Electric current flows through a long rod generating thermal...

Electric current flows through a long rod generating thermal energy at a uniform volumetric rate of $\dot{q}=$ $2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The rod is concentric with a hollow ceramic cylinder, creating an enclosure that is filled with air. The thermal resistance per unit length due to radiation between the enclosure surfaces is $R_{\mathrm{rad}}^{\prime}=0.30 \mathrm{~m} \cdot \mathrm{K} / \mathrm{W}$, and the coefficient associated with free convection in the enclosure is $h=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Construct a thermal circuit that can be used to calculate the surface temperature of the rod, $T_{r}$. Label all temperatures, heat rates, and thermal resistances, and evaluate each thermal resistance. (b) Calculate the surface temperature of the rod for the prescribed conditions.

Electric current flows through a long rod generating thermal energy at a uniform volumetric rate of $\dot{q}=$ $2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The rod is concentric with a hollow ceramic cylinder, creating an enclosure that is filled with air.
The thermal resistance per unit length due to radiation between the enclosure surfaces is $R_{\mathrm{rad}}^{\prime}=0.30 \mathrm{~m} \cdot \mathrm{K} / \mathrm{W}$, and the coefficient associated with free convection in the enclosure is $h=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Construct a thermal circuit that can be used to calculate the surface temperature of the rod, $T_{r}$. Label all temperatures, heat rates, and thermal resistances, and evaluate each thermal resistance.
(b) Calculate the surface temperature of the rod for the prescribed conditions.

Key Concepts

Free Convection Heat Transfer

Free (or natural) convection occurs when fluid motion is induced by buoyancy forces due to density differences arising from temperature gradients. The convective heat transfer coefficient is a key parameter that quantifies the efficiency of this heat transfer mode. By using the convective coefficient, one can estimate the thermal resistance associated with free convection, which in turn is used in the thermal circuit to relate the surface temperature to the ambient conditions.

Radiative Heat Transfer

Radiative heat transfer involves the energy exchange between surfaces due to electromagnetic radiation. In thermal circuit analysis, a radiation thermal resistance can be used to approximate the net effect of radiative exchange between surfaces. This resistance depends on the emissivities, view factors, and geometric configuration of the surfaces, and it characterizes how effectively a surface emits or absorbs radiant energy.

Thermal Circuit Analysis

Thermal circuit analysis treats heat flow problems analogous to electrical circuits by representing temperature differences as voltage differences and heat flow rates as currents. In this approach, various modes of heat transfer, such as conduction, convection, and radiation, are modeled as thermal resistances. This method simplifies the analysis of complex systems by allowing the construction of a network of lumped thermal resistances and sources, enabling the determination of temperatures at different nodes in the system.

Volumetric Heat Generation

Volumetric heat generation refers to the rate at which thermal energy is produced per unit volume within a material. It is often encountered in situations where heat is generated internally, such as in electrical conductors or reactive materials. In thermal analysis, this term serves as the source of heat that is eventually transferred outward through the surrounding thermal resistances, establishing a temperature gradient.

Thermal Resistance Concept

The concept of thermal resistance quantifies the opposition to heat flow through a material or between surfaces. It is analogous to electrical resistance and is defined as the ratio of the temperature difference across a material to the heat transfer rate. In thermal circuits, thermal resistances are combined in series or parallel to model complex heat transfer paths. Evaluating each resistance correctly is crucial for accurately predicting temperature distributions in a system.

Transcript

Please give Ace some feedback