Thermodynamics is a branch of physics
that deals with the energy and work of a system. As aerodynamicists,
we are most interested in thermodynamics in the study of
propulsion systems
The first law of thermodynamics
indicates that the total energy of a system is
conserved. This
includes the potential and kinetic energy, the work
done by the system, and the transfer of
heat
through the system. The
second law of thermodynamics indicates
that, while many physical processes that satisfy the first law are
possible, the only processes that occur in nature are those for which
the entropy (S) of the system either remains constant or
increases. Entropy, like temperature (T) and
pressure (p), is a state variable of the system.
Entropy is defined
to be the heat transfer (delta Q) into the system divided by the
temperature. For a process going from state 1 to state 2:
S2 - S1 = delta Q / T
During a thermodynamic process, the temperature of an object
changes as heat is applied or extracted. So a more correct definition of
the entropy is the differential form that accounts for this
variation.
dS = dQ / T
The change in entropy is then the inverse
of the temperature integrated over the change in heat transfer. For
gases, there are two possible ways to evaluate the change in entropy.
The equations can be formulated in terms of the
internal energy (E) and the work (W)
for a gas. Or, it may be formulated in terms of the
enthalpy (H) of the gas.
We shall present here the enthalpy formulation
because we will use the entropy relations to determine how the temperature
changes for a change in pressure inside a cylinder of the Wright
1903 engine.
Let us begin by deriving a differential form of the first law of thermodynamics.
The first law is given by:
E2 - E1 = Q - W
The differential form would be:
dE = dQ - dW
For a gas, the work is given by
dW = p dV
Substituting:
dE = dQ - p dV
From the definition of enthalpy:
H = E + p * V
dH = dE + p dV + V dp
dE = dH - p dV - V dp
Substitute into the first law equation:
dH - p dv - V dp = dQ - p dV
dH - V dp = dQ
Using the
equation of state, we can modify the second
term of this equation in terms of the gas constant, the pressure,
and the temperature.
p * V = R * T
dH - (R * T) dp / p = dQ
The enthalpy of a
gas is equal to the
specific heat
of constant pressure times the change in
temperature; in differential form:
dH = Cp dT
Cp dT - (R * T) dp / p = dQ
Divide by the temperature and substitute dS = dQ / T :
Cp dT / T - R dp / p = dS
This is a differential equations which we can easily integrate.
S2 - S1 = Cp * ln ( T2 / T1) - R * ln ( p2 / p1)
The "ln" is the natural logarithm function which you find on your calculator and
which is given in tables in math books.
What good is all of this?
There are several processes which take place within an
engine for which there is no transfer of heat
into the system. A
simple compression
is one of these adiabatic processes.
If there is no heat transfer, the change of entropy is zero and
we have a process that is reversible.
The equation derived above relates the change in temperature to the
change in pressure. For a given compression ratio, we can determine how hot
the gas becomes.
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