Mechanical Design October 23, 2009 30 of 50

6.4.5 Magnetic DrivesMagnetic drives eliminate the problems of sealing a rotating shaft by

using magnets to transmit torque from outside a vessel to inside the vessel. Allof the seals required for a magnetic drive are static seals. A motor drives arotating magnet outside a seal can, which turns by magnetic force a shaftinside the seal can. For small reactors, even at high pressures, the magneticdrives provide a simple and effective seal. The problems arise with larger sizedrives. Magnets, even high-strength magnets, cannot transmit large torquesand the magnets need to be as close together as possible. To support the mixershaft, the shaft bearings must be inside the vessel, which exposes them to thevapors from the process. This exposure can be a problem with corrosivematerials. The expense of magnets also is a problem. Even small drives areexpensive compared with typical mechanical seals. The biggest singleadvantage of a magnetic drive is the ability to handle a high pressure withoutthe leakage possibilities associated with a rotating seal.

6.5 Shaft DesignShaft design must accommodate hydraulic and mechanical loads and

must avoid vibration near the natural frequency. Hydraulic loads on the shaftresult from the torque required to turn the impeller(s) and random orsystematic lateral hydraulic loads on the impeller(s). Other sections of thehandbook describe methods for determining impeller power. Shaft design willuse impeller power to calculate torque and hydraulic forces and thus size ashaft within allowable stress limits.

Natural frequency, or critical speed, is the frequency of free vibrationfor the system. At the natural frequency an undamped system with a singledegree of freedom will oscillate after a momentarily displacement. Theoperating speed of the shaft and impeller system must be sufficiently far fromthe system’s natural frequency to prevent undamped vibrations. If deflectionscaused by vibration become sufficiently large, the shaft could bend or break.Although torsional natural frequencies must be examined on very largemixers, the following discussion will be addressed to the lateral naturalfrequencies, which affect the design of all mixer shafts.

6.5.1 Steps in Designing an Appropriate ShaftThe steps necessary to design a mixer shaft first consider strength,

then commercially available material, and finally natural frequency. Thefollowing steps also consider alternatives if natural frequency problems areencountered.

1. Determine the material of construction and the allowable stresses forboth combined shear and combined tensile.

2. Calculate the minimum solid shaft size for an overhung shaft, whichmeets both shear and tensile stress limits. Then round up to thenearest ½-inch increment (or metric equivalent) to obtain a size forcommercially available bar stock.

Mechanical Design October 23, 2009 31 of 50

TP

NQ(max) , 63 025 (6.3)

3. For this standard shaft size, determine the natural frequency of theshaft and impeller system. If the system meets the natural frequencycriterion, the design is complete.

4. If the shaft speed is near the natural frequency, increase the solidshaft size to the next ½-inch larger, or metric equivalent. Then redothe critical speed calculation. Again, if the operating to critical speedratio meets the natural frequency criteria, design is complete. Ifnecessary repeat this process one additional ½-inch increment. Ifmore than 1 inch additional diameter over the calculated diameterfor strength is required to meet the natural frequency criteria, go onto the next step.

5. Select a hollow shaft, usually a standard pipe size, that meets themechanical stress requirements. Compute the critical speed andcompare it with the natural frequency criteria. Typically, a hollowshaft can increase the critical speed about 20%. For strength thehollow shaft diameter is usually not more than twice the solid shaftdiameter. One drawback of hollow shafting is that axially adjustableimpellers are very difficult.

6. If hollow shafting either cannot be made to work or is undesirabledue to axial impeller adjustability, a foot (or steady) bearing willprobably be required. Begin the design with the minimum shaft sizefor strength, and using the formulas for a shaft with a steady bearingto compute the critical speed. Adjust the diameter upward in ½ inchincrements until the natural frequency criterion is met. Adding aninch or less to the minimum diameter for strength should satisfynatural frequency requirements.

The following sections will present methods for calculating strength andnatural frequency for different shaft types. Since all these methods makecertain assumptions about the design, other methods using other assumptionscan be used and usually result in similar results.

6.5.2 Shaft Design for StrengthComputing shaft size for both allowable shear and tensile stress

requires that the designer know the rotational speed of the mixer, plus thestyle, diameter, power, location and service of each impeller. For overhungshafting the maximum torque will occur above the uppermost impeller. Themaximum torque can be determined from the following equation:

Mechanical Design October 23, 2009 32 of 50

MP L f

NDi n H

n

n n

max

,

19 000(6.5)

Table 6.2 Hydraulic Service Factors, fH

P PP

Pi imotor

icalculated

calculated

(6.4)

where TQ is torque, inch-lbs, P is motor power, hp, and N is rotational speed,rpm. To be sure that process upsets or changes do not exceed shaft designlimits, the motor power is used instead of impeller power.

For design calculations, impeller power must be a calculatedquantity, unless power has been measured on a previously built, identicalmixer. Impeller power calculations based on empirical laboratorymeasurements can be used successfully for most mixer design. However, as agood design practice, total impeller should not be more than about 85% or 90%of motor power. Impeller power may be as little as 50% of motor power for aconservative design with uncertain process conditions.

In the following equation for bending moment, individual fractions ofmotor power are needed for each impeller, because the impellers are atdifferent locations on the shaft. The following adjustment will give impellerpower values that will sum to motor power:

The maximum bending moment, Mmax, for an overhung shaft is thesum of the products of the hydraulic forces and the distance from theindividual impellers to the bottom bearing in the mixer drive. The followingexpression computes an empirical hydraulic force related to the impellertorque acting as a load at a distance related to the impeller diameter.

where Mmax is the bending moment, in-lbs, Ln is the distance from the bottomdrive bearing to the nth impeller location, inch, N is the rotational speed, rpm,and Dn is the diameter of the nth impeller. The bending moment also dependson a hydraulic service factor, fH, which is related to the impeller type andprocess operating conditions. Approximate hydraulic service factors for thevarious impellers and conditions can be found in Table 6.2.

Since the bending moment and the torque act simultaneously, theseloads must be combined and resolved into a combined shear stress and acombined tensile stress acting on the shaft. The minimum shaft diameter forthe allowable shear stress can be calculated as follows:

Mechanical Design October 23, 2009 33 of 50

dT M

s

Q

s

16 2 21 3

(max) max

/

(6.6)

s

Q o

o i

T M d

d d

16 2 2

4 4

(max) max(6.8)

d

M T Mt

Q

t

16 2 21 3

max (max) max

/

(6.7)

where ds is the minimum shaft diameter, inch, for the shear stress limit, σs,psi. For 316 stainless steel operating near 100ºF (38 C), the recommendedcombined shear stress is 6,000 psi.

The minimum shaft diameter for the allowable tensile stress iscalculated with a different equation:

where dt is the minimum shaft diameter, inch, for the tensile stress limit, σt,psi. For stainless steel, the recommended combined shear stress is 10,000 psi.

The minimum shaft diameter will be the greater of the two valuespreviously calculated, in (6.6) and (6.7). For practical purposes, most mixershafts are made from bar stock, so standard sizes usually are available in half-inch or one-inch increments. For critical speed calculations, the next largerstandard shaft diameter is usually used.

Limits for shear and tensile stresses depend on shaft material,operating temperature, and chemical environment. Since nearly everychemical system is different, a review by a materials engineer should be madeand appropriate allowable stresses established, especially for new or corrosiveapplications.6.5.3 Hollow Shaft

A hollow shaft, made from pipe, can increase the stiffness and reducethe weight of a mixer shaft in critical speed calculations. Such changes willincrease the natural frequency and extend the allowable shaft length oroperating speed. When determining the appropriate shaft size for the strengthof a hollow shaft, begin with the dimensions for standard available pipe ortube. Then compute the shear and tensile stress values and compare themwith the allowable values. The equations for combined shear and tensile limitsin hollow shafts are respectively:

Mechanical Design October 23, 2009 34 of 50

md x

dtc

dx

dtkx f t

2

2 ( ) (6.10)

t

Q o

o i

M T M d

d d

16 2 2

4 4

max (max) max(6.9)

where do is outside diameter, inch, and di is the inside diameter of the pipe.inch. Because nominal pipe dimensions have tolerances, the minimum wallthickness should be used to determine the inside diameter.

The smallest pipe dimensions which keep the shear stress, σs, and the tensilestress, σt, below allowable limits is probably a good start for furthercalculations.

The process of determining the location of maximum stress for ashaft using a steady bearing can be tedious. Usually the location is just abovethe upper impeller. A conservative approach would be to assume prismsupports at the ends of the shaft. Then apply hydraulic loads to createmoments and sum them at the points shown in Figure 6.35 for the shaft andimpeller system shown in Figure 6.34.6.1.1 Natural Frequency

Natural frequency is a dynamic characteristic of a mechanicalsystem. Of primary concern to mixer design is usually the first naturalfrequency, which is the lowest frequency at which a shaft will vibrate as afunction of length and weight. The first natural frequency is analogous to thevibration of a tuning fork, except on a larger scale.

The concern about natural frequency is that an excitation, such asmixer operating speed, could cause undamped vibrations. Such vibrationscould result in sudden and catastrophic failure of the mixer shaft. Largemixers normally operate below the first natural frequency. Small, portablemixers, which accelerate quickly, often operate above the first naturalfrequency. In either case, operating at or near the natural frequency must beavoided for both mechanical reliability and safety.

The standard vibration equation applies to a mixer shaft:

Where m is mass, cν is called the damping coefficient, k is the effective springconstant for the system, and f(t) is some type of forcing function. The forcingfunction for mixers can be approximated by a sine or cosine function. A mixerdesign must address several issues. The damping coefficient is seldom knownto any degree of accuracy because, the damping coefficient depends on thematerial being mixed, the type and number of impellers, and the size of theimpeller compared with the shaft diameter. To simplify the solution the effect

Mechanical Design October 23, 2009 35 of 50

c c where c kmc c/ 2 (6.11)

08 12. .N N Nc c (6.12)

Figure 6.32 Transmissibility for Various Damping Ratios

of damping is generally represented as the ratio of the damping coefficient tothe critical damping coefficient, cc.

If energy is added to a system more than the amount dissipated throughdamping, the amplitude of vibration will increase. If the energy additioncontinues, the amplitude of vibration can exceed the deflection that will bendthe shaft. The amplification factor depends on the proximity of the operatingspeed to the natural frequency. This relationship is shown in Figure 6.32.

Transmissibility is also often called the force magnification factor. Any appliedforce to a shaft under dynamic conditions will be amplified by thismagnification factor. A side load of 100 units at rest, for a damping ratio of 0.1will behave as a 257-units side load when N/Nc=0.8 and a 388-units side loadwhen N/Nc=0.9. Mixer manufacturers design stress limits are set based on theallowable approach to the first lateral natural frequency, Nc. The worst casescenario is to assume that no damping is present, δ=0. This assumptionensures that even if the mixer is operated in a vessel without liquid present,(no damping) the shaft and impeller system will remain stable.

The other key design assumption is that the support stiffness issufficiently large that the overall stiffness, k, is controlled only by the shaftstiffness. With a stiff support the natural frequency depends only on the shaftstiffness and associated mass. Mixer manufactures generally assume that themixer will be mounted on structure, where a small change in stiffness does notsignificantly affect the natural frequency.

Most structural engineers design primarily for strength. However, tomount mixers properly, stiffness also must be considered. Supports withadequate strength may experience noticeable deflection or movement with thedynamic load from a mixer. In most industrial applications stress levels aremuch less than allowable when appropriate stiffness is provided. Highpressure applications are the exception, where the structure to hold thepressure provides adequate stiffness.

The general rule used to design a mixer shaft and impeller systems isto keep operating speed 20% away from a critical speed:

This rule applies to the first, second, and third natural frequencies. Higherorder natural frequencies are seldom encountered in mixer applications.

Mechanical Design October 23, 2009 36 of 50

W WwL

e n

n

1 4

(6.15)

Nd F

L W L Scm

e b

388 105 2.

(6.13)

FE

Emm

m

316

316

1 2/

(6.14)

Figure 6.33 Shaft and Impeller Schematic

Large mixers running at less than 150 rpm, usually operate belowthe first critical speed. Small mixers operating above 250 rpm usually operatebetween first and second critical, 1.2Nc to 0.8Nc2, where Nc2 is the secondlateral natural frequency. Other frequencies, such as a blade-passingfrequency, four times the operating speed for a four-blade impeller with fourbaffles, can cause mechanical excitations. Structural vibrations at certainfractions of operating speed can also contribute to natural frequency problems.

6.1.1.1 Static Analysis for Natural Frequency of an Overhung ShaftThe elements that determine the lateral natural frequency are: the

magnitudes and locations of concentrated and distributed masses, the tensilemodulus of elasticity of the material, and the moment of inertia of the shaft.Ramsey and Zoller (1976) presented the basic elements of natural frequencyfor a shaft and impeller system. That method uses a lumped mass, statictechnique for computing the critical speed of a shaft and impeller system. Thefollowing equation estimates the first lateral natural frequency of a top-entering mixer with an overhung shaft of a constant diameter, in commonunits (lengths in inches, weight in pounds and speed in revolutions perminute):

For metric units (lengths in meters, weight in kilograms, and speed inrevolutions per second) the coefficient becomes 5.5 × 106. Refer to Figure 6.33for definitions of d, L and Sb. Fm is a material properties factor and is 1.0 for316SS. For other materials the property factor can be determined from:

We, in equation (6.13), is the equivalent weight of all distributed andconcentrated weights of the shaft and impeller system and is defined as:

Mechanical Design October 23, 2009 37 of 50

WD P

Nbi

pitched,

.

0 50 3

(6.16)

WD P

Nbi

pitched,

.

0 30 3

(6.17)

Table 6.3 Impeller Hub Weights

Where Wn is the weight of the individual impellers and w is the weight of theshaft per unit of length. The following section describes a method forestimating the weight of typical industrial impellers.6.1.1.2 Estimating the Weight of Impellers

Impeller weights, even for the same style of impeller, can vary frommanufacturer to manufacturer, so the following calculation is onlyapproximate. The two most common types of impellers used on turbine-stylemixers today are the hydrofoil type and the 45º-pitched blade type. Most ofthese impellers are made of blades bolted to a hub, which is keyed and set-screwed to the shaft. Therefore, the weight of the hub found in the table isadded to the weight of the blades computed from an equation.

Total blade weight for 3-blade, hydrofoil impellers, from15 to 90 inches indiameter can be estimated by the following equation:

Total blade weight for four-blade, 45º-pitched impellers, 15 to 90 inches indiameter can be estimated by the following equation:

Impeller power and speed enter the estimate because mechanical loadsdetermine the blade thickness. Due to the wide variety of hydrofoil impellers,the accuracy of the estimated impeller weight is only about ±25%. The four-bladed 45º-pitched impellers are more easily defined, so the accuracy of theestimated impeller weight is about ± 20%. Adjustment for impellers with asfew as two blades and as many as six blades can be made from these estimates.6.1.1.3 Static Analysis for Natural Frequency of a Steady Bearing Shaft

In many tall tanks designing a mixer with an overhung shaft is noteconomically practical. Often a lower bearing, called a steady bearing or footbearing, is used to provide a more economical design. Such systems are usuallytriple bearing systems: two bearings in the gearbox and the foot bearing, beingthe third bearing. Although not an exact solution, the following equation canbe used to estimate the first natural lateral frequency of a shaft and impellersystem using a steady bearing with a stiffness over 5.00x104 lbf/inch.

Mechanical Design October 23, 2009 38 of 50

d do i4 4 (6.20)

WL

a b We n n n

n

164

2

1

(6.19)

Nd

L w

Nd

L W

NN N F

N N

c

ce

cc c m

c c

16

2

2

26

2

3 2

1 2

12

22

2 44 10

155 10

.

./ (6.18)

Figure 6.34 Shaft and Impellers with Steady Bearing

Figure 6.35 Moment Diagram for Shaft System with Steady Bearing

The equivalent weight, We, for this system is given by:

Refer to Figure 6.34 for definitions of a, b and L.

6.1.1.4 Static Analysis for Natural Frequency of a Pipe ShaftWhen a hollow shaft, such as pipe or tube is used, replace d2 in

equations (6.13) or (6.18) by:

6.1.1.5 Using Stabilizers on Impellers to Improve DampingMost impellers do not need to be stabilized. The idea of stabilizers is

to improve damping, thus minimizing deflection and consequently stressimposed on the shaft. As a general “rule of thumb,” stabilizers are not requiredfor impellers whose diameters exceed the shaft diameter by a factor of 10 ormore. Shaft and impeller systems whose impeller diameters exceed their shaftdiameter by a factor of 10 or more have damping ratios of 0.4 or less. In factmany are over-critically damped. The only exception to the rule is that mixedflow impellers when operating at the liquid level can reduce their deflection byusing stabilizers. Normally designed shaft systems can tolerate operation atthe liquid level for brief periods without the use of stabilizers.

6.1.1.6 Dynamic Analysis for Natural Frequency