Calculating a shaft size uses basic mechanical design equations for a beam with different supports.
The calculation for a bottom support or steady bearing is calculated as a beam with a fixed support at the top and a simple support at the bottom.  The impeller imposes a lateral or sideways load at a point between the two ends.  The load imposed by the impeller is a function of the torque on the impeller and the impeller diameter, along with a hydraulic force factor for different applications.  The hydraulic force factor can be 1.0 for constantly submerged applications and 3.0 for periodic operation at the liquid surface, such as during filling or emptying of the vessel.  The equations for this calculation can be found in the Shaft-Design-Hydrofoil-Steady-Bearing.pdf.  The first calculation is for the torque on the shaft.  The motor power is used in the torque and bending calculations to design for maximum load.  The second calculation is for the lateral or sideways load imposed by the impeller.  The torque and impeller diameter are used to estimate this load, along with the hydraulic force factor.  The impeller force is applied to the shaft at the location calculated from the top of the mixer shaft.  The overall shaft length is the location of the bottom bearing.
The shaft torque and bending loads are combined to calculate a shaft diameter based on the shear stresses in the shaft.  An allowable stress limit, based on metal strength and fatigue limits, is used in the calculation.  A similar calculation for shaft diameter is done based on the tensile strength of the shaft.  The two calculated diameters are compared and the larger of the two diameters is the minimum diameter for the shaft.  The next larger standard bar stock size is normally used for the actual shaft.
A similar series of calculations is done for the case of a middle shaft support, Shaft-Design-Hydrofoil-Middle-Bearing.pdf.  The difference is in the lengths used for the bending moment calculations.  The length from the mixer drive to the middle support and length to the impeller are used to calculate the maximum bending moment on the mixer shaft.
For either calculation, the location of the shaft supports must be precisely aligned with the location where the shaft hangs naturally from the mixer drive.  This location can only be determined after the mixer is mounted on the vessel.  Any secondary shaft support is equivalent to trying to align three bearings on the same shaft.  The alignment is impossible to achieve perfectly, but with care the alignment can be done within acceptable limits.  Any misalignment will result in premature wear of one or more bearings.  The bottom or intermediate bearings are usually the ones to wear quickest.

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