Does a thermal inductor exist?

Question

There is an analogy between electric and thermal circuits.

A voltage difference is equivalent to a thermal difference. $$\Delta V \equiv \Delta T$$

Electric charge $q$ is equivalent to heat $Q$. Electric current is equivalent to heat current

$$I = \dot{q} \equiv \dot{Q}$$

Resistance is equivalent to thermal resistance or 1 / heat conductance $$R \equiv \frac{1}{\sigma}$$ Electric capacitance is equivalent to heat capacitance (specific heat) $$C_{electric} = \frac{q}{\Delta V} \equiv C_{therm} = \frac{Q}{\Delta T}$$

Even Ohms Law can be written for thermal processes as well

$$R = \frac{V}{I} \equiv \frac{1}{\sigma} = \frac{\Delta T}{\dot{Q}}$$

But does a thermal equivalent to an inductance L exist?

Answer 1

The analogy is actually looser than it seems. Imagine a capacitor connected to a battery $V$ (with a resistor $R$ in series). The situation is described by Kirchoff's voltage law:

$$V -R\frac{dq}{dt}-\frac{1}{C}q=0$$

the solution to which is something like $q(t) = CV(1-e^{\frac{-t}{RC}})$. Then the maximum charge stored is $CV$, where $V$ is the potential difference which is held constant.

Now imagine putting an object (your "capacitor") with high thermal mass ($C_v$) between temperature reservoirs held at $T_1$ and $T_2$ held at a constant $\Delta T$. The temperature within the object is then described given by the Heat Equation:

$$\frac{dQ}{dt}=-k\frac{d^2T}{dx^2}\rightarrow \rho C_v\frac{dT}{dt}+k\frac{d^2T}{dx^2}=0$$

where we've assumed a lot about sources, dimensions, etc. What's important is that the system will reach steady state, a time at which points' temperatures are no longer time dependent (conceptually, this is when the "capacitor" no longer absorbs heat). This kills the time dependence in the equations:

$$k\frac{d^2T}{dx^2}=0 \rightarrow k \frac{dT}{dx}=B$$

But this is just Fourier's law ($k \frac{dT}{dx}=Q$) describing constant heat transfer through the object. Saying $R=\frac{\Delta x}{k}$ we've arrived at:

$$Q=\frac{\Delta T}{R}$$

So your "capacitor" starts off impeding the heat flow as it absorbs energy and comes to steady state temperature, but then permits constant heat flux independently of $C_v$ - your "capacitor" is much like the inductor you seek! (But not rigorously).

Main Points

• The "capacitor" in either system behaved differently for a fixed "voltage" - the capacitor analogy is weak for steady heat transfer

• If you use a complicated resistor-capacitor network and a time-varying input voltage (Example) you can actually simulate heat capacity in thermal systems, but its not the simple circuit you'd like it to be

Answer 2

No. There is no thermal equivalent of an inductor.

Think about it, temperatures don't oscillate back and forth between hotter and colder materials. They decay exponentially from higher to lower. No complex poles.

Answer 3

In a recent work I found a thermal inductance given by $L=\frac{\mu_0 c\hbar}{k_BT}$, where $k_B$ is the Boltzman constant. It becomes very important at very low temperature ranges.