10

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?

P3trus
  • 281
  • Interesting question. I'd say probably no in a static system, but maybe considering movable parts (e.g. a piston) you can obtain the equivalent of an inductor. – Massimo Ortolano Dec 27 '16 at 22:10
  • I haven't tried to justify it rigorously, but a "Heat Pipe" might do what you're looking for. – forky40 Dec 29 '16 at 20:25
  • I found a very interesting reference here: https://arxiv.org/abs/1804.06405 - it seems there is an equivalent of an inductor without violating the second law of thermodynamics. – Raoul Vogel May 03 '18 at 21:04
  • you may have a look at https://arxiv.org/abs/1804.06405, where a thermal inductor is described, with oscillating heat currents, that does not violate the second law of thermodynamics – user194597 May 04 '18 at 08:06

3 Answers3

3

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

forky40
  • 761
  • I think you're comparing the wrong models. A material between two temperature reservoirs is not equivalent to a capacitor with a series resistance, because there is a parallel resistance to the capacitance (It conducts heat). – P3trus Dec 29 '16 at 12:01
  • You're correct, my mistake. I'm guessing the proper model would be something like a single temperature reservoir connected to a thermal mass bounded by adiabatic boundaries on all other sides. But imposing adiabatic boundaries (disallowing current) is like introducing an open circuit - this "capacitor" can't be connected anywhere else and serves poorly as a circuit component. – forky40 Dec 29 '16 at 20:19
  • Well, the way you measure heat capacity is similar. You place a sample on a platform of known heat capacity with a thermometer and a heater. This platform is weakly connected to a thermal bath by a wire of known thermal conductivity.Then you switch on the heater and measure the temperature response. The platform + sample temperature response is similar to a capacitor loading curve. – P3trus Dec 30 '16 at 00:12
  • @forky40 You could've considerd an ideal thermal mass which does not radiate energy. – Yashas Feb 22 '17 at 06:21
1

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.

dugz
  • 11
  • 2
    Electric current doesn't oscillate back and forth either unless you have a source whose value varies with time. You can have a source which produces alternating thermal current. Hence, your argument is flawed. – Yashas Apr 08 '17 at 16:32
  • Did you see the article mentioned in the comments to the question? The article establishes that both theoretically and experimentally the flow of heat (and consequently the temperature of body and reservoir) can be made to oscillate without doing work. – Nick Gall Apr 20 '19 at 17:52
0

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.

Maxwell
  • 325