3.5 Balance Laws
127
The second law of thermodynamics states that isolated systems spontaneously
tend to move toward states of greater disorder. Entropy S is a quantitative measure
of disorder. In an isolated system, it can be shown that S increases when heat flows
from warmer to colder regions. After the temperature becomes uniform, the entropy
remains constant at its maximum value. The spontaneous increase in entropy from
its initial value represents the entropy production, which is never negative. If heat is
then added to the system, the disorder (and entropy) increases further as molecules
move around more. This additional entropy is the defined by the entropy input rate
Q. In an open system, the added heat and associated entropy can come from both
internal and external sources.
In general, therefore, the rate of change in entropy in a system is always equal to
or greater than the entropy input rate. Mathematically, this statement becomes
dS
dt
≥ Q,
(3.191)
which is called the entropy inequality principle.
In Eq. (3.191), equality holds (dS/dt = Q) for a reversible process, which is
a process that does not produce entropy and, when reversed, returns the body to its
original state. Loading and unloading an elastic body is an example of a reversible
process. Although real systems are never actually reversible, this sometimes can
be a reasonable first approximation. For example, biological tissues are not really
elastic, but they often are treated as elastic or pseudoelastic (Fung 1993). In
contrast, an irreversible process generates more entropy than is put into the system
(dS/dt > Q). Spontaneous heat flow is one example. Another is loading and
unloading of a viscoelastic material, where the heat generated by internal friction
is lost to increased entropy.
Entropy Principle in 1D
Consider again the 1D element shown in Fig. 3.21. In the current configuration, the
total entropy content is
S = ρη dA dx,
(3.192)
where η is the entropy per unit mass. Entropy is defined thermodynamically as the
ratio of heat to absolute temperature T . Thus, the rate at which entropy is added to
the element follows the rate of heat addition. Modifying Eq. (3.171) yields
Q =
−
∂
∂x
q
T
+
r
T
dA dx,
(3.193)
where q and r have the same definitions as in the energy balance equation (3.173).
Since the mass ρ dA dx is constant, substituting Eqs. (3.192) and (3.193)
into (3.191) gives the entropy inequality
127
The second law of thermodynamics states that isolated systems spontaneously
tend to move toward states of greater disorder. Entropy S is a quantitative measure
of disorder. In an isolated system, it can be shown that S increases when heat flows
from warmer to colder regions. After the temperature becomes uniform, the entropy
remains constant at its maximum value. The spontaneous increase in entropy from
its initial value represents the entropy production, which is never negative. If heat is
then added to the system, the disorder (and entropy) increases further as molecules
move around more. This additional entropy is the defined by the entropy input rate
Q. In an open system, the added heat and associated entropy can come from both
internal and external sources.
In general, therefore, the rate of change in entropy in a system is always equal to
or greater than the entropy input rate. Mathematically, this statement becomes
dS
dt
≥ Q,
(3.191)
which is called the entropy inequality principle.
In Eq. (3.191), equality holds (dS/dt = Q) for a reversible process, which is
a process that does not produce entropy and, when reversed, returns the body to its
original state. Loading and unloading an elastic body is an example of a reversible
process. Although real systems are never actually reversible, this sometimes can
be a reasonable first approximation. For example, biological tissues are not really
elastic, but they often are treated as elastic or pseudoelastic (Fung 1993). In
contrast, an irreversible process generates more entropy than is put into the system
(dS/dt > Q). Spontaneous heat flow is one example. Another is loading and
unloading of a viscoelastic material, where the heat generated by internal friction
is lost to increased entropy.
Entropy Principle in 1D
Consider again the 1D element shown in Fig. 3.21. In the current configuration, the
total entropy content is
S = ρη dA dx,
(3.192)
where η is the entropy per unit mass. Entropy is defined thermodynamically as the
ratio of heat to absolute temperature T . Thus, the rate at which entropy is added to
the element follows the rate of heat addition. Modifying Eq. (3.171) yields
Q =
−
∂
∂x
q
T
+
r
T
dA dx,
(3.193)
where q and r have the same definitions as in the energy balance equation (3.173).
Since the mass ρ dA dx is constant, substituting Eqs. (3.192) and (3.193)
into (3.191) gives the entropy inequality