Impeller Placement Optimization: Mixing Versus Mechanical ...
Transcript of Impeller Placement Optimization: Mixing Versus Mechanical ...
1 Copyright © 2014 by ASME
IMPELLER PLACEMENT OPTIMIZATION; MIXING VERSUS MECHANICAL SHAFT FATIGUE
Sang Jin Lee Analytical Engineer
Robert W. Higbee Senior Analytical, Mechanical Design Engineer
Binxin Wu CFD Engineer
Philadelphia Mixing Solutions, Ltd. 1221 East Main Street
Palmyra, PA 17078, USA
ABSTRACT
Appropriate mixing system design is a balance between
performance and cost. For most mixing systems, all flow in
the mixing vessel is induced by the impeller which creates
predominantly symmetrical circulatory loops and, on average,
does not produce a net horizontal flow impinging against an
agitator assembly. However, in some applications a fluid inlet is
placed adjacent to an impeller which subjects the impeller to a
continuous flow oriented perpendicular to the impeller axis of
rotation. Such side flow adversely affects the life of an
agitator assembly due to fatigue loading. In a particular
commercial waste water treatment mixing application, there
was a desire to place an impeller in a high side flow inlet region
of a basin which would have necessitated an unreasonably large
shaft diameter to prevent premature shaft fatigue failure. Using
a combination of CFD flow analysis and fatigue based shaft
design; the impeller was placed at an appropriate height to both
minimize the fatigue affects of the horizontal inlet flow, as well
as to ensure proper mixing. 3 separate CFD studies are
presented – The originally requested configuration (impeller
next to side flow), impeller situated as high in the vessel as
possible (good fatigue life but poor mixing) and the final
optimum configuration (acceptable fatigue life and acceptable
mixing). Constant Bernoulli side flow forces were computed
from time averaged constant flow velocities determined by the
CFD studies which allowed the computation of mean and
alternating force components whose frequency of application
equaled the shaft rotations per minute. A Goodman fatigue
analysis approach was utilized.
Key Words: Mixing, Computational Methods, CFD,
Fatigue
INTRODUCTION
A mixing system employing a rotating agitator has been
used to make uniform product, help the chemical reaction,
control/modify physical properties, and enhance heat transfer,
etc. [1].
Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014
November 14-20, 2014, Montreal, Quebec, Canada
IMECE2014-36886
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Mixing system design requires mechanical and costing
considerations balanced against mixing process requirements
[2]~[5].
In a typical top entry mixing system, the fluid forces that
are perpendicular to the agitator shaft and act at the impeller
centerline, are a result of transient fluid flow asymmetries
acting on the mixing impeller. These loads are dynamic and are
transmitted from the impeller blades to the mixer shaft and gear
reducer [6], and do not produce a net horizontal flow impinging
against an agitator assembly.
Not much work has been done in the area of high cycle
fatigue that pertains to mixing systems. Research includes
corrosion fatigue [7][8], high cycle thermal fatigue [9][10] and
fatigue of rotating machinery components [11][12].
Karaagac worked on the fatigue analysis of an agitator shaft
[13][14], but his work concentrated on fatigue crack notch
propagation. Therefore, the work here shows a unique study for
the design application of high cycle fatigue to an industrial
mixing system.
For the commercial waste water treatment mixing system
being considered here, the impeller is located close to a high
flow region of a basin (Figure 1). The high speed unidirectional
force, adding to the circulatory forces, is impinging on the
impeller, and causing dynamic shaft loading such that
alternating and mean stresses are induced in the shaft.
Therefore, the originally requested 3 impeller locations (48, 82,
and 108 in off bottom in Figure 2) have been studied in order to
avoid the unnecessarily large shaft diameter that would have
been required to prevent a premature shaft fatigue failure and
which would substantially increase the cost for this mixing
application.
The study here is presented with (i) Computational fluid
dynamics (CFD) for the determination of fluid velocities at the
impeller to calculate Bernoulli forces, (ii) determination of
alternating and mean stress values from fluctuating bending
stress and steady torsional stress on the shaft, (iii) estimate of
fatigue life at each impeller location, based on Goodman
fatigue analysis. The final goal was to determine optimum
impeller location.
Bernoulli forces from CFD study
In a typical wastewater treatment system, a mixer is used
for various reasons, such as Mixing flocculants e.g. (FeCl3 or
Al2(SO4)3) into water, wastewater, or sludge; or for introducing
acid or caustic to control pH [1]. In this study, the mixer is used
for rapid mixing of chemicals into raw water.
Table 1 shows the inputs for the CFD study, which used the
commercial CFD packages Gambit 3.4.6 and Fluent 13.0
(ANSYS-fluent Inc., 2010). X-velocities at the front (left) and
back (right) of the impeller (marked with red dots) from 48 in,
82 in, 108 in off bottom (Figures 4, 5, and 6, respectively) are
checked and the differences of the magnitudes of these velocity
vectors are the inputs for the Bernoulli side force calculation
(Figure 7).
Per Bernoulli, pressure (P) acting on a plate is a function of
the velocity of the flow, the density of the fluid and the drag
coefficient which varies with the shape of the plate. It was
assumed that the flow acting on the impeller was equal to the
flow in the inlet pipe and that was directed at the impeller,
perpendicular to the shaft (Figure 7).
Table 1. Parameters for CFD input
Liquid height 10.5 ft
Tank dimensions Length X Width = 10. 5
X 10.5 ft
Fluid properties Vicosicty 1 cP
Density 1.0 S.G.
Impeller type PBT 4/45
Impeller off the bottom 48 in, 82 in, 108in
Impeller rotational speed 155 rpm
Inlet pipe diameter 36”
Inlet flow rate 33.3 MGD
Inlet pipe off the bottom 30”
Outlet Height X Width = 7 in X
3.5 in
Calculations of Fatigue Loading
The typical high stress points for an agitator shaft are just
below the lower bearing, and at the shaft coupling stress
concentration (Points B and A in Figure 9, respectively). The
dynamic loading acting at these points is considered as
completely reversed bending stress and steady torsion. The
hydraulic side forces, Fhyd in Figure 9, are based upon a PMSL
empirically determined formula, and have been assumed to
induce a mean bending stress for fatigue loading calculations.
The Bernoulli forces, FB in Figures 7 and 9 induce the
alternating bending stresses. The HP and RPM of the prime
mover remained constant throughout. Torque is also assumed as
constant, so a constant shear stress due to torsion was applied to
the agitator shaft. Calculated bending and shear stresses are
shown in Table 2.
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Figure 1. Configuration of water treatment basin
Figure 2. Mixer impeller locations
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Figure 3. Cartesian coordinate system for CFD study
Figure 4. X velocity from 48 in off bottom CFD study
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Figure 5. X velocity from 82 in off bottom CFD study
Figure 6. X velocity from 108 in off bottom CFD study
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Figure 7. Bernoulli force from side flow pressure
calculation
Where, V2 = inlet pipe velocity (ft/s),
Di = impeller diameter for impeller (i) (in),
PH = impeller horizontal projected height (in),
Cd = drag coefficient,
ρ = mass density of fluid (lb s2/ft /ft3),
P = pressure (lb/ft2),
AP = projected area (ft2)
Table 2. Calculated alternating and mean stresses from
bending and torsional load
Figure 8. S-N Curve
Figure 9. Definition of bending moment arm (excerted
from Philadelphia Mixing Solutions, Ltd Outline
Installation Drawing (OID))
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Endurance Limit
When certain materials are exposed to pure, completely
reversed tensile stress (σa) and when σa is plotted against the
fatigue cycle life (N) on log-log paper, a “knee” appears at 106
cycles such that σa vs. N has a negative slope for N < 106 and
has approximately zero (horizontal slope) for N < 106. Figure 8
shows two σa vs. N curves (a.k.a. “S-N” curves)-one typical of
carbon and stainless steels, which exhibits the knee, and one
typical of aluminum where no particular knee is present. For
carbon and stainless steels, the value of σa that corresponds to
the horizontal (> 106 cycles) section of the curve is known as
the “endurance limit” (Sn). Most real world parts are not
subjected to pure completely reversed stress, but experience a
combination of mean stress with a superimposed alternating
stress. For this case an alternate method must be used to
determine fatigue life. The “Goodman” method [15]~[17] is
just such an approach that can be used to determined fatigue
life for various combinations of alternating stress (σa) and mean
stress (σm).
Estimate fatigue life from Goodman diagram
Steel parts subjected to certain combinations of mean and
alternating stress were found by Goodman to have the same
fatigue life. Goodman plotted the mean and alternating
stresses corresponding to 106 cycles fatigue life on a graph
where the Y axis was defined as alternating stress (σa) and the
X axis was defined as mean stress (σm). He discovered that a
straight line starting at (σm = 0, σa = Sn), and ending at (σm = Su,
σa = 0) bounded the lower portion of the data. This line is
commonly known as the Goodman line. If a part is subjected to
a constant σm, σa that plots above the Goodman line, the part
will fail after less than 106 number of cycles. All stress states
that plot below the Goodman line are thought of as having
"infinite" (greater than 106 cycles) life.
On the "Goodman Diagrams" in Figures 10~15, the red dot
corresponds to the equivalent mean stress (σem) and equivalent
alternating stress (σea) for this application. Two lines will be
plotted through (σem, σea). The first line will be defined as the
"equivalent life line". This line starts out at (σm = Su, σa = 0)
and passes through (σem, σea). A part's fatigue life will be
similar for any stress state on this line. The second line will
start at (0, 0) and pass through (σem, σea). This line is called
the "load line". Since the part's stress varies linearly with
power, one can use this line to compute the sustained power
level that would cause the equivalent life line to become
superimposed upon the Goodman line.
The intersection of the equivalent life line with the
Goodman diagram (σa) axis may be considered as the level of
pure alternating stress that will yield the same life as (σem, σea)
and is defined in Figures 10~15 as (S4). In Figures 10~15, the
portion of the S-N curve for > 106 cycles has a PMSL
proprietary negative slope (not horizontal) for the prediction of
fatigue life in excess of 106 cycles. The intersection of S4 and
the PMSL proprietary S-N curve yields a predicted fatigue
cycle life.
Figures 10~15 show the Goodman diagrams and
corresponding curves for each impeller location. S4 is also
shown as a horizontal line on the S-N curves.
Figure 10. Goodman diagram and S/N curve; Impeller
48 in off-bottom, Bending moment at agitator shaft
coupling
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Figure 11. Goodman diagram and S/N curve; Impeller
82 in off-bottom, Bending moment at agitator shaft
coupling
Figure 12. Goodman diagram and S/N curve; Impeller
108 in off-bottom, Bending moment at agitator shaft
coupling
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Figure 13. Goodman diagram and S/N curve; Impeller
48 in off-bottom, Bending moment at lower bearing
Conclusions and Future Work
The fatigue lives determined from the Goodman diagrams
and S-N curves are shown in Table 3. For the 48” off–bottom
(originally requested configuration), the fatigue lives are less
than 3 years, and therefore do not met the requirements for
waste water treatment operation. Moreover, an unreasonably
large-diameter shaft would be needed for this case to improve
the fatigue life and this would be cost prohibitive.
For the highest impeller location at 108” off-bottom, the
impeller location is close to the top of the basin, thus the
performance of the mixing system is poor even though it shows
a very good fatigue life. Therefore, the 82” off-bottom has been
selected as the optimum configuration, which has an acceptable
fatigue life and an appropriate level of mixing.
The mixing process is the primary consideration for the
design of the mixing system, and this study shows that the
consideration of fatigue design adds a useful tool for balancing
mechanical and costing considerations to support the primary
objective.
Figure 14. Goodman diagram and S/N curve; Impeller
82 in off-bottom, Bending moment at lower bearing
Future work would include placing a strain gage sensor
array on the shaft to gain a better understanding of the stress
history for this case where a constant side flow impinges on a
rotating impeller.
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Figure 15. Goodman diagram and S/N curve; Impeller
108 in off-bottom, Bending moment at lower bearing
Table 3. Estimated fatigue lives
Acknowledgement
The authors would like to thank Mr. Ed Gamber, Vice
President of Engineering, Mr. Todd Hutchinson, Vice President
of R&D, and the test lab team members of Philadelphia Mixing
Solutions Ltd. (PMSL) for their idea and guidance throughout
this study.
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