# Internal heating effects of a circuit element?

The heat-up of a solid circuit component in an electric circuit (for which no chemical reactions, phase changes reactions or other stuctural changes take place) may include:

• Joule heating $\dot Q_J$, which is the well-known heat-up when electrons "bump" into the lattice atoms and exchange energy into vibrational energy,
• Fourier's law $\dot Q_F$ of heat conduction, which moves heat from the hot to the colder end, and
• the Peltier effect $\dot Q_P$, which may move heat from one end of a component to the other along with the electron flow.

I should be able to put these together in one combined equation of the total heat flow $\dot Q$ into the colder end:

$$\dot Q = \dot Q_J+\dot Q_F+\dot Q_P$$

All these express internal heating effects (and I here include not only heat generation but also heat that is moved from one end to the other).

Do these three constitute all possible internal heating effects in any such material? Or are there more to consider?

• You are going at this non-classical transport problem from the wrong (classical) end. If you want to understand heat transport in crystalline materials (including semiconductors), then you have to look at phonons. Commented Jun 8, 2015 at 0:29
• @CuriousOne yea, but phonon transport is a form of heat conduction and can be considered included in Fourier's law if I am not mistaken Commented Jun 8, 2015 at 7:22
• Fourier's law is a classical ad-hoc formula and heat conductions doesn't work that way, for one thing all classical heat conduction formulas assume an infinite speed of conduction while in reality phonons have a dispersion relation that is closer to the speed of sound in the material than infinity. Commented Jun 8, 2015 at 8:27

The short answer is that no, these are not the only internal heating processes. Furthermore the Peltier effect is more of a surface than a volume effect, as I'll discuss below. There is a volume analogue, but it is commonly called the "extrinsic Thomson effect" but exists only in non-homogeneous materials.

The longer answer is that the number of internal heats depend on the characteristics of the material. To keep things simple, I consider a macroscopic sample (in very tiny samples, Fourier's law does not necessarily hold. See heat/electrical conduction in mesoscopic systems for more information). The most general case would be a nonhomogeneous anisotropic material. In such a case, there are the Fourier, Joule, Thomson effects as well as a Bridgman effect. Note that there are two "Thomson effects". One is due to the change in the Seebeck coefficient with respect to temperature (the commonly called "Thomson effect", but also less known as "intrinsec Thomson effect"), the other is due to the change in the Seebeck coefficient with respect to position (which is known as the "extrinsec Thomson effect" or "distributed Peltier effect" and occurs for example if the doping of the semiconductor sample is not homogeneous, since $$S$$ may well be position-dependent even if the sample is kept at a well defined temperature).

If you remove the anisotropy, the Bridgman effect disappears. If you have a homogeneous material, the extrinsec Thomson effect disappears, though not the usual/common Thomson effect.

Note: The common Peltier effect (to distinguish from the "distributed Peltier effect") only takes place if the sample is brung into contact with a material of a different Seebeck coefficient. And it would be more of a surface effect than a volume effect. So if you're solving a heat equation, it would be included as a boundary condition; it isn't really a volume effect.

Sources: The paper "Irreversible Thermodynamics of Themoelectric Effects in Inhomogeneous, Anisotropic Media, by Domenicali. The book "Continuum Theory and Modeling of Thermoelectric Elements" by Goupil et al.

There is more to all of this. If you are really careful and take into account the volume expansion/contraction of a heated material, then there is another internal heating/cooling that is proportional to the change in the Seebeck coefficient with respect to volume, and to the absolute temperature. This can be seen if you follow the Domenicali's paper cited above and do not neglect the volume change of the material.

Sidenote: I do not understand the relevance of CuriousOne's comments which I immortalize here:

CuriousOne: You are going at this non-classical transport problem from the wrong (classical) end. If you want to understand heat transport in crystalline materials (including semiconductors), then you have to look at phonons. –

Steeven: yea, but phonon transport is a form of heat conduction and can be considered included in Fourier's law if I am not mistaken

CuriousOne: Fourier's law is a classical ad-hoc formula and heat conductions doesn't work that way, for one thing all classical heat conduction formulas assume an infinite speed of conduction while in reality phonons have a dispersion relation that is closer to the speed of sound in the material than infinity.

Sure, Fourier's law has limitations (you don't even need to go for relativity for that), yet you can still apply it with immense accuracy in everyday materials. Also that critics has nothing to do with the question asked "Do these three constitute all possible internal heating effects in any such material? Or are there more to consider?".