A glass \((k=1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) spherical tank is filled with chemicals undergoing exothermic reaction. The reaction keeps the inner surface temperature of the tank at \(80^{\circ} \mathrm{C}\). The tank has an inner radius of \(0.5 \mathrm{~m}\) and its wall thickness is \(10 \mathrm{~mm}\). Situated in surroundings with an ambient temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the tank's outer surface is being cooled by air flowing across it at \(5 \mathrm{~m} / \mathrm{s}\). In order to prevent thermal burn on individuals working around the container, it is necessary to keep the tank's outer surface temperature below \(50^{\circ} \mathrm{C}\). Determine whether or not the tank's outer surface temperature is safe from thermal burn hazards.

Heat transfer is a fundamental concept in thermodynamics and involves the movement of thermal energy from one place to another due to temperature differences. There are three primary modes of heat transfer: conduction, convection, and radiation. Conduction involves the transfer of heat through solid materials, from higher to lower temperature regions. Convection involves the movement of heat by the physical movement of fluid particles, which could be gases or liquids. Lastly, radiation describes the transfer of heat through electromagnetic waves without the need for a physical medium.

In thermal analysis, we often deal with conduction and convection, especially in the context of industrial applications such as the cooling of a spherical tank in the given exercise. Heat transfer analyses are critical in ensuring the safety and efficiency of thermal systems. Accurate calculations can prevent hazards such as thermal burns from hot surfaces, as seen in our exercise where we must ensure the outer surface temperature of the tank is safe for human contact.

When we analyze heat transfer in objects with curved geometries, like a tank, it’s important to consider the specifically adapted equations for conduction. In spherical coordinates, the heat conduction equation takes into account the radial symmetry of the sphere. This is particularly important when dealing with non-uniform materials or when the rate of heat transfer is not consistent across a surface.

Using the spherical form of Fourier’s law of heat conduction, we're able to solve problems with spherical symmetry by integrating the equation across the volume of the sphere. For instance, in the exercise, the temperature difference across the tank wall is used to calculate the rate of heat transfer through the glass. Understanding this concept allows for proper thermal management of spherical objects and is vital in applications ranging from nuclear reactors to insulated beverage containers.

Convection heat transfer deals with the transfer of heat between a solid surface and a fluid moving over it. It's the heat transfer mode that's most influenced by external conditions, such as fluid speed, its properties, and the configuration of the solid surface. In our exercise, the tank's outer surface heat loss occurs because of air flowing over it.

The convective heat transfer coefficient, denoted by 'h', quantifies the efficiency of the heat transfer process; the higher the value, the more efficient the heat transfer. The equation for convective heat transfer, q = hAΔT, where A is the area and ΔT is the temperature difference, is used to determine whether the cooling provided by airflow is sufficient to maintain a safe outer surface temperature. In engineering, effective convection cooling design is crucial for everything from electronic devices to architectural engineering, ensuring systems operate within safe thermal limits.