fluidistic
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I get a nonsensical result. I am unable to understand where I go wrong.
Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials with vanishing Seebeck coefficients (this simplifies things further). Now both a thermal gradient and electrical currents are passing through this material.
Using thermodynamics relations, one has that the energy flux is worth ##\vec J_U = (ST+\mu)\vec J - \kappa \nabla T##, where ##\vec J## is the electric current, ##\kappa## is the thermal conductivity, and so on.
In steady state, ##\nabla \cdot \vec J_U=0##, which physically mean that the energy flux entering the material must equal the one that leaves it, i.e. there is no accumulation of energy. Great, when I compute this quantity I get a heat equation containing a Joule term, a Thomson term (only in the case where ##S## depends on temperature, which is fine) and a Fourier conduction term, all is fine.
However, this should imply that the energy flux entering a side must equal to the one that leaves at the other side. But when I compute that quantity, I get that it differs, which is impossible.
Indeed, even though the terms ##\mu \vec J## and ##\kappa \nabla T## are the same at the extremities of the material, the Peltier heat ##ST\vec J## differs, solely because the absolute temperature differs, according to which ends we are calculating. This is not consistent with ##\nabla \cdot \vec J_U=0##, but I do not see where I go wrong.
Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials with vanishing Seebeck coefficients (this simplifies things further). Now both a thermal gradient and electrical currents are passing through this material.
Using thermodynamics relations, one has that the energy flux is worth ##\vec J_U = (ST+\mu)\vec J - \kappa \nabla T##, where ##\vec J## is the electric current, ##\kappa## is the thermal conductivity, and so on.
In steady state, ##\nabla \cdot \vec J_U=0##, which physically mean that the energy flux entering the material must equal the one that leaves it, i.e. there is no accumulation of energy. Great, when I compute this quantity I get a heat equation containing a Joule term, a Thomson term (only in the case where ##S## depends on temperature, which is fine) and a Fourier conduction term, all is fine.
However, this should imply that the energy flux entering a side must equal to the one that leaves at the other side. But when I compute that quantity, I get that it differs, which is impossible.
Indeed, even though the terms ##\mu \vec J## and ##\kappa \nabla T## are the same at the extremities of the material, the Peltier heat ##ST\vec J## differs, solely because the absolute temperature differs, according to which ends we are calculating. This is not consistent with ##\nabla \cdot \vec J_U=0##, but I do not see where I go wrong.