A long rod of 20-mm diameter and a thermal conductivity of 1.5 W / m ·K has a uniform internal

step 1 First, writing the given data, given that L is equal to 12070 meter and the temperature gradient

A long rod of 20-mm diameter and a thermal conductivity of...

A long rod of $20-\mathrm{mm}$ diameter and a thermal conductivity of $1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ has a uniform internal volumetric thermal energy generation of $10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The rod is covered with an electrically insulating sleeve of 2-mm thickness and thermal conductivity of $0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. A spider with 12 ribs and dimensions as shown in the sketch has a thermal conductivity of $175 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and is used to support the rod and to maintain concentricity with an $80-$ mm-diameter tube. Air at $T_{\infty}=25^{\circ} \mathrm{C}$ passes over the spider surface, and the convection coefficient is $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The outer surface of the tube is well insulated. We wish to increase volumetric heating within the rod, while not allowing its centerline temperature to exceed $100^{\circ} \mathrm{C}$. Determine the impact of the following changes, which may be effected independently or concurrently: (i) increasing the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and (iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous carbon or quartz). Recommend a realistic configuration that yields a significant increase in $\dot{q}$.

A long rod of $20-\mathrm{mm}$ diameter and a thermal conductivity of $1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ has a uniform internal volumetric thermal energy generation of $10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The rod is covered with an electrically insulating sleeve of 2-mm thickness and thermal conductivity of $0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. A spider with 12 ribs and dimensions as shown in the sketch has a thermal conductivity of $175 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and is used to support the rod and to maintain concentricity with an $80-$ mm-diameter tube. Air at $T_{\infty}=25^{\circ} \mathrm{C}$ passes over the spider surface, and the convection coefficient is $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The outer surface of the tube is well insulated.

We wish to increase volumetric heating within the rod, while not allowing its centerline temperature to exceed $100^{\circ} \mathrm{C}$. Determine the impact of the following changes, which may be effected independently or concurrently: (i) increasing the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and (iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous carbon or quartz). Recommend a realistic configuration that yields a significant increase in $\dot{q}$.

Key Concepts

Geometric Configuration and Design Optimization

The dimensions and number of structural features like ribs or supports affect the thermal performance by altering both conduction paths and surface areas available for convection. Optimizing these geometric parameters, such as increasing rib thickness or altering their number, can significantly reduce thermal resistance and improve cooling efficiency, allowing higher internal heat generation without exceeding the maximum allowable temperatures.

Convection Heat Transfer

Convection is the process by which heat is removed from a surface by a moving fluid (in this case, air). The convection heat transfer coefficient quantifies this effect and is a key parameter; increasing it (e.g., by boosting air speed) can help lower surface temperatures, thereby permitting higher internal heating without breaching temperature limits.

Material Thermal Conductivity

The thermal conductivity of the materials used in the system (such as the rod, insulating sleeve, and supporting ribs) determines how effectively heat is conducted through each component. Materials with higher conductivity can facilitate redistribution of heat, thus influencing the overall thermal performance and safety margins.

Thermal Resistance Networks

This involves representing the composite system (rod, insulating sleeve, support structure, and convection surface) as a series of thermal resistances. Each layer or interface restricts heat flow, and understanding these resistances helps predict overall temperature distributions and identify where design modifications can be most effective.

Conduction in Cylindrical Systems

This concept deals with heat transfer through cylindrical bodies where the geometry significantly affects temperature gradients, especially when heat is generated uniformly inside. It is essential for correctly modeling how temperature rises from the center to the outer surface of round components like rods or tubes.

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