fluidistic
Gold Member
- 3,928
- 272
I am quite confused. I know about the continuity equation, for instance ##\frac{\partial \rho}{\partial t}+\nabla\cdot \vec J = \sigma##. In steady state, ##\frac{\partial \rho}{\partial t}=0## which implies ##\nabla\cdot \vec J = \sigma##. In words, the divergence of the vector field ##\vec J## is equal to the source of that field, ##\sigma##. This means that in steady state, if I focus on an infinitesimal volume element, and I compute the net flux that goes of that volume element, I would get ##\sigma##. Great.
Here comes the problem. In thermoelectricity, the total heat flux is worth ##\vec J_Q = -\kappa \nabla T +ST\vec J## where ##S## is the Seebeck coefficient. The first part of this flux is the usual thermal flux, the second part is what you may call a Peltier flux (it has no name in the literature but it is exactly this). The heat equation of a thermoelectric material has the form $$\nabla \cdot(\kappa \nabla T)+q_\text{Joule}+q_\text{thermoelectric source terms}=0$$. In other words, it is the divergence of the usual heat flux ##-\kappa \nabla T## that's worth the total heat sources. It's not the divergence of the total heat flux that's worth the sum of all heat sources. Why?! I do not understand this.
If you were to compute the divergence of the total heat flux, you'd get some terms that "aren't real" in the sense that they correspond to no measurable heat sources. These terms do not appear in the heat equation, they are not manifest in any experiment, as far as I know. Therefore I do not understand well what the divergence of a field means. What would mean the divergence of the total heat flux in the case of thermoelectricity, for instance?
Here comes the problem. In thermoelectricity, the total heat flux is worth ##\vec J_Q = -\kappa \nabla T +ST\vec J## where ##S## is the Seebeck coefficient. The first part of this flux is the usual thermal flux, the second part is what you may call a Peltier flux (it has no name in the literature but it is exactly this). The heat equation of a thermoelectric material has the form $$\nabla \cdot(\kappa \nabla T)+q_\text{Joule}+q_\text{thermoelectric source terms}=0$$. In other words, it is the divergence of the usual heat flux ##-\kappa \nabla T## that's worth the total heat sources. It's not the divergence of the total heat flux that's worth the sum of all heat sources. Why?! I do not understand this.
If you were to compute the divergence of the total heat flux, you'd get some terms that "aren't real" in the sense that they correspond to no measurable heat sources. These terms do not appear in the heat equation, they are not manifest in any experiment, as far as I know. Therefore I do not understand well what the divergence of a field means. What would mean the divergence of the total heat flux in the case of thermoelectricity, for instance?