NPTEL – Mechanical Engineering – Continuum Mechanics
Module-4: Balance Laws
Lecture-30: The Second Law of Thermodynamics
The second law of thermodynamics is another basic axiom of continuum mechanics. Unlike other basic axioms, the second law is not a conservation law. However, this brings
new information on the direction of heat transfer and also imposes some restrictions on
constitutive relations.
The second law and the entropy:
The first law of thermodynamics states that the energy is conserved and does not put
any restriction on the direction of the process. For example, consider a hot body in
contact with a cold body. Naturally, the hot body releases heat energy and cold body
gains same amount of heat to get thermal equilibrium. The reverse process, i.e., hot
body becoming hotter and cold body becoming colder by exchanging equal amount of
heat energy, is not possible but does not violate the first law. Thus, the second law of
thermodynamics is stated to account the direction of the thermodynamic process. The
second law has an interesting history1 and it has many equivalent statements. Here, we
adopt a form of the second law of thermodynamics called the Clausius-Duhem inequality.
A new thermodynamic quantity called entropy is introduced to account the directionality
of thermodynamic processes.
The entropy physically represents a measure of the disorderness of the system. We know
from basic thermodynamics that any system can possess internal entropy. In addition,
the system can also exchange the entropy with its surroundings through the interaction
of energy and/or mass. Thus, the entropy production of the system can be defined by the
difference between the internal entropy and the entropy influx. The second law asserts that
the entropy production universe (system + surroundings) is nonnegative. Considering
the material volume as system, we can apply the second law of thermodynamics to the
continuous medium.
Mathematical statement of entropy imbalance:
Let Ω be a material volume of continuum body. Let S be net internal entropy of the
material volume Ω. Let J be the rate of entropy flow to the material volume from its
surroundings. Let H be the rate of entropy production. Then
dS
−J.
dt
The second law of thermodynamics states that
H=
H ≥ 0.
(1)
(2)
1
D. Kondepudi and I. Prigogine, “Modern thermodynamics: From heat engines to dissipative structures”, 1998, John Wiley & Sons.
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The non-negativity of entropy production is also known as entropy imbalance. The integral form of entropy imbalance can be stated by introducing field variables.
Integral form of entropy imbalance:
The entropy addition to the material volume can be divided into two parts: (i) entropy
influx/outflux through the boundary (ii) volumetric addition of entropy. Let f be a rate
of entropy outflux. Let ς(x, t) be rate of entropy supply per unit volume. Then the
entropy inflow is defined by
J =−
Z
f · n ∂Γ +
Γ
Z
ς ∂Ω,
(3)
Ω
where Γ is boundary of material volume Ω and n is unit normal field to Γ. The negative
sign appears for the first term on the right hand side of equation as f is outflux (not
influx).
Let η(x, t) be specific entropy, i.e., entropy per unit mass. Then the net internal
entropy of material volume is defined by
S =
Z
(4)
ρη ∂Ω.
Ω
Using entropy imbalance, i.e., Eq. (2), we get
Z
Z
d Z
ρη ∂Ω ≥ − f · n ∂Γ + ς ∂Ω.
dt Ω
Ω
Γ
(5)
This inequality is more general inequality of second law of thermodynamics. We now
specialize the inequality to get Clausius-Duhem inequality.
The Clausius-Duhem inequality:
Let θ be the absolute temperature scale, i.e., Kelvin’s scale of temperature. Then, motivated by classical thermodynamics, we define
f=
q
θ
and
ς=
ρQh
,
θ
(6)
where q is rate of heat flux vector and Qh heat generation per unit mass as defined in
first law. Substituting these relation in Eq. (5), we get the following Clausius-Duhem
inequality:
Z
Z
d Z
ρQh
q·n
ρη ∂Ω ≥
∂Ω −
∂Γ.
(7)
dt Ω
θ
Ω θ
Γ
The equality holds, in the above relation, only for a reversible process.
Using divergence theorem and localization theorem, we get the following differential
form of the Clausius-Duhem inequality:
Dη
Qh 1
q
≥
− ∇x ·
.
Dt
θ
ρ
θ
(8)
Expansion of above relation yields
Dη
Qh
1
1
≥
− ∇x · q + 2 q · ∇x θ.
Dt
θ
ρθ
ρθ
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Free energy imbalance and the dissipation:
We define a scalar field called specific free energy by
ψ(x, t) = e(x, t) − η(x, t) θ(x, t),
(10)
where e(x, t) is specific internal energy. Taking material time derivative, we obtain
Dψ
De Dη
Dθ
=
−
θ−η
.
Dt
Dt
Dt
Dt
(11)
De
De
from the fist law of thermodynamics, i.e., ρ
= τ : D − ∇x · q + ρQh ,
Substituting
Dt
Dt
we get
Dψ
1
1
Dη
Dθ
= τ : D − ∇x · q + Qh −
θ−η
.
(12)
Dt
ρ
ρ
Dt
Dt
Rearranging terms, we obtain
−
Dη
1
Dψ 1
Dθ
θ + Qh − ∇x · q =
− τ :D+η
.
Dt
ρ
Dt
ρ
Dt
Substituting this relation in Eq. (9), we get the following Clausius-Duhem inequality in
terms of free-energy.
Dψ
Dθ 1
q · ∇x θ
+η
− τ :D+
≤ 0.
(13)
Dt
Dt ρ
ρθ
We now define the dissipation δ per unit volume by
Dθ
Dψ
+η
δ =τ :D−ρ
Dt
Dt
!
1
− q · ∇x θ.
θ
(14)
Thus, the dissipation δ ≥ 0.
The Clausius-Duhem inequality in Lagrangian description:
Let ρ0 be the density of reference configuration. Let F (X, t) be deformation gradient
and J be its determinant. Let T (X, t) and S(X, t) be first and second Piola-Kirchhoff
stress tensors. Let θ0 (X, t), ψ 0 (X, t), η 0 (X, t), Q0h (X, t) and q 0 (X, t) be material description of temperature, free-energy, entropy, heat generation per unit mass and heat
flux, respectively. Then we have the following relationships:
ρ0 = ρJ,
1
DF
τ :D=τ :L =
T :
,
J
Dt
q 0 = JF −1 q
∇X · q 0 = J∇x · q
∇X θ 0 = F T ∇x θ
Substituting all the terms in Eq. (13), we get the Clausius-Duhem inequality with respect
to reference configuration
Dψ 0
Dθ0
1
DF
q 0 · ∇X θ 0
+ η0
− T :
+
≤ 0.
Dt
Dt
ρ0
Dt
ρ0 θ 0
(15)
Since the total time derivative and material time derivative are equal, we can also write
∂ψ 0
∂θ0
1
∂F
q 0 · ∇X θ 0
+ η0
− T :
+
≤ 0.
∂t
∂t
ρ0
∂t
ρ0 θ 0
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Let E(X, t) be a Green strain tensor. Then we have the following Clausius-Duhem
inequality in Lagrangian description.
∂θ0
∂ψ 0
1
∂E q 0 · ∇X θ0
+ η0
− S:
+
≤ 0.
∂t
∂t
ρ0
∂t
ρ0 θ0
(17)
The Clausius-Duhem inequality for control volume:
Let Ωc be a fixed control volume in space and Γc be its boundary. Let n be field of unit
outward normals to Γc . Considering the Clausius-Duhem inequality (see Eq. (8)), we can
write
Z
Z
Z
Qh
q
Dη
∂Ω ≥
ρ
∂Ω −
∇x ·
∂Ω.
ρ
θ
θ
Ωc
Ωc
Ωc Dt
Using divergence theorem to second term on the right hand side, we get
Z
Ωc
Z
Z
D(ρη) Dρ
Qh
1
−
η ∂Ω ≥
ρ
∂Ω −
q · n ∂Γ.
Dt
Dt
θ
Ωc
Γc θ
Applying the conservation of mass, i.e.,
Z
Ωc
Since
Dρ
= −ρ(∇x · v), we get
Dt
Z
Z
D(ρη)
Qh
1
ρ
+ ρ(∇x · v)η ∂Ω ≥
∂Ω −
q · n ∂Γ.
Dt
θ
Ωc
Γc θ
∂(ρη)
D(ρη)
=
+ ∇x (ρη) · v, we can write
Dt
∂t
Z
Ωc
Z
Z
∂(ρη)
Qh
1
ρ
+ ∇x (ρη) · v + ρ(∇x · v)η ∂Ω ≥
∂Ω −
q · n ∂Γ.
∂t
θ
Ωc
Γc θ
Substituting the relation ∇x · (ρηv) = ∇x (ρη) · v + ρ(∇x · v)η yields
Z
Ωc
Z
Z
Z
Qh
∂(ρη)
1
ρ
∇x · (ρηv) ∂Ω ≥
∂Ω +
∂Ω −
q · n ∂Γ.
∂t
θ
Ωc
Ωc
Γc θ
Using divergence theorem, we obtain the following Clausius-Duhem inequality for control
volume:
Z
Z
Z
Z
Qh
∂(ρη)
1
ρ
∂Ω +
ρηv · n ∂Γ ≥
∂Ω −
q · n ∂Γ.
(18)
∂t
θ
Ωc
Γc
Ωc
Γc θ
Since Ωc is fixed, we can also write
Z
Z
Z
d Z
Qh
1
ρη ∂Ω +
ρηv · n ∂Γ ≥
∂Ω −
q · n ∂Γ.
ρ
dt Ωc
θ
Γc
Ωc
Γc θ
(19)
We use the inequality statement of second law of thermodynamics to impose some restrictions on constitutive relations.
References
1. C. S. Jog, Foundations and Applications of Mechanics: Continuum Mechanics,
Volume-I, 2007, Narosa Publishing House Pvt. Ltd., New Delhi.
4. M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of
Continua, 2010, Cambridge University Press, New York.
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