Consider an extended surface of rectangular cross section...

Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction. In this problem we seek to determine conditions for which the transverse ( $y$-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid. (a) Assume that the transverse temperature distribution is parabolic and of the form $$ \frac{T(y)-T_{o}(x)}{T_{s}(x)-T_{o}(x)}=\left(\frac{y}{t}\right)^{2} $$ where $T_{s}(x)$ is the surface temperature and $T_{o}(x)$ is the centerline temperature at any $x$-location. Using Fourier's law, write an expression for the conduction heat flux at the surface, $q_{y}^{\prime \prime}(t)$, in terms of $T_{s}$ and $T_{a^{+}}$ (b) Write an expression for the convection heat flux at the surface for the $x$-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio $\left(T_{o}-T_{s}\right) /\left(T_{s}-T_{\infty}\right)$. (c) From the foregoing analysis, develop a criterion for establishing the validity of the onedimensional assumption used to model an extended surface.

Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction.
In this problem we seek to determine conditions for which the transverse ( $y$-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid.
(a) Assume that the transverse temperature distribution is parabolic and of the form
$$
\frac{T(y)-T_{o}(x)}{T_{s}(x)-T_{o}(x)}=\left(\frac{y}{t}\right)^{2}
$$
where $T_{s}(x)$ is the surface temperature and $T_{o}(x)$ is the centerline temperature at any $x$-location. Using Fourier's law, write an expression for the conduction heat flux at the surface, $q_{y}^{\prime \prime}(t)$, in terms of $T_{s}$ and $T_{a^{+}}$
(b) Write an expression for the convection heat flux at the surface for the $x$-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio $\left(T_{o}-T_{s}\right) /\left(T_{s}-T_{\infty}\right)$.
(c) From the foregoing analysis, develop a criterion for establishing the validity of the onedimensional assumption used to model an extended surface.

Key Concepts

Fourier's Law

Fourier's Law states that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the material's thermal conductivity. This fundamental concept underpins the analysis of conduction heat transfer, enabling the derivation of expressions for heat flux based on temperature gradients along specified directions.

Conduction Heat Transfer

Conduction heat transfer refers to the process of heat energy moving through a material due to microscopic collisions of particles and the transfer of energy via vibrations. In the analysis of extended surfaces, it is essential to evaluate the conduction along and across the surface to determine how temperature varies within the material.

Convection Heat Transfer

Convection heat transfer involves the transfer of heat between a surface and a fluid in motion. This mode of heat transfer is characterized by the convection heat transfer coefficient and is used to model the heat exchange at the boundary of an extended surface, thereby setting external thermal conditions.

Extended Surface (Fin) Analysis

Extended surface or fin analysis involves studying structures designed to enhance heat transfer by increasing the surface area beyond what is available on the primary surface. The analysis typically requires balancing conduction within the fin against convection to the surrounding fluid, and it relies on appropriate assumptions concerning the temperature distribution within the fin.

One-Dimensional Temperature Distribution

One-dimensional temperature distribution is a simplifying assumption where temperature variations are assumed to occur predominately in one direction, allowing the neglect of lateral or transverse temperature gradients. This assumption simplifies mathematical models of heat transfer in extended surfaces but must be validated by comparing the magnitude of transverse temperature differences with the overall temperature differences to the environment.

Parabolic Temperature Distribution Assumption

The parabolic temperature distribution assumption posits that the temperature varies as a quadratic function of the transverse coordinate. This assumption is used to model the cross-sectional temperature profile in a fin, providing a tractable means to link the conduction analysis with the convection boundary condition.

Validity Criteria for One-Dimensional Analysis

The validity criteria for one-dimensional analysis determine whether it is acceptable to neglect temperature variations in the transverse directions. This criterion is established by comparing the conduction heat flux derived from a presumed transverse temperature profile with the corresponding convection heat flux at the surface, resulting in a dimensionless parameter. If this parameter indicates that the transverse temperature differences are small compared to the overall temperature differences, the one-dimensional approximation is deemed valid.

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