The conservation of energy is a fundamental concept of physics
along with the
conservation of mass
and the
conservation of momentum.
Within some problem domain, the amount of energy remains constant
and energy is neither created nor destroyed. Energy can be converted from
one form to another (potential energy can be converted to kinetic
energy) but the total energy within the domain remains fixed.
Thermodynamics is a branch of physics
which deals with the energy and work of a system. As mentioned on the
gas properties slide, thermodynamics deals
only with the large scale response of a system which we can observe
and measure in experiments. In rocketry, we are most
interested in thermodynamics in the study of
propulsion systems
and understanding high speed flows.
On some separate slides, we have discussed
the state of a static gas,
the properties
which define the state, and the
first law
of thermodynamics as applied to any system, in general.
On this slide we derive a useful form of the energy conservation equation
for a gas beginning with the first law of thermodynamics.
If we call the internal energy of a gas E, the
work done by the gas W, and the heat transferred into the gas Q,
then the first law of thermodynamics indicates that between state "1" and
state "2":
E2  E1 = Q  W
Aerospace engineers usually simplify a thermodynamic analysis
by using intensive variables; variables that do not depend on
the mass of the gas. We call these variables
specific
variables. We create a specific variable by taking a property whose
value depends on the mass of the system and dividing it by the mass
of the system.
Many of the state properties listed on this slide,
such as the work and internal energy depend on the total mass of gas.
We will use "specific" versions of these variables.
Engineers usually use the lower case letter for the "specific"
version of a variable. Our first law equation then becomes:
e2  e1 = q  w
Because we are considering a moving gas, we add the specific kinetic energy term
to the internal energy on the left side. The normal kinetic energy K of
a moving substance is equal to 1/2 times the mass m times the velocity
u squared:
K = (m * u^2) / 2
Then the specific
kinetic energy k is given by:
k = (u^2) / 2
and the first law equation becomes:
e2  e1 + k2  k1 = q  w
There are two parts to the specific work for a
moving gas. Some of the work, called the shaft work (wsh) is
used to move the fluid or turn a shaft, while the rest of the
work goes into changing the state of the
gas. For a
pressure p and
specific volume v, the work is given by:
w = (p * v)2  (p * v)1 + wsh
Substituting:
e2  e1 + k2  k1 = q  (p * v)2 + (p * v)1  wsh
If we perform a little algebra on the first law of thermodynamics,
we can begin to group some terms of the equations. :
e2 + (p * v)2  e1  (p * v)1 + [(u^2) / 2]2  [(u^2) / 2]1 = q  wsh
A useful additional state variable for a gas is the specific
enthalpy
h which is equal to:
h = e + (p * v)
Simplifying the energy equation:
h2  h1 + [(u^2) / 2]2  [(u^2) / 2]1 = q  wsh
or
h2 + [(u^2) / 2]2  h1  [(u^2) / 2]1 = q  wsh
By combining the velocity terms with the enthalpy terms to form the
total specific enthalpy "ht" we can further simplify the equation.
ht = h + u^2 / 2
The total specific enthalpy is analogous to the total pressure in
Bernoulli's equation; both expressions
involve a "static" value plus one half the square of the
velocity.
The final, most useful, form of the energy equation is
given in the red box.
ht2  ht1 = q  wsh
For a compressor or
power turbine, there is no external heat
flow into the gas and the "q" term is set equal to zero. In the
burner, no work is performed and the "wsh"
term is set to zero.
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