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# Non equilibrium thermodynamics-lecture-6-coupled transport of heat and charge

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Lecture 6

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### Non equilibrium thermodynamics-lecture-6-coupled transport of heat and charge

1. 1. Lecture no. 6 Non-Equilibrium Thermodynamics for Engineers ”Coupled transport of heat and charge” Signe Kjelstrup, Chair of Engineering Thermodynamics Department of Process and Energy TU Delft
2. 2. Lecture no. 6 Transport of heat and charge 1. The entropy production 2. The fluxes 3. The coefficients 4. Electric work from thermal energy in an electrochemical cell:
3. 3. Lecture no. 6 Potential work can be lost by heat conduction in this cell • The energy available for work in the thermoelectric converted cell lies in the temperature gradient. • Conduction will after some time make the system homogeneous, if the heat reservoirs cannot be maintainedTwo semiconductors In a temperature gradient
4. 4. Lecture no. 6 The entropy production of the thermoelectric converter 1 1 q d d J j dx T T dx æ ö æ öf÷ ÷ç çs = + -÷ ÷ç ç÷ ÷ç çè ø è ø Heat is transported in J/s m2. The walll is the frame of reference. The electric potential difference between the two conductors over a distance dx The electric current density does not depend on the frame of reference
5. 5. Lecture no. 6 The coupled flux equations 1 1 1 1 1 qq q q J L L x T T x j L L x T T x f f ff æ ö¶ ¶f÷ç= -÷ç ÷çè ø¶ ¶ æ ö¶ ¶f÷ç= -÷ç ÷çè ø¶ ¶ L Lmf fm= 1 1 qq T J L x x T æ ö¶ ¶ ÷ç=-l = ÷ç ÷çè ø¶ ¶ 1 =j L T x ff ¶f - ¶ Relations to Fourier’s and Ohm’s law: Onsager relations are linking Peltier’s and Seebeck measurements
6. 6. Lecture no. 6 The heat flux • Conduction and charge transfer are superimposed: 2 1 q q q q qq L L LT J L j T L x L f f f ff ff æ ö¶÷ç ÷ç=- - +÷ç ÷÷ç ¶è ø Coupling gives heat transport with the electric current Coupling reduces the thermal conductivity Cooling or heating effect: Peltier showed that 1 1 1 1 1 qq q q J L L x T T x j L L x T T x f f ff æ ö¶ ¶f÷ç= -÷ç ÷çè ø¶ ¶ æ ö¶ ¶f÷ç= -÷ç ÷çè ø¶ ¶
7. 7. Lecture no. 6 The electric work 1 1 1 1 q qq q q J L L x T T x j L L x T T x f f ff æ ö¶ ¶f÷ç=- -÷ç ÷çè ø¶ ¶ æ ö¶ ¶f÷ç=- -÷ç ÷çè ø¶ ¶ L L T T dx j dx x T x Lff é ù¶f p ¶ê úDf = =- -ê ú¶ ¶ê úë û ò ò 0 q q dT J L j L f ff= é ù ê úp = = ê ú ë û Defining the Peltier heat: The electric work in V (for one faraday transferred): We used the Onsager relation! Ohmic potential drop Useful work from the temperature gradient
8. 8. Lecture no. 6 The electric work L L T T dx j dx x T x Lff é ù¶f p ¶ê úDf = =- -ê ú¶ ¶ê úë û ò ò Useful work from the gradient of chemical potential The thermocell has a potential difference of a few mikrovolt per degree temperature difference The Seebeck coefficient and the Peltier coefficient are related (Onsager relations) 0 0j dTT T= = æ ö æ öDf p÷ ÷ç ç=-÷ ÷ç ç÷ ÷ç çè ø è øD But the heat fluxes can be large!
9. 9. Lecture no. 6 How can we understand the reversible heat transport?
10. 10. Lecture no. 6 Are thermoelectric converters useful? 1 L L T dx j dx x T x Lff é ù¶f p ¶ê úDf = =- -ê ú¶ ¶ê úë û ò ò 0 q q dT J L j L f ff= é ù ê úp = = ê ú ë û The electric work from the thermal sources: The Peltier heat is not large, but the converter has no moving parts. An Icelandic company is developing thermoelectric applications for harnessing geothermal power, an abundant energy source in Iceland and many other regions of the world.
11. 11. Lecture no. 6 Are thermoelectric converters useful? • Thermoelectric coolers make use of the Peltier effect to transport heat
12. 12. Lecture no. 6 Summary • The transport of heat and charge have been described by the fluxes and forces in the entropy production • The origin of electric work in systems with transport of heat and charge is the coupling coefficent, or the Peltier coefficient • The coupling coefficient is of the same order of magnitude as the other transport coefficients • We have studied simple thermoelectric converters. Reversible heat transport may be important in industry as well as in biology