Dasgupta, Titas

de Boor, Johannes

Goupil, Christophe

Kolb, Hendrik

Müller, Eckhard

Seifert, Wolfgang

Zabrocki, Knud

Institute of Transport Research:Publications

Transient response of thermoelectric elements and dynamic measurement methods for thermoelectric materials  K. Zabrocki1, W. Seifert2, C. Goupil3,4, H. Kolb1, J. de Boor1, T. Dasgupta1,5, and E. Müller1,6 1Institute of Materials Research, German Aerospace Center (DLR), Köln, Germany  2Institute of Physics, Martin Luther University Halle-Wittenberg, Halle (Saale), Germany 3LIED, CNRS, UMR 8236, Universite Paris Diderot, France 4Laboratoire CRISMAT ENSICAEN, Caen, France 5 IITB, Mumbai, India 6Justus Liebig University Giessen, Giessen, Germany Content 1. Fundamental equations for transient response Continuum theory – thermoelectricity  2. Direct and inverse problems  3. Transient performance calculations  4. Dynamic measurements (thermal conductivity)  - Combined thermoelectric measurement (CTEM)  www.DLR.de  •  Chart 2 - Continuum theory: Description of the properties and measurements of thermoelectrics on a macroscopic level   - Characteristic time and length scales 𝜏 ≥ 10−3 s and  𝑙 ≥ 10−3 m  - Transport of energy and charges – description via differential equations, thermal energy balance equation                  Continuum theory: Two main categories of equations  Thermoelectricity – Continuum approach www.DLR.de  •  Chart 3 Electric field Temperature field 𝐸 𝑟 𝑡 , 𝑡  𝑇 𝑟 𝑡 , 𝑡  Equations independent of the material: kinematic relations of the continuum, loading parameters, balance equations Conservation laws  Equations dependent on the material: Coverage and description of material properties Constitutive equations  Thermoelectricity – Generalized heat equation - Charge conservation    - (thermal) energy conservation  www.DLR.de  •  Chart 4 𝜚d c𝜕𝑇𝜕𝑡+  𝛻 ∙ 𝑗 𝑄 = 𝑗 ∙ 𝐸  𝛻 ∙ 𝑗 = 0 - Generalized Fourier’s law – Heat flux    - Generalized Ohm’s law – Electrical current density   𝑗 𝑄 = −𝜅𝛻𝑇 + 𝛼𝑇𝑗 = 𝑗 𝑄,𝜅 + 𝑗 𝑄,𝜋 𝑗 = 𝜎𝐸 − 𝜎𝛼𝛻𝑇 Conservation equations Constitutive equations Differential form  Onsager‘s linear response theory 𝜚d c𝜕𝑇𝜕𝑡− 𝛻 ∙ 𝜅𝛻𝑇 + 𝜏𝑗 ∙  𝛻𝑇 =𝑗2𝜎  T. C. Harman, J. M. Honig:  Thermoelectric and thermomagnetic effects and applications,McGraw − Hill 1967  Charles A. Domenicali, Irreversible Thermodynamics of Thermoelectricity, Rev.Mod. Phys. 26, 237 −  275 (1954) 𝜏 = 𝑇𝜕𝛼𝜕𝑇… Thomson coefficient, 𝜚d … mass density, 𝑐 … specific heat capacity, 𝜅 … thermal conductivity,  𝛼… Seebeck coefficient, 𝜎… electrical conductivity Solution of heat equation in thermoelectricity www.DLR.de  •  Chart 5 𝜚d c𝜕𝑇𝜕𝑡− 𝛻 ∙ 𝜅𝛻𝑇 + 𝜏𝑗 ∙  𝛻𝑇 =𝑗2𝜎  Direct problem Initial values, boundary conditions, material properties well-posed problem  Inverse problem Not all values are given, Experimental data for estimation of boundary values ill-posed problem  Performance calculation of thermoelectric  devices and systems Determination of parameters in  measurements systems, e.g. material  properties - Initial conditions (IC) Temperature distribution  at 𝑡 = 0: 𝑇0 𝑟 𝑡  - Boundary conditions (BC) - Dirichlet/Neumann/mixed BC  [Boundary value problem (BVP)] Performance calculation of thermoelectric devices www.DLR.de  •  Chart 6 Thermoelectric Generator (TEG) Thermoelectric Cooler (TEC) 𝑇 𝑟 , 𝑡  𝑃el Electrical power output 𝜂 Efficiency 𝑄  Heat flow 𝜑 Coefficient of performance Direct solution of the heat equation material properties IC BC Fixed boundary temperatures Fixed heat  input Variation of 𝑅𝐿 or 𝐼 Transient response in thermoelectricity  - Time dependent fields 𝑇 = 𝑇 𝑟 , 𝑡  and 𝐸 = 𝐸 𝑟 , 𝑡   - Dynamic working conditions:          - Solution of the generalized heat equation: - Steady state ⟹ Ordinary differential equations - Transient response ⟹ Partial differential equations   - Analytical solutions only in particular cases with help of integral transformation (e.g. Laplace) or in a (Fourier) series expansion  - Approximative or numerical solution methods (CPM, FEM, FDM, Circuits...) www.DLR.de  •  Chart 7 H.S. Carslaw/J.C. Jaeger “Conduction of heat in solids”, Oxford Science Publications,1986 J. Crank, “The mathematics of diffusion”, Oxford University Press, 1979    𝑇h = 𝑇h 𝑡   𝑇h 𝑡  BC Ambient 𝑇amb = 𝑇amb 𝑡   𝑇amb 𝑡  𝑇c = 𝑇c 𝑡   𝑇c 𝑡  Material e.g. 𝜅 = 𝜅 𝑇 𝑟 , 𝑡 , 𝑟 , 𝑡  Load/Current 𝑅L = 𝑅L(𝑡) or  𝐼 = 𝐼(𝑡) 𝑅L(𝑡)  𝐼(𝑡) 𝑄 h(𝑡) 𝑄 h = 𝑄 h(𝑡) 𝑄 c(𝑡) - Quasi-stationary processes ⇒ Timescale of changes in the working/boundary conditions much greater than response time of the thermoelectric system ⇒ use of steady state equations for different times  - TEC pulsed supercooling  Stil’bans/Fedorovich, Sov. Phys. Tech. Phys. 3, 460 (1958) Snyder et al., J. Appl. Phys. 92, 1564 (2002) Mao et al., J. Appl. Phys. 112, 014514 (2012)  - Transient TEG  P.E. Gray, The Dynamic Behavior of Thermoelectric  Devices, John Wiley & Sons, Inc., New York, 1960  Montecucco et al., Appl. Therm. Eng. 35 (2012) 177-184 Nguyen/Pochiraju, Appl. Therm. Eng. 51 (2013) 1-9 Meng, J. Power Sources, 245 (2014) 262-269   - AC impedance spectroscopy García-Cañadas/Min (previous talk), J. Electr. Mat. 43, 2411 (2014) C. Goupil (talk yesterday), J. Stockholm (poster yesterday)  - Review: Separate chapter in the book “Continuum theory and modelling of thermoelectric elements” edited by C. Goupil, release date March 2015 Transient performance calculations www.DLR.de  •  Chart 8 𝑇 𝑡 𝑡 𝐼 𝐼opt 𝐼p = 𝑓𝐼opt 𝑇min,ss ∆𝑇p ∆𝑇pp 𝑡min … 𝑡rec - Thermal conductivity for semiconductors often small ⇒  ≈1 W/(m K) - Small samples, mechanically not easy to be processed - Brittleness ⇒ not possible to put in thermocouples in the sample - Hard to realize a good thermal contact via soldering - Specific heat often not known  www.DLR.de  •  Chart 9 Measurement of thermal conductivity Measurement techniques 𝜅 Steady-state/stationary methods Dynamic/transient methods Absolute technique Comparative  technique - Laser Flash Thermal Diffusivity Method - Generalized Ioffe method ⇒ CTEM - 3𝜔 method - time-domain thermoreflectance  S. Reif-Acherman “Early and current experimental methods for determining thermal conductivities of metals” Int. J. Heat Mass Transfer 77 (2014), 542-563 T. M. Tritt “Electrical and Thermal Transport Measurement Techniques for Evaluation of the Figure-of-Merit of Bulk Thermoelectric Materials” Ch. 23, CRC  Thermoelectric Handbook. Macro to Nano (2006)  www.DLR.de  •  Chart 10 Inverse heat conduction problems - Inverse heat conduction problems (IHCP)     - Classification IHCP:  - Material properties determination inverse problems, - Boundary value determination inverse problems, - Initial value determination inverse problems, - Source determination inverse problems, - Shape determination inverse problems  - Unknown thermal conductivity (material property) ⇒ inverse calculation   - Solution of the direct problem to get insights how to solve the IHCP Krzysztof Grysa “Inverse Heat Conduction Problems” Chapter 1 in “Heat Conduction - Basic Research” ed. by V. S. Vikhrenko,  InTech (2011) Temperature 𝑇 measured (at some points, times) Vs IP CTEM – Measurement of thermal conductivity www.DLR.de  •  Chart 11 - Combined thermoelectric measurement (CTEM):  TC Gradient heater Sample - Simultaneous measurement method ⇒  all TE properties including Harman-ZT - Here: focus on thermal conductivity  measurement   - Generalized Ioffe method  Decrease of ΔT Thermal conductivity κ More experimental details on Poster P3.30  H.Kolb today  Heat Transient temperature difference www.DLR.de  •  Chart 12 Relaxation time τ Thermal conductivity κS volumetric heat generation  - Heating of one Cu block - Switching off after reaching ∆𝑇 ≈ 5K - Observing relaxation of temperatures Solution of the direct problem – Simple Ioffe method - Simple Ioffe method (sample and one block): 1. Analytical solution as Fourier series 2. Numerical solution with ANSYS (FEM)   www.DLR.de  •  Chart 13 𝜕𝑇2𝜕𝑡= 𝛼d,2𝜕2𝑇2𝜕𝑥2 𝜕𝑇1𝜕𝑡= 𝛼d,1𝜕2𝑇1𝜕𝑥2 System  of PDE 𝑇1 𝑥 = 0, 𝑡  = 𝑇amb BC 𝜕𝑇2𝜕𝑥𝑥=𝐿= 0  BC 𝑇1 𝑥, 𝑡 = 0 = 𝑇amb  IC 𝑇2 𝑥, 𝑡 = 0 = 𝑇20 > 𝑇 amb Interface 𝑇1 𝑥 = 𝐿1−, 𝑡 = 𝑇2 𝑥 = 𝐿1+, 𝑡  −𝜅1𝜕𝑇1𝜕𝑥𝑥=𝐿1−= −𝜅2𝜕𝑇2𝜕𝑥𝑥=𝐿1+ Sequences of transient simulations www.DLR.de  •  Chart 14 Temperature  Heat flux  𝑡 = 10 −4 s 𝑡 = 10 −1 s 𝑡 = 1s - Short time behavior – heat wave through the sample – non-exponential Relaxation of temperature difference www.DLR.de  •  Chart 15 𝑡 = 10s 𝑡 = 50s 𝑡 = 100s 1 1 2 2 3 3 ∝ exp −𝑡𝜏 - Thermal conductivity after algebraic treatment and Taylor expansion (omitting terms of second order and higher)    - in the example calculation less than 1% approximation error 𝜅1 =𝐿1𝐴c,s𝑀1 𝐶2  1 +𝜅1𝐿23 𝜅2𝐿1+𝐶13 𝐶2  𝜏 = 𝑀1−1  … relaxation time (from the experiment) 𝐶1 = 𝑚1𝑐1 = 𝜚d,1𝑉1 𝑐1…   thermal mass of the sample 𝐶2 = 𝑚2𝑐2 = 𝜚d,2𝑉2 𝑐2…   thermal mass of the block - Relaxation to equilibrium exponential  Influence factors on the relaxation time www.DLR.de  •  Chart 16 Errors Contact resistance at the interface Radiation to the ambient - Fixed values of  - Symmetric for both side  - Emissivity 𝜀 - Ambient temperature 𝑇amb - Significant at high 𝑇 Overestimation of 𝜅 Underestimation of 𝜅 Zabrocki et al., J. Electr. Mat. 42 (7), 2402-2408 (2013) CTEM – Peltier heat www.DLR.de  •  Chart 17 3D Simulation 2D Simulation (axisymmetric) - transient thermal simulation  - No holes for thermocouples - Heat generation in a slice - Contacts – bulk values of a slice Metal block Metal block Sample - DC current through the assembly ⇒ Peltier heat at the contacts - At which side of the contact is the Peltier heat liberated or absorbed? Contacts as a slide www.DLR.de  •  Chart 18 - Contacts through bulk values   - Heat generation in the metal block or sample?  - Galinstan (liquid metal solder)       Volumetric heat generation for a certain time   - Three thin layers: - Metal block - Contact material - Sample Cu block Cu block S Peltier heat from where? www.DLR.de  •  Chart 19 Heat generation/absorbtion Metal Sample - Peltier heat liberated or absorbed at the metal side or the sample side Peltier heat – Qualify contact resistance www.DLR.de  •  Chart 20 Metal Sample - Peltier-heat either liberated/absorbed at metal side or sample side of the contact material ⇒ different behavior at the switch-off - Experimentally observed jump ⇒ Qualification of the thermal contact resistance  Summary - Generalized heat equation in thermoelectricity for transient response   - Direct solution for the determination of the performance of  TE devices   - Inverse problem: Determination of material properties from measurements of temperatures   - Dynamic measurement of the thermal conductivity – CTEM  www.DLR.de  •  Chart 21 Thank you for your attention! www.DLR.de  •  Chart 22 > Vortrag > Autor  •  Dokumetname > Datum The END 

Transient response of thermoelectric elements and dynamic measurement methods for thermoelectric materials

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