Question

What does the effective conductivity of an enclosure represent? How is the ratio of the effective conductivity to thermal conductivity related to the Nusselt number?

Short answer

Answer: Effective conductivity represents the total heat transfer through an enclosure, considering both conduction and convection. The ratio of effective conductivity to thermal conductivity is a dimensionless quantity that indicates the enhancement of heat transfer due to convection. This ratio is directly related to the Nusselt number, which quantifies the enhancement of heat transfer due to convection. In other words, the Nusselt number defines the relative importance of convection and conduction in a given situation, while the ratio of effective conductivity to thermal conductivity demonstrates the combined effects of conduction and convection on heat transfer within an enclosure.

Step-by-step solution

Understanding Effective Conductivity

Effective conductivity represents the total heat transfer through an enclosure, considering both conduction and convection. For an enclosure filled with fluid, the effective conductivity combines the effects of molecular conduction in the fluid and convection caused by fluid motion.

Relating Effective Conductivity to Thermal Conductivity

The ratio of effective conductivity (k_e) to thermal conductivity (k) is a dimensionless quantity that represents the enhancement of heat transfer due to fluid motion within the enclosure (i.e., convection). This ratio is defined as follows: Ratio = k_e / k The higher the value of this ratio, the greater the degree of heat transfer enhancement by convection.

Introducing Nusselt Number

Nusselt number (Nu) is a dimensionless number used in heat transfer studies to characterize the relative importance of convection and conduction in a given situation. It represents the enhancement of heat transfer due to convection and is defined as the ratio of convective heat transfer to conductive heat transfer at a boundary. Nusselt number (Nu) can be mathematically represented as: Nu = hL / k Where: - h is the convective heat transfer coefficient, - L is the characteristic length (distance over which the heat transfer is taking place), and - k is the thermal conductivity of the fluid.

Connecting the Ratio of Effective Conductivity to Nusselt Number

The ratio of effective conductivity to thermal conductivity is directly related to the Nusselt number. The Nusselt number quantifies the enhancement of heat transfer due to convection, while the ratio we derived earlier demonstrates the enhancement of heat transfer through an enclosure due to both conduction and convection. Using the Nusselt number definition and the ratio derived earlier, we can represent their relationship as follows: Nu = k_e / k Where Nu is the Nusselt number, k_e is the effective conductivity, and k is the thermal conductivity of the fluid. In conclusion, the effective conductivity of an enclosure represents the combined effects of conduction and convection on heat transfer within the enclosure. The ratio of effective conductivity to thermal conductivity indicates the enhancement of heat transfer due to convection, and this ratio is directly related to the Nusselt number.