How does the thermal conductivity works? What is the significance of thermal conductivity? I'm newbie to thermal conductivity. I read an article from https://thermtest.com/what-is-thermal-conductivity that thermal conductivity refers to the ability of a material to transfer heat but I don't understand how it works. What is the significance of knowing the thermal conductivity of a material? Hope you can enlighten me.
 A: Thermal energy is in essence thermal "vibration", meaning kinetic energy of the particles in the material. Thermal conductivity is the ability of thermal energy to flow through the bulk of a material.
Is it significant? Definitely!
Temperature $T$ is usually a measure of thermal energy - but temperature is not enough to tell us if someting will be hot or not.
When something feels warm to touch, it is because your fingertips absorb some of the thermal energy (the outer particles of your fingertips absorb some "vibrational" kinetic energy). The temperature tells us that such thermal energy is present - but not how much or how fast that is absorbed into the fingertips. That absorbed energy is quickly conducted further into your body and away from your fingers. They have a thermal conductivity.
But if the hot material delivers thermal energy to your fingertips faster, than they can conduct it away again, then the thermal energy accumulates quickly, and the material feels much warmer. You may even get burned.
This is why thermal conductivity is an important concept: When you touch something, it is not only the temperature of the material that determines how hot it feels, also its thermal conductivity determines it!

*

*That is why you can touch a wooden cutting board that was left in the Sun without getting burned, but not the metal grill - even though their temperatures are equal.

*And that is why bathroom tiles feel cold, while the plastic shelf doesn't - even though their temperatures are the same.

*And that's why you can stir the boiling, 100 degree C warm soup with a wooden or plastic spoon but definitely not with a metal spoon.

What is thermal conductivity physically?
When a particle carrying much thermal "vibration" bumps into one that carries less, it will transfer a bit of it. Such particles can either be moving ("vibrating" while moving) or stationary ("vibrating" on the spot). In general, these two types of thermal energy carrying particles are categorised as: phonons and electrons/charged particles.

*

*Electrons, when flowing through a material due to electric forces, may carry thermal energy along. This is an electric or electron thermal conductivity $\kappa_e$. $\kappa_e$ is thus of course closely tied to electric conductivity $\sigma$ since they both depend on electron motion, and are in general proportional (when one doubles, the other doubles) according to Wiedemann-Franz' law:
$$\frac {\kappa_e} \sigma =LT$$
where $L$ is the Lorentz number. So, good electric conductors are usually also good thermal conductors.


*Phonons on the other hand is a term use for the thermal conduction via neighbouring atoms and particles that are stuck in a lattice or other material structure. They "vibrate", and due to their bonding with neighbour particles, those neighbours will start "vibrating" as well. To quantify this, we can consider this "vibrational" energy transfer as a small "vibration" package or chunk that is being passed on from neighbour to neighbour without any of them are moving themselves. This is termed phonon thermal conductivity $\kappa_{ph}$. The value of $\kappa_{ph}$ depends on the perfectness of the structure - in perfect lattices with no grain boundaries and no distorting impurity atoms to disturb the path for the phonon transport, $\kappa_{ph}$ is high. But in amorphous, unsystematic structures, phonons are scattered a lot, and $\kappa_{ph}$ is low. In for instance layered structures (such as semiconductors for certain applications), phonons have a hard time moving across layers but can more easily move within a layer - so $\kappa_{ph}$ is directional. $\kappa_{ph}$ is thus very material-type and structure dependent.
The thermal conductivity $\kappa$ of a material will be the sum of all such influences: $$\kappa=\kappa_{e}+\kappa_{ph}.$$
The intuition behind it
$\kappa$ can intuitively be thought of as how much energy every second $q$ that over one square metre is transported over the thickness $\Delta x$. The temperature difference $\Delta T$ can be thought of as the "driving force" that pushes the thermal energy to move through the material, while $\kappa$ determines this energy's ability to move.
This is all clear from the thermal conduction formula known as Fourier's law:
$$q=-A\kappa \frac{\Delta T}{\Delta x}.$$
