Sample calculation Polar moment of inertia

the rotor of modern steam turbine.

calculation of steam turbine shaft radius turboset:

assumptions:

power carried shaft 1000 mw; typical large nuclear power plant.
yield stress of steel used make shaft (τyield) is: 250 × 10 n/m².
electricity has frequency of 50 hz; typical frequency in europe. in north america frequency 60 hz. assuming there 1:1 correlation between rotational velocity of turbine , frequency of mains power.

the angular frequency can calculated following formula:

ω
=
2
π
f


{\displaystyle \omega =2\pi f}

the torque carried shaft related power following equation:

p
=
t
ω


{\displaystyle p=t\omega }

the angular frequency therefore 314.16 rad/s , torque 3.1831 × 10 n·m.

the maximal torque is:

t

max


=




τ

max



j

z



r




{\displaystyle t_{\max }={\frac {\tau _{\max }j_{z}}{r}}}

after substitution of polar moment of inertia following expression obtained:

r
=




2

t

max




π

τ

max





3





{\displaystyle r={\sqrt[{3}]{\frac {2t_{\max }}{\pi \tau _{\max }}}}}

the radius 0.200 m. if 1 adds factor of safety of 5 , re-calculates radius maximal stress equal yield stress/5 result radius of 0.343 m, or diameter of 69 cm, approximate size of turboset shaft in nuclear power plant.