Nancy Ma, what interesting questions, at the moment I starting to work with a heat interchange with thermoelectric coolers for harvesting energy. I have identified one or two questions that I can answer now, but I will check all in detail. Thanks for your contribution.

Thanks, Edwin! I really look forward to your comments.

Dear Nancy:

You are right when you said what you need to understand is the Thomson Effect. since when you have one individual material the Thomson Effect is the Thermoelectric Effect which has to be understood to model the system

Is important to say that the both the Seebeck coefficient (S) and the Peltier Coefficient (PI) are both involved in the Thomson Effect. Seebeck Coefficient and Peltier Coefficient are not the same as Seebeck Effect and Peltier Effect. The Seebeck and Peltier Effect are observed just when we have a juction of two different materials (metals or semiconductors in the practice)

But the macroscopic physics of the cobination phenomena of Electric Current transpor and Heat Transport in an individual material (not a junction) should be modeled macroscopically with a set of equations , I'm gonna write the list here, and one of them models the Thomson Effect:

You need to write a system of 4 coupled Linear Partial Differential Eqs :

- Ohms Law (in the Local form)

- Fourier's Law

- The Continuity Eq. in Electromagnetic Theory

- and an Energy Balance in the material

1st. I'm sending you a word document in the form of a secuence of equations. There are the 4 Partial Differential Eqs you would need to solve to model the Themroelectric Effect in a macroscopic body. (there are the four equations enclosed in a rectangle.

and 2nd, is a PPT presentation where I tried to explain the Thomson Effect very generally, details are not showed here though

Trying to answer some of your questions briefly :

(1) Equations are in the word document

(2) The magnitud, or relevance of the Thomson effect in a material depends on the function of the Seebeck Coefficient. Normally the Seebeck Coefficient is a constant parameter in the material along any of the three spatial directions .. S =dV/dT , and that means that the derivative ratio dV/dT doesn't vary in a specific direction in our material ... but, if S is not constante, and this dV/dT has a variation in space .. along any of the interested directions, then there will be a Thomson Coefficient different that zero, since K = T*dS/dT where K (big Kappa) is the Thomson Coefficient.

So it depends on the function of the Seebeck Coefficien across the material, if S is constant, then there is no Thomson Effect, if the derivative dV/dT is not constant along an specific direction, then there is a Thomson Effect and a Thomson Coefficient different thatn zero in that direction.

(3) The Peltier Coefficient is another coefficient inside the Thomson Effect, Pi=ST Pi = the Peltier Coefficient and K =TdS/dV , so the three coefficients are intrinsically related by these two equations (called the Thomson Relations) so the Seebeck, Thomson and Peltier Coefficient are the three part of what we call the Thermoelectric Effect in a material. the three have to be considered , and the three are linked. the Peltier Coefficient has units of W/A or J/C , but also has units of [Volts] ,and it appears on the constitutive equations when the relation Pi=ST is used.

(4) That extra term is due to the Seebeck voltage, which is an external electromotive force (emf), which is an external source of electrical energy generation. Remember again, the Peltier and Seebeck effects are nos observed in individual materials, just in a junction of two different materials. So it's better understood as that S*Delta(T) is due to the Seebeck voltage, and external emf , like when we talk of Faraday electrical induction.

"with the electric current density due to this effect is equal to (– sigma * S * temperature gradient)" , but the correct statement is: J = -sigma*[Grad(V) due to Ohm Law + S*Grad(T)], this is found inside the word document. :)

Again, there is no Seebeck nor Peltier Effect here .. just the Thomson Effect, or if you want to be more general, the Thermoelectric Effect, this effect is composed by the three TE coefficient: S, Pi and Kappa

(5) Is easy to do the Energy Balance for this individual material to consider the Thomson Effect. All the equations in a secuence are in the word document, but without and description of figures ... in the PPT file I drew a very General picture trying to represent the system... , but stablish the system, and the system has to have two Heat fluxes ... one is due to the Thomson Effect and the other is due to the Fourier's Law (as I wrote in the PPT Presentation). Then also include the Density Current due to the Ohm's Law + Thomson Effect ... then Entrances = to Outlets , I'm sure you will be able to reconstruc it from the files I'm sending you.

The main assumptions in the balance are:

i. Steady State condition

ii. No Internal Heat Generation (Q'''v = 0)

iii. No internal Heat accumulation ( Qacum = 0)

iv. No External Heat sources (Qext = 0)

Now that you have stablished your problem and written your system of equations, now what it gets is to define the Boundary Conductions,

The first condition is one by defult with no exceptions: i. Ground voltage at one side

then you need at least two more Boundary Conditions

It could be: ii. The value of the Hot temperature at the Hot side and iii. The value of the Cold Temperature at the Cold side ,

it could also be one of the BC a Constant Heat Flux at the Hot Side ... or a the Constant Density Current at one of the contacts ... depends on what parameters you may know better.

I don't know if you may be able to get access to COMSOL or another FEM simulation software, like ANSYS, but if you can have access to COMSOL , this will help you to understand the problem very quickly, the procces of this simulations is very straightforward in COMSOL.

I'm sending you the .PDF guide of one of these examples in COMSOL. Here you can see the steps to stablish the Boundary Conditions about a problem with Thermoelectric Material.

(7) About the Entropy I don't know so much, for not to say nothing. I have read one or two things, but never worked in this direction or becoming the interest of our group, I don't know much really, but it sounds very interesting for me, so please, if you find something, please share it here.

The only thing I know is that the Entropy of the Heat and Charge Transport in Thermoelectric Materials is intimately (not to say almost equivalent as) related with the Seebeck Coefficient., but I don't know the exact relationship.

Is really interesting but I don't think there is much work done about this specific subject.

Best Regards ! :)

Dear Nancy:

In addition, the Peltier coefficient how much heat is carried per unit charge.

I recommend you to read this article on wikipedia about the Thermoelectric Effect, you will be able to find all the general details of the Seebeck, Peltier and Thomson coefficient, is a recommendable source to start with .

Regards !

:)