Non-Equilibrium Thermodynamics of the Thermoelectric Effects

The thermoelectric effects deals with the interrelations between heat flow and electric conduction in some electric conductor systems (in the thermoelectric devices). From the viewpoint of non-equilibrium thermodynamics, the thermoelectric effects arise from the coupled dependence (as per the Onsager reciprocity relations) of the thermal flow (heat flow) and electrical flow (current) on the generalised thermal force (temperature difference) and the generalised electrical force (potential difference), generally between the two junctions of two different conductors joined together (the Seebeck and the Peltier effects), though there is also the Thomson effect dealing with a kind of thermal flow (heat generation or absorption) associated with current flowing through one conductor having a temperature gradient along its direction of current. However, the effects are varyingly formulated (as detailed below) in forms of interrelations among the two generalised forces (in case of Seebeck effect) or among the two generalised flows (in case of Peltier effect) etc., thus differing from the formal forms of linear phenomenological relations among the generalised forces X_k and the generalised flows J_k.

The Seebeck effect deals with the development of the e.m.f. (potential difference), called the thermoelectric e.m.f., created by the presence of a temperature difference between two junctions (see figure) of two different conductors (of conducting materials A & B) joined together at two ends. This arrangement is called a thermocouple, which is generally used to measure temperatures within various kinds of instruments, and sometimes even for useful conversion of thermal energy to electrical one.

As per Seebeck effect, the potential difference DV generated between the two junctions maintained at two different temperatures T₁ and T₂ (with T₂ > T₁) is given by (where S_A & S_B are the Seebeck coefficients or thermoelectric power of the conductor materials A & B respectively) the relation:
        DV {i.e., V(T₂)V(T₁)} =  {S_B(T)S_A(T)}dT  =  S_AB(T) dT
where S_AB(T) = S_B(T)S_A(T) is the difference in the Seebeck coefficients.
For small temperature-differences, S_A & S_B may be assumed to be constants, so that approximately,
        DV ≈ {S_BS_A}(T₂T₁), or,   DV ≈ {S_BS_A} DT 
For junctions made of copper and constantan (a Cu-Ni alloy), the Seebeck coefficient difference S_AB has one of its largest values of around 41mV/K, so that for a temperature difference of 100 K, the Seebeck potential difference is around 4.1 mV. Thus in general the Seebeck voltages are indeed small quantities, and so large output voltages are sometimes managed by connecting several thermocouples in series forming a thermopile.