Animated Solution for Physics - Properties of Solids and Liquids: Two identical metal wires of thermal conductivities $K_1$ and $K_2$ respectively are connected in series. The effective thermal conductivity of the combination is

Heat as a Current

Imagine water flowing through a pipe, or electrons marching through a wire. Heat flows in a remarkably similar way! When a temperature difference exists across a material, heat energy travels from the hotter end to the colder end. This flow of heat is governed by a principle that perfectly mirrors Ohm's Law in electricity.

In electricity, we have $V=IR$. In thermodynamics, the "voltage" or driving force is the temperature difference $\Delta T$, and the "current" is the rate of heat flow $H=\frac{dQ}{dt}$. The resistance to this flow is called Thermal Resistance, denoted by $R$.

What makes a material resist heat flow? It depends on three factors: the length of the material, its cross-sectional area, and its intrinsic ability to conduct heat, known as thermal conductivity ($K$).

The formula for thermal resistance is:

Notice how this is identical to electrical resistance $R=\rho \frac{l}{A}$, where resistivity $\rho$ is simply the reciprocal of conductivity $K$.

The Obstacle Course

In our problem, we have two identical metal wires connected end-to-end in series. Let's assume they both have a length $l$ and a cross-sectional area $A$.

Because they are in series, heat must flow through the first wire and then entirely through the second wire. Just like electrical resistors in series, the total thermal resistance is the sum of the individual resistances:

Substituting our formula for thermal resistance, we get:

To find the effective thermal conductivity ($K_{eq}$), we must imagine replacing these two wires with a single, equivalent wire that behaves exactly the same way.

What would the dimensions of this equivalent wire be? Since the two original wires are joined end-to-end, the total length becomes $l+l=2l$. The cross-sectional area remains unchanged at $A$.

Therefore, the thermal resistance of this equivalent wire is:

Now, we equate our two expressions for the total equivalent resistance:

Look closely at this equation. The terms $l$ and $A$ appear in every single fraction. This means the geometry of the wires cancels out entirely! Dividing the entire equation by $\frac{l}{A}$, we are left with a beautifully simple relationship:

To solve for $K_{eq}$, we first take the common denominator on the right side:

Finally, we invert both sides and multiply by 2 to isolate $K_{eq}$:

This final result is mathematically known as the harmonic mean of $K_1$ and $K_2$. Whenever you connect identical thermal conductors in series, their effective conductivity will always be their harmonic mean.

Food for thought: What if the wires were connected in parallel instead? In that case, the length would remain $l$, but the area would double to $2A$. The effective conductivity would turn out to be the arithmetic mean: $\frac{K_1+K_2}{2}$. Try deriving it yourself!