ENTROPY PRODUCTION AND NONLINEARITY.
Is entropy production an exclusively nonlinear phenomenon? Must it always
vanish in a linearized model?
Consider simple heat transfer modeled by Fourier's law:
Q̇ = (kA/l)(T1 - T2)
where Q̇ is heat flow rate, k is thermal conductivity, A is a surface area, l is
length and T1 and T2 are absolute temperatures.
Entropy flow and heat flow are related by temperature.
Q̇ = T1Ṡ 1 = T2Ṡ 2
(T1 - T2)
Ṡ 1 = (kA/l) T
1
Ṡ 2 = (kA/l)
Mod. Sim. Dyn. Sys.
(T1 - T2)
T2
Entropy production
page 1
Net entropy production is:
Ṡ net = Ṡ out – Ṡ in = Ṡ 2 – Ṡ 1
= (kA/l)(T1 -
⎛1
T2)⎜⎜T
⎝ 2
⎛(T - T )2⎞
1 ⎞⎟
1
2
- T ⎟ = (kA/l)⎜⎜ T T ⎟⎟
1⎠
1 2 ⎠
⎝
Hence, as dictated by the second law, net entropy production is never negative.
Mod. Sim. Dyn. Sys.
Entropy production
page 2
LINEARIZE
Now linearize the two entropy flow equations. Subscript o denotes operating
point.
∆Ṡ 1 =
(kA/l)T2
(kA/l)
∆T
–
1
T12
T1 o ∆T2
o
(kA/l)T1
(kA/l)
∆Ṡ 2 = T
∆T1 –
∆T2
T22
o
o
2
Net linearized entropy production is:
∆Ṡ net = ∆Ṡ 2 – ∆Ṡ 1
=
⎛1
(kA/l)⎜⎜T
⎝ 2
⎛1
T2 ⎞⎟
T1 ⎞⎟
⎜
– T 2⎟ ∆T1 + (kA/l)⎜T – T 2⎟ ∆T2
1 ⎠o
2 ⎠o
⎝ 1
Net linearized entropy production may be non-zero.
Provided the operating point is not at thermal equilibrium.
Mod. Sim. Dyn. Sys.
Entropy production
page 3
However, if the operating point is at thermal equilibrium,
T1,o = T2,o
⎛1
⎜
⎜
⎝T2
T2 ⎞⎟
– T 2⎟ = 0
1 ⎠o
⎛1
⎜
⎜
⎝T1
T1 ⎞⎟
– T 2⎟ = 0
2 ⎠o
Net entropy production vanishes.
The entropy flow need not vanish, but the input flow must equal the output flow.
Mod. Sim. Dyn. Sys.
Entropy production
page 4
CAUTION!
Another difficulty of the linearized model:
Used sufficiently far from the operating point, the linearized equations
may describe a negative net entropy production.
(remember ∆T1 or ∆T2 may be positive or negative)
That would violate the second law.
The exception when the operating point is at thermal equilibrium, in
which case the model describes no entropy production.
Clearly, linearized models should be interpreted with caution.
Mod. Sim. Dyn. Sys.
Entropy production
page 5