Question

In what medium is the lumped system analysis more likely to be applicable: in water or in air? Why?

Short answer

Answer: The lumped system analysis is more likely to be applicable in water than in air.

Step-by-step solution

Understand Lumped System Analysis and Biot Number

Lumped system analysis is a simplification used in heat transfer problems when the temperature gradient inside a solid object is negligible compared to the temperature difference between the surface and the surrounding medium. It can be applied when heat conduction within the object is much faster than heat convection between the object and the environment. To determine the applicability of lumped system analysis, we can use the Biot Number (Bi), which is defined as: Bi = (h * L)/k where h is the heat transfer coefficient of the fluid, L is the characteristic length of the object, and k is the thermal conductivity of the solid. If Bi << 1, the temperature gradient inside the solid is negligible, and the lumped system analysis is applicable.

Compare heat transfer coefficients in water and air

Heat transfer coefficient (h) is an important parameter to compare, as it directly affects the Biot number. Generally, the heat transfer coefficient is higher in liquids than in gases. As water is a liquid and air is a gas, the heat transfer coefficient for water will be greater than that of air.

Compare thermal conductivity in water and air

Thermal conductivity (k) is another important parameter to consider for this exercise. The thermal conductivity of water is much higher than that of air, which means that heat is conducted through water more efficiently (about 25 times) than in air.

Determine applicability of lumped system analysis in water and air

As water has a higher heat transfer coefficient (h) and thermal conductivity (k) than air, the Biot number (Bi) in water will be smaller than that in air for objects of the same size and material. As a result, the lumped system analysis is more likely to be applicable in water than in air, because the temperature gradient within the solid would be negligible in water more often than in air, as per the condition Bi << 1.