In a natural convection test rig, a glass partition 1 cm thick and 4 cm high separates two cop

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In a natural convection test rig, a glass partition 1 cm...

In a natural convection test rig, a glass partition $1 \mathrm{~cm}$ thick and $4 \mathrm{~cm}$ high separates two copper plates. The lower plate is maintained at $340 \mathrm{~K}$, while the upper plate is maintained at $300 \mathrm{~K}$. The air temperature on each side of the partition is $320 \mathrm{~K}$, and the heat transfer coefficient is $6.0 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. Determine the heat flow at the bottom and the top of the partition (i) assuming one-dimensional conduction. (ii) assuming two-dimensional conduction. Take $k=0.78 \mathrm{~W} / \mathrm{m} \mathrm{K}$ for the glass. You will need the roots of $\mathrm{Eq}$. (3.67), which are given in the following table for an appropriate range of Biot number. $$ \begin{array}{ccccc} \mathrm{Bi} & \lambda_{1} & \lambda_{2} & \lambda_{3} & \lambda_{4} \\ \hline 0.01 & 0.100 & 3.145 & 6.285 & 9.426 \\ 0.02 & 0.141 & 3.148 & 6.286 & 9.427 \\ 0.05 & 0.222 & 3.157 & 6.291 & 9.430 \\ 0.1 & 0.311 & 3.173 & 6.299 & 9.435 \\ 0.2 & 0.433 & 3.204 & 6.315 & 9.446 \\ 0.5 & 0.653 & 3.292 & 6.362 & 9.477 \\ 1.0 & 0.861 & 3.426 & 6.437 & 9.529 \end{array} $$

In a natural convection test rig, a glass partition $1 \mathrm{~cm}$ thick and $4 \mathrm{~cm}$ high separates two copper plates. The lower plate is maintained at $340 \mathrm{~K}$, while the upper plate is maintained at $300 \mathrm{~K}$. The air temperature on each side of the partition is $320 \mathrm{~K}$, and the heat transfer coefficient is $6.0 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.
Determine the heat flow at the bottom and the top of the partition
(i) assuming one-dimensional conduction.
(ii) assuming two-dimensional conduction.
Take $k=0.78 \mathrm{~W} / \mathrm{m} \mathrm{K}$ for the glass. You will need the roots of $\mathrm{Eq}$. (3.67), which are given in the following table for an appropriate range of Biot number.
$$
\begin{array}{ccccc}
\mathrm{Bi} & \lambda_{1} & \lambda_{2} & \lambda_{3} & \lambda_{4} \\
\hline 0.01 & 0.100 & 3.145 & 6.285 & 9.426 \\
0.02 & 0.141 & 3.148 & 6.286 & 9.427 \\
0.05 & 0.222 & 3.157 & 6.291 & 9.430 \\
0.1 & 0.311 & 3.173 & 6.299 & 9.435 \\
0.2 & 0.433 & 3.204 & 6.315 & 9.446 \\
0.5 & 0.653 & 3.292 & 6.362 & 9.477 \\
1.0 & 0.861 & 3.426 & 6.437 & 9.529
\end{array}
$$

Key Concepts

Eigenvalue Methods in Conduction Analysis

Eigenvalue methods are used to solve complex conduction problems, particularly in multi-dimensional or transient heat transfer, where separation of variables leads to eigenvalue problems. These techniques involve finding characteristic roots or eigenvalues that satisfy the boundary conditions, enabling the construction of solutions that accurately represent the temperature distribution within the medium.

Biot Number

The Biot number is a dimensionless parameter that compares the internal thermal resistance of a body to the external convective resistance. It is crucial in determining whether the temperature within an object can be assumed to be uniform (lumped system analysis) or if a detailed spatial temperature distribution must be considered.

Fourier's Law

Fourier's law provides the basic mathematical relationship for conduction heat transfer by linking the heat flux to the temperature gradient and the material's thermal conductivity. It is the starting point for analyzing conduction problems and is applicable in both simplified one-dimensional and more complex multi-dimensional analyses.

Convection Heat Transfer

Convection involves heat transfer between a solid surface and a fluid moving adjacent to it, characterized by a heat transfer coefficient. In conduction problems that involve convection boundaries, the convective heat transfer adds an external resistance to heat flow, which must be combined with conductive resistances to accurately determine the overall heat transfer rate.

Two-Dimensional Heat Conduction

Two-dimensional conduction considers heat transfer in a plane where temperature gradients exist in two perpendicular directions. This analysis is essential when the geometry or boundary conditions do not allow the assumption of a purely one-dimensional temperature distribution, requiring the solution of partial differential equations that account for variations along both axes.

One-Dimensional Heat Conduction

This concept involves analyzing heat transfer through a medium by assuming that the temperature gradient exists in only one spatial direction. It simplifies the governing equations by reducing the multidimensional problem to a single dimension, making it easier to compute the heat flux using basic conduction laws, such as Fourier's law.

Thermal Conduction

Thermal conduction is the mode of heat transfer within a material due to the temperature gradient. It is governed by Fourier’s law, which states that the heat flux is proportional to the negative gradient of temperature and the material's thermal conductivity, making it fundamental for solving both one- and multi-dimensional heat transfer problems.

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