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##### One-dimensional modeling of TE devices

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**One-dimensional modeling of TE devices**Daniel Mitrani and Juan A. Chávez Electrical Engineering Department, Universitat Politècnica de Catalunya Barcelona, Spain. Email: mitrani@eel.upc.edu**Overview**TEM description and formulae Steady-state electrical models and simulations Dynamic electrical model and simulations Conclusions Contents Presentation contents**TEM Description**Ceramic Plates Positive (+) Moisture Protection Thermocouples Negative (-) TE module characteristics • Solid-state devices. • Couples connected electrically in series and thermally in parallel. • Peltier mode: heat pump. • Seebeck mode: electrical power generation. Free standing pellet Single couple unit Tc Th**TEM Formulae (I)**Ohm’s Law Thomson Effect Joule Effect Peltier Effect Seebeck Effect Conduction Convection Radiation 1-D steady-state energy balance equation Interaction between thermal and electrical domains Electrical domain Joule Effect Fourier’s Law Thomson Effect Heat flow per unit area Peltier Effect Fourier’s Law Electric field per unit length Thermal domain Ohm’s Law Seebeck Effect**TEM Formulae (II)**Constant material properties Heat flow per unit area at x=0 and x=L Dirichlet boundary conditions Electrical potential at x=L 1-D temperature distribution**TEM Formulae (III)**Heat released at the hot side Heat absorbed at the cold side • Parallel thermal conductance of the N couples Voltage across the terminals • Serial electrical resistance of the N couples Electrical power • Seebeck coefficient of the N couples**Lumped Electrical Model (I)**Electrical steady-state three-port model Thermal and electrical analogies • Thermal processes described in electrical terms. • Flexible boundary conditions. • Simulation of control electronics and thermal elements. Material parameters are set prior to simulation !!! Model expressions**Lumped Electrical Model (II)**Temperature profile for DTmax case Temperature profile for Qcmax case • Average temperature between hot and cold side • Mean module temperature**Lumped Electrical Model (III)**Material properties are calculated as: Where Tk can be calculated as: • Hot side temperature • Cold side temperature • Average module temperature: Tk • Mean module temperature: Additional VCVS’s are defined as:**Distributed Parameter Electrical Model**• Steady-state equations are accurate as long as the thermoelectric properties do not vary over the region where they are applied. • Divide the pellets of a TEM into many small segments • Each segment would be closer to meeting such criteria**Steady-State Simulation Setup and Results (I)**Th=300 K L=1 mm A=1 mm2**Steady-State Simulation Setup and Results (II)**Temperature Difference vs. Electrical Current**Steady-State Simulation Setup and Results (III)**Cooling Power vs. Electrical Current**Steady-State Simulation Setup and Results (IV)**Spatial profiles for material parameters s(x), r(x), k(x), and z(x)**Dynamic Distributed Parameter Electrical Model**One-dimensional heat flow equation Justification No analytical solution !!! • Start-up and shut-down periods. • Operating conditions are varied with time. • Fast-response heat sources. • Similar TEC and Heat load thermal time constants • Pulse cooling analysis. • I→ Electrical current • → Electrical resistivity • → Thermal conductivity → Thermal diffusivity Proposed distributed parameter transient electrical model**Conclusions**Conclusions • Based on 1-D steady-state analysis we propose • Lumped parameter model • Distributed parameter model • Simulation of electrical and thermal domains with a single tool • Control electronics • Thermal elements • Material parameters chosen according to different module temperatures • Dynamic Distributed Parameter Electrical Model • Start-up and shut down periods • Similar TEM and heat load time constants • Transient cooling operation