1. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 1 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

2. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 2 Overview TEM description and formulae Steady-state electrical models and simulations Dynamic electrical model and simulations Conclusions Contents Presentation contents

3. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 3 TEM Description Negative (-) Positive (+) Moisture Protection Ceramic Plates Thermocouples TE module characteristics Solid-state devices. Couples connected electrically in series and thermally in parallel. Peltier mode: heat pump. Seebeck mode: electrical power generation. Single couple unit Free standing pellet ThTh TcTc

4. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 4 Interaction between thermal and electrical domains 1-D steady-state energy balance equation Thomson Effect Joule Effect Fourier’s Law TEM Formulae (I) Heat flow per unit area Peltier EffectFourier’s Law Electric field per unit length Ohm’s LawSeebeck Effect Electrical domain Thermal domain Peltier Effect Thomson Effect Seebeck Effect Conduction Convection Radiation Joule Effect Ohm’s Law

5. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 5 TEM Formulae (II) Constant material properties Dirichlet boundary conditions 1-D temperature distribution Heat flow per unit area at x=0 and x=L Electrical potential at x=L

6. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 6 Heat absorbed at the cold side Electrical power Heat released at the hot side Voltage across the terminals Parallel thermal conductance of the N couples Serial electrical resistance of the N couples Seebeck coefficient of the N couples TEM Formulae (III)

7. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 7 Lumped Electrical Model (I) Electrical steady-state three-port model Thermal and electrical analogies Thermal DomainElectrical Domain Heal Flow (W)Electrical Current (A) Temperature Difference (K)Voltage (V) Thermal Resistance (K/W) Electrical Resistance (  ) Thermal Mass, (J/K)Electrical Capacitance (F) Model expressions Thermal processes described in electrical terms. Flexible boundary conditions. Simulation of control electronics and thermal elements. Material parameters are set prior to simulation !!!

8. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 8 Temperature profile for Q cmax case Temperature profile for  T max case Average temperature between hot and cold side Mean module temperature Lumped Electrical Model (II)

9. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 9 Mean module temperature: Hot side temperature Cold side temperature Average module temperature: Material properties are calculated as: Where T k can be calculated as: TkTk Additional VCVS’s are defined as: Lumped Electrical Model (III)

10. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 10 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 Distributed Parameter Electrical Model

11. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 11 Steady-State Simulation Setup and Results (I) T h =300 K L=1 mm A=1 mm 2

12. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 12 Temperature Difference vs. Electrical Current Model  T max I  Tmax Relative Error Lumped @ T h 59.8331.214.06% Lumped @ T m 60.5131.342.97% Lumped @ T avg 60.6232.382.78% Lumped @ T c 59.6535.684.34% Distributed (10 FE)62.3432.970.03% Numerical62.3632.99 Steady-State Simulation Setup and Results (II)

13. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 13 Cooling Power vs. Electrical Current ModelQ cmax I Qcmax Relative Error Lump. @ T h =T c =T avg 1.237 W37.74 A0.37% Lumped @ T m 1.230 W36.95 A0.18% Distributed (10 FE)1.232 W37.14 A0.01% Numerical1.232 W37.15 A Steady-State Simulation Setup and Results (III)

14. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 14 Spatial profiles for material parameters s(x),  (x),  (x), and z(x) Steady-State Simulation Setup and Results (IV)

15. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 15 Proposed distributed parameter transient electrical model No analytical solution !!! I → Electrical current  → Electrical resistivity  → Thermal conductivity  → Thermal diffusivity 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. Dynamic Distributed Parameter Electrical Model One-dimensional heat flow equation Justification

16. 16 Pulse cooling simulation analysis examples

17. One-dimensional modeling of TE devices using SPICE International Summerschool on Advanced Materials and Thermoelectricity 17 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