where u is the internal potential energy per unit mass. The left side of the equation is
the rate of increase in the internal potential energy. The right side of the equation
represents the input power that is converted into internal work as mechanical work
and as heat. However, excluding the kinetic energy at macro level, the last term is
negative because it represents heat inflow per unit volume through the boundaries.
We should point out that this equation is very a simple form of the first law of
thermodynamics because it assumes that only W input is due to surface traction
(distributed loads). If we have distributed fields such as electromagnetic and chemical loads, they must be included in this equation. These load cases will be covered in
Chap. 8. Moreover, the first law of thermodynamics given above does not include
energy loss terms because mechanical work terms is defined according to Newtonian
mechanics. In the next chapter, the same equation will be derived for unified
mechanics theory where energy loss will be included.
3.3 Second Law of Thermodynamics
There are several approaches to define the second law; however, we believe KelvinPlanck’s statement is the most concise and classical description. According to this
statement, it is impossible to construct an engine that will produce no other effect
than the extraction of heat from a single heat reservoir and the performance of an
equivalent amount of work. In other terms, we can state that it is not possible to
construct an engine that has an efficiency of 100%, meaning that the input energy
and output energy for the intended work cannot be equal. There will always be
energy loss for unintended work. This energy loss will only occur in positive
direction, meaning that we can only dissipate energy, but cannot gain energy in a
closed system. This law of nature is expressed mathematically as an inequality
stating that the internal entropy production is always nonnegative and is positive
for an irreversible process. This inequality is called Clausius-Duhem inequality.
Simplest definition of entropy is that entropy quantifies how much energy is
unavailable for work.
Another way of stating the second law of thermodynamics is that the entropy of
all natural processes increases. The concept of entropy will allow us to tie probability
and statistics to continuum mechanics.
In the first law of thermodynamics, we saw interconvertibility of energy between
different forms such as heat and mechanical work. In Newtonian mechanics, kinetic
energy and potential energy may be converted from one to another with no energy
loss. The transformation can proceed in either direction, like a pendulum with no
friction that swings to eternity.
On the other hand, the second law of thermodynamics states that such a pendulum
is not possible. Each time the pendulum swings, it loses some of its kinetic energy to
friction; as a result, there is less total energy to continue swinging. Complete reversal
is not possible. The frictional dissipation is an irreversible process.
80
3 Thermodynamics
the rate of increase in the internal potential energy. The right side of the equation
represents the input power that is converted into internal work as mechanical work
and as heat. However, excluding the kinetic energy at macro level, the last term is
negative because it represents heat inflow per unit volume through the boundaries.
We should point out that this equation is very a simple form of the first law of
thermodynamics because it assumes that only W input is due to surface traction
(distributed loads). If we have distributed fields such as electromagnetic and chemical loads, they must be included in this equation. These load cases will be covered in
Chap. 8. Moreover, the first law of thermodynamics given above does not include
energy loss terms because mechanical work terms is defined according to Newtonian
mechanics. In the next chapter, the same equation will be derived for unified
mechanics theory where energy loss will be included.
3.3 Second Law of Thermodynamics
There are several approaches to define the second law; however, we believe KelvinPlanck’s statement is the most concise and classical description. According to this
statement, it is impossible to construct an engine that will produce no other effect
than the extraction of heat from a single heat reservoir and the performance of an
equivalent amount of work. In other terms, we can state that it is not possible to
construct an engine that has an efficiency of 100%, meaning that the input energy
and output energy for the intended work cannot be equal. There will always be
energy loss for unintended work. This energy loss will only occur in positive
direction, meaning that we can only dissipate energy, but cannot gain energy in a
closed system. This law of nature is expressed mathematically as an inequality
stating that the internal entropy production is always nonnegative and is positive
for an irreversible process. This inequality is called Clausius-Duhem inequality.
Simplest definition of entropy is that entropy quantifies how much energy is
unavailable for work.
Another way of stating the second law of thermodynamics is that the entropy of
all natural processes increases. The concept of entropy will allow us to tie probability
and statistics to continuum mechanics.
In the first law of thermodynamics, we saw interconvertibility of energy between
different forms such as heat and mechanical work. In Newtonian mechanics, kinetic
energy and potential energy may be converted from one to another with no energy
loss. The transformation can proceed in either direction, like a pendulum with no
friction that swings to eternity.
On the other hand, the second law of thermodynamics states that such a pendulum
is not possible. Each time the pendulum swings, it loses some of its kinetic energy to
friction; as a result, there is less total energy to continue swinging. Complete reversal
is not possible. The frictional dissipation is an irreversible process.
80
3 Thermodynamics