3.149 Water is heated by submerging 50 -mm-diameter,...

3.149 Water is heated by submerging 50 -mm-diameter, thinwalled copper tubes in a tank and passing hot combustion gases $\left(T_{g}=750 \mathrm{~K}\right)$ through the tubes. To enhance heat transfer to the water, four straight fins of uniform cross section, which form a cross, are inserted in each tube. The fins are $5 \mathrm{~mm}$ thick and are also made of copper $(k=400 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. If the tube surface temperature is $T_{s}=350 \mathrm{~K}$ and the gas-side convection coefficient is $h_{g}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the rate of heat transfer to the water per meter of pipe length?

3.149 Water is heated by submerging 50 -mm-diameter, thinwalled copper tubes in a tank and passing hot combustion gases $\left(T_{g}=750 \mathrm{~K}\right)$ through the tubes. To enhance heat transfer to the water, four straight fins of uniform cross section, which form a cross, are inserted in each tube. The fins are $5 \mathrm{~mm}$ thick and are also made of copper $(k=400 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$.
If the tube surface temperature is $T_{s}=350 \mathrm{~K}$ and the gas-side convection coefficient is $h_{g}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the rate of heat transfer to the water per meter of pipe length?

Key Concepts

Convection Heat Transfer

This concept involves the transfer of heat between a surface and a fluid (gas or liquid) that flows over it. The rate of heat transfer is determined by the convective heat transfer coefficient and the temperature difference between the surface and the fluid. This is a fundamental mechanism in thermal systems used to account for energy exchange in processes like cooling, heating, or heat removal.

Thermal Conduction in Solids

Thermal conduction is the process by which heat is transferred within or between solid materials due to temperature gradients. This concept is pivotal when evaluating the performance of materials like metals, where thermal energy is conducted efficiently across a structure. In heat transfer problems involving tubes or fins, conduction analysis determines how effectively heat travels through the material.

Extended Surface (Fin) Analysis

Extended surfaces, or fins, are used to enhance heat transfer by increasing the surface area available for convection. The analysis of fin efficiency, which compares the actual heat transfer from a fin to the ideal case of a perfect conductor, is crucial in these scenarios. This concept involves understanding the impact of fin geometry, material properties, and convection conditions on the overall heat transfer rate.

Composite Heat Transfer Systems

In many thermal systems, different modes of heat transfer such as conduction and convection work together. Analyzing composite systems involves integrating these mechanisms to predict the overall rate of heat transfer. This requires a consideration of conduction through solid structures and convection at interfaces, which is common in devices that use fins or tubes to enhance heat exchange processes.

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