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Shaft Design
TG8
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The process requirements determine the overall mixer size and configuration, ie. HP,
RPM, number and style of impeller(s) etc.; the next step is to design an appropriately
sized shaft system. The absorbed Horsepower of the impeller(s) creates torsional and
bending stresses on the mixing shaft. The torsional stress is due to the transmitted
torque (BHP at the speed of rotation) and the bending stress is due to the fluid hydraulic
forces acting on the impeller(s).
Pure Torsional Stress
If a mixer shaft and impeller assembly had no fluid hydraulic forces acting at the
impeller we would only be concerned with torsional stress which is the torque
transmitted by the shaft divided by its Polar Section Modulus. The formula for Pure
Torsional Stress is:
TQ
=
=
Z
Where:
HP ( 63025 )
RPM
d3
16
[ ]
lbs
in 2
HP is motor HP
RPM is impeller speed
d is shaft diameter [in.]
Pure Bending Stress
If we were interested in determining the stress in the shaft due only to the unbalanced
hydraulic forces acting on the mixing impellers (ignoring torsional stress) it would be a
matter of determining the magnitude of these hydraulic forces and where along the
shaft they were acting (impeller location).
The hydraulic forces, created by the action between the fluid and the impeller, produce
side loads on the shaft causing this tensile or bending stress. The fluid hydraulic force
acting at an impeller is random in both direction and magnitude, we therefore must
calculate the maximum possible hydraulic load to determine the shaft stress. The
equation for hydraulic loading at an impeller is:
FH =
Where:
24 , 000 ( HP )( CF )
RPM ( D )
[ lbs.]
HP is Impeller Power
CF is a Condition Factor based on impeller style and application (Typ. 1 to 3)
RPM is impeller speed
D is impeller diameter [in.]
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Shaft Design
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After the hydraulic force is found at the impeller(s), the total Bending Moment can be
calculated by using the following:
M = {FH1( L1)} + {FH 2( L 2)}+ ... {FHn( Ln )} [ in . lbs.]
Where:
FH1, FH2 ... FHn are Fluid Hydraulic Forces at each Impeller [lbs.]
L1, L2 ... Ln are the distances from the lower gearbox bearing to the impeller [in.]
Finally the Pure Bending Stress can be calculated by dividing the total bending moment
(M) by the shaft rectangular section modulus.
[ ]
M M
=
= d3
Z
32
Where:
lbs
in 2
M is the Bending Moment calculated above [in.lbs.]
d is shaft diameter [in.]
Combined Stresses
When a mixer is operating the shaft is experiencing both bending and torsional
stresses, therefore the final step in shaft design is to calculate the combined stresses in
the shaft. The combined torsional stress can be found by using the following equation:
=
Max
( )
2
2
+ ( )
[ ]
lbs
in 2
And the combined bending stress can be found by using:
=(
Max
)+ ( )
2
2
2
(
)
+
[ ]
lbs
in 2
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Shaft Design
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Critical Speed
If a shaft and impeller assembly were struck with a hammer, the assembly would begin
to vibrate. The vibratory mode with the lowest frequency is defined as the first natural
frequency of the system, the next highest is the second natural frequency and so on.
Critical Speed is defined as the mixer rotational speed (RPM) which coincides with its
first natural frequency (Hz).
Rotating a mixer assembly at a rate equal to its natural frequency is like hitting it with a
hammer. However, unlike a hammer which provides a momentary influx of energy, the
mixer motor continues to pump energy into the vibrating assembly causing it to oscillate
at higher and higher amplitudes until failure occurs. We therefore must calculate the
first natural frequency of each mixer assembly to ensure we are not operating near this
critical point.
The following equation can be used to determine the first natural frequency (NC) of a
mixer with a uniform solid shaft.
d2
E
NC = 146.4 2
4.13We
L L + a
WB +
L
L
3
Where:
And:
L2
LN
We = W1 + W 2 + ...+ WN
L
L
d is the shaft diameter (in)
L shaft length from lower gearbox bearing (in)
L2 ,... LN is the distance from the lower gearbox bearing to the impeller (in)
a is the gearbox bearing span (in)
W 1, W 2, ... W N are impeller weights (lbs)
W B is the shaft weight (lbs/in3)
E is the Modulus of Elasticity (lbs/in2), ie. E=30,000,000 psi for steel.
Note: All calculations for critical speed assume that the mixer is rigidly
supported.
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Shaft Design
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Deflection
When a mixer is supplied with a shaft seal for closed tank applications or when tank
internals (heating coils, draft tube, limit ring, etc.) are in relatively close proximity of the
mixer wet end, it is necessary to check the shaft deflection due to hydraulic forces at
the impeller(s).
The following equations can be used to check shaft deflection in the seal area or at the
bottom of a uniform solid shaft:
Deflection at seal:
FHX
y =
( 2La + 3LX X 2 )
6EI
Deflection at end of shaft (X=L):
FHL2
( a + L)
y =
3EI
Where,
d4
I=
64
And: L shaft length from lower gearbox bearing (in)
a is the gearbox bearing span (in)
FH is the hydraulic force acting on the impeller (lbs)
E is the Modulus of Elasticity (lbs/in2), ie. E=30,000,000 psi for steel
d is the shaft diameter (in)
y is shaft deflection at point of interest (TIR is two times y)
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Shaft Design
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Mixer Support Design Loads
The mixer support structure is to be designed so that dynamic angular deflection of
drive is limited to 0.25 degrees in any direction.
Where:
D is Impeller Dia. in [In.]
N is Impeller Speed [RPM]
NI is the Number of Impellers
HP is Nameplate Horse Power
LX is distance from impeller to mounting surface [In.]
FHX is the fluid hydraulic force at each impeller (see Pg. 8.01) [lbs.]
TG8
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Shaft Design
TG8
8.06
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Open Tank Mixer Support Structures
For purposes of use during preliminary investigations, information on practical support
construction for mixers mounted above open tanks is outlined here. Suggested beam
sizes are rather conservative, and apply for use with the highest output torque capacity
of each drive and at the longest agitator shaft overhangs practical for each size. For
maximum economy in a specific installation, it is recommended that the Processor or
Engineering Contractor apply his own beam support design standards to the specific
mixer that will be utilized.
Some degree of cross bracing between main beams always represents a sound
engineering approach. The user is encouraged to apply more sophisticated designs of
cross bracing particularly with the larger tanks where cost of supports can become
substantial. It is normal to apply floor plate or grating between the main support beams
which provides walkway access while providing additional rigidity in the support
structure. This information is intended as a guideline and does not relieve the user of
completely analysing the entire mounting system for each mixing application.
BB Dimension (inches)
Drive Size
HRFs
ST-10
ST-11
ST-12
MB-53
MB-54
MB-55
MB-56
MB-57
MB-58
MB-59
LH-9
LH-10
11
10
12
14
12
13
15
17
21
23
23
19
19
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Shaft Design
Tank Diameter (inches)
Drive Size
HRFs
ST-10
ST-11
ST-12
MB-53
MB-54
MB-55
MB-56
MB-57
MB-58
MB-59
LH-9
LH-10
Beam Function
120
180
240
300
360
Main Support
6C10.5
8WF24
10WF33
12WF45
12WF79
Drive Support
4C7.25
5C9.0
7C9.8
8C11.5
8C11.5
Cross Bracing
4C7.25
5C9.0
7C9.8
8C11.5
8C11.5
Main Support
6WF20
10WF25
10WF60
12WF72
12WF120
Drive Support
4C7.25
7C9.8
7C9.8
8C11.5
8C11.5
Cross Bracing
4C7.25
7C9.8
7C9.8
8C11.5
8C11.5
Main Support
6WF25
10WF29
10WF60
12WF79
12WF133
Drive Support
4C7.25
7C9.8
7C9.8
8C11.5
8C11.5
Cross Bracing
4C7.25
7C9.8
7C9.8
8C11.5
8C11.5
Main Support
8WF20
10WF33
12WF53
12WF92
14WF119
Drive Support
5WF16
8C11.5
8C11.5
8C11.5
9C13.4
Cross Bracing
5C6.7
8C11.5
8C11.5
8C11.5
9C13.4
Main Support
8WF31
10WF45
12WF58
14WF87
14WF142
Drive Support
5WF16
8C11.5
8C11.5
9C13.4
9C13.4
Cross Bracing
5C9.0
8C11.5
8C11.5
9C13.4
9C13.4
Main Support
8WF35
10WF49
12WF72
14WF103
14WF167
Drive Support
6WF15.5
6WF15.5
8WF31
10WF33
10WF33
Cross Bracing
6C13
6C13
8C11.5
10C15.3
10C15.3
Main Support
8WF40
10WF60
14WF68
14WF127
14WF202
Drive Support
6WF15.5
6WF15.5
10WF33
10WF33
10WF33
Cross Bracing
6C13
6C13
10C15.3
10C15.3
10C15.3
Main Support
10WF45
12WF58
14WF95
14WF167
14WF264
Drive Support
8WF31
8WF31
10WF33
10WF33
10WF33
Cross Bracing
8C11.5
8C18.75
10C15.3
10C15.3
10C15.3
Main Support
10WF45
12WF58
14WF95
14WF167
14WF264
Drive Support
8WF31
8WF31
10WF33
10WF33
10WF33
Cross Bracing
8C11.5
8C18.75
10C15.3
10C15.3
10C15.3
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