Consider heat transfer between two identical hot solid bodies and their environments. The first solid is dropped in a large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped system analysis more likely to be applicable? Why?

In both situations, heat transfer occurs between the hot solid body and its surroundings. For the solid submerged in water, convection is the dominant mechanism of heat transfer. In the case of the solid body exposed to air, convection and radiation both contribute to the heat transfer process.

The rate at which heat is transferred between the hot solid body and its surroundings depends on the heat transfer coefficient. The higher the heat transfer coefficient, the faster the temperature of the solid will decrease, and the less likely it is that the lumped system analysis will be applicable. In general, the heat transfer coefficient for convection in water is much higher than air due to the higher thermal conductivity of water.

To determine the applicability of lumped system analysis, we consider the Biot number (Bi), which is a dimensionless number that characterizes the relative importance of internal conduction to external convection. It is given by the formula: Bi = \frac{h * L_c}{k} Where: - \(h\) is the convective heat transfer coefficient, - \(L_c\) is the characteristic length of the solid (usually, the volume divided by the surface area), and - \(k\) is the thermal conductivity of the solid. In the lumped system analysis, the Biot number must be much smaller than 1, typically smaller than 0.1, which means that the rate of internal heat conduction is much faster than the rate of heat transfer to the surroundings. In such a case, the temperature within the solid remains uniform at all times.

Since the heat transfer coefficient for convection in water is much higher than convection in air, the Biot number for the solid in water will also be higher than the solid in air, given that the characteristic length and thermal conductivity of the solid are identical.

As the Biot number for the solid body in air is lower than the solid body in water, the lumped system analysis is more likely to be applicable for the solid body cooling in air. This is because, in this case, the rate of internal heat conduction is faster as compared to the heat transfer to surroundings, allowing the assumption of a uniform temperature within the body to be valid.