Let H be the flow of heat per unit time per unit area normal...

Let $H$ be the flow of heat per unit time per unit area normal to the isothermal surface through a point $P$ of the body. Assume the experimental fact $$ \mathbf{H}=-k \nabla T, $$ where $T$ is the temperature and $k$ is the coefficient of thermal conductivity. Finally the thermal energy absorbed per unit volume is given by $c \rho T$, where $c$ is the specific heat and $\rho$ is the density. (a) Make an analogy between the thermal quantities $H, k, T, c, \rho$ and the corresponding quantities $\mathbf{E}, \mathbf{J}, V, \rho$ of steady currents. (b) Using the results of (a) find the heat conduction equation. (c) A pipe of inner radius $r_1$, outer radius $r_2$ and constant thermal conductivity $k$ is maintained at an inner temperature $T_1$ and outer temperature $T_2$. For a length of pipe $L$ find the rate the heat is lost and the temperature between $r_1$ and $r_2$ (steady state).

Let $H$ be the flow of heat per unit time per unit area normal to the isothermal surface through a point $P$ of the body. Assume the experimental fact
$$
\mathbf{H}=-k \nabla T,
$$
where $T$ is the temperature and $k$ is the coefficient of thermal conductivity. Finally the thermal energy absorbed per unit volume is given by $c \rho T$, where $c$ is the specific heat and $\rho$ is the density.
(a) Make an analogy between the thermal quantities $H, k, T, c, \rho$ and the corresponding quantities $\mathbf{E}, \mathbf{J}, V, \rho$ of steady currents.
(b) Using the results of (a) find the heat conduction equation.
(c) A pipe of inner radius $r_1$, outer radius $r_2$ and constant thermal conductivity $k$ is maintained at an inner temperature $T_1$ and outer temperature $T_2$. For a length of pipe $L$ find the rate the heat is lost and the temperature between $r_1$ and $r_2$ (steady state).

Key Concepts

Energy Storage and Heat Capacity

In the context of heat conduction, the thermal energy stored within a material is often expressed in terms of its specific heat and density. This concept describes the amount of energy required to change the temperature of a unit volume of the material and is vital for understanding the dynamic aspects of heat transfer.

Electrical Conduction Analogy

There exists a useful analogy between thermal conduction and electrical conduction, where the temperature corresponds to electrical potential, the heat flux to the current density, and thermal conductivity to electrical conductivity. This analogy allows for the transference of intuition and solution techniques between these two areas of physics.

Thermal Conductivity

Thermal conductivity is an intrinsic property of a material that measures its ability to conduct heat. It quantifies the amount of heat that will flow through a material per unit area per unit time for a given temperature gradient, and plays a central role in Fourier's law.

Fourier's Law

This is the fundamental law of heat conduction stating that the heat flux (or heat flow per unit area) is proportional to the negative temperature gradient. It establishes a direct relationship between how heat flows in a material and the spatial distribution of temperature, with the thermal conductivity as the proportionality constant.

Heat Conduction Equation

Derived from the conservation of energy and Fourier's law, the heat conduction equation describes the distribution and evolution of temperature within a body. In its general form it accounts for heat generation, storage, and conduction, and in the steady state reduces to a spatial differential equation governing the temperature field.

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