Peltier element theory

Definitions
– Th: Temperature of hot side (K)
– Tc: Temperature of cold side (K)
– ΔT: Th-Tc
– Tavg: (Th+Tc)/2

TEC1-12715
This module is the one I ordered and has a detailed datasheet which includes some plots at Th=27°C (300K) and Th=50°C (323K). No formulas and parameters of the module are provided so these were derived from the plots using the fit option of Wolfram Alpha[1].

The Seebeck effect causes an electromotive force, Uemf when a temperature difference is applied to the sides of the module,
Uemf = SΔT,
Where S is the difference between Seebeck coefficients of the n and p material inside the module. Since the module contains 127 elements, S is equal to 127 times the Seebeck coefficient of a single element.

Formula overview
Uemf = SΔT
I = (U-SΔT)/R
U = IR+SΔT
P = UI = U²/R-(USΔT)/R = I²R+ISΔT
QC = QP-P/2-QL
QH = QP+P/2-QL
QP = TavgSI = TavgS(U-SΔT)/R

References
1. https://www.wolframalpha.com/