such as stairs. However, intuition may be
7
Introduction |
Virginia Tech | misleading. In most conventional vehicles having three or more wheels, the radius of the
wheels limits the size of largest single step the vehicle can ascend. This is also true of the
biplanar bicycle, but, since the wheel size is large in relation to the overall vehicle, such
obstacles are relatively easier to overcome. Also, since the “wheelbase” of the biplanar
bicycle is zero, there is less likelihood of the vehicle becoming stranded by an obstacle
wedged under its frame. The reaction mass is vulnerable to interference with the ground,
especially because we would like it to be at the most extreme possible radius. Keeping
the reaction mass as close as possible to the wheels, or even inside the rims of the wheels,
will minimize or eliminate this concern.
The remainder of this thesis examines a simple planar model and its associated dynamics.
Both static and dynamic equilibrium are considered. The issue of wheel slip is
considered in chapter four, and chapter five presents vehicle performance envelops based
on input waveforms. Chapter six introduces the necessary information to complete the
three-dimensional dynamic simulation. The remaining chapters deal with secondary
design considerations in the development of this vehicle class.
8
Introduction |
Virginia Tech | Invented in China, the notorious south-pointing chariot was intended as a navigational
aid. Although its design is similar to that of the roman chariot, its operational intent is
vastly different. The south-pointing chariot uses a simple differential gear train to
maintain a constant pointing direction for the figure on top. This assumes the vehicle will
operate under the dynamic constraint of no-slip so that the dead-reckoning navigation
produced by the differential will be accurate.
Because of the ubiquitous use of two-wheeled carts, chariots, and carriages, we find it
useful to further refine what we intend when describing a vehicle as a “biplanar bicycle.”
Consider the modified “biplanar” to the standard idea of a bicycle.
Figure 2.3: The Standard Bicycle, U.S. Patent 4684143
Figure 2.3 presents the image most commonly conjured by the term bicycle. What is
important to note here is that the two wheels primarily operate within the same spatial
plane, deviating only during turns. Therefore, the biplanar modifier is intended to
exclude the standard bicycle and imply only vehicles containing a two-wheel, single-axle
configuration. Although this restricts the definition considerably, it is not all that is
needed. An inherent feature of the biplanar bicycle not implicit in the name, is the
existence of only two points of ground contact. Reconsider the chariots shown in figures
2.1 and 2.2. Both of these, along with every two wheeled cart ever devised, rely on a
person, pack animal, or other propulsive source to be affixed to the vehicle at a location
in front of or behind the wheel axle. The result of this restriction is the creation of a third
point of contact on the ground. This allows the vehicle to perform in a similar manner as
11
Literature Review |
Virginia Tech | a tricycle. More importantly, it allows the vehicle to maintain a given orientation with
respect to Newtonian ground. When we discuss the biplanar bicycle, we are considering
only vehicles with two points of ground contact and therefore, no orientation reference.
As one may imagine, this restricts our definition to a very small number of vehicles. In
fact, most people have probably never seen a vehicle that falls under this strict definition.
However, some do exist.
In 1998 A. Namngani was awarded a patent for a vehicle intended to move people instead
of cargo. His design can, in every way, be defined as a biplanar vehicle.
Figure 2.4: Vehicle having two axially spaced relatively movable wheels, U.S. Patent 5769441
Figure 2.4 presents Namngani’s design. It is apparent from the diagram that the biplanar
bicycle, when designed for human transport, can be very awkward. Although we have no
confirmation on whether or not this design was physically realized, we are certain that it
would have been, at best, very difficult to manufacture. There is evidence, however, that
successful attempts have been made to build and operate a people-carrying biplanar
bicycle.
12
Literature Review |
Virginia Tech | Chapter 3
Planar Dynamic Model and Control Strategies
In chapter one we introduced the biplanar bicycle and showed that its potential
performance and simplicity of manufacture make it attractive in the field of autonomous
robotics. For this reason, it is important that we understand the dynamic characteristics
of the vehicle. Like any new vehicle, areas of the operational envelope that remain
unknown or misunderstood can eventually lead to unpredicted failure modes. We also
understand from chapter two that no work has been done on the analytical kinematics and
dynamics of this vehicle class. Determining a starting point for this work is therefore
easy. If we know nothing, it is best to begin with a simplified but dynamically
representative model.
3.1 Kinematic Model
The system can be greatly simplified by taking advantage of its inherent geometric
symmetry. By only considering performance within the plane of one wheel, we remove
the non-holonomic constraints normally associated with wheeled vehicles. This concept
will be revisited in chapter six when the complete three-dimensional dynamics are
derived. We may further reduce complexity by imposing constraint conditions of no-slip
and no-bounce. One may argue these assumptions will lead to erroneous results for any
14
Planar Dynamic Model and Control Strategies |
Virginia Tech | real vehicle. To assuage any fears in this regard, the fundamental concepts and validity of
the no-slip and no-bounce constraints will be reconsidered in chapter four. Finally, in
order to leave some generality in the solution, we assume the planar model to be rolling
on an arbitrary incline of b degrees. The idealized planar model of the biplanar bicycle is
shown in figure 3.1.
Im
P, (x,y), M, I
r
Q, m
l
qq
f
b
Re
Figure 3.1: Kinematic Diagram of the Idealized Biplanar Bicycle
Any physical realization of this vehicle will, of course, have mass in all components.
However, we assume here that the mass of link l is negligible relative to the wheel and
reaction masses. This assumption can be validated through a simple thought experiment.
If link l has substantial mass, we can combine it with the reaction mass and resize the
effective link length in order to maintain the correct location for the center or mass
relative to the vehicle body. By doing so, we can once again neglect the link in
subsequent calculations while avoiding any loss of generality. However, we are also
modeling the reaction mass as a point instead of a rigid body. The result of this
assumption is a missing rotational inertia term in the kinetic energy development. The
effects of this assumption, though quantifiable, are considered negligible relative to the
rotational kinetics associated with the wheel. Fortunately, the assumption of zero mass
for link l is of no consequence when analyzing the system’s static and dynamic equilibria.
With the kinematics defined and all assumptions made, we are ready to develop the
planar dynamic model.
15
Planar Dynamic Model and Control Strategies |
Virginia Tech | 3.2 Planar Dynamic Model
The coordinates of the points describing the system geometry with respect to the inertial
complex coordinate system are given by
v
P= xˆ(tˆ)+ jyˆ(tˆ) (3.1)
v
Q= xˆ(tˆ)+ jyˆ(tˆ)- ljejf(tˆ) (3.2)
where “^” indicated dimensional variables, gravity acts in the negative imaginary
v v
direction and P and Q are vectors locating the center of the wheel and the center of the
reaction mass respectively. The underlying goal of creating the simple planar model is to
generate a reasonable dynamic model with as few generalized coordinates as possible.
Here, we impose the assumptions discussed earlier to reduce the system to two
( ) ( )
coordinates: q tˆ for wheel rotation and f tˆ reaction mass angle. The mathematical
representation of the no-slip and no-bounce constraints can be described as
[ ]
xˆ(tˆ)+ jyˆ(tˆ) = rq(tˆ)+ j ejb (3.3)
This completely defines xˆ(tˆ) and yˆ(tˆ) in terms of our desired generalized coordinates.
v v
This result can now be used in our original kinematic definitions for P and Q (eqs. 3.1
and 3.2). We now have enough information to determine the system kinetic and potential
( ) ( )
energy. In terms of q tˆ and f tˆ we find these quantities to be
( ) ( )
T = 1 mQv & (cid:215) Qv & + 1 M Pv & (cid:215) Pv & + 1 Iq&2 (3.4)
2 2 2
(v ) (v )
V = mg Q(cid:215) j +Mg P(cid:215) j (3.5)
16
Planar Dynamic Model and Control Strategies |
Virginia Tech | To make the system more realistic, we must consider possible sources for energy
dissipation. Although aerodynamic drag and rolling resistance will both be present,
neither will be considered here. This is justified by the relatively slow speed
(approximately 5 mi./hr.) applications for which this vehicle type was initially developed.
Instead, we will only consider the damping associated with the pendulum-wheel bearing.
To do so, we assume linear viscous damping and generate a Rayleigh dissipation function
relating damping to the relative rotational velocity between the wheel and reaction mass.
1 ( )
R = Cq&+f& 2 (3.6)
2
With T, V, and R defined we may now solve the left-hand side of the extended Lagrange
equation for the dynamic response (Meirovich, 1970). This is represented as
(cid:230) (cid:246)
d (cid:231) ¶T (cid:247) - ¶T + ¶V + ¶R =Q (3.7)
(cid:231) (cid:247)
dt Ł ¶q& ł ¶q ¶q ¶q& j
j j j j
( ) ( )
where q= q,f and Q = t,t . At this point, the only undefined quantities for the
development of the equations of motion are the generalized forces Q in the Lagrange
j
equations. The only source of external energy is from the actuator used to drive the
vehicle. This actuator is modeled as a simple DC servomotor and its effects transmit to
both the generalized coordinates in the same way. In the development of t we assume a
commanded input voltage resulting from a standard pulse-width-modulation control
signal. Further, we neglect effects from the high-speed pole that results from armature
inductance (Wolovich, 1994). The result is a mathematical description of the motor
voltage as a function of motor parameters and generalized coordinates. We model the
motor as
( )
Vˆ =i R + K q&+f& (3.8)
a a a B
which leads to
17
Planar Dynamic Model and Control Strategies |
Virginia Tech | K K K ( )
t = T Vˆ - B T q&+f& (3.9)
R a R
a a
where K is the back electro-motive-force constant, K is the motor torque constant, and
B T
R is the armature resistance. At this point, all the information necessary to develop
a
complete equations of motion have been derived. However, the resulting equations are
rather complex and have as many as nine variables and parameters for the vehicle
designer to consider while building a workable system. This problem can be greatly
simplified by defining non-dimensional variables and parameters that better describe the
system behavior. In doing so, we also make it a simple matter to compare the relative
importance of nonlinear and damping effects in the system design.
We now introduce the following non-dimensional parameters
l m
L= a =
r M +m
(3.10)
C K K I
z = r2w + R ar2( MB +T m) w m = ( M +m) r2
and non-dimensional variables
ˆ
K V
U = T a and t =tˆw (3.11)
( )
R r2 M +mw2
a
where w = g/r . The final equations of motion may now be written in a much more
tractable form and valuable information about system response and stability can be easily
extracted. The equations of motion are
( ) ( )
1+ m q&&+ Lacos b - f f&&
[ ] (3.12)
( ) ( )
+ z + Lasin b - f f& f&+zq&+sin b =U
18
Planar Dynamic Model and Control Strategies |
Virginia Tech | ( )
( ) ( )
L2af&&+ Lacos b - f q&&+ Lasin f +z q&+f& =U (3.13)
Vehicle response, static and dynamic equilibrium, automatic control, and operating
envelope can now be discussed in terms of non-dimensional descriptors. Furthermore,
Eqs. (3.12) and (3.13) serve as useful tools in the design of this vehicle class. From this
point on, reference to any of these non-dimensional characteristics will be identical in
definition as those presented here. Finally, it is important to note the absence of q in Eqs.
(3.12) or (3.13). This is because q is a cyclic variable and only its derivatives affect the
dynamic response.
3.3 Equilibrium Conditions and Dynamic Stability
Before we can consider design or control of the biplanar bicycle class of vehicles, it
would be wise to investigate the stability on a global level. To do so, we simply consider
Eqs. (3.12) and (3.13). Two states of equilibrium can be easily derived. First, consider
the case in which the vehicle sits stationary on an arbitrary slope of b degrees. In this
limiting case, all angular velocities and accelerations reduce to be identically zero. When
we enforce this condition on the equations of motion, they reduce to
( ) ( )
sin b =U and La sin f =U (3.14)
o o o
where, in the preceding equation, we use notation such that x denotes the equilibrium
o
value of x(t). It stands to reason that there exists a limiting value of slope b after which
the vehicle will be incapable of holding its position. To determine this operational
boundary we solve for f in the previous equations. The resulting solution is
o
Ø sinbø
f =sin- 1 Œ œ (3.15)
o º La ß
19
Planar Dynamic Model and Control Strategies |
Virginia Tech | The inherent limitations on f occur because we cannot allow the angle to be complex.
Clearly, b must be bounded above and below to ensure an inverse sine operand less than
unity:
- sin- 1( La) £ b £ sin- 1( La) (3.16)
Another interesting result from Eq. (3.15) is the implication of two equilibrium values for
f . It will be shown that the solution in the second quadrant is always unstable. Further,
o
the bounds associated with Eq. (3.16) correspond to stable node bifurcations at which the
equilibrium solutions coalesce and disappear. This phenomenon can also be associated
with the dynamic condition of whirling in which the vehicle unsuccessfully attempts to
either remain stationary or climb the slope. Whirling is defined as the dynamic state in
which the reaction mass makes at least one full rotation around the axle.
The second condition for equilibrium is defined by assuming the vehicle maintains a
constant velocity over constant-slope terrain within the previously defined limits. This
condition can also be satisfied by
f&=
0. When this condition is enforced upon the
equations of motion the resulting equation for the equilibrium velocity is
U - sinb
q& = o (3.17)
o
z
while the equilibrium pendulation angle continues to satisfy Eq. (3.15). However,
knowing these equilibrium positions exist is not necessarily enough to understand the
vehicles dynamic characteristics. It would also be helpful to know the stability of the
equilibrium positions.
The stability of this dynamic equilibrium can be demonstrated using Lyapunov’s
linearization method (Slotine and Li, 1991). We enforce the condition for dynamic
equilibrium and impose the following perturbations upon the system:
20
Planar Dynamic Model and Control Strategies |
Virginia Tech | q&=q& +eq& ( t)
(3.18)
o 1
( )
f =f +ef t (3.19)
o 1
where e is a small nondimensional parameter (e << 1). This results in the following
system of equations representing the linearization of the system about the dynamic
equilibrium
( )
Ø 1+m Lacos b - f ø Ø q&&ø Ø z zø Ø q& ø
Œ ( ) o œ Œ 1œ + Œ œ Œ 1œ
º Lacos b - f L2a ß º f&& ß º z zß º f& ß
o 1 1 (3.20)
Ø 0 0 ø Ø q ø
+ Œ ( )œ Œ 1œ =0
º 0 Lacosf ß º f ß
o 1
Note that the sign of the single non-zero term in the stiffness matrix depends only on the
quadrant of f . Thus, our assertion about the stability of the static equilibrium points
o
given by Eq. (3.15) has been demonstrated. Taking the Laplace transform of the system
in Eq. (3.20) yields an eigenvalue problem whose solution is given by the roots of the
characteristic polynomial:
s (cid:238)(cid:237)(cid:236) [( 1+m) s+z] (cid:231) Ł(cid:230) L2 as2 +zs+ Lacosfo(cid:247) ł(cid:246)
} (3.21)
-
s[ Lacos(
b -
fo)
s+z
]2
= 0
We note that the single pole at zero corresponds to the cyclic coordinate q (in control
terms, we have developed a non-minimum state realization). The stability of the
remaining subset of poles may be verified using the Routh-Hurwitz technique. The
analysis demonstrates that, under the conditions of Eq. (3.16), the constant-forward-speed
equilibrium is stable for all physical values of the design parameters.
21
Planar Dynamic Model and Control Strategies |
Virginia Tech | After performing a Routh-Hurwitz stability analysis of this system, one might conclude
that this system is stable. This is to say, any command voltage will eventually result in a
steady state, controlled forward velocity. However, we will see that this is not the case.
This leads to the conclusion that simple linear control of this system may not yield a
sufficiently robust design. It is therefore expected that a more complex, nonlinear control
algorithm will be needed if we expect to control this class of vehicle throughout its
operational envelope of terrain, velocity, and acceleration. These nonlinearities are easily
demonstrated with the numerical solutions to the original, non-dimensionalized equations
of motion.
3.4 Numerical Simulations and Demonstration of Nonlinear Effects
Numerical integration of the governing equations provides an environment for simulation
of the system response. Such a simulation environment is useful for developing
understanding and intuition about the system, and provides a tool that can be applied to
feedback control-law development. Macro-scale influences of the nonlinearities (i.e., far
from equilibrium conditions) are exhibited in the system’s open-loop (i.e., uncontrolled
response). For example, consider upward traversal of a five-degree grade. As might be
expected, higher step-input voltages result in larger pendulation angles f during the
transient (Fig. 3.2). While the steady-state velocity depends only on the voltage, and the
steady-state pendulation angle depends only on the slope, transient overshoot
pendulations are expected because the reaction mass is the source of acceleration. Thus,
increasing the voltage step (that is, the set-point of steady-state velocity) increases the
overshoot. Of course, such an overshoot increase provides improved response only until
the pendulation angle reaches ninety degrees. Any further increase in the applied voltage
will result in a demand for more leveraging resistance than the rotating mass can provide
through gravitational potential. The result is whirling. Once f exceeds the unstable
equilibrium and whirling begins, it is unlikely that direct application of an open-loop
control strategy will result in the vehicle reaching the desired forward-speed equilibrium.
During whirling, a significant fraction of the motor’s energy becomes stored as kinetic
energy of the pendulum mass, leaving the two-wheeled vehicle at the mercy of the slope.
The result is a dynamic equilibrium with a net down-slope velocity. Figure 3.3 illustrates
22
Planar Dynamic Model and Control Strategies |
Virginia Tech | the new, stable, dynamic equilibrium. Even at steady state the reaction mass continues to
whirl (i.e., exhibits a limit cycle behavior), resulting in a similar limit cycle in wheel
velocity, but with a negative DC offset.
We stress that the new equilibrium behavior exhibited by the system does not represent a
destabilizing bifurcation of the steady-forward-speed equilibrium. According to Routh-
Hurwitz analysis, the target equilibrium retains its stability: clearly an alternative solution
exists. The initial conditions and DC voltage amplitude dictate the final equilibrium
response of the vehicle. As Figs. 3.2 and 3.3 indicate, the nonlinear nature of the
problem can result in unexpected responses. The equilibrium’s basin of attraction
boundary generates a sharp contrast in the behavior of the nonlinear system. Figures 3.2
and 3.3 have the same initial conditions and differ by only one ten thousandth in
nondimensional input voltage.
Although the constant-forward-speed equilibrium has been shown to be universally
stable, numerical simulations argue against simple open-loop command. Feedback
control may provide the desired consistency of performance.
3.5 Non-Traditional Rate-Feedback Control
The governing dynamic equations of the two-wheeled vehicle have been shown to be
nonlinear even under the simplifying assumptions of planar operation and slowly varying
terrain. Numerical simulations have demonstrated that such nonlinearities cannot be
neglected. The equilibrium deflection of the reaction mass depends on the slope. This
highly variable parameter proscribes a single linearized approximation to the vehicle
dynamics. This motivates the application of a simple heuristic control law, developed
with the aid of numerical simulation, rather than the application of traditional linear
control theory.
The control law should provide for effective reference tracking. The nature of the stable
steady-forward-speed equilibrium suggests a nonlinear feedback algorithm for
23
Planar Dynamic Model and Control Strategies |
Virginia Tech | developing zero steady-state error. Recall Eq. (3.17), which dictates the required
nondimensional voltage for a particular velocity:
U =zq& +sinb (3.22)
o o
It seems that it may be effective to compose a command voltage of
~
U(t)=zq& +sinb +U(t) (3.23)
ref
~
where U(t) is governed by a feedback control law. It is intended to have a regulating
effect, and it should approach zero as the system approaches equilibrium. Equation
(3.23) has a fundamental flaw, however. It is unlikely that b will be known a priori and
real-time measurement of b is potentially difficult. Equation (3.14) suggests that the
control law can instead be written in terms of the equilibrium value of f. Because f is
o
also unknown, the necessary term in the control signal can be approximated by a
nonlinear state-feedback term:
~
U(t)=zq& +aLsinf(t)+U(t) (3.24)
ref
The closed-loop system exhibits zero steady-state error (assuming its steady state is the
target equilibrium). Tracking, while meeting a design requirement, does not provide the
~
robust performance we seek. The enhancement U(t)to the control signal is composed of
linear feedback terms that seek to improve the transient response of the system (i.e.,
provide regulation). Because our system exhibits zero steady-state error, the primary
objective of the regulation is to prevent the whirling of the reaction mass. Rate feedback
is traditionally used to prevent excessive overshoot through an increase in the effective
linear viscous damping of the system. Rate feedback can be applied through both the
pendulation and wheel-fixed coordinates. System overshoot is controlled by the positive
derivative feedback gain K associated with the pendulum angle f. This gain is increased
d
until the resulting reduction in overshoot no longer justifies the subsequent increase in
24
Planar Dynamic Model and Control Strategies |
Virginia Tech | system response time; recall that some overshoot is intrinsically necessary in the response
to produce accelerations. Thus while a small amount of rate feedback dramatically
reduces overshoot, a large amount has little additional effect.
Experimentation with a proportional feedback gain K applied to the rate of the cyclic
p
variable q shows that this variable does not significantly influence the overshoot. Rather,
it has a moderate effect on the speed of response. Some portion of the increased system
response time can be recaptured through negative rate feedback on this coordinate.
While the speed of response is not as important as the tracking and overshoot criteria, it is
sufficiently important to justify the resulting robustness tradeoff. Here, the robustness in
question is associated with model uncertainty; a sufficiently large error in the modeling
of plant parameters could destabilize the target equilibrium in a Lyapunov sense. The
control law presented here has been developed numerically for generalized non-
dimensional geometry. To aid its use, the dimensioned control algorithm is presented
here as
q& Ø K K ø ml
uˆ = ref Œ C+ (B T )œ + sinf
r2 º R M +m ß (M +m)r
a (3.25)
[ ( ) ] g
+ 0.8 q&- q& - 10f&
ref r
The implementation of this control law demonstrates its ability to moderate pendulum
motion and smooth the vehicle velocity profile. Figure 3.4 shows both open- and closed-
loop dynamic response of the pendulum angle f and wheel rotation angle q as function of
non-dimensional time. All simulations were performed using the following system
geometry:
L=0.8 a =0.9 z =0.2
(3.26)
m =0.066 b =0.0873 K =2
25
Planar Dynamic Model and Control Strategies |
Virginia Tech | Chapter 4
Investigation of Planar Vehicle Slip Conditions
In the previous chapter we developed and analyzed the complete planar dynamic system
of the simplified Biplanar Bicycle model. During that development we imposed certain
kinematic constraints and assumptions to make the underlying mathematics more
tractable. In doing so, we generated useful information regarding dynamic response and
behavioral bifurcation. Further, we were able to distinguish certain operational regimes
in which the vehicle can and cannot operate. These performance envelopes will be
discussed in greater detail in the next chapter. Here, we are concerned with rethinking
the underlying assumptions in the primary planar model.
4.1 Kinematic Model
The most important kinematic constraints imposed in chapter three are those of pure
rolling and no bouncing. From a kinematic standpoint, we effectively reduced the
mobility of the entire system by a single degree of freedom, thus simplifying the dynamic
analysis. Pure rolling is defined as the kinematic relationship of a rigid wheel rolling
across terrain such that the relative velocity between the wheel and the ground along the
common tangent at the contact point remains identically zero. Figure 4.1
diagrammatically demonstrates this concept.
30
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | w
R
V
q
S
Figure 4.1: Kinematic Diagram of Pure Rolling
Note that the constraint, in its pure mathematical form, does not necessitate the wheel and
ground to be in direct contact.; it simply provides a relationship between angular and
linear (along the common tangent) displacement. If the system acts under the no-slip
constraint, the following kinematic relationship governs position in the direction of the
common tangent.
S =Rq V =Rw (4.1)
The no bounce constraint, like that of no-slip, removes a degree of freedom associated
with translation. In this case, the relative velocity of the wheel center and the plane is
identically zero in the direction along the common normal. The constraint associated
with enforcing no bounce is simply that of maintaining a constant distance (radius of the
wheel) between the wheel center and the contact plane. In other words, the wheel must
neither lose contact with nor impinge upon the plane.
Frictional forces are related directly to the no-slip constraint. It is the wheel-plane
frictional interaction that prevents slip in any real system. Likewise, the frictional force
capable of being produced is directly related to the normal force involved in maintaining
the no-bounce condition. Therefore, if we are concerned with whether the biplanar
bicycle will exhibit slip in any given operational scenario, both the frictional and normal
31
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | forces must be accounted for during the dynamic analysis. However, the Lagrangian
approach used in chapter three is designed to ignore internal constraint forces such as
friction and contact forces. We must modify the dynamics in order to see how these
forces are being manifested during operation. The explanation of how this is
accomplished is discussed in the next section. First, we revisit the kinematic definitions
of the planar model as developed in chapter three. The inertial frame in this case is
coincident with the inclined plane. The kinematic diagram of this system is shown in
figure 4.2.
g P, (x,y), M, I
r
Q, m
l
qq
Im
f
Re
b
Figure 4.2: Kinematic Diagram of the Idealized Biplanar Bicycle
The coordinates describing the system geometry with respect to the now rotated inertial
complex coordinate system are presented here as
r ( ) ( )
P= xˆ tˆ + jyˆ tˆ (4.2)
r ( ) ( ) ( )
Q= xˆ tˆ + jyˆ tˆ - lˆ jejf(tˆ)- b (4.3)
r r
where the real axis is now parallel to plane of slope b, and P and Q are vectors locating
the center of the wheel and the reaction mass respectively. Unlike the model in chapter
three, we cannot use no-slip and no-bounce to simplify Eqs. (4.2) and (4.3). Instead, xˆ(tˆ)
and yˆ(tˆ) must remain as two additional generalized coordinates. xˆ(tˆ) is associated with
32
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | the wheel position tangent to the slope face (Re) and yˆ(tˆ) defines the wheel’s position
along the common normal (Im). Instead of constraining these coordinates as was done in
chapter three, they are left free so we can later calculate the force required ensure that
they remain in their previously constrained positions.
4.2 Slip Investigation Dynamic Model
The dynamic system now contains four generalized coordinates: two from the original
model and two new ones associated with the relaxed slip and bounce constraints.
However, the addition of these variables is necessary but not sufficient to yield the
solution of the revised system. We must also add new forces to account for our added
degrees of freedom. We accomplish this by developing Lagrange Multipliers associated
with each new degree of freedom. These will be described in more detail later as we do
not require them to develop the system energy equations.
Unlike the original planar no-slip no-bounce model, we have defined the inertial
reference in a direction such that gravitational acceleration does not coincide with the
imaginary axis. If the plane has slope b, the direction of the increasing potential field is
Gˆ = je- jb (4.4)
r r
We note that P and Q may now be expressed in terms of the fundamental generalized
coordinates q(t), f(t), x(t), and y(t). The resulting kinetic and potential energy is given
by
( ) ( )
T = 1 mQv & (cid:215) Qv & +1 M Pv & (cid:215) Pv & +1 Iq&2 (4.5)
2 2 2
( )
(v ) v
V = mg Q(cid:215) Gˆ + Mg P& (cid:215) Gˆ (4.6)
33
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | Additionally, we can maintain the same Rayleigh dissipation function used in chapter 3 to
model a linear viscous damping effect in the pendulum-wheel bearing.
( )
R= 1 Cq&+f& 2 (4.7)
2
With these energy definitions complete, we must turn attention to the new kinematic
constraints associated with the relaxed slip and bounce conditions. The most efficient
method to check slip without deviating far from the original planar model is to employ
Lagrange multipliers to track internal constraint forces. The constraints associated with
no-slip and no-bounce are
rˆq(tˆ)- xˆ(tˆ) = 0 = C (4.8)
f
( )
yˆ tˆ - r = 0 = C (4.9)
n
respectively. Imposing a Lagrange multiplier on each of the constraint equations and
differentiating with respect to each of the generalized coordinates produces the
generalized forces associated with friction along the plane and contact along the common
normal. The general form of this derivation is
(cid:229) ¶ C ¶ C ¶ C
Q = l i = l f +l n (4.10)
j i ¶ q f ¶ q n ¶ q
i j j j
This calculation results in generalized forces that will supplement the right hand side of
the extended Lagrange equation (Meirovich, 1970). These forces can be shown to be
Q = rl Q = 0 Q = - l Q = l (4.11)
q f f x f y n
34
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | From this point the governing dynamic equations of motion can be derived using the
extended Lagrange equation
d (cid:231)(cid:230) ¶ T (cid:247)(cid:246)
-
¶ T
+
¶ V
+
¶ R
=Q (4.12)
dt (cid:231) Ł ¶ q& (cid:247) ł ¶ q ¶ q ¶ q j
j j j j
( )
where q=( q,f,x,y) and Qj = f t,t,lf ,ln .
The drive torque t is again modeled as a simple DC servomotor including consideration
of both applied voltage and back electromagnetic force. If the armature inductance is
ignored, the applied armature voltage is
( )
Vˆ =i R +K q&+f& (4.13)
a a a B
This leads directly to an expression for the motor torque t.
K K K ( )
t = T Vˆ - B T q&+f& (4.14)
R a R
a a
Where K is the back EMF constant, K is the motor torque constant, and R is the
B T a
armature resistance. Equations (4.8), (4.9), (4.11), (4.12), and (4.14) describe a system
whose input is the applied armature voltage and whose output is the dynamic response of
the four generalized coordinates q(t), f(t), x(t), and y(t). Further, both the normal and
frictional contact forces are found by solving for the two multipliers l and l. These
n f
may be compared with respect to a predetermined static friction coefficient to determine
whether or not the vehicle slips.
4.3 Numerical Simulations
Now that a complete dynamic model has been built, it is important to verify whether the
issue of vehicle slip even enters into a normal operational envelope. We have seen in
chapter three that the whirling phenomenon is a limiting factor on input voltage.
35
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | Therefore, we must only determine if slip is likely at voltages less than that of whirling.
If so, we can conclude that slip has the potential to tighten the operational envelope. A
simple thought experiment can show the potential for slip problems. Imagine the vehicle
at rest on a level plane. If the coefficient of static friction between the wheel and ground
is identically zero, the wheel will slip for infinitesimally small input voltages. In fact,
any commanded input will yield countering motion in both the reaction mass and the
wheel. However, the existence of slip in the limiting case does not predicate a problem in
normal operational environments. Therefore, numerical simulations need to be run if we
expect to prove or disprove slip as a significant effect.
To do so, we return to the simulation model of chapter three. Using identical geometry
and dynamic parameters we can calculate how much friction is required to prevent slip
and how much friction is available from the instantaneous normal force. The only
additional parameter required is the coefficient of static friction. To generate a test case
that is likely to slip, we consider one of the suggested operational applications of the
biplanar bicycle: the autonomous railway inspection vehicle. In this case we would
expect the possibility of having steel wheels on a steel surface with grease at the
interface. The approximate coefficient of static friction in this extreme case is 0.005
[Avallone, 1987].
Finally, we must define a reasonable metric with which to judge the vehicle performance.
In this case, we normalize the friction required to prevent wheel slip with the product of
the frictional coefficient and normal contact force. The resulting value can be considered
an instantaneous percentage of total available frictional effort. If the metric is below
100% at any given time, the system exhibits the no-slip constraint presented earlier. If
the normalized friction is greater than 100%, there is not enough contact force to induce
the friction needed to prevent wheel slip and the no-slip constraint is violated. At the
same time, we must track the normal force acting at the contact point to ensure it never
drops below zero. If it does, the system has violated the no-bounce condition and is
“jumping” off the contact surface.
36
Investigation of Planar Vehicle Slip Conditions |
Virginia Tech | Available and Required Friction
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 5 10 15 20 25 30 35
Time
38
Investigation of Planar Vehicle Slip Conditions
noitcirF
citatS
Required Friction
Available Friction
Excess in Available
Figure 4.2: Constituent elements of the slip condition metric
As seen here, the forces required to maintain the no slip constraint are relatively low
when compared to the dynamic forces being imparted during operation. Many other
simulations similar to those presented here have been run and examined to ensure the
validity of the model results. From these, the conclusion is made that vehicle slip does
not enter into any reasonable biplanar bicycle application. Instead, the whirling effect
remains the definitive boundary to the performance envelope. To verify this conclusion,
the simulation was run using a high-voltage, short-duration pulse input. Although the
input would be large enough to induce whirl, it is cut off before the pendulum can reach
the bifurcation point. As expected, the large initial torque causes the vehicle to slip.
Also as expected, as the input voltage is dropped and duration increased until we are
again using a step input, the whirling condition, not slip, stands out as the limiting
dynamic effect. The complete performance envelope will be discussed in more detail in
chapter five. |
Virginia Tech | Chapter 5
Performance Envelope of the Planar Biplanar Bicycle
In chapter three we saw that non-linear effects in the dynamic behavior of the biplanar
bicycle are significant. In particular, a bifurcation point exists at which the stable and
unstable dynamic equilibrium points coalesce and disappear. It is this phenomenon that
makes a robust control strategy necessary if we expect to maintain control of a real
vehicle. In fact, it was the avoidance of the bifurcation point that motivated the control
strategy presented in chapter three. This chapter considers the whirling problem from a
slightly different perspective. Because we have no analytical descriptor for the
bifurcation point, the control law development of chapter three did not contain a feedback
term based on the proximity of the operational point to that of the bifurcation. However,
it stands to reason that the control effort can be limited if we know how close the current
operational condition is to matching those of the dynamic node. Here, we present a
numerical analysis of the vehicle operational envelope as a function of slope b. In
addition to understanding the general shape of the envelope, an attempt is made to
understand the effects of specific input waveforms on vehicle performance.
39
Performance Envelope of the Planar Biplanar Bicycle |
Virginia Tech | 5.1 Envelope Generation
The numerical solution of the performance envelope is straightforward. The techniques
used to generate solutions are universal to all vehicle configurations. Still, it is
imperative that the designer understands that the results presented here are based on a
single vehicle geometry, and they will change if the non-dimensional descriptors are
modified. Based on several configuration simulations, we have validated the universality
of envelope shape and relative sizes, regardless of changes in vehicle parameters.
To generate a single performance envelope, the equations of motion developed in chapter
three are numerically solved for a given slope and input until the bifurcation point is
located. The slope is increased and another search is initiated until the bifurcation is
located. This process is repeated until the slope and corresponding input voltage equal
the limiting values for static equilibrium as presented in chapter three. As stated before,
this process is used for a variety of waveforms. All waveforms (with the exception of the
step) reach their respective final voltage value in the same period of time. This permits a
better qualitative analysis of their respective performances. Input types investigated are
shown in figure 5.1. The letter T designates the time period over which the input
waveform takes to reach steady-state.
1.2
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5
Time - T = 5
40
Performance Envelope of the Planar Biplanar Bicycle
egatloV
tupnI
dezilamroN
Step
Sin
Ramp
- Cos
Cycloid
Parabolic
Figure 5.1: Waveforms for Vehicle Input |
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