A steam pipe of 0.12-m outside diameter is insulated with a...

A steam pipe of $0.12-\mathrm{m}$ outside diameter is insulated with a layer of calcium silicate. (a) If the insulation is $20 \mathrm{~mm}$ thick and its inner and outer surfaces are maintained at $T_{s, 1}=800 \mathrm{~K}$ and $T_{s, 2}=490 \mathrm{~K}$, respectively, what is the heat loss per unit length $\left(q^{\prime}\right)$ of the pipe? (b) We wish to explore the effect of insulation thickness on the heat loss $q^{\prime}$ and outer surface temperature $T_{s, 2}$, with the inner surface temperature fixed at $T_{s, 1}=$ $800 \mathrm{~K}$. The outer surface is exposed to an airflow $\left(T_{\infty}=25^{\circ} \mathrm{C}\right)$ that maintains a convection coefficient of $h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and to large surroundings for which $T_{\text {sur }}=T_{\infty}=25^{\circ} \mathrm{C}$. The surface emissivity of calcium silicate is approximately $0.8$. Compute and plot the temperature distribution in the insulation as a function of the dimensionless radial coordinate, $\left(r-r_{1}\right) /\left(r_{2}-r_{1}\right)$, where $r_{1}=0.06 \mathrm{~m}$ and $r_{2}$ is a variable $\left(0.06<r_{2} \leq 0.20 \mathrm{~m}\right)$. Compute and plot the heat loss as a function of the insulation thickness for $0 \leq\left(r_{2}-r_{1}\right) \leq 0.14 \mathrm{~m}$.

A steam pipe of $0.12-\mathrm{m}$ outside diameter is insulated with a layer of calcium silicate.
(a) If the insulation is $20 \mathrm{~mm}$ thick and its inner and outer surfaces are maintained at $T_{s, 1}=800 \mathrm{~K}$ and $T_{s, 2}=490 \mathrm{~K}$, respectively, what is the heat loss per unit length $\left(q^{\prime}\right)$ of the pipe?
(b) We wish to explore the effect of insulation thickness on the heat loss $q^{\prime}$ and outer surface temperature $T_{s, 2}$, with the inner surface temperature fixed at $T_{s, 1}=$ $800 \mathrm{~K}$. The outer surface is exposed to an airflow $\left(T_{\infty}=25^{\circ} \mathrm{C}\right)$ that maintains a convection coefficient of $h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and to large surroundings for which $T_{\text {sur }}=T_{\infty}=25^{\circ} \mathrm{C}$. The surface emissivity of calcium silicate is approximately $0.8$. Compute and plot the temperature distribution in the insulation as a function of the dimensionless radial coordinate, $\left(r-r_{1}\right) /\left(r_{2}-r_{1}\right)$, where $r_{1}=0.06 \mathrm{~m}$ and $r_{2}$ is a variable $\left(0.06<r_{2} \leq 0.20 \mathrm{~m}\right)$. Compute and plot the heat loss as a function of the insulation thickness for $0 \leq\left(r_{2}-r_{1}\right) \leq 0.14 \mathrm{~m}$.

Key Concepts

Steady-State Heat Conduction

This concept deals with the analysis of heat flow when thermal energy transfer is balanced, meaning that the temperature distribution does not change with time. In problems of insulation around pipes or other objects, the steady-state assumption allows us to establish and solve differential equations, ensuring that the rate of heat entering any element equals the rate of heat leaving it.

Cylindrical Coordinate Analysis

Heat conduction in cylindrical geometries requires formulating the energy balance in a radial coordinate system. Unlike planar heat conduction, the geometry introduces a radial dependence in the area available for heat transfer. This analysis is crucial for determining temperature distributions and heat flux in pipes and other cylindrical structures where the inner and outer surfaces might be maintained at different temperatures.

Thermal Resistance Concept

The concept of thermal resistance provides a simplified method to model complex heat transfer by considering the resistances to heat flow in series or parallel. In multi-layered systems like insulated pipes, the overall thermal resistance is the sum of the conduction resistance through the insulation and the resistances associated with convection (and possibly radiation) at the boundaries, allowing for straightforward calculations of heat loss.

Convection Heat Transfer

Convection involves the transport of heat by the movement of a fluid (such as air) over a surface. The convective heat transfer coefficient, h, quantifies the efficiency of this process. In insulated systems exposed to ambient air, convection at the outer surface serves as a boundary condition that affects the overall heat loss from the system, making it necessary to account for this mode of heat transfer in thermal analyses.

Radiation Heat Transfer

Radiation is the process by which thermal energy is emitted from a surface in the form of electromagnetic waves. Its effectiveness is influenced by the surface emissivity and the temperatures of the surface and its surroundings. In systems where surfaces are exposed to large surroundings, radiative heat transfer plays a significant role and often has to be integrated with convection calculations for an accurate overall heat loss estimation.

Non-Dimensional Analysis

This concept involves rewriting physical equations in terms of dimensionless variables, which simplifies the analysis and highlights the influence of geometric or material parameters. For temperature distribution and heat loss, using a dimensionless radial coordinate allows for the generalization of results across different insulation thicknesses, facilitating comparisons and the prediction of system behavior without dependence on absolute dimensions.

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