Non equilibrium Thermodynamics Module 8 Non equilibrium Thermodynamics

Lecture 8.1 Basic Postulates

NON-EQUILIRIBIUM THERMODYNAMICS Steady State processes. (Stationary) Concept of Local thermodynamic eqlbm Heat conducting bar define properties Specific property Extensive property

NON-EQLBM THERMODYNAMICS Postulate I Although system as a whole is not in eqlbm., arbitrary small elements of it are in local thermodynamic eqlbm & have state fns. which depend on state parameters through the same relationships as in the case of eqlbm states in classical eqlbm thermodynamics.

NON-EQLBM THERMODYNAMICS Postulate II Entropy gen rate affinities fluxes

NON-EQLBM THERMODYNAMICS Purely “resistive” systems Flux is dependent only on affinity at any instant at that instant System has no “memory”-

NON-EQLBM THERMODYNAMICS Coupled Phenomenon Since Jk is 0 when affinities are zero,

NON-EQLBM THERMODYNAMICS where kinetic Coeff Relationship between affinity & flux from ‘other’ sciences Postulate III

NON-EQLBM THERMODYNAMICS Heat Flux : Momentum : Mass : Electricity :

NON-EQLBM THERMODYNAMICS Postulate IV Onsager theorem {in the absence of magnetic fields}

NON-EQLBM THERMODYNAMICS Entropy production in systems involving heat Flow T1 T2 x dx A

NON-EQLBM THERMODYNAMICS Entropy gen. per unit volume

NON-EQLBM THERMODYNAMICS

NON-EQLBM THERMODYNAMICS Entropy generation due to current flow : I dx Heat transfer in element length

NON-EQLBM THERMODYNAMICS Resulting entropy production per unit volume

NON-EQLBM THERMODYNAMICS Total entropy prod / unit vol. with both electric & thermal gradients affinity affinity

NON-EQLBM THERMODYNAMICS

Analysis of thermo-electric circuits Addl. Assumption : Thermo electric phenomena can be taken as LINEAR RESISTIVE SYSTEMS {higher order terms negligible} Here K = 1,2 corresp to heat flux “Q”, elec flux “e”

Analysis of thermo-electric circuits  Above equations can be written as Substituting for affinities, the expressions derived earlier, we get

Analysis of thermo-electric circuits We need to find values of the kinetic coeffs. from exptly obtainable data. Defining electrical conductivity as the elec. flux per unit pot. gradient under isothermal conditions we get from above

End of Lecture

Thermoelectric phenomena Lecture 8.2 Thermoelectric phenomena

Analysis of thermo-electric circuits The basic equations can be written as Substituting for affinities, the expressions derived earlier, we get

Analysis of thermo-electric circuits We need to find values of the kinetic coeffs. from exptly obtainable data. Defining electrical conductivity as the elec. flux per unit pot. gradient under isothermal conditions we get from above

Analysis of thermo-electric circuits Consider the situation, under coupled flow conditions, when there is no current in the material, i.e. Je=0. Using the above expression for Je we get Seebeck effect

Analysis of thermo-electric circuits or Seebeck coeff. Using Onsager theorem

Analysis of thermo-electric circuits Further from the basic eqs for Je & JQ, for Je = 0 we get

Analysis of thermo-electric circuits For coupled systems, we define thermal conductivity as This gives

Analysis of thermo-electric circuits Substituting values of coeff. Lee, LQe, LeQ calculated above, we get

Analysis of thermo-electric circuits Using these expressions for various kinetic coeff in the basic eqs for fluxes we can write these as :

Analysis of thermo-electric circuits We can also rewrite these with fluxes expressed as fns of corresponding affinities alone : Using these eqs. we can analyze the effect of coupling on the primary flows

PETLIER EFFECT Under Isothermal Conditions a b JQ, ab Je Heat flux

PETLIER EFFECT Heat interaction with surroundings Peltier coeff. Kelvin Relation

PETLIER REFRIGERATOR

THOMSON EFFECT Total energy flux thro′ conductor is JQ, surr Je dx Using the basic eq. for coupled flows

THOMSON EFFECT The heat interaction with the surroundings due to gradient in JE is

THOMSON EFFECT Since Je is constant thro′ the conductor

THOMSON EFFECT Using the basic eq. for coupled flows, viz. above eq. becomes (for homogeneous material, Thomson heat Joulean heat

THOMSON EFFECT reversible heating or cooling experienced due to current flowing thro′ a temp gradient Thomson coeff Comparing we get

THOMSON EFFECT We can also get a relationship between Peltier, Seebeck & Thomson coeff. by differentiating the exp. for ab derived earlier, viz.

End of Lecture

Analysis of thermo-electric circuits  Above equations can be written as Substituting for affinities, the expressions derived earlier, we get

Analysis of thermo-electric circuits We need to find values of the kinetic coeffs. from exptly obtainable data. Defining electrical conductivity as the elec. flux per unit pot. gradient under isothermal conditions we get from above