To determine the temperature at the center of the shaft 20 minutes after the start of the cooling process, we can use the 1-D steady-state heat conduction equation:
q'' = -k*(dT/dx)
where q'' is the heat flux per unit area, k is the thermal conductivity of the shaft, T is the temperature, and x is the distance from the surface of the shaft.
The boundary conditions for this problem are:
T(r=R, t) = 150°C (convective heat transfer coefficient)
T(r=0, t) = 500°C (initial temperature)
We can use the above equation to get the temperature distribution in the shaft, and then integrate it to find the temperature at the center of the shaft.
To determine the heat transfer per unit length of the shaft during this time period, we can use the following equation:
q' = hA(T_surface - T_ambient)
where q' is the heat transfer per unit length, h is the convective heat transfer coefficient, A is the surface area, T_surface is the temperature of the surface and T_ambient is the temperature of the surrounding environment.
Please note that the above information is just an overview of how to solve the problem and is not a complete solution. The above equations and boundary conditions may require some modification according to the given problem, and the exact solution would require a detailed mathematical analysis.