impeller placement optimization: mixing versus mechanical

1 Copyright © 2014 by ASME

IMPELLER PLACEMENT OPTIMIZATION; MIXING VERSUS MECHANICAL SHAFT FATIGUE

Sang Jin Lee Analytical Engineer

[email protected]

Robert W. Higbee Senior Analytical, Mechanical Design Engineer

[email protected]

Binxin Wu CFD Engineer

[email protected]

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


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.


Figure 1. Configuration of water treatment basin

Figure 2. Mixer impeller locations


Figure 3. Cartesian coordinate system for CFD study

Figure 4. X velocity from 48 in off bottom CFD study


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))


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




coupling



coupling



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.



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.




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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impeller placement optimization: mixing versus mechanical

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