Direct Energy, 2018a

284 12.4 Thermodynamic Energy Conversion
instead of an energy [170] [171], but throughout this text Lagrangian is
assumed to represent an energy as described in Ch. 11.
Assume that only one energy conversion process occurs in a device. Also
assume that if we know three (not two) of the four thermodynamic parameters,
we can calculate the fourth. Additionally, assume small amounts
of energy are involved, and the energy conversion process occurs in the
presence of a large external thermodynamic reservoir of energy.
As with the discussion of the previous tables, each column of Table
12.7 details the parameters of calculus of variations for a dierent choice
of generalized path. In order, the columns can be used to describe energy
storage in a gas conned to a nite volume, a material under pressure, a
temperature dierential, or an ordered system. The rows are labeled in
the same way as in the previous tables of this chapter so that analogies
between the systems can be drawn.
Energy can be stored and released from a gas conned to a nite volume
and a gas under pressure. These related energy conversion processes are
detailed in the second and third columns of Table 12.7 respectively. The
second column species parameters of calculus of variations with volume
chosen as the generalized path and pressure as the generalized potential.
The third column species parameters with pressure chosen as the generalized
path and volume as the generalized potential. In reality, energy
conversion processes involving changes in the pressure and volume of a gas
are unlikely to occur without a change in temperature or entropy of the
system simultaneously occurring. Resistive heating, friction, gravity, and
all other energy conversion processes that could simultaneously occur are
ignored. Temperature and entropy are explicitly assumed to remain xed,
and these assumptions are listed in the last row of the table for emphasis.
These columns can apply to energy conversion in liquids and solids in
addition to gases. Using the choice of variables in the second column, the
capacity to store energy is given by V where B is the bulk modulus in units
B
pascals, and it is a measure of the ability of a compressed material to store
energy [103]. Bulk modulus was introduced in Section 8.2. Using volume as
the generalized path, the Euler-Lagrange equation can be set up and solved
for the equation of motion. All terms of the resulting equation of motion
have the units of pressure, and the equation of motion is a statement of
Bernoulli's equation, an idea discussed in Section 10.6.
The fourth and fth columns of Table 12.7 specify parameters of calculus
of variations with temperature and entropy chosen as the generalized
path respectively. A cup of hot liquid stores energy. Similarly, a container
with two pure gases separated by a barrier stores energy. The system is
in a more ordered state before the barrier is removed than after, and it