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Thermal conductivity is a property that indicates how well a material can conduct heat. It is usually denoted by the symbol 'k' and is expressed in units of watts per meter-kelvin (W/m·K). Materials with high thermal conductivity, like metals (for example, aluminum), are able to transfer heat quickly, which makes them great conductors. In contrast, materials with low thermal conductivity, such as glass or plastic, are not as efficient at transferring heat and are considered insulators.

The value of thermal conductivity plays a pivotal role in solving heat transfer problems. For instance, if you were to wrap a hot object in a material with high thermal conductivity, the heat would be lost rapidly, whereas wrapping it in an insulative material with a low 'k' value would help retain the heat longer. In the exercise given, the difference between thermal conductivity of Pyrex glass (\(k_g = 1.1\text{ W/m·K}\)) and aluminum (\(k_a = 237\text{ W/m·K}\)) starkly illustrates why metals feel colder to the touch in cold environments and hotter in warm environments compared to glass; they transfer the heat to or from our skin much more efficiently.

The conduction heat rate, often expressed as \(\dot{Q}_{conduction}\), quantifies the amount of heat transferring through a material by conduction per unit time. It's calculated using the formula \(\dot{Q}_{conduction} = k \times A \times \frac{T_1 - T_2}{R}\) where 'k' is the thermal conductivity, 'A' is the area through which heat is being transferred, \(T_1\) and \(T_2\) are temperatures at different sides of the material, and 'R' is the thermal resistance or thickness of the material.

Understanding this concept is instrumental when designing systems where temperature control is crucial, such as insulation for buildings or cooling systems for electronic devices. In our example with the spherical shell, we see that despite aluminum's high thermal conductivity, the conduction heat rate calculation also depends on the surface area and the temperature gradient, leading to a situation where the midpoint temperature remains unchanged for both Pyrex glass and aluminum.

Temperature distribution refers to how temperature varies within a material or system. In the context of solid materials, temperature distribution is generally not uniform due to the presence of thermal gradients — differences in temperature that lead to heat flow from the hotter areas to the colder ones. Analyzing temperature distribution is key to many engineering applications, from preventing thermal stresses in construction materials to optimizing the performance of heat exchangers.

In the exercise on the spherical shell, the temperature distribution can be visualized as a gradient from the hot inner surface to the cooler outer surface. The temperature at any point within the material depends on the thermal properties of the material and the boundary conditions (in this case, the surface temperatures). The exercise demonstrates that, interestingly, even when a material with vastly different thermal conductivity is chosen, the midpoint temperature can remain the same if the heat rate and the geometry are not altered. This counterintuitive result highlights the intricacies of how temperature distribution is influenced by a combination of factors, not just thermal conductivity alone.