lectures27-28

# lectures27-28 - • Review of irreversible thermodynamics...

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Unformatted text preview: • Review of irreversible thermodynamics • Generalized linear theory for coupled kinetic processes • Thermoelectric phenomena • Thermomigration • Electromigration • Diffusion in multicomponent systems Coupling of Other Processes with Diffusion S d S d dS i e + = T dQ S d e = > S d i Irreversible Thermodynamics Recall Second Law of Thermodynamics where dS is the total entropy change for the system, d e S is the reversible entropy exchange with surroundings, dQ is the reversible heat transfer between the system and surroundings, T is temperature, and d i S is entropy produced inside the system. Let’s consider a simple discrete system with two parts with different uniform temperatures (see figure). The heat transfer between the parts is an irreversible process, so entropy is produced. T 2 T 1 dQ fixed diathermal impermeable isolated dQ T T T dQ T dQ S d i − = + − = 1 2 2 1 1 1 dt dQ T T dt S d i − = = 1 2 1 1 σ The rate of entropy production is (in this case, dQ is the same as dU since there is no mass transfer) Heat Conduction and Mass Diffusion In a continuum temperature gradient, ( ) ( ) Q Q J dV F dt dQ A A dx dx T d = = 1 ) ( 1 σ where A is the cross-sectional area, dx is the infinitesimal distance, F Q is the driving force for heat transfer, J Q is the heat flux current, and dV is the volume of a volume element dQ ( ) dx x T + ( ) x T dx A ( ) Q Q i V J F dt dQ A dx T d dt s d = = = 1 1 σ The entropy production per unit volume is then given by Therefore, the entropy production is a product of a driving force and a flux In a linear theory (strictly only valid near equilibrium), the flux is linearly proportional to the driving force ( ) dx dT T L dx T d L F L J Q Q Q Q Q 2 1 − = = = Compare the above equation with the Fourier’s law dx dT J Q κ − = 2 kT L Q = where k is the heat conductivity i dN ( ) dx x i + µ ( ) x i µ dx A Now consider a one-dimensional system with non-uniform chemical potential for species i , T and p are uniform in the system ( ) ( ) i i i i i dN T x x dN T x S d δ µ µ + − = ∑ = − + + = N i i i dN T dV T p dU T dS 1 1 µ Recall the differential form of entropy in equilibrium thermodynamics ( ) i i i i V J F dt dN A dx T d = − = 1 µ σ Therefore, the entropy increase in the system is The entropy production per unit volume due to mass transfer:...
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lectures27-28 - • Review of irreversible thermodynamics...

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