What is the fundamental difference between the 'Variable State' and the 'Steady State' in heat conduction?

In the Variable State, the temperature at any point in the conductor changes with time as the material absorbs heat. In the Steady State, the temperature at every point remains constant over time because the rate of heat entry equals the rate of heat exit.

How does the series combination of thermal conductors differ from a parallel combination regarding heat current?

In a series combination, the same heat current ($H$) flows through every conductor in the sequence. In a parallel combination, the total heat current is the sum of the currents through each individual path, as they all share the same temperature difference.

Why are metals generally much better thermal conductors than wood or plastic?

Metals possess a high density of free electrons that can move rapidly through the crystal lattice, carrying kinetic energy. Non-metals lack these free electrons and must rely on slower lattice vibrations (phonons) to transfer energy.

What is a common error when calculating the equivalent thermal conductivity of a composite slab in series?

A common mistake is simply averaging the thermal conductivities ($k_1$ and $k_2$). Instead, one must add the thermal resistances ($R = L/kA$) and then solve for the effective $k$ based on the total length and area.

What happens to the heat flow calculation if the units of length are in centimeters but thermal conductivity is given in $W m^{-1} K^{-1}$?

The calculation will be off by a factor of 100 or 10,000 (for area). All spatial dimensions must be converted to meters to maintain consistency with the SI units of thermal conductivity.

Why is it incorrect to assume the temperature is the same everywhere in a rod during steady-state conduction?

Steady state means temperature is constant *at a specific point* over time, but there must be a temperature difference between the ends for heat to flow. If the temperature were uniform, the gradient would be zero, and no conduction would occur.

Define 'Temperature Gradient' and state its SI unit.

Temperature Gradient is the change in temperature per unit length in the direction of heat flow ($\Delta T / \Delta x$). Its SI unit is Kelvin per meter ($K m^{-1}$) or degrees Celsius per meter ($^{\circ}C m^{-1}$).

What is the formula for Thermal Resistance ($R_h$) and what do the variables represent?

The formula is $R_h = \frac{L}{kA}$, where $L$ is the length of the conductor in the direction of heat flow, $k$ is the thermal conductivity of the material, and $A$ is the cross-sectional area perpendicular to the flow.

State Fourier's Law of heat conduction in terms of heat current ($H$).

Fourier's Law states $H = \frac{dQ}{dt} = -kA \frac{dT}{dx}$, where $H$ is the rate of heat flow, $k$ is thermal conductivity, $A$ is area, and $dT/dx$ is the temperature gradient.

According to the Ingen-Hausz experiment, what is the relationship between thermal conductivity ($k$) and the length of wax melted ($l$) on a rod?

The thermal conductivity is directly proportional to the square of the length of the melted wax ($k \propto l^2$). This allows for the comparison of conductivities of different materials by measuring the reach of heat along identical rods.

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Conduction of Heat

Summary

Conduction is the process of heat transfer through a medium where energy is exchanged between adjacent molecules or atoms without any bulk motion of the material itself. It is governed by the material's thermal conductivity and the temperature gradient across it, often modeled using an analogy to electrical circuits for complex systems.

1. Definition & Microscopic Mechanism

Thermal Conduction is the mode of heat transfer in solids where energy moves from a region of higher temperature to a region of lower temperature through molecular collisions and the movement of free electrons.
In non-metallic solids, heat is primarily transferred via lattice vibrations (phonons), where energetic molecules vibrate and collide with their less energetic neighbors.
In metallic solids, free electrons act as the primary carriers of thermal energy, which explains why good electrical conductors are typically excellent thermal conductors.
Unlike convection, there is no macroscopic displacement of the matter; the particles only oscillate about their equilibrium positions.

Diagram of a solid rod showing heat flow from a hot end (T1) to a cold end (T2) across a length L.

2. Steady State vs. Variable State

During the Variable State, the temperature of every cross-section of the conductor increases with time as it absorbs part of the heat flowing through it.
The Steady State is reached when the temperature at any given point in the conductor remains constant over time, meaning no more heat is absorbed by the material; all heat entering one end exits the other.
In the steady state, while the temperature is constant with respect to time, it still varies with respect to position along the length of the conductor.
The rate of heat flow ( $dQ/dt$ ) is uniform across all cross-sections only when the system has achieved a steady state.

3. Fourier's Law & Temperature Gradient

4. Thermal Resistance & Electrical Analogy

5. Composite Slabs: Series and Parallel

6. Exam Strategy & Common Pitfalls