Non-Equilibrium Thermodynamicsfor Engineers

”Why is this field important?”

Signe KjelstrupChair of Engineering Thermodynamics

Department of Process and EnergyTU Delft

A formulation of the second law ofthermodynamics that includes time:

0

0

0

( )

0

irr

irr

L

i ii

S SdS t S Sdt

dS x dxdt

J X

Δ +Δ ≥

Δ =Δ +Δ

=Ω σ

σ= ≥

∫

∑

The second law ofthermodynamics

Transport direction: x

The entropy productionis a product sum of all fluxesand forces

The entropy balance

• Entropy is not conserved in anyelement

• Stationary state, whole system:

0i ii

J Xσ= ≥∑x x+dx

ss Jt x

∂ ∂=− +σ

∂ ∂

( )o iirrs s

dS J Jdt

= − Ω

Transport lawsderived from the second law

• Empirical laws of Fourier, Fick and Ohm

• Fluxes and conjugate forces• Coupled flux equations

• Onsager relations

1,'

1,

1,

1 1 1

1 1 1

1 1 1

Tq qq q q

Tq

Tq

J L L Lx T T x T x

J L L Lx T T x T x

j L L Lx T T x T x

μ φ

μ μμ μφ

φ φμ φφ

⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦

q q q qL L L L L Lμ μ φ φ φμ μφ= = =

' 11 q

dcdT dJ J D jdx dx dx

φ=−λ =− =−κ

Is coupling important?• Water (mass) transport is always

connected with transport of ions (charge) in membranes

• Heat transport can lead to masstransport

• A thermocouple works, because a temperature difference triggers an electric potential difference

• The coupling coefficient gives thepossibility to do work! Examples: Salt power plants, Thermoelectricpower, osmosis plants, frictionelectricity, fuel cells, batteries.

Experiments can be well definedfrom the flux equations

´

0, 0

0, 0

Example: Ohmic conductivities

H om ogeneous, isothermal conductor: /

Stationary state conductor:/

q

dT d

J J

jd dx

jd dx

= μ=

= =

⎡ ⎤⎢ ⎥⎢ ⎥φ⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥φ⎣ ⎦

1,'

1,

1,

1 1 1

1 1 1

1 1 1

Tq qq q q

Tq

Tq

J L L Lx T T x T x

J L L Lx T T x T x

j L L Lx T T x T x

μ φ

μ μμ μφ

φ φμ φφ

⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦⎡ ⎤∂μ⎛ ⎞ ⎛ ⎞∂ ∂φ⎟ ⎟⎜ ⎜⎢ ⎥= + − + −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠∂ ∂ ∂⎣ ⎦

Lost work in exergy analysis

• A process with manyunits

0

0 0

0

0

0 0 0

0

irr

irr

U q p V wU q p V wS S

dS t S Sdt

dSw T t U p V T Sdt

Δ = − Δ +

Δ =− − Δ +

Δ +Δ ≥

Δ =Δ +Δ

= Δ +Δ + Δ − Δ

0irr

lost idealdSw w w T tdt

⎛ ⎞⎟⎜= − = Δ⎟⎜ ⎟⎜⎝ ⎠

0 0idealw U p V T S=Δ + Δ − Δ

Lost work from fluxes and forcesin the system

• A chemical reactor (one process unit) 0

irrlost ideal

dSw w w T tdt

⎛ ⎞⎟⎜= − = Δ⎟⎜ ⎟⎜⎝ ⎠

' 1 1irrq

L

dS G dpr DJ v dzdt T T T dz

⎡ ⎤⎛ ⎞ ⎛ ⎞Δ ⎟ ⎟⎜ ⎜⎢ ⎥= Ωρ − +π Δ +Ω −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∫

Lost work from the entropy balance

• For the reactor, and for any volume element along the path

'qirr

out inaL

JdS S S D dzdt T

= − −π ∫

Summary:What can this theory offer?

1. A more specific formulation of the second law ofthermodynamics

2. The entropy balance – also an equation of thesystem!

3. Transport laws derived from the second law4. A systematic framework that also defines

experiments5. A direct calculation of the lost work6. A better understanding of work and lost work