Concept:
The Sommerfeld number is given by:
\(S = {\left( {\frac{r}{c}} \right)^2}\frac{{μ {n_s}}}{P}\)
Where,
c = radial clearance (mm)
r = radius of journal (mm)
μ = viscosity of the lubricant (Ns/mm2) or (MPa-s)
ns = journal speed (rev/s)
P = unit bearing pressure (N/mm2)
Calculation
Given:
r = 20 mm, L = d = 40 mm, W = 2000 N, N = 2000 rpm, μ = 0.03 Pa-s
\(P = \frac{W}{{Ld}}\)
\(S = {\left( {\frac{r}{c}} \right)^2}\frac{{μ {n_s}}}{P}\)
\(\begin{array}{l} \Rightarrow S =( \frac{{{20 \times {{10}^{ - 3}}}}}{{20 \times {{10}^{ - 6}}}} )^2 \times \frac{{0.03 \times \frac{{2000}}{{60}}}}{{\left( {\frac{{2000}}{{40 \times 40 \times {{10}^{ - 6}}}}} \right)}} = 0.8 \end{array}\)