A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is 5 m and its inner surface temperature is at 120°C. The wall of the vessel has a variable thermal conductivity given as k(T) = k₀ (1 + βT), where k₀ = 1.01 W/m ∙ K, β = 0.0018 K⁻¹, and T is in K. The vessel is situated in a surrounding with an ambient temperature of 15°C, the vessel’s outer surface experiences convection heat transfer with a coefficient of 80 W/m²∙K. To prevent thermal burn on skin tissues, the outer surface temperature of the vessel should be kept below 50°C. Determine the minimum wall thickness of the vessel so that the outer surface temperature is 50°C or lower.