A thin flat plate of length L, thickness t, and width W ⩾L is thermally joined to two large heat

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A thin flat plate of length L, thickness t, and width W ⩾L...

A thin flat plate of length $L$, thickness $t$, and width $W \geqslant L$ is thermally joined to two large heat sinks that are maintained at a temperature $T_{o}$. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of $q_{o}^{\prime \prime}$ (a) Derive the differential equation that determines the steady-state temperature distribution $T(x)$ in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks.

A thin flat plate of length $L$, thickness $t$, and width $W \geqslant L$ is thermally joined to two large heat sinks that are maintained at a temperature $T_{o}$. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of $q_{o}^{\prime \prime}$
(a) Derive the differential equation that determines the steady-state temperature distribution $T(x)$ in the plate.
(b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks.

Key Concepts

Overall Heat Transfer Rate

This concept involves determining the net rate at which heat is transferred through or out of a system by integrating local heat fluxes over the entire surface or boundary. Establishing the relationship between the distributed heat removal and the conduction through the material is critical to understanding the efficiency of the heat transfer process.

Differential Equations and Their Analytical Solution

The temperature distribution in a medium is often governed by differential equations derived from physical balances. Solving these differential equations, typically through integration and the application of boundary conditions, provides an analytical representation of how temperature varies spatially throughout the material.

Boundary Conditions

Boundary conditions are essential specifications that are imposed at the limits of the domain to ensure a unique solution of a differential equation. In the context of heat conduction, these could be known temperatures or prescribed heat fluxes at the surfaces or interfaces, thereby linking the mathematical model to the physical configuration.

Steady-State Heat Conduction

This concept describes the condition in which the temperature field within a material does not change with time. In such systems, the energy entering any region of the medium balances the energy leaving it, which allows the formulation of a mathematical model where temporal derivatives in the heat conduction equation are set to zero.

Energy Balance in a Differential Control Volume

By applying the conservation of energy to an infinitesimally small segment of the medium, one can derive a differential equation that quantifies the balance between heat entering, heat exiting, and any distributed heat loss or gain. This local energy balance is a fundamental approach in analyzing heat transfer processes.

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