The exposed surface (x=0) of a plane wall of thermal...

The exposed surface $(x=0)$ of a plane wall of thermal conductivity $k$ is subjected to microwave radiation that causes volumetric heating to vary as $$ \dot{q}(x)=\dot{q}_{o}\left(1-\frac{x}{L}\right) $$ where $\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$ is a constant. The boundary at $x=L$ is perfectly insulated, while the exposed surface is maintained at a constant temperature $T_{o}$. Determine the temperature distribution $T(x)$ in terms of $x, L, k, \dot{q}_{o}$, and $T_{o}$.

The exposed surface $(x=0)$ of a plane wall of thermal conductivity $k$ is subjected to microwave radiation that causes volumetric heating to vary as
$$
\dot{q}(x)=\dot{q}_{o}\left(1-\frac{x}{L}\right)
$$
where $\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$ is a constant. The boundary at $x=L$ is perfectly insulated, while the exposed surface is maintained at a constant temperature $T_{o}$. Determine the temperature distribution $T(x)$ in terms of $x, L, k, \dot{q}_{o}$, and $T_{o}$.

Key Concepts

Integration of Differential Equations

Solving the steady-state heat conduction equation with internal heat generation involves integrating a second-order ordinary differential equation. Each integration step introduces an unknown constant, which is determined by the imposed boundary conditions. This systematic integration process transforms the second-order differential equation into an explicit expression for the temperature distribution as a function of position.

Thermal Conductivity

Thermal conductivity is a material property that measures its ability to conduct heat. In the heat conduction equation, it appears as a coefficient that moderates the effect of the temperature gradient. A higher thermal conductivity implies that the material can more effectively transfer heat, impacting the overall temperature profile within the domain.

Boundary Conditions

Boundary conditions are essential in specifying the behavior of the temperature field at the edges of the domain. In this type of problem, both Dirichlet conditions (fixed temperature, such as at the exposed surface) and Neumann conditions (insulated, meaning zero heat flux) are used. These conditions ensure the uniqueness and physical relevance of the solution by constraining the integration constants obtained after solving the differential equation.

Volumetric Heat Generation

This refers to the production of heat within a material due to some internal process, such as microwave absorption. In problems where heat is generated within the body, a heat source term is added to the energy equation. The spatial dependence of the heat generation function affects the temperature distribution and must be integrated appropriately within the governing differential equation.

Steady-State Heat Conduction

This concept deals with the differential balance of energy in a material when it has reached a state in which the temperature field does not change with time. The underlying equation is derived from Fourier’s law of heat conduction and includes contributions from internal heat generation. In this context, the second-order ordinary differential equation is set up to represent the balance between conduction and volumetric heating.

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