Based on the cutting data in the following table, please estimate

$($a$)$ the average tool$-$chip interface temperature by Nathan Cook' model $($dimensionless analysis$)$;

$($b$)$ the maximum tool$-$chip interface temperature, the average tool$-$chip interface temperature and average shear plane temperature by Shaw's model;

$($c$)$ the average tool$-$chip interface temperature or the maximum tool$-$chip interface temperature, and average shear plane temperature by Boothroyd's model.

Cutting data for machining steel from Loewen and Shaw $[14]$

$\backslash$table$[[$Work material,SAE B$1113$ steel$],[$Tool$,$K$2$S carbide $20$ rake, $5$ clearance angle$],[$Type of cut,Orthogonal$],[$Cutting speed,$V_c=139\frac{m}{m}in(232c\frac{m}{s})$