The wall of a spherical tank of 1-m diameter contains an...

The wall of a spherical tank of $1-m$ diameter contains an exothermic chemical reaction and is at $200^{\circ} \mathrm{C}$ when the ambient air temperature is $25^{\circ} \mathrm{C}$. What thickness of urethane foam is required to reduce the exterior temperature to $40^{\circ} \mathrm{C}$, assuming the convection coefficient is $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ for both situations? What is the percentage reduction in heat rate achieved by using the insulation?

Key Concepts

Convective Heat Transfer

Convection refers to the heat transfer between a surface and a moving fluid, such as air. The convection coefficient quantifies how effectively heat is removed or delivered via fluid motion. This concept is important for assessing the heat loss from external surfaces and for setting boundary conditions in thermal analysis.

Conduction Through Solids

Conduction is the mechanism by which heat energy is transmitted through materials due to a temperature gradient. It depends on the material's thermal conductivity and the thickness of the material. This concept is fundamental when analyzing how insulation materials reduce heat transfer by slowing the rate at which heat moves through them.

Thermal Resistance and Composite Insulation

Thermal resistance is a measure of a material's opposition to heat flow, analogous to electrical resistance. When several layers or modes of heat transfer (such as conduction through insulation and convection from a surface) are involved, their resistances add up to form a composite thermal resistance. This approach simplifies the analysis of complex thermal systems, allowing calculation of overall heat transfer rates.

Steady-State Heat Transfer

Steady-state heat transfer assumes that the temperatures in the system remain constant over time, which simplifies calculations by allowing the use of constant thermal resistances. This assumption is critical when designing insulation to achieve a target external surface temperature based on known internal conditions and environmental factors.

Percentage Reduction in Heat Rate

The percentage reduction in heat rate is a measure of the effectiveness of insulation, representing how much the heat transfer is decreased due to the application of the insulation. It provides a quantitative assessment of energy savings by comparing the heat flux with and without insulation.

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