Thermodynamics of Thermocouple

Consider a thermocouple consisting of two meals A and B. Let T and T + dT be the temperatures of the cold and hot junctions respectively. Let π and π + d π be the Peltier coefficient for the pair at the cold and hot junctions. Let σ_a and σ_b be the Thomson coefficient for the metals A and B respectively, both taken as positive. When a charge flows through the thermocouple, heat will be absorbed and evolved at the junctions due to Peltier effect and all along the metal due the Thomson effect.

Let 1 coulomb of charge flow through the thermocouple in the direction from A to B at the hot junction.

Heat energy absorbed due to Peltier effect at the hot junction = (π + d π) joules.
Heat energy absorbed due to Peltier effect at the cold junction = π joules.
Heat energy absorbed in the metal A due to Thomson effect = σ_a dT joules
Heat energy evolved in the metal B due to Thomson effect = σ_b dT joules
. : Net heat energy absorbed in the thermocouple
        = (π + dπ – π) + (σ_bdT - σ_bdT)
        = (dπ + (σ_a - σ_b) dT
The energy is used in establishing a P.D. dE in the thermocouple
        dE = dπ + (σ_a - σ_b) dT